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# 7.3.5 Analytic Excited-State Hessian in TDDFT

(February 4, 2022)

To carry out vibrational frequency analysis of an excited state with TDDFT,702, 703 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 analytic 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 ${\cal{O}}({3mN^{2}})$, 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 $rem variable CPSCF_NSEG is always needed. In the calculation of the analytic TDDFT excited-state Hessian, one has to evaluate a large number of energy-functional derivatives: the first-order to fourth-order functional derivatives with respect to the density variables as well as their derivatives with respect to the nuclear coordinates. Therefore, a very fine integration grid for DFT calculation should be adapted to guarantee the accuracy of the results. Analytic TDDFT/C-PCM Hessian has been implemented in Q-Chem. Normal mode analysis for a system in solution can be performed with the frequency calculation by TDDFT/C-PCM method. The$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. Example 7.7 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
$end$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  View output Example 7.8 Geometry optimization for a low-lying excited state of cyclopentadienone in methanol, using TDDFT/C-PCM $molecule
0 1
C   -0.0000000    0.6920860    1.4656691
C   -0.0000000   -0.6920860    1.4656691
C   -0.0000000   -1.1528931    0.1065000
C    0.0000000   -0.0000000   -0.7957576
C    0.0000000    1.1528931    0.1065000
O    0.0000000   -0.0000000   -2.0301721
H   -0.0000000    1.3254423    2.3427356
H   -0.0000000   -1.3254423    2.3427356
H   -0.0000000   -2.1834532   -0.2231979
H    0.0000000    2.1834532   -0.2231979
$end$rem
JOBTYPE             opt
EXCHANGE            b3lyp
CIS_N_ROOTS         5
CIS_SINGLETS        true
CIS_TRIPLETS        true
CIS_STATE_DERIV     1   Lowest TDDFT state
BASIS               6-311G*
XC_GRID             3
SOLVENT_METHOD      pcm
THRESH              12
$end$pcm
Theory      CPCM
Method      SWIG
Solver      Inversion
$end$solvent
Dielectric 32.613
$end  View output Example 7.9 Hessian calculation, using the optimized geometry from the previous example. $molecule
0 1
C     0.0000000000    0.6940558365    1.4635362645
C     0.0000000000   -0.6940558367    1.4635362652
C     0.0000000000   -1.1539902580    0.1063088532
C     0.0000000000    0.0000000000   -0.7890068343
C     0.0000000000    1.1539902569    0.1063088524
O     0.0000000000    0.0000000000   -2.0333287471
H     0.0000000000    1.3287019844    2.3394236351
H     0.0000000000   -1.3287019837    2.3394236361
H     0.0000000000   -2.1861329696   -0.2193590119
H     0.0000000000    2.1861329678   -0.2193590131
$end$rem
JOBTYPE           freq
EXCHANGE          b3lyp
CIS_N_ROOTS       5
CIS_SINGLETS      true
CIS_TRIPLETS      true
CIS_STATE_DERIV   1   Lowest TDDFT state
BASIS             6-311G*
XC_GRID           3
SOLVENT_METHOD    pcm
MEM_STATIC        4000
MEM_TOTAL         24000
CPSCF_NSEG        3
THRESH            12
$end$pcm
Theory      CPCM
Method      SWIG
Solver      Inversion
$end$solvent