hybrid QM/MM methods, junction between subsystems, APSLG wave function

The hybrid QM/MM schemes for calculating large molecular systems acquire an increasing popularity []. In the literature there exist a large variety of hybrid QM/MM approaches differing by the QM and MM methods chosen and by specific forms of junction between classical and quantum subsystems. The common objects for applying the QM/MM schemes are large biological systems. Also other problems in the area of computational chemistry are related to the QM/MM setting: these are the problem of embedding in the cluster calculations on solids and surfaces; the problem of description of solute/solvent effects for reactions in condensed media; the pseudopotential descriptions of molecular and atomic electronic structure to mention several most important.

The purpose of the present work is to elucidate the precise form of the junction between the QM and MM regions in the hybrid QM/MM methods. It is more or less covered by the polarization effects when it takes place ''through space'' and there is no covalent bond between the subsystems. An opposite less transparent situation arises when chemical bonds exist between the QM and MM regions like in the case of reactions of large organic molecules.

It is now generally accepted that interregion boundary must not cut bonds because it leads to large fluctuations of electron density between subsystems causing numerous computational problems. Theoretical substantiation of such a choice of the interregion boundary has been given in Ref. TokmTchMis. The most popular way to perform the junction is based on using link atoms for saturation of free valence in the QM subsystem []. The link atom correction is, however, a poorly defined contribution to the energy. It leads, for example, to difference between potential and total energies, to collapse of link atoms with the neighbour QM atoms and to unphysical polarization of the boundary atoms. Also the question arises how to set the parameters for the link atoms.

Another group of methods uses localized orbitals [] coming from SCF procedures. These orbitals can follow the changes of molecular geometry on the basis of purely geometrical criteria. The main problem of these methods is the freezing of boundary one-electron states which does not allow to cover the effect of changes of the local environment upon them. Additional uncertainty is caused by poorly defined procedures of ''tail cutting'' of localized one-electron states. Also two other important features seem to be missed in the literature: (i) the renormalization of the MM part of the system due to variations of the electronic structure parameters (ESPs) on the boundary atom due to its participation in the QM subsystem, and (ii) the need that the adjustment of the QM residing hybrid orbitals (HOs) follows the variational principle for the energy when the geometry of the MM part changes. Here we present the formulae describing the linear response version of the theory resulting in the renormalization of the QM and MM parameters.

To perform sequential derivation of hybrid QM/MM schemes some kind of
derivation of the MM itself from
specially designed (underlying) QM method is necessary. The method used is
based on the
trial wave function in the form of antisymmetrized product of strictly
localized geminals (APSLG). Each geminal represents either a chemical bond or
an electron lone pair. This method requires special choice of the Arai
subspaces in
which the geminals are constructed. These subspaces are formed by two (or one
in the case of a lone pair) HOs *t*_{m} which in their turn are the result of an
orthogonal transformation of the valence (s-, p-) AO basis for each heavy
(non-hydrogen) atom:

| (0) |

| (0) |

| (0) |

| (0) |

Semiempirical implementation of this method using the MINDO/3
type of the Hamiltonian results in the following expression for the energy
[]:

| (0) |

| (0) |

It was shown [] that the precision of results obtained by
the APSLG-MINDO/3 method for
alkanes, alyphatic amines and alcohols is somewhat better than that obtained
by the standard SCF-MINDO/3 method. Moreover, the APSLG-MINDO/3
method possesses some special attractive features: (i) it operates with
intuitive chemical concepts; (ii) it is a special case of *O*(*N*)-scaling
methods and (iii) it has correct behaviour of the trial wave function for
the whole range of interatomic separations.

The first feature and MM-like representation of the energy Eq. (5)
allows one to use the APSLG-MINDO/3 method as a starting point to
theoretically derive additive schemes including the MM. The
derivation
of deductive molecular mechanics can be done by fixing the ESPs corresponding
to the geminals which perfect transferability was proven [].
In the vicinity of the energy minimum the linear response relation between
the variation of
geometry parameters *q* and the reaction of the ESPs *x* to the latter holds:

| (0) |

In the case of sp^{3} nitrogen the explicit form
of the pyramidalyzation potential as a function of pyramidalization angle
d is obtained:

| (0) |

| (0) |

The problem of describing the boundary atoms in the QM/MM junction can be solved in the present framework by noticing that the boundary atom keeps some of its HOs in the MM region and thus the geminal ESPs for them are fixed, whereas other orbitals are lended to the QM region.

Let us consider a boundary sp^{3} atom with one HO pointing to the QM region
and others related to the MM one. The one- and two-electron density matrix
elements for the QM residing HO are evaluated by the QM
procedure of choice and thus differ
from the fixed ESPs values accepted in the MM
region. Setting *m* = 1 for the HO belonging to the QM subsystem and assuming the
atom under consideration to be the ''right-end'' atom for the QM bond
the perturbation to the one-center energy for the boundary
atom due to participation of one of its HOs in the QM subsystem can be
written as:

| (0) |

| (0) |

In the QM subsystem the bond orders vary as well. The atoms in
the QM part have nonvanishing off-diagonal elements
of the one-electron density matrix between arbitrary one-electron states
ascribed to the QM subsystem. The corresponding contribution to the energy
reads:

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

It is possible that the boundary atom serving as the QM/MM junction is the sp^{3} nitrogen atom supplying its lone pair to the QM subsystem whereas three
chemical bonds remain in the MM subsystem. In this case the same happens as
above: the one-
and two-electron density matrix elements change which produces the pseudo-
and quasitorques acting on the nitrogen hybridization tetrahedron. The
effect of these pseudo- and quasitorques is quite different since the second
derivatives matrix Ñ_{[(w)\vec] }^{2}*E* must be now calculated for
the sp^{3} nitrogen atom. We
omit intermediate derivation because it is rather cumbersome and give here
only the explicit expression for the deformation (pyramidalization) momentum:

| (0) |

Any deformation in the MM system results in the
variation of the pseudo- and quasirotation angles by Eq. (7).
The shifts of the
positions of the MM neighbours of the boundary atom result in quasi- and
pseudotorques acting upon its hybridization tetrahedron. This
produces variations of both one-center parameters correspoding to the QM
residing HO and of the resonance parameters for the QM residing HO and all
other orbitals in the QM region. The variation of the one-center matrix
elements of the Hamiltonian corresponding to the QM HO is:

| (0) |

| (0) |

This work has been performed with financial support of RAS through the grant # 6-120 dispatched by its Young Researchers' Commission. Financial support for AMT on the part of the Haldor Topsoe A/S is acknowledged as well.

- []
- CECAM-NSF Meeting on QC/MM methods,
*International Journal of Quantum Chemistry***60**(1996) No 6, Special Issue. - []
- A.M. Tokmachev, A.L. Tchougréeff and I.A. Misurkin,
Effective electronic Hamiltonian for quantum subsystem in hybrid QM/MM
methods as derived from APSLG description of electronic structure of
classical part of molecular system,
*Journal of Molecular Structure*(*Theochem*)**506**, 17-34(2000). - []
- D. Bakowies and W. Thiel, Hybrid models for combined
quantum mechanical and molecular mechanical approaches,
*Journal of Physical Chemistry***100**, 10580-10594 (1996). - []
- D.M. Philipp and R.A. Friesner, Mixed
*ab initio*QM/MM modeling using frozen orbitals and tests with alanine dipeptide and tetrapeptide,*Journal of Computational Chemistry***20**, 1468-1494 (1999). - []
- A.M. Tokmachev and A.L. Tchougreeff, Semiempirical
implementation of strictly localized geminals approximation for analysis of
molecular electronic structure,
*Journal of Computational Chemistry***22**, 752-764 (2001). - []
- A.L. Tchougréeff and A.M. Tokmachev, Submitted.
- []
- A.M. Tokmachev and A.L. Tchougréeff, Generic molecular
mechanics as based on local quantum description of molecular electronic
structure,
*International Journal of Quantum Chemistry***88**, 403-413 (2002).

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On 28 May 2002, 18:01.