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Page | 1
Examining the Impact of Harmonic Correlation on Vibrational
Frequencies Calculated in Localized Coordinates
Magnus W. D. Hanson-Heine
School of Chemistry, University of Nottingham, University Park, Nottingham NG7 2RD.
magnus.hansonheine@nottingham.ac.uk
ABSTRACT: Carefully choosing a set of optimized coordinates for performing vibrational
frequency calculations can significantly reduce the anharmonic correlation energy from the
self-consistent field treatment of molecular vibrations. However, moving away from normal
coordinates also introduces an additional source of correlation energy arising from mode-
coupling at the harmonic level. The impact of this new component of the vibrational energy is
examined for a range of molecules, and a method is proposed for correcting the resulting self-
consistent field frequencies by adding the full coupling energy from connected pairs of
harmonic and pseudoharmonic modes, termed vibrational self-consistent field (harmonic
correlation). This approach is found to lift the vibrational degeneracies arising from
coordinate optimization, and provides better agreement with experimental and benchmark
frequencies than uncorrected vibrational self-consistent field theory without relying on
traditional correlated methods.
Page | 2
Introduction
Theoretical spectroscopy has grown to become both indispensible and routine for a wide
range of applications.
1-8
However, the accurate prediction of vibrational frequencies remains
challenging for even moderately sized molecules due to the high computational cost. Such
calculations are made tractable through approximations to the solution of the nuclear
Schrödinger equation. The most common of these is the harmonic approximation,
9
where the
total vibrational energy is considered as the sum of a harmonic energy term and an
anharmonic correction, which is either assumed to be negligible or treated by scaling,
=
+
.
1
Within this model, harmonic vibrational frequencies and normal modes can be computed
routinely for large molecules from the eigenvalues and eigenvectors of the mass-weighted
Hessian matrix, (H), corresponding to the nuclear Cartesian displacements for each atom,
=
1
2
()
,
2
where x
i
represents a Cartesian displacement coordinate of an atom with mass m
i
.
As additional computational power becomes available and the size of the molecules being
studied increases, anharmonic effects such as band reordering, overtone bands, and transition
specific anharmonicity become vital for the accurate interpretation of experimental spectra.
At the same time, the non-local nature of the nuclear Schrödinger equation and the
Page | 3
exponential scaling of the complete nuclear potential energy surface (PES), causes the gap
between the time taken for harmonic and anharmonic calculations to widen rapidly.
In order to address this, additional approximations are commonly used when computing
vibrations beyond the harmonic approximation. A wide range of methods exists, including
taking harmonic solutions in normal coordinates and then adding anharmonic effects as
perturbations using vibrational perturbation theory (VPT),
10-13
or its transition optimized
shifted Hermite (TOSH)
14
or Van Vleck (VVPT) modifications,
15, 16
or by using vibrational
configuration interaction (VCI) based on harmonic excitations.
17, 18
Alternatively, vibrational
self-consistent field theory (VSCF)
19-24
can be used to generate variational modal
wavefunctions, and correlations between them are taken into account through the use of
perturbation (VMP2),
25-28
configuration interaction (VSCF-CI),
29-33
or coupled cluster (VCC)
theories.
34, 35
The VSCF method is the preferred way of calculating anharmonic corrections for large
molecules where the dimensions of the vibrational wavefunction make calculating the
correlation energy computationally demanding.
24, 36-38
However, a common bottleneck across
all of these methods is the significantly increased sampling of the PES required for chemical
accuracy. This is particularly true when ab initio methods are used to model the electronic
structure.
39
The complete nuclear potential energy surface can be approximated using grid based
methods, or by expanding the surface as a truncated Taylor series where derivatives of the
energy with respect to nuclear displacement are evaluated up to the fourth order (quartic force
field, QFF) around an optimized geometry for which the gradient of the energy is zero.
40
As a
further approximation, energy derivatives involving more than a specific number of normal
Page | 4
modes can also be excluded, reducing the mode-coupling in the PES to an n-mode
representation (nMR).
14, 25, 40
More recent studies have tried to address the issue of increased PES sampling by using
coordinates other than normal coordinates to describe the PES. Jacob and Reiher have
calculated unitarily transformed harmonic normal modes and frequencies associated with
bands in the vibrational spectra of α-helical (Ala)
20
and several other polypeptides,
41-43
such
that the spatial localization of the normal coordinates and harmonic modes was maximized
according to criteria developed from localized molecular orbital theory.
41, 44, 45
Following
earlier work by Thompson and Truhlar,
46
Yagi, Keçeli, and Hirata also developed a scheme
for using optimized normal coordinates that minimize the ground-state energy during a VSCF
calculation (oc-VSCF),
47
and Yagi and Otaki used vibrational quasi-degenerate perterbation
theory (oc-VQDPT) to account for the degeneracies that arise when making use of these
optimized coordinates.
48
Cheng and Steele recently proposed that by expressing the
anharmonic force field in the spatially localized coordinates of Jacob and Reiher during a
VSCF calculation (L-VSCF), the increased spatial decay of the mode-coupling elements
could be exploited to reduce the size of the potential energy surface.
49
Their work showed
that local modes centred more than 5 to 8 a
0
apart could be removed with only a slight
reduction in the calculations accuracy, and that energy derivatives involving more than two
modes at a time also became less important. Panek and Jacob independently suggested an
equivalent procedure, noting that VSCF-CI frequencies expressed in localized coordinates, L-
VCI, converged more quickly with respect to expansion of the excitation space than
equivalent calculations using normal coordinates.
50
Christiansen and co-workers have also
extended the VCC methodology to use the optimized coordinates of Thompson and Truhlar
(oc-VCC),
51, 52
and determined that significant improvements in accuracy are possible over
Page | 5
normal coordinate calculations for small and weakly interacting systems, and that optimized
coordinates can provide a reduction in mode-mode correlations present within the VCC
wavefunction.
Using optimized or localized coordinates shows a wide range of potential benefits for
anharmonic calculations. However, following coordinate optimization, the mass-weighted
Hessian matrix is no longer diagonal. The choice of coordinates for use within VSCF and
subsequent post-VSCF correlated methods is therefore complicated, as significant differences
between the optimized and normal modes increases the coupling between modes at the
harmonic level and introduces an additional error into the mean-field calculation. The same
differences also increase the likelihood that useful improvements can be made to the
anharmonic part of the calculation.
While the effects of this compromise can be significant for molecules of any size, they are
expected to become more pronounced for larger systems where the spatial delocalization in
normal modes becomes more apparent.
41
However, errors relating to harmonic coupling
effects have so far received little attention in the literature owing to the limited size of the
systems studied and the relatively small differences observed between harmonic and
optimized mode frequencies.
46, 47, 49-52
In particular, the VSCF ground state minimum energy
coordinates used during oc-VSCF and its correlated variations have been shown to give
pseudoharmonic frequencies in close agreement with the normal mode frequencies for small
or weakly interacting systems,
46, 47, 51, 52
while the differences between pseudoharmonic and
normal mode frequencies associated with the localized modes of Jacob and Reiher have been
found to be more significant.
49, 50
L-VSCF frequencies have therefore been used throughout
this work in order to highlight the effects of harmonic coupling during VSCF calculations.
Page | 6
Theory
Harmonic Correlation Energy
Excluding rotations, the nuclear vibrational Hamiltonian can be written as
=
1
2
2
2
=1
+
1
, ,
,
3
where
1
, ,
represents the Born-Oppenheimer nuclear PES expressed in normal
coordinates, assumed here to be represented by a Taylor series expansion up to fourth-order
around a minimum geometry,
=
1
2
2
()
,
+
1
6
3
()
,,
+
1
24
4
()
,,,
,
4
In the VSCF approximation, the vibrational wavefunction is described by the Hartree
product of single mode wavefunctions, referred to as modals,
Ψ
s
1
, . . . ,
M
=
(
)
=1
,
5
where
i
is a modal along normal coordinate Q
i
, with the associated quantum number s
i
.
Varying the normalized modals so as to minimize the expectation value of the nuclear
Page | 7
Hamiltonian operator leads to the VSCF energy where each mode sees an averaged potential
from the other modes.
The total vibrational energy of a given state (Eq. 6) can therefore be considered as the sum of
three different terms; the harmonic energy, the difference between the harmonic energy and
the VSCF energy,
(Eq. 7), and the remaining difference between the VSCF energy
and the total energy, termed vibrational correlation,
(Eq. 8)
=
+
+
,
6
=
,
7
=
.
8
After the nuclear Hamiltonian has been transformed into localized coordinates, the VSCF
potential for each mode now contains additional terms reflecting the mean-field treatment of
the mode-coupling between the localized modes at the harmonic level. The VSCF procedure
has been described in detail elsewhere,
24, 39
and therefore only the additional harmonic
coupling terms are included here,
,
(0)
=
1
2
,
,
9
and
,
(1)
=
,
10
where
,
(0)
and
,
(1)
are changes to the zeroth- and first-order terms in the expansion of
the one-mode potential of the mth local mode caused by harmonic coupling between the
localized modes, F
ij
is the harmonic force constant involving the ith and jth local modes,
Page | 8
is shorthand notation for
, and the sums run over distinct local modes excluding
the mth mode.
The VSCF expression for the total energy in Eq. 6 can then be rewritten as the sum of a
pseudoharmonic term derived from the diagonal elements of the mass-weighted Hessian
matrix, ′
, the difference between the pseudoharmonic energy and the L-VSCF energy,
′
(Eq. 12), and the correlation energy associated with these new coordinates, ′
(Eq. 13),
= ′
+ ′
+ ′
,
11
′
= ′
′
,
12
′
=
′
.
13
As the total vibrational energy is the same in both sets of coordinates, it follows from Eqs. 6
and 11 that the complete harmonic energy is progressively recovered during the L-VSCF and
post-L-VSCF calculations. The VSCF and correlation energy corrections to the
pseudoharmonic frequencies can therefore be further split into an anharmonic and a harmonic
component,
′
=
+
,
14
′
=
+
,
15
= ′
+
+
,
16
where the harmonic local mode-coupling energy is progressively reintroduced in a self-
consistent, and then correlated form, as ′
and ′
in Eq. 16, respectively. This
Page | 9
additional component of the vibrational correlation energy, termed harmonic correlation, can
lead to significant errors in the L-VSCF frequencies, which are the focus of this work.
By rearranging Eq. 16, the harmonic correlation energy may be expressed as
′
=
′
,
17
where the harmonic and pseudoharmonic energies are determined from the square root of the
diagonal elements of the Hessian matrix (
) in normal or localized coordinates using Eq. 18,
(′)
=
+
1
2
,
18
and
are the harmonic or pseudoharmonic mode occupation numbers.
Calculating ′
is complicated by the interaction between the harmonic and anharmonic
parts of the wavefunction and Hamiltonian. However, previous studies using optimized
coordinates have reported that degeneracies formed between modes as a result of
optimization are not noticeably lifted at the VSCF level when reported to within wavenumber
accuracy, and instead require single quanta VCI excitations or higher forms of correlation to
remove.
46, 47, 49-52
Eqs. 9 and 10 also suggest that the energy term is expected to be small for
low-lying vibrational states. In lieu of a more rigorous definition of ′
, the harmonic
coupling energy can be used as an approximation for the harmonic correlation,
′
,
19
This correction, denoted (HC), can be added to the L-VSCF energy, and has been calculated
for the L-VSCF frequencies of the systems studied.
Page | 10
Computational Details
The L-VSCF and L-VCI methods, along with their normal mode counterparts, were applied
to water, oxygen difluoride, hydrogen cyanide, methanol, and 9-fluorenone, together with
calculations including the harmonic coupling energy as a post-VSCF energy correction. The
triatomic molecules were chosen as they are naturally localized systems for which highly
correlated benchmark calculations are feasible, methanol represents a more flexible system
with a more complex set of harmonic couplings between the localized and normal modes, and
9-fluorenone is used for comparing against experiment.
VSCF-CI calculations were performed using VSCF modals in normal coordinates, and are
referred to simply as VCI(n) from this point forward, where n denotes the vibrational
excitation rank included in the VCI wavefunction expansion.
The equilibrium structures, normal coordinates, and harmonic frequencies were determined
using the 6-311G(d,p) basis set and Euler-Maclaurin-Lebedev EML-(128,590) grid
combination for H
2
O, F
2
O, and HCN, using the 6-31G(d,p) basis set and EML-(75,302) grid
for methanol, and using the 6-311G(d,p) basis set and SG-1 grid for 9-fluorenone. Detailed
discussions of the integration grids can be found elsewhere.
53-55
Anharmonic third- and
fourth-order derivatives of the energy with respect to nuclear displacements were used within
in a Taylor series expansion of the QFF PES, calculated using numerical differentiation of the
Hessian matrices with a step size of 0.5291 Å along normal modes and a step size of 0.4 Å
used for the localized modes. The anharmonic frequencies were calculated from a 2MR
representation of the QFF for 9-fluorenone and a 4MR representation for all of the other
molecules. The geometry and Hessian calculations were all performed using the B97-1
exchange-correlation functional,
56
and were carried out using a developmental version of the
Q-Chem software package.
57
The (L-)VSCF calculations were carried out with a basis set
Page | 11
consisting of the 20 lowest harmonic oscillator wavefunctions along each of the normal
modes and a convergence threshold of 1×10
-6
cm
-1
. Anharmonic derivatives associated with
harmonic modes below 300 cm
-1
were excluded from the calculation of 9-fluorenone, and the
lowest energy mode of methanol was removed due to possessing a high degree of internal
rotational character. The (L-)VSCF frequencies reported for the triatomic molecules studied
were obtained from separate VSCF calculations on the ground and excited states, while the
(L-)VSCF frequencies reported for methanol and 9-flurenone were obtained from the
eigenvalues of the ground state VSCF modals so as to avoid issues relating to variational
collapse.
The localized modes used throughout this work used the localization transformation proposed
by Jacob and Reiher,
41
= ,
20
where the transformation, (U), is based on the molecular orbital localization scheme of Pipek
and Mezey,
45
and the transformation maximizes sum of the squares of the atomic
contributions to the normal modes,
=
,
2
= ,,
2
,
21
where μ and i are normal modes and nuclei, respectively, M is the number of normal modes,
and N is the total number of nuclei.
The unitary transformations were determined by Jacobi sweeps with a convergence criterion
of 1×10
-6
ξ
at
, and the resulting localized modes were then used to transform the mass-
Page | 12
weighted Hessian matrix from atomic Cartesian displacement coordinates into localized
mode coordinates using
=
.
22
The normal mode localization for 9-fluorenone was limited to the diagnostic region between
1200 and 1800 cm
-1
, and simulated spectra were generated by representing each vibrational
band as a Gaussian function with an area proportional to the intensity. The intensity for each
band was calculated from the derivative of the dipole moment with respect to displacement
along the normal coordinate associated with that frequency. The Gaussian bandwidths were
determined using the intensities, with bandwidths of 2, 3, 4, 5, and 6 cm
-1
used for intensities
in the ranges < 10, 10-20, 20-30, 30-150, and > 150 km mol
-1
, respectively.
58
Results and Discussion
Triatomic Molecules
The effects of the harmonic coupling energy have been examined for the L-VSCF
frequencies of the three triatomic molecules H
2
O, F
2
O, and HCN, through comparison with
highly correlated VCI(10) benchmark calculations, which are shown in localized coordinates
in Table 1 and in normal coordinates in Table 2.
In the case of H
2
O, the L-VSCF method gives a mean absolute deviation (MAD) of 28 cm
-1
from the converged VCI result. This improves to 9 cm
-1
once the harmonic coupling
correction has been added. In the localized mode coordinates the VCI frequencies converge
to within a 6 cm
-1
MAD after the inclusion of just single quanta excitations, while 4-quanta
excitations are required for a similar level of convergence in normal coordinates. Although
the bending mode has the same frequency in both sets of coordinates, the majority of the
Page | 13
anharmonic contribution to this mode is already accounted for at the L-VSCF level. Overall,
L-VSCF(HC) provides a superior match with full VCI than either VCI(2) or VCI(3).
For F
2
O, L-VSCF(HC) has a MAD of 2 cm
-1
from the converged result, which is roughly
equivalent to VCI(2) in normal coordinates, and outperforms L-VCI(4) in localized
coordinates. Unlike the other two triatomics studied, the bending mode of F
2
O has a
harmonic coupling correction due to a reduction in the partial stretching character included in
the mode following localization.
For H
2
O and F
2
O, the average harmonic coupling energies are 33 and 87 cm
-1
, respectively.
However, for HCN these differences are larger, with 219 cm
-1
observed for the CN stretching
mode. As a consequence there is significantly greater harmonic coupling present between the
local mode coordinates than for these two molecules, and the frequencies are slower to
converge with respect to the VCI excitation level. Furthermore, the frequencies are not fully
converged by VCI(10) in either set of coordinates. In local coordinates, L-VSCF(HC) is
closer to the L-VCI(10) result than either L-VSCF or L-VCI(3), while the MAD of 83 cm
-1
is
also superior to VSCF and VCI(3), which have MADs of 85 and 87 cm
-1
when compared to
VCI(10) in normal coordinates, respectively.
Overall, the inclusion of harmonic coupling energy in L-VSCF provides a significant
improvement over uncorrected L-VSCF with no significant increase in computational time,
and can lead to better agreement with the L-VCI(10) result by up to 219 cm
-1
despite not
accounting for any additional anharmonicity. Frequencies calculated using L-VSCF(HC) can
also be more accurate than incomplete VCI across a range of different excitation levels. This
effect is expected to become more significant for larger systems where the extent of the
harmonic coupling increases, while a combination of lower nMR representations of the PES
Page | 14
and lower VCI(n) representations of the wavefunction become necessary for efficient
calculations.
Connecting Vibrational States
Calculating the harmonic coupling energy is straightforward for low-lying transitions in small
rigid molecules where the normal modes are naturally localized, or for vibrational ground
states, where the occupation numbers are zero in both sets of coordinates. However, as the
systems become larger and more flexible, and the normal modes become more delocalized,
an excitation in one of the local modes can no longer be intuitively related to a corresponding
excitation in the normal modes. In these cases, the definition of related harmonic and
pseudoharmonic energies becomes more involved, and a protocol must be established to
relate the harmonic and pseudoharmonic states.
Zou, Kalescky, Kraka, and Cremer have recently proposed a connection scheme for relating
normal modes with their corresponding adiabatic local coordinate modes by scaling the
coupling terms linking the two.
59, 60
A similar procedure is proposed here. In optimized
coordinates the diagonal part of the mass-weighted Hessian matrix contains the square
pseudoharmonic frequencies, and the off-diagonal terms relate them to the square harmonic
frequencies. Using a connection factor, λ, the Hessian matrix can be expressed as the sum of
these diagonal and off-diagonal terms,
=
+
,
23
where the subscripts “o” and “d” denote the off-diagonal and diagonal elements, respectively.
Page | 15
Diagonalizing the resulting matrix returns the square of the harmonic frequencies in the limit
that λ = 1, and returns the square of the pseudoharmonic frequencies in the limit that λ = 0.
Solving the eigenvalue equation,
+
=
,
24
for an increasing value of λ therefore progressively reintroduces the coupling between the
localized modes, connecting them to their corresponding normal modes. Each mode is
followed by comparing the current frequency with the closest frequency at the previous λ
value, unless a two-point extrapolation indicates that two modes undergo reordering, at which
point the order of the modes at that step is chosen so as to maintain maximum overlap with
the modes from the previous level of coupling, determined by Eq. 25,
=
3
,
25
where O
μν
is the overlap between the μth mode at the current level of coupling and the νth
mode at the previous level. This procedure allows for differentiation between modes in cases
where the frequencies become degenerate or undergo energetic reordering as they couple
together.
The coupling connection scheme is analysed using a step size of 1×10
-5
for methanol and is
shown in Figure 1. The mode-coupling scheme of methanol is significantly more complicated
than for the triatomic molecules studied, due to both the additional degrees of freedom and
near degeneracy in several of the localized modes.
Page | 16
Figure 1. The coupling connection scheme relating the localized and normal mode
frequencies for methanol.
The harmonic modes ω
5
and ω
7
with pseudoharmonic frequencies at 2013 cm
-1
reorder after
ca. 87 % coupling has been reintroduced. The more energetic of these two modes then
crosses ω
6
after ca. 98 % coupling. These three local modes then become associated with the
harmonic frequencies at 1387, 1507, and 1497 cm
-1
, respectively. The two modes ω
9
and ω
10
,
with pseudoharmonic frequencies at 2597 cm
-1
also reorder after ca. 23 % coupling, to
become the harmonic modes at 2981 and 3030 cm
-1
.
This reordering highlights the importance of accurate local-harmonic mode assignments, as
even the normal mode VCI transitions being used as a benchmark are dependent on the
assignment of the harmonic and pseudoharmonic states.
1000
1500
2000
2500
3000
3500
4000
0.0 0.2 0.4 0.6 0.8 1.0
Wavenumber / cm
-1
Scaling factor λ
Page | 17
The calculated frequencies show that even relatively small cases, such as methanol, can show
large differences between L-VSCF and benchmark frequencies unless the choice of
optimized coordinates is carefully selected to minimize the harmonic mode-coupling. Several
of the localized modes in Figure 2 show hybrid bending and stretching character as the
bending and stretching mode subspaces were not explicitly separately selected during the
localization procedure.
Figure 2. The localized vibrational modes for methanol, shown with their associated
pseudoharmonic frequencies in cm
-1
.
The L-VSCF modes were found to vary between -934 to +537 cm
-1
from the VCI(4) result
(Table 3), with a MAD of 454 cm
-1
. The MAD decreased significantly to 36 cm
-1
, after
inclusion of the harmonic coupling energy. The largest harmonic coupling, seen for the
localized form of mode ω
8
, causes a 951 cm
-1
redshift in the frequency compared with an
anharmonic shift of just 46 cm
-1
. The total 997 cm
-1
correction amounts to more than 67 % of
the overall VCI(4) transition energy, and the frequencies reported for the ω
8
transition differ
from the VCI(4) result by 934, 235, and 17 cm
-1
when calculated using L-VSCF, L-VCI(1),
and L-VSCF(HC), respectively.
Page | 18
Due to the difficulty in assigning VCI states that fall intermediate between substantially
different sets of localized and normal modes, L-VCI states have been included with up to
single excitations, and have been assigned by comparing the fundamental transitions in order
of ascending energy. The harmonic coupling converges rapidly with respect to the L-VCI
excitation rank, indicated by the MAD falling to 149 cm
-1
following inclusion of single
quanta excitations. However, where the L-VSCF reference is significantly different from the
target state, high excitations are likely to be needed for convergence, and single reference or
perturbation methods may fail to converge all together. The error in L-VCI(1) for methanol is
still more than four times greater than for L-VSCF(HC).
Harmonic correlation errors are significantly more important for methanol than for the
simpler triatomic molecules studied, indicating that uncorrected L-VSCF frequencies may be
much further from accurate theoretical or experimental frequencies for larger or more flexible
molecules where the energy surface is more complex and increased spectral density and line
broadening effects make higher accuracy more important.
Experimental Comparison
One of the main purposes of vibrational calculations is to aid in the assignment of
experimental spectra. This is particularly true for molecules of medium or larger size. The
harmonic, VSCF, L-VSCF, and L-VSCF(HC) spectra of 9-fluorenone have therefore been
calculated, and are compared with the experimental infrared spectrum in Figure 4. The
experimental spectrum has been adapted from Ref.
61
so as to provide as close a match as
possible to the in vacuo calculations.
Page | 19
Figure 3. The molecular structure of 9-fluorenone.
Page | 20
Figure 4. The gas phase experimental infrared spectrum of 9-fluorenone adapted from Ref.
61
,
shown with the calculated infrared spectra using the L-VSCF(HC), L-VSCF, VSCF, and
harmonic approximations.
The two highest frequency vibrational bands at 1734 and 1609 cm
-1
contain a high degree of
carbonyl stretching and out-of-phase ring carbon stretching, respectively. For these two
modes VSCF and L-VSCF(HC) provide very similar results. However, L-VSCF(HC)
provides a slightly better match with the experimental data for the lowest energy vibrational
Page | 21
band. The MADs from experiment are 22 cm
-1
and 23 cm
-1
for L-VSCF(HC) and VSCF
across the four modes in Table 4, respectively, and the differences between them are small
enough to fall within the error for anharmonic calculations.
14
These two methods can
therefore be considered roughly equivalent for 9-fluorenone.
In contrast, uncorrected L-VSCF provides a much worse visual match with the experiment
than the harmonic prediction, with a MAD of 78 cm
-1
for L-VSCF compared to 38 cm
-1
for
the harmonic calculation. While the harmonic calculation is also in reasonably good
agreement with the experimental data, as anharmonic effects tend to be relatively small and
systematic in this region of the spectrum, this fact only serves to highlight the detrimental
effects that harmonic coupling can produce. When calculated using L-VSCF, the carbonyl
mode is 5 and 4 cm
-1
closer to the experimental value than seen for L-VSCF(HC) and VSCF,
respectively. However, all of the remaining bands are clustered in the region between 1490
and 1426 cm
-1
.This kind of qualitative error can lead to experimental misassignment if
uncorrected L-VSCF is used as a standalone method.
Overall, the large magnitude and unpredictable sign of the harmonic correlation errors can
make L-VSCF significantly worse than fully coupled harmonic calculations. This is true even
when a limited localization subspace is used. Uncorrected L-VSCF frequencies are therefore
not recommended for comparison with experiment unless the harmonic correlation is known
to be suitably small in the chosen set of coordinates.
Conclusions
The results presented above show that even for relatively small systems with limited normal
mode localization, such as methanol or the three triatomics examined, optimized coordinates
Page | 22
need to be carefully selected in order to avoid introducing significant errors from the
incomplete treatment of harmonic mode-coupling with the VSCF and truncated VCI
wavefunction approaches.
If the normal and localized modes are allowed to vary significantly, then the harmonic
coupling can grow to become significantly larger than the normal mode anharmonicity
correction by many hundreds of wavenumbers. This effect is expected to grow with the
system size, and has the potential to severely limit the widespread use of L-VSCF as either a
standalone method or as the zeroth-order approximation for correlated calculations.
Including the harmonic coupling energy as a first-order approximation for the harmonic
correlation in L-VSCF(HC) has been shown to be an efficient way of adding a post-VSCF
correction in localized coordinates, significantly reducing the error with a fractional increase
in the computational cost. Scaling the mass-weighted Hessian matrix allows for the
pseudoharmonic frequencies of the localized modes to be related to the normal mode
harmonic frequencies through a coupling connection scheme so as to define pairs of related
localized and normal mode harmonic states in complex systems.
These approaches are particularly useful for larger molecules where other more expensive
post-VSCF correlation approaches become less practical, and the normal mode localization is
less constrained by system size. Harmonic correlation errors in the L-VSCF frequencies lead
to qualitative errors in the infrared spectrum of 9-fluorenone, and provides a significantly
worse match for the experimental spectrum than provided by a standard harmonic
calculation. This increases the likelihood of experimental misassignment. However,
incorporation of the harmonic coupling energy gives corrected frequencies that are in better
overall agreement with the experimental spectrum than VSCF.
Page | 23
Although the harmonic coupling energy provides an approximate way of correcting L-VSCF
frequencies in cases where harmonic correlation errors can be significant, care must still be
taken to ensure that the harmonic and pseudoharmonic states are correctly matched and that
the coordinates chosen have a negligible amount of harmonic coupling at the SCF level until
a more rigorous definition of the mean-field harmonic coupling can be defined.
Acknowledgements
I would like to thank the University of Nottingham and EPSRC for funding (EP/L50502X/1).
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Page | 28
Table 1. Fundamental anharmonic vibrational frequencies calculated using L-VSCF, L-VSCF(HC), and L-VCI(n) (n = 1-10). Frequencies are in cm
-1
.
Molecule
L-VSCF
L-VSCF(HC)
L-VCI(1)
L-VCI(2)
L-VCI(3)
L-VCI(4)
L-VCI(5)
L-VCI(6)
L-VCI(7)
L-VCI(8)
L-VCI(9)
L-VCI(10)
H
2
O
1584
1584
1587
1585
1582
1581
1579
1579
1579
1579
1579
1579
3742
3692
3700
3706
3706
3705
3704
3704
3704
3704
3704
3704
3743
3792
3790
3786
3785
3785
3783
3783
3783
3783
3783
3783
F
2
O
581
481
508
527
487
482
480
479
479
479
479
479
902
862
865
885
865
861
860
860
860
860
860
860
903
996
1017
1027
997
997
994
994
994
994
994
994
HCN
763
763
808
806
802
754
726
712
698
684
667
644
763
763
808
806
802
754
726
712
698
684
667
644
2373
2154
2202
2228
2192
2177
2170
2168
2167
2165
2161
2170
3111
3260
3292
3392
3364
3330
3349
3335
3332
3329
3320
3340
Page | 29
Table 2. Fundamental anharmonic vibrational frequencies calculated using VSCF and VCI(n) (n = 1-10). Frequencies are in cm
-1
.
Molecule
VSCF
VCI(1)
VCI(2)
VCI(3)
VCI(4)
VCI(5)
VCI(6)
VCI(7)
VCI(8)
VCI(9)
VCI(10)
H
2
O
1585
1588
1585
1599
1581
1579
1579
1579
1579
1579
1579
3750
3751
3717
3731
3709
3705
3705
3704
3704
3704
3704
3808
3904
3816
3828
3785
3784
3783
3783
3783
3783
3783
F
2
O
480
480
479
481
479
479
479
479
479
479
479
863
873
864
865
860
860
860
860
860
860
860
1001
1002
996
997
994
994
994
994
994
994
994
HCN
765
810
779
797
751
723
710
697
682
665
640
765
810
779
797
751
723
710
697
682
665
640
2165
2164
2170
2195
2178
2170
2169
2168
2168
2166
2168
3254
3275
3328
3353
3326
3335
3326
3325
3323
3315
3343
Page | 30
Table 3. Fundamental vibrational frequencies for the normal and localized harmonic modes of methanol together with their
differences (E
corr
), and several localized anharmonic approximations. Δ values represent differences from VCI(4). Modes are
numbered in order of ascending harmonic energy. Frequencies are in cm
-1
.
Mode
VCI(4)
Harm.
Local
E
corr
ΔAnhar.
ΔL-VSCF
a
ΔL-VSCF(HC)
a
ΔL-VCI(1)
ω
2
1045
1065
1185
-120
-20
-125
-5
-67
ω
3
1083
1101
1461
-360
-18
-320
40
-250
ω
4
1151
1179
1461
-282
-29
-252
29
-197
ω
5
1356
1387
2013
-626
-31
-602
24
-151
ω
6
1461
1497
2107
-610
-36
-587
23
-73
ω
7
1459
1507
2013
-506
-48
-499
7
-137
ω
8
1478
1525
2476
-951
-46
-934
17
-235
ω
9
2826
2981
2597
384
-155
307
-77
-137
ω
10
2901
3030
2597
433
-129
382
-50
-105
ω
11
3000
3119
2643
476
-120
451
-26
-86
ω
12
3701
3865
3227
638
-165
537
-101
-205
MAD
72
454
36
149
a
Frequencies calculated using virtual (unoccupied) modals.
Page | 31
Table 4. The vibrational frequencies of four prominent bands in the infrared spectrum of 9-fluorenone together with the
differences from their closest visual matches calculated at the harmonic, VSCF, L-VSCF(HC), and L-VSCF levels of theory.
Frequencies are in cm
-1
.
Exp.
a
ΔHarm.
ΔVSCF
ΔL-VSCF(HC)
b
ΔL-VSCF
b
1732
65
42
43
38
1615
36
19
20
-119
1455
24
9
9
18
1301
28
21
15
134
MAD
38
23
22
78
a
See Ref.
61
b
Frequencies calculated using virtual (unoccupied) modals.
