7.5.1 Introduction

Systems such as transition metals, open-shell species, and molecules with highly-stretched bonds often exhibit multiple, near-degenerate solutions to the SCF equations. Multiple solutions can be located using SCF metadynamics (Section 4.9.2), but given the approximate nature of the SCF calculation in the first place, there is in such cases no clear reason to choose one of these solutions over another. These SCF solutions are not subject to any non-crossing rule, and often do cross (i.e., switch energetic order) as the geometry is changed. Consequently, the lowest energy state may switch abruptly with consequent discontinuities in the energy gradients. It is therefore desirable to have a method that treats all of these near-degenerate SCF solutions on an equal footing and might yield a smoother, qualitatively correct potential energy surface. This can be achieved by using multiple SCF solutions (obtained, via e.g. SCF metadynamics) as a basis for a configuration interaction (CI) calculation. Since the various SCF solutions are not orthogonal to one another — meaning that one solution cannot be constructed as a single determinant composed of orbitals from another solution — this CI is a bit more complicated and is denoted as a non-orthogonal CI (NOCI).¹¹²⁰

NOCI can be viewed as an alternative to CASSCF within an “active space” consisting of the SCF states of interest, and has the advantage that the SCF states, and thus the NOCI wave functions, are size-consistent. In common with CASSCF, it is able to describe complicated phenomena such as avoided crossings (where states mix instead of passing through each other) as well as conical intersections (whereby via symmetry or else accidental reasons, there is no coupling between the states, and they pass cleanly through each other at a degeneracy).

Another use for a NOCI calculation is that of symmetry restoration. At some geometries, the SCF states break spatial or spin symmetry to achieve a lower energy single determinant than if these symmetries were conserved. As these symmetries still exist within the proper electronic Hamiltonian, its exact eigenfunctions should preserve them. In the case of spin, this manifests as spin contamination, while for spatial symmetries it usually manifests as artefactual localization. To recover a (yet lower energy) wave function retaining the correct symmetries, one can include these broken-symmetry states (with all relevant symmetry permutations) in a NOCI calculation; the resultant eigenfunction will have the true symmetries restored, as a linear combination of the broken-symmetry states.

A common example occurs in the case of a spin-contaminated UHF reference state. Performing a NOCI calculation in a basis consisting of this state, plus a second state in which all $\alpha$ and $\beta$ orbitals have been switched, often reduces spin contamination in the same way as the half-projected Hartree-Fock method,⁷⁶⁴ although there is no guarantee that the resulting wave function is an eigenfunction of ${\widehat{S}}^2$. Another example consists of using a UHF wave function with $M_S=0$, along with its spin-exchanged version (wherein all $\alpha \leftrightarrow \beta$ orbitals are switched), resulting in two new NOCI eigenfunctions, one with even $S$ (a mixture of $S=0,2,\mathrm{\dots }$), and one with odd $S$ (mixing $S=1,3,\mathrm{\dots }$). These may be used to approximate singlet and triplet wave functions.

NOCI can be enabled by specifying CORRELATION = NOCI, and will automatically use all of the states located with SCF metadynamics. Two spin-exchanged versions of a UHF wave function can be requested simply by not turning on metadynamics. For more customization, a $noci input section can be included having, e.g., the following format:

$noci
   1 2 -2 4
   2
$end

In this particular case, the first line specifies that states 1, 2, and 4 are to be included in the NOCI calculation, along with state “$-$2”, which indicates the spin-exchanged version of state 2. The second (optional) line indicates which eigenvalue is to be returned to Q-Chem, with the convention that 0 indicates the lowest state so the $noci input section above is requesting the third state.

Analytic gradients are not available for NOCI but geometry optimizations (if requested) will be performed automatically using finite-difference gradients.