Karpov Institute of Physical Chemistry

Vorontsovo pole 10, Moscow 103064 RUSSIA

**Key words**: hybrid QM/MM methods, perturbation expansion, effective
Hamiltonian, potential energy surfaces, strictly localized geminals.

The hybrid quantum mechanical/molecular mechanical (QM/MM) methods became
very popular in modern research on large molecular systems (particularly,
biological ones) []. These approaches are based on the well
justified experimental observation: chemical transformations are usually
local, *i.e.* restricted to a small region of the entire molecular
system called reaction center (RC). For that reason it seems to be practical
to use approximations of different level in order to describe different
parts of the system when its potential energy surface (PES) is considered.
The smaller part of the molecular system with actually transforming
electronic structure (RC) must be described by a thorough (preferrably,
highly correlated) quantum chemical approach while the rest of the molecular
system can be described either by molecular mechanics or by a faster quantum
chemical method capable to reproduce only general features of the PES. The
physical reason of that difference is the relative importance of electron
correlations for correct description of the processes of the bond formation
and bond cleavage - those where the uncorrelated SCF approach gives wrong
asymptotic behavior of the electronic wave function of the system.

Two important questions immediately arise for such hybrid schemes. The first one is about the margin between the QM and MM subsystems. The important restriction must be set on the subsystems. Elaborated quantum chemical approaches (as well as the molecular mechanical schemes) work properly only being applied to the objects with an integer number of electrons. Therefore, distributing electrons between the subsystems is restricted by the condition of small fluctuations of numbers of electrons in the subsystems.

The second question arises from the fact that the classical part of the
molecule modifies the PES of the reactive (quantum) subsystem. The
interaction between subsystems can not be neglected. It leads to various
forms of the junction between quantum and classical subsystems accepted in
many hybrid schemes implemented by now in the literature
Aqvist,Bak,Carmer,Field,Field1,Gao,Gao1,Gao2,Luzhkov,Maseras,Stanton,Thompson,Tunon,Truong,Morokuma,Bernardi. At the same time it is noteworthy that the form of the junction in these
approaches is usually taken *ad hoc*, without proper substantiation. For
example, authors of Ref. [] represent the junction between
subsystems as a sum of electrostatic and van der Waals interactions with
adjusted parameters; in Ref. [] ''junction dummy atoms'' are
introduced; in Ref. [] the intersystem Coulomb and exchange
integrals are taken as linear combinations of exponential functions with
subsequent parameterization.

From out point of view the correct form of the junction between the quantum
and classical parts of the molecular system must be constructed on the basis
of the quantum chemical description of the whole system and of the
sequential separation of electronic variables related to respective
subsystems. This approach was originally proposed in Ref. []. It
is based on the perturbation expansion and uses effective electron
Hamiltonian technique. The margin between quantum and classical subsystems
is chosen so that the chemical bonds are not broken and can be assigned only
to one subsystem. The construction of hybrid schemes in Ref. []
supposed that the parameters of the MM method must be renormalized due to
interaction with quantum subsystem. Another construction of hybrid schemes
was proposed in Ref. []: MM method remains constant and
interactions modify only Hamiltonian of quantum subsystem. The effective
Hamiltonian of quantum subsystem was explicitly constructed in Ref.
TheoChem. The main idea of all these works is to single out the subset of
electronic variables responsible for a chemical transformation to be
described with use of a QM approach and to construct (using the Löwdin
partition technique []) the effective Hamiltonian for the
reactive subsystem which takes into account the interactions between
subsystems perturbatively. To perform this program *i.e.* to construct
the correction to the bare Hamiltonian for the RC it is necessary to assume
some form of the electronic wave function of the inert environment
underlying its MM description. This function has been chosen to be of the
form of antisymmetrized product of strictly localized geminals (APSLG)
Surjan,Weeny,ZhFizKhim,JCC. The APSLG based approach operates by local
quantities such as chemical bonds and electron lone pairs, describes the
ground states of the organic molecules with closed electron shells and high
excitation energies, and provides the natural starting point for a
transition from the quantum mechanics to the molecular mechanics (to be
described elsewhere []). The effective Hamiltonian of the
reactive subsystem in the presence of ''classical'' subsystem described by
the APSLG method has been explicitly constructed in Ref. [].
The present work is aimed to construct the PES of the molecular system
described by the hybrid method with the APSLG wave function underlying the
MM description of the classical subsystem.

Let us consider the application of the effective electron Hamiltonian method
to the separation of the electronic variables which is aimed to constructing
the PES for molecular systems in the hybrid QM/MM methods. We denote the
reactive subsystem (described by quantum mechanics) by subscript *R* and the
inert subsystem (described by MM) by subscript *M*. The Hamiltonian for the
whole molecular system can be divided into parts corresponding to separate
subsystems and to the interaction between them:

| (0) |

The wave function for the whole system is then approximated by the
antisymmetrized product of the wave function of the reactants (which can be
taken as a linear combination of configurations) F_{k}^{R} (where *k* is
the number of the electronic state of the *R*-subsystem) and the ground
state wave function of the inert environment to be calculated without
reactants F_{00}^{M}:

| (0) |

The wave function F_{00}^{M} is the approximation to the lowest
eigenstate of the Hamiltonian *H*_{0}^{M}. It is assumed to be obtained in the
framework of the APSLG method [,,,]. Therefore,
it has the form:

| (0) |

To construct these geminals a unitary transformation must be performed from
the initial atomic basis set to the hybrid one for each heavy (non hydrogen)
atom:

| (0) |

Each geminal is a linear combination with varying amplitudes of three
possible singlet states (two ionic and one covalent - Heitler-London type)
constructed from the hybrid spin-orbitals | *r*_{ms}
ñ
and | *l*_{ms}
ñ :

| (0) |

The approximate wave function Eq. (2) can be obtained from
the exact wave function of the whole system

| (0) |

The projections imply the transition from the exact Hamiltonian for the
entire molecular system to the relevant effective Hamiltonians. The
construction of the effective Hamiltonian is performed to fulfill the
requirement that the eigenvalues of the effective Hamiltonian acting in the
restricted subspace coincide with those of the original Hamiltonian acting
in the full space. First projection yields the effective Hamiltonian of the
form:

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

The energy of the *k*-th state of the combined system can be obtain by
averaging the effective Hamiltonian over the approximate wave function Eq. (2):

| (0) |

| (0) |

| (0) |

To perform the averaging the explicit form of the operators of the Coulomb
and resonance interactions is needed. The operator of the Coulomb
interaction is taken to conserve the number of electrons in the subsystems
and thus can be written as:

| (0) |

| (0) |

The operator of resonance interaction describes the one-electron transfers
from the *M*-subsystem to the *R*-subsystem or *vice versa*:

| (0) |

| (0) |

| (0) |

| (0) |

The Coulomb electron interaction between the subsystems contributes to the
energy:

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

Using these notations we obtain the contribution to the energy from the
operator *V*^{rr}(*q*,*E*):

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

The contribution to the energy of the second order in operators *V*^{R}(*q*) does not change the relative energies of the states in the *R*-subsystem and, therefore, is *k*-independent, where *k* is the number of
electronic state in the *R*-subsystem. The operator *V*^{R}(*q*) can be
written in the form:

| (0) |

| (0) |

| (0) |

The cross term between the *V*^{c}(*q*) and *V*^{R}(*q*) equals to:

| (0) |

| (0) |

The next two contributions of Eq. (40), including the
operator *V*^{rr}(*q*,*E*), correspond to the third order in the perturbation
expansion. Therefore, they may be only roughly estimated. The main
assumptions are the ZDO approximation and that the energies of an electron
transfers between the subsystems are replaced by average ones depending only
on the number of bond taking part in the electron transfer. The cross term
between the *V*^{c}(*q*) and *V*^{rr}(*q*,*E*) is:

| (0) |

Analogously, the cross term between *V*^{R}(*q*) and *V*^{rr}(*q*,*E*) equals
to:

| (0) |

Large molecular systems are very important from the practical point of view.
Polymers and many biological molecules contain hundreds and thousands atoms.
Modern *ab initio* methods do not allow to calculate the electronic
structure of such systems because the computational time for these methods
grows as *N*^{4}¸*N*^{7}, where *N* is number of one-electron basis functions.
Even in the case of the standard semiempirical methods based on the
Hartree-Fock approximation this growth is proportional to *N*^{3}. It explains
the especial role of the hybrid QM/MM schemes in the computational
chemistry. There is a large diversity of the hybrid QM/MM schemes proposed
in the literature due to the form of junction between quantum and classical
subsystems. It leads to divergence of results obtained with different
junctions. The results of the calculations seems to be ambiguous due to
uncontrolled influence of the junction on the calculated PES of the
molecular system.

We try to justify the form of the junction between subsystems. The complete
neglect of the junction, *i.e.* the use of the bare (''*ab initio*'') Hamiltonian for the *R*-subsystem and standard molecular mechanical
procedure without renormalization leads to the energy *E*_{k0}^{R}+*E*_{0}^{M}
instead of the correct value *E*_{k}. Taking into account the interaction
between the system (junction) results in a correction to the PES, which can
be represented as a sum of contributions:

| (0) |

One of us (A.M.T.) acknowledges financial support from the Haldor Topsø e A/S.

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