7.3 Time-Dependent Density Functional Theory (TDDFT)

7.3.5 Analytical Excited-State Hessian in TDDFT

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 Cx 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 𝒪(3mN2), 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.

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