7.2.2 Configuration Interaction with Single Substitutions (CIS)

The derivation of the CI-singles (CIS) energy and wave function ²⁹⁴ Del Bene J. E., Ditchfield R., Pople J. A.
J. Chem. Phys.
(1971), 55, pp. 2236. Link ^{, 370} Foresman J. B. et al.
J. Phys. Chem.
(1992), 96, pp. 135. Link begins by selecting the HF single-determinant wave function as reference for the ground state of the system:

where $n$ is the number of electrons, and the spin orbitals

are expanded in a finite basis of $N$ atomic orbital basis functions. Molecular orbital coefficients $\{c_{\mu i}\}$ are usually found by SCF procedures which solve the Hartree-Fock equations

where S is the overlap matrix, C is the matrix of molecular orbital coefficients, $\epsilon$ is a diagonal matrix of orbital eigenvalues and F is the Fock matrix with elements

involving the core Hamiltonian and the anti-symmetrized two-electron integrals

On solving Eq. (7.3), the total energy of the ground state single determinant can be expressed as

where $P^{\mathrm{HF}}$ is the HF density matrix and $V_{\mathrm{nuc}}$ is the nuclear repulsion energy.

Equation (7.1) represents only one of many possible determinants made from orbitals of the system; there are in fact $n(N-n)$ possible singly substituted determinants constructed by replacing an orbital occupied in the ground state ($i$, $j$, $k,\mathrm{\dots }$) with an orbital unoccupied in the ground state ($a$, $b$, $c,\mathrm{\dots }$). Such wave functions and energies can be written

where we have introduced the anti-symmetrized two-electron integrals in the molecular orbital basis

These singly excited wave functions and energies could be considered crude approximations to the excited states of the system. However, determinants of the form Eq. (7.7) are deficient in that they:

• •

  do not yield pure spin states

• •

  resemble more closely ionization rather than excitation

• •

  are not appropriate for excitation into degenerate states

These deficiencies can be partially overcome by representing the excited state wave function as a linear combination of all possible singly excited determinants,

where the coefficients $\{a_{ia}\}$ can be obtained by diagonalizing the many-electron Hamiltonian, A, in the space of all single substitutions. The appropriate matrix elements are:

According to Brillouin’s, theorem single substitutions do not interact directly with a reference HF determinant, so the resulting eigenvectors from the CIS excited state represent a treatment roughly comparable to that of the HF ground state. The excitation energy is simply the difference between HF ground state energy and CIS excited state energies, and the eigenvectors of A correspond to the amplitudes of the single-electron promotions.

CIS calculations can be performed in Q-Chem using restricted (RCIS), ²⁹⁴ Del Bene J. E., Ditchfield R., Pople J. A.
J. Chem. Phys.
(1971), 55, pp. 2236. Link ^{, 370} Foresman J. B. et al.
J. Phys. Chem.
(1992), 96, pp. 135. Link unrestricted (UCIS), or restricted open-shell ⁸⁶¹ Maurice D., Head-Gordon M.
Int. J. Quantum Chem.
(1995), 29, pp. 361. Link (ROCIS) spin orbitals. See Section 7.2.4 for a list of CIS-related job control variables.

While CIS excitation energies are relatively inaccurate, with errors of the order of 1 eV, CIS excited state properties, such as structures and frequencies, are much more useful. This is very similar to the manner in which ground state Hartree-Fock (HF) structures and frequencies are much more accurate than HF relative energies. Generally speaking, for low-lying excited states, it is expected that CIS vibrational frequencies will be systematically 10% higher or so relative to experiment. ¹²⁰³ Stanton J. F. et al.
J. Chem. Phys.
(1995), 103, pp. 4160. Link ^{, 1462} Zilberg S., Haas Y.
J. Chem. Phys.
(1995), 103, pp. 20. Link ^{, 425} Gittins C. M., Rohlfing E. A., Rohlfing C. M.
J. Chem. Phys.
(1996), 105, pp. 7323. Link If the excited states are of pure valence character, then basis set requirements are generally similar to the ground state. Excited states with partial Rydberg character require the addition of one or preferably two sets of diffuse functions.

Q-Chem includes efficient analytical first and second derivatives of the CIS energy, ⁸⁶³ Maurice D., Head-Gordon M.
Mol. Phys.
(1999), 96, pp. 1533. Link to yield analytical gradients, excited state vibrational frequencies, force constants, polarizabilities, and infrared intensities. Analytical gradients can be evaluated for any job where the CIS excitation energy calculation itself is feasible, so that efficient excited-state geometry optimizations and vibrational frequency calculations are possible at the CIS level. In such cases, it is necessary to specify on which Born-Oppenheimer potential energy surface the optimization should proceed, and care must be taken to ensure that the optimization remains on the excited state of interest, as state crossings may occur. (A state-tracking algorithm, as discussed in Section 9.8.5, can aid with this. ¹⁴³⁵ Zhang X., Herbert J. M.
J. Chem. Phys.
(2015), 143, pp. 234107. Link )

Sometimes it is precisely the crossings between Born-Oppenheimer potential energy surfaces (i.e., conical intersections) that are of interest, as these intersections provide pathways for nonadiabatic transitions between electronic states. ⁸⁵⁸ Matsika S., Krause P.
Annu. Rev. Phys. Chem.
(2011), 62, pp. 621. Link ^{, 525} Herbert J. M. et al.
Acc. Chem. Res.
(2016), 49, pp. 931. Link A feature of Q-Chem that is not otherwise widely available in an analytic implementation (for both CIS and TDDFT) of the nonadiabatic couplings that define the topology around conical intersections. ³⁵⁴ Fatehi S. et al.
J. Chem. Phys.
(2011), 135, pp. 234105. Link ^{, 1433} Zhang X., Herbert J. M.
J. Chem. Phys.
(2014), 141, pp. 064104. Link ^{, 1434} Zhang X., Herbert J. M.
J. Chem. Phys.
(2015), 142, pp. 064109. Link ^{, 959} Ou Q. et al.
J. Chem. Phys.
(2015), 142, pp. 064114. Link Due to the analytic implementation, these couplings can be evaluated at a cost that is not significantly greater than the cost of a CIS or TDDFT analytic gradient calculation, ¹⁴³³ Zhang X., Herbert J. M.
J. Chem. Phys.
(2014), 141, pp. 064104. Link and the availability of these couplings allows for much more efficient optimization of minimum-energy crossing points along seams of conical intersection, as compared to when only analytic gradients are available. ¹⁴³³ Zhang X., Herbert J. M.
J. Chem. Phys.
(2014), 141, pp. 064104. Link These features, including a brief overview of the theory of conical intersections, can be found in Section 9.8.1.

For CIS vibrational frequencies, a semi-direct algorithm similar to that used for ground-state Hartree-Fock frequencies is available, whose computer time scales as approximately $𝒪(N^3)$ for large molecules. ⁸⁶² Maurice D., Head-Gordon M.
J. Phys. Chem.
(1996), 100, pp. 6131. Link The main complication associated with analytical CIS frequency calculations is ensuring that Q-Chem has sufficient memory to perform the calculations. Default settings are adequate for many purposes but if a large calculation fails due to a memory limitation, then the following additional information may be useful.

The memory requirements for CIS (and HF) analytic frequencies primarily come from dynamic memory, defined as

dynamic memory = MEM_TOTAL $-$ MEM_STATIC   .

This quantity must be large enough to contain several arrays whose size is $3N_{\mathrm{atoms}}{N}_{\mathrm{basis}}^2$. Meanwhile the value of the $rem variable MEM_STATIC, which obviously reduces the available dynamic memory, must be sufficiently large to permit integral evaluation, else the job may fail. For most purposes, setting MEM_STATIC to about 80 MB is sufficient, and by default MEM_TOTAL is set to a larger value that what is available on most computers, so that the user need not guess or experiment about an appropriate value of MEM_TOTAL for low-memory jobs. However, a memory allocation error will occur if the calculation demands more memory than available.

Note:  Unlike Q-Chem’s MP2 frequency code, the analytic CIS second derivative code currently does not support frozen core or virtual orbitals. These approximations do not lead to large savings at the CIS level, as all computationally-expensive steps are performed in the atomic orbital basis.

The Random Phase Approximation (RPA), ¹³⁸ Bouman T. D., Hansen A. E.
Int. J. Quantum Chem. Symp.
(1989), 23, pp. 381. Link ^{, 479} Hansen A. E., Voight B., Rettrup S.
Int. J. Quantum Chem.
(1983), 23, pp. 595. Link also known as time-dependent Hartree-Fock (TD-HF) theory, is an alternative to CIS for uncorrelated calculations of excited states. An RPA calculation is requested using the RPA $rem variable that is described in Section 7.2.4.1.

RPA offers some advantages for computing oscillator strengths, e.g., exact satisfaction of the Thomas-Reike-Kuhn sum rule, and is roughly comparable in accuracy to CIS for singlet excitation energies, but is inferior for triplet states. RPA energies are non-variational, and in moving around on excited-state potential energy surfaces, this method can occasionally encounter singularities (arising from triplet instabilities in the underlying reference state) that prevent numerical solution of the RPA equations. ²⁵¹ Cordova F. et al.
J. Chem. Phys.
(2007), 127, pp. 164111. Link ^{, 532} Herbert J. M.
Phys. Chem. Chem. Phys.
(2024), 26, pp. 3755. Link In Q-Chem, these instabilities are generally accompanied by an error message to the effect that an imaginary root has been detected in the RPA equations. This is mathematically impossible in CIS calculations, which decouple the excitation energy problem from the stability problem.

DFT/CIS ⁴⁵⁴ Grimme S.
Chem. Phys. Lett.
(1996), 259, pp. 128. Link is a semi-empirical method that uses Kohn-Sham orbitals instead of Hartree-Fock orbitals with a CIS-like formalism. This leads to added correlation energy from the Kohn-Sham orbitals and, perhaps more importantly, orbital energy differences ${\epsilon }_a-{\epsilon }_i$ that better resemble excitation energies, and bound virtual orbitals that better approximate localized excited states as compared to Hartree-Fock virtual orbitals, which are typically unbound. ⁵³² Herbert J. M.
Phys. Chem. Chem. Phys.
(2024), 26, pp. 3755. Link Along with certain semi-empirical corrections, DFT/CIS significantly improves conventional Hartree–Fock-based CIS. Two parameterizations of the DFT/CIS Hamiltonian are available in Q-Chem: Grimme’s original one, ⁴⁵⁴ Grimme S.
Chem. Phys. Lett.
(1996), 259, pp. 128. Link which is designed for use with the B3LYP functional, and a new parameterization designed for CAM-B3LYP. The latter (CAM-B3LYP/CIS) implementation has specific parameterizations aimed at accurately reproducing X-ray spectra, whereas B3LYP/CIS is suitable only for valence excitations. The nature of the parameterization limits use of DFT/CIS to the aforementioned two functionals, and primarily to two basis sets. The CAM-B3LYP/CIS method is parameterized for use with def2-TZVPD while B3LYP/CIS is parameterized for use with TZVP.

The DFT/CIS method is controlled by the DFTCIS and DFTCIS_PARAMS job control variables that are described in Section 7.2.4.1.