Prof.-Pirlet-Str. 1, 52074 Aachen, Germany

Vorontsovo pole 10, 105064 Moscow, Russia

on the occasion of his 70-th birthday.

Molecular mechanics (MM) [,] is currently a versatile and very
popular tool in laboratory and industrial practice. It remains the most
practical way of analyzing potential energy surfaces (PES) for large molecules even though linear scaling *O*(*N*) methods
of quantum chemistry are becoming available [,]. The nature of the MM as a
combination of totally empirical classical force fields allows to realize its
main drawbacks (for example, inapplicability to highly correlated systems) and
advantages (fast evaluation of the total energy and its gradient and high
accuracy of molecular geometries obtained) and to characterize
them as the most preferable according to the "quality/cost" criterion but with
the field of application limited to certain combination of properties,
processes, and classes of molecules.

Despite its long history and a wealth of successful applications the MM approach remains not substantiated theoretically. It is clear that the molecular PES can be expanded up to the second order in nuclear displacements near the equilibrium geometry and this fact is often considered as a general substantiation of the possibility to use the MM-type expansions for the total energy []. However, it is very questionable since (i) the transferability of the elements of the dynamic (second derivatives) matrix between molecules is not proven, and (ii) the geometry variables (bond lengths and valence angles) used in the definitions of the force fields do not diagonalize the dynamic matrix even approximately, whereas the standard MM PESs are by construction diagonal in the bond-length/valence-angle representation.

Substantiation of MM is not merely an academic question. The last
years have demonstrated a growing interest [,,,] to hybrid
quantum mechanical/molecular mechanical (QM/MM) schemes where different parts
of the entire molecule are treated by different methods. The
relatively small part of the molecule (reactive center) is treated by an
adequate quantum mechanical (QM) method while the rest (inert environment) is
described by classical force fields. Thus constructed QM/MM methods combine the universality
of the QM description and the low cost of the MM that
makes them an optimal tool for analysis of the structure of large molecules
including those of biological importance. Though the QM/MM methods shift the
limits for numerical applications they require more insight into the
electronic structure underlying the MM schemes since it is not clear
how to construct the junction between quantically and classically treated
subsystems. Though many *ad hoc* recipes for the junction construction are
proposed in the literature (see, for example, reviews
[,,]), the problem has not been yet solved. Meanwhile, the
absence of sequential derivation for the intersubsystem junction leads to
numerous artefacts in QM/MM schemes reviewed in Ref. [].
The general formal solution of this problem proposed in Refs.
[,] is based on the assumption that a generic MM scheme
can be obtained from a suitable QM method by a series of approximations.
Thus the prospect of
development of effective QM/MM schemes constitutes the practical aspect of the
general task of the MM substantiation.
An important prerequisite for the announced derivation
of MM from QM is
the choice of the quantum chemical method underlying the MM description.
By taking a proper form of the trial wave
function having some common features (or resemblance) with the MM schemes we
can significantly simplify the task. The
representation of the total energy of the covalently bound molecule in the
current MM schemes is tightly connected to the concept of
two-center bonds. The energy is taken as a sum of
intrabond, bending, torsion and non-bonding contributions. The specific forms
of force fields depend on the implementation and quite a lot of schemes with
systems of parameters fitted to reproduce various characteristics of different
classes of molecules have been proposed [,,,,,,]. In
the course of the evolvement of the MM approach increasingly sophisticated
contributions are added to the original simplistic picture [] which
allow to extend the MM approach to more and more complex and not transparently
tractable classes of the molecules [,,]. Thus an
acceptable substantiation of the MM should not be reduced to deriving or
validating any specific MM scheme. By contrast, a generic mechanistic picture
must be obtained on the basis of the adequate QM description.

The standard QM methods in that or another manner based on the delocalized picture of molecular electronic structure provided by the SCF approximation are not quite suitable for the purpose of the MM substantiation. First, the general SCF energy expression contains contributions of the same form for all pairs of atoms and separation of the energy into bonding and non-bonding contributions is not built in to the SCF methods. Second, even applying different localization techniques intended to reconcile the SCF picture of the electronic structure with the chemist's view relying upon the bonds concept leads not to very much progress due to "tails" of the localized orbitals which are difficult for handling and absolutely non-transferable (it is also impossible to prove theoretically the transferability of the MO LCAO coefficients or, even, of the SCF density matrix elements).

The above discussion allows to specify some
criteria for selecting a QM method suitable to serve as a starting point for the MM
derivation: discrimination and different
treatment of bonding and non-bonding contributions to the energy, strict
locality of one-electron states, variational determination of the electronic
structure parameters (ESPs). Failure to fulfil any of these criteria leads
to a necessity to take whichever of the mentioned features as *assumptions*
(as for example, in Ref. [], where the analysis was based on the PCILO
method []).

The unsuitabilty of the SCF approximation as a basis for the additivity concept in general
(and thus for the MM in particular) was clear even on very early stage of the
theory development. Yet then (see []) it had been proposed to use a geminal
based description for substantiating the additive methods and the bond energy concept.
However, the one-electron carrier spaces to be used for geminal consctuction had not been
sufficiently specified and the entire geminal-based scheme had not been explored
to a due extent by adequate numerical studies. Only
recently we proposed [,] a semiempirical method which satisfies
the suitability criteria
formulated above. It is based on the trial wave function
in the form of antisymmetrized product of strictly local geminals (SLG).
Variational determination of strictly local one-electron states and of the
density
matrix elements provides the flexibility of the wave function necessary to
describe the electronic structure and properties of organic compounds.
Numerical estimates based on the MINDO/3 parameterization of the semiempirical
Hamiltonian [,] and three (MNDO, AM1, and PM3) parameterizations
of the NDDO family [] have shown that the method proposed
supercedes the SCF one in description of the heats of formation, molecular
geometries, and ionization potentials of organic molecules. Incidentally, it
provides a good basis for description of molecular electronic structure in
chemical terms. The method is well suited for description of large molecules
since it belongs to a class of *O*(*N*) methods and can be used for constructing
hybrid schemes []. Also the approximate transferability of the
ESPs of the SLG approach was numerically demonstrated in Ref. [],
where it was shown that the energy calculated with some ESPs "fixed" and
characteristic for a particular class of atoms and for bonds is close to that
obtained by direct minimization of the SLG energy. More refined treatment was proposed in
Refs. [,], where linear response relations for the form of
one-electron states were used to derive explicit expressions for the angular
dependence of the energy in the case of *sp*^{3} hybridized carbon and nitrogen
atoms. The numerical estimates
of the MM force field parameters
obtained in [] on the basis of the analytical expressions
for the constants of force fields are close
to those accepted in the MM. Special attention was paid to
off-diagonal force fields and to possible sources of the angular dependence of
the energy [] as well as those of piramidalization potential in
nitrogen-containing compounds []. The development of these ideas has
led to the formulation of the "deductive molecular mechanics" [],
which represents atoms by their hybridization tetrahedra (see below) of different shapes
adjusted according to geometry and valence state variations rather than by
harmonically interacting point masses ("balls-and-springs") used in standard MM.
All these results are obtained within the
*assumptions* of (i) the perfect transferability of ESPs characterizing
chemical bonds and (ii) of the validity of linear response relations for
hybridization with respect to geometry variations. In the present paper we
address the problem of transferability of the density related ESPs, explore
the precision and the validity limits for the linear response formulae for the
shapes of hybridization tetrahedra and consider the possibility of recovering the
standard MM description from the
deductive molecular mechanics.

The paper is organized as follows: in Section we briefly review the main features of the SLG method (with the MINDO/3 semiempirical parameterization) relevant to constructing the MM description. In Section the ESPs characterizing the density distribution in chemical bonds and lone pairs as they appear in the SLG method are analyzed and the features assuring their transferability are singled out. In Section we briefly discuss the structure of the hybridization manifold and give expressions for the variations of the hybridization related ESPs in response to geometry changes. Results of numerical experiments are given in these Sections to support our theoretical derivation of transferability and of linear response relations. In Section we discuss different possible approximate descriptions of the ESPs and energy leading to various versions of the deductive MM theory. In Section we consider the possibility of deriving the standard MM picture from the DMM by projecting out excessive (from the MM point of view) variables of the latter and give theoretical expressions for the force fields' parameters of the standard MM. Finally we discuss the relation between the transferability of the density related ESPs and that of the MM force fields.

Constructing the SLG trial wave function according to
[,] requires the following moves. First, the one-electron basis
of the strictly local
hybrid orbitals (HOs) must be contructed. These orbitals are obtained by an
orthogonal transformation of the *s* and *p* AOs for each "heavy"
(non-hydrogen) atom. These transformations are represented by
4×4 orthogonal matrices *h*^{A} Î *O*(4) for each heavy atom *A*. All the
HOs are assigned either to respective two-electron chemical bonds or to
electron lone pairs. Each chemical bond refers to two such HOs -
| *r*
ñ and | *l*
ñ (right- and left-end ones,
respectively). Each lone pair is formed by one HO only (a right one for the
sake of definiteness).

Chemical bonds and lone pairs are described by singlet two-electron functions
- geminals
[] taken in the form originally proposed by Weinbaum [].
With use of the second quantization notation they are written as:

| (0) |

| (0) |

| (0) |

The wave function of electrons in the molecule is then taken as the antisymmetrized
product of the geminals given by Eqs. (1), (2):

| (0) |

The SLG energy is a function of the intrabond matrix
elements of spinless one- and two-electron density matrices:

| (0) |

| (0) |

If the Hamiltonian of the MINDO/3 form [] in the HO basis is used
with the SLG trial wave function the total energy can be written in a form
somewhat close to the MM energy with interactions between bonded and non-bonded
atoms treated in different ways and closely relates it to that given in Ref.
[] in the context of analysis
of a variety of additive schemes of molecular energy:

| (0) |

| (0) |

| (0) |

Molecular integrals entering the above expressions depend on molecular geometry
and on the orthogonal matrices *h*^{A} for all non-hydrogen atoms *A*. The
expressions for the matrix elements in the HO basis are given in Ref.
[].

The energy expression Eq. (7) depends on the
amplitudes of bond geminals through the values of *P*_{m}^{tt¢} and
G_{m}^{tt¢}, and on the form of HOs through the molecular
integrals.
The amplitudes of two ionic (*u*_{m}, *v*_{m})
and one covalent (Heitler-London type, Ö2*w*_{m})
configurations
in Eq. (1) for each
geminal are
determined with use of the variational principle as well as the matrices *h*^{A}
of transformation of the AO basis to the HO one.

This comprises the essence of the semiempirical SLG method.

In the previous Section we reviewed briefly the semiempirical implementation of the SLG method for analysis of electronic structure and expressed the total molecular energy in the form Eq. (7) which allows the representation of the molecular PES as a sum of local increments. These increments depend on the ESPs of two classes (i) those defining the hybridization of atomic basis sets and (ii) the intrabond density matrix elements. In this Section we concentrate upon the proof of transferability of the electron density matrix elements, related to the geminal amplitudes, and on the structure of the hybridization manifold as it appears in the SLG approximation.

As it is reported in Section 1 the energy in the SLG approximation is a function of one- and two-electron density matrices. Their matrix elements are in turn expressed through the geminal amplitudes, appearing while diagonalizing the effective bond Hamiltonians. Thus any analysis of the properties of the density ESPs starts from description of the latter.

Within the original SLG approach [,] the geminals are
characterized by the amplitudes (see Eq.
(1) *u*_{m}, *v*_{m}, and Ö2*w*_{m} = *z*_{m}, which
simplifies the normalization condition Eq. (3) for the amplitudes
to: *u*_{m}^{2}+*v*_{m}^{2}+*z*_{m}^{2} = 1. To determine them the effective Hamiltonians for each bond
geminal are constructed. The optimal values of these amplitudes are the
solutions of the eigenvector problem (see also []):

| (0) |

The matrix elements of the effective bond Hamiltonians are defined as (with the MINDO/3
Hamiltonian):

| (0) |

| (0) |

The calculations of Refs. [,] performed on organic compounds
of different classes (alkanes, alcohols, amines *etc.*) have demonstrated a
remarkable stability of all the geminal related ESPs. The values of the
polarity *P*^{rr}_{m} - *P*^{ll}_{m} do not exceed 0.07 by absolute value for the compounds
containing carbon, nitrogen, and hydrogen atoms (for the situation with oxygen
and fluorine see below). Also the ionicity *u*^{2}_{m} + *v*^{2}_{m} for a rich variety of bonds
has a stable value about 0.4.
The bond orders 2*P*^{rl}_{m} *all* acquire values between 0.92 and 1.0.
These features though not completely unexpected,
since the transferability of the parameters of the single bonds in organic
compounds is well known experimentally, require a theoretical explanation.

In order to provide the required explanation we notice that the effective
Hamiltonians for the bond geminals can be represented as a sum of the
unperturbed part which when diagonalized yields an invariant, *i.e.*
exactly transferable, values of the ESPs and of a perturbation responsible for
specificity of different chemical compositions and environments.

**Pseudospin operator of the bond geminal **

Let us introduce a pseudospin [^(t)]_{m} operator corresponding to the
pseudospin value t_{m} = 1. The matrices of its components
in the basis of the configurations defining the geminal are given by:

| (0) |

| (0) |

| (0) |

| (0) |

**Perturbative estimate of ESPs with respect to noncorrelated
bare Hamiltonian **

One can try to estimate the optimal values of the ESPs specific for each bond and
molecule perturbatively by using the linear response approximation []. According
to the latter the response d
á *A*
ñ of a quantity described
by the operator *A* to the
perturbation l*B* of the
Hamiltonian (where l is the parameter characterizing the intensity of
the perturbation) has the form:

| (0) |

| (0) |

| (0) |

**Perturbation of the density matrix elements for correlated ground state **

In order to overcome the above failure of the perturbative estimation of the
two-electron density and of the bond orders let us consider
a symmetric bond. This would correspond to a different
decomposition of the effective bond Hamiltonian than that of Eq. (15).
We assume that the contribution to the effective bond Hamiltonian which is
proportional to [^(t)]_{zm}^{2} is included into the unperturbed (zero
order) Hamiltonian. The problem then reduces to a 2×2 matrix
diagonalization. The ESPs, as they appear from solution of this
problem, are:

| (0) |

| (0) |

| (0) |

Now, when the total ionic contribution to the geminal is calculated exactly
(variationally), the bond polarity can be estimated perturbatively in the
linear response approximation, but with the correlated ground state of the
symmetric effective bond Hamiltonian taken for evaluating the Green's function. It
can be conveniently done with use of a dimensionless asymmetry parameter:

| (0) |

| (0) |

The expression for the bond polarity coincides with that of Eq. (24)
even if the second-order perturbation correction to the wave function is used
(*i.e.*, the contribution to the bond polarity proportional to m_{m}^{2}
is absent). At the same time the second-order corrected bond ionicity and bond
order have the following form:

| (0) |

Another archetypical form of two-electron group is the lone pair. As it is
mentioned above the lone pair is described by a degenerate geminal containing
the contribution of only one ionic configuration. For the sake of definiteness
we set it to be the right-end ionic configuration of the corresponding
degenerate bond (the amplitude *u*_{m} becomes equal to unity, see Eq.
(2)). The ESPs related to the lone pair can be readily evaluated:

| (0) |

The above analytical results must be controlled by numerical estimates in order
to get a feeling of the real sense of the "first" and ßecond" orders. Table
represents
the results of calculations on the ESPs
á [^(t)]_{zm}
ñ ,
á [^(t)]_{zm}^{2}
ñ ,
and
á [^(t)]_{+m}
ñ
by the SLG method (Eq. (10)) and by the approximate formulae Eqs.
(19), (20), (22), (24), and
(25) for some characteristic bonds in small molecules. The results
show that in the case of bonds with small polarity all the formulae perform
very well. The most precise approximations Eqs. (24) and
(25) give results which perfectly coincide with the exact (SLG-MINDO/3)
ones even for very polar O-H and F-H bonds. Also estimates according
to the asymptotic (z_{m} >> 1) formulae Eq. (20) give reasonable
results for the ESPs of the bonds in not too polar molecules at their
equilibrium geometries. The main source of stability of the bond order values
is the validity of the above limit which in its turn takes place due to the
fact that the difference between one- and two-center electron-electron
repulsion integrals (Dg_{m}) at interatomic separations
characteristic for chemical bonding
is much smaller than the resonance interaction at the same distance. The data of
Table illustrate the
difference between that
which may be called MM atom types. For example, the primary C-H bonds in the
ethane and propane molecules have very similar ESPs at the same time somewhat
differing from those for the secondary C-H bonds in the propane molecule.

Further analysis of the quantities m_{m} allows to single out two types of
factors loaded upon this parameter: those related to the bond itself (which are
again hybridization dependent) and the rest describing the environment of the
bond. These factors contribute additively:

| (0) |

| (0) |

The contribution
to the bond asymmetry
coming from the environment of the bond is:

| (0) |

In the framework of the SLG scheme the structure of one-electron basis states
is defined by orthogonal transformations of AOs for each atom with an
*sp*-valence shell. The energy expression Eq. (7) is the function of the
parameters defining these transformations. The 4×4 *O*(4) matrix *h*^{A}
of transformation from the AO to the HO basis set on the atom *A* depends on
six angular variables. Three of them (pseudorotation angles [(w)\vec] _{b} = (w_{sx},w_{sy},w_{sz}) with subscripts indicating pairs of
basis AOs mixed by the corresponding 2×2 Jacobi rotations) define the
structure of the HOs (*s*-/*p*-mixing and relative directions of the HOs) while
other three (quasirotation angles [(w)\vec] _{l} = (w_{yz},-w_{xz},w_{yz})) define the *SO*(3) matrix performing rotation of the set of
four HOs as a whole (prefix *quasi* refers to the fact that no physical body
rotates under its action, only the system of HO's). Generally, the transformation
of orbitals caused by a
pseudorotation forms a set of HOs which is known as hybridization pattern (like
*sp*^{3}, *sp*^{2} *etc.*) which is more or less stable, while the set of
quasirotation angles is totally non-transferable, depends on the relative
placement of bonded atoms and, obviously, is governed by the resonance
contribution to the energy since only the latter depends on the directions of
the HOs.

The mathematical description of hybridization is based on employing the algebraic group structure of the hybrids'
manifold. Due to the latter any small variation of HOs in a vicinity of a given
set of HOs represented by a 4×4 orthogonal matrix *h* can be expressed
with use of the *SO*(4) matrix *H* close to the unity matrix:

| (0) |

| (0) |

The analysis of the properties of the ESPs pertinent to the SLG approximation
performed in Sections 0.1 and 0.1 allows to
rewrite the energy Eq.
(7) as follows:

| (0) |

Further
components of the description are those related to the HOs. The latter enter into
the theory through the Hamiltonian matrix elements in the HO basis.
The matrix elements entering Eqs. (7), (32) are either
invariant with respect to basis transformations (the interatomic Coulomb
interaction g_{AB}) or can be uniquely expressed through contributions
of *s*-AO to the HOs (the one-center matrix elements). The only class of
molecular integrals depending on the whole structure of the HOs (including
directions) is that of the resonance integrals. As we mentioned in Section 0.1
each *sp*^{x}-HO can be considered as a normalized quaternion
(*s*,[(*v*)\vec]). Following [] we represent the entire system of HOs at
any given atom by four vector parts [(*v*)\vec] _{m} of the corresponding orthonormal
quaternions. Even this representation is superfluous since only six Jacobi angles
suffice to describe the system of HOs of each given atom completely. Nevertheless,
usage of the vector parts is visual. If the latter are assumed to have the
corresponding nucleus as their common origin the tetrahedral shape thus obtained
contains (with an excess) all necessary information about the system of HOs
of the given atom. In [] such a construct was called the hybridization
tetrahedron of the heavy atom at hand. Using the hybridization tetrahedra as elements
of the theoretical construct allows further discrimination of possible approximations.
Due to the mentioned dependencies of the molecular integrals on the Jacobi angles both
the FA and TA approximations to the energy Eq. (32) depend
on the relative orientation of the hybridization tetrahedra through the bond
resonance integrals b_{rmlm}^{RmLm}. The resonance integrals depend also on
the shapes (relative weights of the *s*- and *p*-contributions to the HOs which
ultimately define the interhybrid angles) of the hybridization tetrahedra. All other
terms in Eq. (32) in the FA and TA approximations depend only on the shapes
of the hybridization tetrahedra. This leads to the possibility to either *fix* the
relative weights of the *s*- and *p*-orbitals (FO *i.e.* *fixed orbitals*
approximation) at *sp*^{n} (n = 1 ¸3) or any other allowable values and by
this fix the shapes of the hybridization tetrahedra which thus become interacting rigid
bodies or to allow the relative weights of the *s*- and *p*-*orbitals* to be *tuned*
thus leading to the TO - *tuned orbitals* - picture of the flexible
hybridization tetrahedra. Whichever combination of the FA or TA treatments for the
density matrix elements on one hand with the FO or TO treatments for the HOs on the
other hand results in a representation of the molecular energy Eq. (32)
as such of the system of tetrahedral bodies (rigid or flexible) whose interactions and
self energies depend on distances between their centers, their shapes and relative
orientations. For example, the energy variation due to small pseudo- and quasirotations
of the hybridization tetrahedron in the vicinity of the
equilibrium for *sp*^{3}-hybridized carbon atom in the FA approximation is given
by a diagonal quadratic form:

| (0) |

The content of the deductive molecular mechanics as formulated in Ref.
[] and above is a description of the
molecular energy in the form of Eq. (32) as a function
of shapes and mutual orientations of the hybridization tetrahedra and of
geometry parameters. On the other hand the standard MM can be qualified as a
scheme directly parameterizing the molecular energy as a function of molecular geometry
only. From this point of view the Jacobi angles variables [(w)\vec] _{b}, [(w)\vec] _{l}
describing the shapes and orientations of hybridization tetrahedra
are superfluous
and must be excluded. This can be done by finding the
response of the corresponding ESPs to the variations of bond lengths and valence angles
with use of linear response relations between different
subsets of variables pertinent to the DMM picture. To do so let us consider
a minimum of the energy with respect to both geometry and the ESPs.
In the vicinity of a minimum
the energy can be expanded upto second order with respect to
nuclear displacements *q* and variations of the ESPs *x*:

| (0) |

| (0) |

The main use of the formulae Eq. (35) is for exclusion of the
angular variables
describing the hybridization tetrahedra from the mechanistic picture.
Now we estimate the precision of the linear response relations (Eq. (35))
between geometry and
hybridization variations themselves by numerical study of
elongation of one C-H bond and
deformations of valence angles. We
consider tetrahedral methane molecule as a reference (its parameters then correspond
to subscript 0 in Eqs. (34), (35)). First of all,
we notice that the Ñ_{x}Ñ_{x}*E* matrix further simplifies
for methane
since
*s*^{Lm}_{m} = 1 and,
therefore, simple analytical expressions become possible. Also
we remark that the FA
approximation is
adequate here since, for example, even very large elongation of one C-H bond by 0.1
Å leads to changes of the bond geminal amplitudes *u*,*v*, and *w* not exceeding
0.003. The same applies to the averages of the pseudospin
([^(t)]) operators.

**Linear response of hybridization to bond elongation **

Let us consider first the relation between hybridization
and elongation of the C-H bond. For this end we need the mixed second order
derivatives coupling the bond
stretching with the hybridization ESPs. For every C-H bond in methane we can
introduce diatomic coordinate frame with the z axis directed along the
bond and express the resonance integral as:

| (0) |

| (0) |

Formula Eq. (37) gives the
analytical expression for the coupling coefficient between the bond elongation
(in Å ) multiplied by unit vector of this bond direction and changes of
pseudorotation angles d[(w)\vec] _{b} in methane (in radians). Its numerical
value *C*_{1} is 0.2764 rad·Å^{-1}. This distortion corresponds
to the following form for the matrix of small transformation of the whole set of HOs
(matrix *H* in Eq. (30)):

| (0) |

**Linear response of hybridization to valence angle deformation **

The linear response relations between the molecular shape and the shape of hybridization
tetrahedron are rather tricky due to complex structure of the
hybridization manifold.
The molecular shape can be formally characterized by unit vectors with origin
at an atom considered, taken as a center, and pointing to those bonded to the
central one.
In the case of methane
the deformations
of thus defined coordination polyhedron are small rotations of unit vectors [(*e*)\vec]_{m}
directed from the carbon atom to hydrogen atoms.
Their small rotations d[(j)\vec] _{m} form an 8-dimensional space which decomposes to
a direct sum of two subspaces: one 3-dimensional corresponding to rotations
of the molecule as a whole and another 5-dimensional corresponding to independent
variations of valence angles. The former one is precisely mapped on the 3-dimensional
space of quasirotations d[(w)\vec] _{l}
while the latter (5-dimensional) one is mapped on the
3-dimensional space of pseudorotations d[(w)\vec] _{b} corresponding to
changes of the shape of
the hybridization tetrahedron [].
Due to very general theorems of linear algebra [] there exists at
least a two-dimensional kernel in the space of deformations of molecular shape which
maps to zero deformation of hybridization tetrahedron. In [] the term
"hybridization incompatible" has been coined for the deformations from this kernel. The
structure of
deformations laying in the kernel of the mapping is quite simple: they
are produced by equal variations of opposite (spiro) valence angles. In
contrast, the variations which correspond to increase of one valence angle by
dc and decrease of its spiro counterpart by the same value fall into "coimage"
of this mapping *i.e.* to the subspace which one-to-one maps to the space of
pseudorotations d[(w)\vec] _{b}. The deformations in the coimage
can be
called "hybridization compatible". It is clear that only these latter
variations should be considered. It is also clear that any variation of the valence
angle is a sum of equal amounts of
hybridization compatible and hybridization incompatible deformations.
The denominator in the linear
response relation Eq. (35) is the same as for Eq.
(37) while the relevant block of the Ñ_{x}Ñ_{q}*E* matrix
(with *q* taken as a difference of two opposite valence angles) is proportional
to b_{zs}^{CH}. Applying the linear response technique to the
"hybridization compatible" variation of two spiro valence angles we obtain:

| (0) |

The coupling coefficient between the change of the pseudorotation vector and
totally hybridization compatible deformation of valence angles can be easily
found by Eq. (39). Its numerical value *C*_{2} is -0.20734 for the equilibrium
interatomic distance. The considered distortion produces the following HO
transformation matrix (matrix *H* in Eq. (30)):

| (0) |

The smallness of the coupling coefficient *C*_{2} even for the totally
hybridization compatible deformations allows to qualitatively understand
certain features of the electronic structure of cyclopropane as it
appears in the SLG approach: a very large distortion of the C-C-C valence angle from
the tetrahedral to 60^{°} one leads only to a
relatively small distortion of the corresponding interhybrid angle.
We
model this process by
strongly deforming the methane molecule. Simple estimate is based on Eqs.
(39), (40) and runs as follows.
The valence angle deformation when going from methane to cyclopropane is of
49.5^{°} (=109.5^{°}-60^{°}); only one half of it is hybridization compatible;
after multiplying by *C*_{2} this
yields the value of the
interhybrid angle between the HOs corresponding to the üntouched" C-H bonds of
114.5^{°} (*i.e.* the angle variation amounts only 5^{°}). From the
energy minimum condition for hydrides it
follows that the C-H bonds indeed
must follow the directions of the HOs. Numerical experiments performed with
use of the SLG-MINDO/3 method show that if one of the H-C-H valence angles
is fixed at the
cyclopropane value of 60^{°}
the energy minimum corresponds to its spiro counterpart
of 115^{°}. These results can be directly compared
with the experimental H-C-H valence angle in the cyclopropane molecule which equals to
115.1^{°}.

Analogous estimate can be applied to cyclobutane. In this case we consider
the distorted methane molecule with angle 90^{°}. The response of
the HOs to the deformation is proportional in our model to deviation of the
valence angle from the tetrahedral one. The deviation of the
C-C-C angle from the tetrahedral one in cyclobutane
(19.5^{°}) amounts 40% of that in cyclopropane.
Therefore, we can expect that about the same ratio will be
observed for the deviations of the H-C-H valence angle from the tetrahedral one
in the cyclobutane and cyclopropane molecules. In fact, this ratio in the SLG-MINDO/3
numerical experiment is about 39%.

Announced transition from the DMM model of molecular PES to a model dependent on molecular
geometry is formally obtained by inserting Eq. (35) to Eq. (34) which yields

| (0) |

| (0) |

| (0) |

We consider in more details the energy curve for the C-H bond. The curve
corresponding to the *sp*^{3} hybridization of carbon atom and to the symmetric
TA picture (Eq. (20)) is given by Fig. 3. It has correct qualitative
behaviour for all interatomic separations. The minimum depth on this curve is
approximately -0.23 a.u. and can be considered as the "pure" energy of the
C-H bond. It is generally accepted in the literature that the energy of
C-H bond is approximately 0.15 a.u. At the same time the latter value is
thermodynamic one while our value is obtained by extracting the contributions
to the energy intrinsic to this bond and excluding the interaction between the
bonds. The difference between the thermodynamic value for the bond energy and
that obtained from the SLG energy in the FA picture can be explicitly written
in quite simple form:

| (0) |

It is interesting to compare the form of the bond energy curve Fig. 3 with the
Morse potential. To this purpose we tried to approximate the curve of Fig. 3 by
the Morse function *D*_{0}[1-exp(-*a*(*r*-*r*_{e})/*r*_{e})]^{2} by minimizing the area
between two curves in the interval from 0.72 Å to 2.50 Å . With the
parameters *D*_{0} and *r*_{e} fixed at the values equal to the
minimum depth and position on the curve (0.2295 a.u. and 1.078 Å ) the optimal value
of parameter *a* is then 2.306 but with these parameters two curves are in fact quite
different (the area between curves is almost 11% of area between the bond energy
curve and the abscissa). If we optimize all three parameters of the
Morse curve they become slightly modified *D*_{0}=0.2333 a.u., *r*_{e}=1.045 Å ,
and *a*=2.295. This reduces the area between the curves by 30%. It should be
concluded that the energy profile in the TA approximation is not partcularly well
reproduced by whatever Morse curve.

In order to estimate the parameters of harmonic
force fields we consider the symmetric correlated single bond, where the
energy can be obtained without any reference to its
environment. In our case the derivative of the bond energy with respect to
a geometry parameter *q* has the form:

| (0) |

The same concepts can be used to determine the elasticity constant for the
bond stretching by taking the second derivative of the energy with respect
to the bond length. In the FA picture we get:

| (0) |

| (0) |

Analogous treatment of the energy terms quadratic in
valence angles' deformations yields the bare estimate for the harmonic
bending constant in the form:

| (0) |

In the previous Section we provided the exclusion of the angular variables characterizing the shapes and orientations of the hybridization tetrahedra from the mechanistic DMM model of molecular PES. This results in a model announced in the Introduction, which is similar to the standard MM models but is obtained by the sequential derivation from the QM (SLG) model of molecular electronic structure. As it is mentioned the transferability of the ESPs characterizing chemical bonds in molecules and linear response relations for hybridization ESPs are main components of deriving MM theory of molecular PESs from corresponding QM theory. Both these features have been mathematically derived and numerically checked in Section 1.

Despite its long history the very term "transferability" remains somewhat vaguely defined synonym of äll the best" in parameterization schemes, referring largely to their capacity to be used without change for any molecule in a sufficiently wide class of similar ones. From quantitative point of view this concept have got some attention in two related areas. First we mention the estimates of transferability given in Ref. [] where that of the semiempirical quantum chemical parameters has been related to the fact that the corresponding quantities remain the same for all molecules of similar structure upto the second order with respect to overlap integrals between AOs residing at neighbour atoms. That allowed to define the transferability for the quantum chemical parametrs (ultimately, for the Hamiltonian matrix elements) as invariance of some quantity to a given order of precision with respect to a small parameter. Analogously in Ref. [] the problem of constructing transferable dynamical matrices in relation to analysis of vibrational spectra has been considered. The stability of the dynamical matrix was analyzed with respect to small parameter of relative mass variation under the isotope substitution in a series of related molecules.

The importance of the transferability of the geminals has been pointed out yet in
[]. It was stated that the assumption of the transferability of the
geminal amplitudes is a prerequisite for that of the bond energy. However, in
[] the geminal transferability had not been shown and the authors
concentrated on the statements equivalent to the transferability of the MM bond
stretching force fields.
The transferability of the MM force fields must be considered as an important
chracteristic of this approach.
The reasons to treat that or another force
field as a transferable between two specific molecules or classes of molecules
are either purely pragmatic or this question is solved on the school-wise
grounds [].
As far as we know in the literature there were no attempts
reported to *prove* this property of the force fields
from any general point of view. Here we present a step towards quantitative
analysis of the
transferability of the MM force fields by proving the transferability of the
density matrix elements - equivalent to that of the geminal amplitudes, but
more directly related to the energy. Under the assumptions given by Eqs.
(15),
(16) the averages of the pseudospin operators (and thus all the
bond ESPs) are invariant in that sense that they do not depend on environment
of the bond under consideration and even on particular composition of the bond,
*i.e.* on the nature and the hybridization of the atoms the bond connects.
This corresponds to the FA
picture (see above and Ref. []).
It is important that the invariance (at the established level of precision) of the
density matrix
elements can be proven only for the basis of the variationally determined HOs -
a specific characteristic of the SLG approach [,].
In the basis of AOs the
density matrix elements are not invariant even approximately. Though the
approximation sufficient to obtain formally these invariant results (the SCF approximation)
is very crude it, nevertheless, breakes only at large interatomic separations
which normally are not covered by any MM-like approximation. This result
allows to pose further questions: to what extent the density ESPs' invariance may stand
further improvements of the description and whether it is possible
to relate the invariance of the density matrix elements with the transferability of the MM
force fields.
To answer these questions we notice that the invariant values of ESPs
can be improved by perturbative corrections (the TA picture) reflecting all diversity of
chemical
compositions and environments the bond may occur in.
Nevertheless, all the variety of perturbations is
characterized by two small dimensionless parameters: z_{m}^{-1} Eq.
(21) and m_{m} Eq.
(23). Both parameters depend
on the atoms connected by the bond, their separation, and their hybridization.
The perturbative treatment allows to
estimate the precision of transferability. For
example, using Eqs. (22), (24), and (25) we
conclude that the bond order is the quantity transferable upto second order
with respect to both z_{m}^{-1} and m_{m}; the ionicity (the total weight
of the ionic configurations) is
transferabile upto second order with respect to m_{m} and upto first order
with respect to z_{m}^{-1}; the bond polarity is transferable upto first
order with respect to both z_{m}^{-1} and m_{m}. The second order
transferability of bond orders
explains to certain extent the success of the concept of ßingle bond" suitable for
a large variety of chemical bonds. Note that the second order
transferability takes place for the bond orders also in the case when we employ
the SLG bond wave function with the correct asymptotic behavior despite the fact
that the transferable numerical value itself is obtained from the SCF
wave function which does not have the correct asymptotic
behavior. Within this picture all specific
characteristics of the force field are loaded into parameters of the (effective)
Hamiltonian, which are numbers specific either for a given atom in certain
hybridization state or for a pair of such hybrid states of atoms - ends of the
bond.
Although the density ESPs can be considered as constants independent on any details
of molecular composition or geometry, the force fileds which are basically sums of
products of ESPs by matrix elements of molecular Hamiltonian are geometry
dependent and composition specific. The force fileds thus obtained are expected
to be the same for the same composition of the bond and to depend weakly (to the
extent of the variance of the m_{m1} parameters) on the environment.
These properties are basically much more
than necessary for substantiating whichever MM-like description.

In the present paper we discussed the problem of deriving the MM representation
of the molecular PES from a relevant QM description. Using the SLG wave function
we analyzed the ESPs related
to bond geminals and to hybridization tetrahedra. It was shown that the
bond-related parameters can be represented as functions of parameters of the
MINDO/3 Hamiltonian in the HO basis, transferable from one molecule to another.
The functional form of the ESPs found is valid at arbitrary interatomic
separations. At the interatomic separations close to the equilibrium bond
lengths characteristic for the MM-like treatments two approximations, both
suitable for substantiation of the ESPs transferability were considered. One is
the *fixed* geminal *amplitudes* approximation which results in
perfectly transferable numbers referring to the ESPs in question. Another, more
exact is the *tuned* geminal *amplitudes* approximation which takes
into account small corrections to the invariant ESPs. Two small parameters
characterizing specificity of the bond and effects of its environment were
introduced. By this the whole manifold of quantum chemical parameters defining
the effective bond Hamiltonian boils down to only two relevant parameters
z_{m} ^{-1}
and m_{m}. The
presence of such only two-dimensional manifold and smallness of the parameters
for a wide range of bonds in quite different environments essentially explains
the transferability of the density related ESPs.
This allows for a family of mechanistic models describing molecular PESs in terms of
hybridization tetrahedra with interactions dependent on distances between
their centers and on mutual orientations.
Linear response relations for variation of
hybridization parameters due to elongation of chemical bonds or specific
changes of valence angles are considered.
They allow to exclude the angular variables describing the shapes and orientations
of hybridization tetrahedra and to represent the molecular
energy in both the FA and TA approximations
as that of the system of interacting point masses [] ("balls-and-springs" picture)
depending on the molecular geometry only. This energy has a form of a sum of
local (bond) increments corresponding to the force fields of the standard MM.
The estimates of the paramemters of these force fields coming from the analytical
expressions are
compared with those obtained in numerical experiments showing the high accuracy of
analytical estimates. The reasons for this possibility are both the
transferability of the bond-related ESPs and the linear response relations for
the hybridization tetrahedra are numerically tested for their precision.

This work was performed with financial support of the RFBR through the grants 02-03-32087, 04-03-32146, and 04-03-32206. It has been completed during the stay of A.M.T. in the RWTH, Aachen in the frame of the Alexander von Humboldt Postdoctoral Fellowship which is gratefully acknowledged as is the kind hospitality of Prof. R. Dronskowski. A.L.T. gratefully acknowledges valuable discussions with Profs. N.F. Stepanov, A.A. Levin, and I. Mayer.

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Molecule | Bond | z^{-1} | m |
á [^(t)]_{zm}
ñ |
á [^(t)]_{zm}^{2}
ñ |
á [^(t)]_{+m}
ñ | ||||||||

(10) | (19) | (24) | (10) | (19) | (20) | (25) | (10) | (20) | (22) | (25) | ||||

H_{2} | H-H | 0.127 | 0.000 | 0.000 | 0.000 | 0.000 | 0.437 | 0.436 | 0.437 | 0.437 | 0.992 | 0.992 | 0.992 | 0.992 |

CH_{4} | C-H | 0.181 | 0.093 | 0.065 | 0.095 | 0.065 | 0.414 | 0.410 | 0.411 | 0.414 | 0.982 | 0.984 | 0.984 | 0.982 |

NH_{3} | N-H | 0.172 | 0.097 | 0.069 | 0.098 | 0.069 | 0.419 | 0.414 | 0.415 | 0.419 | 0.983 | 0.986 | 0.985 | 0.983 |

H_{2}O | O-H | 0.161 | 0.341 | 0.244 | 0.345 | 0.247 | 0.464 | 0.419 | 0.420 | 0.466 | 0.959 | 0.987 | 0.987 | 0.959 |

HF | F-H | 0.285 | 0.540 | 0.309 | 0.562 | 0.308 | 0.451 | 0.357 | 0.363 | 0.457 | 0.925 | 0.962 | 0.959 | 0.929 |

C_{2}H_{6} | C-C | 0.193 | 0.000 | 0.000 | 0.000 | 0.000 | 0.405 | 0.404 | 0.405 | 0.405 | 0.982 | 0.982 | 0.981 | 0.982 |

C-H | 0.181 | 0.071 | 0.050 | 0.072 | 0.050 | 0.413 | 0.410 | 0.411 | 0.413 | 0.983 | 0.984 | 0.984 | 0.983 | |

C-C | 0.193 | 0.016 | 0.011 | 0.016 | 0.011 | 0.405 | 0.404 | 0.405 | 0.405 | 0.982 | 0.982 | 0.981 | 0.982 | |

C_{3}H_{8} | C1-H | 0.181 | 0.074 | 0.052 | 0.075 | 0.052 | 0.413 | 0.410 | 0.411 | 0.413 | 0.983 | 0.984 | 0.984 | 0.983 |

C2-H | 0.180 | 0.051 | 0.035 | 0.051 | 0.035 | 0.412 | 0.410 | 0.411 | 0.412 | 0.984 | 0.984 | 0.984 | 0.984 | |

Cyclo- | C-C | 0.199 | 0.000 | 0.000 | 0.000 | 0.000 | 0.402 | 0.400 | 0.402 | 0.402 | 0.981 | 0.981 | 0.980 | 0.981 |

propane | C-H | 0.180 | 0.084 | 0.059 | 0.086 | 0.059 | 0.414 | 0.410 | 0.411 | 0.414 | 0.983 | 0.984 | 0.984 | 0.983 |

N_{2}H_{4} | N-N | 0.290 | 0.000 | 0.000 | 0.000 | 0.000 | 0.361 | 0.355 | 0.361 | 0.361 | 0.960 | 0.960 | 0.958 | 0.960 |

N-H | 0.176 | 0.092 | 0.065 | 0.093 | 0.065 | 0.416 | 0.412 | 0.413 | 0.416 | 0.983 | 0.985 | 0.984 | 0.983 | |

N-C | 0.219 | 0.025 | 0.016 | 0.026 | 0.016 | 0.393 | 0.390 | 0.393 | 0.393 | 0.977 | 0.977 | 0.976 | 0.977 | |

CH_{3}NH_{2} | N-H | 0.173 | 0.098 | 0.069 | 0.099 | 0.069 | 0.419 | 0.412 | 0.415 | 0.419 | 0.983 | 0.985 | 0.985 | 0.983 |

C-H | 0.187 | 0.078 | 0.053 | 0.079 | 0.053 | 0.410 | 0.406 | 0.408 | 0.410 | 0.982 | 0.983 | 0.982 | 0.982 | |

O-C | 0.218 | 0.276 | 0.178 | 0.282 | 0.179 | 0.420 | 0.391 | 0.394 | 0.421 | 0.964 | 0.977 | 0.976 | 0.964 | |

CH_{3}OH | O-H | 0.158 | 0.328 | 0.236 | 0.332 | 0.240 | 0.462 | 0.421 | 0.422 | 0.464 | 0.962 | 0.988 | 0.988 | 0.961 |

C-H | 0.182 | 0.077 | 0.053 | 0.078 | 0.053 | 0.413 | 0.409 | 0.411 | 0.413 | 0.983 | 0.984 | 0.984 | 0.983 | |

CH_{3}F | F-C | 0.355 | 0.417 | 0.212 | 0.442 | 0.208 | 0.382 | 0.322 | 0.333 | 0.383 | 0.930 | 0.942 | 0.937 | 0.932 |

C-H | 0.180 | 0.089 | 0.062 | 0.091 | 0.062 | 0.414 | 0.410 | 0.411 | 0.414 | 0.982 | 0.984 | 0.984 | 0.982 |

Molecule | Bond | m_{0} | m_{1} |
á [^(t)]_{zm}
ñ |
á [^(t)]_{zm}^{2}
ñ |
á [^(t)]_{+m}
ñ | |||

(10) | (24) | (10) | (25) | (10) | (25) | ||||

H_{2} | H-H | 0.000 | 0.000 | 0.000 | 0.000 | 0.437 | 0.437 | 0.992 | 0.992 |

CH_{4} | C-H | 0.075 | 0.018 | 0.065 | 0.052 | 0.414 | 0.413 | 0.982 | 0.983 |

NH_{3} | N-H | 0.091 | 0.006 | 0.069 | 0.065 | 0.419 | 0.419 | 0.983 | 0.984 |

H_{2}O | O-H | 0.343 | -0.002 | 0.244 | 0.249 | 0.464 | 0.466 | 0.959 | 0.959 |

HF | F-H | 0.540 | 0.000 | 0.309 | 0.308 | 0.451 | 0.457 | 0.925 | 0.929 |

C_{2}H_{6} | C-C | 0.000 | 0.000 | 0.000 | 0.000 | 0.405 | 0.405 | 0.982 | 0.982 |

C-H | 0.068 | 0.003 | 0.050 | 0.047 | 0.413 | 0.413 | 0.983 | 0.983 | |

C-C | 0.007 | 0.009 | 0.011 | 0.005 | 0.405 | 0.405 | 0.982 | 0.982 | |

C_{3}H_{8} | C1-H | 0.068 | 0.006 | 0.052 | 0.047 | 0.413 | 0.413 | 0.983 | 0.983 |

C2-H | 0.061 | -0.010 | 0.035 | 0.042 | 0.412 | 0.413 | 0.984 | 0.983 | |

Cyclo- | C-C | 0.000 | 0.000 | 0.000 | 0.000 | 0.402 | 0.402 | 0.981 | 0.981 |

propane | C-H | 0.092 | -0.007 | 0.059 | 0.064 | 0.414 | 0.415 | 0.983 | 0.982 |

N_{2}H_{4} | N-N | 0.000 | 0.000 | 0.000 | 0.000 | 0.361 | 0.361 | 0.960 | 0.960 |

N-H | 0.094 | -0.002 | 0.065 | 0.066 | 0.416 | 0.417 | 0.983 | 0.983 | |

N-C | 0.033 | -0.007 | 0.016 | 0.021 | 0.393 | 0.393 | 0.977 | 0.977 | |

CH_{3}NH_{2} | N-H | 0.102 | -0.005 | 0.069 | 0.073 | 0.419 | 0.419 | 0.983 | 0.983 |

C-H | 0.078 | -0.001 | 0.053 | 0.054 | 0.410 | 0.410 | 0.982 | 0.982 | |

O-C | 0.297 | -0.021 | 0.178 | 0.193 | 0.420 | 0.425 | 0.964 | 0.962 | |

CH_{3}OH | O-H | 0.346 | -0.018 | 0.236 | 0.253 | 0.462 | 0.469 | 0.962 | 0.958 |

C-H | 0.093 | -0.016 | 0.053 | 0.065 | 0.413 | 0.414 | 0.983 | 0.982 | |

CH_{3}F | F-C | 0.431 | -0.015 | 0.212 | 0.215 | 0.382 | 0.387 | 0.930 | 0.932 |

C-H | 0.092 | -0.003 | 0.062 | 0.064 | 0.414 | 0.415 | 0.982 | 0.982 |

r_{0}^{CH} | k_{CH} | k_{HCH} |

Å | mdyn/Å | mdyn/deg |

FA: 1.069 | 8.30 | 0.509 |

TA_{symm}: 1.078 | 7.77 | |

TA_{pert}: 1.096 | 7.17 | |

Standard MM: | ||

[]: 1.113 | []: 4.5 ¸ 4.7 | []: 0.549 |

[]: 1.105 | []: 5.31 | []: 0.508 |

[]: 1.090 | []: 7.90 | []: 0.493 |

^{1}AvH Postdoctoral Fellow, on leave from the
Karpov Institute of Physical Chemistry, Moscow, Russia

File translated from T

On 18 Aug 2004, 13:36.