Calculation of the Interaction Energy

Two theoretical methods can be used for calculating the interaction energy (A E). The supermolecular variation method which determines AE as the difference between the energy of the cluster (supermolecule) and the energies of the isolated systems. The perturbational method gives A £ directly as the sum of physically distinct contributions such as electrostatic, induction, dispersion, and exchange-repulsion. Both the methods have advantages and drawbacks. The supermolecular approach is theoretically able to provide an interaction energy at any level of accuracy, providing that a sufficiently high percentage of the correlation energy is covered. The advantage of this approach is that it is simple and straightforward and any quantum chemical code can be used. The advantage of the perturbational method is that AE is obtained directly and not as the energy difference. It is, however, supermolecular approach which nowadays is used almost exclusively. The perturbation approach, in the form of symmetry-adapted per turbation theory [18], is used for highly accurate calculations providing benchmarks for supermolecular calculations.

In the following parts various theoretical supermolecular procedures will be used for evaluation of interaction energy and other properties of molecular clusters.

3.3.3.1 Nonempirical ab initio Variational Method

The interaction energy of the complex is evaluated as a sum of the SCF (self consistent field) interaction energy (A£SCF) and the correlation interaction energy (AE"m)

The former contribution covers electrostatic, induction and exchange-repulsion terms. The dispersion energy, which plays important role especially in case of interaction of very large (biological) systems, is not included in the AZsSCF term. To include this contribution, the supermolecular treatment must include energy terms originating in the correlation of electron movements. AECOR in Eq. (1) covers not only the dispersion energy but also other less important contributions.

Correlation interaction energy is always important and cannot be neglected; evaluation of this term is much more tedious and time-demanding than the evaluation of A£SCF. Among suitable methods, the M0ller-Plesset (MP) perturbational theory is now in common use. The second-order MP theory (MP2), which is easily applicable even to large clusters (having up to 100 atoms), gives surprisingly good estimates of A£COR. MP2 covers contributions from the double-electron excitations in the second pertur-bative order. The gopd performance of MP2 is, however, due to the partial compensation of higher-order contributions.

More accurate values of A£C0R result if the MP theory is performed through the 4th order. The double electron excitations are described in the 2nd, 3rd, and 4th order. In the 4th order also single, triple, and quadruple electron excitations are covered; among them, triple excitations play a dominant role. Unfortunately, their evaluation is extremely time consuming, much more than that of the other contributions at the 4th perturbative order. The next step in the accuracy of A£'COR results from the use of the coupled-cluster (CC) method.

Basis set

Choice of the basis set is very important and the quality of the basis set needed depends on the nature of the cluster, specifically on the role of the correlation interaction energy. If the stabilization energy is properly described by the Hartree-Fock interaction energy (true for ionic, and H-bonded clusters), then relatively small basis sets give accurate values of the stabilization energy. On the other hand, the basis sets for London clusters, where all the stabilization comes from the correlation interaction energy, must be considerably larger.

Calculating the correlation energy even at the lowest, perturbational level (MP2), brings a limitation to the size of the cluster, and clusters having more than about 200

atoms are prohibitively large. The SCF interaction energy itself can, however, be evaluated for much larger clusters.

3.3.3.2 Density Functional Theory

In density functional theory (DFT) the exact Hartree-Fock exchange is replaced by a more general expression - the exchange-correlation functional [19]. The DFT energy thus includes terms accounting for both the exchange and correlation energies. Let us recall that the HF theory covers only the exchange energy and the additional evaluation of the correlation energy is tedious and time consuming.

Encouraging results were obtained with DFT for isolated molecules by incorporating nonlocal, density-gradient terms in the exchange and correlation.functional [20]. The use of DFT for molecular clusters is, however, limited for the following reasons [21]: i) the dispersion energy is not included and consequently no minimum is found for London clusters; ii) the stabilization energy of charge-transfer clusters is strongly (about ten times) overestimated; iii) in the case of H-bonded clusters, the DFT may incorrectly predict the structure of the global minimum. It can thus be concluded that the use of the DFT method in the realm of vdWand biological clusters in rather limited and cannot be recommended.

3.3.3.3 Semiempirical Methods

The use of semiempirical methods of quantum chemistry for vdW clusters cannot be recommended. None of the methods including those recently developed (AMI and PM3) led to reliable results for various types of vdW clusters. The most complicated task is imposed by London clusters; semiempirical methods are not able properly to evaluate the dispersion energy. On the other hand, semiempirical methods can be used under some conditions for H-bonded clusters. It is, however, necessary first to confirm nonempirically the validity of semiempirical results for several H-bonded clusters of the type considered (e.g., H-bonded DNA base-pairs). Without nonempirical verification, the semiempirical methods cannot be used, even for the H-bonded clusters.

3.3.3.4 Empirical Procedures

Molecular mechanics methods, also called semiempirical force field or empirical potentials differ basically from the nonempirical or semiempirical methods of quantum chemistry because they are not based on solving Schrodinger equation. These methods treat molecules as systems composed of atoms held together by bonds, and deal with the contributions to a molecule's electronic energy from bond stretching, bond bending, van der Waals attraction and repulsion between nonbonded atoms, electrostatic interaction due to polar bonds, and energy changes accompanying internal rotation about single bonds. Empirical procedures were developed and parametrized for specific, rather narrow classes of systems. The methods are mostly successful within these classes of system and theoretical results agree with the experimental features of systems in the ground electronic state.

Widely used potentials include AMBER [22] and CHARMM [23], both parametrized for proteins and nucleic acids, and MM2 and MM3 [24], designed for hydrocarbons. The BIOSYM force fields calledCVFF and CFF91 [25], both parametrized for peptides and proteins, represent the most recent development in the field.

Parametrization of these force fields was based on various experimentaldata like geometry, conformation, various heats (formation, vaporization, sublimation, solvation, adsorption), and coefficients (virial, viscosity, transport). The basic idea of use and application of empirical procedures is based on believe that parameters of force fields are transferable, i.e., transferable from the systems used for parametrization to the system under study (not included in the parametrization). Evidently, the narrower is a set (in the sence of structural types) of systems used for parametrization, the larger chance to obtain reliable results for system studied (belonging to this "narrow" set of systems). An alternative to empirical potentials parametrized on the basis of experimental quantities are potentials derived from theoretical quantum chemical calculations. The main advantage of the latter procedure is that it can be applied to any type of interacting systems.

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