Contents
Index:
Basis sets
In general, the basis sets used to calculate intermolecular interaction energies need to be large and augmented with diffuse functions. In part, this is because of the second-order energies which are response energies. Consequently, the basis set has to be good enough to describe not only the charge density accurately (particularly in the region of the density tail), but also the //response// of the density to perturbations. These requirements are adequately met only by basis sets of the augmented triple-$\zeta$ kind (and larger), such as aug-cc-pVTZ and Sadlej-pVTZ.
However, when the overlap of the charge densities of the interacting monomers becomes significant (typically at around the equilibrium separation) the basis set must be flexible enough to describe the intermolecular electron-electron cusp as well as the intermolecular charge-transfer (CT). The former effect is manifested as part of $E_{\rm disp}^{(2)}$ and the latter as part of $E_{\rm ind}^{(2)}$.
It is often quite difficult to converge $E_{\rm disp}^{(2)}$ with basis set. This is because it is hard to describe the intermolecular electron-electron cusp in the bonding region using basis functions located on atomic sites only 1. In this case, fairly high-angular-momenta functions are needed in order to do so. However, this leads to very large basis sets and a consequent increase in computational requirements.
An alternative is to use basis sets augmented with the so-called //mid-bond// functions: a small set of basis functions located in the bonding region 2. The dispersion energy is not so sensitive to the exact composition of the mid-bond basis, which is usually chosen to consist of a set of 3 $s$, 2 $p$ and one $d$ diffuse functions 3. A convenient choice for the location of the mid-bond set has been given in 4.
An alternative to mid-bond functions is to use the basis-set extrapolation scheme of 5. For intermolecular interactions, this involves calculating $E_{\rm disp}^{(2)}$ using two correlation-consistent basis sets, say the aug-cc-pV$n$Z and the aug-cc-pV$(n+1)$Z Dunning bases. This energy can be fitted to the form $a + b/X^3$ where $X=n$ and $n+1$ and the constants determined. The complete basis set (CBS) estimate of \Edisp{2} is obtained by extrapolating $X \rightarrow \infty$, i.e., it is the constant $a$. This scheme has the advantage that it applies equally well to small as well as large molecules, for which the mid-bond scheme is potentially ambiguous. However, two electronic structure calculations need to be performed, so there is an increase in computational cost.
Charge transfer poses a very different problem. This subject is discussed more completely in \citet{StoneM09a,Stone:book:96}. The CT energy is the part of the short-range induction energy that involves excitations from the occupied orbitals on one molecule into the virtual orbitals of another. Consequently, in \citet{Stone93} it was suggested that the CT energy could be calculated as the difference of \Eind{2} calculated in the dimer and monomer basis. The understanding here is that the monomer basis is localized, so excitations into the virtual space of the interacting partner should be suppressed. In any case, if CT excitations are to be correctly included, the induction energy should be calculated using the dimer basis. In practice this is rather excessive and it has been shown \cite{WilliamsMSJ95} that only a subset of the basis of the interacting partner need be included. This is typically taken to be just the functions of $s$ and $p$ symmetry. Thus, when calculating the induction energy of A in the presence of B, i.e. $E_{\rm ind}^{(2)}(A)$, we use in addition to the basis set of A, the $s$ and $p$ symmetry functions of B located where the atomic sites of B would be. These ghost functions are known as the {\em far-bond} functions.
The combined basis consisting of functions on the molecule and the mid-bond and far-bond functions is referred to as the {\em monomer centred plus} or MC+' basis type. The +' indicating the presence of the additional basis functions. A typical SAPT(DFT) calculation of the interaction energy would involve two such bases; one for each of the two interacting molecules.
The additional functions are required for the second-order energies. If all that is needed are the first-order energies, the MC basis can be used, that is, basis functions need be included only on the monomer. First-order electrostatic and exchange-repulsion energies calculated in such a basis are generally very close to those calculated in the larger MC+ type of basis if a triple-$\zeta$-quality basis is used.
Finally, if each molecule is described using the dimer basis we obtain the DC or DC+ basis types; the latter additionally including the mid-bond set. These basis types must be used for supermolecular calculations of the interaction energy.