Subtraction of these two equations leads to a recipe for the charge transfer due to the external potential.

Since the potential depends on the charges at all sites, this must be solved iteratively.

Once the iterations have converged, the electrostatic energy is given by the simple Coulomb form in eq. (2.20).

The fluctuating charge model is a simple way of introducing a coupling between the electrostatic energy and geometry, but it should only be considered as a first approximation, as it is unable to account for example for the charge polarization of a planar molecule where the external field is perpendicular to the molecule plane. An explicit incorporation of polarization can be done by including an atomic polarization tensor,25 and this also implicitly accounts for some of the geometry dependence of the atomic charges. The polarization contribution to the electrostatic interaction is at the lowest order given by a dipolar term (mind) arising from the electric field (F = df/dr) created by the electric moments at other sites multiplied by the polarizability tensor (a).26

Note that since the (atomic) hardness is inversely related to the average polarizabil-ity, the charge transfer in eq. (2.28) is essentially the average polarizability times the potential. As each of the atoms contributes to the electric field at a given position (eq. (2.29) with additional contributions from dipole moments), the set of atomic dipoles must be solved self-consistently by iterative methods. For molecular dynamics simulations, the change in the induced dipoles or charges with geometry can be treated by an extended Lagrange method (Section 14.2.5), where fictive masses are assigned to the dipoles or charges, and progressed along with the other variables in a simulation.24 When adding multipole and/or polarizability terms, a decision has to be made on which interactions to include and which to neglect, and either the multipole order or the distance dependence of the interaction can be used as a guiding criterion. The distance dependence on the interactions between multipoles is given in Table 2.2.

Table 2.2 Distance dependence of multipole interactions

If all interactions between multipoles up to quadrupoles are included, then only some of the interactions having Rr4 and Rr5 distance dependencies are accounted for. Alternatively, if the distance dependence is taken as the deciding factor, then quadrupoles are required for including all interaction of order Rr3 or lower, but the dipole-quadrupole and quadrupole-quadrupole interactions should not be included.

The polarizability in eq. (2.30) corresponds to an electric field inducing a dipole moment, but higher order polarizabilities giving induced quadrupole and octopole moments are also possible, although these will usually be significantly smaller. It is at present unclear how many multipole moments, which interaction and level of polarizability should be included for a balanced description. The charge-induced dipole interaction has a distance dependence of Rr4, while the dipole-induced dipole interaction is Rr6, suggesting that the former should be included when quadrupole moments are incorporated. There is also no clear picture of what kind of improvement for the calculated results can be obtained by including these higher order effects.

Incorporation of electric multipole moments, fluctuating charges and atomic polar-izabilities significantly increases the number of fitting parameters for each atom type or functional unit, and only electronic structure methods are capable of providing a sufficient number of reference data. Electronic structure calculations, however, automatically include all of the above effects, and also have higher order terms. The data must therefore be "deconvoluted" in order to extract suitable multipole and polarization parameters for use in force fields.27 A calculated set of distributed dipole moments, for example, must be decomposed into permanent and induced contributions, based on an assigned polarizability tensor. Furthermore, only the lowest non-vanishing multipole moment is independent of the origin of the coordinate system, i.e. for a non-centrosymmetric neutral molecule the dipole moment is unique, but the quadrupole moment depends on where the origin is placed.

It should be noted that the transfer of polarization data from gas-phase calculations to a condensed phase may lead to errors. The close spatial arrangement of the molecules in a condensed phase will display quantum mechanical exchange phenomena, which will reduce the effective polarization. The possibility of charge transfer between molecules, however, can lead to an enhancement of the polarization relative to the gasphase result.28 It is therefore likely that polarizable force fields will need to be re-tuned to reproduce experimental results, unless of course the quantum mechanical effects are incorporated directly.

The addition of multipole moments increases the computational time for the electrostatic energy, since there now are several components for each pair of sites, and for multipoles up to quadrupoles the evaluation time increases by almost an order of magnitude. If bond midpoints are added as multipole sites, the number of non-bonded terms furthermore increases by a factor of ~4 over only using atomic sites. Inclusion of polarization further increases the computational complexity by adding an iterative procedure for evaluating the induced dipole moments, although recent advances have reduced the computational overhead to only a factor of 2 over a fixed charge model.29 Advanced force fields with a description beyond fixed partial charges of the electrostatic energy have consequently only seen limited use so far. Nevertheless, the neglect of multipole moments and polarization is probably the main limitation of modern force fields, at least for polar systems, and future improvements in force field techniques must include such effects.

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