Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.7 Theoretical and computing modeling 479
The third term in eq. [8.112] is the interaction energy between solvent molecules. It is
usually approximated by the molecular mechanics force field, which in general contains
bond stretching, angle bending, dihedral torsion and non-bonded terms (see Section 8.4.5).
Once the Hamiltonian of the system is defined, the total energy of an instantaneous
configuration sampled during an MC or MD simulation is determined as:
E = ΦHΦ = E + E + E
� [8.114]
M MS SS
More details on the procedure can be found in literature. 78
The forces used to integrate Newton’s equation of motion in MD simulations are determined
by differentiating eq. [8.114] with respect to nuclear coordinates.
In the practical QM/MM calculation, the solvent S is equilibrated first using simulations
(or a RISM version of the integral equation approach, see Section 8.7.1.1), and then
the electronic structure of the solute M is modified via an iterative QM procedure in the
presence of a fixed potential of the S component. The procedure is repeated iteratively; at
the next step the S distributions is determined again, now taking into account the modified
description of M. This sequence of steps is repeated until convergence. A problem of such a
methodology is that changes in the internal geometry of M must be treated apart, scanning
point-by point the relevant PES, not being available analytical expressions for first and second
derivatives of the energy with respect to nuclear coordinates, which would be necessary
to take into account any modification in the solute geometry.
To overcome problems arising from the finite system size used in MC or MD simulation,
boundary conditions are imposed using periodic-stochastic approximations or continuum
models. 75 In particular, in stochastic boundary conditions the finite system is not
duplicated but a boundary force is applied to interact with atoms of the system. This force is
set as to reproduce the solvent regions that have been neglected. Anyway, in general any of
the methods used to impose boundary conditions in MC or MD can be used in the QM/MM
approach.
8.7.3 CONTINUUM MODELS
Originally, continuum models of solvent were formulated as dielectric models for electrostatic
effects. In a dielectric model the solvent is modeled as a continuous medium, usually
assumed homogeneous and isotropic, characterized by a scalar, static dielectric constant ε.
This model of the solvent, that can be referred to the original work by Born, Onsager and
Kirkwood 60-80 years ago, assumes linear response of the solvent to a perturbing electric
field due to the presence of the molecular solute.
This simple definition then has been largely extended to treat more complex phenomena,
including not only electrostatic effects; and nowadays continuum solvation models
represent very articulate methodologies able to describe different systems of increasing
complexity.
The history of, and the theory behind, continuum solvation models have been described
exhaustively in many reviews 31a,80,81 and articles 82-84 in the past, so we prefer not to
repeat them here. In addition, so large and continuously increasing is the amount of examples
of theoretical developments on one hand, and of numerical applications on the other,
that we shall limit our attention to a brief review of the basic characteristics of these models
which have gained wide acceptance and are in use by various research groups.