Time-dependent DFT
******************

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 [1] [2] as applications of time-
dependent DFT (TDDFT); see [3] for a review.

New in ADF2004.01 is the calculation of circular dichroism (CD)
spectra, and the calculation of the optical rotation (dispersion).

Starting from the ADF2005.01 version it is possible to calculate
excitation energies for open-shell systems with TDDFT, including spin-
flip excitation energies. New in ADF2005.01 is the possibility to use
time-dependent current-density functional theory (TDCDFT).

New in ADF2006.01 is the possibility to calculate excitation energies
for closed-shell molecules including spin-orbit coupling.

New in ADF2008.01 is the possibility to calculate lifetime effects in
(dynamic) polarizabilities (AORESPONSE key).

The input description for these properties is split into three parts:
(a) general advice and remarks, (b) excitation energies, and (c)
frequency-dependent (hyper) polarizabilities (two alternative
implementation: RESPONSE key and AORESPONSE key) and related
properties.

Tip:

  See also the ADF Advanced TDDFT tutorial: TDDFT Study of 3 different
  Dihydroxyanthraquinones


General remarks on the 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"
   The current implementation often 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.

"Open-shell"
   Excitation energies can be obtained for open-shell systems in a
   spin-unrestricted TDDFT calculation. Spin-flip excitation energies
   can only be obtained in a spin-unrestricted TDDFT calculation.

"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 even-
   tempered basis sets are included in the ET/ directory in
   "$AMSHOME/atomicdata/ADF" for the elements H-Kr. Somewhat older
   basis sets can be found in the Special/Vdiff directory in
   "$AMSHOME/atomicdata/ADF". For other atoms, the user will have to
   add diffuse basis and fit functions to the existing database sets.
   It is not necessary to start from basis V as was done for the basis
   sets in Special/Vdiff. For example, for heavier elements it may be
   a good idea to start from the ZORA/QZ4P basis sets. It may be
   expected that even 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 LINEARSCALING 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 LINEARSCALING) 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 been incorporated
   only in this part of the code (excitation energies). 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 an XC potential with
   correct asymptotic behavior (approaching -1/r as r tends to
   infinity). Finally, several asymptotically correct XC potentials
   have been implemented in ADF, such as the LB94 potential [4] and
   the statistical average of orbital potentials SAOP [8] [6]. SAOP is
   recommended. Because of the correct asymptotic behavior the SAOP
   potential (and the LB94 potential) can describe Rydberg states
   correctly. The potentials based upon the so-called GRadient
   regulated seamless connection of model potentials (GRAC) for the
   inner and the outer region [5] [7] have similar performance to
   SAOP, but have the disadvantage that the ionization energy of the
   molecule has to be used as input.

   With the SAOP and GRAC functionals for the potential (as well as
   for LB94), the XC potential is computed from the exact charge
   density for reasons of stability and robustness (whereas for other
   functionals 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 these
   potentials, the final (bonding) energy of such calculations uses
   another GGA and hence the energy result is not (exactly) consistent
   with the SCF procedure. Note, finally, that these potentials have
   been found to be not suitable for geometry optimizations because
   they may not be sufficiently accurate in the bonding region, see
   the discussion of the XC input key. Applications with SAOP to
   (hyper)polarizabilities and excitation energies, also for Rydberg
   transitions, can be found in [6] and with SAOP and Becke-Perdew-
   GRAC in [5] [7]. Applications with the old LB94 potential to
   response calculations can be found in refs. [9] (polarizabilities),
   [20] [21] [22] (hyperpolarizabilities), [10] (high-lying excitation
   energies), [11] (multipole polarizabilities and dispersion
   coefficients).

"XC kernel"
   In most cases the adiabatic local density approximated (ALDA)
   kernel is used in the TDDFT functionality. In case of calculating
   excitation energies with hybrid functionals the Hartree-Fock
   percentage times the Hartree-Fock kernel plus one minus the
   Hartree-Fock percentage times the ALDA kernel is used (hybrid ALDA
   kernel). For the LibXC range separated functionals, like CAM-B3LYP,
   starting from ADF2016.102 the kernel consists of range separated
   ALDA plus the kernel of the range separated exact exchange part. In
   ADF2016.101 the kernel for LibXC range separated functionals, like
   CAM-B3LYP, was using a 100% ALDA plus range separated exact
   exchange kernel (the ALDA part was not range-separated corrected).

   For excitation energy calculations in some cases the full (non-
   ALDA) kernel can be evaluated, see the full XC kernel description.
   The Full kernel cannot be used in combination with symmetry or
   excited state geometry optimizations.

"COSMO"
   The COSMO model for solvation is a cheap method to include
   solvation effects in the TDDFT applications, see the SOLVATION key.
   Note that in TDDFT calculations one may have to use two dielectric
   constants. The reason is that the electronic transition is so fast
   that only the electronic component of the solvent dielectric can
   respond, i.e., one should use the optical part of the dielectric
   constant. This is typically referred to as non-equilibrium
   solvation. The optical dielectric constant can be obtained from the
   (frequency dependent) refractive index n of the solvent as:
   \epsilon_opt  = n^2 . One can use the argument NEQL in the subkey
   SOLV of the key SOLVATION to set a value for the optical dielectric
   constant. No dielectric constant in the response might be closer to
   the optical dielectric constant than using the full dielectric
   constant. This can be achieved if one includes the subkey NOCSMRSP
   in the block key SOLVATION, such that the induced electronic
   charges do not influence the COSMO surface charges. However, if one
   does geometry optimization then the full dielectric constant should
   be used in the TDDFT simulations since the solvent dielectric now
   has time to fully respond. The default is that the full dielectric
   constant is used in the TDDFT calculations.

"Accuracy check list"
   We strongly recommend the user to build experience with aspects
   that may affect the accuracy of TDDFT results. In particular we
   advise to 'experiment' with

   * Varying basis sets

   * Varying numericalquality

   * Varying the TOLERANCE values in an Excitation calculation

   * Using diffuse functions in case of small molecules

   * Using the DEPENDENCY key in case of large basis sets

   * Applying the ZORA relativistic corrections for molecules
     containing heavy nuclei

   * Using an asymptotically correct XC potential such as SAOP


Analysis options for TDDFT (excitation energies and polarizabilities)
=====================================================================

Several options are available to obtain more detailed results than a
few bare numbers for excitation energies, oscillator strengths,
transition dipole moments, and (hyper)polarizabilities. For a zero-
order understanding of which occupied and virtual orbitals play an
important role in the polarizability or intensity of an absorption
peak,

it may be useful to know the values of the dipole matrix elements
between (ground-state) occupied and virtual Kohn-Sham orbitals. If
these dipole matrix elements are large for a particular occupied-
virtual orbital pair, then this pair is almost certainly of great
importance for the whole spectrum or polarizability. This information
can be obtained by specifying somewhere in the input file (but NOT
inside the RESPONSE or EXCITATIONS block keys):

   PRINT DIPOLEMAT


Time-dependent Current DFT
==========================

The time-dependent current-density-functional (TDCDFT) implementation
is built entirely upon the normal TDDFT implementation. Therefore all
general remarks that are made for the TDDFT part of the program are
also valid for TDCDFT. Only the polarizability and excitation energies
of closed shell molecules can be calculated with TDCDFT in the present
implementation.

If TDCDFT is used together with the ALDA functional (NOVK option) it
will give the same results for the polarizability and excitation
energies as TDDFT in a complete basis set. TDCDFT in ADF by default
uses the VK functional [12] [13], since this is the only current
dependent functional that is known presently. Many aspects of the
functional are still unknown and the functional should therefore be
used with caution. The user is referred to the references for more
information on when the VK functional gives good results and when not.

For more information on the implementation and applications of the
TDCDFT and the VK functional please read the references [23] [24] [25]
[26]. For more details on the theory and implementation in ADF see:
[14].

To activate TDCDFT and the VK functional one should add the following
block key to the input file:

   CURRENTRESPONSE
   END

To calculate the polarizability the keyword Response can be used with
the following options:

   RESPONSE
     ALLCOMPONENTS
     Frequencies freq1 freq2 ... freqN
   END

The block key EXCITATIONS can be used with all of its options.

In ADF2012 the block keyword AORESPONSE can also be used with the
Vignale-Kohn functional. The current-density is generated on the fly
but otherwise the computation is based on the time dependent density
response.

By default the VK functional will be applied where the NCT
parametrization [15] is chosen for the transverse exchange-correlation
kernel for the polarizability and singlet excitation energies (giving
the best results for the systems studied so far). For triplet
excitation energies the only available parametrization will be used
[16]. This option is not tested much and the results are in general
much worse than ALDA [14]. It is therefore suggested that VK is not
used to calculate triplet excitation energies.

In the output the polarizability tensor (in case of an ALLCOMPONENTS
calculation) has a different shape, the results are printed in the
more intuitive order x, y, z, instead of y, z, x that the TDDFT
implementation uses.

The following subkeys are available within the datablock of
CURRENTRESPONSE

   CURRENTRESPONSE
     QIANVIGNALE
     NOVK
   END

"QIANVIGNALE"
   The QV parametrization [16] will be used for the transverse
   exchange-correlation kernel instead of NCT.

"NOVK"
   TDCDFT will be applied with the ALDA functional instead of the VK
   functional. In a complete basis this will give the same results as
   a TDDFT calculation.


Magnetic properties within TDCDFT
---------------------------------

It is now possible to calculate several magnetic properties via
TDCDFT, among which magnetizabilities, rotational g-tensors, NMR
shielding constants, specific rotations and circular dichroism. This
alternative method is based on the diamagnetic-current sum rule, which
consists of rewriting the diamagnetic current in terms of the response
functions. This is the magnetic analogue of the conductivity sum rule,
which is invalid when an external magnetic field is applied.

Details on the method can be found in these references: [17] [18].
Further details regarding the implementation can be found in the
following PhD thesis as well: [19].

In order to obtain these properties, the following subkeys can be
used:

   CURRENTRESPONSE
     MAGNET
     GTENSOR
     NMRSHIELDING
     CDSPEC
     DAMPING Damping
   END

Depending on which property is calculated, the program chooses a
reference point for the external vector potential (see references).
For the magnetizability, the center of electronic charges is chosen,
while for the rotational g-tensor, the center of nuclear charges is
picked up. This choice has no influence in the static case, and has no
influence neither for the specific rotations nor for the circular
dichroism. It actually only affects the dynamical magnetizabilities
(the rotational g-tensor being always computed at zero frequency).
Below is a brief description of each subkey and how to use them.

"MAGNET"
   The (static or dynamical) magnetizability tensor.

"GTENSOR"
   The rotational g-tensor, calculated directly from the
   magnetizability.

"NMRSHIELDING"
   The NMR shielding tensor. Care should be taken when using this
   keyword, though, as it was observed that the results failed to
   match experimental and other theoretical studies when using
   standard basis sets except for the lightest elements. Only when
   using very large basis sets should the results be trusted.

"CDSPEC"
   Calculates both the specific rotation and the circular dichroism.
   This keyword should be used in combination with DAMPING if one
   wants to simulate CD spectra. The output spectrum will exhibit
   relatively sharp peaks. Linear-interpolating the results with a
   third-tier program is then necessary. This is done on purpose, so
   that the final spectrum could be broadened easily (only the
   absolute amplitudes of the peaks will be affected by this
   broadening).

"DAMPING Damping"
   Adds a (fixed) damping factor to the response function. This allows
   one to avoid divergences that may arise from the response function
   when close to resonance, hence helping the self-consistent cycle
   converge. The value for Damping should be given in eV. A value of
   0.1 eV seems to be reasonable.

[1] S.J.A. van Gisbergen, J.G. Snijders and E.J. Baerends,
    *Implementation of time-dependent density functional response
    equations*, Computer Physics Communications 118, 119 (1999)

[2] S.J.A. van Gisbergen, *Molecular Response Property Calculations
    using Time Dependent Density Functional Theory*, in *Chemistry*.
    1998, Vrije Universiteit: Amsterdam. p. 190.

[3] E.K.U. Gross, J.F. Dobson and Petersilka, in *Density Functional
    Theory*, R.F. Nalewajski, Editor. 1996, Springer: Heidelberg.

[4] R. van Leeuwen and E.J. Baerends, *Exchange-correlation potential
    with correct asymptotic behavior*, Physical Review A 49, 2421
    (1994)

[5] M. Grüning, O.V. Gritsenko, S.J.A. van Gisbergen and E.J.
    Baerends, *Shape corrections to exchange-correlation Kohn-Sham
    potentials by gradient-regulated seamless connection of model
    potentials for inner and outer region*, Journal of Chemical
    Physics 114, 652 (2001)

[6] P.R.T. Schipper, O.V. Gritsenko, S.J.A. van Gisbergen and E.J.
    Baerends, *Molecular calculations of excitation energies and
    (hyper)polarizabilities with a statistical average of orbital
    model exchange-correlation potentials*, Journal of Chemical
    Physics 112, 1344 (2000)

[7] M. Grüning, O.V. Gritsenko, S.J.A. van Gisbergen and E.J.
    Baerends, *On the required shape correction to the LDA and GGA
    Kohn Sham potentials for molecular response calculations of
    (hyper)polarizabilities and excitation energies*, Journal of
    Chemical Physics 116, 9591 (2002)

[8] O.V. Gritsenko, P.R.T. Schipper and E.J. Baerends, *Approximation
    of the exchange-correlation Kohn-Sham potential with a statistical
    average of different orbital model potentials*, Chemical Physics
    Letters 302, 199 (1999)

[9] S.J.A. van Gisbergen, V.P. Osinga, O.V. Gritsenko, R. van Leeuwen,
    J.G. Snijders and E.J. Baerends, *Improved density functional
    theory results for frequency-dependent polarizabilities, by the
    use of an exchange-correlation potential with correct asymptotic
    behavior*, Journal of Chemical Physics 105, 3142 (1996)

[10] M.E. Casida, C. Jamorski, K.C. Casida and D.R. Salahub,
     *Molecular excitation energies to high-lying bound states from
     time-dependent density-functional response theory:
     Characterization and correction of the time-dependent local
     density approximation ionization threshold*, Journal of Chemical
     Physics 108, 4439 (1998)

[11] V.P. Osinga, S.J.A. van Gisbergen, J.G. Snijders and E.J.
     Baerends, *Density functional results for isotropic and
     anisotropic multipole polarizabilities and* C_6 , C_7 , and C_8
     *Van der Waals dispersion coefficients for molecules*, Journal of
     Chemical Physics 106, 5091 (1997)

[12] G. Vignale and W. Kohn, *Current-Dependent Exchange-Correlation
     Potential for Dynamical Linear Response Theory*, Physical Review
     Letters 77, 2037 (1996)

[13] G. Vignale and W. Kohn, in *Electronic Density Functional Theory:
     Recent Progress and New Direction*, ISBN13: 9780306458347, J. F.
     Dobson, G. Vignale, and M. P. Das, Editors. 1998, Plenum: New
     York.

[14] M. van Faassen, Time-Dependent Current-Density-Functional Theory
     for Molecules, PhD thesis, Rijksuniversiteit Groningen, 2004.

[15] R. Nifosi, S. Conti and M. P. Tosi, *Dynamic exchange-correlation
     potentials for the electron gas in dimensionality D=3 and D=2*,
     Physical Review B 58: p. 12758 (1998)

[16] Z.X. Qian and G. Vignale, *Dynamical exchange-correlation
     potentials for the electron liquid in the spin channel*, Physical
     Review B 68, 195113 (2003)

[17] N. Raimbault, P.L. de Boeij, P. Romaniello, and J.A. Berger,
     *Gauge-Invariant Calculation of Static and Dynamical Magnetic
     Properties from the Current Density*, Physical Review Letters
     114, 066404 (2015)

[18] N. Raimbault, P.L. de Boeij, P. Romaniello, and J.A. Berger,
     *Gauge-Invariant Formulation of Circular Dichroism*, Journal of
     Chemical Theory and Computation 12, 3278 (2016)

[19] N. Raimbault, Gauge-invariant magnetic properties from the
     current, PhD thesis, Université Paul Sabatier, LCPQ, Toulouse,
     2015

[20] S.J.A. van Gisbergen, J.G. Snijders, and E.J. Baerends, *Time-
     dependent Density Functional Results for the Dynamic
     Hyperpolarizability of* C_60, Physical Review Letters 78, 3097
     (1997)

[21] S.J.A. van Gisbergen, J.G. Snijders and E.J. Baerends,
     *Calculating frequency-dependent hyperpolarizabilities using
     time-dependent density functional theory*, Journal of Chemical
     Physics 109, 10644 (1998)

[22] S.J.A. van Gisbergen, J.G. Snijders and E.J. Baerends, *Accurate
     density functional calculations on frequency-dependent
     hyperpolarizabilities of small molecules*, Journal of Chemical
     Physics 109, 10657 (1998)

[23] M. van Faassen, P. L. de Boeij, R. van Leeuwen, J. A. Berger and
     J. G. Snijders, *Ultranonlocality in Time-Dependent Current-
     Density-Functional Theory: Application to Conjugated Polymers*,
     Physical Review Letters 88, 186401 (2002)

[24] M. van Faassen, P. L. de Boeij, R. van Leeuwen, J. A. Berger and
     J. G. Snijders, *Application of time-dependent current-density-
     functional theory to nonlocal exchange-correlation effects in
     polymers*, Journal of Chemical Physics 118, 1044 (2003)

[25] M. van Faassen and P. L. de Boeij, *Excitation energies for a
     benchmark set of molecules obtained within time-dependent
     current-density functional theory using the Vignale-Kohn
     functional*, Journal of Chemical Physics 120, 8353 (2004)

[26] M. van Faassen and P. L. de Boeij, *Excitation energies of* \Pi
     *-conjugated oligomers within time-dependent current-density-
     functional theory*, Journal of Chemical Physics 121, 10707 (2004)
