To carry out vibrational frequency analysis of an excited state with TDDFT,583, 584 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 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 , where 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.
Example 7.7 A B3LYP/6-31G* optimization in gas phase, followed by a frequency analysis for the first excited state of the peroxy radical
$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
Example 7.8 The optimization and Hessian calculation for low-lying excited state of 9-Fluorenone + 2 methanol in methanol solution using TDDFT/C-PCM
$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