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# 6.20.2 Theory

(February 4, 2022)

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

 $E=E_{\mathrm{ref}}+E_{C}=E_{\mathrm{ref}}+\sum_{i}\epsilon_{i}+\sum_{j (6.67)

where each $\epsilon_{X}$ term denotes an increment of correlation energy and $i,j,k$ refer to bodies of the expansion. Incremental correlation energies are defined as

 $\displaystyle\epsilon_{i}$ $\displaystyle=$ $\displaystyle E_{C}(i)$ (6.68) $\displaystyle\epsilon_{ij}$ $\displaystyle=$ $\displaystyle E_{C}(ij)-\epsilon_{i}-\epsilon_{j}$ (6.69) $\displaystyle\epsilon_{ijk}$ $\displaystyle=$ $\displaystyle E_{C}(ijk)-\epsilon_{ij}-\epsilon_{ik}-\epsilon_{jk}-\epsilon_{i% }-\epsilon_{j}-\epsilon_{k}$ (6.70) $\displaystyle\cdots$

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 $\epsilon_{X}$ value is computed by solving CAS-CI for $2n$ electrons in $N_{v}+n$ orbitals. For example, $n=1$ performs CAS($2,N_{v}+1$)-CI to give the value of $E_{C}(i)=E(\mathrm{CAS}(2,N_{v}+1))$. Proceeding likewise for higher $n$, CAS($2n,N_{v}+n$)-CI produces each $E_{C}(X)$.

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 $|H_{ij}c_{i}|>\varepsilon_{i}$, where $\varepsilon_{i}$ is the energy cutoff and $c_{i}$ are determinants in the HBCI subspace.

Truncation of incremental terms is performed by considering natural orbital (NO) occupancy cutoffs, $\eta^{(m)}$, where

 $\displaystyle\epsilon_{i}^{(m)}$ $\displaystyle=$ $\displaystyle E_{C}(i;\eta^{(m)})$ (6.71) $\displaystyle\epsilon_{ij}^{(m)}$ $\displaystyle=$ $\displaystyle E_{C}(ij;\eta^{(m)})-\epsilon_{i}^{(m)}-\epsilon_{j}^{(m)}$ (6.72) $\displaystyle\cdots$

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

 $\zeta>|\epsilon^{(m+1)}-\epsilon^{m}|$ (6.73)

with units of $10^{-\zeta}$ E${}_{\mathrm{h}}$. Further truncation in $n\geq 3$ can be performed by utilizing the $\zeta$ parameter and a screening cutoff, $\mathcal{C}_{n}$, in the form

 $\mathcal{C}_{n}=10^{-\zeta}\times\mathcal{S}_{n}$ (6.74)

where $\mathcal{C}_{n}$ is in E${}_{\mathrm{h}}$ and $\mathcal{S}_{n}$ is a scalar. This screening is performed by selecting $n-1$ body correlation energy contributions that are above $\mathcal{C}_{n}$. See Ref. 947 for more details. $\mathcal{S}_{n}$ 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.