Probably the sim plest ab initio method is to calculate the energy o f an entire system, Emn o f two molecules, as well as the energies o f the isolated molecules, Em and E^. By (3.1), the interaction energy, Umn, is sim ply the difference between the dimer and the sum o f monomer energies. There are many program s available for such calculations and the only required input is the list o f atomic coordinates and a specification o f the approximations {i.e. basis set and level o f theory) to use in solving for the wavefunction. The calculation for the dim er is not treated differently from a single m olecule calculation, so is termed a supermolecule calculation. M ost ab initio calculations are variational, m eaning that the calculated energy o f the system is always an upper bound for the system's true ground state energy. Because o f the variational principal, if a change is made that lowers the calculated energy, then the calculation is in some way better. In this way, the true ground state energy can be approached systematically. The problem is that the variational principle says nothing about the quality o f energy differences. Even if the energies Emn, Em, and E ^ are all improved, for exam ple by using a bigger basis set, there is no guarantee that the resulting interaction energy is closer to the true value. Besides the non-variational nature o f the energy differences, the technical difficulties with the supermolecule approach are basis set superposition error and size consistency.
Basis set superposition error (BSSE) is a result o f using a linear combination o f a finite {i.e. incomplete) set o f functions to describe the wavefunction. Essentially, when the tw o molecules are brought together, the basis functions on molecule M are available to molecule N to variationally lower its energy and vice versa, leading to an erroneously low energy for the superm olecule and an
L attice D ynam ical S tu d ies o f M olecular C rystals G raem e M . D ay w ith A pplication to Polym orphism and Structure P rediction 2003
Chapter 3. Theory and Modelling o f Intermolecular Forces_____________________________________ 45 exaggerated binding energy. There are corrections for this e r r o r , s o BSSE simply complicates the calculation. A more serious problem is with the size consistency o f computational m ethods, which affects some methods for including electron correlation in ab initio calculations. One o f the m ost common correlated methods is configuration interaction, where excited configurations o f the wavefunction are mixed into the SCF ground state configuration. Unless all possible configurations are included, then the energy o f a pair o f molecules at infinite separation is not equal to the sum o f m onom er energies. This is a serious hindrance to calculating interaction energies. Thankfully, other methods for including electron correlation, such as M oller-Plesset perturbation theory {e.g. M P2 calculations) do not suffer from this problem.
The technical problems with supermolecule calculations for the intermolecular interaction energy can be overcome and more and more accurate calculations are becoming accessible w ith increasing com puter speed and memory. However, the number o f geometries that m ust be sampled to fit a model potential to such calculations is normally very high to sample all relative orientations o f the molecules. One last drawback is that the calculation gives a value for the energy that is not broken down into the constituent contributions {e.g. repulsion, dispersion, electrostatic, etc.) to the interaction, so cannot be used to calibrate terms in a model potential separately. M orokum a and Kitaurat^ -^^‘^^1 suggested an approach to the decomposition o f the interaction energy, but it runs into problems o f basis set superposition error.
3.2.2.2 P e rtu rb a tio n m ethods
Another approach for the computation o f the interaction energy uses perturbation theory, as introduced in Section 3.1. For a general method that is applicable at both long- and short-range, the problem o f antisymmetrisation o f the product w avefunction |/nn) must be addressed - the simple product o f individual wavefunctions that we used in Section 3.1 is not antisymmetric with respect to exchange o f electrons and this is important at short range. Two general approaches have been used in place o f the standard Rayleigh-Schrôdinger perturbation theory - 'symm etric' and 'sym m etry adapted' perturbation theories.
In the former, the unperturbed state is the set o f antisymmetrised products o f the individual wavefunctions. These are non-orthogonal, so cannot be eigenstates o f any Hermitian operator, i.e. there can be no zeroth order Hamiltonian that has these antisym m etrised products as eigenstates. One formalism, called intermolecular perturbation theory (IM PT), has been developed by Hayes and Lattice D ynam ical S tudies o f M o lecu lar C rystals G raem e M . D ay w ith A p plication to Polym orphism and Structure P rediction 2003
Chapter 3. Theory and Modelling o f Intermolecular Forces 46 Stonet^-^2’^^1 for using this non-orthogonal basis in perturbation theory, expanding the perturbation about the leading diagonal o f the Hamiltonian and giving the interaction between SCF wavefunctions. In the other approach, symmetry adapted perturbation theory (SAPT), the unperturbed states are taken as the simple products o f the monomer states (as in the long-range treatment, Section 3.1.1), without antisymmetrisation, essentially assigning each electron to one o f the molecules in the unperturbed state. The antisymmetrisation is then applied at each order o f perturbation theory. The approach can be more accurate because the m onomer wavefunctions can include electron correlation.
The first order energy for all perturbation theories is the energy difference between the zeroth order energy and the expectation value o f the Hamiltonian for the antisymmetrised wavefunction without excitations | mn) = | OO) ; this is the electrostatic (including penetration) plus exchange- repulsion energy. The separation o f the latter into exchange and repulsion is somewhat a choice o f the method and, in Hayes and Stone's symmetric intermolecular perturbation theory (IM PT) scheme,[^-^^l the exchange energy is taken as the additive attractive component, while the repulsion energy is the energy contribution that is always repulsive and has an explicit overlap dependence (and is, therefore, non-additive).
The energy at second order is dependent on the perturbation approach because different expansions and choices o f the unperturbed state lead to different distributions o f the energy contributions at different orders o f perturbation. We focus on the IM PT scheme o f Hayes and Stonel^ -^^1 because this is what we use to calibrate model potentials in Chapter 8. The scheme has been implemented to second order including only single and double excitations. Using a finite basis set, m olecular orbitals can be unam biguously assigned to one o f the two interacting molecules, which allows a convenient separation o f energy contributions. The single excitations are classified as polarisation if the occupied and virtual orbitals belong to the same molecule and charge-transfer if the electron is excited from one molecule to the other. In the limit o f a complete basis set, the basis functions o f molecule M can completely describe the wavefunction o f molecule N and vice versa, so all single excitations can be described as polarisation and the charge-transfer term becomes zero. Thus, the split o f induction and charge-transfer energies is basis set dependent. The latter is seriously contaminated by basis set superposition error, but can be corrected to be virtually free o f BSSE e f f e c t s . E n e r g y terms involving double excitations give contributions to the dispersion energy, a 'charge-transfer correlation'
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Chapter 3. T heory and M odelling o f Intermolecular Forces 47 and /«rrom olecular correlation energy, the last o f which hardly contribute to the interaction energy after correction for BSSE.t^-^^1