Time-dependent DFT: Excitation Energies, (Hyper) Polarizabilities

• General remarks on the use of the TDDFT Response and Excitation functionality
• Excitation Input
• Applications of the Excitation feature in ADF
• Input description for the Response functionality

Excitation energies, frequency-dependent (hyper) polarizabilities, Van der Waals dispersion coefficients, higher multipole polarizabilities, Raman scattering intensities and depolarization ratios of closed-shell molecules are all available in ADF [46, 47] as applications of time-dependent DFT (TDDFT); see [48] for a review.

The input description for these properties is split in three parts: (a) general advice and remarks, (b) excitation energies, and (c) frequency-dependent (hyper) polarizabilities and related properties.

General remarks on the use of the TDDFT Response and Excitation functionality

Symmetry

As in calculations without TDDFT the symmetry is automatically detected from the input atomic coordinates and need not be specified, except in the following case: infinite symmetries cannot be handled in the current release (ATOM, C(lin), D(lin)). For such symmetries a subgroup with finite symmetry must be specified in the input. The usual orientation requirements apply.
If higher multipole polarizabilities are required, it may also be necessary to use a lower subgroup (the program will stop with an error message otherwise). For verification of results one can always compare to a NOSYM calculation.

Closed-shell only

The current implementation supports only closed-shell molecules. If occupation numbers other than 0 or 2 are used the program will detect this, (but only at a later stage of the calculation) and abort. All "RESPONSE" calculations must be spin-restricted.

Atomic coordinates in a RAMAN calculation

Atomic coordinate displacements in a RAMAN calculation must be Cartesian, not Z-matrix. Furthermore, the current implementation does not yet support constrained displacements, i.e. you must use all atomic coordinate displacements.

Use of diffuse functions

The properties described here may require diffuse functions to be added to the basis (and fit) sets. Poor results will be obtained if the user is unaware of this. As a general rule, diffuse functions are more important for smaller than for larger molecules, more important for hyperpolarizabilities than for normal polarizabilities, more important for high-lying excitation energies (Rydberg states) than for low-lying excitations, more important for higher multipole polarizabilities than for dipole polarizabilities. The user should know when diffuse functions are required and when they are not: the program will not check anything in this respect. For example, in a study on low-lying excitation energies of a large molecule, diffuse functions will usually have little effect, whereas a hyperpolarizability calculation on a small molecule is pointless unless diffuse functions are included. Diffuse basis sets are included in the database (in the Vdiff/ directory of the database), but only for a few atoms. For some other atoms diffuse basis sets may be available at the web site http:tc.chem.vu.nl/~vgisberg. For other atoms, the user will have to add diffuse basis and fit functions to the existing data base sets. It is not necessary to start from basis V as was done for the basis sets just mentioned. All this means, of course, that usage of the TDDFT functionality is still somewhat experimental. It may be expected that more extensive basis sets will come available in the future, when usage and experience increase.

Linear dependency in basis

If large diffuse basis sets are used, or if diffuse functions are used for atoms that are not far apart the calculation may suffer from numerical problems because of (near-) linear dependencies in the basis set. The user should be aware of this danger and use the DEPENDENCY key to check and solve this.

The TAILS input keyword

For reasons of numerical robustness and safety rather strict defaults apply for the neglect of tails of basis and fit functions (see the key TAILS) in a Response or Excitation calculation. This may result in longer CPU times than needed for non-TDDFT runs, in particular for larger molecules. Possibly this precaution is not necessary, but we have not yet tested this sufficiently to relax the tightened defaults.

Relativistic effects

The Response and Excitations options can be combined with scalar relativistic options (ZORA or Pauli). The one-electron relativistic orbitals and orbital energies are then used as input for the property calculation. Spin orbit effects have not yet been incorporated in this part of the code. In case of a ZORA calculation, the so-called "scaled" orbital energies are used as default.

Choice of XC potential

For properties that depend strongly on the outer region of the molecule (high-lying excitation energies, (hyper) polarizabilities), it may be important to use a XC potential with correct asymptotic behavior (approaching -1/r as r tends to infinity). In ADF the LB94 potential has been implemented for this purpose [24].

With this particular XC functional, the XC potential is computed from the exact charge density for reasons of stability and robustness (whereas for other functions the (cheaper) fit density is used). This implies that computation times may be longer. Another `side effect' is that, since there is no energy expression corresponding to the LB94 potential, the final (bonding) energy of a LB94 calculation uses another GGA and hence the energy result is not (exactly) consistent with the SCF procedure. Note, finally, that the LB94 potential is not suitable for geometry optimizations because it is rather inaccurate in the bonding region, see the discussion of the XC input key.
Applications with the LB94 potential to response calculations with ADF can be found in [49] (polarizabilities), [50-52] (hyperpolarizabilities), [53] (high-lying excitation energies), [54] (multipole polarizabilities and dispersion coefficients)

Accuracy check list

As mentioned before, the TDDFT module is relatively new and not extensively tested for a wide range of applications. Therefore, we strongly recommend the user to build experience about aspects that may affect the accuracy of TDDFT results. In particular we advise to `experiment' with

- Varying integration accuracy

- Varying the SCF convergence

- Varying the ORTHONORMALITY and TOLERANCE values in an EXCITATION calculation

- Varying the TAILS parameters

- Using diffuse functions

- Using the DEPENDENCY key

- Applying the ZORA relativistic corrections for molecules containing heavy nuclei

- Using an asymptotically correct XC potential such as LB94

Excitation Input

You can perform a calculation of singlet-singlet and singlet-triplet excitation energies of a closed-shell molecule by supplying in the input file the block key EXCITATION

EXCITATION

{ OPTION }

{ OPTION }

...

END

Several options can be addressed with subkeys in the data block. This functionality is based on TDDFT and consequently has a different theoretical foundation than the SCF techniques described elsewhere in this User's Guide. Two possible ways are available to solve the eigenvalue equation from which the excitation energies and oscillator strengths are obtained, of which the iterative Davidson procedure is the default. In this case, the program needs to know how many excitation energies are needed per irrep, what accuracy is required, and what type of excitation energies are required (singlet-singlet or singlet-triplet). Suitable defaults have been defined for all of these. Each of these points is discussed below.

Exact diagonalization vs. iterative Davidson procedure

The most straightforward procedure is a direct diagonalization of the matrix from which the excitation energies and oscillator strengths are obtained. Since the matrix may become very large, this option is possible only for very small molecules. It can be activated by specifying the word EXACT as one of the subkeys in the EXCITATIONS data block. The default is the iterative Davidson method. A few of the lowest excitation energies and oscillator strengths are then found within an error tolerance. An advantage of the EXACT option is that additional information is produced, such as the Cauchy coefficients that determine the average dipole polarizability.

Singlet versus triplet

By default, the singlet-singlet and singlet-triplet excitation energies are both calculated. The singlets are handled first, then the corresponding triplet excitation energies. One can skip one of these two parts of the calculation by specifying either ONLYSING or ONLYTRIP as a subkey in the data block.

Which excitation energies and how many?

The user can specify how many excitation energies per irrep should be calculated. If no pertaining input is available the program determines these numbers from the smallest differences between occupied and virtual Kohn-Sham orbital energies. By default it looks at the 10 lowest orbital energy differences. This number can be modified, by specifying inside the EXCITATION block key, for example:

LOWEST 30

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