![Adiabatic stretching force constants of NF bonds in dependence of equilibrium bond lengths r(NF) calculated at B3LYP/6-31G(d,p) [139]. The general exponential decay of ka(NF) for increasing r(NF) is not fulfilled by three fluoroamine classes: In these cases, the NF stretching force constant and thereby the bond strength increases with increasing bond length.](https://www.researchgate.net/profile/Dieter-Cremer/publication/233490757/figure/fig7/AS:668349608521730@1536358372425/Fig-3-Adiabatic-stretching-force-constants-of-NF-bonds-in-dependence-of-equilibrium_Q320.jpg)
![Adiabatic bond stretching frequencies a (CO) and a (CF ) given as a function of the corresponding bond lengths (bottom) for 27 structures (top) (1-16: F-substituted carbocations; 17-19: reference compounds with normal C-F bond; 20-23: fluoronium ions; 24-27: transition states with loose or normal CF bonds; 28-42: C=O analogs of 1-16; 43-46: reference compounds with C-O(H) bond some of which being isoelectronic with 20-22. Correlation coefficient R 2 for a (CO) : 0.999, and for a (CF ) : 0.993 [48].](https://www.researchgate.net/profile/Dieter-Cremer/publication/233490757/figure/fig8/AS:668349608517643@1536358372555/Fig-4-Adiabatic-bond-stretching-frequencies-a-CO-and-a-CF-given-as-a-function-of_Q320.jpg)
![Adiabatic CF and CO stretching force constants of structures 1 -46 (Fig. 4) expressed in form of an exponential relationship of the corresponding bond lengths. Correlation coefficient R 2 for ka(CO): 0.998, and for ka(CF): 0.982. If both sets of data are correlated with the bond length, R 2 is 0.985. B3LYP/6-31(d,p) calculations [48].](https://www.researchgate.net/profile/Dieter-Cremer/publication/233490757/figure/fig1/AS:340804762652674@1458265595419/Fig-5-Adiabatic-CF-and-CO-stretching-force-constants-of-structures-1-46-Fig-4_Q320.jpg)
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1524 Current Organic Chemistry, 2010, 1 4, 1524-1560
1385-2728/10 $55.00+.00 © 2010 Bentham Science Publishers Ltd.
From Molecular Vibrations to Bonding, Chemical Reactions, and Reaction
Mechanism
Dieter Cremer* and Elfi Kraka
Department of Chemistry, Southern Methodist University, 3215 Daniel Ave, Dallas, Texas 75275-0314, USA
Abstract: The vibrational motions of a molecule in its equilibrium or during a chemical reaction provide a wealth of information about
its structure, stability, and reactivity. This information is hidden in measured vibrational frequencies and intensities, however can be
unraveled by utilizing quantum chemical tools and applying the Cal-X methods in form Vib-Cal-X. Vib-Cal-X uses the measured
frequencies, complements them to a complete set of 3N-L values (N number of atoms; L number of translatio ns and rotations), derives
experimentally based force constants, and converts them into local mode stretching, bending, and torsional force constants associated
with the internal coordinates describing the geometry of the molecule. This is done by utilizing the adiabatic vibrational mode concept,
which is based on a decomposition of delocalized normal vibrational modes into adiabatic internal coordinate modes (AICoMs) needed to
describe bonding or changes in bonding. AICoM force constants relate to the intrinsic bond dissociation energy (IBDE) of a bond and,
accordingly, are excellent descriptors for bond order and bond strength. It is shown that bond dissociation energies, bond lengths, or bond
densities are not directly related to the bond strength because they also depend on other quantities than just the bond strength: the bond
dissociation energy on the stabilization energies of the fragments, the bond length on the compressibility limit distance between the
atoms, the bond stretching frequency on the atom masses, etc. The bond stretching force constants however lead directly to bond order
and bond strength as has been demonstrated for the bonds in typical organic molecules. Using this insight, the generalized vibrational
frequencies of reacting molecules are used to obtain insight into the chemical processes of bond breaking and forming. An elementary
chemical reaction based on these processes is characterized by a curved reaction path. Path curvature is a prerequisite for chemical
change and directly related to the changes in the stretching force constants as they respond to the bond polarizing power of a reaction
partner. The features of the path curvature can be used to partition the reaction path and by this the reaction mechanism in terms of
reaction phases. A reaction phase is characterized by an elementary structural change of the reaction complex leading to a chemically
meaningful transient structure that can convert into a real transition state or intermediate upon changing the environmental conditions or
the electronic structure (substituents, etc.) of the reaction complex. A unified approach to the study of reaction mechanism (URVA:
Unified Reaction Valley Approach) is discussed that is based extensively on the analysis of vibrational modes and that is aimed at
detailed understanding of chemical reactions with the goal of controlling them.
Key Words: Cal-X methods, vibrational frequency, bond strength, chemical bond, adiabatic internal Coordinate mode (AICoM), unified
reaction valley approach (URVA), reaction path curvature, reaction path.
1. COLLABORATION BETWEEN EXPERIMENT AND
THEORY: THE CAL-X METHODS
In the last four decades quantum chemistry has become a
valuable and indispensable tool of chemistry that is peer to any
spectroscopic or analytic tool of chemical research [1-3]. Quantu m
chemical calculations can be utilized in three different ways. First,
they can lead to chemical information not amenable to experiment
(possibility I). This possibility is exploited whenever principle
reasons h inder chemical experiments, because they are too costly,
time consuming, or too dangerous. Secondly, quantum chemistry is
also often used to complement, confirm, or correct experimental
results (possibility II). Nowadays, this is done on a routine basis
and there are many experimentalists who take advantage of this
possibility without considering themselves as being quantum
chemists. The third possib ility (possibility III) is based on a
combination of experimental and computational tools with the
objective of extracting additional chemical information from
experimental measurements, especially information that is hidden
in experimental results obtained from spectroscopy, structure
analysis, and other chemo-physical measurements. Possibility III of
Address correspondence to this author at the Department of Chemistry, Southern
Methodist University, 3215 Daniel Ave, Dallas, Texas 75275-0314, USA; Tel: (214)
768-1300; E-mail: dcremer@smu.edu
utilizing quantum chemistry has been less frequently exploited
although a number of examples can be found in the literature (see
below). Unfortunately, different and sometimes misleading terms
have been used in connection with this approach. Therefore, we
will use in this article a terminology that clearly distinguishes this
use of quantum chemistry from possibilities I and II. We will speak
of the Cal-X combination methods, which use Calculations to
obtain important additional information from eXperiment, for
example measu red NMR chem ical shifts and spin-spin coupling
constants to determine the geom etry of molecules in solution
(NMR-Cal-X combination method), measured vibrational
frequencies to determine molecular structure and stability (Vib-Cal-
X combination method), measured infrared intensities to determine
atomic charges (Vib-Cal-X), etc.
etc
.
1a
1b
+
etc
.
7
1
1c
+
+
+
+
+
+
An examp le for an NMR-X approach concerning the structure
of the homotropenylium cation (Scheme 1) in solu tion is shortly
described here [4-6]. Early investigations had provided rather
From Molecular Vibratio ns to Bonding, Chemica l Reactions, and Rea ction Current Organic Chemistry, 2010, Vol. 14, No . 15 1525
confusing descriptions of the equilibrium geometry of this cation
[7-11]. Some investigations described the molecule as possessing a
rather short C1C7 bond of 1.6 Å typical of a bicyclic structure (1a)
whereas others suggested a long C1C7 interaction distance of 2.4 Å
typical of a monocyclic cation (1b). These observations had to be
considered critically because the structural information of
substituted homotropenylium cations referred to the solid state,
which did not provide a reliable basis for predicting the solution
phase structure of the cation. A direct structure determination of the
homotropenylium cation in solution would have provided a direct
answer to the question of its electronic nature, however was not
possible because of principle reasons. In this situation, the
dependence of NMR chemical shifts on the molecular geometry
was exploited computationally to determine the geometry of the
cation and its electronic nature in solution [4]. For a series of fifteen
1,7-distances increasing from 1.44 to 2.54 Å the geometry of the
cation was optimized and the 13C NMR chemical shifts were
calculated with the IGLO (individual gauge for localized orbitals)
method [12-14]. The minimum of the mean deviation of calculated
and solution phase measured NMR chemical shifts determined the
molecular geometry as possessing a 1,7-distance of 1,97 Å in
solution (1c, Scheme 1), which confirmed homoaromatic character
(delocalization of a 6 electron system bridging the methylene
group through-space corresponding to a non-classical carbocation
structure) [4]. This finding could later be confirmed by high level
ab initio calculations performed at the MP4 level of theory [4,6,15].
Similar studies were carried out in the sense of this early example
of an NMR-Cal-X approach to determine the geometry of donor-
acceptor complexes (BH3NH3, [5] F3SiONMe2 [16]), silyl cations in
solution, [17-21] carbo cations [4,5,22-24] or carbonyl oxides [25]
where some of this work is summarized in two review articles
[6,15].
The key feature of the NMR-Cal-X approach (originally coined
NMR/ab initio/IGLO or NMR/DFT/IGLO according to the
quantum chemical methods used [4-6,16-25]) is the dependence of
NMR chemical shifts on molecular geometry, i.e. the measured
NMR chemical shift values carry beside magnetic and electronic
structure information on a molecule also hidden geometric
information. If there is a large variation of the molecular geometry
in dependence of the environment, the changes in the chemical
shifts will be large enough to be unraveled with the help of the
quantum chemical calculations. This is the essence of the Cal-X
methods in general, which have the advantage to operate with
measured rather than computed values because the latter provide for
these methods just auxiliary quantities. Therefore, the question of
calculational accuracy is of minor concern. For example, the
problem of directly calculating the 1,7-distance of the
homotropenylium cation with highly correlated ab initio methods
such as coupled cluster with triple excitations (e.g., CCSD(T) [26])
was avoided at a time when these calculations were too costly. All
what was needed in the case described above was a reasonable
description of the 13C chemical shifts with the moderately co stly but
sufficiently reliable HF/IGLO method [14].
In this review article, the use of molecular vibrations in the
sense of a Cal-X method are discussed. The normal vibrational
modes of a molecule are investigated by infrared and/or Raman
spectroscopy and the information obtained from measured
vibrational spectra is provided in form of normal mode frequencies
and intensities [27-30]. Depending on its geometry, conformation,
and electronic structure, each molecule has typical vibrational
spectra, which are used to identify and characterize new chemical
compounds [29]. In this connection, experimental work is strongly
supported by quantum chemistry. If a given new compound is
formed in a mixture of different compounds, its identification, for
example by infrared spectroscopy, may be problematic. The
quantum chemical calculation of the infrared spectrum in question
clarifies which infrared bands at what frequencies with what
intensities can be expected thus facilitating the identification. The
combination of experiment and quantum chemical calculations has
been utilized numerous times, especially in connection with the
matrix isolation spectroscopy [31] of labile compounds such as
biradicals, [32-38] carbenes, [39,40] carbonyl oxides or dioxiranes
[41-43]. A highlight of this work was the first identification and
characterization of m-benzyne in the matrix by Sander and co-
workers using the CCSD(T) calculations of the biradical by Kraka
[33]. This however can be considered as a standard
complementation of experimental work by quantum chemical
calculations, which as such has little to do with the Vib-Cal-X
(calculation of Vibrational spectra to obtain additional information
from eXperimental spectra) approach discussed in Section 2.
Vib-Cal-X is based on the fact that the vibrational spectra of a
molecule contain detailed, although hidden, information on
bonding, geometry, charge distribution, and stability [44-49]. In so
far, vibrational spectroscopy should lead to the most complete
characterization of molecules, which in reality is far from being
accomplished. Most experimental work on medium-seized and
larger molecules uses vibrational spectra just for the rapid
identification of the functional groups of a molecule or a
fingerprint-type comparison of different molecules. In this review,
we will describe and summarize research that mak es it possible to
fully exploit the information content of measured vibrational
spectra utilizing quantum chemistry in form of the Vib-Cal-X
approach. Vib-Cal-X is based on the adiabatic mode concept of
Cremer and co-workers [50-54] and we will sketch the contents of
the latter in Section 2. In Section 3, we will apply Vib-Cal-X to
describe the bond strength of typical organic bonds. This will imply
some basic considerations on the nature of the chemical bond and
how this is best characterized utilizing experimental quantities.
Section 4 will lead from vibrating molecules to chemical reactions.
The physical connection between the force constant of a vibrating
bond, its changes under the impact of a reaction partner, and the
curving of the reaction path is drawn. The path curvature will be the
basis for the dissection of the reaction mechanism into reaction
phases. In Section 5, it will be discussed how the various phases
can be described using the properties of the reaction complex.
Important results, conclusions, and outlook of the mechanistic
analysis based on the vibrations of the reaction complex will be
presented in the last section.
2. THE VIB-CAL-X METHOD AND THE ADIABATIC MODE
CONCEPT
Modern vibrational spectroscopy has shifted its focus more in
the direction of macromolecules, polymers, materials, and living
matter using advanced techniques such as imaging, 2D correlation
spectroscopy or Operando spectroscopy [29,30]. Vib-Cal-X,
however, refers to the part of vibrational spectroscopy that aims at
the characterization of isolated, small to medium-seized molecules
(N 100 atoms) and that has already been considered as becoming
superfluous [55] because rivalling analytical methods such as NMR
or mass spectrometry lead to much more detailed information on
molecular composition and structure. In so far this article aims at
triggering a renaissance of traditional vibrational spectroscopy
carried by the combination of spectroscopic and computational
1526 Current Organic Chemistry, 2010, Vol . 14, No. 15 Cremer
and Kraka
work in the form of Vib-Cal-X. On the quantum chemical side,
there is no need to expedite and improve the computational
techniques used to calculate vibrational spectra, for example by
developing an easier and more accurate way of determining
anharmonic corrections to harmonic vibrational frequencies either
by explicitly calculating cubic and quartic force constants [56-58]
or implicitly obtaining correct normal mode frequencies with ab
initio molecular dynamics calculations [59-61]. (For a review on
the calculation of vibrational frequencies, see Ref. [62].) It is an
advantage of the Vib-Cal-X method that it is based exclusively on
measured normal mode frequencies whereas calculated frequencies
are only used as an analysis tool, which does not require high
accuracy . This makes it possible to apply Vib-Cal-X to many
interesting molecules using standard calculational techniques at
moderate costs. The unraveling of structural and stability
information from the vibrational data and a detailed insight into the
electronic structure of a target molecule is the rew ard of applying
traditional vibrational spectroscopy and its analytical tool Vib-Cal-X.
For the purpose of attaining the objectives of Vib-Cal-X, three
prerequisites have to be fulfilled: 1) Vibrational frequencies depend
both on the electronic structure of a molecule and the vibrating
mass, which leads to different values in the case of isotopomers and
makes a comparison of different bond types with regard to their
properties difficult. Therefore it is desirable to obtain from
measured vibrational frequencies the corresponding vibrational
force constants which depend exclusively on the electronic nature
of the chemical bonds of a molecule. The problem to be solved is
how to obtain force constants from measured vibrational
frequencies. - 2) Often the complete set of Nvib=3NL (N: number
of atoms of a molecule; L: number of translational and rotational
motions) cannot be determined in the experiment. In this situation,
a way has to be found to complement the experimental frequencies
in a reliable manner so that added frequencies are largely
independent of the way they have been generated. - 3) Vibrational
normal modes are in many cases delocalized modes that include a
larger number of atoms of the molecules. If one wants to describe
the bonds of a molecule with the help of its vibrational modes, these
have to be converted in some way into localized bond stretching
vibrations that do no longer couple to other vibrational modes of the
molecule. In the following we will present the so lutions to these
three problems.
2.1. Obtaining Vibrational Force Constants from Measured
Vibrational Frequencies
Vibrational frequencies are derived from the vibrational
eigenvalues of the standard vibrational equation [49, 63, 64]
FD = G
1D
(2.1)
where F is th e force constant matrix and D contains the normal
mode vectors dμ (μ=1,…,Nvib) given as column vectors. Both
matrices are expressed in terms of internal coordinates. Matrix G is
the Wilson matrix [27] (kinetic energy of the vibrating molecule)
and matrix is a diagonal matrix with the vibrational eigenvalues
μ
=4
2
c
2
μ
2
on the diagonal where
μ
is the (harmonic) vibratio-
nal frequency of mode μ and c the speed of light. The number of
vibrations Nvib determines the size of the four square matrices
involved.
From Eq. (2.1) it is obvious that the vibrational eigenvalues μ and
by this the vibrational frequencies
μ
are connected to the force
constants via the vibrational eigenvectors collected in D. These are
not available from experiment. The Wilson G matrix, however is
easily determined from the mass matrix M and the B matrix
G = B M
1B† (2.2)
with
Bni =
qn(X)
xi
x=x
e
(2.3)
and qn being an internal coordinate. Once G and a reasonable
approximation of matrix D are known, one can use first order
perturbation theory to derive from calculated force constants the
corresponding analogues associated with the measured vibrational
frequencies
μ
exp . For this purpose, the differences between
experimental and calculated frequencies,
μ
exp
μ
cal , and
correspondingly
μ
exp
μ
cal
are used to set up a correction matrix
associated with a correction matrix F. Furthermore, one
assumes that the perturbation leaves D unchanged so that the
new vibrational Eq. (2.4) has to be solved:
D(F0 +
F)D† =
+
(2.4)
It holds that
D† F0D =
(2.5)
because the normal mode eigenvectors collected in matrix D are
normalized with regard to G, i.e. DD†=G. The corresponding
equation for the first order correction is given by Eq. (2.6)
D†
FD =
(2.6)
which leads to
F = (D
1)†
(D)
1 = G
1D
D†G
1 (2.7)
and an experimental force constant matrix
Fexp = F0 +
F (2.8)
corresponding to the measured vibrational frequencies
μ
exp . Force
constants are not measurable quantities (frequencies are) and
therefore it is not problematic that the force constants of Eq. (2.8)
are harmonic. They absorb th e anharmonicities (described by cubic
and quartic force constants [56]) needed in theory to calculate
normal mode frequencies corresponding to the measured ones. It is
appropriate to speak in this case of effective force constants leading
directly to the correct vibrational frequencies measured by infrared
or Raman spectroscopy.
2.2. Complementing Measured Vibrational Frequencies to a
Complete Set
In the case that the set of Nvib vibrational normal modes is not
complete, frequencies, which have not been experimentally
observed, can be taken from calculated spectra after appropriate
scaling. Calculated vibrational frequencies are mostly based on the
harmonic approximation, which yields frequency values that are too
large by 5 - 15 %, especially in the case of vibrational modes
dominated by bond stretching motions (less problematic are the
frequencies of the framework deformation modes). Apart from
these deficiencies one has to consider that calculated vibrational
frequencies are in accurate because of the use of an incomplete basis
set or electron correlation errors of the method employed. For
From Molecular Vibratio ns to Bonding, Chemica l Reactions, and Rea ction Current Organic Chemistry, 2010, Vol. 14, No . 15 1527
molecules with heavy atom s relativ istic errors may also play a role.
A priori one cannot expect that a single scaling factor can account
for the different error sources, however in reality the harmonic
approximation causes by far the largest deviation from
experimental frequency values provided a reliable quantum
chemical method with a sufficiently large basis set is used. In the
last 25 years, scaled quantum mechanical force fields, [65] direct
scaling of force constants, [66] frequency scaling, [67] the effective
scaling frequency factor method based on the diagonal coefficients
of the potential energy distribution (PED) matrix, [68,69] or the
wavenumber-linear scaling method [70,71] have been used to adjust
calculated harmonic frequencies to experimental normal mode
frequencies. Also, a numerical procedure has been suggested to
derive force field scaling factors directly for the Cartesian force
constant matrix avoiding the introduction of internal coordinates
[72].
Utilizing one of the improved scaling methods one can reduce
average deviations from experimental frequencies from originally
close to 15 cm1 to currently just 3 cm1 wave numbers in selected
cases [73]. For 19, most frequently used methods and basis sets,
average scaling factors are available based on a reference set of 122
small molecules with 1062 different frequencies [67]. These scaling
factors are directly applied to all calculated frequencies. Although
the scaling procedures in use cannot reproduce experimental
frequencies exactly, they are sufficiently accurate to complement
the frequencies of an experimental vibrational spectrum where the
scaling is done in a small fraction of the time that would be needed
to obtain anharmonic corrections with the perturbation theory
approach in use today [56].
2.3. Localizing Vibrational Normal Modes
Normal vibrational modes are delocalized modes where the
degree of delocalization increases for lower frequency modes.
Delocalization can result from d ifferen t causes. Vibrational modes
which are equivalent because of symmetry (e.g. the CH stretching
modes of a CH3 group) mix with each other and lead to delocalized
modes. Coupling phenomena such as Fermi resonance or Darling-
Dennison resonance coupling [74,75] leads to mixing and a change
in vibrational frequen cies. Apart from these special cases, the
degree of delocalization can be assessed by the following
considerations based on Eq. (2.1) expressed in 3N Cartesian
coordinates:
fl
μ
=
μ
2
Ml
μ
,
μ
=l,…,N
vib
(2.9)
where f is the force constant matrix, lμ the μth vibrational
eigenvector, and M the mass matrix of the molecule in question. It
may be assumed that lμ is identical to the local mode vm associated
with the internal coordinate qm. This assumption will not be
fulfilled if one of two or both coupling mechanisms suggested by
Eq. (2.9) occur: i) An electronic coupling involving the potential
energy part of the vibrational problem according to
vn
†fvm
(2.10)
or a mass coupling involving the kinetic energy part
vn
†Mvm
(2.11)
The delocalized nature of vibrational normal modes is a result
of electronic and mass coupling as reflected by the fact that neither
the force constant matrix nor the Wilson G matrix are diagonal. The
off-diagonal force constants reflect the electronic coupling and the
off-diagonal elements of G the mass coupling. Naively, one could
say that only if all off-diagonal elements are zero, local modes are
enforced for the molecule in question. This however would
correspond to the unrealistic situation that the molecule would be
an union of diatomic (bond stretching vibrations), triatomic
(bending vibrations), tetratomic (torsional vibrations), and higher
entities whose interplay would not be clear. Any partitioning into
local modes would require that for a linear N-atom molecule N-1
local bond stretching vibrations are complemented by N-2 angle
bending, and N-3 torsion vibrations. Sizable bending (torsion) force
constants would imply an interaction between the bonds, which
would be excluded by the picture of local, non-interacting bonds
needed for the description of local bond stretching modes. Such a
situation may be partly given in a van der Waals complex
containing two monomers (e.g. two H2 molecules), however cannot
be adopted for the electronic structure of a molecule that is
characterized by the interplay of bonding and non-bonding
interactions. These considerations reveal that it is not possible to
partition delocalized normal modes into local modes. However, it
should be possible to project local modes out from normal modes or
to transform th e latter into the former where of course the question
arises whether the resulting modes will be physically reasonable. It
will be essential to clarify whether the modes in question lead to a
local mode frequency, force constant, mass, and intensity that
describe the local electronic structure. Furthermore, it is important
that normal modes can be described in terms of local modes, which
implies the definition of a weighting factor or amplitude. Finally, it
must be possible that local modes can be derived from measured
vibrational modes to guarantee a general applicability of the local
mode concept in question. Apart from listing these requirements it
may be useful to further discuss the nature of a local mode in
somewhat more detail.
If a local AB bond stretching vibration is initiated, local mode
behavior would imply that all atoms bonded to either atom A or
atom B should follow this motion without resistance. This rigid
vibration would truly be localized since no other vibration will be
triggered. Of course, the rigid local bond stretching vibration would
imply a much too strong energy increase since the molecule would
not be able to adjust to the displacements of atoms A and B and the
periodic change in the bond length. The stretching force constant of
the rigid, local motion would be too large and when used to
describe the strength of bond AB would predict a much too strong
bond. A more realistic situation would result if the rest of the
molecule could always adjust to the changes in the bond length AB
so that the molecular energy would always be minimal for any
displacement r(AB). The relaxation of the molecular geometry for
a given vibrational motion does not imply that the latter is
delocalized within the rest of th e molecule. The other atoms still
follow the AB stretching motion, however now in a non-rigid
fashion. This is the principle of the adiab atic internal coordin ate
modes (AICoMs), [50-54] which will be discussed in Section 2.4.
However before doing so, other approaches aimed at the
determination of local vibrational will be summarized.
Averaging of Frequencies: Intrinsic Frequencies
The OH stretching modes of water split up into a symmetric
and an asymmetric stretching mode possessing vibrational
frequencies of 3657 and 3756 cm1 (Fig. 1), [76] neither of which
presents a measure of the strength of the OH bond in water. One
could assume that the arithmetic mean of the two frequen cies,
1528 Current Organic Chemistry, 2010, Vol . 14, No. 15 Cremer
and Kraka
3706.5 cm1, is such a measure. A theoretical approach based on
this idea was suggested by Boatz and Gordon [77], who derived the
intrinsic frequencies
n
in
as representatives of local mode frequen-
cies asso ciated with an internal coordinate qn.
n
in
()
2
=P
nm
μ
m=1
N
Parm
μ
=1
N
vib
μ
2
(2.12)
where
P
nm
μ
is an element of the PED density matrix measuring the
contributions of the internal vibrational modes cm associated with
the internal coordinate qm [77] and NParm defines the number of
internal parameters used in the set of internal coordinates. The
matrix C that collects all vectors cm relates normal mode
eigenvectors expressed in Cartesian coordinates, lμ, to those
expressed in internal coordinates, dμ, according to
l
μ
= Cd
μ
(2.13)
Considering that D relates internal coordinates q to normal
coordinates Q according to
q = DQ, (2.14)
a local mode driven by internal coordinate qm would require that the
corresponding mode vector dμ fulfills the requirement dμ,m= m,μ for
all its elements dμ,m thus yielding lμ=cm. Local modes expressed in
form of c-vector modes will not couple if all off-diagonal force
constant and G-matrix elements are zero, which leads as pointed
out above to an unrealistic electronic structure. Hence, they couple
as is reflected by Fn,m and Pn,m 0. Averaging of vibrational
frequencies leads to different values for different sets of internal
coordinates (unless redundant coordinate sets are used), a large
scattering of frequencies for similar bonds. Negative frequencies
may result, which prohibit an electronic structure analysis because
they cannot be associated with a vibrational mode and a force
constant.
Projecting Out Local Modes
Local modes are the appropriate tool to analyze delocalized
vibrational modes in biomolecules. Therefore various attempts have
been made to calculate local vibrational modes. Jacob and Reiher
[78] recently suggested a procedure that leads to modes which are
maximally localized with respect to a suitably defined criterion.
They chose two localization criteria in analogy to mathematical
procedures used for the localization of molecular orbitals. The
authors showed how the application of localized modes to the
vibrational spectrum of the alpha helix of (Ala)20 polypeptide
makes it possible to visualize which atoms dominantly contribute to
each normal mode leading to a characteristic p attern for each
residue. The coupling between the local modes defines the
interaction between the residues. Choi and Cho [79] describe the
extent of delocalization and the vibrational properties of amide
normal modes in dipeptides by amide local mode frequencies and
intermode coupling constants utilizing suitable local vibrational
modes. The latter are derived by a generalized form of the Hessian
matrix reconstruction method [80]. In this procedure, the mass-
weighted Hessian defined in atomic Cartesian coordinates is
divided into submatrices associated with the peptide fragments.
Diagonalization of the fragment submatrices leads to the
frequencies of vibrations localized at the two fragments. The
diagonal elements of the submatrices correspond to the force
constants of the fragments and the off-diagonal elements describe
their coupling. This method can be applied to other systems with a
repeating fragment unit such as nucleic acids. In each case, the
Fig. (1). Schematic representation of the three normal modes of the water molecule (from left to right): symmetric OH stretching, asymmetric OH stretching,
and HOH bending mode (top); adiabatic OH stretching and HOH bending modes (bottom). Experimental normal [76] and adiabatic mode frequencies are
given in cm1. Displacement arrows are given not to length scale to illustrate atom movements. Dashed arrows indicate movement of H atoms due to adiabatic
relaxation.
From Molecular Vibratio ns to Bonding, Chemica l Reactions, and Rea ction Current Organic Chemistry, 2010, Vol. 14, No . 15 1529
degree of localization of the reference modes can be determined
according to worked out procedures [81].
Local Vib rational Modes in Crystals
In solid state physics, crystal defects or impurities lead to a
lowering of the lattice symmetry and the appearance of additional
vibrational modes [82]. These modes possess frequencies
significantly separated from the lattice frequencies provided the
mass of the impurity atom (H, C, etc.) is smaller than that of the
lattice atoms. Therefore, one speaks of local vibrational modes that
can be used to localize and analyze the lattice impurity [82-84]. The
use of the term local vibrational mode differs in this case clearly
from that used in molecular vibrational spectroscopy, since the
former modes couple with the modes of the surrounding lattice and
are only in so far localized as they do not involve lattice positions
beyond the immediate surrounding.
Site-specific Modes from Nuclear Resonance Vibrational
Spectroscopy
Related to the local vibrational mode spectroscopy of solid state
physics is nuclear resonance vibrational spectroscopy (NRVS) [85-
88] in so far as it leads to site-specific vibrational modes. NRVS
uses synchrotron radiation to excite Mössbauer active isotopes such
as 57Fe and measures beside the zero-phonon resonance also
transitions corresponding to the complete set of vibrational modes
involving the excited nucleus. Hence, NRVS can be considered as
Mössbauer spectroscopy with vibrational sidebands. Localized
vibrational modes are obtained, however no real local modes
because the modes measured exhibit a significant degree of
delocalization [88].
Isolated Stretching Modes
Local modes in the sense of the AICoMs to be discussed in
Section 2.4 are the isolated XH stretching modes originally
investigated by McKean [89-95]. The CH stretching mode
undergoes a Fermi resonance with the first overtone of the CH3 and
CH2 bending modes, which leads to a CH stretching frequency that
does no longer reflect the true nature of the CH bond [44]. A way of
suppressing Fermi resonances is to replace all H atoms but the
targeted H atom by deuterium. This has also the advantage that in
CH3 or CH2 groups mixing of CH stretching modes is suppressed so
that the stretching mode of the remaining CH bond is largely
localized. McKean investigated D-isotopomers of many organic
compounds and presented the corresponding CH stretching
frequencies [89]. They nicely correlated with known r0(CH) bond
lengths and could be used to predict unknown values with a
precision of 0.001 Å [89]. Larsson and Cremer [44] calculated
isolated CH stretching frequencies and compared them with the
corresponding AICoM CH frequencies. Their work revealed that
AICoMs are the theoretical equivalences of McKean’s isolated
stretching modes: The two types of local CH stretching modes
overlap by more than 98%, which is reflected by a linear
relationship between them with R2 = 0.997. The only exceptions
were found for CH stretching in alkines where the isolated CH
modes were still contaminated due to some coupling with the CC
stretching motion [44].
The isolated stretching frequencies are based on a suppression
of mass coupling, Fermi resonances, and other mode-mode
coupling effects. Due to a doubling of the hydrogen mass,
frequencies for C-H/C-D stretching vibrations are significantly
shifted to lower values thus eliminating a mixing with the
remaining CH stretching motion. Isolated stretching frequencies,
despite of reducing mass coupling, still suffer from electronic
coupling since the CH stretching force constants are not affected by
D substitution. Also, an extension of the original concept of the
isolated stretching frequencies to other than just XH bonds [96, 97]
is difficult since a doubling of the nuclear mass is not possible for
any other element frequently used in organic chemistry.
Local Vibrational Modes from Overtone Spectroscopy
Information on local XH stretching modes can also be obtained
from the overtone spectra of these vibrational modes [98]. Henry
has shown that the higher overtones of a XH stretching mode can
be described with an anharmonic potential of a quasidiatomic
molecule [98]. Higher overtones (v 3) possess considerable local
mode character [99-101]. For overtones with v = 5,6 one observes
mostly one band for each unique XH bond, even if there are several
symmetry equivalent XH bonds in the molecule. In fundamental
and lower overtone modes, there is always a splitting of the
frequency into, e.g., a symmetric and an antisymmetric mode
frequency of two symmetry equivalent XH stretching modes,
whereas this splitting virtually disappears for overtones with v
5. In general, the different linear combinations of symmetry
equivalent XH stretchings become effectively degenerate for the
higher overtones.
The local mode behavior of the fifth overtone (v=6) of CH
stretching modes can be verified by comparison with the
corresponding AICoM frequency [49]. There is a linear relationship
between the two quantities (correlation coefficient R2 = 0.990),
which again confirms that AICoMs are suitable local vibrational
modes. The use of overtone spectroscopy as a means of obtaining
information on local vibrational modes and their properties is
limited to terminal bonds, of which so far only XH (X = C, N, O, S,
etc. [102,103]) bonds were investigated. An extension of the local
mode description by overtone spectroscopy is rather limited.
Local Mode Information from Compliance Force Constants
Another way of obtaining local mode information seems to be
provided by the compliance force constants [104-110]. The latter
are obtained by expressing the potential energy of a molecule in
terms of generalized displacement forces rather than internal
displacement coordinates [104, 105]:
V(g)=1
2qFq
(2.15)
V(g)=1
2gqCgq
(2.16)
where the elements of the compliance matrix C are given as the
partial second derivatives of the potential energy V with regard to
forces fi= gi and fj= gj:
C
ij
=
2
V
f
i
f
j
(2.17)
The gradient vector gq of Eq. (2.16) can be obtained by
differentiation of Eq. (2.15):
gq = Fq (2.18)
thus yielding
V(g)=1
2q
†
F
†
CFq
(2.19)
Hence, the compliance matrix C is identical with the inverse of the
force constant matrix:
1530 Current Organic Chemistry, 2010, Vol . 14, No. 15 Cremer
and Kraka
C = F
1 (2.20)
From Eq. (2.18) one sees that
q = F
1g (2.21)
The diagonal compliance force constant Cii gives the
displacement of internal coordinate qi under the impact of a unit
force whereas all other forces are allowed to relax [105]. This leads
to the fact that off-diagonal elements of C are largely reduced
suggesting a suppression of electronic coupling. There is no
indication that also mass coupling is reduced. In addition, the
compliance force constants are force constants without a vibrational
mode. Nevertheless, there is need in view the increasing use of
compliance force constants to investigate to which extend local
mode information is provided by the compliance force constants.
2.4. The Adiabatic Mode Concept
In Section 2.3, it has been described how the molecule should
move under the impact of a local vibrational motion. Considering
the similarity of internal coordinates and internal coordinate force
constants between molecules of the same type, one can expect that
electronic coupling is small. However, there is no guarantee that
mass coupling is also small. Accordingly, it is important to
eliminate mass coupling to obtain an internal vibrational mode vn
associated with the in ternal coordinate qm that describes a molecular
fragment m (bond length rm for a diatomic entity, bending angle for
a triatomic entity, etc.) that leads to atom movements exactly as
described in Section 2.3. Clearly, a suppression of mass coupling
could be accomplished if all m asses but those of the fragment m
would be zero. This is the idea of the AICoMs as first realized by
Konkoli and Cremer [50]. For the purpose of deriving the AICoMs
the Euler-Lagrange equations of a molecule are written in the
modified form of Eq. (2.22).
p
i
=
V
x
i
,i=1,…,3K
m
(2.22a)
0=
V
x
j
,j=3K
m
+1,…,3N
(2.22b)
where it is assumed that fragment m has Km atoms and is described
by 3Km Cartesian coordinates (a formulation in internal coordinates
has been given in the original literature [50-54]. The masses of the
other nuclei (Km+1,…,N) are zero, which leads to p
•
j = 0 (j = 3Km +
1,…,3N) for the momenta related to the Cartesian coordinates of the
nuclei outside fragment m, where
pi=
L
xi
=mi
xi,i=1,…,3N
(2.23)
and,
pj = 0, j = 3Km + 1,…,3N (2.24)
Eqs. (2.22) can be solved by rewriting them in the form of (2.25)
i
=
V
x
i
,i=1,…,3K
m
(2.25a)
0=
V
x
j
,j=3K
m
+1,…,3N
(2.25b)
with
p
i
=
i
,i=1,…,3K
m
(2.26)
Using (2.25),
x
k
=x
k
1
,…,
3K
m
()
,k=1,…,3N
(2.27)
and combining (2.23), (2.26) and (2.27) the time dependence of i
and thereby xk can be found [50]. In the harmonic approximation,
the solution of dynamical Eqs. (2.24) leads to the vibrational
problem and an eigenvalue equation, which determines the set of
local mode vectors
v
n
u
a
n
u
(u = 1,…,3KmLm) that describe the
AICoM vibrations of the fragment m of the molecule.
If one uses Eq. (2.22b) instead of Eq. (2.27), one can obtain,
x
j
=x
j
x
1
,…,3K
m
()
,j=3K
m
+1,…,3N
(2.28)
which reveals that nuclei not belonging to fragment m follow
exactly the motions of the fragment nuclei due to their zero mass,
i.e. the fragment nuclei determine th e motion of the whole molecule
for the A ICoMs determin ed. This fulfills exactly the requirement
for a local vibrational mode formulated in Section 2.3.
The second requirement for the local vibrational modes implies
that for any displacement of the nuclei belonging to m, the other
parts of the molecule are relaxed so that the molecular en ergy
always adopts a minimum. This is expressed in Eq.s (2.29).
V(x) = min (2.29a)
xi = const, i = 1,…,3Km (2.29b)
where th e coord inates xi give the displacements of the fragment
nuclei from their equilib rium value xi=0. These displacements are
kept frozen whereas those of the other nuclei are relaxed until the
molecular energy adopts a minimum. Using the method of
Lagrange multipliers with 3Km multipliers i leads to the Eqs.
(2.29), which are equivalent to Eqs. (2.25) [50]. Actually, Eqs.
(2.29) describe the motion of a molecule where the nuclei leading
the motion move infinitesimally slowly, which has led to speaking
of adiabatic motions. Hence, the AICoMs fulfill via Eqs. (2.28) and
(2.29) both requirements of a local vibrational mode.
Properties of Adiabatic Internal Coordinate Modes
Each AICoM is driven by a specific internal coord inate. In this
work, we will focus on the AICoMs associated with bond lengths,
i.e. the bond stretching modes. Hence, a molecule is considered as
the union of quasi-diatomic units. The adiabatic internal mode
a
m
Q
associated with internal coordinate qm and expressed in terms of
normal coordinates is given by Eq. (2.30):
Q
μ
(m)=(am
Q)
μ
qm
*
(2.30)
where
qm
*
is the internal coordinate driving the AICoM by adopting
a fixed displacement value as indicated by the *. The elements of
vector
a
m
Q
are given by Eq. (2.31):
am
Q
()
μ
=
Dm
μ
k
μ
Dmv
2
kv
v=1
N
Vib
, (2.31)
The AICoM
a
m
Q
can be transformed into an AICoM expressed
in Cartesian coordinates, am, with the help of the L-m atrix
From Molecular Vibratio ns to Bonding, Chemica l Reactions, and Rea ction Current Organic Chemistry, 2010, Vol. 14, No . 15 1531
containing the normal mode eigenvectors lμ expressed in Cartesian
coordinates.
a
m
=La
m
Q
(2.32)
so that AICoMs are completely specified no matter whether internal
coordinates, normal coordinates or Cartesian coordinates are used.
Each AICoM possesses a force constant
k
m
a
k
m
a
=a
m
†
fa
m
(2.33)
and a mass
m
m
a
m
m
a
=(b
m
†
a
m
)
2
b
m
†
M
1
b
m
=G
mm
(2.34)
where Gmm is a diagonal element of the G matrix, vector bm
corresponds to the mth column of the B matrix, and
b
m
†
a
m
=1
(2.35)
since the AICoMs are properly normalized. Hence, the AICoM
mass is based on a generalization of the reduced mass to internal
coordinates connecting more than two atoms. With the AICoM
force constant and the AICoM mass, it is straightforward to obtain
the AICoM frequency
m
a
=(a
m
+
fa
m
G
mm
)
1/2
=(
k
n
a
m
m
a
)
1/2
(2.36)
The force constant, frequency, and mass associated with a given
AICoM for internal coordinate qm fully ch aracterize it. This
information can be used to investigate normal modes by
considering them as being composed of AICoMs. If one knows the
decomposition of a normal mode in terms of AICoMs, then one can
clarify whether the normal modes are more or less delocalized and
what electronic or geometric inform ation they contain.
The AICoMs are the dynamic counterparts of the internal
coordinates, i.e. the latter are used to describe the equilibriu m
geometry of a molecule (static property) whereas the former
describe the dynamic behavior of a molecule including the
vibrational motions and the translational motion of a reaction
complex during a reaction (dynamic behavior of a molecule). The
dissection of the normal modes into AICoMs cannot be done in the
sense of a simple partitioning. Instead, amplitudes Amμ are
determined that reflect the contribution of an AICoM to a given
normal mode according to three essential criteria: [52,53] 1)
Symmetry criterion: symmetry equivalent AICoMs have the same
amplitude in a normal mode provided the normal mode retains this
symmetry. 2) Stability criterion: AICoM amplitudes are not
influenced by the choice of the internal coordinate set. The
amplitudes should not change for a normal mode if they are
calculated with different redundant internal coordinate sets and the
differences in the parameter sets only concerns coordinates
irrelevan t to the normal mode. 3) Dynamic criterion: There is a
relationship between the amplitude Amμ of an AICoM contained in a
normal mode and the difference
m
μ
=m
μ
in the w ay that a
small difference implies large amplitudes while large differences
lead to very small amplitudes. In other words, the scattering of
points Amμ in dependence of
m
μ
=m
μ
should be enveloped
by a Lorentzian curve. - B ased on these three criteria the A ICoM
amplitudes Amμ are given by Eq. (2.37):
A
m
μ
=
l
μ
|f|a
m
2
l
μ
|f|l
μ
a
m
|f|a
m
(2.37)
Eq. (2.37) provides the basis of a Characterization of Normal
Modes (CNM) in terms of AICoMs and as such it is the counterpart
of the PED analysis [111-114]. Contrary to the CNM approach, the
PED analysis suffers from several deficiencies that can lead to non-
physical results as has been demonstrated elsewhere [49,52,53].
Advantages of AICoMs
AICoMs have the following advantages that distinguish them
from other attempts of defining local vibrational modes:
1) They are derived from a clear dynamic principle, namely the
leading parameter principle, which points out that a single internal
coordinate qm (in general, a single internal parameter) defines the
displacements of the nuclei from their equilibrium positions and, by
this, leads the internal mode am. This leading parameter principle
[50,51] implies a new set of Euler-Lagrange equations because the
generalized momenta for all other internal coordinates qn (nm)
become zero [50], which can be pictured in the way that all atomic
masses outside the molecular fragment are considered as massless
points. This is equivalent to requiring that the potential energy V is
minimized for a geometry perturbation under the constraint that the
perturbation is defined by
q
m
*
[50].
2) All vibrational mode properties (adiabatic force constant,
adiabatic mass, and adiabatic frequency) are clearly defined.
3) AICoM amplitudes Amμ lead to the CNM analysis of normal
modes in a more sound and physically meaningful way than
provided, for example, by the PED analysis. The CNM analysis
provides an easy way of analyzing vibrational spectra and
quantitatively specifying the degree of delocalization of each
vibrational mode.
4) AICoMs and the CNM analy sis simplify the correlation of
the vibrational spectra of different molecules.
5) AICoM intensities can be used to investigate the charge
distribution in a molecule [54].
6) AICoMs are easily calculated utilizing measured vibrational
frequencies as described in Section 2.1. In this way experimental
information can be used in the sense of a Cal-X method thus
leading to Vib-C al-X.
7) Harmonic AICo Ms can be scaled to resemble experimental
AICoMs. In this way, they provide the basis of a powerful general
scaling procedure.
8) AICoM bond stretching force constants can be used to define
bond order and bond strength for any bond type [48,49].
9) AICoMs can also be defined for and applied to a reacting
molecule. In this case, the AICoMs are based on generalized
vibrational modes. They can be used to analyze the changes of the
vibrational modes along the reaction path, the curvature and the
Coriolis coupling constants, energy transfer and energy dissipation
as well as other properties of the reaction complex along the
reaction path [115-124].
10) Special AICoMs can be defined to analyze the translational
motion of the reaction complex along the reaction path, which leads
1532 Current Organic Chemistry, 2010, Vol . 14, No. 15 Cremer
and Kraka
to an analysis of the reaction path direction and its curvature [115-
124].
There is a large application spectrum for AICoMs, which is
currently exploited. This includes the automated analysis and
comparison of vibrational spectra [125], their use in connection
with the Vib-Cal-X approach to obtain geometry, bonding, and
electronic structure information, the set up of force fields based on
AICoM force constants, and the systematic investigation of the
mechanism of chemical reactions. Some of these application
possibilities will be discussed in Section 3. We will show that
AICoM force constants are suitable descriptors for the strength of a
chemical bond.
3. A BASIC VIEW ON THE CHEMICAL BOND AND ITS
DESCRIPTION BY MEASURABLE QUANTITIES
Both stru cture, stability , and reactivity of a molecule can be
predicted if the nature of its chemical bonds is understood [126-
132]. One of the central questions in chemistry is how to find easily
accessible bond properties that reflect the strength of the chemical
bond. Frequently mentioned in this connection are bond length
r(AB), bond dissociation energy (BDE) and bond energy (BE) of
bond AB, bond density (AB), one-bond spin-spin coupling
constant J(AB), bond stretching frequency
(AB), and bond
stretching force constant k(AB). One has also defined the bond
order n (number of bonding - number of antibonding electrons) as a
simple measure of the bond strength. The bonding character of the
electrons is determined via the molecular orbitals they occupy. The
orbital-based definition of the bond order will work only if the
orbital in question is localized in the bond and can be identified as
being bonding or antibonding via its nodal surfaces. In polyatomic
molecules, especially when hetero atoms are involved, these
requirements are seldom fulfilled. However, even if the
requirements would be fulfilled, the bond order defined in this way
could only be used to describe different bonding situations for one
type of bond, e.g. just for CC bonds. Use of a bond order to
compare the strength of different bonds is problematic.
In general, the strength of the chemical bond is the result of a
covalent and a polar (ionic) contribution [126,127]. In
perturbational MO theory [127,128], the covalent part can be
assessed by the overlap of the fragment orbitals forming the
bonding MOs and the stabilization of the latter, which is also
related to the energies of the fragment o rbitals forming the MOs. If
both bonding and antibonding orbitals are occupied, all stabilizing
and destabilizing contributions have to be considered where the
antibonding MO is always more destabilized than the bonding MOs
stabilized. The ionic character of the bond always leads to an
increase of the bond strength [126]. It has to be assessed via the
electronegativity difference between the atoms involved in bonding
and the resulting charge transfer from the more electropositive to
the more electronegative atom.
3.1. Assessing Bond Strength from Bond Energy (BE) or Bond
Dissociation Energy (BDE)
BE and BDE are the two quantities that are often used to
describe the strength of a chemical bond [45]. The BE accounts for
both covalent and ionic contributions to the bond strength.
However, the BE is a model quantity that cannot be measured.
Normally, one uses the atomization energy (AE) to derive BEs. For
example, the BE(XH) of a highly symmetrical molecule XHn is just
the nth part of the corresponding AE. However, if there are
Fig. (2). Schematic representation of a Morse potential for the dissociation of the bond AB in HpABHq. Bond dissociation energy BDE, intrinsic bond
dissociation energy IBDE, fragment stabilisation energy ES=EDR+EGR, density reorganization energy EDR, geometry relaxation energy EGR, and compressibility
limit distance dc are indicated.
From Molecular Vibratio ns to Bonding, Chemica l Reactions, and Rea ction Current Organic Chemistry, 2010, Vol. 14, No . 15 1533
different bonds in a molecule it is no longer clear how to split up
the BE between the individual bonds. One has to revert to
assumptions that are in general difficult to make in a reasonable,
insight providing way. Therefore, there is currently no reliable BE
model by which the strength of the bond or a bond order n can be
determined in an unambiguous, straightforward way.
It is still a wid e-spread custom to utilize measured BDE values
to describe the bond strength. This is however misleading since the
BDE depends on two quantities as is illustrated in Fig. (2) where for
a molecule HpABHq the Morse potential
V(r)=De(1 ea(rre))2De
(3.1)
is schematically drawn. De, re, a denote the BDE at equilibrium, the
equilibrium bond length, and the Morse constant, respectively,
which reflects the anharmonicity (”width”) of the potential function
V(r) for dissociation of the AB bond. Besides the bond strength of
bond AB being broken, also the stabilization energies (SEs) of the
fragments generated determine the magnitude of the BDE (Fig. 2).
If the SE values are large because of geometry relaxation effects
(pyramidal to planar, bent to linear, etc.) and density reorganization
(conjugation, hyperconjugation, or anomeric delocalization), the
BDE becomes small (see Fig. 2), perhaps disguising the fact that
the actual bond strength is large. Hence, the BDEs do not provide a
reliable measure for bond strength.
In Fig. (2), it is indicated that the sum of BDE and SEs,
henceforth called the intrinsic bond dissociation energy, IBDE =
BDE + SE, would provide a reliable measure of bond strength. Of
course, IBDEs are model quantities, which cannot be measured
[45]. Nevertheless, it is advantageous to estimate the magnitude of
the IBDE. It corresponds to an electronic situation of the fragments
that exactly mirrors the situation in the molecule, i.e. geometry,
charge distribution, hybridzation are kept frozen in the situation of
the molecule and in this way all SE effects are excluded. This
hypothetical situation is difficult to model and even more difficult
to calculate. An estimate however can be giv en in the case of
methane. In the atomization process, C(3P) and four H(1S) atoms
are formed that possess different electron density distributions than
C and the H atoms in the methane molecule. The electronic
situation for carbon being bound in methane can be modeled by
promoting a 2s electron into the empty 2p orbital thus yielding the
C(5S) or C(3D) state and then mixing s and p orbitals to form four,
tetrahedrally oriented sp3 orbitals. The energy needed for these two
processes is about 100 + 60 kcal/mol, [45] which leads to a 40
kcal/mol contribution to the BE(CH) value of methane being AE/4
= 104 kcal/mol. Hence the intrinsic BE (IBE) of methane is at least
104 + 40 = 144 kcal/mol because the actual density distribution in
methane cannot be remodeled utilizing a hybridization model and
the energy required to restore the exact density distribution of
methane (anisotropic density around the H nuclei, density paths
extending into the directions of the 4 H atoms) is not known. A
distortion of the density in this way leads to a non-negligible
increase in energy even at the H atoms so that IBE(CH) values
between 150 - 170 kcal/mol are likely.
The IBE(CH) value of methane can be used to obtain the
IBDE(CH) value for the dissociation of methane into methyl radical
and H atom since in both cases the same reference densities and
geometries are used (which is no longer the case for the methyl,
methylene, or methine dissociation that start from different
references with relaxed geometry and electron density). Contrary to
the AE of m ethane, which is equal to the sum of the four different
BDEs, the sum of the four IBDE values will not be equal to
intrinsic atomization energy (IAE) because for each of the four
intrinsic processes a new reference state of lower energy will be set
(frozen CH3 vs. relaxed CH3, frozen CH2 vs. relaxed CH2, etc.) so
that the sum of the IBDE should become smaller than the IAE value
(but still larger than AE ).
For diatomic molecules AB, th e BE is identical to the BDE and
one tends to consider the BDE as a direct measure of the bond
strength [133]. We note that the BDE, even in this case, is not equal
to the IBDE (keeping the electron density distribution of A and B
frozen at that of the molecule AB). Hence, the comparison of the
BDE(AB) values of different AB molecules does provide only
qualitative rath er than quantitative answers to th e relative strength
of different bonds AB. Clearly, relaxation of the electron density of
a Li atom upon dissociation of Li2 is clearly smaller than that of N
upon dissociation of N2. Accordingly, the SE value is larger for N2
and the use of its BDE as bond strength indicator leads to an
underestimation of the bond strength as compared to that of Li2.
This has to be considered when using listed BDE values to describe
bonding in diatomic molecules [133]. We conclude that neither BE
nor BDE values provide a reliable measure for the bond strength
because they always depend on both bond strength and the
fragment stabilization energies.
3.2. Assessing Bond Strength from the Bond Length
In chemistry, it is largely accepted that a shorter (longer) bond
length is indicative of a stronger (weaker) bond, especially if the
same bond type AB is considered [126,127]. One might argue that
it is difficult to define the actual bond length. Bonds can be bent as
found in small ring molecules [131,132,134,135]. In such a case the
internuclear connection line between two bonded atoms does no
longer coincide with the bond path and it is the question how to
experimentally determine the bending of the bond. It has been
argued that the maximum path of electron density connecting the
atoms is an image of the bond [131,132] and therefore leads to a
reasonable definition of the bond length. Bending of bonds, if
described via electron density paths, is actually also relevant for
acyclic molecules since there is indication for small molecules that
the path of maximum electron density does generally not follow the
internuclear connection line although deviations are small
[131,134,135]. These considerations reveal that a strict definition of
bond length seems to be only possible within a given model or
concept (electron density model, orbital model, geometric model,
etc.), which of course is a result of the fact that the chemical bond
itself is just a model quantity rather than an observable quantity,
which leads in consequence to the fact that none of the so-called
”bond properties” are observable. There is no experiment that could
measure a bond length. Measurable are (vibrationally averaged)
atomic positions from which one can derive bond lengths using
some model of the chemical bond. Common bond models used in
chemistry are applicable only in a limited sense. They fail when
bonded and non-bonded situations have to be distinguished in
border cases.
For the moment, we set aside th ese basic considerations and
examine whether at least in a semiquantitative sense bond
lengthening (shortening) is an indicator of bond strength. The bond
distance is the result of two different contributions. First of all, the
size of the atoms connected by a bond determines the magnitude of
the atom, atom distance. Often this dependence is estimated on the
basis of known atomic radii, covalent radii, or ionic radii depending
on the type of bonding. Clearly, these radii depend on the number
of core shells of an atom A, i.e. with increasing period within a
1534 Current Organic Chemistry, 2010, Vol . 14, No. 15 Cremer
and Kraka
given group the atom-specific radii increase thus leading to larger
bond lengths. Since atoms are polarizable and compressible, it is
useful to define a situation of highest atom compressibility
(compressibility limit) given by the steep increase of the potential V
of Eq. (3.1), which accordingly can be rewritten in the form of Eq.
(3.2) (see also Fig. 2):
V(r>r
c
)=D
e
(1 e
a
e
(rr
e
)
)
2
D
e
(3.2a)
V(r
c
)=V
c
=D
e
(1 e
a
e
(r
c
r
e
)
)
2
D
e
(3.2b)
For r-values smaller than the compressibility limit value, the
potential takes the value V(r < rc)=. The atom-specific
compressibility limit radius rc of an atom is more useful than any
other atom radius to specify the influence of atomic size on the
bond length. The value of rc is about half the magnitude of the
covalent radius and is either calculated or empirically determined
[136-139]. The sum of the compressibility limit radii of atoms A
and B determines the distance dc identical to a minimum contact
distance (see Fig. 2), which turns out to be largely independent of
the nature of A and B. Badger [136,137] showed that in a first
approximation the distance dc for all bonds AB with A being
located in period i of the periodic table and B in period j exclusively
depends on i and j, i.e. dc(AB)=dc(i,j) for all atoms A with the same
i and all atoms B with the sam e j.
If the minimum contact distance dc(i,j) of a given bond AB is
known, the difference Reff = redc(i,j) provides an effective bond
length Reff that is directly related to the bond strength of AB. The
smaller Reff the stronger is the bond. Since effective bond lengths
have been determined for many different bonds, the geometry of a
molecule may provide a simple basis to discuss the strengths of its
bonds provided one sets aside all principle qualms on the definition
of the bond. In practise however, effective bond lengths determined
via universal compressibility distances turn out to be too inaccurate
to be reliable bond strength descriptors. The compressibility limit
radius critically depends on the charge situation of a bonded atom.
A cation A+ has clearly a smaller, and an anion A clearly a larger
compressibility radius than A itself. In molecules, electronegative
(electropositive) substituents generate partial positive (negative)
charges on A. For example, DFT calculations reveal that in
fluoroamine radicals and fluoroamines of the formula NFnH2n
(NHF, NF2), NFnH3n (NH2F, NHF2, NF3), and CH3NFnH2n
(CH3NHF, CH3NF2) the shorter NF bonds are weaker than the
longer ones (see Fig. 3). This effect is significant and confirmed by
structural and spectroscopic measurements in the case of the amines
NFnH3n [140-143]. The compound with the largest number of
fluorine atoms is always the one with the shortest NF bond length.
In this case, the F atoms strongly withdraw electron density from
the N atom they are attach ed to. Consequently, the N atom
approaches the situation of an N
+ ion, which possesses a smaller
compressibility (and covalent) radius than N itself. With in creasin g
F substitution, the NF bonds become shorter because they
correspond more to N
+F
rather than NF bonds. The bond
shortening causes an increase in electron pair repulsion between the
lone pairs at F and N. Despite the bond shortening, destabilizing
four electron interactions reduce the bond strength. The NF
stretching force constants decrease with decreasing bond length and
reflect the weakening of the NF bonds (Fig. 3). This observation
seems to be in conflict with common chemical thinking that expects
bond strengthening upon bond shortening. However when adjusting
the compressibility limit radius of the N atom to the value of an N+
ion, the effective bond length Reff becomes larger with increasing F
substitution confirming the general observation that (effectively)
longer bonds are weaker bonds.
A similar effect is observed for the OF bonds in OF2 and HOF,
FOF [144-146] however not in the series ONFnH3n
(ONH2F,
ONHF2, ONF3; included in Fig. 3) nor for the CH bonds in the
series CFnH4n (CH3F, CH2F2, CHF3, CF4) since there are no vicinal
Fig. (3). Adiabatic stretching force constants of NF bonds in dependence of equilibrium bond lengths r(NF) calculated at B3LYP/6-31G(d,p) [139]. The
general exponential decay of ka(NF) for increasing r(NF) is not fulfilled by three fluoroamine classes: In these cases, the NF stretching force constant and
thereby the bond strength increases with increasing bond length.
From Molecular Vibratio ns to Bonding, Chemica l Reactions, and Rea ction Current Organic Chemistry, 2010, Vol. 14, No . 15 1535
lone pair-lone pair repulsions in these molecules. In the case of the
anions CFnH3n
(CH2F, CHF2
, CF3
), there are vicinal lone pairs
which repel each other. However, the lone pair at C is diffuse and,
accordingly, repulsion is rather small so that there is only a small
weakening of the CF bond. The CF bond length is decreased from
1.520 Å(CFH2
) to 1.486 Å(CF2H ) and 1.445 Å(CF3
). Contrary
to molecules CH3NFnH2n the NF stretching force constant becomes
larger with decreasing bond length (indication of bond
strengthening, see below) since lone pair-lone pair interactions do
not dominate in this case.
Computed lengths of bonds involving heavy transition metal
atoms as for example mercury or gold decrease when relativistic
effects are included into the quantum chemical calculation [147-
149]. This is often connected to a relativistic 6s-orbital contraction
although the actual bond length contraction is due to changes in the
one-electron contributions caused by the relativistic mass-velocity
effect. The quantum chemical calculations reveal that in most cases
the shorter bonds are weaker than the longer bonds. In the case of
HgX (X = halogen, chalcogen), this is due to reduced charge
transfer from the mercury 6s-orbital to X (consequence of the
relativistic 6s-orbital contraction) thus causing a lower bond
polarity (bond weakening) and an increase in lone pair repulsion
(consequence of bond shortening) [147-149] Again, an adjustment
of the compressibility limit radius of the transition metal atom
leading to a longer effective bond length clarifies the qualitative
relationship between bond length and bond strength.
There is no unique way of determining compressibility
distances dc for individual bonds in dependence on their electronic
environment. Those parameters dc given in the literature have been
derived empirically for certain bond types (period 1 - period 2
bonds, period 2 - period 2 bonds, etc.) [49,136,137]. The resulting
effective bond lengths are far too inaccurate to provide reliable
bond strength indicators. We conclude that the bond length is not a
useful starting point for deriving bond order or other bond strength
parameters, which, apart from the fact that there are principal
difficulties to define a bond length, predominantly results from the
dependence of re on both the size of the bonded atoms
(compressibility limit radii) and the bond strength, which are are
difficult to separate.
3.3. Assessing Bond Strength from the Bond Density
Bader’s work on the virial partitioning of the molecular electron
density distribution (r) [131] suggests that a much better
descriptor of bond strength should be directly derived from the
bond density. However, the bond density is a quantity that lacks a
rigorously defined bond region. The virial partitioning approach
leads to a definition of atoms in molecules separated by zero-flux
surfaces of the electron density [131]. The point of largest electron
density in a zero-flux surface is the bond critical point rbc, which is
at the same time a point of minimum density along a path of
maximum electron density (MED path) perpendicular to the zero-
flux surface connecting atoms A and B [131]. There have been
attempts to derive from the density bc at rbc relationships for the
bond order of a bond AB [132,150]. For closely related bonds (for
example, different CC bonds) with chemically well-accepted
references (the CC bonds in ethane and ethene defining n = 1 and n
= 2) these relationships lead to the prediction of useful bond orders
as measures for bond strength. However, a generally defined bond
order on the basis of bc is difficult to justify when considering the
following. A single bond-density value, even if determined at a
characteristic and unique location of the bond region, is an
insufficient measure for the bond strength as was shown in several
investigations [45,48]. The bond strength depends on the total
electron density in the bond region, which of course can only
vaguely be described as the region between the bonded atoms. In
addition, it includes also to some extend the lone-pair regions (lone
pair repulsion) and core regions of the atoms (core-valence electron
polarization) as well as regions of substituents attached to the
bonded atoms that indirectly contribute to the bond strength via
steric and electronic effects between the substituents.
A better assessment of the bond density was made by Cremer
and Gauss [151] who determined the density in the zero-flux
surface between the bonded atoms by appropriate integration. The
zero-flux surface density is a measure for the number of electrons
involved in the formation of the bond AB. Theory shows that by
dividing the zero-flux surface density by the square of the
internuclear distance an energy quantity is obtained that reflects the
bond strength in the case of simple hydrocarbons [151]. For CC and
CH bonds in hydrocarbons, the covalent part of bonding dominates
because the bond polarity is rather low. The zero-flux surface
density is a measure for the covalent part of the bond strength
whereas ionic (polar) contributions are not considered. Bond
polarity is reflected by the position of rbc (shifted from the bond
midpoint towards the more electropositive atom) or the charge
transfer from A to B using calculated virial charges. There is no
reliable and generally applicable way of including contributions
from bond polarity into the description of the bond strength so that
a generalization of the Cremer-Gauss approach has so far not been
pursued.
Apart from the question how both covalent and polar part of the
bond strength can be determined, there is also the problem how to
assess other bond strength-relevant features of the bond density. For
example, density may be contracted toward one nucleus thus
shielding it effectively, reducing nuclear repulsion, and increasing
the bond strength. Such an effect is not necessarily reflected by the
zero-flux surface density, which underlines the need of considering
all density in the bond region. A solution to this problem has not
been found yet.
In the work of Cremer and Gauss, [151] an energy quantity was
directly obtained from the calculated density utilizing quantum
mechanical properties of the zero-flux surface. Alternatively, the
energy density H(r) can be determined along the MED path or for
the zero-flux surface and integrated [151-153]. Cremer and Kraka
used H(r) to distinguish covalent from non-covalent bonding
[152,153]. It should be possible to extend their work to obtain
useful bond strength descriptors. So far, no attempts have been
made in this direction. In summary, the bond density parameters
discussed do not provide a reliable account on the bond strength
and the progress reported was restricted to a limited number of
bonds mostly found in organic molecules.
3.4. Assessing Bond Strength from the NMR Spin-Spin
Coupling Constant
NMR spin-spin coupling constants are sensitive probes for the
electronic structure, geometry, and conformation of molecules.
Therefore, it has repeatedly been tried to relate the one-bond
13C,13C, spin-spin coupling constant with the strength of the CC
bond, e.g. in ethane, ethene, and acetylene [154]. We note that these
attempts cannot be successful because the indirect spin-spin
1536 Current Organic Chemistry, 2010, Vol . 14, No. 15 Cremer
and Kraka
coupling constant depends on the first order rather than the zeroth
order density (the latter is responsible for bonding), probes spin
polarization and density currents, and strongly depends on the
environment of the bond [155]. Spin-spin coupling is not just the
result of the transport of spin information along the bond path, but
also includes transport along bond paths involving substituents and
through-space interactions. There has been only one attempt to
determine the -character of CC bonding in typical hydrocarbons
with the help of the spin dipole and paramagnetic spin orbit terms
of the one-bond spin-spin coupling constant, which however did not
lead to any quantitative insights with regard to the bond strength
[156].
3.5. Assessing Bond Strength from the Vibrational Bond
Stretching Frequency
Vibrational spectroscopy is a frequently used tool to identify
and characterize a molecule with the help of its vibrational modes.
Depending on its geometry, conformation, and electronic structure,
Fig. (4). Adiabatic bond stretching frequencies
a
(CO)
and
a
(CF )
given as a function of the corresponding bond lengths (bottom) for 27 structures (top)
(1-16: F-substituted carbocations; 17-19: reference compounds with normal C-F bond; 20-23: fluoronium ions; 24-27: transition states with loose or normal
CF bonds; 28-42: C=O analogs of 1-16; 43-46: reference compounds with C-O(H) bond some of which being isoelectronic with 20-22. Correlation coefficient
R2 for
a
(CO)
: 0.999, and for
a
(CF )
: 0.993 [48].
From Molecular Vibratio ns to Bonding, Chemica l Reactions, and Rea ction Current Organic Chemistry, 2010, Vol. 14, No . 15 1537
each molecule has typical vibrational spectra, which are measured
with the help of infrared or Raman spectroscopy [27-31]. For
example, an infrared band at 1700 cm1 is typical of a carbonyl
stretching frequency or one at 1250 cm1 of a Si-C bond stretching
frequency. However, this kind of information does not qualify as
bond strength descriptor since the vibrational modes are delocalized
and contain besides the contribution from a given stretching mode
also other stretching, bending, etc. contributions. The AICoMs [50-
54] solve this dilemma by providing bond specific vibrational
information in form of AICoM frequencies. However, the use of
AICoM frequencies is hampered by the fact that they depend again
on two quantities, one of them being the bond strength and the other
being the adiabatic mass of the bond. This may not matter in the
case of the isolated stretching frequencies, which were originally
exclusively investigated for CH bonds. However, as soon as one
wants to compare different bonds, vibrational frequencies are more
an obstacle than the answer to the problem. This was demonstrated
by Kraka and Cremer in the case of the CF and the CO bonds [48].
These two types of bonds were first investigated utilizing measured
vibrational frequencies in the sense of a Vib-Cal-X approach [47]
(for other Vib-Cal-X studies, see Ref. [49]). Correlating CF and CO
AICoM bond stretching frequencies with the corresponding bond
lengths [47] or bond densities (rbc) [48] two different
relationships, one for the CF and one for the CO bonds, resulted.
Later it was found that this was just a result of the different masses
of O and F atoms [48]. In view of the fact that the compressibility
limit radii of the two atoms are almost identical, one relationship
resulted for both CF and CO bonds irrespective of the fact whether
isoelectronic C=O and C=F
+ bonds were compared or the non-
isoelectro-nic C-O(H) and C-F bonds. Although vibrational
frequencies, contrary to vibrational force constants can be measured
it is of advantage to use the latter rather than the former for the
purpose of assessing the strength of a chemical bond. This is in line
with the original work of Badger and many other investigations
who focused on the stretching force constant of a bond rather than
on its stretching frequency.
3.6. Assessing Bond Strength from the Vibrational Bond
Stretching Force Constant
IBDEs are difficult to determ ine because the geometry and the
density distribution of the undissociated molecule must be kept
frozen during the dissociation process so that the fragments still
have the electronic structure of the undissociated molecule. An
alternative to th e dissociation process is an infinitesimally small
shift of the dissociation coordinate causing such a small change in
the electron density distribution that it can be considered to be
preserved. This shift leads to an increase in the potential energy V,
which can be determined by expanding V into a Taylor series. At
the equilibrium, there is no first order correction to V so that the
correction is dominated by second order term. Hence, the
magnitude of the correction is reflected by the curvature of V in the
direction of the dissociation coordinate as described by the
corresponding stretching force constant. The stretching force
constant is in this way directly related to the IBDE and, therefore, it
can replace the latter as a suitable measure for the bond strength.
One might argue that there should not be any difference as to
whether the bond stretching vibration is expressed in terms of
AICoMs or c-vector modes, i.e. wheth er force constants ka or kc are
Fig. (5). Adiabatic CF and CO stretching force constants of structures 1 - 46 (Fig. 4) expressed in form of an exponential relationship of the corresponding
bond lengths. Correlation coefficient R2 for ka(CO): 0.998, and for ka(CF): 0.982. If both sets of data are correlated with the bond length, R2 is 0.985.
B3LYP/6-31(d,p) calculations [48].
1538 Current Organic Chemistry, 2010, Vol . 14, No. 15 Cremer
and Kraka
used to determine the strength of the bond. The latter do not include
a relaxation of the geometrical parameters of the molecule upon
bond stretching. Therefore, kc force constants are always larger than
ka force constants [48]. The adiabatic stretching mode, contrary to
the c-stretching mode, initiates bond dissociation and accordingly
provides a natural measure for the strength of the bond. As has been
shown, kc force constants in general can lead to unphysical values
(kc<0 kc not complying with symmetry, etc. [45,50,51]) and
therefore their use is not recommended. The following discussion
will be ex clusively based on adiabatic stretching force constants ka
as the most reliable bond strength descriptors.
When the calculated ka(CF) and ka(CO) values for the
structures shown in Fig. (2) are related to the corresponding bond
lengths the two relationships observed for the adiabatic stretching
frequencies (Fig. 4) merge into just one exponential relationship
despite the different nature of CO and CF bonds. The scattering of
data points is moderate, which suggests that the variation in the
compressibility limits of C, O, and F atom in dependence of their
substituents is small. This is observed for the C=F
+ ion, the normal
C-O single bonds (43 - 46), and some of the transition states, i.e. in
those situations where the positive charge on C is different from
that found for the fluoro carbocations and the keto structures. Apart
from this, the results found for CO and CF bonds clearly support
the view that adiabatic stretching force constants provide a measure
for bond strength and that it is therefore reasonable to use them to
define a common bond order n (Fig. 5). The definition of a bond
order implies the use of suitable reference values, which were
chosen to be n=1 and n=2 for the CO bonds in methanol (43) and
formaldehyde (28) in agreement with general chemical
understanding. In addition, it was required that for a vanishing
stretching force constant the bond order is equal to zero. In this way
the power relationship shown in Fig. (6) was obtained (some
selected values are shown in Fig. 7). The calculated bond orders
reveal that the bond in carbon monoxide has less than triple bond
character in agreemen t with the measured and calculated BD E
values of methanol, formaldehyde, and carbon monoxide [48].
Furthermore, the CF
+ ion is found to have just a double bond
because F is too weak a donor to share more than one electron
lone pair with th e positively charged carbon atom. Otherwise F-
substituted carbocations possess significant double bond character
(Fig. 7) with bond orders between 1.3 and 1.7. The magnitude of
the bond order sensitively depends on the presence of electron-
donating substituents (methyl, phenyl groups) that by charge
donation to C+ reduce -donation of F and by this the double bond
character. The opposite effect is found for electron-withdrawing
groups (acyl) that increase the CF
+ double bond character. Hence,
the CF
+ double bond character can be used as a sensitive antenna
for the electronic environment of a carbo cation.
Fluoronium, cations have rather weak CF -bonds (n = 0.3 -
0.6; BDE values between 14 and 47 kcal/mol; Fig. (7) and Ref.
[48]). Although they are isoelectronic with CO single bonds, the
electron-deficient F atom is no longer sufficiently shielded and
therefore strong nuclear, nuclear repulsion leads to lengthening and
weakening of the CF bond. The CF bonding situation in the
transition states investig ated (FH elimination reactions, Fig. 4, top)
can be assessed by comparison with the different types of CF
bonding (CF, C=F
+, CF
+, Fig. 7) investigated. In Section 4, we
will discu ss the use of vibrational info rmation in the situation of a
chemical reaction.
Fig. (6). CF and CO bond orders n based on the adiabatic CF and CO stretching force constants of structures 1 - 46 shown in Fig. 4. Reference points are given
by the black, solid circles at n = 0, n = 1 (CH3OH), and n = 2 (CH2=O). All other bond orders n were calculated using the available adiabatic stretching force
constants and the power relationship given in the upper part of the diagram.
CF
H
H
CF
H
H
F
C
C
F
CH
3
H
3
C
From Molecular Vibratio ns to Bonding, Chemica l Reactions, and Rea ction Current Organic Chemistry, 2010, Vol. 14, No . 15 1539
4. FROM VIBRATING MOLECULES TO CHEMICAL
REACTIONS
Thermally triggered chemical reactions are mostly started by an
energy transfer during a collision. This energy transfer leads to
vibrational excitation, for example, of a bond stretching vibration in
an unimolecular dissociation. One can say that the chemical
reaction is the consequence of the favorable interaction of two
events, a molecular collision at an angle that enables energy transfer
and a vibrational excitation that leads to bond loosening and
cleavage. Chemical reactions start with a vibrational motion that is
forced to adopt larger and larger vibrational amplitudes until bond
cleavage occurs. The situation is similar for a photolytically
triggered reaction, however with the difference that an excitation to
a higher electronic state takes place. This state is dissociative when
a vibrational mode converts into a translational motion thus leading
to bond cleavage. Molecular vibrations always play an important
role in chemical reactions and the purpose of this section is to
describe this role. However, before doing so some useful terms in
connection with the mechanism of chemical reactions have to be
clarified.
4.1. Chemical Processes, Reaction Complex, and Energy
Landscape of a Reaction
The understanding of chemical reactivity and the mechanism of
chemical reactions is the primary objective of chemical research.
This understanding is essential because it enables chemists to
control chemical reactions and to synthesize materials with desired
properties. In a chemical reaction, one or more reactants are
interconverted into new products. Normally, this implies one or
more chemical processes each of which leads to a change in
chemical bonding. More specifically, we define bond breaking or
bond forming as chemical processes to distinguish them from other
processes such as van der Waals processes, conformational
processes, (de)solvation processes, self-assembly (self-organized)
processes, electron-transfer processes, excitation processes, or
ionization processes. All these processes can occur in the course of
a chemical reaction and therefore are investigated by chemists
although they are often more physical than chemical in nature. For
example biochemists investigate folding processes of proteins,
which include predominantly conformational but also solvation
processes. Medicinal chemists and drug developers study docking
processes, which involve van der Waals, conformational, and
solvation processes. Compared to these processes, chemical
processes are at the core of chemistry and the study of reaction
mechanism focuses primarily on these processes.
What is a Chemical Bond?
Since chemical interconversions imply changes in the bonding
pattern there is a need to distinguish between bonding and
nonbonding situations. This can only be done if one defines a
chemical bond [126-133]. Most chemists would not claim that a
dispersion-stabilized van der Waals complex is kept together by
chemical bonds although the term van der Waals bond is sometimes
used. Considering a weakly bonded donor-acceptor complex it
becomes more problematic to distinguish between a bonding or
non-bonding situation. The actual reason for all problems in
connection with specifying the nature of a chemical bond results
from the fact that the chemical bond is a model quantity rather than
an observable quantity developed within a given concept to
Fig. (7). Some selected adiabatic CF stretching force constants in mdyne/Å (bold face), bond lengths in Å (italics), and bond orders (normal print) of F-
substituted carbocations [48].
1540 Current Organic Chemistry, 2010, Vol . 14, No. 15 Cremer
and Kraka
understand molecular structure and stability (see Section 3). There
is no experiment that could measure properties of the chemical
bond and by this help to clearly distinguish between a bonding and
a non-bonding situation. It is only possible within a given model to
define the chemical bond and bond forming / bond breaking, i.e. the
chemical processes and the chemical reaction caused by the
chemical processes. This may be illu strated by an example: Ethane-
1,2-diol is stabilized in a gauche conformation due to H bonding
between the OH groups. A rotation at the CC bond leads to a
cleavage of the H-bond and a change in energy by just 2.0 kcal/mol,
[46] which is definitely not typical of the energy changes
accompanying normal bond ruptures. Without a specific model of
the chemical bond also applicable to H bonding, it is not possible to
distinguish in this case between a chemical reaction and a
conformational process although most chemists would emphasize
the latter in this case. Hence, we have to clarify this question in the
following.
Reaction Complex and Reaction Path
One characterizes chemical reactions according to their
molecularity (unimolecular, bimolecular or a few trimolecular
processes). In the quantum chemical study of reaction mechanism
one simplifies by focusing on a reaction complex that is the union
of all reactants. By determining the properties of the reaction
complex one unravels the reaction mechanism. It is a non-trivial
problem to describe the vibrating and rotating reaction complex
during the reaction. For this purpose two conceptual simplifications
are introduced. First, the Born-Oppenheimer approximation
separating nuclear and electronic motion is applied. Second, the
changes of the reaction complex are modeled on the potential
energy (hyper)surface (PES) V(q), which is spanned by the 3N-L
internal coordinates qm of the reaction complex, i.e. the reaction is
described in a model space rather than in real space. Although any
function depending on more than 2 variables is difficult to
visualize, one determines internal coordinates that describe major
deformations of the reaction complex and makes two-dimensional
cuts through the PES to represent energy changes in dependence on
these two coordinates in form of contour line diagrams or
perspective drawings. In addition, one uses the geographical terms
of a mountain region to describe the multi-dimensional energy
landscap e of the PES. In a chemical reaction, the reaction complex
follows a minimum energy path (reaction path) from a minimum
occupied by the reactants up an energy valley called the reaction
valley, to an energy pass between energy mountains (the transition
state of the reaction determining the energy barrier and
corresponding to a first order saddle point of V(q)). From the
energy pass, the reaction complex follows down another energy
valley to an energy minimum occupied by the product(s).
4.2. Reaction Step, Reaction Phase, and Reaction Mechanism
A reaction from reactants to products via a single transition
state is called a one-step or elementary reaction. A two step reaction
passes through an intermediate minimum occupied by a chemical
reaction intermediate. The kinetic stability of the intermediate
decides whether the compound in question can be experimentally
intercepted. Th ere are intermediates of low stability on the PE S
which vanish on the Gibbs energy surface G(q). From an
experimental point of view, these pseudo-intermediates can be
ignored. However, they are of high importance for the mechanistic
analysis because they indicate the possibility of obtaining stable
intermediates if the electronic situation and/or the environment is
changed in a way that can be assessed from an analysis of the
pseudo-intermediate. Clearly, the first and most important step of
any mechanistic analysis has to focus on electronic structure
features of the reaction complex as they relate to the features of the
PES. Features of the Gibbs energy surface along the reaction path
can be analyzed to determine entropy, temperature and other
effects. However, they will not be subject of this review that
focuses exclusively on electronic structural features of the reaction
complex.
When Does One Speak of a Reaction Mechanism?
Experimentalists use the term reaction mechanism when
discussing the elementary reactions leading from reactants to
products. This is distinguished from a reaction sequence, which is a
proposed reaction mechanism based on incomplete experimental
data. It is unusual in experimental chemistry to use the term
reaction mechanism in the case of a one-step reaction with or
without transition state. In a quantum chemical investigation of a
reaction, one describes the properties of the reaction complex at any
transient point along the reaction path to obtain a detailed analysis
of the chemical processes. For decades, it was a common practice
among quantum chemists to study the reaction complex just at the
stationary points (reactant or product minima and transition states),
which leads to an incomplete and sometimes even misleading
picture of the reaction mechanism as will b e shown in Section 4.5.
A continuous description of the reaction complex along the reaction
path has the advantage that all changes in its properties can be
registered and their influence on the course of the reaction assessed.
In such a description, it makes sense to speak of the reaction
mechanism of a one-step reaction even if this is proceeding without
barrier and transition state. For example, th e dissociation of H2 is an
elementary reaction without transition state but with a distinct
reaction mechanism consisting of two reaction phases (Fig. 8). The
first is characterized by a ”reaction complex” with continuously
elongated H-H bond. Any bonding interaction ceases when the
electron pair responsible for bonding undergoes spin decoupling to
an open-shell singlet. Then, Coulomb repulsion facilitates
lengthening of the H,H interaction distance. A second reaction
phase starts, which is characterized by H atoms quickly drifting
apart. It ends when the energy asymptote representing two
completely free H atoms is reach ed. In th e first reaction phase, the
potential energy V(r) possesses a positiv e curvature (d2V/dr2), i.e.
for each step r of the dissociation reaction the energy increases
stronger as predicted by the gradient dV/dr. The reaction complex
resists the bond breaking motion. This changes as soon as the
electron pair of H2 is spin-decoupled. The function V(r) becomes
concave with a negative second derivative, i.e. the reaction complex
does no longer resist an elongation of the HH distance. the reaction
force is n egative, and th e predicted en ergy ch anges for an in crease
r are larger than actually given by the energy curve and the
reaction force.
The two-phase mechanism of the H2 dissociation, although
rather simple, leads to several questions: 1) Does th e inflection
point (IFP, change from positive to negative derivative d2V(r)/dr2;
Fig. 8) of the V(r) potential relate to the distance r, at which spin
decoupling occurs? 2) Is spin decoupling a sudden or a gradual
process and what triggers it? 3) How is the position of the IFP
related to the values of BDE and IBDE of H2? 4) Which part of the
BDE is needed for spin decoupling and which part for complete
H,H separation? 5) Does the stretching vibration support or hinder
spin decoupling? - Answering these questions leads to the
establishment of a dissociation mechanism, which passes through at
least two different reaction phases. The physical importance of the
From Molecular Vibratio ns to Bonding, Chemica l Reactions, and Rea ction Current Organic Chemistry, 2010, Vol. 14, No . 15 1541
IFP as the point that separates two reaction would be clarified if the
IFP is also the point of spin decoupling. However, it is more likely
that spin decoupling is not a sudden but a slowly developing
process that depends on bond strength and the (spin) polarizability
of the bond density. Several authors have agreed to the special role
of the IFP and connected it with other properties of the reaction
complex [157-159].
The H2 example underlines that transient points along the
reaction path, not identical with the stationary points, can be
mechanistically relevant. This is taken care of by the phase model
of the reaction mechanism [115,117,119-124], which dissects the
elementary reactions into different phases. In general, one can
expect that a bimolecular elementary reaction with transition state
proceeds in five phases: i) a contact or van der Waals phase
between the reactants where first van der Waals interactions
(possibly leading to a van der Waals complex) result in a preferred
configuration of the reaction complex that will influence the
stereoch emistry of the reaction; ii) a preparation phase, in which the
reaction complex distorts to prepare for the actual chemical
processes; iii) a conversion or transition state phase, in which the
chemical processes tak e place; iv) an adjustment phase, in which
the reaction complex adjusts to the electronic structure, geometry,
and conformation of the products; v) a separation phase, in which
weak van der Waals interactions are dissolved. - Clearly, contact
and separation phase, preparation and adjustment phase reverse
their roles for the reverse reaction. There is no general ru le that
requires that the en ergy transition state is always located in the
conversion phase. If this would be the case, the energy barrier was
primarily a result of the bond breaking or forming processes. If
however the barrier is due to the preparation process of the reaction
complex, the energy transition state can also be positioned in the
preparation (adjustment) phase. These considerations already reveal
that the transition state location and its energy may disclose little
about the detailed reaction mechanism.
Macroscopic and Atomistic Descriptions of Chemical Reactions
Clearly one has to distinguish between a macroscopic and
atomistic (microscopic) description of the chemical reaction both
adding to the understanding of the reaction mechanism as it is
needed to control the reaction. The macro scopic description
includes the energetics as given by the free reaction energy RG
and free activation energy G‡, the reaction temp erature and
pressure, solvent, reaction rate, yield, etc. The micro scopic
description includes the geometric distortion of the reaction
complex, its energy change, changes in the electron density
distribution, in electric and magnetic properties, all given along a
representative reaction path. Important are also the vibrations of the
reaction complex as they change in the course of the reaction. Since
the reaction is a dynamic process described as a translational
motion along the reaction path, it may be connected to other
dynamic processes such as the vibrational and rotational
movements of the reaction complex. For reasons of simplicity, we
do not consider here the rotation of the reaction complex (although
it may be relevant as for example in unimolecular reactions
[160,161]) and instead focus on the vibrations of the reaction
complex.
4.3. Vibrations of the Reaction Complex
Chemical reactions are normally triggered by some uptake of
energy in a collision or excitation process. However, the actual
chemical process starts by a vibrational motion that is converted
into a translational motion along the reaction path. This is often
accompanied by strong coupling to other vibrational modes of the
reaction complex and therefore the mechanism of a chemical
reaction can only be clarified by analyzing the vibrations of the
reaction complex and their interplay. Upon conversion of one
vibrational mode into a translational movement, the remaining 3N-7
vibrations orthogonal to the reaction path reflect in a characteristic
way the changes in the reaction complex. Bond stretching
vibrations of bonds being broken vanish and those of bonds being
formed appear, by this giving an account on structural and stability
changes of the reaction complex. Also, certain vibrations of the
reaction complex can couple with its translational motion thus
making an energy transfer between these two dynamic processes
possible [162,163]. Alternatively, different vibrational modes of the
reaction complex can couple, which facilitates the energy transfer
Fig. (8). Two-phase mechanism of an A-A single bond dissociation (left side). Five-phase mechanism of the reaction AB+CA+BC (right side). The reaction
phases are separated by vertical dashed lines. Note that the phase borders are not specified on the right side.
1542 Current Organic Chemistry, 2010, Vol . 14, No. 15 Cremer
and Kraka
between the vibrational modes and by this the energy dissipation in
the reaction complex. Energy dissipation can occur whenever the
eigenstates of two vibrational modes approach each other and
undergo an avoided crossing [115,118,162].
As was pointed out in Section 2, normal vibrational modes of a
molecule are always delocalized and this also holds for the reaction
complex. For the purpose of facilitating the analysis of the reaction
complex vibrations as they change along the reaction path, they can
be decomposed into AICoMs where the AICoMs are led by the
internal coordinates used to describe the geometry of the reaction
complex [115,117,119-124]. In this way, the forming and breaking
of new bonds can be followed in detail without reference to a
specific bond model. Any developing bonding interaction can be
detected when the adiabatic frequencies or force constants are
displayed as a function of the reaction path coordinate. The force
constant values, which are finally reached in the products, relate to
the strength of the bond generated because the adiabatic force
constants provide reliable descriptors of bond strength (see Section
3) [44, 45, 48, 49, 54].
4.4. Choice of the Representative Reaction Path
The reaction path of a chemical reaction is lik e the chemical
bond a model quantity. Therefore, many different reaction paths can
be chosen when investigating the mechanism of a chemical
reaction. It turns out that the mechanistic study can be significantly
simplified if a representative path is used for the analysis. In the
case of elementary reactions with transition state, the floor line of
the reaction valley is suitable as representative path of the reactio n
complex. There are basically two different strategies for
determining reaction paths: i) moving downhill, i.e. starting at the
transition state in the direction of the negative curvature and tracing
down the steepest decent path toward both the reactant and the
product minimum; i) moving uphill, i.e. starting at the m inimum
occupied by the reactants (products) and following the valley floor
line up to the transition state [164, 165]. The most commonly used
method belonging to the first category is Fukui’s intrinsic reaction
coordinate (IRC) approach [166]. The IRC is the steepest descend
path starting at the transition state and being expressed in mass-
weighted or mass-scaled Cartesian Coordinates [166]. The union of
the two steepest descent paths in en trance and exit valley leads to a
unique path connecting reactant minimum with product minimum
via the transition state [167]. Usually the IRC path tends to follow
the valley floor-line, especially in the convex part of the PES
between minimum and IFP about halfway up to the transition state
[168]. It has been pointed out that in the concave region between
IFP and transition state, the IRC path may deviate from the valley
floor line [169]. However, small deviations from the floor line are
of little consequence for the mechanistic discussion and therefore
the IRC path is the path of choice for reactions with a barrier.
Representative Paths for Reactions without Barrier
An IRC path does no longer exist for chemical reactions
without a barrier. In this case, one can use an uphill strategy for
determining a representative reaction path. For example, gradient
extremals [170] can trace out such a path along the uphill going
floor line. The floor line fulfills two criteria: i) The energy increases
along all directions perpendicular to the direction of the valley floor
line. ii) The curvature of the PES along the floor line must be
minimal compared to any other direction [171]. Hence, the gradient
of the floor line is the eigenvector of the Hessian matrix associated
with the lowest eigenvalue. Ruedenberg called lines following the
least uphill curv ature gradient extremals [172-174]. Quapp and co-
workers showed that IRC path and gradient extremal are identical
close to the reactant or product minimum (convex PES parts),
however when the IRC path becomes curvilinear in the transition
state region IRC path and gradient extremal deviate slightly to
coincide again at the transition state [169].
Alternatively, Newton trajectories [174] can be used for
deriving a representative path in the case of barrierless reactions.
Newton trajectories are defined by the condition that a preselected
gradient direction is kept by all points on the path, which can be
considered as a generalization of the distinguished reaction
coordinate approach [175]. Since any gradient direction can be
chosen, Newton trajectories do not lead to a unique path. Instead, a
family of Newton trajectory curves connects stationary points with
a differen ce of order 1, for example minimum and tran sition state
[174]. Quapp, Kraka, and co-workers [122] showed that Newton
trajectories bundle in the exit channel of the reaction valley before
merging onto the energy plateau. This makes it possible to
determine a central point in the valley from where a representative
downhill path can be started. There is also the possibility that the
reaction valley opens up to a broad bowl as in case of a cirque
created by a glacier. In this case the Newton trajectories can be used
to describe size and boundaries of the cirque, which again makes it
possible to find a suitable starting point for a representative down
hill path. The latter situation was encountered for the chelotropic
addition of methylene to ethene yielding cyclopropane [122,123].
There are other definitions of reaction paths, which are however
less suited as representative paths becau se they are used in
connection with the statistical averaging of trajectories for the
calculation of reaction rates [176,177].
4.5. Reaction Path Splitting
When following the reaction valley, it can happen that it splits
into two or even more valleys and the reaction path bifurcates,
trifurcates, etc. [178-180]. Two different situations are encountered
depending whether path splitting occurs in the exit valley, i.e. for
the downhill path (starting from the transition state), or in the
entrance valley, i.e. for the uphill path (starting from th e reactant
minimum). In the first case, valley splitting precedes path
branching. On the path down through the valley all curvatures
orthogonal to the gradient are positive corresponding to generalized
vibrational frequencies larger than zero. If beyond a certain path
point, generally called a valley-ridge inflection (VRI) point
[181,182], one curvature becomes negative, the eigenvalue of the
Hessian matrix corresponding to this curvature must be zero at the
VRI. By this condition, the VRI is not a stationary point and
therefore there is no path splitting at this point although the valley
splits at this point. In this connection, it may be noted that
according to the McIver-Stanton rule [183] the reaction complex
can change its symmetry only at a stationary point, otherwise it
must follow the reaction path. As pointed out by Ruedenberg [181],
the reaction path follows after the VRI no longer the floor line of a
valley but the crest line of an energy ridge flanked by two (or more)
energy valleys or troughs. The curvature from the ridge down to the
flanking valleys is negative corresponding to an imaginary
generalized frequency. The path follows the ridge until a stationary
point (energy gradient equal to zero) is reached which corresponds
to the path branching or bifurcation point. In most cases, VRI points
and branching points are adjacent points along the path separated
by an unspecified path distance. Each of the new paths starts in the
direction of the n egative curvature downhill until the two valleys
From Molecular Vibratio ns to Bonding, Chemica l Reactions, and Rea ction Current Organic Chemistry, 2010, Vol. 14, No . 15 1543
are reach ed where the reaction path can again follow a floor line to
one of the product minima. Two products are generated that may or
may not be equivalent by symmetry [178].
VRI points and branching points appear more often than is
generally assumed as the work of Caramella and co-workers
demonstrates [184-187] (for a recent review, see Houk and co-
workers [178]). Examples for reaction valley and reaction path
branching processes include many substitution reaction as e.g. the
reaction CH3+H2CH4+H (trifurcation by van der Waals
interactions), [117] isomerization of methoxy radical to
hydroxymethylene (VRI, bifurcation by rotation) [188-191], bond
shifting in cyclooctatetraene (bifurcation by ring puckering) [192],
isomerization of cyclooctatetraene to semibullvalene (bifurcation
by alternative bond formation) [193], ring-opening of
cyclopropylidene to allene (bifurcation by rotation) [194], cope
rearrangements of cis-1,2-divinylcyclo-propane (bifurcation by ring
inversion), [195] and HF addition to ethene (bifurcation by rotation)
[120]. All these examples are characterized by a symmetric
branching leading to equivalent products. However, there is also the
possibility that the reaction path bifurcation leads to different
products and as a consequence the valley splitting is no longer
symmetric. Into this category fall for example the Cope
rearrangement of 1,2,6-heptatriene leading to the concerted
rearrangement product or a biradical intermediate (bifurcation by
alternative bond or no-bond formation) [178,196], ene reactions of
substituted allenes leading either to the cycloaddition product or a
biradical intermediate [197], the cycloaddition of cyclopentadiene
with diphenylketene leading to both [4+2] and [2+2] cycloadducts
sharing a common transition state (alternative bond formation in
transition state) [198].
Many of the path splitting reactions involve symmetry breaking
in the reaction complex often as a result of the formation of instable
products that can stabilize by equivalent conformational processes
(clockwise and counterclockwise rotation, inversion, ring
puckering) or van der Waals relaxation processes. In the latter case,
bond allineation of the separating group (atom) or movement into
corner, edge, or face arrangements can occur where the multiplicity
of branching is given by the highest symmetry element of the
reaction complex before the VRI point. For example, C2v symmetry
can lead to bifurcation, C3v to trifurcation, C4v tetrafurcation, etc. In
most of these cases, the reaction path along the valley and the
adjoined ridge is clearly representative because valley splitting is a
result of non-chemical processes of relatively low energy.
However, as pointed out above, there are also reactions involving
branching into different chemical processes. These reactions require
that, starting from the branching point, two or more new
mechanistic studies have to be carried out.
When valley splitting occurs on an uphill path mechanistically a
new situation is encountered because the two valleys lead to
different transition states associated with different reactions. If the
latter are symmetry equivalent only one of them has to be
investigated. Usually there is no ridge between the different valleys.
If one uses gradient extremals to determine an uphill branching of
valley and reaction path, complications may occur due to the
existence of turning points at which the nature of the gradient
extremal changes from a floor line into a flank line being parallel to
an energy ridge. Hence neither VRI nor branching point can be
found in this situation. Mathematical recipes have been discussed to
get around this problem although no floor line solution has been
found so far for these cases [169].
5. CALCULATION OF REACTION COMPLEX
PROPERTIES ALONG THE REACTION PATH
For the computational elucidation of the reaction mechanism
one has to consider two aspects: i) A reliable calculation of the
energetics of a reaction requires high level methods such a couple
cluster or multireference configuration interaction, large augmented
basis sets, and the inclusion of vibrational and thermal corrections
to relate to quantities that are obtained by measurements.
Calculational co sts set a limit to the size of the reaction complex.
For example, nowadays CCSD(T) calculations up to 20 non-
hydrogen atoms are feasible on a larger scale provided vibrational
corrections are obtained at a low er level of theory. ii) The analy sis
of the reaction mechanism can also be carried out at a lower level of
theory because the sequence of chemical processes is (at least
qualitatively) correctly described at all levels of theory. In so far the
mechanistic analysis resembles the qualitative molecular orbital
analysis, which is mostly correct even when using the simplest
quantum chemical methods. Taking both considerations into
account, a dual level approach is most suitable for th e
computational description of the reaction mechanism. In the first
stage of the analysis, all stationary points are described as
accurately as possible. In this way, the energetics of the reaction is
1544 Current Organic Chemistry, 2010, Vol . 14, No. 15 Cremer
and Kraka
correctly assessed. Then at a lower level of theory the topology of
the PES is clarified, for example by connecting reactant and product
minima and the associated transition states by reaction path s. This
becomes important when there is a large number of reaction
possibilities of a reaction complex so that it becomes difficult to
determine the relationship between transition states, intermediates,
and products. For example, Cremer and co-workers have shown
that such means are necessary to unrav el the sequence of reactions
taking place in the ozonolysis of acetylene and other alkines [199].
The same holds if one wants to explore all possible reactions on a
given PES. This was for example done in the case of the C4H4 PES
for which a total of 106 stationary points (52 minima and 54
transition states) being connected by 32 reactions and reaction
sequences (55 elementary reactions in total) were calculated
[200,201]. The chemical relationship between these many
stationary points could only be assessed by extensive IRC
calculations.
The mechanistic investigation that is presented by the Unified
Reaction Valley Approach (URVA) of Kraka and Cremer [115-124]
is based on representativ e paths such as IRC, grad ient extremal,
Newton trajectory paths, or down hill paths from an energy plateau
through a valley or in a cirque situation. In addition, URVA
determines the (harmonic) reaction valley as reflected by the
generalized vibrational modes of the reaction complex, their
frequencies, and force constants and their changes along the
representative reaction path. URVA collects a large number
properties of the reaction complex where especially energy,
geometry, electron density distribution, and vibrational motions are
analyzed . Common basis of all URVA analyses is the Reaction Path
Hamiltonian (RPH) of Miller, Handy, and Adams (Section 5.1)
[202], the adiabatic mode concept of Cremer and Konkoli ([49-54],
see also Section 2.4), and the vibrational mode ordering of Konkoli,
Cremer, and Kraka [118].
5.1. Reaction Path Hamiltonian, Adiabatic Mode Concept, and
URVA
Once the PES is known for a reaction system, one can apply
Newton dynamics to treat the movements of the reaction complex
on this surface and calculate reaction rate and other dynamic
properties. Apart from the fact that a quantum system should be
described with the help of quantum dynamics, there are two major
obstacles to proceed in this way. First, there are the high,
prohibitive computational costs of determining a PES for a reaction
complex with more than 3 atoms. Assuming that about 300 PES
points are needed for a reliable description in each direction of the
surface, a total of 3003NL points have to be calculated, which
becomes no longer feasible for a large polyatomic system.
Secondly, the nuclear dynamics become also costly with increasing
size of the reaction complex. Out of this computational dilemma,
the idea of a one-dimensional cut through the PES following the
reaction path was born [203,204]. The question however remained
how to carry out reaction path dynamics that would lead to
reasonable rate constants, energy distributions, coupling quantities,
kinetic isotope effects, etc. Miller, Handy, and Adams [202]
answered this question by developing the RPH. The RPH is a
classical Hamiltonian, which facilitates the computational task
tremendously when calculating rate constants, but which has to be
considered when describing quantum effects.
The configuration space of the RPH is partitioned into the one-
dimensional reaction path part and the multi-dimensional valley
part orthogonal to the path. The one-dimensional part of the RPH is
expanded as a function of the arclength s of the reaction path (Fig.
9) and the conjugate momentum ps. Normally, the zero of s is taken
as the location of the transition state and positive s values are used
for the exit, negative s values for the entrance channel of a chemical
reaction. The potential energy of the RPH is given by (i) the term
V0(s), describing the change in energy along the path and (ii) a
second term describing the energy change in the 3NL1=Nvib1
(for L = 6: 3N-7)-dimensional harmonic valley, spanned by the
Nvib1 normal coordinates Qμ(s) of as many generalized harmonic
modes with frequencies
μ
(s):
V[s,{Q
μ
}] =V0(s)+1
2
μ
2(s)Q
μ
2(s)
μ
N
ib1
(5.1)
The simplification to a harmonic valley is justified as long as
the reaction complex moves close to the minimum energy path, i.e.
the floor line of the valley. For the purpose of determining the
multidimensional harmonic valley, at each path point s the
generalized force constant matrix has to be calculated after
projecting out translational and rotational motions and solving the
vibrational equation for the remaining degrees of freedom to obtain
frequencies
μ
and normal mode coordinates Qμ [202]. Despite this
extra calculation al work, the potential energy part has a simple
form, which is no longer the case for the kinetic energy contribution
to the RPH. Again, this splits up in a kinetic energy portion for the
movement along the reaction path depending on the conjugate
momentum ps and an effective mass of the reaction complex meff(s)
and another part corresponding to the kinetic energy associated with
the vibrational motions orthogonal to the path, which depends on
Qμ(s) and the conjugate momenta Pμ(s). The complex structure of
the kinetic energy part results from the fact that it involves also
coupling terms between vibrational motions as well as the coupling
between the translational motion along the path and vibrational
motions (see Fig. 9). The magnitude of these couplings is given by
mode-mode coupling coefficients (Coriolis coefficients) Bμ,(s) and
the curvature coupling coefficients Bμ,s(s): [115,202,205]
T[s,p
s
,{Q
μ
},{P
μ
}] =1
2
[p
s
B
μ
,
(s)Q
μ
(
N
ib
1
μ
s)P
(s)]
2
[1 +B
μ
,s
(s)Q
μ
μ
N
ib
1
(s)]
2
+1
2P
μ
2
(s)
μ
N
ib
1
(5.2)
where
B
μ
,
(s)=l
(s)
†
dl
μ
(s)/ds =B
,
μ
(s)
(5.3)
B
μ
,s(s)=t(s)†dl
(s)/ds =l
μ
(s)†dt/ds =l
μ
(s)†k(s) (5.4)
(lμ(s): mass-weighted generalized normal mode vector of mode μ at
path point s; t(s): reaction path tangent vector at s giving the path
direction; k(s) is the curvature vector at point s.) The numerator of
the first part corresponds to a generalized momentum of the
movement of the reaction complex along the path whereas the
nominator corresponds to the effective mass of the reaction
complex. The curvature coupling coefficients will be large if the
reaction path is strongly curved (Fig. 9, see also Eq. (5.4)) and this
will have an impact on the effective mass meff(s). Curvature is an
essential feature of the reaction path, reflects the mechanism of a
reaction, is associated with energy transfer and energy dissipation,
and reveals at the same time limits of the RPH [115].
Summarizing, the RPH was developed to solve the
dimensionality problem of calculating a Nvib-dimensional PES for N
From Molecular Vibratio ns to Bonding, Chemica l Reactions, and Rea ction Current Organic Chemistry, 2010, Vol. 14, No . 15 1545
4 and carrying out Newton dynamics (or quantum dynamics) for
the reaction complex on this PES to obtain reaction rates and other
dynamic reaction properties. The simplifications and assumptions
used for the RPH include: 1) It is a classic Hamiltonian based on a
reaction path where mostly the IRC path is used, i.e. the mass-
weighted or mass-scaled steepest descent path. - 2) It is designed
for reactions dominated by a single translational motion of the
reaction complex. If this is not the case, a reaction surface
Hamiltonian or even more complex Hamiltonians have to be used
[115]. - 3) It is assumed that the vibrational motions orthogonal to
the path can be described in the harmonic approximation, which is
reasonab le as the translational motion follows largely the IRC path.
- 4) Vibrational adiabaticity is assumed, e.g. the Nvib1 vibrations
remain in the same adiabatic quantum state throughout the reaction,
which for most reactions w ill be the state v = 0. This assumption is
based on the fact that the movement of the reaction complex along
the path is slow compared to the transverse vibrational motions. - 5)
Rotations of the reaction complex are not considered and therefore
a total angular momentum of zero is assumed. - 6) Vibrational
eigenstates of modes possessing the same symmetry must not cross
along the reaction path (avoided crossing theorem), which is a
consequ ence of the use of an adiabatic Hamiltonian. (For a diabatic
reaction path Hamiltonian, see Ref. [206]).
Especially, point 6) has to be fulfilled to determine curvature
and Coriolis couplings correctly, which in turn determine
mechanism and energy transfer. In this connection, it is useful to
point out that the scalar curvature is closely related to the curvature
coupling coefficients via Eq. (5.5):
K(s)=|| k(s)||=[k
†
(s)k(s)]
1/2
=[B
μ
,s
2
μ
N
ib
1
]
1/2
(5.5)
Curvature and curvature coupling constan ts are the primary
tools used to analyze the mechanism of a chemical reaction and
therefore they will be discussed in detail in Sections 5.2 and 5.6.
5.2. Reaction Path Curvature as a Prerequisite of a Chemical
Reaction
The reaction path is a curved line in the Nvib-dimensional
configuration space of the reaction complex. If we consider a
reaction AB+CA+BC in which the bond AB is broken and the
new bond BC is formed, the configuration space of the reaction
complex will be of dimension 3 for the non-linear case where
however essential features of the chemical processes can be
described in a 2-dimensional space spanned by the internal
parameters r(AB) and r(BC) as done in Fig. (10). If the reactio n
path would not be curved, for example, by following the dashed
line directly leading from the reactants in the upper left corner of
the diagram of Fig. (10) to the products located in the lower right
corner, the length r(AB) of the bond to be broken would be a linear
function of r(BC), the length of the bond to be formed.
Furthermore, this would imply that the forces acting in the
processes of bond cleavage and bond formation depend linearly on
the bond length r, which is a contradiction in itself because these
forces are dominated by the Coulomb law and its 1/r-dependence.
Hence, a straight, non-curved reaction path can be excluded.
A clear indication of the start of the bond breaking process is a
lengthening of bond AB. Four different situations could be
hypothetically distinguished (Fig. 10): 1) Path 7-6-5-4-3-2-1 (path
P1): The bond AB keeps its length until C has approached B so
much that the value of the bond length BC is adopted. The bond
switches from A to C and A moves away from the new molecule
BC (increasing r(AB)). For path P1, the chemical processes of bond
cleavage and bond formation would be catastrophic ones (an
infinitesimally small change in the distance AB would lead from
bonding to non-bonding) without gradually developing mutual
polarization of the electron density of the reactants that prepares
them for the chemical processes. Clearly, the catastrophic
description is not correct and therefore a reaction path P1 suddenly
changing its direction at the transition state (point 4) is not correct.
In the same way, any sudden (catastrophic) change in the reaction
path direction is hardly likely, unless the chemical processes are
coupled to excitation, ionization or electron transfer processes. - 2)
Path 7-6-5-S-E-3-2-1 (path P2): The catastrophic event of P1 is
avoided by a smooth curving of the reaction path from S to E. The
beginning of P2 curving at point S indicates the beginning of AB
bond lengthening and thereby AB bond cleavage. The end of P2
curving at E indicates that the actual bond length BC will no longer
change thus suggesting that the bond BC is finalized. Although a
unique definition of chemical bonding is difficult, path P2 (as well
as P3, P4, etc.) suggests a dynamic definition of chemical bonding
based on the curvature of the reaction path: The curvature identifies
Fig. (9). Direction and curvature of the reaction path as given by the tangent vector t(s) and the curvature vector k(s). The normal mode vectors lμ are
positioned in a plane perpendicular to the path direction. Large curvature couplings require that vectors lμ and k(s) are (anti)collinear and have large
magnitudes.
1546 Current Organic Chemistry, 2010, Vol . 14, No. 15 Cremer
and Kraka
changes in the bond and thereby its cleavage or formation. This
leaves two questions: a) When does the bond AB cease to exist (at
M, E or any intermediate point)? - b) When is the new bond first
formed? - Actually th ese two questions are of little relevance for
the dynamic processes of the reaction. Investigation of the reaction
complex in the path region S-M-E reveals th at the reaction is
initialized by a polarization of the density of bond AB leading to a
gradual change in the bond properties. Similarly, the newly formed
bond adjusts gradually to its equilibrium situation. Therefore, the
path curvature increases first slowly from zero, adopts in the middle
part close to M a curvature maximum, and then decreases to zero,
which is reached close to E. Hence P2 is characterized by one
curvature peak. Such a situation is found in degenerate bond
processes where the bond cleaved and the bond formed are of the
same type as in the reaction AB+AA+BA. The curvature peak will
be high (low) and its width small (large) if bond AB is strong
(weak) and the polarizing power of B is small (large). The situation
becomes more complex when one considers that a strong, highly
polar bond is difficult to polarize whereas a strong nonpolar
multiple bond can easily be polarized. This should also influence
magnitude and shape of the path curv ature. H ence, the curvature
contains detailed information on the chemical processes, but it has
to be unraveled.
Fig. (10). Schematic representation of different reaction paths with different
curvatures for the reaction AB+CA+BC shown in the 2-dimensional space
spanned by distances r(AB) and r(BC) (see text). Arrows indicate the
direction of movement of the reaction complex.
3) Path 7-6-5-2-1 (path P3): In the general case of C being
unequal to A and the bonds AB and BC being different with regard
to both strength and polarizability, a path such as P3 may result.
The reaction path curvature at point 5 will be larg er than that at
point 2 if bond AB is stronger and less well to polarize than bond
BC and C has a lower polarizing power than A. Between points 5
and 2 the path curvature is likely to adopt smaller values or may
vanish totally so that two rather than just one curvature peak result,
the first for the cleavage of bond AB and the second for the
formation of BC. The curvature of the reaction path will give more
specific information on the nature of the reactants if it becomes
possible to eliminate one or two of the contributing effects (bond
strength, bond polarizability, polarizing power) by investigating a
series of BX bonds for varying X and constant B all being attacked
by the same atom A to form the bond AB and cleaving the bond BX
(for example substitution reactions A
+CR3XACR3+X
). For path
P3’(7-6-3-2-1), the situation of P3 is reverted and an equiv alent
description holds. - 4) Paths 7-6-5-3-2-1 and 7-6-2-1 (symmetric
paths P4 and P5). These paths have equal curvature in the AB
cleaving (points 5 or 6) and BC forming process (points 3 or 2).
Although such cases cannot be excluded, identical curvatures are
rare for bonds AB and BC (reaction partners C and A) being
different. However, the space between th e curvature areas, being
for P5 larger than for P4, gives further insight into the ch emical
processes in so far as this also reflects lability (stability) of the
bonds involved. In this connection, it must not be overlooked that
the length of the IRC path depends on the number N of reaction
complex atoms and their masses in the way that an increase in N
and/or the atomic mass lengthen the path (because of mass-
weighting) and thereby also the region between the locations of
path curvature.
The region of the path curvature locations is the location of the
chemical processes. This however does not imply that the curvature
region is necessarily the region of the energy transition state. For P2
(path of a degenerate reaction), the transition state will be located
on the diagonal r(AB) = r(BC), which will also be the position of
the curvature maximum. For P3 or P3’, however the location of the
transition state can b e anywhere between or at the curv ature
locations or even outside this range. There are possibilities to relate
the position of the energy transition state to the shape and position
of the curvature peaks [115, 117].
Physical Meaning of the Reaction Path Curvature
The observations made with regard to the reaction path
curvature and schematically shown in Fig. (10) can be generalized:
A chemical reaction that is a result of at least two chemical
processes (bond cleavage and bond formation) requires a curving of
the corresponding reaction path. An exclusively straight reaction
path with zero curvature indicates a physical rather than chemical
process. All vibrational motions of the reaction complex in a
physical system are conserved and consequently both Coriolis and
curvature couplings are nil. Only slight variations in the electronic
structure of the reaction complex may take place because of van der
Waals interactions or interactions with an electromagnetic field. A
chemical reaction requires that at least one bond stretching motion
vanishes and a new one appears. For the reaction AB+CA+BC,
the force constant ka(AB)(s) (altern atively
[
a
(AB)(s)]
2
decreases
to zero whereas th e force constant ka(BC)(s) (alternatively
[
a
(BC)(s)]
2
increases from zero to its equilibrium value. Hence,
one might conclude that curvature reflects the change in the sum of
the force constants, which however is not the case. In Fig. (11), the
vibrational frequencies of the reaction CH3+H2CH4+H are given
in dependence of s. The bond AB corresponds here to the HH bond,
which is broken to form the new bond CH (corresponding to bond
BC). The vibrational modes change their character along the
reaction path, especially at the avoided crossing AC1, AC2, and
AC3 (Fig. 11). For example, mode 11 corresponds in the entrance
channel to the HH stretching vibration. At point AC1, this role is
taken over by mode 8 and finally at point AC2, HH stretching
becomes a translational motion. From AC2 on, mode 8 adopts the
character of a CH stretching mode, which switches at AC3 to mode
11. Fig. (11) indicates that close to the positions of curvature peaks
K2 and K3 (indicated by the vertical lines, top half of figure), there
are inflection points of frequency functions
(HH)(s) and
(CH)(s), which means that
'
(HH)(s) and
'
(CH)(s) adopt
maximal v alues. Hence, the larg est changes in the stretching
From Molecular Vibratio ns to Bonding, Chemica l Reactions, and Rea ction Current Organic Chemistry, 2010, Vol. 14, No . 15 1547
vibrations of breaking and forming bond are responsible for the
curvature maxima.
As is shown in Fig. (11) for the reaction CH3+H2CH4+H, the
curvature is related to the change in the vibrational frequencies
rather than their magnitude or the difference of the magnitudes.
Frequency
(HH) decreases slower than
(CH) increases and
this can b e directly related to the fact that curvature peak K2
associated with the breaking of the HH bond is smaller and broader
than curv ature peak K3 associated with the formation of the CH
bond. Since the force constant is directly related to the frequency
and has the advantage of being mass independent, it is
advantageous to use AICoM stretching force constants of cleaving
bond AB and forming bond BC to describe the path curvature.
Again, the change in the force constant in dependence of the path
parameter s, i.e. the first derivatives dka(AB)(s)/ds=ka'(AB)(s) and
dka(BC)(s)/ds=ka'(BC)(s) determines any path curving. Exact
relationships can be obtained by starting from the definition of the
curvature in terms of first and send derivatives of the energy with
regard to internal coordinates and then solving for the force
constants. In this way, it can be shown that there is a direct
relationship between scalar curvatu re and the sum of the first
derivatives, k’(AB)(s) + k’(BC)(s).
In the general case of a reaction AB+CA+BC, the maximal
changes in the stretching force constants for cleaving bond AB and
forming bond BC are most likely to occur at different positions of
the reaction path, which means that two rather than just one
curvature peak are found (see also Fig. 10). For the purpose of
Fig. (11). Top: Variation of the HH (first mode 11; at AC1 mode 8) and CH stretching frequency (from AC2 on mode 8; at AC3 mode 11) in the reaction
CH3+H2CH4+H. The slope of
(HH)(s) and
(CH)(s) is given at the inflection points close to s = -0.3 and 0.4 amu1/2 Bohr. Bottom: Curvature diagram
(bold line) with curvature peaks K1, K2, and K3. Curvature coupling coefficients are indicated by dashed lines. Reaction phases are separated by vertical
dashed lines. Symmetry of vibrational modes are given. The energy transition state is located at s = 0 amu1/2 Bohr [117].
1548 Current Organic Chemistry, 2010, Vol . 14, No. 15 Cremer
and Kraka
understanding this situation, its is important to provide further
reasoning why the change in rather than the magnitude of the force
constant determine the curving of the reaction path. The derivative
k’(s) reflects the polarizability of a bond, which itself depends on
the polarizing power of the attacking reactant. This confirms the
essence of the discussion in connection with the path situations
shown in Fig. (10): The shape of the curvature peaks (see bottom of
Fig. 11) reflects the ease by which the bond is broken by an agent:
A broader smaller peak indicates a smooth cleaving of the bond
starting early whereas a large narrow curvature peak has to be
associated with a rapid bond cleavage starting after some sudden
short-range polarization of the bond. Clearly, the magnitude of the
curvature peaks does not exclusively relate to the strength of the
bond. It is important to note in this connection that in the case of a
polyatomic reaction complex bond cleavage may lead to geometry
relaxation effects in other parts of the reaction complex and these
changes may add to the path curvature. These contributions can
become large when electronic effects such as bond conjugation
causing strong geometry adjustments play a role. In turn, there are
situations where just partial bond breakage, for example in the
course of -bond cleavage leaving at the same time the -bond
intact, leads to smaller curvature enhancements. These details can
be unraveled with the help of the curvature-coupling coefficients
also shown in Fig. (11).
The CH bond in CH4 possesses a bond dissociation energy De
of 110.9 kcal/mol, which is of similar magnitude as that of H2 (De =
109.2 kcal/mol) [207,208]. Since bond dissociation energies include
also the stabilization energies of the dissociation products, it is
better to refer to the adiabatic stretching force constants, which are
5.75 and 4.90 mdyne/Å for H2 and CH stretching in CH4,
respectively. These values reflect the fact that the bond energy of
the CH bond in methane is just 99 kcal/mol compared to the bond
dissociation enthalpy of H2 being 104.8 kcal/mol. The longitudinal
static dipole polarizability of the HH bond is 0.80 Å3 and by this
larger as the corresponding value of the somewhat polar CH bond
in methane ( = 0.65 Å3 [209]. Apart from this, the methyl radical
with its anisotropic charge distribution at the C atom has a much
stronger polarizing power than the H atom does. Accordingly, the
polarization of the HH bond by H3 starts earlier (at a larger
distance between the reactants) and proceeds easier than the
polarization of a CH bond in methane by a H atom. The H atom has
to approach the CH bond much closer before polarization starts and
the latter is followed directly by bond rupture so that these two
electronic events take place in a relatively short part of the reaction
path. Hence, the curvature is larger and the scalar curvature
maximum K3 clearly exceeds that of K2 by 80% (Fig. 11) Despite
the fact that the HH bond is stronger (as clearly reflected by the
AICoM force constants), the change in k(HH) close to curvature
peak K2 is slower than that of k(CH) at K3, which is confirmed by
Fig (11) (see tangent of the curve
(HH)(s) as compared to that of
curve
(CH)(s). This confirms that the bond strength is not the
decisive factor for bond cleavage. Bond polarizability and
polarizing power of attacking agents have a strong impact on
k’(AB)(s), which in turn determines (together with k’(BC)(s)) the
path curvature.
Since each curvature peak is associated with a specific chemical
process, it is useful to partition the reaction path in different phases
using the curvature peaks (or other important features of K(s)). The
minimum of the curvature between two peaks (corresponding to
minimal chemical activity) indicates the transition from one
reaction phase to the next. Each phase is characterized by one
curvature peak or curvature enhancements. In this way, 6 phases
can be recognized for the reaction CH3+H2CH4+H as will be
discussed in Section 5.6
5.3. Energy and Force Analysis
Changes of the energy of the reaction complex in dependence
of the path parameter s determine the energetics of the reaction. The
maximum of E(s) at the transition state (corresponding to a first
order saddle point on the PES), is usually taken as distinct point that
provides information of the reaction mechanism. Despite of
common chemical thinking, in most cases the chemical processes
do not take place at the transition state. If the preparation of the
reaction complex requires much energy, the energy transition state
is positioned on the reaction path before the location of the actual
chemical processes. For example, in the case of the Diels-Alder
reaction, equalization of single and double bonds and the associated
distortion of the reaction complex requires about 23 kcal/mol and
thereby determines the reaction barrier [125]. The formation of the
new CC bonds leading to the cyclic structure of cyclohexene
happens 56 kcal/mol downhill from the transition state, far out in
the exit channel, shortly before the product is formed.
Use of the Term Transition Sta te
It is appropriate to point out the various uses of the term
”transition state.” According to transition state theory and general
chemical understanding, [210] a transition state corresponds to the
highest energy point of the reaction path, which connects reactant
and product minimum. Alternatively, one can consider enthalpy or
Gibbs energy changes along the reaction path and define the
transition state location as that corresponding to the highest
enthalpy or Gibbs energy along the path. In view of the fact that
neither energy nor enthalpy reveal mechanistic details, it has
become customary to speak in connection with the reaction
mechanism of a transition state region as that reaction phase in
which the chemical processes of bond breaking and forming take
place [115,117,119-124]. This is in line with the use of the term in
transition state spectroscopy, which is a part of femtosecond
spectroscopy [211-213]. Laser spectroscopists define the transition
state as the full set of transient configurations which are traversed
by the reaction complex in entrance and exit channel between
reactants and products. Transition state spectroscopy exploits the
fact that the electronic processes, which accompany bond breaking
and bond formation are parallel to substantial changes in the
internuclear separation. If one assumes that the atoms move during
a chemical reaction by about 10 Å and if one further assumes that
they move with a speed of 104 - 105 cm/s, then a period of about 10
ps will be sufficient to observe transition state configurations.
Hence, femtosecond spectroscopy can detect transient reaction
complex configurations along the reaction path and observe even
path bifurcations. There are different approaches to obtain
information about transition state configurations. Since the
transition state is spectroscopically seen an unbound state, its
absorption spectrum will be most meaningful if an electronic
transition takes place into a bound state. An alternative way of
getting information is the ’clocking’ of transition states. Utilizing
ultrashort laser pulses one is able to define a zero of time and
follow the elementary dynamics of a chemical reaction [214]. Laser
spectroscopy may even invoke vibrationally driven reactions, which
includes both enhancement of reaction rates (see Section 5.6),
manipulation of energy disposal and promotion of a certain product
channel by mode selective excitation [214]. All the information
gained by these approaches refer to a loose definition of the
transition state region somewhere between reactants and products.
From Molecular Vibratio ns to Bonding, Chemica l Reactions, and Rea ction Current Organic Chemistry, 2010, Vol. 14, No . 15 1549
This use of the term is different from wh at is meant by energy
transition state in transition state theory (first order saddle point of
E) and transition state region or phase in mechanistic th eory
(region, in which the chemical processes of bond formation and
cleavage take place as indicated by the corresponding curvature
peaks).
Analysis of the Reaction Force
Terms su ch as transition state zone (zone between the IFPs
before and after the energy transition state [157,158]) are based on
the trends in the reaction force F(s) = dE(s)/ds and the changes in
the reaction force along the path as reflected by the curvature of
E(s), d2E(s)/ds2. At the IFP of E(s) before (after) the transition state,
the reaction force F(s) adopts a minimum (maximum) and its
changes vanish (curvature of E(s) is zero). In the case of an
elementary reaction with transition state, Toro-Labbe [157,158]
partitions the reaction path in four zones (Fig. 12) where these
zones have no obvious connection with the reaction phases based
on the path curvature. (It has to be noted that more than two IFPs
appear along the reaction path of an elementary reaction with
Fig. (12). Partitioning of the energy profile along the path of an elementary
reaction with transition state (TS) into zones utilizing the reaction force F(s)
= - E’(s). The reaction path parameter s (arc length) is set to zero at the
transition state; s 0: entrance channel; s 0: exit channel; IFP: inflection
point of E(s) with F(s) being extremal and F’(s) = E”(s) = 0. Reaction zones
according to Ref. [157] (see text). Functions E(s), F(s), and F’(s) are not
drawn to scale.
barrier when E(s) possesses shoulders). In zone 1 (reactant zone),
the reaction complex ”resists” any deformation caused by the
chemical reaction. The retarding force (F(s) <0) becomes
increasingly negative toward the IFP and then increases until it
reaches a zero value at the tran sition state. This is reflected by the
fact that in the reactant zone more energy as predicted from the
reaction force is needed to move the reaction complex from path
point s to path point s+ds: E(s)=E(s+ds) E(s)>E(s)+F(s)ds. The
addition al energy is related to the curvature d2E(s)/ds2 which,
increasing from a minimum value, achieves a maximum before th e
IFP and is exactly zero at the IFP (Fig. 12). After the IFP,
deformations of the reaction complex are “accelerated” because less
energy, as predicted from the reaction force F(s), is needed to move
the reaction complex from s to s+ds: E(s)=E(s+ds)E(s)<E(s)+F(s)ds.
The retarding fo rce decreases and becomes zero at the transition
state. Tentatively, one might relate the IFP w ith the position of spin
decoupling in a simple bond dissociation reaction (Section 4.2)
since the required force for driving the reaction becomes smaller
after the spin decoupling occurs. However, this assumption does not
hold in most reactions since the transition state region is not
necessarily the region where the chemical processes take place. In
the reactions investigated so far, the chemical processes often start
before the energy IFP is reached and in the exit channel they end
after zone 4 (product zone, Fig. 12) has been entered, i.e. the
relevant curvature regions (chemical processes) can be located
before and after the two IFPs. We conclude that an E-based
dissection of th e mech anism into reaction zones misses m echanistic
details. The energy is a cumulative quantity that absorbs all energy
changes caused by geometry deformations and electronic structure
adjustment of the reaction complex and accordingly, often disguises
mechanistically relevant features such as bond breaking or forming.
Internal Coordinate Forces
It is more useful to monitor internal coordinate forces exerted
on parts of the reaction complex such as bonds or functional groups
along the reaction path rather than analyzing a collective reaction
force F(s). The internal coordinate forces can be repulsive or
attractive thus indicating whether the reaction complex resists or
accepts a given structural change. Those forces, which are
associated with internal coordinates dominating the path direction
are especially interesting because they reveal which structural
changes are responsible for what energy increase (decrease) along
the reaction path [115,117,119-121]. There is also the possibility of
relating changes in the forces to ch anges in the electron density
distribution of the reaction complex (see Section 5.5).
5.4. Geometry Analysis
The most direct and detailed description of the reaction should
be provided by the changes of the internal coordinates of the
reaction complex. In Fig. (13) this is shown for the reaction
CH3+H2CH4+H. In general, there are Nvib internal coordinates
q(s), i.e. in the case of the CH3+H2 reaction 12 coordinates that
because of C3v-symmetry of the reaction complex reduce to just 4:
the HH distance R1, the CH distance R2, the bond length R3 of the
CH spectator bonds (not involved in the reaction), and the HsC…H
angle (alternatively, the HsCHs angle can be taken). The bond
length R3 does not substantially change along the path and can be
ignored because it is associated with the three CH spectator bonds
of the methyl group. The changes in the geometry of the reaction
complex caused by the chemical processes are described by just
three parameters: R1, R2, and angle (Fig. 13). These internal
coordinates reveal characteristic changes along the reaction path as
indicated by arrows and associated with features of the scalar
curvature shown in Fig. (11). There is an obvious relationship
between the path curvature K(s) and changes in the q(s) functions,
which can also be determined via the curvature d2q(s)/ds2. There are
large curvature values for R1 at s = - 0.1 amu1/2 Bohr and for R2 at
s = 0.6 amu1/2 Bohr, which are exactly the positions of the curvature
peaks K2 and K3 (Fig. 11). Therefore, one could argue th at a
mechanistic analysis of a ch emical reaction is best carried out by
studying the changes of all q(s) along the reaction path. Several
problems prevent this possibility: i) The number of internal
coordinates q(s) becomes prohibitively large for larger N. For
example, in the case of the Diels-Alder reaction, N = 16 and Nvib =
42 [119]. ii) With increasing N, it is no longer possible to predict
which internal coordinates are most important at what s-values for
1550 Current Organic Chemistry, 2010, Vol . 14, No. 15 Cremer
and Kraka
the description of the chemical processes. iii) Which changes in the
internal coordinates are significant? For example there is a
significant change in the curvature of R2(s) at s = - 0.1 amu1/2 Bohr
(upper arrow in Fig. 13) that is difficult to see and to analyze
because of its small magnitude. - In view of i) - iii), it is difficult to
assess the importance of the changes in all q(n) along the path and
to relate th em to electronic structure changes and mechanistic
detail. A lso, th ere are different sets of {qn(s)} possible for the
description of the reaction complex. A priori, it is not clear, which
of those sets is the most suitable one. Also, different coordinate sets
may be needed to appropriately describe entrance and the exit
channel. For the purpose of avoiding all these difficulties, a
procedure has been worked out that identifies those internal
coordinates, which describe the chemical processes and thereby the
reaction mechanism in a meaningful way (see section 5.7).
5.5. Electron Density Analysis
A direct insight into the changes in the electronic structure of
the reaction complex during the reaction is provided by the electron
density distribution. This can be obtained by analyzing the total
electron density distribution (r) with the help of the topological or
viral analysis, [131,150] difference density distributions (r),
[115,117] Laplace concentrations of the electron density
distribution,
2(r)
[120] population densities such as the
Natural Bond Orbital (NBO) populations and Natural Atomic
Orbital (NAO) populations [123,215], electron localization
functions [216-218], or pair population densities [219,220]. The
density analysis can be supplemented by pictorial representations of
density changes along the reaction path [117,120]. Since the MED
paths provide images of the bond paths within an electron density
model of the chemical bond (Section 3.3) they can be used to
follow the breaking of a chemical bond [132]. Cremer and Kraka
have established necessary and sufficient conditions for covalent
bonding by referring to the existence of a bond critical point (first
order saddle point of the (r) of the MED path and a negative
(stabilizing) energy density H(r) at this point. Bond path and bond
critical point vanish in the course of a bond cleavage process
catastrophically (infinitesimally small change in bond length leads
to vanishing of bond critical point) [132]. In most cases, the
location of the density catastrophe along the reaction path does not
coincide with that of the energy transition state but with the region
of large path curvature caused by the bond cleavage. Future work
has to show how the topological and virial analysis of the reaction
complex can complement, detail, or confirm results of the URVA
analysis.
Analysis of Difference Electron Density Distributions
Difference electron density distributions taken for the reaction
complex along the reaction path and represented in form of contour
line diagrams provide a sensitive visual tool of assessing the degree
of reactant density or, more specifically, bond density polarization
preceding bond cleavage. The difference density distribution is
defined by
(r,s)=
(r,s)[reaction complex]
(r,s)[reference complex]
(5.6)
where the choice of an appropriate reference density distribution of
a reference complex is difficult to make. In case of the CH3 + H2
reaction, the reference den sity was chosen as the average
0.5*{(r,s)[CH3] + (r,s)[H2] + (r,s)[CH4] + (r,s)[H]} calculated
for the noninteracting molecular subunits at the geometries they
adopt within the reaction complex at point s. Taking snapshots of
the difference density distribution in entrance and exit channel, the
gradual polarization of the electron density distribution of the
reaction complex could by demonstrated [117]. Clearly, such an
approach cannot be generalized because the reference complex is
not uniquely defined in this way. It suffers from the disadvantage
that the reaction complex and the parts of the reference complex
have to be calculated with different methods, which makes
Fig. (13). Changes in the internal coordinates q(s) of the reaction complex of the reaction CH3+H2CH4+H as a function of the path parameter s [117].
Curvature in the function q(s) is indicated by an arrow and relates to the curvature peaks K1, K2, and K3 as well as the shoulder behind K3 (see Fig. 11).
Subscript s denotes a spectator atom or bond.
From Molecular Vibratio ns to Bonding, Chemica l Reactions, and Rea ction Current Organic Chemistry, 2010, Vol. 14, No . 15 1551
calculated difference densities method-dependent and contaminated
by incompatible contributions. Also, if the same reaction complex
geometry is used for molecular situations with little distortions
(e.g., early in the en trance channel for CH3···H2) and at the same
time for molecular situations with strong distortion (H4C···H),
calculational problems aggravate or even prevent the determination
of a difference density. However, Cremer and co-workers [45] have
offered an alternative approach of calculating difference densities in
the case of an adiabatic mode, which can b e easily generalized for
the motion along the reaction path.
Analysis of Orbital Populations
Orbital population analyses have been used to study chemical
reactions since decades, however their usefulness has to be
questioned in view of the arbitrary nature of most of the original
population analysis methods. NAO and NBO analyses are
nowadays widely accepted since they are based on mature quantum
chemical concepts (with the limitation that they favor L ewis
structures where non-classical structures are more appropriate [15]).
Therefore, NBO values have been calculated to document the
electronic structure of the reaction complex in different reaction
phases. For example, Cremer and co-workers [123] showed that in
case of the methylene addition to ethene calculated NBO charges
help to v isualize the different phases of the reaction. - Electron
Localization functions (ELF) basins [216] have also been utilized to
distinguish between reaction domains as domains with different
numbers or types of ELF basins. For example, Santos and co-
workers [217] showed that the ELF analysis of the acetylene
trimerization leads to five different domains describing 1) closed-
shell repulsion between acetylene moieties, 2) deformation of the
reactants under the impact of increasing repulsion, 3) preparation of
the reaction complex for CC bond formation 4) simultaneous
(concerted) formation of the CC -bonds, and finally 5)
development of -aromaticity. There has also been a successful
attempt to combine the ELF analysis w ith Bader’s virial
partitioning of the electron density into atomic basins in the case of
transition metal complexes [218] with the far-reaching goal of
getting more detailed density descriptions for chemical reactions. -
Ponec [219,220] has applied a pair population analysis to study
electron reorganisation in the course of pericyclic reactions.
Symmetry-allowed reactions could be described as a simple cyclic
shift of electron pair bonds whereas symmetry-forbidden reactions
imply complex changes in electron structure.
5.6. Analysis of Vibrational Motions
Monitoring the vibrational frequencies of the reaction complex
along the reaction path (Fig. 11, top) provides already insight into
essential features of the reaction mechanism. In the case of the
CH3+H2 reaction, the path regions where the chemical processes
take place can be identified. Also, the splitting of the reaction v alley
is indicated by the fact that the frequencies of the 1e vibrational
mode becomes imaginary (indicated in Fig. (11) by a negative value
of
(1e)). More details of the form and the partitioning of the
reaction valley into phases are difficult to assess directly from the
vibrational modes and their frequencies. Additional information
however can be obtained by a more advanced analysis involving 4
steps: i) Decomposition of normal modes into AICoMs; ii) Analysis
of curvature in terms of normal mode curvature couplings; iii)
Analysis of mode-mode coupling; iv) Analysis of curvature in
terms of adiabatic mode curvature couplings.
Decomposition of Normal Modes into Adiabatic Modes
Each vibrational mode is decomposed into AICoMs driven by
the internal coordinates q(n) of the reaction complex. This is done
in Fig. (14) where vibrational mode 11(3a1) is decomposed into the
4 AICoMs of the reaction complex {CH3…H2}, of which only
R1(HH), R2(CH), and R3(CHs) contribute. Mode 11 abruptly
Fig. (14). Decomposition of vibrational mode 11(3a1) of the reaction complex of CH3+H2CH4+H in terms of AICoMs given as a function of the path
parameter s [117]. AC: avoided crossing points; see Fig. (11). CHs denotes the spectator bonds.
1552 Current Organic Chemistry, 2010, Vol . 14, No. 15 Cremer
and Kraka
changes its character at the avoided crossing points AC1 and AC3
(see Fig. 11). It starts as a HH stretching vibration, becomes at AC1
the CH stretching vibration of the three CH spectator bonds, and
then changes again at AC3 to become the CH stretching mode of
the new CH bond of CH4 mixed with HH stretching of the
dissociating van der Waals complex CH4…H. The CH/HH
stretching admixture in mode 11 is a direct consequence of the
linear C3v-symmetrical H3C…H…H arrangement. Any movement
of the central H triggers a CH and HH stretching motion.
Interesting is the admixture of CH stretching to the HH stretching
mode at s = -2.5 amu1/2 Bohr. This is the position of curvature peak
K1 and without analyzing the curvature at this path position it is
difficult to rationalize this CH-AICoM admixture on common
chemical understanding (see below).
Analysis of the Path Curvature in terms of Normal Mode
Curvature Couplings
Each curv ature peak or enhancement is decomposed into
curvature coupling coefficients Bμ,s(s) given in terms of normal
vibrational modes of the reaction complex. This is shown in Fig.
(11) (bottom) and is the basis of the Laser-spectroscopic technique
of mode-selective rate enhancement [221]. It exploits the fact that
energy can be pumped at the beginning of a reaction into
vibrational modes that couple with the translational motion of the
reaction complex along the reaction path. A priori it is difficult to
predict which of the vibrational modes are suitable for this purpose.
Symmetry arguments can be used to exclude all those vibrational
modes that do not comply with the symmetry of the translational
mode vector, i.e. that of the reaction complex. Common chemical
sense can also help to identify those modes that should influence
the rate of a chemical reaction. However, all common chemical
sense will not help if Coriolis coupling leads to a dissipation of
vibrational energy before an energy transfer into the translational
motion and thereby rate enhancement can take place. Accordingly,
it is appropriate to combine analysis ii) with analysis iii), i.e. the
analysis of the coupling between different vibrational modes in
terms of the Coriolis coupling coefficients Bμ,(s).
Analysis of Coriolis Couplings
The calculation of curvature and Coriolis coupling constants
provides the only predictive analytical approach to the question of
mode selective rate enhancement. A number of combined
computational and spectroscopic investigations have gone this way:
In the reaction of CH3 with H2 it has been demonstrated (Fig. 11,
bottom) th at the a1-symmetrical CH stretching mode of CH3 (mode
8) couples strongly with the translational mode shortly before the
energy transition state [117]. Despite of this fact it is advantageous
to pump energy into mode 11, which undergoes an efficient
Coriolis coupling with mode 8 at AC1 so that via curvature
coupling the Laser energy is selectively channeled into the
translational motion thereby enhancing the rate of reaction [117]. -
Riedel and co-workers [222] reported on a reactive scattering
experiment of Cl atoms with mode-selected, vibrationally selected
dideutero-methane. They found that the vibrational excitation of
either the symmetric or antisymmetric CH2 stretching mode leads to
a nearly identical enhan cement factor in total reactivity, which
again is a result of Coriolis coupling and energy dissipation.
Throughout the reaction Cl+CH2D2HCl+CHD2 the reaction
complex should conserve Cs-symmetry (collinear Cl
…
HC
configuration). Ab initio calculations reveal that there is a strong
curvature coupling involving the symmetric C-H stretching mode
whereas the asymmetric C-H stretching mode is of b symmetry and
therefore cannot couple directly to the translational motion of the
reaction complex. However, energy which is pumped into the
Fig. (15). a) Decomposition of the scalar curvature (bold line) into adiabatic curvature coupling coefficients (dashed lines) for the reaction CH3+H2CH4+H.
Vertical lines indicate the borders of 5 of the 6 reaction phases. The vertical dashed line at s = 0 amu1/2 Bohr denotes the position of the energy transition state.
- b) Decomposition of the reaction path direction given by vector t(s) into adiabatic internal coordinate contributions. - c) Repulsive and attractive internal
coordinate forces exerted on the atoms of the reaction complex. Bold lines indicate forces for which the associated internal coordinate dominates the direction
of the reaction path.
From Molecular Vibratio ns to Bonding, Chemica l Reactions, and Rea ction Current Organic Chemistry, 2010, Vol. 14, No . 15 1553
asymmetric stretching mode can wind up in the symmetric
stretching mode and by this in the translational motion via an
effective Coriolis coupling between the two modes, which is
possible due to the twisting of the vibrational mode vectors around
the path direction according to Eq. 5.4 (scalar product between
mode vector and derivative of another mode vector) and the
resulting mutual overlap despite of different starting symmetries. -
Similar observations were made by Duncan and co-workers [223]
who discussed ab initio dynamics studies of thermal and
vibrational-state selected rates of the H abstraction reactio n
CH4+ClCH3+HCl. - Liu and co-workers [224] showed in a
dynamics study of the reaction CH+H2CH2+H that vibrational
excitation of the HH stretching mode leads to notable enhancement
of reaction rates whereas excitation of the CH stretching mode is
insignificant. This is reflected by a large HH stretching curvature
coupling coefficient in the entrance channel of the reaction. -
Formic acid decomposes via competing reactions to either H2O+CO
(1) or H2+CO2 (2). The measured reaction rate for path (2) is
smaller by a factor of 40 than that for path (1) despite the fact that
the calculated activation energies are comparable for both paths.
Takahashi and co-workers used calculated curvature and Coriolis
coupling coefficients to derive a rationale for these findings [225].
Both paths possess a comparable curvature pattern in the entrance
channel. However, the Coriolis couplings are more pronounced in
case of path (2), thus supporting energy dissipation rather than
energy transfer into the translational motion, which consequently
reduces the reaction rate of (2) compared to that of (1). It seems that
mode selective enhancement of reaction rates does not only take
place in the gas phase but also in surface chemistry as the HCD3
decomposition to HCD2+D on bulk nickel in heterolytic catalysis
indicates [226]. - Analyses ii) and iii) lead to an understanding of
the energy flow along the reaction path as it is reflected by
processes of energy transfer and energy dissipation. They also
provide the basis to determine the translational energies left with
the products of a reaction.
Analysis of the Path Curvature in Terms of Adiabatic Mode
Curvature Couplings
The decomposition of the curvature peaks (enhancements) in
terms of cu rvature coupling coefficients is repeated, but now in
terms of localized AICoMs rather than delocalized normal
vibrational modes (Am,s(s) rather than Bμ,s(s) is analyzed). In Fig.
(15), the scalar curvatu re K(s) is decomposed in terms of AICoM-
curvature coupling co efficients. Curvature peaks K2 and K3 are
associated with the HH stretching and the CH stretching modes,
respectively where positive coefficients indicate that the local
vibrational motion in question supports the curving of the reaction
path and in this way the ch emical reaction whereas negative
coupling coefficients indicate that the corresponding vibrational
motion resists the curving of the path. The maxima and minima of
the AICoM coupling coefficients are located where th e functions
q(s) possess their maximal curving as reflected by the extremal
values of d2q(s)/ds2 (Fig. 13). For example, angle (HCHs)
determines the position of curvature peak K1 because at this point
van der Waals interactions between methyl radical and hydrogen
molecule cause th e former to change from a planar to a pyramidal
form (increase of from 90° to larger values; alternatively changes
in the angle =HsCHs can be connected with K1). Pyramidalization
couples with the R2 approach motion and therefore K1 is also
effected by a R2 curvature coupling. This finally explains the R2
contribution to mode 11 at the position of K1 (see Fig. 14). The HH
vibration is at a certain contact distance R2 no longer free and starts
involving th CH3 radical so that a CH vibration mixes in with th e
HH vibration (Fig. 13).
Curvature peak K2 is a consequence o f the change in parameter
R1(HH) (Fig. 13) that via adiabatic curvature coupling AR1,s is
responsible for the strong curving of the path. Local mode R2(CH)
resists the curving of the p ath (AR2,s<0, Fig. 15, see also Fig. 13)),
which results from the exchang e repulsion between the methyl
radical and the hydrogen molecule. For curvature peak K3, the
situation is reversed: Now the AICoM asso ciated with R2(CH)
couples with the path motion because a curving of the function
R2(s) (Fig. 13) dominates the path curvature. The AICoM driven by
R1 resists this path curving, which has to do with exchange
repulsion between CH4 and H atom in the reverse reaction. When
the process of CH bond formation is finalized, the C3v-symmetrical
reaction complex with angles <109.47º adjusts to the tetrahedral
angle of CH4, which leads to the shoulder in the curvature diagram
(Fig. 15, see also Fig. 13).
The adiab atic curv ature coupling coefficients relate any change
in the path curvature K(s) to a structural change of the reaction
complex and by this to the electronic events driving the reaction. In
this way, they provide the physical background to the partitioning
of the reaction mechanism into reaction phases. It becomes possible
to identify the positions of m inimal path curving with structures of
the reaction complex that are chemically meaningful. In the case of
the reaction CH3+H2, there are six phases (Fig. 15: phase 1: contact
phase, not shown in Fig. (15); phase 2: reactant preparation phase;
specifically pyramidalization and HH bond polarization phase;
phase 3 + phase 4: transition state ph ase with the ch emical
processes (phase 3: HH bond loosening and initialization of CH
bonding; phase 4: finalization of CH bond formation and HH bond
cleavage); phase 5: product adjustment phase, especially adjustment
from C3v- to Td - symmetry; phase 6: separation phase. Between the
phases, at positions of low curvature (not necessarily a curvature
minimum; see transition from phase 4 to phase 5), three chemically
relevant transient structures can be identified: 1) van der Waals
complex between pyramidal CH3 and polarized H2; 2) nonclassical
•CH5 radical with 3-center 3-electron C
…
H
…
H interaction; 3) van
der Waals complex between CH4 and H atom. By identifying the 6
reaction phases and the 3 transient structures, the reaction
mechanism is established.
5.7. From Reaction Complex to Reaction Path: Strategy of an
URVA Analysis
The analysis of the reaction mechanism based on URVA
utilizes th e close relationship between reaction complex an d
reaction path to overcome dimensionality problems and the
problem of quantitatively identifying among the multitude of
geometrical parameters the few that drive important structural or
electronic changes of the reaction complex [115-124]. The analysis
of 3N-L geometry parameters q(s) in the way indicated in Fig. (13)
and the identification of those changes in q(s), which are relevant
for the reaction mechanism, becomes tedious and ambiguous.
URVA solves these problems by a shift in the analysis from
reaction complex to reaction path as indicated in Fig. (16). The
changes in the geometry of the reaction complex during the reaction
are manifested in the form of the reaction path , which is also
defined in 3N-6 dimensional space and on first sight does not seem
to offer any adv antages for the mechanistic analysis. However, th e
reaction path can be considered as a smooth, curved line, which is
fully characterized by just three rather than 3N-L quantities at each
path point s. These are the path direction given by the tangent
1554 Current Organic Chemistry, 2010, Vol . 14, No. 15 Cremer
and Kraka
vector t(s), the path curvature given by the curvature vector k(s),
and the path torsion given by (s). The latter parameter is of little
relevance because helical or loop-type paths have not been
observed. Hence, it is sufficient to analyze just path direction and
path curvature as a function of s (Fig. 16).
In Figs. (9 and 16), it is schematically shown that tangent vector
and curvature vector describe complementary properties of the
reaction path. URVA decomposes the translational motion in path
direction and the vibrational motions orthogonal to the path
direction into adiabatic modes driven by internal coordinates that
are used to describe the reaction complex. Accordingly, if there is
an intern al coordinate q that dominates the path direction it will
provide only a small or no contribution to the transverse vibrational
modes of the reaction complex and vice versa. Therefore, there are
two ways of analyzing the reaction mechanism: i) via the path
direction; ii) via the path curvature. Sin ce the latter is much more
sensitive and physically related to the changes in the adiabatic force
constants (i.e. the first derivative of ka(s) with regard to s), which in
turn measures the polarizability of a bond, URVA is primarily
based on the analysis of the path curvature and uses the path
direction only to obtain complementary information. By analyzing
the curvature in term s of adiab atic curv ature coupling coefficients a
connection to the changes in the geometry of the reaction complex
is made because the internal coordinates q(s) that drive the
adiabatic modes with large curvature coupling coefficients
determine in this way also the curvature. Use of the adiabatic
curvature coupling coefficients has the advantage of i) singling out
those few internal coordinates q(s) that reflect important structure
changes of the reaction complex and ii) also identifying q(s)
changes that are not detectable in a diagram such as that of Fig.
(13). Normal mode curvature and Coriolis coupling coefficient are
used to determine energy transfer and energy dissipation whereas
adiabatic curvature coupling coefficients provide the basis to
elucidate the reaction mechanism.
Fig. (15b) contains a diagram with the amplitudes of the
adiabatic modes dominating the direction of the reaction path. In
phase 2, where AICoM R1 dominates the cu rvature, AICoM R2
does the same with regard to the path direction whereas in phase 3
the situation is reverted. Once it is clear which internal coordinates
dominate the path direction, one can calculate the corresponding
internal coordinate forces (Fig. (15c): bold lines). They can b e
interpreted using changes in the electron density distribution and
atomic population values. The sum of all forces leads to the
reaction force discussed in connection with Fig. (12).
6. CONCLUSIONS AND OUTLOOK
Vibrational spectroscopy provides a wealth of data that helps to
describe bonding, structure, and stability of molecules in their
equilibrium as w ell as in the situation of a chemical reaction.
Utilizing the Vib-Cal-X method as one of the Cal-X methods one
can use measured vibrational frequencies and derive from them a
complete set of normal mode force constants and their normal mode
eigenvectors. With the help of the adiabatic mode concept it is
possible to derive from the normal modes local modes that are
associated with the intern al coord inates describing the molecule in
question. The AICoMs are based on a dynamic principle and
possess properties such as frequency, mass, and force constant. It is
the advantage of the Vib-Cal-X approach th at in this way
experimental frequencies can be directly used for the determination
of AICoM bond stretching frequencies and force constants.
Contrary to bond length, bond density, and bond dissociation
energy, which all depend on two or more molecular quantities, the
bond stretching force constant is the only quantity that directly
relates to the intrinsic strength of a bond and therefore provides a
useful starting point for defining bond order or intrinsic bond
dissociation energies.
The information contents of the molecular vibrations can be
also exploited when studying a reacting molecule. The vibrations of
a reaction complex span the reaction valley, determine its shape, the
Fig. (16). Analysis scheme of URVA. The reaction path is analyzed in terms of path direction and path curvature to reduce the Nvib-dimensional problem of
analyzing the reaction complex to a M-dimensional problem (M < N) with the help of adiabatic vibrational modes and adiabatic curvature coupling
coefficients.
From Molecular Vibratio ns to Bonding, Chemica l Reactions, and Rea ction Current Organic Chemistry, 2010, Vol. 14, No . 15 1555
curving of the floor line of the valley, bi- or trifurcations of the
valley, and the conversion of a local vibrational movement into a
translational motion along the reaction path. Any reaction if
properly triggered starts with a vibration of increasing amplitude.
Since normal mode vibrations are delocalized, the analysis and
understanding of their properties require also for the reaction
complex a decomposition into local AICoMs. It has been shown in
this article that chemical reactions that involve at least two
chemical processes imply a curving of the reaction path. Paths
without curvature describe physical rather than chemical processes.
The path curvature is related to the change in the force constants of
the cleaving and forming bonds as reflected by the derivatives
ka
'(s).
The derivative of the force constant reflects both the
polarizability of the changing bond and the polarizing power of the
reaction partner. Since the peaks of the scalar curvature K(s)
correspond to the chemical processes, their number and sequence
determines the number and sequ ence of chemical even ts. From the
magnitude and shape of a curvatu re peak, one can draw conclusions
with regard to bond strength, polarizability, and reaction partner.
Due to the chem ical importance of the path curvature, the curvature
diagram K(s) can be used for the partitioning of a reaction into
reaction phases. For an elementary reaction with transition state,
van der Waals or contact phase, reactant preparation phase,
transition state phase, product adjustment phase, and separation
phase can be distinguished. Attempts to obtain a similar
mechanistic partitioning with the help of the reaction force are
limited by the fact that the energy is a cumulative quantity th at does
not enable one to distinguish between individual chemical events.
This is possible in the case of the scalar curvature because the latter
can be expressed with the help of the curvature coupling
coefficients, which in turn can be related to AICoMs and adiabatic
curvature coupling coefficients driven by the internal coordinates
used to describe the geometry of the reaction complex. Hence, by
shifting the analysis from the reaction complex to the reaction path,
describing the latter by path direction and path curvature, and
decomposing them into adiabatic contributions, each of which is
driven by an internal coordinate, a complicated Nvib-dimensional
problem is reduced to a much simpler M-dimensional problem (M
< N). This is the essence of the URVA mechanistic analy sis (Fig.
16).
A number of reaction systems have been studied so far utilizing
URVA and the insight provided by the vibrations of the reaction
complex [115-124]. Important outcomes of these studies concern
the dynamic definition of a chemical bond, the partitioning of the
reactions into reaction phases, and many other aspects of chemical
reactions. Noteworthy are the following findings.
1) In a study of HX and XX (X: halogen) cycloaddition
reactions to CC double bonds [120] the role of van der Waals
complexes was illuminated: They determine via exchange,
dispersion, inductive, and electrostatic forces the first electronic
structure chang es in th e reaction complex. Especially, van der
Waals interactions can determine the steric arrangement of the
reaction partners which is often kept throughout the reaction. In
these cases, it is justified to say that the van der W aals complex
determines the fate of the reaction complex for the whole reaction.
The URVA analysis reveals furthermore that even without the
formation of a stable van der Waals complex any van der Waals
interactions can be detected and their consequences for the
chemical changes of the reaction complex determined. - 2) In larger
reaction systems, the breaking of a bond can involve several
individual steps such as charge transfer or bond polarization,
rehybridization at the atomic centers involved, conformational
changes, -bond shifts, etc. All these changes can be identified with
the help of the local vibrational mod es and their coupling with th e
curvature vecto r. Especially, it is possible to determine the degree
of (a)synchronicity of coupled bond breaking and forming
processes [117,121,123]. - 3) Energy transition state and transition
state region, i.e. the region where the chemical processes take place
are not necessarily related. The preparation of a reaction complex
for the actual ch emical processes can require so much energy that it
leads to the highest point of energy along the reaction path. The
chemical processes take place in such a situation down hill long
after the en ergy transition state. An example for such a discrepancy
between energy transition state and transition state region is the
Diels-Alder reaction [119]. - 4) The partitioning of the reaction
mechanism into reaction phases unravels similarities between
seemingly different reaction types that were so far not known. For
example, the cycloaddition of HX (X = halogen) to CC multiple
bonds resembles in the first part of the mechanism a substitution
reaction and in the second part a simple ion combination reaction.
This can be used for an advanced classification of elementary
chemical reactions [120]. - 5) The shape and magnitude of the
calculated curvature peaks provide direct insight into the ease of
breaking or forming a chemical bond. It is possible to determine the
interplay between the strength of a bond being changed and the
bond polarizing ability of a reaction partner. Future work is aimed
at setting up quantitative relationships so that the outcome of
chemical reactions becomes predictable and controllable. - 6) By
studying the avoided vibrational crossings of the reaction complex
along the reaction path it becomes possible to make qualitative
predictions with regard to energy dissipation (between vibrational
modes) and energy transfer (between vibrational and translational
motions). In this w ay, mechanistic pred ictions with regard to mode-
selective rate enhancement can be made even in non-trivial cases
[117]. In many cases, the possibility of rate enhancement is lost as a
consequence of energy dissipation between vibrational modes. A
detailed understanding of the reaction mechanism makes it possible
to change the reaction complex in such a way (isotope substitu tion,
substituent variation, etc.) that energy transfer is suppressed
because of extended energy dissipation.
Previous work with URVA has focused on two major
mechanistic topics: i) To understand the dynamic reality of the
Woodward-Hoffmann rules and to unravel the mechanistic
differences between symmetry-allowed and symmetry-forbidden
reactions [119-121]. Results obtained so far clearly show that
forbidden reactions involve the drastic change of just a few bonds
of a reaction complex leaving all other structural parameters largely
unaffected. Symmetry-allowed reactions mostly possess a
preparation phase in which many structural parameters collectively
and smoothly change to some degree. This requires some energy,
however not as much as would be required for the drastic change of
just a few parameters in a symmetry-forbidden reaction. In the latter
case, there are additional reaction phases that develop into distinct
reaction steps as soon as environmental factors lead to the
generation of an intermediate and a second transition state. ii) These
observations have led to the concept of hidden transition states and
intermediates [120,123]. By partitioning the reaction mechanism
into reaction phases with the help of th e path curvatu re one can
identify transient structures of the reaction complex in regions of
minimal curvature that relate to real structures in form of radicals or
1556 Current Organic Chemistry, 2010, Vol . 14, No. 15 Cremer
and Kraka
biradicals, carbenes or ions, nonclassical structures with 3-center
bonds, transition state similar structures, etc. It has been shown that
by changing the reaction environment (temperature, pressure,
solvent) or by introducing stabilizing factors such as substituents,
rings, hetero atoms or units with electron-delocalization
possibilities the transient stru ctures can become real intermed iates
and real transition states. Therefore the terms hidden intermediates
and hidden transition states have been coined. It was shown for the
strongly exothermic barrierless chelotropic cycloaddition of
methylene to ethene yielding cyclopropane that its mechanism
comprises four reaction phases, two hidden intermediates and one
hidden transition state, which by variation of the methylene (either
by substituent variation or by exchange of the central atom) can be
converted into real intermediates and real transition state. The
investigation further suggested that by the mechanistic description
of a prototypical reaction the mechanism of other chelotropic
carbene (silylene, etc.) additions can be reliably predicted [123].
Clearly, the identification of hidden intermediates and hidden
transition states is a step forward to both the understanding and
controlling of chemical reaction mechanism.
There is a broad spectrum of possible applications for both the
Vib-Cal-X approach and URVA analysis of chemical reactions. A
systematic application of the former approach will lead to a more
comprehensive understanding of chemical bonding that will provide
detailed scales of bond strength as reflected by adiabatic force
constants and bond orders and intrinsic bond dissociation energies
derived therefrom. It will be possible to assess the various facets of
weak bonding reaching from 3-electron situations to H-bonding,
agostic bonds, and noncovalent interactions in general. It will be
challenging to establish bond order relationships that can be applied
throughout a chemical reaction thus offering simple parameters that
measure the degree to which a chemical process has been
developed and that help to compare transient structures of different
or the same reaction. With such a tool at hand, it will be also
possible to design bonds of extreme strength where especially
transition metal combinations are suitable candidates for high bond
energies. It will be another challenge to determine intrinsic bond
dissociation energies and fragment stabilisation energies to
complement the description of chemical bonding. - So far the
URVA analysis has provided advanced insight into chemical
reaction mechanism. However, there is still a long way to go to
detail all factors th at influence the sequence of reaction steps,
mutations of the reaction complex, energetics, stereochemistry, and
the environmental dependence of a chemical reaction. Primary goal
of the current work is i) to exploit the informational content of
vibrational spectra using Vib-Cal-X and ii) the understanding of
enzymatic reactions and chemical catalysis in general. By fulfilling
these goals it will be possible to control chemical reactions and to
design effective chemical catalysts.
ACKNOWLEDGMENT
This work was financially supported by the National Science
Foundation, Grant CHE 071893. We thank SMU for providing
computational resources. Proofreading and useful comments by
Robert Kalescky are acknowledged.
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Received: 26 December, 2009 Revised: 11 February, 2010 Accepted: 11 February, 2010
