6.20 Incremental Correlation Methods

6.20.1 Theory

(May 16, 2021)

Electronic energy is retrieved by iFCI using an n-body expansion of the form

E=Eref+EC=Eref+iϵi+j<iϵij+k<j<iϵijk+ (6.67)

where each ϵX term denotes an increment of correlation energy and i,j,k refer to bodies of the expansion. Incremental correlation energies are defined as

ϵi = EC(i) (6.68)
ϵij = EC(ij)-ϵi-ϵj (6.69)
ϵijk = EC(ijk)-ϵij-ϵik-ϵjk-ϵi-ϵj-ϵk (6.70)

where terms n>1 subtract lower-order increments to avoid double counting. Terms represent n-body additions to the correlation energy from 2n electrons in the mean field of the remaining 2(N-n) electrons, where each ϵX value is computed by solving CAS-CI for 2n electrons in Nv+n orbitals. For example, n=1 performs CAS(2,Nv+1)-CI to give the value of EC(i)=E(CAS(2,Nv+1)). Proceeding likewise for higher n, CAS(2n,Nv+n)-CI produces each EC(X).

Heat-bath CI (HBCI) is utilized to solve each CAS-CI Hamiltonian, performing selected CI computations according to determinants, j, coupled to the CI wave function in the form |Hijci|>εi, where εi is the energy cutoff and ci are determinants in the HBCI subspace.

Truncation of incremental terms is performed by considering natural orbital (NO) occupancy cutoffs, η(m), where

ϵi(m) = EC(i;η(m)) (6.71)
ϵij(m) = EC(ij;η(m))-ϵi(m)-ϵj(m) (6.72)

Doing so reduces the size of the virtual space by only including virtual orbitals with sufficiently large NO eigenvalues. Convergence for each iFCI increment is reached when

ζ>|ϵ(m+1)-ϵm| (6.73)

with units of 10-ζ Eh. Further truncation in n3 can be performed by utilizing the ζ parameter and a screening cutoff, 𝒞n, in the form

𝒞n=10-ζ×𝒮n (6.74)

where 𝒞n is in Eh and 𝒮n is a scalar. This screening is performed by selecting n-1 body correlation energy contributions that are above 𝒞n. See Ref. 916 for more details. 𝒮n is a parameter in the input.

iFCI requires a high-spin perfect pairing (PP) reference, where NOs are localized as local bonding-antibonding pairs, or geminals.