7.2.6 CIS Analytical Derivatives

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.^{867, 1070, 308} 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,^{625, 626} 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 10.6.5, can aid with this.)

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 non-adiabatic transitions between electronic states.^{622, 374} A feature of Q-Chem that is not otherwise widely available in an analytic implementation (for both CIS and TDDFT) of the non-adiabatic couplings that define the topology around conical intersections.^{258, 1049, 1050, 694} 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, 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.¹⁰⁴⁹ These features, including a brief overview of the theory of conical intersections, can be found in Section 10.6.1.

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.⁶²⁴ 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.