Deriving a mechanistic model for potential energy surface of
coordination compounds of nontransition elements
A.L. Tchougréeff
Division of Electrochemistry, Department of Chemistry,
Moscow State University,
and
Karpov Institute of Physical Chemistry, Moscow, Russia
Abstract
Molecules of coordination compounds (those formed by a central atom 
largely by a transition or nontransition metal ion, but also by
nontransition elements like sulfur, phosphorus etc. and by atoms
or groups surrounding it  the ligands) for decades represent a significant
problem for any "classical" description given in terms of empirical force
fields of molecular mechanics (MM) due to diversity of coordination
polyhedra resulting in the numerosity of the parameters necessary to
describe the objects of interest within such a setting. This situation is
further toughen by the specific collection of effects known as mutual
influence of ligands, of which the transeffect in transition metal
complexes is the most known. A feature particularly complicating
understanding the ligand influence is its qualitative dependence on the
nature of the central atom.
The real source of these problems is of course the specificity of the
electronic structure of the coordination compounds. If compared with usual
organic molecules for which the MM in its classical form is rather
successful the major distinction characteristic for coordination compounds
is the absence of fairly distinguisheable twocenter twoelectron bonds
incident to its central atom (ion).
Using a methodology called deductive molecular mechanics (DMM) we have
recently shown that for örganic" molecules it is possible to sequentially
derive the form of the MM force field departing from a simple but
intuitively transparent model of electronic structure of a single chemical
bond. Here we present an analogous derivation for the force fields
describing coordination compounds. It is based on the analysis of electronic
structure of the closest ligand shell of coordination compounds and on that
of its relation with the geometry changes induced by chemical substitution
(ligand influence) performed recently by Levin and Dyachkov. By using the
elements of the oneelectron density matrix as an economical set of
electronic structure variables for the closest ligand shell the DMM model of
coordination compounds is constructed. Next, by excluding the electronic
structure variables the effective elastic elements describing the force
fields concordant with the mutual ligands' influence in coordination
compounds are constructed and their dependence on the state of the central
ion is established.
Introduction
Constructing a mechanistic model for potential energy surfaces (PES) of
coordination compounds, which would be consistent with the ligand influence
effects in this class of molecules for decades remains actual for the
respective area of computational chemistry []. Modern
progress in quantum chemistry allows to make reliable predictions for the
PES of polyatomic molecular systems including coordination compounds on the
basis of direct calculations of their electronic structure. However,
highlevel methods, like ab initio and DFT, are computationally too
demanding when dealing with coordination compounds and incidentally do not
provide any theoretical insight into physical reasons controlling the
observable (and hopefully numerically reproducible) behavior of these
species. On the other hand qualitative concepts, such as pointsonasphere
(POS) [] or metalligand sizematch selectivity [], playing important rôles in theoretical coordination chemistry of both
transition and nontransition metals, miss any reliable semiquantitative
structureenergy relation. Thus, effective numerical tools suitable for
semiempirical modeling of coordination compounds and reproducing the
qualitative aspects of their stereochemistry (including ligand influence)
are strongly in demand. Recent studies reviewed in [] are
devoted to analysis of chemical bonding in various coordination compounds.
They are, however, mainly focused on truely covalent compounds with more or
less well distinguisheable twocenter bonds. By virtue of this the
electronic structure of compounds considered in [] differs on
an intuitive level from the picture of unsaturable and undirected
coordination bonds characteristic for
complexes of metals with organic ligands with donor atoms []. Here we assume to consider namely this latter class of compounds from the
point of view of possibility of constructing a mechanistic model of their
PESs based on a sequential quantum model of electronic structure of their
closest coordination spheres. Before plunging into this we explain what has
to be expected from the theory to be constructed here.
It is known from the literature [] that in coordination
compounds due to mutual ligand influence effects the MM parameters involving
the central atom (ion) are that numerous that any mechanistic description of
them with an acceptable result makes the entire enterprize eventualy
senceless. There would be no chance to get any insight to this diversity
from the scratch. Fortunately, A.A. Levin and P.N. Dyachkov being based in
the I.B. Bersuker theory of vibronic interactions []
yet in 1980ies performed an exhaustive qualitative analysis of the
interplay between chemical substitutions and deformations of the complex
geometry i.e. of the ligand influence in those of both transition
and nontransition elements [] for the most widespread
coordination polyhedra: octahedron, tetrahedron, and planar square. By this
they established basic relations between the characteristics of electronic
structure of coordination compounds and their geometry. Analysis performed
in [] reduces to qualitative reasoning on the properties
of the solutions of the MO LCAO method in the restricted basis of functions.
Formally it is a special case of applying the M.J.S. Dewar's theory of
perturbations of MOs [] to the special class of the MOs
residing in the closest ligand shell (CLS) of a coordination compound. Our
purpose in the present paper is to develop such a formal representation to
the results of [] which could be recast into a mechanistic
view upon the PES of coordination compounds in terms of classically looking
force fields derived from the quantum mechanical picture of molecular
electronic structure. By this we theoretically derive an MMlike description
covering coordination compounds and give an explanation and introduce
systematization into diversity of their parameters.
Ligand influence from MM perspective
In the standard MM [] setting the substitution
effects upon molecular geometry (ligand influence) can be only awkwardly
described. The PES in the MM setting is the sum
E = E_{b}+E_{ang}+E_{tors}+E_{nb}, 
 (0) 
of the bond stretching ( E_{b}) , valence angle
bending ( E_{ang}) , torsional ( E_{tors}) , and nonbonding ( E_{nb}) contributions
(force fields), which are all explicit functions of nuclear coordinates. The
nonbonding force field is mainly contributed by the Coulomb interactions of
some effective charges and by the van der Waals forces:
E_{nb} = E_{Coul}+E_{vdW} 

The equilibrium nuclear coordinates q_{eq} coming from the MM
approximation are defined by the equilibrium conditions which can be
conveninetly reformulated as those of the evanescence of the forces acting
on the atoms (in the MM context using this form of the equilibrium condition
dating back to sir Isaac Newton himself was suggested in Ref. Pletnev00). In the standard MM setting the equilibrium interatomic
distances are close to some ideal values q_{0} (which are part of the
parameters' sets describing respective force field) specific for the bonds
of each considered type. In the vicinity of these ideal values the energy
can be safely assumed (and it is frequently but not always done in the MM
setting) to be quadratic in the nuclear coordinate shifts so that the actual
values of the equilibrium interatomic distances (and of other geometry
parameters) appear as a result of the responce of the system to the
presumably small forces coming from the nonbonding part of the molecular
force field:


 (0)  

 Ñ_{q}( E_{tors}+E_{Coul}+E_{vdW})  _{q = q0} \notag 
 (0) 

where by Ñ_{q} we understand the first derivative (gradient) with
respect to all the nuclear coordinates and D^{1} is the inverse of the
dynamic matrix (that of the second derivatives of the energy) derived from
the harmonic bonding contribution (bond stretching and valence angle
bending) to the MM energy only. This approximation to the MM energy yields
the diagonal dynamical matrix if the coordinates are taken as the bond
lengths' and valence angles' shifts, so that D^{1} is also diagonal and
can be easily calculated. Below we shall restrict ourselves to a model of
octahedral complex where no quadruplet of atoms sequentially connected by
bonds appear. That means that no proper torsion terms E_{tors} actually appear in the expression for the energy eq. (1). For
that reason we do not consider them hereinafter. Under these conditions
whatever coupling between the individual bond stretching and valence angle
bending modes is only possible through the mediation of the nonbonding force
fields unless it is introduced into the model explicitly. We are not going
to to insert these missing couplings "by hand" rather to derive them
sequentially. Hence we first explore what the nonbonding force fields can
give in this respect.
Coulomb contribution to ligand influence
Let us explore the possibility to ascribe the ligand influence to the
nonbonding part of the classical force field starting from the leading
Coulomb contribution to the energy. In the octahedral complex ML_{6} a single substitution ML_{6}® ML_{5}X manifests itself in changing the ideal bond length of one of the
ML bonds to that of the MX bond. Under the
formulated conditions this does not affect directly the lengths of other ML bonds (the offdiagonal terms are absent), but changes the
potentials of the force fields felt by other atoms L. Let us
assume for the sake of definiteness that the replaced ligand occupies the
apical position i.e. takes a positive value of its zcoordinate
in the coordinate frame centered at the M atom with the axes
directed along the ML bonds. Next, we can set the ideal value of
the MX bond r_{MX} to differ from that for the ML bonds r_{ML} by the quantity dr_{XL} :
and assume that the ideal position of the X substituent does not
deviate from the zaxis. We also assume that the effective charge of the X substituent differs from that of the ligand L by an
amount dQ_{XL}. This value is not, however, absolutely
independent since we assume that the effective charge of the central atom
under the substitution changes by the same quantity dQ_{XL}
in the opposite sence. Under these assumptions one can write the variation
of the electrostatic potential caused by the substitution in the point r of the threedimensional space, which is enough distant from the
substituent X. Retaining the terms of the first order with
respect to either dr_{XL} or dQ_{XL} we
get:
dj_{Coul}( r) = dQ_{XL} 
æ ç
è


1  rR_{L}

 
1  rR_{M}

ö ÷
ø

+Q_{L} 
( rR_{L},dR_{XL})  rR_{L} ^{3}

. 
 (0) 
The vectors involved in the above formula refer to the complex geometry and
its variation:
Applying the standard formulae of electrostatics we can write the extra
field excerted at the point r due to the above potential
variation eq. (4):


 (0)  

dQ_{XL} 
æ ç
è


rR_{L}  rR_{L} ^{3}

 
rR_{M}  rR_{M} ^{3}

ö ÷
ø


 (0)  

Q_{L} 
dR_{XL}  rR_{L} ^{3}

+Q_{L} 
( rR_{L},dR_{XL}) ( rR_{L})  rR_{L} ^{5}

\notag 
 (0) 

The ideal position for the transligand L is:
r = R_{L}^{trans} = ( 0,0,r_{ML}) 
 (0) 
and the electrostatic force additional to that it had to experience in the ML_{6} complex reads:
df_{Coul}( R_{L}^{trans}) = 
1 4


Q_{L}^{2} r_{ML}^{2}


æ ç
è


dr_{XL} r_{ML}

+3 
dQ_{XL} Q_{L}

ö ÷
ø

( 0,0,1) 
 (0) 
Then according to approximation given by eq. (1) one can
conclude that since the force felt by the transligand in the substituted
molecule differs by df_{Coul}( R_{L}^{trans}) the coordinate of the transligand
differs by the same force divided by the elasticity constant characteristic
for the nuclear shift along the given direction. Incidentally the required
elasticity constant equals to the elasticity constant for the ML
bond stretching in the classical MM force field, so that:
dR_{L}^{trans} = 
df_{Coul}( R_{L}^{trans}) K_{ML}

. 

This result can be qualitatiely undestood as follows: two factors contribute
to the transinfluence. First is the charge redistribution. Indeed, if for
the substituent ligand X the effective charge decreases by its
absolute value (the negative Q_{L} becomes less negative i.e. dQ_{XL} > 0) this results in an effective attractive
force. However, if for the substituent ligand X its effective
charge increases by absolute value as compared to that of the ligand L (the negative Q_{L} becomes more negative i.e. dQ_{XL} < 0) this contributes an effective repulsive force
acting on the transligand. Nevertheless, due to the assumption that the
charge brought to the substituent X is taken from the central
atom M the latter acquires additional charge dQ_{XL} which is located much closer to the transligand than the substituent
and produces a stronger force. The overall effect of the charge
redistribution is opposite to that excerted by the charge transferred to the
substituent and it is the transML bond contraction for the more
negative substituent ligand X and the transML bond
elongation for the less negative substituent ligand X. Even this
already rather complicated picture can significanly change since the net
effect depends also on the geometry shift of the substituent ligand X relative to the ideal position of the ligand L. If dr_{XL} > 0 (the substituent is further away from the central atom) the
repulsion of the transligand from the substituent decreases which results
in an effective attraction force acting on the transligand and by this
reducing the ML bond length. The overall transinfluence in the
Coulomb approximation is controlled by the quantity

dr_{XL} r_{ML}

+3 
dQ_{XL} Q_{L}

. 

If it is positive the substitution results in an effective attraction of the
transligand (and thus in the shortening of the bond in the transposition
to the substituent) and vice versa.
The situation with the ligands in the cisposition to the substituent is
even more complicated. Taking for the sake of definiteness the cisligand at
r = R_{L}^{cis} = ( r_{ML},0,0) 
 (0) 
which is one of the four equivalent positions we see that the additional
electrostatic force acting on it is:
df_{Coul}( R_{L}^{cis}) = 
1 2Ö2


Q_{L}^{2} r_{ML}^{2}


æ ç
è


dr_{XL} r_{ML}


æ ç
è

 
3 2

,0, 
1 2

ö ÷
ø

+ 
dQ_{XL} Q_{L}

( 12Ö2,0,1) 
ö ÷
ø



which (depending on the relation between [(dr_{XL})/(r_{ML})] and [(dQ_{XL})/(Q_{L})]) may result either in approach of the cisligand to the
central atom of the complex or in its going further from the latter. The
elongation of the apical bond (dr_{XL} > 0) always tends to
shorten the bond in the cisposition to it. On the other hand the
substitution by a less electronegative substituent (dQ_{XL} > 0 ) results in an effective repulsion of the cisligand (lengthening of
the bond in the cisposition to the substituent). But this effect is
expected to be somewhat less pronounced than the bondlength variation for
the transligand:

 df_{Coul}( R_{L}^{cis})   df_{Coul}( R_{L}^{trans})

= 
4 3


2Ö21 2Ö2

» 0.86 < 1 

although the difference should not be very significant (only 15%). In all
cases (which is expectable) there is a component of the force acting along
the shift direction of the substituent X.
Van der Waals contribution to ligand influence
Although one may think that on the scale of the Coulomb interactions the
effect of other (van der Waals) nonbonding interactions may be neglected we
notice that in the present context it goes not about the absolute magnitude
of the forces exserted by different nonbonding components of the classical
force fields upon the idealized structure of the complex but about the
variations of the forces exserted by these fields. With this precaution in
mind we address the effect of monosubstitution upon the forces exserted by
the substituent within the simplest possible model of the van der Waals
force field  the LennardJones (612) potential. In this approximation the
contribution to the potential energy of the vdW interaction of a particle L at the point r with the particle at R_{X} has the from:
V_{LX}( r) = e_{LX} 
æ è


~ y

2

2 
~ y

ö ø



where

~ y

= 
æ ç
è


d_{LX}  rR_{X}

ö ÷
ø

6



In this notation the minimum of the LJ potential curve corresponds to [y\tilde] = 1. Using the Berthelot combination rules we get for the
substituted case:


 Ö

e_{LL}e_{XX}

=  Ö

e_{LL}( e_{LL}+de_{XL})

» e_{LL} 
æ ç
è

1+ 
1 2


de_{XL} e_{LL}

ö ÷
ø


 (0)  


1 2

( d_{LL}+d_{XX}) = d_{LL}+ 
1 2

dd_{XL} \notag 
 (0) 

Inserting the linear approximation for the distance between the point r and the position of the substituent R_{X}
like in the case of the Coulomb interaction:
 rR_{X} =  rR_{L}dR_{XL} »  rR_{X} 2( rR_{L},dR_{XL}) 
 (0) 
and performing somewhat long but simple algebra we arrive to an estimate of
the substitution stipulated variation of the potential energy of the
particle L located in the point r:
dV_{vdW}( r) » 
1 2

de_{XL}( y^{2}2y) +12e_{LL} 
æ ç
è


dd_{XL} d_{LL}

+ 
( rR_{L},dR_{XL})  rR_{L} ^{2}

ö ÷
ø

( y^{2}y) 
 (0) 
where
y = 
æ ç
è


d_{LL}  rR_{L}

ö ÷
ø

6



The additional forces have the form:


ÑdV_{vdW}( r) = de_{XL}( y1) Ñy \notag 
 (0)  

12e_{LL}( y^{2}y) Ñ 
( rR_{L},dR_{XL})  rR_{L} ^{2}

 
 (0)  

12e_{LL} 
æ ç
è


dd_{XL} d_{LL}

+ 
( rR_{L},dR_{XL})  rR_{L} ^{2}

ö ÷
ø

( 2y1) Ñy \notag 
 (0) 

Performing necessary algebra yields rather cumbersome formulae (not given
here) which, however, allow for the qualitative understanding. Like in the
case of the Coulomb interaction the additional forces in the case of the van
der Waals interaction appear from two sources  the variation of the
parameters of the LennardJones (LJ) potential and the variation of the
idealized geometry. In the accepted first order approximation these two
sources contribute independently. The variations of the potential parameters
(de_{XL} and dd_{XL}) are even
more uncertain than the values of e_{LL} and d_{LL} themselves. If one neglects their contribution the result can
be understood from the analysis of the LJ potential curve. For the
transligand two situations are thinkable: one is that the ideal LL separation (equal to 2r_{ML}) falls onto the repulsive segment
of the LJ curve ([(d_{LL})/(2r_{ML})] < 1).
Then, getting the MX bond longer than the ML bond
results in abstraction of the repulsive wall from the trans L
ligand which manifests itself in the effective attraction and in the
shortening of the transbond. However, if the trans ligand appears on the
attractive segment of the LJ curve ([(d_{LL})/(2r_{ML})] > 1) the situation is not that simple. For shorter distances
(closer to the potential minimum) the restoring force (the first derivative
of the potential) increases when the interatomic separation increases. In
this case getting the MX bond longer than the ML
bond results in the increase of the restoring force and thus to an effective
extra attraction of the trans L ligand. This has to result in a
shortening of the trans ML bond as well. On the other hand, the
restoring force on the attractive segment passes through its maximum since
at larger distances it decreases and completely vanishes at the infinite
separation. The larger and shorter distance ranges on the attractive segment
of the LJ potential are separated by the value:
If y < y^{*} the elongation of the MX bond as compared to
the ML bond results in the decrease of the restoring force and
thus to an effective extra repulsion of the trans L ligand.
The situation with the cisligands is even more complicated. Performing a
shift of the X ligand as compared to the original position of the
L ligand produces an additional force acting on the cisligand
which has two components: one directed along the xaxis of the complex
coordinate frame and that directed along the zaxis. Both of these
contributions may be either of attractive or repulsive character i.e. in the case of the xcomponent of the force may be directed either to
the central atom or out of it, whereas the zcomponent of the force may be
pointing to the same direction as the shift of the X ligand
relative to the ideal position of the L ligand. The actual
direction of the extra force depends on the precise position of the
cisligand relative to the substituted one. For example for the more
distant X ligand, the ycomponent of the force is directed away
from the central atom for y < y^{*} = [4/7] and
toward it for shorter distances meaning larger values of the variable y.
On the other hand the zcomponent of the force is directed opposite to the
shift direction of the X ligand for y < y^{**} = [1/2].
Ligand influence as derived from nonbonding force fields vs experiment
From the above treatment one can derive two conclusions: (1) The diversity
of the types of behavior which in principle can be ascribed to different
nonbonding force fields is impressive. By assigning these or those values to
the parameters characterizing the force field variation under substitution
one can reproduce eventually all thinkable modes of mutual influence of the
ligands in the coordination sphere. (2) The parameters' values possibly
needed to reproduce the observed behavior cannot be systematized and their
values will probably remain the result of a play of uncontrollable factors.
This reduces the validity of the entire picture since certain type of
systematization is possible in chemical terms (see below). Here we are going
to discuss it briefly relying basically upon the review given in Ref. LevinDyachkov which will be compared with the above theoretical sketches.
The ligand influence in coordination compounds of nonmetals in higher and
lower oxidation states differs significantly both in magnitudes and signs of
the effects. An instructive example is provided by the substitution in the
perhalogenated complexes which can be described as L_{6}M® L_{5}XM where the rôle of L is taken
by a halogen anion, X stands for a ligand less electronegative
than the halogen L, and M stands for a nontransition
element like J, S, P, Sn, etc. in a higher oxidation state. In some
cases the substitution does not cause any difference between the cis and
transbondlenghts or only a marginal one (compounds of S(VI)). However, in
other cases the substitution results in remarkable deformations of the
coordination octahedron (compounds of J(VII)). In the latter case the cis
JF bonds are significantly shorter than the JF bond in the transposition
to the presumably less negatively charged oxygen substituent. This situation
can be characterized by some values dQ_{XL} > 0, Q_{L} < 0, and dr_{XL} < 0 so that the entire picture
qualitatively fairly agrees with the Coulomb model of transeffect described
in Section 0.1. Turning to the beginning of the
Periodic row one can see that namely the bond in the transposition to the
substituent turns out to be significantly shorter than those in the
cispositions for the complexes of P(V), As(V), Sn(IV), Pb(IV), Ga(III).
Although in all these cases one could expect a significant contribution of
the charge redistribution effect upon the complex geometry through the
mediation of the Coulomb forces, the picture is inverted as compared to the
predictions based on the Coulomb contributions to the MM energy (and to the
case of J(VII) complexes). Remarkably enough that this contradiction most
probably cannot be cured by referring to the bond length variations dr_{XL} which are mostly negative since one can expect that the XM bonds in the examples reviewed in Ref. [] are
shorter than the LM ones.
On the other hand in the case of complexes with nonmetallic central atom of
a lower oxidation degree the electropositive substitution (in complexes of
Te(IV) and Se(IV)) results in a significant increase of the bondlength in
the transposition to the substitution as compared to the the lengths of the
cisbonds. Similar picture is observed in the coordination compounds of
transition metals. So we see that the diversity of the observed types of
behavior is too large to try to squeeze the available experimental data to
the picture relying upon the Coulomb forces acting between the effective
charges of restricted mobility. Doing so one drives in the situation when
the demands to the relative magnitudes of the charges' and ideal
bondlengths' variations ([(dr_{XL})/(r_{ML})] and [(dQ_{XL})/(Q_{L})]) become very difficult to satisfy. Much more important is that the
quantities dQ_{XL} playing a key rôle in the picture of
the ligand influence based on the Coulomb forces as the bare charges Q_{L} themselves must be found from some kind of quantum chemically
based procedure since otherwise one cannot guarantee the correct behavior of
them. At the same time any attempt to reload the specific behavior observed
in different complexes upon the parameters of the LJ potentials brings up a
necessity to assume the dependence of these parameters on the nature of the
"third"  central atom, which is clearly not desirable. One can try to get
around these problems by ascribing the values of different signs to the
offdiagonal stretchingstretching constants when it goes about the cis and
transbonds. The signs of these constants may be made different depending on
the nature (and the oxidation state) of the central atom. Thus the necessary
values will have to be assigned on the basis of more and more refined system
of atomic types as it is done in the "classical" MM, but this brings back
the problem of enourmous growth of the number of necessary parameters and of
setting physically sound limits to their values. Our purpose in the present
paper is ultimately to sequentially derive the expression for the required
offdiagonal terms and to obtain estimates for their values. This will be
done in further Sections on the basis of analysis of interrelations between
the perturbations of the electronic structure of the coordination compound
and its geometry, having the theory of the ligand influence by Levin and
Dyachkov (which successfully described the cases reviewed above) as a
benchmark to be reached by the scheme under construction.
Account of deductive molecular mechanics
General setting
The methodology designed for deriving mechanistic picture of PES from one
based on a suitable quantum description of molecular electronic structure
has been proposed in Ref. [] under the name of deductive molecular
mechanics (DMM). It has been applied in Refs. [,,] to
the örganic" molecules  the main object of the classical MM with a
considerable success. The result of these works summarized in Ref. TchDScThesis ia that a sequential derivation of whatever mechanistic model
of PES for each specific class of molecules consists of the following steps:
 Groups of electrons responsible for the observed features of molecular
energy and geometry to be reproduced in the target MM model must be
identified.
 Approximate methods sufficient for description of the responsible
electron groups and reproducing the necessary features must be identified
and formalized in the structure of the trial wave function for the target
class of molecules.
 The electronic structure variables (ESVs) describing the relevant
electron groups in the sufficient approximations must be identified.
 In terms of these sets of ESVs one constructs an intermediate (DMM)
model of the PES.
 The intermediate ESVs can be excluded for example by using a linear
responce theory in order to get the target MM model of PES.
For the molecules of interest in the present paper  the complexes of
nontransition elements the results first three stages can be extracted by
analysing some previous works. As we mentioned previously the LD theory
serves as a benchmark here since it allows to establish and by comparision
with experiment to verify relations between some elements of molecular
electronic structure and molecular geometry of coordination compounds.
Constructing DMM for octahedral complexes
As any phenomenological theory the LD theory of the ligand influence tacitly
assumes the existence of an effective Hamiltonian describing certain group
of electrons responsible for the experimentally observed behavior. The
orbitals responsible for the binding of the central atom with donor ligands
can be reasonably identified with the valence AOs of the corresponding
central atoms and with the hybrid orbitals (HOs) of the lone pairs (LPs) of
the ligand donor atoms. These orbitals and electrons residing in the central
atom and in the closest vicinity of the latter can be termed as the closest
ligand shell. In their terms the explanation of experimental behavior have
been given in Ref. []. The problem is how to sequentially
define the orbitals to be used to span the CLS in a polyatomic system like a
coordination compound with organic ligands. Paper [] gives a
tentative answer to this question. There we performed comparative study of
electronic structures of simple amines and ethers on one hand and of their
polycyclic counterparts on another hand by the semiempirical SLGMNDO method
[,]. The results given in Ref. [] show that the
relevant characteristics of electronic structure (the bond orders, electron
densities on the bonding orbitals of the donor atoms, and the weights of the
sfunctions in the LPs) of the lowmolecular amines and ethers and their
polycyclic analogs are fairly close. For this reason we can assume that the
LP HOs required for the CLS construction can be taken from some the
strcictly local geminal (SLG) based procedure for free ligands and are
subsequently only slightly modified due to complexation.
Next question to be answered relates to the form of the wave function F_{CLS} of the group of electrons in the carrier space defined
above and to the acceptable approximation to be used for obtaining it. The
LD theory of ligand influence had been constructed with use of the Hückel type procedure. By this the Coulomb interaction within the considered
subset of oneelectron states was not taken into account although (as it is
shown in Section 0.1) it affects the process of
the charge (electron density) redistribution in the CLS and can give a
considerable contribution to the energy (see below). The HFR aproximation
takes into account the necessary Coulomb terms and by this the true energy
operator for the CLS group is the effective Fock operator. This reduces the
problem to solving the system of HartreeFock equations for the occupied and
vacant MOs in the carrier space of the CLS electron group. For the case of
an octahedral complex the problem is further consideraby simplified by
symmetry. Only the following symmetry adapted linear combinations first
introduced in Refs [,,] are allowed
to serve as either occupied or vacant canonical MOs of the CLS:

y(e_{g}c) = 
1

(2c_{z}+2c_{z}c_{x}c_{x}c_{y}c_{y}) 

y(e_{g}s) = 
1 2

(c_{x}+c_{x}c_{y}c_{y}) 

y^{a}(a_{1g}) = x_{a1g}f_{s}+ 
y_{a1g} Ö6

(c_{x}+c_{y}+c_{z}+c_{x}+c_{y}+c_{z}) 

y^{b}(a_{1g}) = y_{a1g}f_{s}+ 
x_{a1g} Ö6

(c_{x}+c_{y}+c_{z}+c_{x}+c_{y}+c_{z}) 

y^{a}(t_{1u}g) = x_{t1u}f_{g}+ 
y_{t1u} Ö2

(c_{g}c_{g}) 

y^{b}(t_{1u}g) = y_{t1u}f_{g}+ 
x_{t1u} Ö2

(c_{g}c_{g}) 



 (0) 
where the supersripts a and/or b refer to the antibonding or bonding
linear combination of the symmetry Gg (here G stands
for the irreducible representation and g for its row). The basis
functions f are the oneelectron states f_{s}  the sorbital
of the central atom and f_{g} (g = x,y,z)  three porbitals of the latter directed long the coordinate axes, and the functions
c_{g}, c_{g} are the LP HOs directed along the gaxis (g = x,y,z) of the Cartesian coordinate system centered
respectively at the donor atom of the ligand located on the positive and
negative semiaxes g. There is only one instance of the MO
transforming according to either row (c or s) of the e_{g} symmetry.
In the 12electron complexes (in the present context it goes about the
number of electrons in the CLS electron group) which will be considered
below they are occupied. Coefficients x_{G} and y_{G} = Ö{1x_{G}^{2}} describe the mixing between the central ion AOs and the
ligand HOs and are to be determined from the secular equations of the SCF
MO LCAO method. This all reduces the number of the variables describing the
electronic structure of the octahedral complexes (its CLS) to only two,
which contain all information necessary to describe the octahedral CLS,
e.g.: x_{a1g} and x_{t1u}. In the octahedral symmetry the
orbitals of each Gg appear no more than twice. For that reason
the problem of defining variables x_{a1} and x_{t1u} (or their
equivalents  see below) reduces to diagonalization of the 2×2
Fockian blocks corresponding to the respective irreducible representations:
The exact definition of the matrix elements of the Fockian for an
SCFtreated grup of electrons in the presence of other groups is given in
Refs. [,].
The oneelectron density matrix corresponding to the solution of the
HartreeFock problem in the CLS is as any HartreeFock density matrix an
operator (matrix) P projecting to the occupied MOs:

P = x_{a1}^{2} a_{1}^{0}ñáa_{1}^{0}+y_{a1}^{2} sñás+x_{a1}y_{a1}(  sñáa_{1}^{0}+ a_{1}^{0}ñás )+ 
å
g = c,s

 e_{g}^{0}g
ñ
á e_{g}^{0}g + 

+ 
å
g = x,y,z

[ x_{t1u}^{2} t_{1u}^{0}g
ñ
á t_{1u}^{0}g+y_{t1u}^{2} p_{g}ñáp_{g}+x_{t1u}y_{t1u}(  p_{g}
ñ
át_{1u}^{0}g + t_{1u}^{0}g
ñ
á p_{g} ) ] 



 (0) 
where the quantities x_{G},y_{G} are defined after eq. () and the orbitals with the superscript "0" refer to the
symmetry adapted combinations of the LP HOs c_{g} in the right
hand side of eq. (21). The above expression can be further
simplified by noticing that the normalization condition for the quantities x_{G},y_{G} can be absorbed in a rational function of another
(single) electronic structure variable for each G. Indeed, a
twodimensional operator projecting onto onedimensional subspace has the
form:
P_{G} = 
æ ç
è



ö ÷
ø

= 
1 1+v_{G}^{2}


æ ç
è



ö ÷
ø

. 

The projection operstor eq. (23) is a direct sum of
the 2×2 projectors with the appropriate values of v_{G}
(inparticular v_{eg} = 0) taken in the required numed of instances (one
for each row of the irreducible representation G). The projection
operstor eq. (23) is one for the 12electron
complex. If it goes about a 14electron complex the P_{a1} in the
direct sum has to be replaced by the 2×2 identity matrix thus
reducing the number of ECVs to only one: v_{t1u}.
Insering the ground state projection operator in the HartreeFock expression
for the energy of the CLS electron group we get:


( 2\limfuncSph^{eff}P+\limfuncSpPS[P]) ,provided 
 (0)  

 (0) 

where h^{eff} is the oneelectron part of the Fock operator and S[P] is the selfenergy part representing the electrostatic
field induced by electrons in the CLS group upon each other we arrive to an
explicit expression for the energy in terms of the ESVs v_{G}. This
is the closed expression for the energy required by the DMM methodology (the
molecular geometry enters through the respective dependence of the Fockian
matrix elements). Moreover it is rational function of the ESVs involved.
This expression can be efficiently searched for minimum with respect to the
relevant variables yielding the equilibrium geometry and corresponding
electronic structure. For example the effective charges  key quantities
for the considerations of Section 0.1 appear as
averages of P over the corresponding AOs or HOs.
It is possible however to obtain analytical estimates for the equilibrium
values of ESVs which posess rather interesting properties. The simplest
analytical expression represetning the solution can be written for the
product x_{G}y_{G} which is expressed through the single
parameter z_{G}:
condencing all the necessary information:
x_{G}^{2}y_{G}^{2} = 
1 4


æ ç
è

1 
1 1+z_{G}^{2}

ö ÷
ø


 (0) 
If one is interested in the complex formation then the limit z_{G} << 1 has to be considered. In this case:
x_{G}^{2}y_{G}^{2} » 
1 4

z_{G}^{2} 

The opposite limit z_{G} >> 1 describes the situation close to
the equilibrium. In it the following estimate holds:
x_{G}^{2}y_{G}^{2} » 
1 4


æ ç
è

1 
1 z_{G}^{2}

ö ÷
ø


 (0) 
These results for surely known for decades as far as we know have never been
considered from the point of view of possible transferability of the
offdiagonal density among different molecules. This latter property is
however a key to constructing any mechanistic model of PES as it is shown in
Ref. [].
The situation described by the formula eq. (27)
differs in an important respect from analogous results of Ref. []
proven for isolated twocenter twoelectron bonds characteristic to organic
species. In the örganic" domain the transferability of the offdiagonal
element of the oneelectron density matrix immediately brought up the
transferability of the corresponding Coulson bondorder directly involved in
the expression for the bond energy. The formula eq. (27), however, applies to the density matrix element in
the basis of the symmetry adapted linear combinations of the LP HOs. They
are not related to individual ML bonds which are not even
öbservable" elements of molecular electronic structure in the sense
proposed by Ruedenberg [] (in oppsition with the
twocenter twoelectron bonds in örganics"). By contrast the stable (upto
the second order in the presumably small parameters z_{G}^{1})
values of the oneelectron density matrix elements refer to a completely
different elements of the construction: to the threedimensionally
delocalized CLS group of electrons whose ESVs themselves posess necessary
transferability properties which makes it an öbservable" component of the
molecular electronic structure in the sence of Ref. [].
Pragmatic outcome of this might be in replacing in the vicinity of
equilibrium of the ESVs either by transferable value of v_{G} = 1 (G = a_{1},t_{1u}) or by inserting the estimates eq. (27) and by this arriving to the PES as a function of
the nuclear coordinates only. The described result applies however to the
octahedral complexes only. The major task is to extend this treatment to the
complexes of lower symmetry which will be done in subsequent Sections.
Perturbative analysis of the DMM model of CLS and its relation to
LD theory of ligand influence
Now let us consider what is going to happen to the above DMM picture under
the variation of composition (chemical substitution) and/or geometry both
reducing the symmetry of the CLS. An interplay between these two types of
perturbation was the main concern in the LD theory of ligand influence.
This theory evolves in terms of two key objects: the substitution operator
and the electronvibration (vibronic) interaction operator. These two
perturbations are applied to the matrix representation of the CLS Fockian
written with respect to formally the same set of oneelectron states
(central atom AOs and LP HOs). In this setting the dependence of the Fockian
on the chemical composition of the species involved reduces to the
corresponding dependence of its matrix elements. Analogously the geometry
dependence becomes that of the matrix elements of the Fockian.
DMM on nonsymmetrical coordination compounds
Whatever Fock operator can be represented as a symmetric one and a
perturbation of the latter, which includes both the dependence of the matrix
elements on nuclear shifts from the equilibrium positions and the transition
to a less symmetric environment due to the substitution. To pursue this way
employing the above ESVs we introduce first some notations. Let h^{¢}
be the supervector of the first derivatives of the matrix of the Fock
operator with respect to nuclear shifts dq counted from a
symmetrical equilibrium configuration. By a supervector we understand here a
vector whose components numbered by the specific nuclear shifts are
themseves 10×10 matrices of the first derivatives of the Fock
operator with respect to the latter. Then the scalar product of the vector
of all nuclear shifts  dq) and of the supervector h^{¢} yields a 10×10 matrix of the corrections to the Fockian
linear in the nuclear shifts:
( h^{¢}  dq) = 
å
i


¶h ¶q_{i}

dq_{i}. 
 (0) 
Next, let h^{¢¢} be the supermatrix of the second derivatives
of the matrix of the Fock operator with respect to the same shifts. As
previously we refer here to supermatrix indexed by the pairs of nuclear
shifts in order to stress that the elements of this matrix are themselves
the 10×10 matrices of the corresponding second derivatives of the
Fockian with respect to the shifts. The contribution of the second order in
the nuclear shifts can be given the form of the (super)matrix average over
the vector of the nuclear shifts:
( dq h^{¢¢} dq) = 
å
ij


¶^{2}h ¶q_{i}¶q_{j}

dq_{i}dq_{j}. 
 (0) 
Supplying this all with the 10×10 matrix of the substitution
operator
h^{S} = F_{S}^{CLS} = F_{MLnXYZ...}^{CLS}F_{ML6}^{CLS} 
 (0) 
we get the "bare" perturbation of the effective Fockian in the CLS carrier
space as:
( h^{¢}  dq) + 
1 2

( dqh^{¢¢} dq) +h^{S}. 
 (0) 
This does not form the entire ("dressed") perturbation since in case the
electron density changes to the first order in the above perturbation the
Fockian acquires additional perturbation through the variation of its
selfenergy part which leads to the selfconsistent perturbation. Thus the
perturbed Fockian can be written as:
F = F_{0}[P_{0}]+( h^{¢}  dq) + 
1 2

(dq h^{¢¢} dq)+h^{S}+S[DP] 
 (0) 
Here DP stands for the correction to the unperturbed projection
operator P_{0} to the occupied MOs which in case to the octahedral
complexes equals to P given by eq. (23). This
serves as a prerequisite for performing two remaining steps of the recipe of
Section 1 of constructing a DMM description of coordination
compounds of arbitrary (low) symmetry and of the linear response theory
based on it and leading to strictly mechanistic description of this class of
compounds.
To proceed further we look what is the perturbed density matrix. It was
assumed to have the form

 
P_{0}+DP = P_{0}+ 
å
n > 0

P^{(n)} 



 (0) 
where the correction DP can be expanded in terms of the matrices V
satisfying the conditions:

P_{0}V = 0;VP_{0} = V;(1P_{0})VP_{0} = V; 

P_{0}V^{+} = V^{+};P_{0}V^{+}(1P_{0}) = V^{+};V^{+}P_{0} = 0 




as follows []:


 

 

 


 
V^{+}VV^{+}VVV^{+}VV^{+}, 





which can be continued. The matrices V are 4×6 matrices for
12electron complexes and 3×7 matrices for 14electron complexes
which organize into a single entity independent ESVs of the problem  the
first order transition densities between the occupied and empty MOs of the
unperturbed problem. One can check that only the even terms of the above
expansion contribute to the effective charges residing on the atoms (orbital
populations) of the CLS.
Inserting the expansion eq. (33) rewritten in terms of
matrices V in the energy expression eq. (24) with the
perturbed Fockian eq. (32) yields a DMM model of the
coordination compound of an arbitrary symmetry since the transition
densities V take account of all possible perturbation of electronic
structure keeping the CLS a separate entity. The series eq. (33) in fact appears by expanding the closed expression for
the pojection operator:
P = (P_{0}+V)(1+V^{+}V)^{1}(P_{0}+V^{+}), 

which involves the inversion of a 10×10 matrix and nowadays is not a
great computational problem. On the other hand it is possible to restrict
oneself with certain power in the expansion eq. (33)
getting to polynomial model of electronic structure of required accuracy.
It is easy to analyse the above model keeping the terms of the total order
not higher than two in dq and V simultaneously and taking into
account that under the spur sign the argument of the selfenergy part S of the Fockian can be interchanged with the matrix multiplier McWeenyBook. Using these moves we arrive to:

E_{CLS} = 
2\limfuncSp[h_{0}P_{0}]+\limfuncSp[P_{0}S(P_{0})] = E_{0}

+2\limfuncSp[F_{0}( V+V^{+}) ]+ 

+2\limfuncSp[( h^{¢}  dq) P_{0}]+ 

+2\limfuncSp[( h^{¢}  dq) ( V+V^{+})]+\limfuncSp[( V+V^{+}) S(V+V^{+})]+ 

+\limfuncSp[( dq h^{¢¢} dq) P_{0}]+2\limfuncSp[F_{0}( VV^{+}V^{+}V) ]. 



 (0) 
At the equilibrium the terms linear in dq and V+V^{+} vanish so
that the electronic energy becomes:


+2\limfuncSp[( h^{¢}  dq) ( V+V^{+})]+\limfuncSp[( V+V^{+}) S(V+V^{+})]+ 

+\limfuncSp[( dq h^{¢¢} dq) P_{0}]+2\limfuncSp[F_{0}( VV^{+}V^{+}V) ], 



 (0) 
which is a quadratic form with respect to the nuclear shifts and the ESVs V. The average of the second derivatives of the oneelectron part of the Fock
operator with the operator P_{0} projecting to the occupied MOs of the
unperturbed system:
( dq 2\limfuncSp[h^{¢¢}P_{0}]dq) = ( dq D_{0} dq) 

is nothing, but the bare harmonic potential of the symmetric complex with
the square dynamic matrix D_{0} acting on the nuclear shifts. Analogously
the second order energy corrections with respect to V  the variation of
ESVs describing oneelectron density matrix:
2\limfuncSp[F_{0}( VV^{+}V^{+}V) ]+\limfuncSp[(V+V^{+}) S(V+V^{+})] = 
1 2

á
áV L  V
ñ
ñ 
 (0) 
turns out to be the quadratic form giving the electronic energy as a
function of the variation of the one electron density matrix. The quantity L can be considered as a superoperator (supermatrix) acting in the
space of the 10×10 matrices taken as elements of a linear space (the
Liouville space). The supermatrix L has four indices running
through oneelectron states in the carrier space of the CLS group. Then the
formula
á
á A  B
ñ
ñ = \limfuncSp( A^{+}B) 

defines a scalar product in the Liouville space which ultimately allows the
notation used in eq. (36). Next move consists in
forming a direct sum of the the Liouville space of the of matrices V which
can be expanded over the basis formed by the matrix unities bñáa with a and b running over all basis
states of the CLS carries space and of the space spanned by the nuclear
shifts. Extending the definion of the scalar product to this new space
allows to rewrite the spurs in eqs. (34), (35) as scalar products in this vector
space. Then the two types of perturbations introduced above couple by the
bilinear term:
2\limfuncSp[( h^{¢}  dq) ( V+V^{+})] =
á
á V  h^{¢}dq) +( dq h^{¢} V
ñ
ñ . 
 (0) 
This is nothing but the electronvibration interaction in the chosen
notation. We remind that the quantities h^{¢} and h^{¢} are
the three index supermatrices; they act, respectively, to the right on the
vector of nuclear shifts producing a 10×10 matrix next forming a
Liouville scalar product with matrix V, and on the variations V of the
density matrix, producing a vector to be convoluted with that of nuclear
shifts dq. With use of this set of variables the energy in the
vicinity of the symmetric equilibrium point becomes:
E_{CLS} = E_{0}+ 
1 2




ê ê
ê



ê ê
ê



, 
 (0) 
which is a quadratic form with respect to both the nuclear shifts and the
ESVs. The substitution operator gives additional terms which also can be
recast into the form of the scalar products in the Liouville space:


 (0) 
2\limfuncSp[h^{S}( V+V^{+}) ] 


á
áV  w
ñ
ñ +
á
á w  V
ñ
ñ \notag 
 (0) 

With this notation the energy of the CLS becomes:


E_{0}+
á
á V  w
ñ
ñ +
á
á w  V
ñ
ñ + 
 (0)  

 (0) 

This can be treated as the minimal order of the DMM picture for the PES of
the coordination compounds of nontransitional elements. It perfectly
condences all the necessary elements of the LD theory of the ligand
influence and of the theory of vibronic interactions. The specificity of the
"class" of compounds is fixed by the presence of the CLS group. The
specificity of a ßubclass" within this class is controlled by the number of
electrons in the CLS which defined the specific form of the quantities P_{0} and L . Both the geometry and the electronic structure of
the substituted or/and deformed complex can be obtained (in the
harmonic approximation) by taking
derivatives of the above expresion with respect to dq and V and
setting these former equal to zero. Doing that we see that the fixed
deformation  dq) and the substitution w result in
the modification of the electronic structure as compared to the symmetric
undeformed complex. The amount of the modification bringing the system back
to the new equilibrium is given by the formula:
 V
ñ
ñ = L ^{1} h^{¢} dq) +w
ñ
ñ . 
 (0) 
It is remarkable that the supermatrix L ^{1} is nothing PupyshevPrivate but the polarization propagator P for the CLS
subsystem calculated for the symmetric molecule. With this we get:
V = P h^{¢} dq)+w
ñ
ñ . 
 (0) 
This performs the announced program of obtaining a closed expression for the
energy of the coordination compound (or at least of its CLS) in terms of its
geometry and electronic structure variables.
PES of coordination compound as derived from DMM
Now we can turn to deriving a true mechanistic (MMlike) model for
coordination compounds of nontransition element by excluding the ESVs V.
Inserting eq. (44) in eq. (41) we get for the energy:



1 2

( ( dq D dq)+( dq
á
á h^{¢}  P h^{¢}
ñ
ñ dq)+
á
á w  P w
ñ
ñ + 
 (0)  

(dq
á
á h^{¢} P w
ñ
ñ +
á
á w  P h^{¢}
ñ
ñ dq) . \notag 
 (0) 

This expression in a condenced form contains all the results which is in
details are obtained in [], namely the theory of ligand
influence which can be considered as a responce of molecular geometry to the
chemical substitution. For example, optimizing the above expression with
respect to  dq) yields the response of the complex
geometry to the substitution of the ligands. One easily gets the close
expression for it:
 dq) = D^{1}
á
áh^{¢}  P w
ñ
ñ ) . 

Of course, within such a formulation the effect of the substitution does not
reduce to modification of the nonbonding potentials felt by the ligands
(Section 0.1). By contrast the substitution
affect the very substance of what is going on. Different ligands are
characterized by their specific contributions to the Fockian for the CLS
group. In the simplest approximation adopted in Ref. []
the ligand is characterized by its diagonal Foclian matrix element which is
a true parameter of the model. The semiempirical SLG theory as applied to
isolated ligands allows to estimate these quantities related to the LPs and
even provides formulae describing their dependence on the deformations of
the örganic" bonds incident to the donor atom. However, it is important to
mention already now that replacing one ligand by another in a coordination
compound (local perturbation) produces a nonlocal effect in that sense that
it does not necessary decrease with the distance from the perturbation
location (as it will be formally described below).
The MMlike model of complexes of nontransition elements requires even less
than it is given by eq. (45): only the first and the second
term in the first row. They represent the bare harmonic dependence of the
energy on the nuclear shifts and the renormalizations of the respective
harmonic constants due to adjustment of the electronic structure to these
shifts:
D = D_{0}+
á
á h^{¢}  P h^{¢}
ñ
ñ . 

As we mentioned previously the specifics of the central atoms in
coordination compounds is determined by the structure of the supermatrix P, which is in its turn predefined by the structure of the carrier space
of the CLS group and by the number of electrons in it. Indeed, the
supermatrix P of the polarization propagator is particularly simple in
the basis of the eigenstates of the Fock operator F_{0}. Its matrix
elements are:
P_{ii¢jj¢} = 
d_{ii¢}d_{jj¢} e_{i}e_{j}



where the subscripts ii^{¢} run over all occupied MOs and the
subscripts jj^{¢} run over the vacant ones. In this basis the
elements V_{ji} of the matrix V and of its conjugate by definition
represent the transition densities between the ith occupied and the jth empty MO. They are numerical coefficients at the matrix unities  j
ñ
á i being the basis
vectors of the Liouville space. In terms of the Liouville space the
superoperator P can be written:
P = 
å
\substack i Î occ j Î vac


 i® j
ñ
ñ
á
á i® j  e_{i}e_{j}



( i® j
ñ
ñ is the
Liouville space notation for the matrix unity  j
ñ
á i ) which allows the straitforward use of the
scalar product formulae with notion that:
á
á i® ji^{¢}® j^{¢}
ñ
ñ =
á ii^{¢}
ñ
á jj^{¢}
ñ = d_{ii¢}d_{jj¢}. 

The simplest approximate description of P corresponds to what is known
as the frontier orbitals approximation where only the highest occupied and
lowest unoccupied MOs (HOMO and LUMO, respectively) are involved. Within it
one gets:
P_{hh¢ll¢} = d_{hh¢}d_{ll¢}( e_{H}e_{L}) ^{1} 

where subscripts hh^{¢} run over the orbitals in the HOMO manifold
(they may be degenerate in the highly symmetric case) and ll^{¢} do
the same in the possibly degenerate LUMO manifold.
The given formulae contain all necessary results, but cannot be easily
qualitatively interpreted. The necessary interpretation had been done by
Levin and Dyachkov and is based on clarifying the interplay of the effects
produced by substitution and vibronic operators upon the solution of the Hückellike problem in the 10dimensional orbital carrier space with use
of symmetry considerations. This will be done in the next Section.
Symmetry adapted formulation
For the purposes of the present paper the symmetry analysis of Ref. LevinDyachkov can be reformulated as follows. The deformation of the
molecule of a coordination compound  dq) is a
vector with the components referring to the individual nuclear shifts:
For a symmetric (say, octahedral) molecule it may be rewritten with use of
the symmetry adapted nuclear shifts:
 dq) = 
Å
Gg

dq^{Gg} Gg) 

where G and g refer respectively to the irreducible
representation of the symmetry group and its row (in the case of a
degenerate irreducible representation). In an octahedral complex if only the
shifts leading to the ML (MX) bond lengths
variation are concerned the symmetry classification suffice to label all
possible collective shifts which can be either of a_{1g}, e_{g}, or t_{1u} symmetry. They can be explicitly written through the nuclear shifts
of the individual ligands according to:



1 Ö6

[ x_{Lx})  x_{Lx}) + y_{Ly}) y_{Ly}) + z_{Lz}) z_{Lz}) ] , 
 


1 2

[  x_{Lx}) x_{Lx})  y_{Ly}) +y_{Ly}) ] , 
 


1 2Ö3

[ 2z_{Lz}) 2 z_{Lz})  x_{Lx})+ x_{Lx})  y_{Ly}) +y_{Ly}) ] , 
 


1 Ö2

[ g_{Lg}) + g_{Lg}) ] . 


The meaning of notation for the individual nuclear shifts is that  g_{Lg}) represents a unit shift in the
positive direction along the g axis of the ligand located at the ±g semiaxis of the coordinate frame.
A remarkable feature is that the derivative of oneelectron part of the
Fockian with respect to the symmetry adapted nuclear shift dq^{Gg} (an operator acting on the oneelectron states in the CLS carrier
space) itself transforms according to the irreducible representation G and its row g. That means that applying the deformation  Gg) to a complex results in a perturbation of
the Fock operator having the same symmetry Gg. This allows to
write the vibronic operator in a symmetry adapted form:
( h^{¢}  dq) = 
å
Gg

dq^{Gg}( h_{Gg}^{¢}  Gg) 

Finally, the substitution operator can be expanded as a sum of symmetry
adapted components. For example, in the octahedral complex single
substitution ML_{6}® ML_{5}X defined
in Section 0.1 results in the substitution
operator:

 

1 Ö6

h_{a1g}^{S}+ 
1 Ö3

h_{egc}^{S}+ 
1 Ö2

h_{t1uz}^{S} 




As we see for the symmetric system all the elements of the present picture
are classified according to irreducible representations of the relevant
symmetry group  O_{h}. For example, the energies defining the
polarization propagator depend on G_{H} and G_{L}, but not on the rows g_{H} and g_{L} of the involved irreducible representations. Using the symmetry
notation for the polarization propagator allows simply realize its rôle
as a selection mechanism for interaction of different perturbations. As we
mentioned, in the frontier orbitals approximation the only energy parameter
is the energy gap e_{H}e_{L}. The
polarization propagator thus acquires the form
P = ( e_{H}e_{L})^{1} 
å
g_{H},g_{L}

 g_{H}® g_{L}
ñ
ñ
á
á g_{H}®g_{L}  

It is obvious that the superoperator P acts as a projection operator in
the Liouville space cutting out those components of the 10×10
transition density matrices which mix g_{H} state with the g_{L} state, which is only possible if the symmetries of the
perturbations both the symmetry of deformation G_{def} and
the symmetry of substitution G_{S} satisfy the selection
rule:
G_{def},G_{S} Ì G_{H}ÄG_{L} 

i.e. both enter in the expansion of the tensor product of the irreducible
representations of the frontier orbitals.
Offdiagonal elastic constants for stretchings of bonds incident
to the central atom
Upto this point our main concern was to reformulate the results of the LD
ligand influence theory in the DMM form. Its main content was the symmetry
based analysis of the possible interplay between two types of perturbation:
substitution and deformation, which is controlled by the selectrion rules
incorporated in the polarization propagator of the CLS. The mechanism of
this interplay can be simply formulated as follows: substitution produces
perturbations of different symmetries which are supposed to induce
transition densities of the same symmetries. In the frontier orbital
approximation only those densities among all possible ones can actually
appear which have the symmetry which enters into decomposition of the tensor
product G_{H}ÄG_{L} to the
irreducible representations. These survived transition densities then induce
the geometry deformations of the same symmetry as those former.
The deformation (nuclear shifts) may play the same rôle as the
substitution. Inducing a deformation of some symmetry leads to appearence of
the transition densities of the corresponding symmetry. The same selection
rule as that for the substitution makes only the symmetry component entering
into decomposition of the tensor product G_{H}ÄG_{L} to survive and to induce the deformation of the same
symmetry. For example: the zshift of the apical ligand expands as:
 z_{Lz}) = 
é ê
ë


1 Ö6

a_{1g}) + 
1 Ö3

 e_{g}c) + 
1 Ö2

 t_{1u}z) 
ù ú
û



Thus it may produce the transitional densities of the a_{1g}, e_{g}c,
and t_{1u}z symmetries. At this point selection rules pertinent to the
frontier orbitals approximation enter: for the 12electron complexes the
symmetries of the frontier orbitals are G_{H} = e_{g} and G_{L} = a_{1g}, the tensor product G_{H}ÄG_{L} = e_{g}Äa_{1g} = e_{g} contains only the
irreducible representation e_{g} so that the selection rules allow only
the density component of the e_{g}c symmetry to appear. In its turn this
density induces additional deformation of the same symmetry. That means that
in the frontier orbitals approximation only the elastic constant for the
vibration modes of the symmetry e_{g} is renormalized. This result worths
to be undertood in terms of individual nuclear shifts of the ligands in the
trans and cispositions relative to the apical one. They, respectively, are:


 
é ê
ë


1 Ö6

a_{1g}) + 
1 Ö3

 e_{g}c)  
1 Ö2

 t_{1u}z) 
ù ú
û


 


é ê
ë


1 Ö6

a_{1g})  
1 2Ö3

 e_{g}c) + 
1 2

 e_{g}s)  
1 Ö2

 t_{1u}x) 
ù ú
û




Combining this all we obtain for the offdiagonal constant coupling the
indifidual shifts of the ligands in the transpositions to each other as:

1 3

( e_{g}c
á
áh_{egc}^{¢}  P h_{egc}^{¢}
ñ
ñ  e_{g}c) 

and for the offdiagonal constant coupling the indifidual shifts of the
ligands in the cispositions to each other we get
 
1 6

( e_{g}c
á
áh_{egc}^{¢}  P h_{egc}^{¢}
ñ
ñ  e_{g}c) . 

By contrast for the 14electron complexes (nontransition nonmetals) the
symmetries of the frontier orbitals are: G_{H} = a_{1g} and G_{L} = t_{1u} and the tensor product G_{H}ÄG_{L} = a_{1g}Ät_{1u} = t_{1u} so that only the
transition density corresponding to the representation t_{1u} survive.
Analogous moves allow to conclude that the offdiagonal elastic constant for
stretching the transbonds has the form:
 
1 2

( t_{1u}z
á
áh_{t1uz}^{¢}  P h_{t1uz}^{¢}
ñ
ñ  t_{1u}z) , 

whereas that for the cisbonds vanishes.
This allows to make some predictions concerning the offdiagonal elastic
constants depending on the electron count in their CLS. Due to different
symmetry properties of the polarization propagator in these two cases (and
according to the LD picture which ultimately explains the qualitative
difference in the stereochemistry of the 12 and 14electron complexes) the
offdiagonal constant coupling the shifts of the ligands in the trans and
cispositions to each other in the 12electron case is expected to have
different sign. The sign of the offdiagonal coupling of the
transpositioned ligands in the 14electron case is expected to be the same
as that for the cispositioned ligands in the 12electron case, whereas the
coupling of the shifts of the cispositioned ligands in the 14electron case
is expected to be small.
Medium range offdiagonal elastic constants
In the above Subsection we obtained some esitmates for the offdiagonal
harmonic terms coupling the stretchings of different ML bonds
incident to the central atom. The employed treatment can be extended to
other types of offdiagonal terms. They originate as well from the h^{¢}Ph^{¢} term in the general energy expression. The
traditional MM picture tends to avoid the appearance such offdiagonal terms
and tries to represent the energy as a sum of force fields attributed to
local elements of molecular structure like bonds, etc. This implies
the strictly local character of the underlying electronic structure. It is
easy to understand also from a pragmatic point of view since including
longrange typespecific terms in addition to those already introduced makes
the entire parameterization too complicated. On the other hand in case when
the electronic structure is physically formed by not that local elements as
twocenter bonds this must be reflected in the corresponding force fields.
Incidentally, the coordination compounds posess such delocalized structure
elements  the CLS  where oneelectron states are extended over all atoms
forming it. In such a situation one has to expect some medium range
offdiagonal harmonic couplings i.e. specific effective coupling
between the deformations occurring at the separations usually not included
in the MMlike consideration. With use of the developed technique it is
possible to get estimates of such
offdiagonal elements of the harmonic molecular potential
with participance of the metal atom the very existence of which in the PES
expansion is difficult to imagine if not only to stick to an informationless
idea that all must be included . As an
illustrative example we consider a twocoordinated linear complex. The
chemical examples are provided by the compounds of Cu^{+}, Ag^{+}, or Hg^{2+}. In the context of the standard MM analysis it is assumed that the
interactions between the atoms separated by more than three bonds are not
specific and must be taken into account as nonbonded
fields with use of the LennardJones potentials.
Meanwhile using the technique presented above it can be easily shown that in
the case of the above metal complexes there are specific interactions of
noticeable magnitude which according to standard scheme must be classified
as the 15 interactions (those between the atoms separated by four bonds).
Let us consider a (metal) ion bearing as previousy four vacant (one s and
three p) orbitals. As previously we assume that ligand molecules are
represented by one LP each. In the case of linear coordination (zaxis is
the molecular axis) and assuming that in the equilibrium state the LPs are
directed along the bonds between the donor atoms and the metal atom the
symmetry adapted combinations of the LPs have the form:
 a_{±}^{(0)}
ñ = 
1 Ö2

( u
ñ ± l
ñ ) 

According to [] the LP HOs  u
ñ and  l
ñ (upper and lower with respect to the zaxis) are composed of s and porbitals of the donor atom which are directed
along the unit vectors [e\vec]_{u} and [e\vec]_{l}:


s s_{u}
ñ +  Ö

1s^{2}

 p_{[e\vec]u}
ñ 
 (0)  

s s_{l}
ñ +  Ö

1s^{2}

 p_{[e\vec]l}
ñ \notag 
 (0) 

(with the obvious sence of s as of a coefficient of the corresponding sorbital in the expansion of the corresponding HO). With use of these
definitions and of the symmetry considerations it is easy to identify
nonvanishing matrix elements of the Fock operator acting in the CLS:

á s h a_{+}^{( 0)}
ñ = Ö2 
æ è

b_{ss}^{DM}s+b_{zs}^{DM}  Ö

1s^{2}

ö ø

¹ 0, 

á z h a_{}^{( 0)}
ñ = Ö2 
æ è

b_{sz}^{DM}s+b_{zz}^{DM}  Ö

1s^{2}

ö ø

¹ 0, 




where b_{ss}^{DM}, b_{zs}^{DM},b_{sz}^{DM}, and b_{zz}^{DM} are the resonance
(oneelectron hopping) integrals in the diatomic coordinate frame for the
pairs metaldonor atom and where we denote by s and z
respectively the s and pstates of the metal and donor atoms, having
the s symmetry with respect to the molecular axis (linear
coordination).
The nontrivial oneelectron eigenstates of the effective Fock operator for
this CLS have the form:


 a_{+}
ñ = y_{+}s
ñ +x_{+} a_{+}^{( 0) }
ñ, a_{}
ñ = y_{} z
ñ+x_{} a_{}^{( 0) }
ñ 
 (0)  

 a_{+}^{*}
ñ = x_{+} s
ñ +y_{+} a_{+}^{(0) }
ñ , a_{}^{*}
ñ = x_{} z
ñ +y_{} a_{}^{(0) }
ñ . \notag 
 (0) 

Two more states of the psymmetry ( x
ñ and  u
ñ ) on the metal ion remain unchanged as in
the free metal ion and both are empty. The frontier orbitals here are the  a_{}
ñ (HOMO) and those in the pmanyfold ( x
ñ and  u
ñ 
LUMO).
Now let us assume that the LPs belong to polyatomic ligands. Then a valence
angle MDX with a vertex at a donor atom D is one of the geometry variables
of the molecules in the standard MM setiing. We shall estimate the
magnitude of the indirect (CLS mediated) interactions between variations of
these valence angles. Further consideration evolves as follows. We assume
the LPs to be rigidly attached to the ligands. Then changing the valence
angle MDX by dc_{u} (dc_{l}) yields the
corresponding nonvanishing angle between the vector [e\vec]_{u} ([e\vec]_{l}) and molecular axix. It respectively turns on the resonance
interaction between this LP and the  x
ñ state of
the metal atom (we assume that either of the ligand LPs and the metal atom
itself stay in the (xz) plane). The corresponding matrix elements
are:

á x h u
ñ = b_{pp}^{DM}  Ö

1s^{2}

sindc_{u} 

á x h l
ñ = b_{pp}^{DM}  Ö

1s^{2}

sindc_{l} 




where b_{pp}^{DM} is the resonance (oneelectron hopping)
parameter for the pair of states of the metal and donor atoms which have psymmetry with respect to molecular axis. The derivatives of these
matrix elements (and of the Fockian itself) with respect to dc_{u} and dc_{l} are:


á

x 
ê ê
ê


¶h ¶c_{u}

ê ê
ê

u 
ñ

ê ê
ê

dc_{r} = 0

= 
á

x 
ê ê
ê


¶h ¶c_{l}

ê ê
ê

l 
ñ

ê ê
ê

dc_{l} = 0

= b_{pp}^{DM}  Ö

1s^{2}




á

x 
ê ê
ê


¶h ¶c_{l}

ê ê
ê

u 
ñ

= 
á

x 
ê ê
ê


¶h ¶c_{u}

ê ê
ê

l 
ñ

= 0 




The deformation coordinates  dc_{u}) and  dc_{l}) apparently transform according to the xth row of the representation p and can be further combined into
the symmetric and antisymmetric adapted coordinates with respect to the
plane perpendicular to the molecular axis:

 dc_{+}) = 
1 2

(  dc_{u}) + dc_{l}) ) 

 dc_{}) = 
1 2

(  dc_{u})  dc_{l}) ) 




The individual deformation coordinates recover from the relations:

 dc_{u}) = (  dc_{+}) + dc_{}) ) 

 dc_{l}) = (  dc_{+})  dc_{}) ) 




Assembling the relevant terms (those producing the antisymmetric xtransition densities in the CLS) we get for the offdiagonal interaction of
two valence angles the following expression:


K µ 
b_{pp}^{2}x_{}^{2}(1s^{2}) 4( e_{p}e_{L})






whose numerical value can be estimated as follows: for the sp^{3} of the
donor atoms s^{2} = [1/4], the weight x_{}^{2} of the
antisymmetric combination of the ligand LP states in the corresponding HOMO
can be safely estimated as [2/3] so that with the
energy gap ( e_{p}e_{L}) of about 5 eV and
the same value of b_{pp} we arrive to the estimate for K of
0.7 eV/rad^{2} which can be treated as if not a large, but noticeable
specific contribution of the 15 type.
Discussion
It is a widespread point of view in the MM community that the latter
represents a "practical" alternative to standard quantum chamical treatments
of molecular structure. On this basis the quantum mechanical models are
taken as excessively complex and superfluous as compared to the problems to
be solved. The problem, however, is that in the absence of such models it is
difficult to estimate to what extent each specific problem possibly fits to
some may be adjusted MM scheme or by contrast requires some essentially
quantum mechanical approach to be solved. On the other hand just practical
needs stipulate the interest in developing some MMlike models for wider
classes of moleculae as compared to örganic" ones for which the standard MM
treatment is by many examples proven to be valid. The key point is that in
fact behind any "classical" MM picture there is always covered a fairly
quantum view of molecular electronic structure. As it was shown previously
[] it is possible to imagine and to successfully construct more
general mechanistic models of molecular potentials (PES) than usually
accepted ballsandsprings models of
the standard MM. The derivation in Ref. [] is based on the
concept of electron group dating back to McWeeny [] and on
the ßemiobservability" of these groups introduced by Ruedenberg Ruedenberg1962. In these terms one can state that classical MM of organic
molecules implies that twoelectron groups desribing bonds are
ßemiobservable" i.e. well defined stable groups spanning the
molecular electronic structure. Then the moves described in Section 1 result in a fairly mechanistic picture of interacting atomic
tetrahedra representing the sets of orthogonal HOs which can be further to
the standard MM with the externally i.e. independently defined force field
parameters. The problems faced when extending any MMlike description to
other classs of molecules is the lack of understanding of the pertinent
electronic group structure of the wavefunctions characteristic for the new
classes of compounds to be included in the MM domain. At a first glance all
the characteristics of the coordination bonds: mutual influence of the
ligands, charge redistribution, dependence of magnetic properties on tiny
details of molecular geometry and composition (these latter not addressed in
the present paper)  all have too much of quantum origin so that no
mechanistic model of these properties is possible. This point of view seems
to be however an opposite extreme. Finally the MM is quite a flexible tool,
not limiting anyhow either the complexity of the force fields to be used or
other characteristics of the model. In the present paper we employed the
representation of the electronic structure of coordination compounds in the
form of the group function product recently formalized in Ref. []
and developed a mechanistic picture of their PES involving some necessary
elements of the electronic structure description through the ESVs v_{G} and V. This approach can be qualified as a deductive molecular
mechanics (DMM) of the CLS group of electrons specific for the octahedral
environment. For other types of coordination analogous picture can be
developed which may be useful provided the electronic structure of the
molecule at hand can be described with use of the corresponding CLS group.
Then using the perturbation theory the EVSs have been excluded from the
consideration thus yielding the estimates for the parameters of the force
fields of more traditional form.
The models thus built remain mechanistic ones, but they naturally take into
account those important features of the electronic structure, which in a
standard formulation would require innumerable parameterizations for more
and more tricky force fields whoce form any way remains without any
fundamental basis. For example, thus obtained offdiagonal elastic constants
do not assume the angular dependent form like
proposed in Ref. [] (q stands for the valence angle
between the bonds incident to the central ion), but suggest an existence of
some more or less stable ratio between the constants describing coupling of
the cis and transpositioned ligands. Also the obtained estimates allow to
relate the sign and other characteristics of these offdiagonal constants
with the chemical nature of the central atom, which is a complex problem for
classical MM itself.
The performed analysis shows the weakness of all tentative attempts to
include metals into "classical" MM. Within the classically looking picture
possible influence effects are attributed to charge redistributions among
other possibilities. In fact the charge variations are the quantities of the
second order in the ESVs V, whereas the energy in the DMM picture depends
already on the first power of V. This affects the entire structure of the
theory. where the polarization propagator supermatrix becomes the key player
defining the generalized elastic properties of molecular electronic
structure expressed in terms of the ESVs V in the harmonic approximation.
Of course this treatment is parallel to the random phase approximation (see
e.g. []). It is also fair to say that polarization
propagators were in use when analyzing the substitution effects in the
coordination compounds at a pretty early stage of these studies (see Refs.
[,,]). However, in these papers the
polarization propagator was used within the reactivity indices paradigm:
i.e. in order to estimate some elements of the density matrices considered
as ïndices of influence" rather the molecular energy itself. The general
vibronic approach of Ref. [] adopted in Ref. LevinDyachkov stressed the possibility of explicit expression for the PES
of substituted compounds, but did not underline the importance of the
polarization propagator. This is done in the present paper.
Conclusion
In the present paper we developed a strcuture of possible molecular
mechanics of coordination compounds analysing the electronic structure of
the CLS of this class of molecules. The obtained expressions can be used
either as a standalone theory of the DMM style or as a source of independent
estimates of the relevant force fields in classical MM of coordination
compounds.
Acknowledgments
The author is thankful to Dr. I.V. Pletnev who some time ago has drawn the
author's attention to the problem of mechanistic description of coordination
compounds and to Prof. A.A. Levin for valuable discussions. This work is
supported by the RFBR grants Nos 040332146, 040332206, 050790067, and
070301128.
References
 []
 R.D. Hancock. Prog. Inorg. Chem., 37 (1989) 187.
 []
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 []
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systems: bonding, shapes and isomer stabilities. Taylor and Francis. NY,
London. 2002. 2003.
 []
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106 (2006) 571587.
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326376.
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 V.I. Pupyshev, Private Communication.
 []
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Mechanics, 2nd Edition, AP, London, 1992.
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Russian].
The adequate form of the ESVs also was suggested: these are elements of the
matrices V eq. (). The zero approximate projection operator
P_{0} to the filled subspace for the octahedral arrangement is
constructed. Thus there is only a little left to do: to construct a DMM
description with use of the above mentioned ESVs and of the linear response
theory and then eventually to exclude these ESVs from the consideration 
getting
where x and y are the expansion coefficients on the occupied orbital
normalized to unity: x^{2}+y^{2} = 1. Then identifying eqs.(), () we get:
In terms of the matrix elements of the Hamiltonian eq. (22) the
exact solution is:
v = 
1 2z


æ è

 Ö

1+4z^{2}

1 
ö ø

. 

Explicit solution
In order to apply the above formulae to the ligandmetal interaction the b
parameter of eq. (22) must be identified with the matrix element of
the effective Fockian b_{Gm} between the symmetry adapted
orbital  Gg
ñ and the metal AO m
ñ . The c and a parameters are the corresponding diagonal
matrix elements of the Fockian. The formula for the z corresponding
to the considered symmetry is then:
z_{Gm} = 
F_{Gm} e_{m}e_{L}


 (0) 

where the subscript G stands for the symmetry type of the linear
combination of the LP orbitals and m is a metal AO index. At this point we
neglect the interaction (splitting) of the LP orbitals due to direct
oneelectron hopping between them and for this reason the energy of their
linear combination coincide with the diagonal matrix element of the Fockian
over a single LP orbital:
e_{L} =
á c_{g} F_{CLS}^{eff} c_{g}
ñ 
 (0) 

and does not depend on the symmetry of the overlap (a_{1} or t_{1u}).
Offdiagonal matrix elements of the symmetry adapted Fockian are expressed
in terms of the bare resonance integral and its renormalization due to
exchange term coming from the HartreeFock approximation. For the octahedral
coordination:

F_{a1s} = Ö6(b_{sL}+g_{sL}dP_{sL}) 

F_{t1ugpg} = Ö2(b_{pgL}+g_{pgL}dP_{pgL}) 



 (0) 

For each symmetry type of the MO's can be formulated as a 2×2
problem similar to that we considered above. The parameters x_{G}
eq.(21) for each symmetry type are expressed through l_{G} (G = a_{1}, t_{1u}):
x_{G}y_{G} = 
l_{G} 1+l_{G}^{2}



The diagonal matrix elements of the 2×2 symmetry blocks of the
symmetry adapted Fockians depend on metal and ligand lone pairs occupancies
(see eq.()) so obtaining operators P_{a1} and P_{t1u} from eqs. ()  () has to be considered as a
step in the selfconsistent procedure leading to the oneelectron density
matrix characterizing the symmetrized closed ligand shell.
Then we have to apply the symmetry adapted perturbation technique to two
perturbations simultaneously  the the vibronic and substitution operators.
Taking into account that


6y 
rR_{L}  rR_{L} ^{2}

and 
 (0) 
Ñ 
( rR_{L},dR_{XL})  rR_{L} ^{2}





dR_{XL}  rR_{L} ^{2}

 
2( rR_{L},dR_{XL}) ( rR_{L})  rR_{L} ^{4}

\notag 
 (0) 

and
carrier space spanned by of the those of the LP
Our calculations on cyclic chelating ligands have been performed at more or
less arbitrary conformation of the molecules at hand (NH_{3}, Me_{3}N, Et_{3}N, MeEtNH, 18ane(N)_{6}). We found [] that the dispersion of the
values of all the variables describing the electronic structure (electronic
structure variables  ESVs) related to donor atoms entering the cyclic
chelating ligands is always smaller than the dispersion of the same values
in a series of differently substituted ethers or amines ranging from water
or ammonia to the corresponding alkyl di or trisubstitutes, respectively.
Thus the SLG form (together with its semiempirical implementation) seems to
be a relevant approximation for treating free chelating agents like crown
ethers or cyclic polyamines which allows to define the carrer space for the
CLS group.
Now we turn to determinig the set of ESVs to be used in the course of
constructing the DMM picture of the PES of coordination compounds. To do so
we notice that the theory [] evolves in terms of
oneelectron wave functions: eigenstates of the CLS effective Fock operator.
We notice that for the symmetric (octahedral) arrangement of the equivalent
ligands around the central ion the involved quantities z_{a1} and z_{t1u} are characteristic functions of interatomic separation
(more generally  molecular geometry) and of the chemical composition and
hybridizations of the LPs included in the CLS.
This expression demonstrates that the substitution leads to the appearance
of the terms linear in the ESVs in the energy expression.
In its turn the energy in a vicinity of the symmetric equilibrium geometry
expands as:
E( dq) = 
1 2


å
Gg


å
nn^{¢}

( dq_{n}^{Gg} D_{nn¢}^{Gg} dq_{n¢}^{Gg}) 

Somewhat simplified approach may be based on a
soft selection rule in this case is that In this case the
denominator in the expression eq. () is minimal so that under other equal
conditions the observed effects must be dominated namely by this mixing.
The final element of the setting of the LD theory can be called the electron
count or frontier orbital principle. Indeed, above we mentioned all the
selection rules controlling the the orbital interactions induced by two
types of perturbations relevant to the problem at hand. However, there is
one more element providing somehting which can be called a
soft selection rule The idea is again
very simple. Whatever of the perturbations mentioned above results in
modification of the electronic structure of the complex. We described this
Two questions have to be answered before we can turn directly to
constructing a mechanistic model of complexes. It remains unclear what
should be done if the perturbation is not weak (like in the case of the bond
breaking, see above). Also the effective Fock operator to be used to find
the covalency parameters x_{a1g} and x_{t1u}. This task is solved
in Section where we separate the electronic variables
relevant for the description of the close ligand shell as a separate
electron group defined by the carrier space eq. (21). Also
in the work [] an important aspect is ignored the electron
correlation in the dshell. Nonbonding isolated delectrons are treated
in [] on the common grounds although their physical regime
(see chapter ) is quite different. Despite
these drawbacks from the qualitative point of view the description of
complex stereochemstry (largely nontransition elements) given in LevinDyachkov is complete. It is only pity that (as far as we know) these
results have never been employed in the conext of constructing MM models of
the metal complexes. Nevertheless, in that or another form the results LevinDyachkov are necessary for success of any mechanistic model of
geometry and stereochemistry of the complexes since only in its framework it
is possibly to qualitatively correctly describe characteristc features of
their behavior known as mutual influence of the ligands. Any numerical model
must somehow incorporate these features.
The selection rules for the oneelectron matrix elements of the symmetry
adapted substitution operators are the same as for vibronic operator. The
major difference comes from the fact that any "chemical" substitution when
symmetrized results in a nontrivial expansion which always contains several
symmetry adapted components.
Also the matrix elements of V in the basis of the eigenstates of the zero
order Fock operator F^{(0)} can be classified by symmetry:
V_{n¢G¢g¢® nGg}^{[`(G)][`(g)]} 

which means that the component of the oneelectron transitional density
between the one electron states  n^{¢}G^{¢}g^{¢}
ñ and  nGg
ñ itself transforms according to the irreducible repesentation
[`(G)] (and its row) [`(g)]. The selection rules reduce to
the tables of the generalized vector coupling coefficients for the point
groups (octahedron group in this case) can be formulated. The same
classification takes place also for the matrix elements of the substitution
operator h_{S}.
w_{n¢G¢g¢® nGg}^{[`(G)][`(g)]} 

From this consideration the rôle of the electron counting rules for
describing the qualitative features of geometry response of the
corresponding electron group to the perturbations induced by the
substitution and other geometry variation becomes obvious.
Octahedral complexes of nontranstion elements which have twelwe electrons in
the respective CLS there are six such independent excitation energies.
resonance (oneelectron hopping) integrals in the diatomic coordinate frame
(one with the zaxis directed along the interatomic axis) between the sorbitals of the corresponding atoms referred to as sones and the p_{z}ones referred to as zones. We use as parameters the resonance
Since each LP enters the complex with two electrons their total count in the
considered CLS equals to four and for that reason two eigenstates  a_{+}
ñ and  a_{}
ñ are
doubly occupied thereas other two sstates remain empty.
Eight excitation energies define the structure of the polarization
propagator of this system:

 a_{+}
ñ ®  a_{+}^{*}
ñ ; 
 
 a_{+}
ñ ®  a_{}^{*}
ñ ; 
 
 
 



Within this setting it is easy to consider the ligand influence effects. As
within the LD theory we assume that the the substitution operator equals to Da r
ñ
á r  the
diagonal matrix element for the  r
ñ LP state is
shifted by Da. This expands into the sum of the symmetric and
antisymetric substitution operators:
Da r
ñ
á r =  
Da 2

(  a_{+}^{(0)}
ñ
áa_{+}^{(0)} + a_{}^{(0)}
ñ
áa_{}^{(0)} + a_{}^{(0)}
ñ
áa_{+}^{(0)} + a_{+}^{(0)}
ñ
áa_{}^{(0)} ) 
 (0) 

The first two terms are symmetric the other two asymmetric. This produces an
expected effect on the geometry of the system: increasig electronegativity
of the right ligand (Da > 0) leads to ... of the bondlength of
this ligand. However, according to our consideration the transbond
length is sensitive to this variation as well so that:
These excitations correspond to the population of the transition densities
of the symmetries a_{+}, p_{x},a_{},p_{u}.
this type of argumentation is taken to be excessively sophisticated. One may
also argue that a correctly designed theory may operate with the observable
quantities only, whereas the wave function is not an observable. Somewhat
more important is the necessity to include into the theory the effects of
charge redistribution. For that reason we subsequently reformulate the LD
theory in terms of observables  the oneelectron density matrix (which
incidentally yields the effective atomic charges) and the polarization
propagator of the CLS.
Then we get in the equilibrium:
File translated from
T_{E}X
by
T_{T}H,
version 2.67.
On 3 May 2007, 12:50.