Content uploaded by Shant Shahbazian
Author content
All content in this area was uploaded by Shant Shahbazian on Oct 08, 2015
Content may be subject to copyright.
1
Toward the multi-component Quantum Theory of
Atoms in Molecules: A variational derivation
Mohammad Goli and Shant Shahbazian*
Faculty of Chemistry, Shahid Beheshti University, G. C. , Evin, Tehran,
Iran, 19839, P.O. Box 19395-4716.
Tel/Fax: 98-21-22431661
E-mail:
(Shant Shahbazian) chemist_shant@yahoo.com
* Corresponding author
2
Abstract
The general formalism of an extended quantum theory of atoms in molecules dealing
with the multi-component quantum systems, composed of various types of quantum
particles, is disclosed in this contribution. This novel methodology, termed as the multi-
component quantum theory of atoms in molecules (MC-QTAIM), is able to deal with
non-adiabatic ab initio wavefunctions extracting atoms in molecules quantifying their
properties. It can also be applied to elucidate the AIM structure of exotic species; bound
quantum systems consisting of fundamental elementary particles like positrons and
muons. The formalism is based on the previously disclosed density combination idea that
is extended to derive the multi-component subsystem hypervirial theorem as well as the
extended subsystem energy functional. Through the extended subsystem variational
procedure, inspired from Schrödinger's original variational principle, the surface terms
containing the flux of the current property densities are derived. Accordingly, the
extended Gamma field is introduced during this variational procedure that is used as the
basic scalar field in the topological analysis yielding atoms in molecules and their real
space boundaries. The Gamma field is central to the MC-QTAIM replacing the usual
one-electron density employed in the orthodox QTAIM and corresponding topological
analysis. Through the multi-component hypervirial theorem various regional theorems
are derived which are then used to quantify the mechanical properties of atoms in
molecules; these include the force, virial, torque, power, continuity and current theorems.
In order to demonstrate the capability of the formalism, isotopically asymmetric
hydrogen molecules, HD, HT and DT as well as YX systems (Y = 6Li, 7Li; X = H, D, T)
composed of electrons and two different nuclei, all treated equally as quantum waves
instead of clamped particles, are analyzed within context of the MC-QTAIM. The
resulting computational analysis demonstrates that the MC-QTAIM is able to yield
reasonable topological structures similar to those observed previously for diatomic
species within context of the orthodox QTAIM. The asymmetrical nature of these
species, inherent in their non-Born-Oppenhiemer wavefunctions, manifests itself clearly
in the MC-QTAIM analysis yielding two distinguishable atomic basins with different
properties. These differences are rationalized generally by the observed electron transfer
from one basin to the other. Finally, some possible future theoretical extensions are
considered briefly.
Keywords
Non-adiabatic wavefunctions, Real space quantum subsystems, Non-Born-Oppenhiemer
systems, Subsystem variational procedure, Multi-component quantum theory of atoms in
molecules
3
1. Introduction
Recently an extended formulation of the quantum theory of atoms in molecules
(QTAIM) has been developed [1-7], which goes beyond the realm of the orthodox
QTAIM [8-10]. This extended formulation termed the two-component QTAIM (TC-
QTAIM) is able to extract atoms in molecules and their properties from ab initio
wavefunctions of positronic molecules as well as non-adiabatic wavefunctions containing
one type of nuclei as quantum particles. Thus, the TC-QTAIM is able to deal with the
AIM analysis of molecular systems beyond the Born-Oppenhiemer (BO) paradigm.
Within this methodology, the clamped nuclei approximation is of no relevance to the
AIM analysis and the vibrational nuclear dynamics of ground and excited states is
introduced into the analysis from the outset [1]. On the other hand, its was demonstrated
that the TC-QTAIM "encompasses" the orthodox QTAIM since in the limit of infinite
nuclear mass the TC-QTAIM analysis recovers the usual results derived from the
orthodox QTAIM [5-7]. This observation confirms that the orthodox QTAIM is just the
"asymptotic" formulation of the TC-QTAIM or in other words, the latter contains the
former. Previous computational investigations within this extended framework revealed
novel features; the most notable: the atomic basins of isotopes are distinguishable [1] and
local positron affinity of atomic basins is quantified [3].
However, in contrast to these achievements, the main drawback of the TC-
QTAIM is the fact that only two kinds of involved particles are considered as quantum
waves; positron and electrons in positronic systems [2-4], proton/deuteron/tritium and
electrons in hydrides [1,5]. This renders many interesting "multi-component" systems
out of reach of the AIM analysis; exotic three (or many) body systems as well as
4
molecular systems containing various hydrogen isotopes are just some examples [11]. To
circumvent this undesirable limitation, as also emphasized previously [1,4,7], an
extended multi-component QTAIM (MC-QTAIM) is of urgent need. This contribution is
a first step toward such goal. Accordingly, in contrast to the strategy used for introducing
the TC-QTAIM [1-6], Schrödinger's variational approach is directly employed to
introduce the basic ingredients of the MC-QTAIM based on the extended subsystem
variational procedure (SVP). Alternative and more sophisticated variational approaches
and their extension to real space quantum subsystems namely, Schwinger's and
Feynman's delicate methodologies [8], will be considered in a second complementary
contribution.
The "combining" strategy for construction of property fields/currents is elaborated
throughout the text as the basic guideline for proper formulation of the MC-QTAIM.
Also, it is assumed from outset that the target set of wavefunctions correspond to WF1
family [5], thus the translational/rotational invariance is of no concern (see reference 5
for a comprehensive discussion on the classification of non-BO ab initio wavefunctions
and corresponding proposed terminology).
The paper is organized as follows: First, the mathematics and the physical
content of the SVP are considered in section 2. This section is a detailed and critical
reinvestigation of the SVP which was proposed and elaborated by Bader and coworkers
[12-18], in light of recent considerations [19-25]. In section 3 the proper energy
functional for a multi-component SVP is introduced and corresponding variational
procedure is verified. Accordingly, in this section the extended Gamma field is
constructed and the surface terms describing the flux of the property current densities are
5
derived. The extended subsystem hypervirial theorem and the relevant regional
theorems, in recently proposed tensor notation [6], are also discussed in this section. In
section 4, as illustrative examples, the AIM structure of isotopically asymmetric
hydrogen molecules, HD, HT and DT as well as 6LiX and 7LiX (X = H, D, T) systems is
disclosed within the framework of the MC-QTAIM.
2. Theoretical Background: Digging the Nature of the SVP
In original literature of the QTAIM, the SVP is usually conceived as the most
basic methodology capable of deriving key ingredients of the orthodox QTAIM [8,12-
18], however its precise role has been the matter of some controversies and exchanges
[26-31] (see references 19-21 and 24 for a detailed review on the whole discussion).
Accordingly, in order to shed some light on the very nature of the SVP, in a series of
studies, its mathematical and physical content have been scrutinized recently [19-25].
These studies revealed delicate issues regarding the nature of the SVP that does not seem
to be noticed or at least properly emphasized in the original literature. In this section,
before extending the SVP for the multi-component systems, these delicate issues are
disclosed revealing a different/alternative picture of the SVP. Particularly, in a less
formal language the "theoretical content" of the SVP is reorganized excluding any
ambiguity regarding its role within context of the MC-QTAIM. Based on this new
"reinterpretation" the SVP possesses a less key role within the MC-QTAIM, than
emphasized in original literature of the QTAIM, and hopefully this reinterpretation
prohibits any new misunderstanding. Then, in subsequent section, the whole machinery
is extended to the multi-component systems.
2.1. Variational principles: Why and how they work
6
The least action principles (better to be called extremum principles) have a rich
and sometimes controversial history in recent centuries and have been the main
motivation behind development of the calculus of variations [32]. Their role in modern
formulation of classical/quantum/relativistic mechanics and many other branches of
theoretical physics/chemistry is undeniable [33]. Probably the most illustrative example,
elaborated in detail elsewhere [8,32], is Hamilton's least action principle in classical
mechanics which is based on the variation of the action integral yielding Lagrange's
equations of motion. One may claim that the essence of Hamilton's principle is its
elegant competence to uniquely pick the path traversed by a classical particle taking into
account certain constraints (fixing the initial and final points of the path, the nature of
operative forces, etc.). In other words among infinite possibilities (a set of real and
virtual paths), nature prefers the path that minimizes/extremizes the action integral. In
order to extract Schrödinger's equation the same variational strategy is also used in
elementary quantum mechanics but in this case the energy functional is varied and then
extremized with respect to trial normalized wavefunctions [8]. Accordingly,
Schrödinger's variational principle is capable of picking the eigenfunctions of
Schrödinger's equation uniquely and is practically used under the title of the variational
theorem in quantum chemistry. There are numerous other examples but, without going
into technical details, two common features are always ubiquitous in variational (or least
action) principles:
1- As an axiom, a fundamental/characteristic functional is assumed that upon
proper variations with respect to one/some function(s), subject to certain constraints, is
extremized or in more concrete language its first order variations are null.
7
2- During the corresponding extremization procedure one/some
conditions/equations must be satisfied that uniquely pick the proper "member(s)" (a
classical path, a wavefunction, etc.) from the target "set" (the set of real plus all virtual
paths connecting two fixed points, etc.).
Based on these common features, in the variational calculus, various abstract
mathematical procedures have been elaborated without need for elucidating the relevant
physical content; the well-known Euler-Lagrange variational theorem is an illustrative
example [34]. Particularly, the uniqueness of picking the proper member(s) is central to
variational principles since they are alternatives for the basic equations that are capable of
deriving the proper member(s). Thus, without these two features, one can not claim that
a variational principle is established. As is demonstrated in the rest of this section the
SVP of the QTAIM does not satisfy these two conditions consequently we are not faced
with a "subsystem variational principle".
2.2. The SVP: Why it is not a variational principle
The explicit form of the subsystem energy functional for a multi-electron system
in an external electric field has it roots in the original Schrödinger's energy functional
[8,19]:
Ne
i
iie EVmdrdI ˆ
2,, 2
1
(1)
In this functional e
m and e
N are the mass and the number of electrons respectively,
while V
ˆ and
E
are the potential energy operator and the total energy respectively. Also,
d implies summing over spin variables of all electrons and integrating over spatial
coordinates of all electrons except an electron that is arbitrarily labeled as 1. For
8
subsequent discussions it is convenient to rewrite the subsystem energy functional as
follows:
11
,, rIrdI
,
1Ldrd
EVmL Ne
i
iie ˆ
2, 2
(2)
The general expression for the variation of the subsystem energy functional includes
variation with respect to (
) and its derivatives i
(
i
) as well as surfaces
delineating real space subsystems [8,19]. The following well-known recipe is applied
during variation ( cc stands for complex-conjugate):
Ne
i
i
i
LL
drdI
1
ccrSLdrdS
11 ,,,
(3)
The proper mathematical manipulations have been discussed in the previous
contributions and are not reiterated here [19-21], however, it is worthwhile to review the
relevant constraints:
1- Schrödinger's equation is satisfied,
0
ˆ EH (
0
ˆ
EH ). In other
words, it is tacitly assumed that Schrödinger's variational principal holds for total system,
0,, 3 RI
.
2- The usual boundary conditions of a quantum system, namely, 0, at
infinitely remote regions are satisfied.
9
3- The real space variational constraint,
0
1
2
1
rrd
, must be satisfied
where
dr1
. As is demonstrated subsequently, the nature of emerging real
space subsystems is tightly tied with the uniqueness dilemma of subsystems (vide infra).
4- The basic variations,
, are not arbitrary rather produced entirely by
Schwinger's recipe through the act of generators of the infinitesimal unitary
transformations,
gi ˆ
(
gi ˆ
), ( g
ˆ is a one-particle hermitian
generator while
is an unspecified infinitesimal); corresponding delicate mathematical
problems originating from this type of variations have been detailed recently [24] and are
not reiterated in this contribution.
The final outcome of the relevant mathematical manipulations is stated
introducing the current property density,
1
rJg
:
nrJrdSI g
11
,Re,,
ggdimrJ eg ˆˆ
2
1
(4)
The right-hand side of the variational equation is the product of an infinitesimal and the
surface term namely the flux of the current density of an observable g
ˆ through the
boundaries of real space subsystem ( n
is a unit vector normal to the boundary of
subsystem).
In the original literature, it was usually claimed that the SVP is a "generalization"
of the variational principle held for total quantum system. From a purely mathematical
viewpoint this seems to be the case since variation of the energy functional of total
10
system is done assuming 3
R
in equation (1); the third term in the right-hand side of
equation (3) that is variation of the boundaries of subsystem then disappears. However,
the price one pays for this generalization is the fact that the subsystem energy functional,
in contrast to total system, is not extremized as is evident from equation (4),
0,, I
[24]. On the other hand, since the subsystem energy functional is not
extremized during the variational procedure, Schrödinger's equation is assumed as a
constraint because it is not derivable from the procedure itself. In other words, one
tacitly assumes that
0,, 3 RI
thus the validity of variational principle for total
system is "granted" rather than "derived". This subtle issue calls the fact that without
assuming the validity of variational principle for total system, it is not feasible to derive
equation (4) as the final result of the SVP; although equation (4) seemingly recovers the
variational principle assuming 3
R
, its derivation is based on the validity of this
principle. Therefore, from a physical viewpoint, it does not seem legitimate to claim that
equation (4) is a generalization of the variational principle of total system to real space
subsystems.
It was also claimed that the SVP yields the topological atoms as the proper real
space subsystems uniquely. This claim also does not seem to be legitimate since
constraint 3 is satisfied not only by a partitioning of total space into topological atoms,
but also any other exhaustive real space partitionings [24]. The reasoning is simple since
for any exhaustive partitioning the integration of the flux integral in each partition
(assuming no cusps in the Laplacian field of the one-electron density) yields a constant
finite value,
.
1
2
1constrrd
, thus satisfying constraint 3. Accordingly, one may
11
claim that equation (4) is not a specific trait of topological atoms where 0. const but
common feature to be held for all conceivable regions emerging from the partitioning of
real space by 2D surfaces. In other words, constraint 3 is not capable of yielding subset
of topological atoms uniquely as a member of the general set of all exhaustive
partitionings; this was termed the neutrality of the SVP [24]. This observation indicates
that the logic of employing constraint 3 must be reversed: As an axiom, topological
atoms are assumed to be the proper real space subsystems satisfying constraint 3; in this
framework the set of topological atoms is "granted" rather than "derived".
It is timely to emphasize on the central role of the variational constraints in above
discussions. The fact that one is not faced with a variational principle for real space
subsystems implies delicate consequences for the role of the variational constraints that
are rather strange from the viewpoint of the orthodox variational calculus. Accordingly,
in contrast to variational principles in which associated constraints have only an auxiliary
role imposed during variation reaching a core condition/equation, in the case of the SVP
constraints are not auxiliary, rather are the key/informative ingredients as discussed
above.
If the SVP is not a variational principle and is not able to pick the set of
topological atoms from other conceivable sets of 3D partitionings, then what is its
significance within context of the QTAIM? It seems that the main message of this
procedure is the revelation of the surface term (the right-hand side of equation (4)). This
surface term that also appears in the subsystem hypervirial theorem (vide infra) is the
main difference between introduction of mechanical properties of topological atoms, as
the open quantum subsystems, and those of total closed quantum systems. Therefore, the
12
SVP is just an alternative for realization/derivation of the fact that mechanical properties
of topological atoms are dependent on the current property densities operative on the
boundaries. This fact is scrutinized in subsequent section.
3. The multi-component SVP
The previous computational studies on some positronic species as well as
hydrides with hydrogen nuclei treated as quantum waves demonstrated that the TC-
QTAIM yields reasonable results and is promising for novel applications [1-4].
However, theoretical reasoning behind formulation of the TC-QTAIM was based on
direct derivation of the two-component subsystem hypervirial theorem and associated
regional theorems, i.e. force, virial, torque, power, current and continuity theorems [5].
The main guideline for the proper extension was the strategy of the combination of
densities. In following subsection it is demonstrated that this strategy has been
employed, albeit tacitly, even in the formulation of the orthodox QTAIM demonstrating
that the TC-QTAIM is the "natural" extension of the orthodox QTAIM to the two-
component systems. Subsequently, the same strategy is used for deriving the multi-
component subsystem hypervirial theorem as well as for constructing the proper
subsystem energy functional. The resulting extended SVP not only yields the multi-
component surface term that is compatible with the one computed independently from the
multi-component subsystem hypervirial theorem, but also reveals interesting clues for the
scalar field that is used to derive atoms in molecules in the multi-component systems.
3.1. The strategy of the combination of densities
The strategy of the combination of densities was implemented for both the
property densities of electrons and positively charged particles constructing the joint
13
density within context of the TC-QTAIM,
qMqMqM
~
[4,5]. However, if one
implements this strategy for each quantum particle of system separately, not just for
groups of indistinguishable particles, then, it is easy to see how it is also in the heart of
the QTAIM formalism.
The subsystem hypervirial theorem within context of the orthodox QTAIM is as
follows [8]:
nqJqdSGHi G
,Re
ˆ
,
ˆ
Re (5)
In this equation is the basin of a topological atom, G
ˆ stands for the sum of
one-particle operators (vide infra) and
GHdqdGH ˆ
,
ˆ
ˆ
,
ˆ
, in which qd
denotes the spatial coordinate of any arbitrary electron. For a multi-electron system
Ne
i
i
gG
1
ˆ
ˆ and equation (5) is rewritten in a new form:
nrJrdSghi ii
Ne
i
ii
Ne
ii
g
,Re
ˆ
,
ˆ
Re
11
(6)
In this equation i
h
ˆ is the part of Hamiltonian,
H
ˆ, that contains all variables of the i-th
electron; these include the kinetic energy operator, i
t
ˆ, and all involved potential energy
terms, i
v
ˆ, iii vth ˆ
ˆ
ˆ (note that in general
Ne
ii
hH
1
ˆ
ˆ) while
iiiii ghdrdgh ˆ
,
ˆ
ˆ
,
ˆ
and
iiiiei ggdimrJ i
g
ˆˆ
2
.
14
Assuming
iii mghi ˆˆ
,
ˆ where i
m
ˆ is a hermitian operator as well as introducing the
density of this operator,
iii mdrm ˆ
, one may simplify equation (6):
nrJrdSrmrd ii
Ne
i
Ne
i
iii i
g
,ReRe
11
(7)
Now, in this equation each term contains the coordinate of a single electron and the both
sides are just summations on electrons' contributions. To clarify this fact, it is illustrative
to rewrite equation (7) in a more proper form:
nqJqdSqmqd i
g
Ne
i
Ne
i
i
11
,ReRe (8)
Although this equation is equivalent to equation (5), it is particularly enlightening since
in both sides the one-particle property densities are first combined yielding the total
property and current densities,
Ne
i
iqmqM
1
and
Ne
i
GqJqJ i
g
1
, and then the
basin integration is done (in previous studies to emphasize on this combination a notation
based on the "transformers" was introduced) [4,5]. Since in the case of multi-electron
systems one is faced with indistinguishable particles, in both of these summations all the
one-particle terms are equal thus one arrives:
qmNqM e
and
qJNqJ geG
; by
substituting these expressions into equation (8), equation (5) is re-derived. Equation (8)
is a proper starting point to extend the orthodox QTAIM to the multi-component systems.
First, the two-component case is considered.
Let’s assume that in a hypothetical situation e
N electrons of the system are
colored such that two subsets are emerged: 1
N "blue" and 2
N "red" electrons,
15
21 NNNe . In this case the indistinguishability of the whole set of electrons is ruined
and two distinguishable subsets are emerged; blue(red) electrons are indistinguishable in
their own subset but distinguishable from the members of red(blue) subset. Equation (8)
can be utilized for this hypothetical situation however in this case the combination of the
property/current densities are as follows:
qmNqmNqmqmqmqM N
Ne
Nk
N
jkj
Ne
i
i
1
1
1
1
1
1
211
1
qJNqJNqJqJqJqJ N
g
e
N
Nk g
N
jk
g
j
g
i
g
Ne
i
G
1
1
1
11
1
1
21
1
(9)
Substituting these equations into equation (8) yields:
nqJqJqdSqMqMqd GG
21
21 ,ReRe (10)
In the left-hand side of this equation
qmNqM
1
11
and
qmNqM N
1
1
22
while in
the right-hand side
qJNqJ g
G
1
1
1 and
qJNqJ N
g
G
1
1
2
2
. This is comparable with
equation (5) though instead of a single particle/current density, it contains two such
densities, one for blue and one for red electrons; if electrons lose their hypothetical
"color" this equation reduces to equation (5). Instead of the two hypothetical blue-red
subsets of electrons it is possible to employ equation (10) for real two-component
systems including two distinguishable set of quantum particles since no restriction is
imposed on the Hamiltonian used in equation (5); because of the varied mass or charge of
particles, iii vth ˆ
ˆ
ˆ is generally different for each subset. In other words generally
16
qJqM G
1
1/ and
qJqM G
2
2/, in contrast to the case of colored electrons, are not the
same. Thus, equation (10) deserves to be termed as the two-component subsystem
hypervirial theorem. Recently, the relevant regional theorems have been derived and
discussed in detail and are not reiterated here [5,6]. Extension to the multi-component
systems is straightforward since one may simply extend the two-color metaphor to a
multi-color analog. The resulting extended subsystem hypervirial theorem is as follows:
nqJqdSqMqd n
G
P
n
n
P
n
11
,ReRe (11)
In this equation n enumerates each group of distinguishable quantum particles while
qqnnn ghdiNqM ˆ
,
ˆ
and
qqqqnnn
n
GggdimNqJ
ˆˆ
2
Also, n
d
implies summing over spin variables of all quantum particles and integrating
over spatial coordinates of all quantum particles except one arbitrary particle belonging to
the n-th subset. This equation vividly shows that the property/current densities of each
type of quantum particles are combined to yield the total densities,
P
n
nqMqM
1
~ and
P
n
n
GG qJqJ
1
~
[5,7]. In subsequent subsection the same strategy of density
combination is first used to re-derive the subsystem energy functional used in the
orthodox QTAIM and then to introduce the extended energy functional of the MC-
QTAIM.
3.2. The multi-component energy functional
The usual starting point for the SVP is equations (1) and (2) [8]. However, before
starting the variation, one may interpret the subsystem energy functional as the basin
17
integral of an energy density,
1
rI
. Accordingly, a new subsystem energy functional
compatible with combination strategy is proposed:
qIqdNqIqdrIrdS e
N
i
eiii
Ne
i
11
1,, (12)
It is evident from this equation that the variation of this functional, because of the
indistinguishability of electrons, reduces to the variation of
,,
I:
nqJqdSS G
,Re,,
1
ggdimNqJ ee
Gˆˆ
2
(13)
The fourfold imposed constraints are equivalent to the orthodox case apart from the fact
that the real space variational constraint is now:
0
2
qqd
, where
dNq e
is the usual one-electron density. Although, the results are trivial
in the case of orthodox QTAIM, the methodology is capable of yielding the proper
subsystem energy functional for the multi-component systems. Let's first consider the
two-component systems.
Based on the metaphor of colored electrons the subsystem energy functional for a
mixture of blue-red electrons is as follows:
qIqdNqIqdNqIqdqIqdrIrdS N
Ne
Nk
N
jkj
Ne
i
ii
1
1
1
1
1
1
211
1
2,,
(14)
18
The variation of this functional yields non-trivial result as demonstrated in the special
case of positronic systems [4]. For the general multi-component case the following
subsystem energy functional emerges upon applying the multi-color metaphor:
,,,,
11 1
P
n
nn
P
nn
N
i
iin INqIqdNrIrdS
n
qIqdI nn
,, ,
n
N
i
ninin
P
n
nn EVmdqI ˆ
2
2
1
(15)
In this equation N stands for the total number of quantum particles,
P
n
n
NN
1
. Since
no specific assumption regarding the mass and the form of the potential energy terms has
been imposed in equation (15), this functional is quite general thus proper to be applied
for a large set of the multi-component systems. However, before considering the
extended SVP, it is demonstrated that this functional also satisfies the known criteria of a
proper energy functional [4]. These include:
1- The proposed subsystem energy functional converts to the total energy
functional in the limit of total system, 3
R , satisfying the variational principle of total
system yielding Schrödinger's equation for the multi-component systems. Accordingly
(
ddqd n
):
Pn
N
n
nini
i
n
N
i
iin EVmdNrIrdRS
R11
2
1
3ˆ
2,,
3
0,, 3 RSn
0
ˆ
orEHext ,
P
n
n
N
i
innext VmH
11
22 ˆ
2
ˆ (16)
19
2- The proposed subsystem energy functional converts to the known orthodox
subsystem energy functional, equation (1), when the numbers of all quantum particles
except electrons, 1nand e
mm
1, are zero. Accordingly:
0
n
N,2n
,,
1
n
S
,,
INe (17)
3- The proposed subsystem energy functional converts to the known subsystem
energy functional for positronic systems when the numbers of all quantum particles
except electrons and positrons are zero; 2,1
n and e
mmm
21 [4]. Accordingly:
0
n
N , 3n
,,
2
n
S
qINqINqd
2211
21
1
2
2,12,1 ˆ
2
NN
j
jjekk EVmdqI
(18)
Since the proposed energy functional satisfies these conditions, one may proceed to the
basic machinery of the extended SVP.
3.3. The variation of the multi-component subsystem energy functional
The variational manipulations of the proposed functional has the same
mathematical stages considered in detail within contexts of both the orthodox QTAIM as
well as the QTAIP (Quantum Theory of Atoms In Positronic molecules) [4,8,19-21].
Therefore, only a brief sketch of mathematical manipulations is presented in this
subsection.
One starts the subsystem variational procedure as follows:
,,Re2,,
1
P
nnnnn n
INSSS
(19)
20
To proceed further, each term in the bracket is varied separately and to do so, the
variation of n
I,
nnn III
, is considered at first step:
,,, extnn LdqdI
EVmL n
N
i
nini
P
n
next ˆ
2,
11
2
(20)
Following recipe is applied during the variation:
n
N
P
i
ni
ni
ext
n
ext
nn LL
dqdI
11
qSLdqdS extn
,,,
(21)
This equation is comparable with equation (3) term by term. After some mathematical
manipulations [4,8,19], and taking the constraint 2 into account, ( 0,
), one
arrives at:
EHdqdI extnn ˆ
qSLnmdqdS extqnn
,,2, 2
(22)
Recalling Schrödinger's equation for the multi-component systems, equation (16), the
first term on the right-hand side of equation (22) vanishes (the extended form of the
constraint 1), and the variational expression reduces to:
n
I
ccqSLnmdqdS extqnn
,,2, 2
(23)
To proceed further, the integrand,
,
ext
L, must be converted into an appropriate form
employing the following identity:
21
n
N
P
i
nini
n
n
m
1
22
1
24
n
N
P
n
N
P
i
ni
n
n
i
nini
n
nmm
1
2
1
2
11
242
(24)
Applying this identity, it is straightforward to demonstrate the following equation:
n
N
P
i
i
n
nextextext mEHHL
1
2
1
24
ˆˆ
21,
(25)
Since the total system obeys multi-component Schrödinger's equation, this equation
simplifies to:
n
N
P
i
i
n
next mL
1
2
1
24, (26)
Incorporating equation (26) into the second term on the right-hand side of equation (22)
and applying Gauss's theorem [4,8,19], one obtains:
qSLdqdS extn
,,,
qSqqdSNm nqnn
,,4 22
(27)
where,
nnn dNq
is the one-particle density of the n-th subset of quantum
particles. Accordingly, equation (23) transforms into:
n
I
ccqSqNndmqdS nqnqnn
,212, 22
(28)
22
In the next step, the variation of
qqd nq
2 is considered in order to eliminate the term
of surface variation,
qS
,
, which appears as the second term on the right-hand side
of equation (28).
qSqqdSqqdqqd nqnqnn
,, 222
(29)
Also, the first term on the right-hand side of equation (29) is expanded as follows:
qqqnnnq dqdNqqd
2 (30)
Incorporating equations (29) and (30) into equation (28) converts the latter into:
ccqqdNmnqJqdSiNI nqnn
n
nn
22 4,2 (31)
where,
qJ n
qqnnn dimN
2 is the current density of n-th subset
of quantum particles (alternatively this is derived assuming 1
ˆ
g
in
qJ n
G
). At this stage
of development, it is appropriate to incorporate equation (31) into equation (19) pursuing
the main variational procedure. It is straightforward to derive the following equation
after some manipulations:
nqJqdSiSn
~
,2
ccqmmqdm nq
n
n
P
2
1
11
24 (32)
The first term in the right-hand side contains the varied combined current density
P
n
nqJqJ
1
~
while the second term is a combination of the masses and the one-
23
particle densities. In order to proceed further, the real space boundary condition must be
introduced to eliminate the second term on the right-hand side of equation (32)
discriminating the very nature of real space quantum subsystems.
3.4. The extended Gamma field
One of the pivotal ingredients of the orthodox QTAIM is the topological analysis
of the one-electron densities [8]. From this analysis the theory of molecular
form/structure within context of the QTAIM is emerged. Accordingly, the Gamma is
replaced by the one-electron density in the TC-QTAIM which is a combination of the
one-particle densities of electrons and positively charged particles scaled with
corresponding mass ratio,
qmmqq
. It is evident that the strategy of
density combination is also operative in introducing the Gamma field. The previous
computational studies demonstrated that the topological analysis of this scalar field in
positronic species and hydrides yields reasonable results [1-5]. On the other hand, it was
shown that if one sets
m, then
qq e
, demonstrating that the Gamma field
reduces to the usual one-electron density in the case of clamped nuclei [5,6].
For the multi-component systems the extended Gamma field is proposed as
follows [7]:
qmmqq n
P
n
n
P
2
11 (33)
In this equation
q
1
is the one-particle density of the lightest quantum particles (usually
electrons).
P
is deserved to be called the cardinal number of the extended theory; 1
P
yields the orthodox single-component QTAIM while 2
P
yields the general TC-
QTAIM; the QTAIP is a special example of the latter, e
mmm
21 . Thus,
q
in the
24
TC-QTAIM is now re-termed, in this new terminology, to
q
2
. The following axiom,
in the spirit of a previous axiomatic presentation of the TC-QTAIM [5], is proposed to
disclose the nature of real space subsystems within context of the MC-QTAIM:
Every molecular system is partitioned exhaustively,
i
iR3
, into disjoint
regions, ji Ø, where each partition, i
, is the 3D basin of attraction of the vector
field
q
P
. These partitions are called topological atoms.
Accordingly, the (3, -3) critical points (CPs) of the
q
P
discriminate the attractors of
the field while a subset of the zero-flux surfaces of the field, surfaces emerging
from
0 nq
p
and going through the (3, -1) CPs but not (3, -3) CPs, delineates the
boundaries of the attraction basins; these are topological atoms/real space subsystems and
inter-atomic surfaces. The cases considered computationally in this study as well as a
more comprehensive study that will be presented in a future contribution verify the
capacity of
q
P
as the proper field for discriminating the AIM structure in the multi-
component systems. Also, as detailed elsewhere [5,6], if one attributes a single localized
s-type Gaussian function to each nucleus of a molecule, as a ground state nuclear
vibrational wavefunction, it is straightforward to demonstrate that
qq e
P
m
lim . This observation clarifies that apart from some exotic situations
that a molecular system is in a superposition of quantum states, e.g. systems with
significant intramolecular proton tunneling, upon approaching the clamped nuclei regime,
q
P
approaches to the basic scalar field of the orthodox QTAIM, namely
q
e
.
Based on this background, equation (32) is rewritten as:
25
ccqqdmnqJqdSiS P
qn
2
1
24
~
,2
(34)
Since the topological atoms are characterized by following equation:
0
2
qdq p
,
the second term in the right-hand side vanishes. On the other hand, based on the
constraint 4 (vide supra), the unspecified variational elements,
(
), are now
specified as Schwinger's variational recipe,
q
gi ˆ
(
q
gi ˆ
). This
transformers equation (34) into its final form:
nqJqdSS Gn
~
,Re
(35)
The right-hand side of this equation is equivalent to equation (13) apart from the fact that
combined total property current density replaces the property current density of electrons.
Also, in line with discussions of section 2, the current term emerging from the extended
subsystem variational procedure is completely equivalent to the surface term computed
independently from the extension of the subsystem hypervirial theorem to the multi-
components systems; compare the right hand sides of equations (11) and (35). This
observation points to theoretical self-consistency in the construction of the MC-QTAIM.
3.5. The regional/local theorems
The multi-component subsystem hypervirial theorem, equation (11), delivers the
regional theorems upon employing proper generators, q
g
ˆ; this procedure has been
detailed in both the orthodox QTAIM as well as the TC-QTAIM [5,7,35]. Based on
recent contributions [5,6], it is straightforward to derive the main regional/local theorems
in both dyadic and tensor notations though, as emphasized recently [6], the latter is more
26
informative for theoretical as well as computational applications. Before proceeding
further, it is proper to restate the local version of the multi-component subsystem
hypervirial theorem as follows [6]:
qJqM G
~
~
(36)
After the regional integration of this equation, taking just the real part, the original multi-
component subsystem hypervirial theorem, equation (11), is emerged. If one evaluates
the left-hand side of this equation then the following quantities with well-known
counterparts in both the orthodox QTAIM and the TC-QTAIM are derived:
Force density:
qF
~
VdN qnn
P
n
ˆ
1
Kinetic energy density:
qT
~
22
1
2Re qnnn
P
n
dmN
Basin Virial density:
qV B
~
VqdN qn
P
n
nˆ
1
Torque density:
q
~
VqdN qn
P
n
nˆ
1
Power density:
qP
~
VdimN qqqnnn
P
n
ˆ
2
1
(37)
Using the right-hand side of equation (36) it is straightforward to derive following
theorems, written in tensor notation [6], after some mathematical manipulations:
The force theorem, pgqˆˆ
ji
j
iq
qF
~
~
ijijijij
n
n
n
n
ji qqqqqqqq
d
m
N
q
4
2
(38)
27
The virial theorem, pqgqˆˆˆ
qLqVqT T
~
~
~
2
qLqT jiij
~
21
~
21
~
,
jiijjij
i
ji
j
i
SBT q
qq
qqVqVqV
~~~
~~~
qmqL P
2
1
24
~
(39)
The torque theorem, pqgqˆˆˆ
jk
j
li q
qq kli
~
~
(40)
The power theorem, nq mpg 2
ˆˆ 2
j
j
ij
ij qqq
qP ~
2
~
4
2
jijijiji
n
n
n
n
n
ij
n
ij qqqqqqqq
d
im
N
m
qj
q
2
2
jiijii
n
n
n
n
n
j
n
jqqqqqq
d
im
N
m
qt
q
2
2
(41)
The continuity theorem,
qrgq
ˆ
0
~
i
iJ
q
ii
n
n
n
n
iqq
d
im
N
qJ
2
(42)
The current theorem, qgq
ˆ
0
~
ji
iqJ
q (43)
In all equations the subscripts kji ,, are used to enumerate the three main components of
vectors or the nine components of tensors while ij
and ijk
are Kronecker delta and
Levi-Civita symbol, respectively [36]. Also,
jij
i
Sq
q
qV
~
~
is the surface virial
density while for compactness all equations have been written employing Einstein's
summation convention [36]. In line with the previously introduced convention,
28
components of tensor and vector fields are introduced as follows:
qMqM P
n
n
jiji
1
~ and
qMqM P
n
n
ii
1
~. Equations (37) to (43) or their regional counterparts are the basis for
both computational and analytical studies within context of the MC-QTAIM. It must be
noted that in the cases of 1
P
/2
P
all these theorems are reduced to their counterparts
derived previously for the orthodox QTAIM/TC-QTAIM [5,6,35].
4. Computational Study
In order to demonstrate the capability of the developed formalism, in this section
the MC-QTAIM analysis is performed on nine three-component systems namely HD,
HT, DT, 6LiX and 7LiX (X = H, D, T) treating all constituent particles as quantum waves.
This set first considered in detail by Tachikawa
and Osamura employing their ab initio FV-MC_MO (Fully Variational Multi-
Component Molecular Orbital) methodology [37,38]. This non-BO methodology, used
also in our previous computational studies [1,5], produces non-BO wavefunctions that are
members of the WF1 family of wavefunctions; assuming a fixed molecular frame from
outset in their construction [5]. The results of this exploratory analysis are just briefly
considered concentrating on general trends leaving a more comprehensive computational
MC-QTAIM analysis, exploiting a larger set of species, for a future contribution.
4.1. Computational procedure
The FV-MC_MO code developed and described in detail previously was utilized
producing non-BO ab initio wavefunctions [1,5]. A [5s:1s] Gaussian basis set was used
expanding all the considered wavefunctions (since there are two nuclei in each
considered molecule, the actual number of employed basis functions is doubled). The
29
used masses for nuclei throughout calculations are: e
mH 15267247.1836,
e
mD 4829654.3670, e
mT 9215269.5496, e
mLi 983866.10961
6 e
mLi 492587.12786
7.
All variables of basis functions namely exponents, centers and SCF coefficients were
simultaneously optimized in a non-linear optimization procedure described in detail
previously [5]. In order to verify the quality of the optimization algorithm, total energy,
optimized variables of basis functions, mean nuclear distance and virial ratio of the usual
hydrogen molecule, H2, were computed using various basis sets and then compared with
those reported by Tachikawa and Osamura [37]. The MC-QTAIM analysis has been
performed with further development of the code used previously for the TC-QTAIM
analysis of positronic species [3] as well as hydrides [1]. Apart from carefully chosen
internal checks [1,3], the sum of basin properties derived from basin integrations was
verified to be equated with those derived for total molecule from ab initio calculations.
In order to ensure the numerical precession, the parameters of numerical integration were
varied until the flux integral of each basin,
qqdm q
32
1
24
, was equal or less than
5
10 in atomic units. Because of full variational optimization of the variables of the basis
functions, the virial theorem is automatically satisfied thus the computed virial ratio of all
species are close to its exact value,
)101(2
ˆˆ 6
TV , relegating the need for the
usual ad hoc virial scaling of basin energies [3].
4.2. Topological analysis of Γ(3)
Evidently, for the considered species, the required cardinal number is 3P so the
topological analysis is done on
qmmqmmqq ee
3322
3
(the minus
subscript throughout the rest of this subsection is used for the electronic properties while
30
always 23 mm ). The topological analysis of
q
3
for all species immediately reveals
the usual topological structure of a diatomic species namely, two (3, -3) CPs and a single
(3,-1) CP in between; the zero-flux surface going through the (3, -1) CP divides the 3D
space into two atomic basins. Table 1 offers some typical data derived from the relevant
topological analysis. This analysis presents the primary clues regarding the
"asymmetrical" nature of the considered hydrogen molecules since the length of the two
bond paths as well as the value of
q
3
and
q
32 at the (3, -3) CPs are not the
same in each species; this observation is further strengthened taking into account the fact
that the properties of the two basins are not same (vide infra). Also, the values of
q
3
and
q
32 at the bond critical points (BCPs), (3, -1) CPs, are not same in these three
species demonstrating that there are delicate differences in the bonding modes. One
observes similar trends in bonding modes of lithium hydrides since hydrogen isotopes
impose their fingerprint on the topological indexes at the BCPs (for a comprehensive
discussion see [1]). Replacing one lithium isotope with the other, fixing the hydrogen
isotope of hydride, seems to affect the properties of BCPs much less; the latter
observation is in line with the general expectation that mass variations are less influential
on the topological analysis of nuclei more massive than hydrogen isotopes [5]. The only
relatively pronounced difference is more intense accumulation of the Gamma around 7Li
in comparison to 6Li. Deriving the effective size of nucleus distribution, termed the ESD
[1,5], for all quantum nuclei demonstrates that in all considered species,
qq
32 /
are
completely contained in associated atomic basins thus at the BCPs for all practical
proposes
qq
3. It is timely to emphasize that
q
is not the usual one-
electron density,
q
e
, used in the orthodox QTAIM [8-10]; literally speaking, the
31
former is a dynamic electron density containing the fingerprint of nuclear vibrational
dynamics while the latter is a static electron density introduced just within the clamped
nucleus paradigm.
4.3. Basin integration
Some typical results of basin integrations are gathered in Tables 2-4. At first step,
considering the fact that there are no clamped nuclei in this analysis, in order to reveal the
identity of each atomic basin, populations of nuclei are derived,
nn Nqqd
; in
line with the fact that the ESDs are completely contained in a single basin, this
calculation demonstrates that each basin encompasses the whole population, ~1.000, of
just one type of nuclei. In hydrogen molecule series, electron distribution is
asymmetrical and electron population of basins,
Nqqd
, containing the
heavier isotope in each molecule is always larger than that containing the lighter one.
This observation is in line with previous observation, within context of the TC-QTAIM
[1], that the electronegativity of hydrogen depends on nuclear mass; basins containing
heavier nuclei are more electronegative. Accordingly, it is legitimate to confirm the
initial proposal of Tachikawa and Osamura depicting these species as
D
H
,
T
H
and
T
D
beyond the clamped nucleus paradigm [37]. In the case of lithium hydride
series, slight variation of hydrogen basin’s electron population upon the isotopic changes
also conforms to the same electronegativity trend while for lithium basins the mass
dependence is less pronounced. It is tempting to extrapolate that the variation of nuclear
mass, upon isotopic changes, less affects electron population of basins containing nuclei
heavier than hydrogen isotopes. In contrast to the “fine structure” of electronic
32
distribution in lithium hydride series, the gross picture emerging from electronic
population conforms well to ionic bonding,
XLi .
As a complementary analysis, various components of the electric dipole moments
are also considered using following equations [6]:
tttt
CT
tPPdd
2 , 3,2
t
GACT RNd 222 )1(
t
tttt rrrdP
,
t
t
tt
t
t
t
ttt
trrrdZP
(44)
In these equations CT is an abbreviation for Charge Transfer while t
PP
/
and t
Z stand
for the first electric moments of electrons/nuclei and the nuclear charge, respectively.
The vectors locating electrons and nuclei from the center of an arbitrary coordinate
system are
r
and t
r
which are decomposed employing the (3, -3) CPs of
q
2
and
q
3
, located by the vectors GA
t
R
, as the centers of the “local” coordinate systems. In
the latter coordinate systems one observes
t
GA
trRr
, t
t
GA
tt rRr
where
t
r
and t
t
r
are the vectors locating electrons and nuclei from the local coordinate systems,
respectively. Also, all charge transfer dipoles are computed assuming the (3, -3) CP of
q
3
as the origin. Based on this background, the first moment of nuclei are
null,
0
t
P
, since a single s-type spherical Gaussian function is used to describe each
nucleus in total wavefunction. Inspection of Table 2 demonstrates that polarizations of
electronic distribution in hydrogen basins are unequal and counteracting; first moments
are appreciably larger than charge transfer dipoles and although they tend to cancel each
other, even their net effect is non-negligible thus the “magnitude” of total ab initio
33
electric dipoles are not quantitatively reproducible neglecting basin polarizations.
Nevertheless, in a more qualitative view one may claim that the "direction" of total
electric dipole in these species is generally dictated by charge transfer dipole. It is timely
to emphasize that the orthodox QTAIM is unable to distinguish between H2, D2, T2 and
HD, HT, DT always yielding symmetric atomic basins with no charge separation for all
isotopic combinations of hydrogen molecule. On the other hand, inspection of Tables 3
and 4 demonstrates that in lithium hydride series first moments of lithium basins are
small and almost constant throughout the series revealing a spherical electronic
distribution whereas those of hydrogen basins are much larger disclosing strong
polarization; polarization dipoles of atomic basins of all species are accumulative.
However, in contrast to the asymmetric hydrogen molecules both the magnitude and the
direction of the total electric dipoles are dominated by charge transfer dipoles and the role
of polarization dipoles are just marginal (for a comprehensive discussion see [1]).
Accordingly, this observation conforms to emerging ionic bonding regardless of the
nature of considered isotopic combination.
In the final part of the analysis, based on the regional virial theorem, disclosed in
previous subsection, basin energies and also contribution of each type of particles in the
basin energies are considered. Since each heavy particle is confined to a single basin, the
following equation is used:
ttttttt TTEEE
~
, 3,2
t (45)
It is evident from Table 2 that in the hydrogen molecule series the electronic energy
contribution of the basin containing the heavier nucleus is lower/more negative than that
of the contribution of basin containing the lighter one. The reverse trend is observed for
34
nuclear energy contribution to basin energies but overall, total basin energies are lower
for basins containing heavier nucleus. The lower electronic energy of basins containing
heavier nucleus originates probably from both the larger electronic population of these
basins as well as the larger "aggregation" of electrons around their more localized nuclear
distribution; roughly speaking, electrons have orbits with smaller mean radius around the
more massive nucleus thus acquiring larger kinetic energies [1,5]. At current state of
knowledge a precise quantitative discrimination of these two effects, electron transfer and
electronic redistribution, is not feasible. However, if one plots the difference between the
basins electronic populations,
233,2 NNN , versus the difference
between basins electronic energies,
233,2 EEE , an almost linear
graph,
3,23,2 . EvsN , emerges in these species suggesting that charge
separation is the more influential factor. The observed trend for the nuclear energy
contribution is easily rationalized using the previously derived analytical equation linking
this energy contribution and mass of nucleus ( m) [6]:
21
21 43
mkE tt ; for
heavier isotopes, since the force constant ( k) is almost isotope independent, the
associated energies are less negative. Tables 3 and 4 reveal similar trends for hydrogen
basins in the lithium hydride series therefore they are not reiterated however, it is worth
mentioning that the total as well as electronic and nuclear energy contributions of these
basins are virtually insensitive to isotopic change of lithium nucleus. The electronic
contributions of lithium basins are almost constant in 6LiX and 7LiX triples whereas
nuclear contribution is completely constant in each triplet. In line with previous
discussions, the electronic contribution of the basin containing heavy isotope is more
negative than the basin containing the lighter isotope whereas the reverse is true for
35
nuclear contribution; overall, the electronic contribution overwhelms and the total basin
energies are lower for the basins containing heavy isotope. One may conclude
reemphasizing that all these regularities are observed just within context of the MC-
QTAIM and the orthodox QTAIM analysis is not capable of revealing such delicate
effects.
5. Prospects
This contribution is a natural extension of the previous studies that started from
formulating the QTAIP for positronic species [2-4] and then introducing the more general
TC-QTAIM thus incorporating nuclear/exotic particles' dynamics into the AIM analysis
[1,5,6]. The resulting MC-QTAIM is quite general capable of being used for a large set
of the multi-component species however, new theoretical developments are also of urgent
need to widen the framework of the AIM analysis bypassing some limitation of the MC-
QTAIM. Some of these directions, currently under consideration in our laboratory, are
mentioned in this section.
The whole analysis in the present and previous contributions was done
considering the electric fields (generally but not exclusively Coulombic) as the sole
origin of interactions in corresponding Hamiltonian. However, weak internal magnetic
fields operative on usual electronic systems and more importantly the act of external
magnetic fields, recently shown to be the source of new bonding mechanism [39], must
be taken into account in a comprehensive AIM theory. Based on field theoretic
variational methodologies developed by Schwinger and Feynman [40,41], this issue will
be considered in a subsequent contribution. It is particularly desirable to re-derive the
whole theory in the presence of external electromagnetic fields.
36
Although it is possible to consider excited state nuclear dynamics within context
of the MC-QTAIM, as demonstrated previously [1], the whole developed formalism is
intrinsically a zero-temperature scheme, 0
T. This stems from the fact that only pure
quantum state(s) of a system is amenable to be analyzed by the MC-QTAIM/orthodox
QTAIM however, for applications in solid state physics and crystallography, one
inevitably encounters an ensemble of systems not in a single pure state. In such cases
finite-temperature extension of the MC-QTAIM is clearly desirable since it incorporates
the complex phonon dynamics into the AIM framework; such framework is inevitably
based on density matrix formalism taking statistical/thermal factors explicitly into
account (Shahbazian, under preparation).
The extended Gamma field itself also deserves for more thorough future
investigations. For instance, the ingredients of the extended Gamma field,
q
n
and n
m,
are also the basic ingredients of the extended Hohenberg-Kohn theorem stated within
context of the multi-component density functional theory (MC-DFT) [7,42]. Thus, it is
tempting to speculate regarding a link between the former and the MC-QTAIM.
Additionally, according to a recent proposal by Ayers on formulating bi-functional DFT
based on the charge and mass densities [43], one may seek for a new version of DFT
based on the extended Gamma field as one of the underlying densities. The
"experimental" derivation of the Gamma using various scattering data, e.g. X-ray,
electron and neutron scatterings, is also of interest for future experimental
implementation of the MC-QTAIM in crystallographic studies. These and other
possibilities are all now under consideration in our laboratory.
37
From a computational viewpoint, absence of general user-friendly codes
performing relevant ab initio calculations as well as the MC-QTAIM analysis on the
multi-component species is an obstacle for widespread use of the latter methodology.
Hopefully, potential applications of the MC-QTAIM may also catalyze future activities in
these directions. Indeed, recent developments in ab initio methodologies that treat
all/part of nuclei from outset as genuine quantum waves, bypassing the usual BO
paradigm, is truly promising (for a bibliography see references cited in ref. 5) [44-53].
Acknowledgments
The authors are grateful to the Research Council of Shahid Beheshti Univestity
(SBU) for their financial support. Shant Shahbazian is grateful to Masume Gharabaghi,
Cina Foroutan-Nejad and Shahin Sowlati for a detailed reading of a previous draft of this
paper and their fruitful comments and suggestions.
38
References
1. Goli M, Shahbazian Sh (2011) Theor Chem Acc 129: 235
2. Nasertayoob P, Goli M, Shahbazian Sh (2011) Int J Quantum Chem 111: 1970
3. Goli M, Shahbazian Sh (2011) Int J Quantum Chem 111: 1982
4. Heidar Zadeh H, Shahbazian Sh (2011) Int J Quantum Chem 111: 1999
5. Goli M, Shahbazian Sh (2012) Theor Chem Acc 131: 1208
6. Goli M, Shahbazian Sh (2013) Theor Chem Acc DOI: 10.1007/s00214-013-1362-8
7. Shahbazian, Sh (2013) Found Chem DOI 10.1007/s10698-012-9170-0
8. Bader RFW (1990) Atoms in Molecules: A Quantum Theory. Oxford University
Press, Oxford
9. Popelier PLA (2000) Atoms in Molecules an Introduction. Pearson, London
10. Matta C, Boyd RJ (2007) Quantum Theory of Atoms in Molecules: From Solid
State to DNA and Drug Design. Wiley-VCH, Weinheim
11. Armour EAG, Richrad J-M, Varga K (2004) Phys Rep 493: 1
12. Srebrenik S, Bader RFW (1975) J Chem Phys 63: 3945
13. Srebrenik S, Bader RFW, Nguyen-Dang TT (1978) J Chem Phys 68: 3667
14. Bader RFW, Srebrenik S, Nguyen-Dang TT (1978) J Chem Phys 68: 3680
15. Bader RFW, Nguyen-Dang TT (1981) Adv Quantum Chem 14: 63
16. Bader RFW (1988) Pure & Appl Chem 60: 145
17. Zou PF, Bader RFW (1992) Int J Quantum Chem 43: 677
18. Bader RFW (1994) Phys Rev B 49: 13348
19. Nasertayoob P, Shahbazian Sh (2008) Int J Quantum Chem 108: 1477
20. Nasertayoob P, Shahbazian Sh (2009) Int J Quantum Chem 109: 726
39
21. Nasertayoob P, Shahbazian, Sh (2010) Int J Quantum Chem 110: 1188
22. Heidar Zadeh F, Shahbazian Sh (2011) Int J Quantum Chem 111: 2788
23. Heidar Zadeh F, Shahbazian Sh (2011) Theor Chem Acc 128: 175
24. Shahbazian Sh (2011) Int J Quantum Chem 111: 4497
25. Anderson JSM, Ayers PW, Rodriguez Hernandez JI (2010) J Phys Chem A 114:
8884
26. Cassam-Chenaï P, Jayatilaka D (2001) Theor Chem Acc 105: 213
27. Bader RFW (2001) Theor Chem Acc 105: 276
28. Mohallem JR (2002) Theor Chem Acc 107: 372
29. Kryachko ES (2002) Theor Chem Acc 107: 375
30. Delle Site L (2002) Theor Chem Acc 107: 378
31. Bader RFW (2002) Theor Chem Acc 107: 381
32. Yourgrau W, Mandelstam S (1968) Variational Principles in Dynamics and
Quantum Theory, 3rd edition. Dover, New York
33. Nesbet RK (2004) Variational Principles and Methods in Theoretical Physics and
Chemistry. Oxford University Press, Cambridge, UK
34. Courant R, Hilbert D (1953) Methods of Mathematical Physics, Vol. 1. Wiley
Interscience, New York
35. Bader RFW, Popelier PLA (1993) Int J Quantum Chem 45: 189
36. Arfken G (1985) Mathematical Methods for Physicists. Academic Press, San
Diego
37. Tachikawa M, Osamura Y (2000) Theo Chem Acc 104: 29
38. Ishimoto T, Tachikawa M, Nagashima U (2009) Int J Quantum Chem 109: 2677
40
39. Lange KK, Teligren EI, Hoffmann MR, Helgaker T (2012) Science 337: 327
40. Toms DJ (2007) The Schwinger Action Principle and the Effective Action.
Cambridge University Press: Cambridge
41. Feynman RP, Hibbs AR (1965) Quantum Mechanics and Path Integrals.
McGraw-Hill, USA
42. Capitani JF, Nalewajski RF, Parr RG (1982) J Chem Phys 76: 568
43. Ayers PW (2006) Theor Chem Acc 115: 253
44. Hoshino M, Nishizawa H, Nakai H (2011) J Chem Phys 135: 024111
45. Nishizawa H, Hoshino M, Imamura Y, Nakai H (2012) Chem Phys Lett 521: 142
46. Nishizawa H, Imamura Y, Ikabata Y, Nakai H (2012) Chem Phys Lett 533: 100
47. Kamarchik E, Mazziotti D (2007) Phys Rev A 75: 013203
48. Kamarchik E, Mazziotti D (2009) Phys Rev A 79: 012502
49. Kato T, Yamanouchi K (2009) J Chem Phys 131: 164118
50. Sirjoosingh A, Pak MV, Hammes-Schiffer S (2011) J Chem Theory Comput 7:
2689
51. Mátyus E, Hutter J, Müller-Herold U, Reiher M (2011) J Chem Phys 135: 204302
52. Mátyus E, Reiher M (2012) J Chem Phys 137: 024104
53. Bubin S, Pavanello M, Tung W-C, Sharkey KL, Adamowicz L (2013) Chem Rev
113: 36
41
Table 1- The results of the topological analysis all offered in atomic units.*
Property/species HD
HT
DT
p-(3, -3) (3, -1) d-(3, -3) p-(3, -3) (3, -1) t-(3, -3) d-(3, -3) (3, -1) t-(3, -3)
Position** 0.675 0 -0.704 0.666 0 -0.711 0.687 0 -0.687
Gamma 0.328 0.2444 0.348 0.329 0.2466 0.358 0.349 0.2514 0.360
Lap. of Gamma -13.829 -1.1655 -19.221 -13.853 -1.1844 -22.346 -18.323 -1.1969 -22.460
Property/species 6LiH
6LiD
6LiT
p-(3, -3) (3, -1) li-(3, -3) d-(3, -3) (3, -1) li-(3, -3) t-(3, -3) (3, -1) li-(3, -3)
Position** 1.725 0 -1.366 1.714 0 -1.363 1.709 0 -1.362
Gamma 0.253 0.0375 11.301 0.266 0.0379 11.301 0.273 0.0381 11.300
Lap of Gamma -9.677 0.1450 -7965.130 -12.369 0.1493 -7964.414 -14.108 0.1510 -7962.539
Property/species 7LiH
7LiD
7LiT
p-(3, -3) (3, -1) li-(3, -3) d-(3, -3) (3, -1) li-(3, -3) t-(3, -3) (3, -1) li-(3, -3)
Position** 1.725 0 -1.366 1.714 0 -1.362 1.709 0 -1.361
Gamma 0.253 0.0375 11.377 0.266 0.0379 11.376 0.273 0.0381 11.376
Lap. of Gamma -9.678 0.1449 -8334.750 -12.367 0.1492 -8333.941 -14.104 0.1509 -8332.559
* Lap. stands for Laplacian while p, d , t, and li are abbreviations for proton, deuteron, tritium and lithium, respectively.
** The center of coordinate system is fixed on (3, -1) CP while the z-axis goes through the both (3, -3) CPs.
42
Table 2- The results of the basin integrations for hydrogen isotopomers all offered in atomic units.*
HD
HT
population of particles population of particles
Basins p d e Basins p t e
H 1.000 0.000 0.984
H 1.000 0.000 0.975
D 0.000 1.000 1.016
T 0.000 1.000 1.025
total 1.000 1.000 2.000
total 1.000 1.000 2.000
electric dipoles
electric dipoles
Basins CT
first
moments total Basins CT
first
moments total
H 0.023 -0.098 -0.075 H 0.036 -0.102 -0.066
D 0 0.085 0.085 T 0.000 0.081 0.081
total 0.023 -0.013 0.010 total 0.036 -0.021 0.015
ab initio
0.010 ab initio 0.015
Energies Energies
Basins electronic p d total Basins electronic p t total
H -0.50504 -0.01837 0 -0.52341 H -0.50289 -0.01839 0 -0.52128
D -0.52625 0 -0.01360 -0.53985 T -0.53563 0 -0.01138 -0.54701
total -1.03129 -0.01837 -0.01360 -1.06326 total -1.03853 -0.01839 -0.01138 -1.06830
ab initio -1.06326 ab initio -1.06829
DT
population of particles
Basins d t e
D 1.000 0.000 0.998
T 0.000 1.000 1.002
total 1.000 1.000 2.000
electric dipoles
Basins CT
first
moments total
D 0.002 -0.092 -0.089
T 0.000 0.094 0.094
total 0.002 0.002 0.004
ab initio 0.004
Energies
Basins electronic d t total
D -0.52431 -0.01364 0 -0.53795
T -0.52995 0 -0.01141 -0.54136
total -1.05426 -0.01364 -0.01141 -1.07931
ab initio -1.07931
* The symbols p, d , t, e and CT are abbreviations for proton, deuteron, tritium, electron and charge transfer, respectively.
43
Table 3- The results of the basin integrations for 6LiX (X=H, D, T) all offered in atomic units.*
6LiH
6LiD
population of particles population of particles
Basins li p e Basins li d e
Li 1.000 0.000 2.102
Li 1.000 0.000 2.101
H 0.000 1.000 1.898
D 0.000 1.000 1.899
total 1.000 1.000 4.000
total 1.000 1.000 4.000
electric dipoles electric dipoles
Basins CT
first
moments total Basins CT
first
moments total
Li 0 0.016 0.016 Li 0 0.015 0.015
H -2.788 0.387 -2.401 D -2.776 0.373 -2.403
total -2.788 0.403 -2.385 total -2.776 0.389 -2.388
ab initio
-2.385 ab initio -2.387
Energies Energies
Basins electronic li p total Basins electronic li d total
Li -7.11519 -0.08034 0 -7.19552 Li -7.11557 -0.08034 0 -7.19591
H -0.56494 0 -0.01630 -0.58124 D -0.57855 0 -0.01206 -0.59061
total -7.68012 -0.08034 -0.01630 -7.77676 total -7.69412 -0.08034 -0.01206 -7.78652
ab initio -7.77678 ab initio -7.78653
6LiT
population of particles
Basins li t e
Li 1.000 0.000 2.101
T 0.000 1.000 1.899
total 1.000 1.000 4.000
electric dipoles
Basins CT
first
moments total
Li 0 0.015 0.015
T -2.770 0.367 -2.403
total -2.770 0.382 -2.388
ab initio -2.388
Energies
Basins electronic li t total
Li -7.11573 -0.08034 0 -7.19607
T -0.58484 0 -0.01007 -0.59490
total -7.70057 -0.08034 -0.01007 -7.79097
ab initio -7.79099
* The symbols li, p, d, t, e and CT are abbreviations for lithium, proton, deuteron, tritium, electron and charge transfer, respectively.
44
Table 4- The results of the basin integrations for 7LiX (X=H, D, T) all offered in atomic units.*
7LiH
7LiD
population of particles population of particles
Basins li p e Basins li d e
Li 1.000 0.000 2.102
Li 1.000 0.000 2.101
H 0.000 1.000 1.898
D 0.000 1.000 1.899
total 1.000 1.000 4.000
total 1.000 1.000 4.000
electric dipoles electric dipoles
Basins CT
first
moments total Basins CT
first
moments total
Li 0 0.016 0.016 Li 0 0.015 0.015
H -2.787 0.387 -2.400 D -2.775 0.374 -2.401
total -2.787 0.403 -2.384 total -2.775 0.389 -2.386
ab initio
-2.384 ab initio -2.386
Energies Energies
Basins electronic li p total Basins electronic li d total
Li -7.13262 -0.07485 0 -7.20747 Li -7.13300 -0.07485 0 -7.20785
H -0.56495 0 -0.01630 -0.58125 D -0.57857 0 -0.01206 -0.59062
total -7.69757 -0.07485 -0.01630 -7.78872 total -7.71157 -0.07485 -0.01206 -7.79848
ab initio -7.78872 ab initio -7.79848
7LiT
population of particles
Basins li t e
Li 1.000 0.000 2.101
T 0.000 1.000 1.899
total 1.000 1.000 4.000
electric dipoles
Basins CT
first
moments total
Li 0 0.015 0.015
T -2.769 0.367 -2.402
total -2.769 0.382 -2.387
ab initio -2.387
Energies
Basins electronic li t total
Li -7.13317 -0.07485 0 -7.20802
T -0.58485 0 -0.01007 -0.59492
total -7.71802 -0.07485 -0.01007 -7.80293
ab initio -7.80293
* The symbols li, p, d, t, e and CT are abbreviations for lithium, proton, deuteron, tritium, electron and charge transfer, respectively.
