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(May 16, 2021)

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

$$ | (6.67) |

where each ${\u03f5}_{X}$ term denotes an increment of correlation energy and $i,j,k$ refer to bodies of the expansion. Incremental correlation energies are defined as

${\u03f5}_{i}$ | $=$ | ${E}_{C}(i)$ | (6.68) | ||

${\u03f5}_{ij}$ | $=$ | ${E}_{C}(ij)-{\u03f5}_{i}-{\u03f5}_{j}$ | (6.69) | ||

${\u03f5}_{ijk}$ | $=$ | ${E}_{C}(ijk)-{\u03f5}_{ij}-{\u03f5}_{ik}-{\u03f5}_{jk}-{\u03f5}_{i}-{\u03f5}_{j}-{\u03f5}_{k}$ | (6.70) | ||

$\mathrm{\cdots}$ |

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 ${\u03f5}_{X}$ 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 $|{H}_{ij}{c}_{i}|>{\epsilon}_{i}$, where ${\epsilon}_{i}$ 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

${\u03f5}_{i}^{(m)}$ | $=$ | ${E}_{C}(i;{\eta}^{(m)})$ | (6.71) | ||

${\u03f5}_{ij}^{(m)}$ | $=$ | ${E}_{C}(ij;{\eta}^{(m)})-{\u03f5}_{i}^{(m)}-{\u03f5}_{j}^{(m)}$ | (6.72) | ||

$\mathrm{\cdots}$ |

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

$$\zeta >|{\u03f5}^{(m+1)}-{\u03f5}^{m}|$$ | (6.73) |

with units of ${10}^{-\zeta}$ E${}_{\mathrm{h}}$. Further truncation in $n\ge 3$ 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. 916 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.