# Differences

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time-dependent_density_functional_theory [2022/08/05 20:00] – [TDDFT program] hesselmann | time-dependent_density_functional_theory [2022/08/16 14:15] (current) – external edit 127.0.0.1 | ||
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+ | ===== Time-dependent density functional theory ===== | ||

+ | |||

+ | Excitation energies and linear response properties can be calculated utilising the time-dependent density functional theory (TDDFT) method. The program should normally be called after a Kohn-Sham or Hartree-Fock calculation because it looks for the most recent orbital dump record to read in the MO coefficients and orbital energies. Further settings like the functional type and quadrature grid are then adopted from the previous ground-state calculation, | ||

+ | |||

+ | ==== TDDFT program ==== | ||

+ | |||

+ | Assuming that the molecule has $C_{2v}$ symmetry, a typical input for calculating the 6 lowest roots of the Hessian for IRREp 1 and 3, respectively, | ||

+ | |||

+ | < | ||

+ | | ||

+ | | ||

+ | </ | ||

+ | The type of the xc kernel is usually adjusted automatically using the parameters from the previous ground-state KS calculation. However, various user inputs are available to change this, see below. Currently, | ||

+ | the kernel can not (easily) be derived) may be combined with existing adiabatic LDA (ALDA) xc kernels. | ||

+ | |||

+ | Excitation energies for spin-unrestricted wave functions can be computed, too, using, e.g. | ||

+ | |||

+ | < | ||

+ | uhf | ||

+ | | ||

+ | </ | ||

+ | for time-dependent Hartree-Fock (TDHF) or | ||

+ | |||

+ | < | ||

+ | | ||

+ | | ||

+ | </ | ||

+ | for time-dependent DFT. | ||

+ | |||

+ | The program can be run in various integral transformation modes using | ||

+ | |||

+ | < | ||

+ | tddft, | ||

+ | </ | ||

+ | with '' | ||

+ | stored on disk. '' | ||

+ | in each TDDFT Davidson iteration step. While this option is faster than '' | ||

+ | |||

+ | While both, '' | ||

+ | |||

+ | < | ||

+ | df-tddft, | ||

+ | </ | ||

+ | in which case only 3-indexed integrals are used to compute the Hessian-vector products. While this requires an additional auxiliary basis set to be defined, the program can usually detect the fitting basis | ||

+ | set that corresponds to the given orbital basis automatically, | ||

+ | |||

+ | < | ||

+ | df-tddft, | ||

+ | </ | ||

+ | with '' | ||

+ | |||

+ | Note that density-fitting can also be used in conjunction with '' | ||

+ | integrals. | ||

+ | |||

+ | |||

+ | The following list summarises the keywords available in the TDDFT program: | ||

+ | |||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * | ||

+ | |||

+ | Parameters which influence the behaviour of the Davidson solvers that can be used (should normally only be modified if convergence is hampered for some reason): | ||

+ | |||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * | ||

+ | |||

+ | A number of options from the list above can be given separately in the TDDFT command group, e.g.: | ||

+ | |||

+ | < | ||

+ | | ||

+ | </ | ||

+ | |||

+ | Some of them are exclusive, namely: | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | |||

+ | Default values for all parameters may also be looked up in the '' | ||

+ | |||

+ | |||

+ | ==== Analysing the spectrum ==== | ||

+ | |||

+ | By default, Molpro prints oscillator strengths and the transition moments correponding to the excitations computed at the end of the calculation in a summary table. | ||

+ | Using | ||

+ | |||

+ | < | ||

+ | tddft,...; gnuplot,< | ||

+ | </ | ||

+ | |||

+ | where '' | ||

+ | For example, for the water molecule (PBE0 functional) the resulting plot for the absorption spectrum is {{ : | ||

+ | Note that the shape of the plot can be adapted by modifying the value of sigma in ''' | ||

+ | for the Gaussian approximations for the respective transitions. | ||

+ | |||

+ | The excitation vectors can be exported to formats for visualisation by using either | ||

+ | |||

+ | < | ||

+ | tddft,...; molden,< | ||

+ | </ | ||

+ | which exports the excitation densities to a file which can be read with the program [[https:// | ||

+ | coefficients, | ||

+ | |||

+ | < | ||

+ | tddft,...; cube,< | ||

+ | </ | ||

+ | in which case for each excitation a Gaussian cube file is created which can be visualised with various external programs, e.g., [[http:// | ||

+ | |||

+ | For ' | ||

+ | problem (see, e.g., [[https:// | ||

+ | to obtain reliable results for the transition energies for such systems. Note that the failure for Rydberg excitations for standard DFT methods can be | ||

+ | resolved using asymptotically corrected xc potentials, see section [[the density functional program# | ||

+ | and the paper by Handy and Tozer [[http:// | ||

+ | |||

+ | In order to be able to distinguish between the different types of excitations, | ||

+ | values of $\Lambda$ described in the paper by Peach, Benfield, Helgaker and Tozer [[https:// | ||

+ | |||

+ | < | ||

+ | | ||

+ | </ | ||

+ | |||

+ | The values are defined as | ||

+ | $$\Lambda=\frac{\sum_{ia}\kappa_{ia}O_{ia}}{\sum_{ia} \kappa_{ia}}$$ | ||

+ | with | ||

+ | $$O_{ia}=\langle|\phi_i||\phi_a|\rangle=\int d{\bf r} |\phi_i({\bf r})||\phi_a({\bf r})|$$ | ||

+ | where $\kappa_{ia}$ is a coefficient of the excitation vector and $O_{ia}$ corresponds to the spatial overlap between an occupied orbital $\phi_i$ and an unoccupied orbital $\phi_a$, | ||

+ | see [[https:// | ||

+ | |||

+ | |||

+ | |||

+ | ==== Reponse properties from the coupled perturbed Kohn-Sham method ===== | ||

+ | |||

+ | The linear response to (frequency-dependent) perturbations can be calculated with the TDDFT program using | ||

+ | (use '' | ||

+ | |||

+ | < | ||

+ | | ||

+ | </ | ||

+ | where ''< | ||

+ | [[program control# | ||

+ | A list of frequencies ($\omega$) can be specified by '' | ||

+ | the response to perturbations oscillating at imaginary frequencies is calculated. Real and imaginary input values can be mixed | ||

+ | arbitrarily. If static response properties are requested, too, the value of '' | ||

+ | list (note that if the list of $\omega$' | ||

+ | |||

+ | For the calculation of multipole-multipole polarisabilities it is possible to use the short-hand input variant | ||

+ | |||

+ | < | ||

+ | | ||

+ | </ | ||

+ | in which case all $l=1,2,3$ rank responses (dipole, quadrupole and octopole) are computed without having to | ||

+ | insert all individual cartesian components. If all components for a given rank are given, the program performs | ||

+ | a transformation to the corresponding spherical harmonics representation and prints the results in the output | ||

+ | as well. | ||

+ | |||

+ | === Dispersion coefficients === | ||

+ | |||

+ | Dispersion coefficients can be calculated using | ||

+ | |||

+ | < | ||

+ | tddft, | ||

+ | </ | ||

+ | where '' | ||

+ | integral over imaginary frequencies with the Casimir-Polder integral transform. Normally, values | ||

+ | of the order of '' | ||

+ | quadrature scheme as described in the paper by [[https:// | ||

+ | is used in the response program. | ||

+ | |||

+ | The isotropic leading order $C_6$ dispersion coefficient between two monomers $A$ and $B$ is given | ||

+ | by | ||

+ | $$C_6^{AB} = \frac{3}{\pi}\int_0^\infty d{\omega}~\alpha^A(i\omega)\alpha^B(i\omega) $$ | ||

+ | with $\alpha^A$ and $\alpha^B$ denoting the isotropic dipole-dipole polarisabilities | ||

+ | of the two monomers. The $C_6$ coefficients are computed automatically if '' | ||

+ | is used. Higher order coefficients $C_8$ and $C_{10}$ can be computed as well | ||

+ | if quadrupole-quadrupole and dipole-octopole polarisabilities are available ('' | ||

+ | Note that the $C_{10}$ coefficients are computed even if no octopole moments are given in the input file. | ||

+ | The values then contain only the quadrupole-quadrupole $\times$ quadrupole-quadrupole polarisability | ||

+ | contributions. | ||

+ | |||

+ | For obtaining accurate results: use asymptotically corrected xc potentials ' | ||

+ | with ' | ||

+ | |||

+ | The following options can be set in the TDDFT response program: | ||

+ | * **'' | ||

+ | * **'' | ||

+ | |||

+ | === Rotatory strengths === | ||

+ | |||

+ | Rotatory strengths for the excitations requested can be calculated with | ||

+ | < | ||

+ | {tddft, | ||

+ | </ | ||

+ | These can be used to simulate CD spectra for optically active molecules. Since the conventional length representation of the underlying transition moments is not gauge-invariant (and so can strongly deviate from the cbs result for small basis sets), also the velocity representation for the electric transition dipole moments are computed and printed in the output, see, e.g., works by Autschbach //et al.// [[https:// | ||

+ | |||

+ | Rotatory strenghths can be currently computed only for the spin-restricted TDDFT case. | ||

+ | |||

+ | |||

+ | |||

+ | |||

+ | ==== Excited state gradient calculations ===== | ||

+ | |||

+ | For performing TDDFT gradient calculations and geometry optimisations for excited states the value of the | ||

+ | '' | ||

+ | < | ||

+ | tddft, | ||

+ | </ | ||

+ | Currently this is possible for closed-shell singlet excited states using LDA, GGA, hybrid-GGA or range-separated hybrid functionals. | ||

+ | |||

+ | In TDDFT gradient calculations it is recommended to set tighter thresholds for the convergence of the excited state vectors | ||

+ | and for the iterative solution of the Z-vector equation. E.g., for small systems the following settings | ||

+ | would be reasonable: | ||

+ | < | ||

+ | tddft, | ||

+ | </ | ||

+ | while for larger systems the threshold values may be set larger but should normally be more tight compared | ||

+ | to conventional TDDFT energy calculations, | ||

+ | The TDDFT program then computes the excited state density matrix as well as the excited state contribution to | ||

+ | the overlap lagrangian and writes these to internal records for later use. | ||

+ | The gradient can then be computed with the force program, see chapter [[energy_gradients|Energy gradients]] for more details: | ||

+ | < | ||

+ | tddft, | ||

+ | force | ||

+ | </ | ||

+ | and geometry optimisations can be done with the [[geometry_optimization_optg|optg]] command: | ||

+ | < | ||

+ | tddft, | ||

+ | optg | ||

+ | </ | ||

+ | Note that by default the gradient is computed for the lowest excitation of a given symmetry | ||

+ | (calculations for multiple symmetries are not allowed for gradient calculations). This can be | ||

+ | changed by setting the parameter '' | ||

+ | < | ||

+ | tddft, | ||

+ | </ | ||

+ | will compute the gradient for the 4th excited state of the second symmetry. It is necessary, of course, that the number | ||

+ | of states requested by the '' | ||

+ | an eigenvector-following method like is implemented in the EOM-CCSD program is not yet available for | ||

+ | TDDFT geometry optimisations. Because of this it might happen that the initial | ||

+ | starting guess is not followed strictly when there are crossings with other close excitations on | ||

+ | the PES. This will be improved in a future version of the program. | ||

+ | |||

+ | There are a few special parameters which can influence the behaviour of tddft gradient calculations. Most of them are | ||

+ | for debugging purpose and should normally not changed compared to their default values set in the registry file. | ||

+ | |||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | * **'' | ||

+ | |||

+ | |||

+ | ==== Nonadiabatic coupling matrix elements ===== | ||

+ | |||

+ | First order nonadiabatic coupling matrix elements (NACME' | ||

+ | $$\tau^\xi_{nm}=\langle \Psi_n|\frac{\partial}{\partial \xi}|\Psi_m\rangle $$ | ||

+ | where $\xi$ is a cartesian nuclear coordinate and $\Psi_n$ and $\Psi_m$ are the wave functions of the two states. In order to derive an expression for the first-order NACME from time-dependent reponse theory one considers the time evolution of the imaginary matrix element | ||

+ | $$C_\lambda^\xi(t)=\langle \Psi_\lambda(t)|\frac{\partial}{\partial\xi}|\Psi_\lambda(t)\rangle$$ | ||

+ | under the influence of a monochromatic one-particle perturbation $\hat{W}$ and finds that the NACME is given by the residues of $\overline{C}^{\xi(1)}$ at the excitation energies $\omega_n$ | ||

+ | $$\tau_{0n}^\xi = \frac{2\pi i\mbox{Res}[\overline{C}^{\xi(1)}(\omega); | ||

+ | |||

+ | Following the work of Send and Furche [[https:// | ||

+ | |||

+ | In Molpro, first-order NACME' | ||

+ | < | ||

+ | {tddft, | ||

+ | </ | ||

+ | The option '' | ||

+ | |||

+ | First-order NACME' | ||

+ | |||

+ | |||