7.2.3 CIS Methods with Extended Excitation Manifolds

The motivation for XCIS stems from the fact that ROCIS and UCIS are less effective for radicals than CIS is for closed shell molecules. Using the attachment/detachment density analysis procedure, ⁴⁹⁹ Head-Gordon M. et al.
J. Phys. Chem.
(1995), 99, pp. 14261. Link ^{, 532} Herbert J. M.
Phys. Chem. Chem. Phys.
(2024), 26, pp. 3755. Link the failing of ROCIS and UCIS methodologies for the nitromethyl radical was traced to the neglect of a particular class of double substitution which involves the simultaneous promotion of an $\alpha$ spin electron from the singly occupied orbital and the promotion of a $\beta$ spin electron into the singly occupied orbital. The spin-adapted configurations

are of crucial importance. (Here, $a,b,c,\mathrm{\dots }$ are virtual orbitals; $i,j,k,\mathrm{\dots }$ are occupied orbitals; and $p,q,r,\mathrm{\dots }$ are singly-occupied orbitals.) It is quite likely that similar excitations are also very significant in other radicals of interest.

The XCIS proposal, a more satisfactory generalization of CIS to open shell molecules, is to simultaneously include a restricted class of double substitutions similar to those in Eq. (7.12). To illustrate this, consider the resulting orbital spaces of an ROHF calculation: doubly occupied ($d$), singly occupied ($s$) and virtual ($v$). From this starting point we can distinguish three types of single excitations of the same multiplicity as the ground state: $d\to s$, $s\to v$ and $d\to v$. Thus, the spin-adapted ROCIS wave function is

The extension of CIS theory to incorporate higher excitations maintains the ROHF as the ground state reference and adds terms to the ROCIS wave function similar to that of Eq. (7.13), as well as those where the double excitation occurs through different orbitals in the $\alpha$ and $\beta$ space:

XCIS is defined only from a restricted open shell Hartree-Fock ground state reference, as it would be difficult to uniquely define singly occupied orbitals in a UHF wave function. In addition, $\beta$ unoccupied orbitals, through which the spin-flip double excitation proceeds, may not match the half-occupied $\alpha$ orbitals in either character or even symmetry.

For molecules with closed shell ground states, both the HF ground and CIS excited states emerge from diagonalization of the Hamiltonian in the space of the HF reference and singly excited substituted configuration state functions. The XCIS case is different because the restricted class of double excitations included could mix with the ground state and lower its energy. This mixing is avoided to maintain the size consistency of the ground state energy.

With the inclusion of the restricted set of doubles excitations in the excited states, but not in the ground state, it could be expected that some fraction of the correlation energy be recovered, resulting in anomalously low excited state energies. However, the fraction of the total number of doubles excitations included in the XCIS wave function is very small and those introduced cannot account for the pair correlation of any pair of electrons. Thus, the XCIS procedure can be considered one that neglects electron correlation.

The computational cost of XCIS is approximately four times greater than CIS and ROCIS, and its accuracy for open shell molecules is generally comparable to that of the CIS method for closed shell molecules. In general, it achieves qualitative agreement with experiment. XCIS is available for doublet and quartet excited states beginning from a doublet ROHF treatment of the ground state, for excitation energies only.

Spin-flip extended CIS (SF-XCIS) ¹⁸⁵ Casanova D., Head-Gordon M.
J. Chem. Phys.
(2008), 129, pp. 064104. Link is a spin-complete extension of the spin-flip single excitation configuration interaction (SF-CIS) method. ⁶⁸⁹ Krylov A. I.
Chem. Phys. Lett.
(2002), 350, pp. 522. Link The method includes all configurations in which no more than one virtual level of the high spin triplet reference becomes occupied and no more than one doubly occupied level becomes vacant.

SF-XCIS is defined only from a restricted open shell Hartree-Fock triplet ground state reference. The final SF-XCIS wave functions correspond to spin-pure $M_S=0$ (singlet or triplet) states. The fully balanced treatment of the half-occupied reference orbitals makes it very suitable for applications with two strongly correlated electrons, such as single bond dissociation, systems with important diradical character or the study of excited states with significant double excitation character.

The computational cost of SF-XCIS scales in the same way with molecule size as CIS itself, with a pre-factor 13 times larger.

Spin-Adapted Spin-Flip CIS (SA-SF-CIS) ¹⁴³⁵ Zhang X., Herbert J. M.
J. Chem. Phys.
(2015), 143, pp. 234107. Link is a spin-complete extension of the spin-flip single excitation configuration interaction (SF-CIS) method. ⁶⁸⁹ Krylov A. I.
Chem. Phys. Lett.
(2002), 350, pp. 522. Link Unlike SF-XCIS, SA-SF-CIS only includes the minimal set of electronic configurations needed to remove the spin contamination in the conventional SF-CIS method. The target SA-SF-CIS states have spin eigenvalues one less than the reference ROHF state, i.e., if singlet states ($S=0$) are targeted then the reference state should be a triplet ($S=1$), or if doublet states ($S=1/2$) are targeted then the reference state should be a quartet ($S=3/2$). The SA-SF-CIS approach uses a tensor equation-of-motion formalism, ¹⁴³⁵ Zhang X., Herbert J. M.
J. Chem. Phys.
(2015), 143, pp. 234107. Link such that the dimension of the CI vectors in SA-SF-CIS remains exactly the same as that in conventional SF-CIS. A DFT correction to SA-SF-CIS (i.e., SA-SF-TDDFT) can be added. ¹⁴³⁵ Zhang X., Herbert J. M.
J. Chem. Phys.
(2015), 143, pp. 234107. Link As with other SF-TDDFT methods, ¹⁴³⁶ Zhang X., Herbert J. M.
J. Chem. Phys.
(2021), 155, pp. 124111. Link the BH&HLYP functional has become something of a de facto standard choice. ⁵²⁵ Herbert J. M. et al.
Acc. Chem. Res.
(2016), 49, pp. 931. Link