L.Ya. Karpov Institute of Physical Chemistry,

Vorontsovo pole 10, Moscow, 103064, RUSSIA;

Center for Computational Chemistry at

the M.V. Keldysh Institute for Applied Mathematics of RAS

The molecular mechanics (MM) [,] numerical schemes are widely used for modeling potential energy surfaces (PES) of large molecular systems. The practical usefulness of the MM schemes is obvious - they provide the equilibrium geometries, dynamic matrices and other features of the systems of large size when the quantum chemical procedures becomes inapplicable due to enormous computational costs. The MM results are sufficiently accurate for many purposes and for large objects MM schemes are the best considering the cost-efficiency ratio.

Despite the intense employment of the MM schemes they remain purely
empirical and are only supported by the agreement of calculated and
experimentally observed quantities. Such level of substantiation means that
the success of the MM schemes in general must be considered as an
experimental fact. This fact in its turn requires theoretical explanation
which as any sequential description of molecular structure and properties
must be based on quantum mechanics (QM). The question of the interrelations
between the MM and QM approaches is well-known in the literature (see, for
example, []). It is, however, not only of academic
interest. Additional attention to the foundations of the MM schemes is
caused by increasing interest to the hybrid QM/MM schemes []
where a chemically active part of the whole system is described by precise
quantum chemical methods while the rest (more or less inert environment) is
represented by an appropriate MM method. The hybrid QM/MM schemes use a
combination of sometimes high quality QM and of an empirical MM procedure.
The status of the combined approach thus remains unclear as is the form of
the junction between the quantum and classical subsystems in the hybrid
QM/MM schemes which is commonly chosen *ad hoc* since the relations
between the QM and MM approaches are not clearly defined. To derive
successively the explicit form of the junction it is necessary to assume
also that of the wave function underlying the MM description. In this
context the very concept of the MM must be clarified. The common feature of
all MM schemes is that the contributions to the energy from the bonded and
non-bonded atoms are treated differently, *i.e.*, the bonding is
somewhat externally assumed. Different MM schemes employ different specific
forms of potentials and numerous systems of their parameters fitted to
reproduce a variety of characteristics [,,,] for more
or less wide class of molecules. Also during the evolution of the MM
approach itself some increasingly sophisticated contributions are added to
the original simple picture which allow to extend the MM approach to
increasingly complex and not transparently tractable classes of the
molecules where the metal containing complexes and other molecules with
significant electron redistribution must be mentioned
MMGK,CombaHambley,Kozelka. For the latter cases the modern MM schemes do
not represent the energy as an explicit function of the geometry parameters
but include some kind of optimization scheme (for example, in Ref.
Kozelka the scheme adjusting the charges on atoms on the ground of some
simple criterion is employed). We see that the definition of MM becomes even
more diffuse. Summarizing, we can define the MM scheme as one representing
the energy as a combination of bonding and non-bonding contributions which
are either explicit functions of molecular geometry structure or can be
obtained from the latter by simple non-iterative optimization procedure. In
the present paper we suggest a derivation and a numerical test of a scheme
of this type - a local parametric expression for the total energy of
organic molecules on the basis of a quantum description which uses a special
form of the trial wave function of electrons, namely that of the
antisymmetrized product of strictly local geminals (APSLG).

The reason to choose the APSLG form of the underlying wave function is some
intimate similarity in the energy expressions of the APSLG and MM schemes.
In our previous works [,] we have derived the required
form of the junction between the QM and MM subsystems in an assumption that
the inert part of the combined system can be represented by the wave
function in the APSLG form [,,]. At the same time the APSLG
based QM method is rather complicated self-consistent iteration procedure
with numerous diagonalizations of effective Hamiltonian matrices though of
small dimensionality. Nevertheless, the APSLG approach seems to be
appropriate for obtaining the MM-type schemes for different reasons: the
APSLG approach provides the local description of the molecular electronic
structure; it represents the energy as the sum of intrabond and interbond
contributions and thus can be considered as a natural starting point for
construction of different additive schemes. Since the MM scheme is itself a
parametric empirical procedure, it is unnecessary to use sophisticated
nonempirical description for deriving the latter. It is more logical to use
a semiempirical method as a starting point. A semiempirical implementation
of the APSLG trial wave function has been recently developed with use of the
MINDO/3 form of the Hamiltonian and resulted in an *O*(*N*)-scaling method
suitable for describing organic molecules. This method and its
parameterization are described in details in Refs. [,,].

The present paper is organized as follows. In the next Section we first briefly summarize the main features of the APSLG approach. Next the one-step procedures for optimization of geminal amplitudes and hybridization matrices are discussed. In the Section 3 we propose a series of approaches for evaluation of the electronic energy of molecules in the APSLG approximation and evaluate their accuracy. Next we discuss the possibility of treating these approaches as generic MM-type descriptions. The last Section contains the summary of results.

The APSLG wave function of electrons in the molecule has the form:

| (0) |

| (0) |

| (0) |

| (0) |

The above function is applied to find an estimate for the system energy with
the Hamiltonian of the MINDO/3 approximation. In the HO basis the latter can
be rewritten as a sum of one- and two-center contributions:

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

The electronic energy is then the sum of four terms:

| (0) |

| (0) |

| (0) |

The nuclear-nuclear (core-core) repulsion in the MINDO/3 approximation
differs from the pure Coulomb repulsion. It has the form:

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

The optimal values of *u*_{m}, *v*_{m}, and *z*_{m} are the solutions of the
eigenvalue problem:

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

According to Ref. [] each hybridization matrix can be
represented by the product of six Jacobi rotation matrices acting in the
two-dimensional subspaces of the whole four-dimensional space. Therefore it
depends on six angles w_{sx}^{A}, w_{sy}^{A}, w_{sz}^{A}, w_{xy}^{A}, w_{xz}^{A}, and w_{yz}^{A}. These six angles,
however, play different roles: the first triple determines the hybridization
itself and must be at least approximately transferable while three others
rotate the set of the HOs attached to the atom *A* as a whole and can not
possess any transferability. It is congenial to the modern MM schemes to
consider the six angles determining hybridization matrices as quantities
which are determined by minimizing a simple functional:

| (0) |

The important feature of the APSLG-MINDO/3 method [] is a remarkable
transferability of its relevant ESPs - the geminal amplitudes and
hybridization angles. It distinguishes the APSLG method from others (based,
for example, on the SCF approximation where the corresponding ESPs - MO
LCAO coefficients - are not transferable). The geminal amplitudes vary
slightly when going from one molecule to another: for example, the
quantities *u*_{m}, *v*_{m}, and *w*_{m} for the C-H bond in methane are 0.4897,
0.4178, and 0.5412, respectively, while in ethane the analogous
quantities are 0.4805, 0.4272, and 0.5416. These amplitudes also
remain approximately the same in the case of the C-H bonds in significantly
more different environment (for example, in methylamin molecule these
amplitudes are 0.4814, 0.4223, and 0.5431). That degree of
transferability is even more accurate than one could guess.

At this point we are in a position to formulate a generic MM scheme with certain QM foundation. Indeed, if we assume the expression Eq. ( 19) for the energy to be an exact QM one, then schemes fitting into a diffuse concept of MM can be suggested. The key point with them is to accept specific procedures for defining the ESPs entering Eq. ( 19).

Within such a setting the parameters of the method can be subdivided in two groups: the first one comprises the parameters of the Hamiltonian of the underlying semiempirical QM approach (in our case those of the APSLG-MINDO/3 []). They do not require special adjusting; other parameters reflects the transferability of the electronic structure characteristics obtained by the APSLG approach and can be taken from calculations of simple molecules by the APSLG approach or specially adjusted.

Using such definition of the HO matrices we can define zero-order
approximation to the energy Eq. (19) by the following
procedure: the amplitudes *u*_{m}, *v*_{m}, *w*_{m} are chosen to be constant for
each type of the bond and the angles w_{sx}^{A}, w_{sy}^{A}, w_{sz}^{A}, determining the hybridization are also constant for each
type of the atom (type of the atom can be determined in the way analogous to
that of MM: it depends on the closest environment). In other way, the values
of the hybridization angles define the type of the atom in terms of its
hybridization. The quantities w_{xy}^{A}, w_{xz}^{A}, w_{yz}^{A} are chosen on the basis of purely geometric correspondence between
directions of HOs and those of the chemical bonds. The above assumption of
perfect transferability of ESPs *u*_{m}, *v*_{m}, *w*_{m} and w_{sx}^{A}, w_{sy}^{A}, w_{sz}^{A} allows to represent the total energy as an
explicit function of geometric structure. At the same time this
approximation may turn out to be rather crude because the transferability of
the zero approximation ESPs is good only for similar geometric structure and
the large distortions of the molecule can impair the quality of this simple
model. In fact the transferability is characteristic only for the
interatomic separations close to the equilibrium bond lengths. If the bond
becomes elongated the ionic contributions diminish while the amplitude (*w*_{m}) of the homeopolar configuration tends to 1/Ö2. In this case the
transferability of the zero approximation amplitudes is broken and they must
be readjusted. Analogously, if the valence angles are far from the
equilibrium ones the degree of transferability of the hybridization angles
is questionable as well. Incidentally, the quality of the approximation to
the amplitudes *u*_{m}, *v*_{m}, *w*_{m} can be significantly improved without
loss of the explicit character of the energy expression.

The general idea of constructing local parametric expressions based on the APSLG description of the molecular electronic structure can be verified by numerical estimates of the validity of different schemes for obtaining the ESPs. The results of previous Sections allow to propose a range of schemes for estimating the total energy by Eq. (19) differing by approximations used for obtaining the ESPs:

- the geminal amplitudes and hybridization angles w are taken
as perfectly transferable parameters and are not additionally tuned. This
scheme can be termed as FAFO (fixed amplitudes and fixed orbitals);
- the geminal amplitudes are additionally adjusted by using formulae
Eq. (28) while the HOs remain fixed at their transferable values
- the TAFO scheme (tuned amplitudes and fixed orbitals);
- only the HOs are adjusted by minimizing the functionals Eq. (
29) for each heavy atom but the geminal amplitudes are taken equal to
their transferable values - the FATO scheme;
- the geminal amplitudes are adjusted first and the HOs are corrected
for the adjusted values of the bond amplitudes - the TATO scheme;
- analogous to the previous scheme but the order of two adjustment procedures is changed - the TOTA scheme.

In the present paper we performed test calculations on some organic
molecules with use of the above five schemes. As the first test we applied
the above schemes to constructing the energy profile for the process of
variation of one of the C-H bonds length in the methane molecule. The
transferable hybridization matrix is taken to correspond to the *sp*^{3}-hybridization and the zero approximation C-H bond amplitudes are taken from
a calculation on the methane molecule at its experimental equilibrium
geometry. When such a choice is made all the five schemes give in the
equilibrium point the electronic structure (and other properties, for
example, heat of formation) coinciding with those obtained by the original
APSLG approach. Figure 1 represents the differences between heats of
formation calculated by each of the five above MM-type schemes and by the
APSLG method as a function of the C-H bond length. This Figure shows that
all the five schemes work well in the vicinity of the equilibrium geometry.
It can be concluded that both the minimum position and the heat of formation
in the minimum remain the same as in the original QM APSLG approach
JCC. Since the APSLG approach reproduces the experimental equilibrium
geometry of methane with perfect accuracy the MM-type schemes do the same as
well. At the same time the results of the zero-order approximation (FAFO)
deviate from the precise APSLG one rather strongly for interatomic distances
far from the equilibrium. This level of approximation can, however,
satisfactorily cover the PES within a ±0.1 Å range near the
equilibrium position. It is important to find the quantitative measure of
deviation between PES calculated by different computational methods. In our
case the difference between an MM-type PES and the APSLG PES must be
characterized. As a convenient and representative measure in the
one-dimensional case we choose the area (integral) between two energy
profiles in the definite interval of variation of the geometric parameter.
Table 1 contains such data for three different ranges of variation of the
C-H bond length in methane for all five schemes. The methods TAFO and FATO
somewhat improve the results of the FAFO scheme but the range of their
validity is not strongly extended as compared to that of the FAFO scheme.
The methods combining the optimization of HOs with adjustment of the geminal
amplitudes work satisfactorily for all interatomic distances studied. At the
same time the TOTA method yields the numerical values which are closer to
those obtained by the underlying APSLG approach than the TATO scheme. The
maximal deviation of the TOTA scheme from the exact APSLG one does not
exceed 0.2 kcal mol^{-1} in the whole considered range of geometry
parameters.

The above example is not representative enough because in the methane
molecule the *sp*^{3} hybridization can be a good approximation even for very
large geometry distortions. The hybridization can be significantly perturbed
when less symmetric molecule is considered. To show how these schemes work
under larger deviation of the hybridization from one calculated by the APSLG
at the equilibrium geometry, we consider as the next example distortions of
the water molecule taken with the *sp*^{3} hybridization for the oxygen atom
as the zero approximation. The amplitudes *u*_{m}, *v*_{m}, and *w*_{m} of the O-H
bond were taken from the APSLG-MINDO/3 calculation of the water molecule: 0.5946, 0.3317, and 0.5179, respectively. The distortions considered
were (i) the variation of the length of one O-H bond, (ii) simultaneous
variation of the lengths of both O-H bonds, (iii) variation of the valence
angle H-O-H (these three cover all the possible distortions of the water
molecule). Figures 2-4 represent for these processes the deviations of the
heats of formation calculated by MM-type schemes and the underlying QM APSLG
method for these processes. Analogously, Table 1 contains the integral
characteristics of these deviations.

These data show the importance of the quality of the zero approximation hybridization employed. The FAFO and TAFO methods result in very large deviations of the heats of formation from the APSLG method even in the case of geometries close to the equilibrium. Also in the cases of variation of the lengths of O-H bonds the minimum position is significantly displaced, and in the case of the variation of the valence angle the shift of the minimum position is very large. Much better results are obtained with use of the FATO scheme. This scheme in this case works even better than the TATO scheme. At the same time the best of these schemes is the TOTA scheme. It gives the results which are in perfect agreement with those of the underlying QM APSLG approach itself. The data of Figures 1-4 and Table 1 unambiguously demonstrate that only the TOTA scheme can be successfully and surely applied for description of the large portions of molecular PESs. Only this scheme is applied in our further analysis.

The molecules considered previously have only one type of bonds. Now we
consider how the TOTA scheme works with different types of bonds present.
Table 2 shows the results of calculations on the heats of formation for the
ethane molecule as a function of the C-C interatomic separation. The zero
approximation amplitudes *u*_{m}, *v*_{m}, and *w*_{m} for the C-H bond were taken
from the APSLG calculation on the methane molecule in its equilibrium
geometry and the analogous quantities for the C-C bond were taken from the
calculation on the ethane molecule (0.4502, 0.4502, and 0.5453). The
data in Table 2 show that the ethane molecule is well described by the TOTA
scheme for all interatomic distances considered. Moreover, also for the
large C-C distances the difference between the APSLG and the TOTA heats of
formation becomes very small. The position of the minimum on the PES remains
unchanged.

Table 3 contains the results of calculations on a series of organic
molecules (for their experimental geometry parameters) by the QM
APSLG-MINDO/3 method and by the TOTA scheme for the two sets of zero
approximation ESPs (geminal amplitudes). The first type of ESPs contains no
adjusted parameters. The geminal amplitudes for the C-H, C-C, O-H and C-O
bonds are taken from the APSLG calculations on the methane, ethane, water,
and methanol molecules at their experimental geometry parameters. The *sp*^{3}
hybridization was taken as a zero approximation one for all heavy atoms.
These results show that the TOTA scheme provides the heats of formation well
corresponding to those of the APSLG approach. The deviation from the
underlying APSLG scheme is significantly smaller than the precision of this
scheme itself. It allows one to make a conclusion of a validity of the
MM-type TOTA scheme for calculations of the heats of formation and
equilibrium geometry structures of organic compounds with well defined
chemical bonds and lone pairs. It can be noted that this parameterization
takes zero approximation ESPs from calculations of the simplest
representatives of organic molecules. At the same time these ESPs are
somewhat different from those characteristic for more large molecules
because the simplest molecules have somewhat specific structure in the class
of homologues. Therefore, it can be reasonable to improve the results on the
heats of formation by slight change of the parameters and thus, the second
procedure is an attempt to improve the results of calculations by changing
the initial amplitudes for C-H bond. They were taken as follows: 0.4739, 0.4306, and 0.5431. The change of the parameterization allows to improve
the agreement between the heats of formation calculated by the APSLG and
TOTA schemes. At the same type this improvement is not very significant and
the initial geminal amplitudes can be left unadjusted.

The present work is aimed to constructing parametric expressions for the energy of molecules by making start from an underlying QM description of molecular electronic structure. Such a construct could serve as a generic form of MM. The APSLG trial wave function was proven to be the proper choice for underlying the desirable MM description. In such a way a range of different MM-type schemes has been constructed for description of the total molecular energy as a function of molecular geometry. These generic MM schemes can be characterized as the QM APSLG-MINDO/3 method [] accompanied by special procedures of selection of the relevant electronic structure parameters - geminal amplitudes and hybridization angles. Within such a setting the parameters of generic MM schemes are (i) those of the underlying QM Hamiltonian, and (ii) the electronic structure parameters (ESPs). The latter are calculated by the APSLG-MINDO/3 method for some simple molecules and are used further according to a procedure adopted. The deformations of methane and water molecules have been used to discriminate the quality of different schemes for selecting the ESPs. The TOTA scheme results in a reliable agreement with the underlying QM APSLG method. The performed calculations on the organic molecules of different classes confirm this conclusion. An attempt to change the initial electronic structure parameters leads to slight improvement of the results.

The authors gratefully acknowledge valuable discussions with Dr. I.V. Pletnev. This work is performed under partial financial support on the part of the Federal Target Program of Russia ''Integracia'' (grant No A0078) and on the part of the RAS grant program for younger researchers No 6 (Grant No 120). Financial support for AMT from the Haldor Tops oe A/S is acknowledged as well.

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Variation range for a | |||||

geometry parameter | FAFO | TAFO | FATO | TATO | TOTA |

(Å or deg.) | |||||

change of one C-H bond length in methane | |||||

0.994¸1.194 | 0.031 | 0.021 | 0.013 | 0.004 | 0.0006 |

0.794¸1.594 | 2.119 | 1.241 | 1.075 | 0.246 | 0.038 |

0.794¸3.094 | 73.973 | 17.618 | 56.527 | 2.121 | 0.207 |

change of one O-H bond length in water | |||||

0.880¸1.060 | 3.131 | 2.676 | 0.047 | 0.405 | 0.037 |

0.800¸1.200 | 7.904 | 6.732 | 0.229 | 1.112 | 0.113 |

change of two O-H bond lengths in water | |||||

0.880¸1.060 | 3.039 | 2.835 | 0.028 | 0.409 | 0.006 |

0.800¸1.200 | 9.325 | 7.978 | 0.273 | 1.230 | 0.014 |

change of angle H-O-H in water | |||||

101.0¸108.0 | 115.23 | 98.48 | 0.32 | 13.67 | 0.30 |

97.0¸112.0 | 246.64 | 211.00 | 0.81 | 29.29 | 0.66 |

r(C-C), Å | APSLG | TOTA | D |

1.370 | -8.456 | -8.158 | 0.298 |

1.420 | -15.585 | -15.383 | 0.202 |

1.450 | -17.902 | -17.037 | 0.865 |

1.470 | -18.739 | -18.044 | 0.695 |

1.490 | -19.065 | -18.519 | 0.546 |

1.510 | -18.923 | -18.508 | 0.415 |

1.520 | -18.689 | -18.333 | 0.356 |

1.530 | -18.354 | -18.051 | 0.303 |

1.536 | -18.106 | -17.833 | 0.273 |

1.540 | -17.922 | -17.668 | 0.254 |

1.550 | -17.397 | -17.188 | 0.209 |

1.560 | -16.785 | -16.615 | 0.170 |

1.580 | -15.312 | -15.209 | 0.103 |

1.600 | -13.538 | -13.484 | 0.054 |

1.620 | -11.493 | -11.471 | 0.022 |

1.650 | -7.977 | -7.973 | 0.004 |

1.700 | -1.138 | -1.087 | 0.051 |

Molecule | APSLG | TOTA(1) | D(1) | TOTA(2) | D(2) |

CH_{4} | -8.414 | -8.414 | 0.000 | -8.397 | 0.017 |

H_{2}O | -55.648 | -55.605 | 0.043 | -55.605 | 0.043 |

C_{2}H_{6} | -18.106 | -17.833 | 0.273 | -17.844 | 0.262 |

C_{3}H_{8} | -23.438 | -22.805 | 0.633 | -22.869 | 0.569 |

C_{4}H_{10} | -26.951 | -26.200 | 0.751 | -26.295 | 0.656 |

C_{5}H_{12} | -32.093 | -31.157 | 0.936 | -31.283 | 0.855 |

cyclopropane | 18.520 | 18.877 | 0.357 | 18.806 | 0.286 |

CH_{3}OH | -47.677 | -47.014 | 0.653 | -47.001 | 0.666 |

C_{2}H_{5}OH | -59.112 | -57.905 | 1.207 | -57.997 | 1.115 |

C_{3}H_{7}OH | -63.811 | -62.386 | 1.425 | -62.520 | 1.291 |

File translated from T

On 19 Jul 2001, 17:56.