6.20 Incremental Correlation Methods 6.20.1 Introduction 6.20.3 Job Control for iFCI

6.20.2 Theory

(April 13, 2024)

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<i}% \epsilon_{ij}+\sum_{k<j<i}\epsilon_{ijk}+\cdots

(6.69)

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

$\displaystyle\epsilon_{i}$	$\displaystyle=$	$\displaystyle E_{C}(i)$	(6.70)
$\displaystyle\epsilon_{ij}$	$\displaystyle=$	$\displaystyle E_{C}(ij)-\epsilon_{i}-\epsilon_{j}$	(6.71)
$\displaystyle\epsilon_{ijk}$	$\displaystyle=$	$\displaystyle E_{C}(ijk)-\epsilon_{ij}-\epsilon_{ik}-\epsilon_{jk}-\epsilon_{i% }-\epsilon_{j}-\epsilon_{k}$	(6.72)
$\displaystyle\cdots$

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

$\displaystyle\epsilon_{i}^{(m)}$	$\displaystyle=$	$\displaystyle E_{C}(i;\eta^{(m)})$	(6.73)
$\displaystyle\epsilon_{ij}^{(m)}$	$\displaystyle=$	$\displaystyle E_{C}(ij;\eta^{(m)})-\epsilon_{i}^{(m)}-\epsilon_{j}^{(m)}$	(6.74)
$\displaystyle\cdots$

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}|

(6.75)

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}

(6.76)

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. 1035 Rask A. E., Zimmerman P. M.
J. Phys. Chem. A
(2021), 125, pp. 1598. Link 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.

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6.20.2 Theory