To carry out vibrational frequency analysis of an excited state with
TDDFT,^{579, 580} an optimization of the excited-state geometry
is always necessary. Like the vibrational frequency analysis of the ground
state, the frequency analysis of the excited state should be also performed at
a stationary point on the excited state potential surface. The *$rem* variable
CIS_STATE_DERIV should be set to the excited state for which an
optimization and frequency analysis is needed, in addition to the *$rem*
keywords used for an excitation energy calculation.

Compared to the numerical differentiation method, the analytical calculation of
geometrical second derivatives of the excitation energy needs much less time
but much more memory. The computational cost is mainly consumed by the steps to
solve both the CPSCF equations for the derivatives of molecular orbital
coefficients C${}^{\mathrm{x}}$ and the CP-TDDFT equations for the derivatives of
the transition vectors, as well as to build the Hessian matrix. The memory
usages for these steps scale as $\mathcal{O}(3m{N}^{2})$, where $N$ is the number of basis
functions and m is the number of atoms. For large systems, it is thus essential
to solve all the coupled-perturbed equations in segments. In this case, the
*$rem* variable CPSCF_NSEG is always needed.

In the calculation of the analytical TDDFT excited-state Hessian, one has to evaluate a large number of energy-functional derivatives: the first-order to fourth-order functional derivatives with respect to the density variables as well as their derivatives with respect to the nuclear coordinates. Therefore, a very fine integration grid for DFT calculation should be adapted to guarantee the accuracy of the results.

Analytical TDDFT/C-PCM Hessian has been implemented in Q-Chem.
Normal mode analysis for a system in solution can be performed with the
frequency calculation by TDDFT/C-PCM method. The *$rem* and *$pcm*
variables for the excited state calculation with TDDFT/C-PCM included
in the vertical excitation energy example above are needed. When the properties
of large systems are calculated, you must pay attention to the memory limit. At
present, only a few exchange correlation functionals, including Slater+VWN,
BLYP, B3LYP, are available for the analytical Hessian calculation.

$molecule 0 2 C 1.004123 -0.180454 0.000000 O -0.246002 0.596152 0.000000 O -1.312366 -0.230256 0.000000 H 1.810765 0.567203 0.000000 H 1.036648 -0.805445 -0.904798 H 1.036648 -0.805445 0.904798 $end $rem JOBTYPE opt EXCHANGE b3lyp CIS_STATE_DERIV 1 BASIS 6-31G* CIS_N_ROOTS 10 CIS_SINGLETS true CIS_TRIPLETS false XC_GRID 000075000302 RPA 0 $end @@@ $molecule read $end $rem JOBTYPE freq EXCHANGE b3lyp CIS_STATE_DERIV 1 BASIS 6-31G* CIS_N_ROOTS 10 CIS_SINGLETS true CIS_TRIPLETS false XC_GRID 000075000302 RPA 0 $end

$molecule 0 1 C -1.9872490 0.6997110 0.0805830 C -1.9871870 -0.6995370 -0.0805190 C -0.5980490 -1.1489320 -0.1312990 C 0.2825460 0.0001600 0.0001370 C -0.5981390 1.1492190 0.1314790 O 1.5605680 0.0001590 0.0002090 O 3.3231500 2.1192220 0.1254540 H 2.6693090 1.3896420 0.0843860 C 3.6669020 2.4893960 -1.2082390 H 4.3975510 3.2984440 -1.1513100 H 4.1162820 1.6546500 -1.7594860 H 2.7950880 2.8493370 -1.7682060 H 2.6692050 -1.3893820 -0.0843430 O 3.3229890 -2.1190060 -0.1256200 C 3.6664120 -2.4898980 1.2079740 H 4.3969660 -3.2990230 1.1507890 H 4.1158000 -1.6554850 1.7597300 H 2.7944320 -2.8500010 1.7675930 H -2.8223711 1.3019694 0.1504396 H -2.8222558 -1.3018612 -0.1504451 H -0.2904548 -2.1276870 -0.2428714 H -0.2906211 2.1279968 0.2430616 $end $rem JOBTYPE opt EXCHANGE b3lyp CIS_N_ROOTS 10 CIS_SINGLETS true CIS_TRIPLETS true CIS_STATE_DERIV 1 Lowest TDDFT state RPA true BASIS 6-311G* XC_GRID 000099000590 SOLVENT_METHOD pcm THRESH 14 $end $pcm Theory CPCM Method SWIG Solver Inversion Radii Bondi $end $solvent Dielectric 32.613 $end @@@ $molecule read $end $rem JOBTYPE freq EXCHANGE b3lyp CIS_N_ROOTS 10 CIS_SINGLETS true CIS_TRIPLETS true RPA true CIS_STATE_DERIV 1 Lowest TDDFT state BASIS 6-311G* XC_GRID 000099000590 SOLVENT_METHOD pcm MEM_STATIC 4000 MEM_TOTAL 24000 CPSCF_NSEG 3 THRESH 12 $end $pcm Theory CPCM Method SWIG Solver Inversion Radii Bondi $end $solvent Dielectric 32.613 $end