Selected-CI method

The HF method is a variational but since it is an ”uncorrelated” method, HF determinant (a single determinant) is usually not good enough to approximate the true wave function. Furthermore, in some cases due to a particular symmetry of a system the true wave function can not be approximated by a single determinant but a multi-determinant wave function (i.e., non-dynamic correlation effects).

The Full Configuration Interaction (Full CI) method might build an exact wave function as a linear combination of the HF determinant and all excited determinants (i.e., the full spectrum of the particle-hole expansion) in a complete basis set (CBS) limit. However, there are a few drawbacks in the application of Full CI. First of all, unfortunately, the CBS limit is currently not possible; truncated basis sets are employed. Second, the number of excited determinants grows exponentially with system size and the number of basis functions. Therefore, the determinant space (or configuration space) also needs to be truncated, resulting in the truncated CI. Third, in either the Full CI or the truncated CI, the hamiltonian matrix is diagonalized to find the optimum linear combination of the determinants [79]. This full matrix diagonalization is a significantly time-consuming step of the method and can be another obstacle to considering a larger space of determinants. Finally, following a CI calculation, a space with a large number of determinants is generated. The construction of a trial wave function in QMC by using this s determinant expansion is done usually by selecting the determinants according to the absolute values of their (normalized) coefficients, not according to the energy drop that they cause.

On the other hand, although the Selected-CI method also suffers from the finite basis set bias, it provides more flexibility in generating the excited determinant space than the truncated CI. Furthermore, the determinants are selected according to the energy drop they cause which may be effective in keeping the trial wave function at a high level of quality and also as compact as possible.

The Selected-CI method also provides more flexibility in generating the excited determinant space. Furthermore, the determinants are selected according to the energy drop they cause, which may be an effective feature in keeping the trial wave function high quality and also compact as much as possible.

Let|Ψ(n)ibe the wave function at the iterationnand let{|Dii}be the determinants. Also let

K(n). That is

|Ψ(n)i= X i

|Dii∈K(n)

c(_in)|Dii (7.2)

which is a solution of an eigen value equation of the form:

HΨ =EΨ (7.3)

where

H ←Hij =hDi|H|Dji f or Di, Dj ∈K(n). (7.4)

The Selected-CI algorithm can be summarized as follows:

i) Generate a space of excited determinants; a universal determinant space. Let it be denoted

as Ku. One may apply restriction in the generation of determinants since the number may

reach an prohibitive level. Alternatively, one can generate the determinant space dynamically (i.e, on-the-fly). That is, for instance, one may generate single and double excitations only from those determinants that are already in the current subspace,K(n).

Moreover, one has more options restricting the generation of determinants such as choosing active occupied and virtual orbitals, choosing level of excitations and even choosing symmetry of excitations (e.g., choosing perfect pair type excitations).

Before beginning iterations, one has an initial wave function, Ψ(0), originating from an initial subspace, K(0), and a universal determinant space, Ku which is either pre-defined or will be defined dynamically.

ii) Scan each determinant in Ku which does not exist in K(n) (for the iteration n); find the coupling of each determinant with|Ψ(n)i:

Hin=hDi|H|Ψ(n)i (7.5)

whereDi ∈/ K(n)butDi∈Ku. If the coupling is non-zero, then the energy change is calculated:

∆Ei = hDi|H|Ψ(n)i2 E(n)_−E ii = H 2 in E(n)_−E ii (7.6)

where Eii = hDi|H|Dii and E(n) = hΨ(n)|H|Ψ(n)i are the self-energies of the particular de- terminant and the current wave function, respectively. Please note that the determinants are normalized, i.e., hDi|Dji = δij. Also it is noteworthy that Hij = hDi|H|Dji is calculated ac- cording to Slater-Condon rules (see Sec. (2.4.3)).

iii) The determinant with the lowest ∆Ei is taken into the subspace:

K(n)+|Dζi →K(n+1)

whereDζ is the determinant such that ∆Eζ=min({∆Ei}).

iv) The hamiltonian matrix,Hij, is formed and diagonalized with the determinants in the new subspaceK(n+1) which yields a new wave function Ψ(n+1) and the energy E(n+1)

|Ψ(n+1)i= X i

|Dii∈K(n+1)

c(_in+1)|Dii (7.7)

E(n+1) =hΨ(n+1)|H|Ψ(n+1). (7.8)

v) Go to step ii) and repeat the steps until the size of the wave function or convergence in the energy reaches the desired value.

It is worthy to note that in step iii) one can also choose a number of best determinants for an iteration.

So far we have only pronounced determinants. However, a determinant is not a pure spin state, therefore, a selection scheme based on determinants may result in spin contamination. In order to avoid spin contamination, Configuration State Functions (CSFs) (see Sec. (2.4.4)) which are appropriate linear combinations of a number of determinants yielding pure spin states may be preferred. Switching to CSFs will also lower the number of functions in the universal determinant (now CSF) space.