# 6.21.2 Theory

(May 16, 2021)

If we have a wave function $|\Psi\rangle=\displaystyle\sum\limits_{i}C_{i}|D_{i}\rangle$ (where $|D_{i}\rangle$ are Slater determinants with coefficients $C_{i}$) as an eigenstate of the Hamiltonian, then

 $C_{i}=\frac{\sum_{j\neq i}H_{ij}C_{j}}{(H_{ii}-E)}\;,$ (6.75)

where $H_{ij}=\langle D_{i}|\mathbf{H}|D_{j}\rangle$ is the Hamiltonian matrix element between determinants $i$ and $j$, and $E$ is the energy of the eigenstate $|\Psi\rangle$. This exact relationship can be generalized to a metric to predict the expected weight of a determinant $|D_{i}\rangle$ in a CI expansion, by how it connects to other determinants in an approximate trial wave function. This metric is also used in Epstein-Nesbet Perturbation theory as coefficients for the determinants in the first order wave function.

In the ASCI method, all determinants $|D_{i}\rangle$ that are single or double excitations away from the most important determinants (as ranked by magnitude of coefficients) in the trial wave function $|\psi_{k}\rangle$ are assigned an estimated importance $A_{i}$ given as

 $A_{i}=\frac{\sum_{|D_{j}\rangle\in|\psi_{k}\rangle}H_{ij}C_{j}}{(H_{ii}-E_{k})% }\;,$ (6.76)

where $E_{k}$ is the energy of the trial wave function $|\psi_{k}\rangle$. The search and selection is only done in the space spanned by determinants connected to the top $c$ determinants in $|\psi_{k}\rangle$ because unimportant determinants are unlikely to be the sole generator for a top ranked determinant, and this pruning of the search space greatly accelerates the algorithm. The top $t$ determinants (as ranked by magnitude of $A_{i}$) connected to $|\psi_{k}\rangle$ are used to form the new wave function $|\psi^{k+1}\rangle$ by exact diagonalization within that Hilbert subspace.

Once several cycles of ASCI has been completed, the wave function will contain all (or very nearly all) of the largest weight determinants in the FCI wave function and the remaining determinants not included should be of small weight. The effect of these many small remaining determinants are estimated by second order Epstein-Nesbet perturbation theory (PT2). , This final PT2 correction gives extremely accurate results, often within a kcal/mol of the absolute FCI energies even when only a tiny fraction of the Hilbert space is included in the ASCI wave function. An extrapolation of the variational energy against the PT2 correction (to the FCI limit of of zero PT2 correction) can also be carried out to generate more accurate estimates, and predict a metric for error in the final estimate. Indeed, it has been shown that linear or quadratic fits are quite accurate for extrapolation of SCI energies against the PT2 correction. ,, We observe essentially linear behavior in the case of ASCI.

ASCI may be used as the full-CI solver for a CASSCF calculation, permitting the extension of CASSCF to active spaces of $\approx 50$ electrons in $\approx 50$ orbitals. The resulting method is termed ASCI-SCF  . See section 6.19 for details on CASSCF job control.