Excitation energies: UV/Vis, X-ray, CD, MCD¶

Excitation energies, UV/Vis spectra¶

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. See the next sections for settings of technical parameters, the calculation of excitation energies for open shell molecules, inclusion of spin-orbit coupling, and the calculation of CD spectra.

EXCITATIONS
 EXACT &
   IRREP1 N1
   IRREP2 N2
 SUBEND
 DAVIDSON &
   IRREP3 N3
   IRREP4 N4
 SUBEND
 ALLOWED
 ONLYSING
 ONLYTRIP
 LOWEST nlowest
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. The EXACT option can not be used in unrestricted calculations.

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.

In case of a calculation including spin-orbit coupling one can not separate the singlet-singlet and singlet-triplet excitations. The subkeys ONLYSING and ONLYTRIP are misused in this case to do a spin-restricted calculation, or a spin-polarized calculation, respectively. One should in fact only use the results of the spin-polarized calculation.

Dipole-allowed versus general excitations.

If you are interested in the optical absorption spectrum, you may not want to compute singlet-triplet excitation energies, nor singlet-singlet excitation energies which, by symmetry, have zero oscillator strengths. This subkey should not be used in case of spin-orbit coupling. The subkey ALLOWED tells ADF to treat only those irreducible representations for which the oscillator strengths will be nonzero. Of course, the oscillator strengths may still be negligibly small. The ALLOWED subkey automatically implies ONLYSING. The simplest, fastest, and recommended way to obtain information about the ten lowest dipole-allowed excitation energies would be:

EXCITATIONS
 ALLOWED
 LOWEST 10
END

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

One should be aware that this procedure does not guarantee that the lowest 10 (or 30) excitation energies will actually be found, since the orbital energy difference approximation to the excitation energy is rather crude. However, if the program decides on the basis of this procedure to calculate 4 excitation energies in a certain irreducible representation, these 4 excitation energies are certainly the lowest in that particular irrep.

The user has more control when the number of excitations per irrep is explicitly specified within the EXCITATION block key by the Davidson subkey:

DAVIDSON &
 E'' 5
 T1.u 2
SUBEND

The DAVIDSON sub key is a general (simple or block type) subkey. For usage as block type it must be followed by the continuation character &. Its data block may contain any number of records and must end with a record SUBEND. In the subkey data block a list of irreps, followed by the number of requested excitation energies is specified. Note that the irrep name may not be identical to the usual ADF name. For example E’’ is called EEE in ADF. The Excitation code will skip an irrep if the label is not recognized. For multidimensional irreps, only the first column is treated, because the other would produce identical output. This implies that the oscillator strengths for E-irreps have to be multiplied by 2 and the oscillator strengths for T-irreps by 3. The ALLOWED subkey should not be used if irreps are specified with the Davidson block subkey, however, the subkey ONLYSING (or ONLYTRIP) can be used in this case.

The EXACT subkey, mentioned already above, can also be used as a block type subkey to treat only a few irreps instead of all. The number of excitation energies does not have to be specified then.

TDA

XC
 ...
 XCFUN
End
EXCITATIONS
 ...
 FullKernel
END

XC
 LibXC ...
End
EXCITATIONS
 ...
 FullKernel
END

EXCITATIONS
 VECTORS 40
 TOLERANCE 1e-6
 ORTHONORMALITY 1e-8
 ITERATIONS 200
 KFWRITE 3
END

Excitation energies for open-shell systems¶

Excitation energies can be obtained for open-shell systems in a spin-unrestricted TDDFT calculation [154]. To perform an open-shell TDDFT calculation one just needs to do an unrestricted SCF calculation and use the EXCITATION keyword. Presently the excitation energies can only be found with Davidson’s procedure. In case of spin-orbit coupling, see the section on approximate spin-orbit coupled excitation energies open shell molecule.

The printed symmetry in the output in TDDFT calculations is actually the symmetry of transition density. For closed-shell systems, the symmetry of the excited state is the same as the symmetry of the transition density, while for open-shell systems, the symmetry of the excited states is the direct product between the symmetry of the transition density and the ground state symmetry. Note that the ground state symmetry of an open shell molecule is not necessarily A1.

For degenerate representations such as the 2-dimensional E-representations or the 3-dimensional T-representations, the occupation should be either fully occupied or zero. For example, for an orbital in an E-representation the \(\alpha\) and \(\beta\) occupation number should be either 2 or 0. The \(\alpha\) occupation number can of course be different from the \(\beta\) occupation number.

As for the spin-state, the general rule is that if the excited state mainly results from transitions from the singly occupied orbitals to virtual orbitals or from fully occupied orbitals to the singly occupied orbitals, the spin state of the excited state should roughly be the same as that of the ground state. However, if the excited state mainly comes from transitions from fully occupied orbitals to virtual orbitals, the spin state of the excited state are usually a mixture since TDDFT can only deal with single excitations within adiabatic approximation for the XC kernel [155]. Sometimes we just suppose the spin state of this kind of excited states to be the same as that of ground state [154]. In the MO → MO transitions part for the excitations of the output file, the spin of each molecular orbitals are also specified to help assign the spin state of the excited states. The transitions are always from \(\alpha\) spin-orbital to \(\alpha\) spin-orbital or from \(\beta\) spin-orbital to \(\beta\) spin-orbital.

Spin-flip excitation energies¶

Spin-flip excitation energies [156, 157] can only be obtained in a spin-unrestricted TDDFT calculation. This can not be used in case of spin-orbit coupling. At present, the spin-flip excitation energies can only be calculated with Tamm-Dancoff approximation (TDA) [158] and Davidson’s method.

To calculate spin-flip excitation energies, one must specify two keys:

SFTDDFT
TDA

anywhere in the input file in addition to the EXCITATION block keyword.

In spin-flip TDDFT, the XC kernel can be calculated directly from the XC potential. To use the LDA potential for the XC kernel, which roughly corresponds to the ALDA in ordinary TDDFT, one must specify the key

FORCEALDA

anywhere in the input file. Only calculations using the LDA potential in the SCF are fully tested. Using other GGA potentials in the SCF and using the FORCEALDA key at the same time may introduce unreasonable results, while using LB94 or SAOP potential in the SCF without the FORCEALDA key may give unstable results. Unstable results have been reported for the PW91 functional.

For open-shell molecules, spin-flip transition can result in transition to the ground state with a different S_z value, while the symmetry of the transition density is A1. The excitation energy of this transition should be zero and this can be used to test the reliability of spin-flip TDDFT.

The symmetry of the excited states can be determined in the same way as that in spin-unrestricted TDDFT calculations. As for the spin state, similar to that in the spin-unrestricted TDDFT calculations, some states may be more or less pure spin states, others may just be mixtures. The users can interpret the excited state through the transitions that contribute to this state. Note that the transitions are always from \(\alpha\) spin-orbital to \(\beta\) spin-orbital in spin-flip calculations, or from \(\beta\) spin-orbital to \(\alpha\) spin-orbital.

Select (core) excitation energies, X-ray absorption¶

Two methods can be used to reduce the computational costs of, for example, core excitation energies, or some other high lying excitation energy. In the state selective method scheme a guess vector for the orbital transition has to be provided. An overlap criterion is used to follow the wanted eigenvector. In this scheme the one-electron excited state configuration space remains complete, see Ref [346]. In the second scheme, the range of excitations that are calculated is modified, which means that the one-electron excited state configuration space is reduced to the interesting part, see Ref. [169]. The calculated excited states are more accurate with the state selective method if convergence is reached, however, the second scheme is more robust, and it is easier to find convergence.

These selection methods can, for example, also be used in case one calculates excitation energies as Kohn-Sham orbital energy differences, see subkey SINGLEORBTRANS of the key EXCITATIONS.

For X-ray spectra it can be important to include quadrupole intensities.

SELECTEXCITATION
   OscStrength oscstrength
   UseOccVirtRange elowoccvirt ehighoccvirt
   UseOccVirtNumbers nrlowoccvirt nrhighoccvirt
   UseOccRange elowocc ehighocc
   UseVirtRange elowvirt ehighvirt
   UseOccupied
      irrep orbitalnumbers
      irrep orbitalnumbers
      ...
   SubEnd
   UseVirtuaL
      irrep orbitalnumbers
      irrep orbitalnumbers
      ...
   SubEnd
   UseScaledZORA
end

MODIFYEXCITATION
   OscStrength oscstrength
   UseOccVirtRange elowoccvirt ehighoccvirt
   UseOccVirtNumbers nrlowoccvirt nrhighoccvirt
   UseOccRange elowocc ehighocc
   UseVirtRange elowvirt ehighvirt
   UseOccupied
      irrep orbitalnumbers
      irrep orbitalnumbers
      ...
   SubEnd
   UseVirtuaL
      irrep orbitalnumbers
      irrep orbitalnumbers
      ...
   SubEnd
   SetOccEnergy esetocc
   SetLargeEnergy epsbig
   UseScaledZORA
end

EXCITATIONS
 SingleOrbTrans
END

EXCITATION
  ...
  XAS
  {ALLXASMOMENTS}
  {ALLXASQUADRUPOLE}
END

XES: X-ray emission spectroscopy¶

For the calculation of XES (X-ray emission spectroscopy) spectra the approach of Ref.[386] is implemented in the ADF program by Atkins et. al., see Ref.[387]. This is a frozen orbital, one-electron ΔDFT approach which uses orbital energy differences between occupied orbitals to model the X-ray emission energies. Even though it is the simplest possible approximation for the calculation of XES spectra, it has been shown to work well for V2C-XES (valence-to-core X-ray emission spectroscopy) spectra of transition metal complexes.

The XES keyword initiates the calculation of X-ray emission energies to a core orbital. In addition to dipole oscillator strengths this keyword also triggers the calculation of the higher order moment integrals and the calculation of the quadrupole oscillator strengths. By default it calculates the emission to the first orbital in the first symmetry, often the deepest core orbital, and only prints the total oscillator strength and the excitation energy. Can not be used in combination with spin-orbit coupling.

XES
   {COREHOLE irrep number}
   {ALLXESMOMENTS}
   {ALLXESQUADRUPOLE}
END

COREHOLE irrep number

This allows the selection of the acceptor orbital for the calculation of the emission oscillator strengths. For example ‘COREHOLE A1 2’ calculates oscillator strengths to the orbital 2 in irrep A1.

ALLXESMOMENTS

Use ALLXESMOMENTS within the XES block. This will print out all the individual transition moments used within the calculation of the total oscillator strength.

ALLXESQUADRUPOLE

Use ALLXESQUADRUPOLE within the XES block. This will print out the individual oscillator strength components to the total oscillator strength.

Excitation energies and Spin-Orbit coupling¶

Spin-orbit coupling can be included in the TDDFT calculation of excitation energies for closed-shell molecules. Two methods can be used in ADF. The first one includes spin-orbit coupling as a perturbation to a scalar relativistic calculation of excitation energies. The second one includes spin-orbit coupling self-consistently in the ground state calculation. If spin-orbit coupling is large, the second one is more accurate, but is also more time-consuming.

The results of these spin-orbit coupled TDDFT calculations include the calculation of the zero field splitting (ZFS) of triplet excited states and the calculation of radiative rate constants, which could be used to calculate radiative phosphorescence lifetimes.

SOPERT {NCALC=ncalc} {ESHIFT=eshift}
RELATIVISTIC SCALAR ZORA
EXCITATIONS
END

SOPERT {NCALC=ncalc} {ESHIFT=eshift}
PRINT SOMATRIX

GSCORR

RELATIVISTIC SPINORBIT ZORA
EXCITATIONS
  {ALSORESTRICTED}
END

STCONTRIB

Relativistic spinorbit ZORA
Unrestricted
NonCollinear
Symmetry Nosym
TDA
Excitations
End

CV(n)-DFT: Constricted Variational DFT¶

In the constricted nth order variational density functional method (CV(n)-DFT) [393, 394, 413] the occupied excited state orbitals are allowed to relax in response to the change of both the Coulomb and exchange-correlation potential in going from the ground state to the excited state. This theory is not time-dependent nor is it based on response theory. It is instead variational in nature and has been termed constricted variational DFT or CV(n)-DFT.

Due to bugs in older versions it is important to use ADF2016.105 or later (ADF2016.105 might not be released yet). CV(n)-DFT icw GGA’s was not working. The CV_DFT and SCF_CV_DFT methods gave incorrect numbers. ALLPOINTS was required for calculations that use symmetry, now automatically set in case of CV(n)-DFT. EXACTDENSITY was required for precision reasons, now automatically set in case of CV(n)-DFT. Thus if one is using ADF2016.104 or older, this method will probably not work.

The CVNDFT block key regulates the execution of the CV(n)-DFT code, which calculates the singlet electronic excitations for the closed shell molecules. The triplet excitations are not yet included in the present ADF version. The parameter n in (n) describes the order of this theory. There are two limiting cases implemented in the CV(n)-DFT code: n=2 and n= \(\infty\). Since CV(n)-DFT coincides in the second order with TDDFT, CV(n)-DFT code is designed as an extension of the EXCITATIONS module. Therefore, the EXCITATIONS block has to be present in the input file together with CVNDFT. CVNDFT takes all the information about the number of excitations and their symmetry from the EXCITATIONS block. Moreover, the TDDFT transition density vectors are used as an initial guess for the CV(\(\infty\))-DFT calculations.

There are a few approximations within the CV(\(\infty\))-DFT approach, therefore there are a few corresponding sub-blocks in the CVNDFT block. In the simplest case the TDDFT transition density U-vector is substituted into the infinite order CV(\(\infty\))-DFT excitation energy. This approximation corresponds to the CV_DFT sub-block, so the input fragment would look like this:

CVNDFT
  CV_DFT &
  SUBEND
END

In general, the U-vector has to be optimized with respect to the infinite order CV(\(\infty\))-DFT excitation energy. It is accomplished iteratively in the SCF-CV(\(\infty\))-DFT method [412], which is invoked when the SCF_CV_DFT sub-block is present. Therefore, there are input parameters that control the SCF procedure, i.e. the total number of iterations and tolerance. The corresponding input fragment would look like this:

CVNDFT
  ITERATION iter
  TOLERANCE tol
  SCF_CV_DFT &
    DAMPING mix
  SUBEND
END

iter

iter is the maximum number of iterations

tol

tol is the convergence criterion, i.e. the SCF-CV(\(\infty\))-DFT procedure stops when the given accuracy is achieved

mix

mix is the relative weight of the new U-vector that is added to the one from the previous iteration (default value is mix=0.2).

Another feature of CV(\(\infty\))-DFT is a possibility to optimize (or relax) the molecular orbitals for the particular excitation. This is accomplished in the R-CV(\(\infty\))-DFT method, see for an aplication Ref.[411]. In the last case, the transition U-vector is frozen, while the orbitals are relaxed. For example, the input fragment which employs the U-vector from TDDFT would look like this:

CVNDFT
  ITERATION iter
  TOLERANCE tol
  R_CV_DFT &
    RELAXALPHA start_a_iter
    RELAXBETA  start_b_iter
    DAMPORBRELAX mix_relax
  SUBEND
END

start_a_iter

start_a_iter is the SCF cycle number at which the relaxation of \(\alpha\) orbitals starts

start_b_iter

start_b_iter is the SCF cycle number at which the relaxation of \(\beta\) orbitals starts

Since there is no optimization of the U-vector in R-CV(\(\infty\))-DFT, the relaxation of orbitals (either $alpha$ or $beta$ or both) should be turned on at the very first iteration.

mix_relax

The mix_relax parameter defines the relative weight of the new relaxation vector that is added to the one from the previous iteration (the default value is mix_relax=0.2).

Note that all tree methods can be used in the input file. The corresponding input fragment would look like this:

CVNDFT
  ITERATION iter
  TOLERANCE tol
  CV_DFT &
  SUBEND
  SCF_CV_DFT &
    DAMPING mix
  SUBEND
  R_CV_DFT &
    RELAXALPHA start_a_iter
    RELAXBETA  start_b_iter
    DAMPORBRELAX mix_relax
  SUBEND
END

In this case the U-vector will be optimized with respect to the infinite order CV(\(\infty\))-DFT excitation energy by the SCF-CV(\(\infty\))-DFT code and then supplied to the R-CV(\(\infty\))-DFT code.

Recommended is to use all an electron basis set. A reasonable choice is only to use the R-CV(\(\infty\))-DFT method for the calculation of excitation spectra containing charge transfer transitions. This will avoid the possibly expensive optimization of the U vectors. The input will then be something like:

Basis
  ...
  Core None
End
Excitations
  onlysing
End
CVNDFT
  R_CV_DFT &
  SUBEND
End

CVNDFT prints in the output the excitation energies as well as the maximum value of the lambda parameter \(\lambda\). This number corresponds to the largest singular value of the singular value decomposition of the U matrix/vector (i.e. the U-vector can be considered as a matrix with nvir rows and nocc columns, where nvir is the number of virtual orbitals and nocc is the number of occupied orbitals). If this number is close to \(\pi / 2\) (i.e. \(\lambda_{max} \approx 1.57\)), then the corresponding electronic excitation can be approximated by a single natural transition orbital (NTO) to another single NTO transition. This type of transitions are obtained in \(\Delta\) SCF. Therefore, this information can be used, for example, to compare the CV(\(\infty\))-DFT results to the \(\Delta\) SCF excitation energies.

TD-DFT+TB¶

The basis idea of TD-DFT+TB [424] is to use the molecular orbitals from a DFT ground state calculation as input to an excited state calculation with TD-DFTB coupling matrices. TB means tight binding. If many excitations are required, this method will speed up the calculation drastically in comparison to the standard time needed for TDDFT calculations of excitation energies. This method is best suited if a (meta-)GGA or LDA is used in the SCF.

SYMMETRY NOSYM
EXCITATIONS
 TDDFTB
END

One can use this, for example, in combination with the MODIFYEXCITATION key to use only single-orbital transitions that have a minimal oscillator strength:

SYMMETRY NOSYM
EXCITATIONS
 TDDFTB
 ONLYSING
END
MODIFYEXCITATION
  OscStrength 0.001
End

In TD-DFT+TB reducing the space of single-orbital transitions with the key MODIFYEXCITATION will reduce the memory needed in ADF. Starting from ADF2016.102 for TD-DFT+TB only the lowest (in energy) 100000 single orbital transitions are included by default. One can override this by using a different value for the subkey UseOccVirtNumbers of the key MODIFYEXCITATION.

MODIFYEXCITATION
 UseOccVirtNumbers 1 100000
END

The value for the subkeyword KFWRITE in the EXCITATIONS block key is set by default to 0 in case of TD-DFT+TB. If kfwrite is 0 then do not write contributions, transition densities, and restart vectors to TAPE21, since this can lead to a huge TAPE21, especially if many excitations are calculated.

sTDA, sTDDFT¶

The simplified Tamm-Dancoff approach (sTDA) [389] and simplified time-dependent DFT approach (sTDDFT) [390] by Grimme et al. are implemented in ADF. In these approaches the time-dependent DFT equations are simplified by the evaluation of the two-electron integrals as short-range damped Coulomb interactions between (transition) charge density monopoles and a truncation of the single excitation expansion space. These methods are best suited if a (meta-)hybrid or a range-separated-hybrid is used in the SCF. These methods will speed up the calculation drastically in comparison to the standard time needed for TDA or TDDFT calculations of excitation energies for hybrids.

For sTDA use:

SYMMETRY NOSYM
EXCITATIONS
 sTDA
END

For sTDDFT use:

SYMMETRY NOSYM
EXCITATIONS
 sTDDFT
END

For global hybrids ADF will use the parameters \(\alpha\) and \(\beta\) that depend on the amount of exact exchange \(a_x\) as:

where \(\alpha_1 = 1.42\), \(\alpha_2 = 0.48\), \(\beta_1 = 0.2\), \(\beta_2 = 1.83\) are used in ADF. These values are fitted parameters by Grimme [389]. For range-separated hybrids one should set the parameters \(\alpha\), \(\beta\) and \(a_x\) in the input of ADF with the keyword MODIFYEXCITATION. See Ref. [391] for parameters that are used by Risthaus et al. for a few range-separated functionals. However the \(\alpha\) and \(\beta\) parameters are mixed up in Ref. [391]. Thus use:

For example, for CAM-B3LYP use

MODIFYEXCITATION
  GrimmeAlpha 0.9
  GrimmeBeta 1.86
  GrimmeAex 0.38
END

If one uses the keyword MODIFYEXCITATION one may also set more parameters that are used in the sTDA and sTDDFT approach.

MODIFYEXCITATION
  GrimmeAlpha Alpha
  GrimmeBeta Beta
  GrimmeAex Aex
  GrimmeDEmax DEmax
  GrimmeTPmin TPmin
  GrimmePertC|NoGrimmePertC
END

GrimmeAlpha Alpha

To set the \(\alpha\) parameters, should be used icw GrimmeBeta and GrimmeAex.

GrimmeBeta Beta

To set the \(\beta\) parameters, should be used icw GrimmeAlpha and GrimmeAex.

GrimmeAex Aex

To set \(a_x\) that is used in the sTDA or sTDDFT part of the calculation, should be used icw GrimmeAlpha and GrimmeBeta.

GrimmeDEmax DEmax

Single orbital transitions that have an orbital energy difference less than DEmax (in Hartree) are included. Default value DEmax = 0.4 Hartree. See also the meaning of \(E_{max}\) in Grimme [389].

GrimmeTPmin TPmin

Single orbital transitions that would have a cumulative perturbative energy contribution larger than TPmin is included. Default value TPmin = \(10^{-4}\). See also the meaning of \(t_p\) in Grimme [389].

GrimmePertC|NoGrimmePertC

In case of GrimmePertC, which is the default, if the cumulative perturbative energy contribution for a single orbital transition is smaller TPmin, this contribution is used to modify the diagonal value of a matrix that is used in the sTDA or the sTDDFT method. If NoGrimmePertC is included such contributions are neglected. See Grimme [389] for more details on the truncation of the single excitation space.

Starting from ADF2016.102 for sTDA or sTDDFT only the lowest (in energy) 10000 single orbital transitions are included by default. One can override this by using a different value for the subkey UseOccVirtNumbers of the key MODIFYEXCITATION.

MODIFYEXCITATION
 UseOccVirtNumbers 1 10000
END

The value for the subkeyword KFWRITE in the EXCITATIONS block key is set by default to 0 in case of sTDA or sTDDFT. If kfwrite is 0 then do not write contributions, transition densities, and restart vectors to TAPE21, since this can lead to a huge TAPE21, especially if many excitations are calculated.

NTO: Natural Transition Orbitals¶

Natural Transition Orbitals [396] come from a transformation of the transition density matrix. They are the closest you can get to visualizing an excitation as a one-electron excitation from 1 orbital to another. They give insight in the localization of excitations. Calculation of Natural Transition Orbitals (NTO’s) will be performed if the subkey NTO is included in the key EXCITATIONS.

EXCITATIONS
  ...
  NTO
End

For (meta-)hybrids and range-separated functionals NTO’s will be calculated only if one uses the Tamm-Dancoff approximation (TDA).

CD spectra¶

Circular dichroism (CD) is the differential absorption of left- and right-handed circularly polarized light. Starting from ADF2010 Hartree-Fock and hybrids can also be used to calculate CD spectra.

EXCITATIONS
 CDSPECTRUM
 ANALYTIC
 VELOCITY
End

CDSPECTRUM

If the subkey CDSPECTRUM is included in the key EXCITATIONS the rotatory strengths for the calculated excitations are calculated, in order to simulate Circular Dichroism (CD) spectra [80, 81]. Interesting for chiral molecules. This subkey should not be used in case of spin-orbit coupling. For accuracy reasons you should also use the subkey ANALYTIC in the block key EXCITATIONS, otherwise the results may be nonsense.

ANALYTIC

If the subkey ANALYTIC is included the required integrals for the CD spectrum are calculated analytically, instead of numerically. Only used in case of CD spectrum.

Velocity

If the subkey VELOCITY is included ADF calculates the dipole-velocity representation of the oscillator strength. If applicable (use of subkey CDSPECTRUM) the dipole-velocity representation of the rotatory strength is calculated. Default the dipole-length representation of the oscillator strength and rotatory strength is calculated.

MCD¶

MCD or magnetic circular dichroism is the differential absorption of left and right circularly polarized light in the presence of a magnetic field. MCD intensity is usually described in terms of different contributions called A, B and C terms, see Refs. [273, 274]. A further parameter D is often discussed in MCD studies. D is proportional to the intensity of an absorption band and is closely related to the oscillator strength. A and B terms for closed and open-shell molecules and C terms of open-shell molecules induced by spin-orbit coupling can be calculated. Starting from ADF2010 C terms related to spatially degenerate states, i.e. breaking of degeneracies can be calculated.

For MCD calculations for molecules that have C(2) or D(2) symmetry use SYMMETRY NOSYM.

Input options

EXCITATIONS
   MCD {options}
   ONLYSINGLET
   {SELECT transition number}
   {DTENSOR {Dxx Dxy Dyy Dxz Dyz|D E/D}}
End

ALLPOINTS
{RELATIVISTIC ZORA}
{SOMCD}
{ZFS}

MCD

If the subkey MCD is included in the key EXCITATIONS the MCD parameters of some or all of the excitations considered in the TDDFT procedure are calculated [275-278]. This subkey should not be used with spin-orbit coupling (but, see below). Several other keywords could be important.

ALLPOINTS: required for an MCD calculation.

ONLYSINGLET: this keyword should be used in combination with a MCD calculation.

RELATIVISTIC ZORA: required for a calculation of temperature-dependent C terms. In this case the keyword SOMCD must also be added as a key by itself, and the calculation must be unrestricted. If only A and B terms are calculated then ZORA is not needed but can be included if desired.

ZFS: If the ZFS keyword and MCD with SOMCD are also included then the influence of the calculated zero-field splitting (ZFS) on the temperature-dependent MCD is evaluated. The MCD in the presence of ZFS is described as anisotropic in the output because the Zeeman splitting becomes orientation dependent in the presence of ZFS.

In ADF2010 the temperature-dependent MCD due to the breaking of degeneracies of excited states by spin-orbit coupling can be calculated. Although all temperature-dependent MCD is typically called “C terms”, the parameters associated with the MCD are labeled “CE” to distinguish them from the MCD due to mixing between states caused by spin-orbit coupling that is labeled “C”. The CE terms have a derivative shape like A terms. They have the same temperature-dependence as normal C terms. If they are present, CE terms are calculated automatically along with C terms if the keyword SOMCD is included in the input.

MCD {options}

Options include NMCDTERM, NMIX, DCUTOFF, MCDOUT, CGOUT, NANAL, NANAL2, FULLOMEGA, NOAB, NODIRECT, NOCG, CONVCG, ITERCG, ITER2CG, BMIN, BMAX, TMIN, TMAX and NTEMP.

NMCDTERM=nmcdterm

Number of excitations for which MCD parameters are to be calculated. The nmcdterm lowest energy excitations are treated. The default is the number of transitions considered in the TDDFT calculation.

NMIX=nmix

Number of transitions allowed to mix in a SOS calculation. Default is the number of transitions considered in the TDDFT calculation.

DCUTOFF=dcutoff

MCD parameters will only be calculated for transitions with sufficient intensity. Each Cartesian component of each transition is considered separately. If the dipole strength D of that component is below dcutoff then the MCD is not calculated. The default is 1.0e-6.

MCDOUT=mcdout

Number that determines the amount of output to be printed about the MCD calculation. Higher means more output. Possible values are 0, (orientationally averaged and Cartesian components of MCD parameters only) 1 (as for 0 but with the addition of a short analysis) or 2 (as for 1 but with the addition of a lengthy analysis). Theoretical analyses of MCD parameters are presented in several places including Refs. [273-274, 276-278]. The default for MCDOUT is 0.

CGOUT=cgout

The perturbed transition densities used to evaluate the B and C term parameters can be obtained through an iterative conjugate-gradient procedure. Convergence information of the conjugate-gradient algorithm is printed every cgout iterations. Default is 10.

NANAL=nanal, NANAL2=nanal2

If MCDOUT is set to 2, a detailed analysis of the B and/or C term parameters in terms of which states mix and how much MCD each mixing causes, is presented. The parameters NANAL and NANAL2 determine how many contributions are included in the analyses. Defaults are 10 for NANAL and 5 for NANAL2.

FULLOMEGA

A standard TDDFT calculation involves the solution of an eigenvalue equation to obtain the excitation energies and transition densities of interest. ADF can solve this eigenvalue equation two ways: through diagonalization of the full Omega matrix or through the Davidson procedure where Omega is never explicitly constructed. Construction of the complete Omega matrix is generally only feasible for smaller problems. The matrix Omega appears again in the equations solved to obtain MCD. Here again Omega can be built or only the products of Omega with a vector can be used as is the case in the Davidson procedure. The default is to not construct Omega. If the keyword FULLOMEGA is included then Omega is constructed. Note that the choice of FULLOMEGA is completely independent of whether EXACT or DAVIDSON is chosen in the earlier TDDFT calculation.

NOAB

If this keyword is included then A and B terms are not calculated. NOAB only makes sense if SOMCD is included in the input otherwise no MCD will be calculated at all.

NODIRECT

The perturbed transition density needed to evaluate B and C term parameters is obtained through the solution of a large system of equations. This system of equations is solved in two ways: through a sum-over-states (SOS) type approach where the solution is expanded in a known set of transition densities or through the direct solution of the system of equations by the conjugate gradient procedure. The SOS method is much faster but also less accurate, particularly for larger systems. By default MCD parameters are evaluated through both approaches. If the NODIRECT keyword is included then only the SOS calculation is performed.

NOCG

The conjugate gradient procedure is first used in combination with a preconditioner that generally speeds up convergence significantly. If no solution is found in a reasonable number of iterations then the procedure is restarted without the preconditioner. If the NOCG keyword in included then the preconditioner is never used.

CONVCG

Convergence criterion for the CG iterative methods. The default value of 0.01 is probably good enough for most applications. This choice seems to produce B and C terms that are converged to 3 significant figures. Except for small systems, it is not recommended that CONVCG be set to a much smaller number as this will probably cause a large number of convergence failures.

ITERCG=itercg

Number of iterations before failure in the first (preconditioned) CG solver. This solver either succeeds quickly or not at all so the default value is 30.

ITER2CG=iterc2g

Number of iterations before failure in the B of C term parameter calculation of the unconditioned CG solver. This solver is often slow so the default value is 200.

BMIN=bmin, BMAX=bmax, NBFIELD=nbfield, TMIN=tmin, TMAX=tmax, NTEMP=ntemp

Temperature dependent MCD intensity often varies nonlinearly with T and B when T is small and/or B is large. It may therefore be of interest to evaluate the MCD intensity over a range of temperatures and/or magnetic fields. This can be achieved through the use of the BMIN, BMAX, NBFIELD, TMIN, TMAX and NTEMP keywords. The MIN and MAX keywords give the maximum values of B or T. NBFIELD and NTEMp indicate how many values are to be considered. Note that magnetic fields are assumed to be given in Tesla and temperatures in Kelvin. For example, BMIN=1, BMAX=5, NBFIELD=5 means that fields of 1,2,3,4 and 5 T will be considered. Defaults are BMIN=BMAX=1, TMIN=TMAX=5 and NBFIELD=NTEMP=1.

SELECT nselect1 nselect2 nselect3...

Rather than selecting the first nmcdterm transitions for consideration individual transitions can be selected through the SELECT keyword. The transitions of interest are listed after the SELECT keyword. Note that the numbering follows that given in the summary table at the end of the TDDFT calculation. To consider a degenerate transition only the first component need be included. Note that it makes no sense to use both the SELECT and NMCDTERM keywords together.

DTENSOR Dxx Dxy Dyy Dxz Dyz DTENSOR D E/D

As noted earlier, if the ZFS keyword is included with MCD and SOMCD then the influence of zero-field splitting on temperature-dependent MCD will be evaluated. As an alternative to the ZFS keyword the D-tensor parameters can be entered directly through the DTENSOR keyword in the EXCITATIONS block. Two input formats are possible. Five real numbers Dxx Dxy Dyy Dxz Dyz can be entered. These five numbers are sufficient to define the traceless tensor D. Alternatively, the two parameters D and E/D can be entered. In this case the coordinate system chosen to define the molecular geometry must be the the principle axis system of the D-tensor. D, Dxx, Dxy, Dyy, Dxz and Dyz should be given in wavenumbers (cm-1).

Notes

If an MCD calculation is run, the transition densities obtained in the TDDFT calculation are saved to TAPE21. For large molecules this can result in a very large TAPE21 file.

An MCD calculation relies on the excitation energies and, in particular, the transition densities that result from the preceding TDDFT calculation. If the results of the TDDFT calculation are poor then it is likely that the results of the MCD calculation will be poor. It therefore should be kept in mind that most TDDFT calculations will make use of the Davidson method for finding the eigenvalues and eigenvectors of the TDDFT equation. The Davidson approach involves some approximations that can lead to some variation in results with the applied parameters. The most important example of this is the fact that the results vary depending on how many eigenvalue/eigenvector pairs are calculated, ie how many transitions are selected through the LOWEST keyword. The variation is small for the eigenvalues (excitation energies) but can be significant for the eigenvectors (transition densities). A variation in the transition densities leads to variation in the transition dipoles which can significantly impact calculated MCD parameters. The moral of this story is that when calculating MCD parameters it is best to choose one value of LOWEST and stick with it.

The most time-consuming part of an MCD calculation is the solution of the system of equations through the conjugate-gradient solver. The solver can fail so be aware of warnings concerning convergence in the output. A few hints to improve convergence are: a) choose a value of LOWEST that is at least double the number of transitions for which you desire MCD parameters. This helps to improve the SOS calculation which provides an initial guess for the conjugate gradient solver. The solver is sensitive to the initial guess so changing LOWEST by a small amount may help (or hinder) convergence significantly. Keep the previous note in mind when playing with LOWEST however. b) The preconditioned conjugate gradient solver is usually fast but does not converge monotonically to the correct answer. The unpreconditioned solver is much slower but tends to converge monotonically. If the preconditioned solver fails but leaves a fairly well converged result for the unpreconditioned solver the latter usually converges quickly. If the preconditioned solver does not leave a fairly well converged result it may be worth changing the number of iterations it uses since a few iterations earlier or later may provide a much better converged answer. c) The SELECT keyword can be used to work on the remaining transitions for which converged results have not been obtained.

All MCD parameters are presented in au. To convert A and C terms to the alternative unit D² (Debye squared) the value in au should be multiplied by 6.46044. To convert the B term to the alternative unit of D² /cm-1 the value in au should be multiplied by 2.94359e-05.

The A, B and C terms are defined through the equation suggested by Stephens (equation 1 in [278] and also see [273-274, 107]). This equation assumes that MCD intensity varies linearly with applied magnetic field and that the temperature-dependent component varies linearly with temperature as 1/T. For the most part, these assumptions are reasonable. An exception is that the temperature-dependent part varies from linearity when T is very small. To allow for this situation a temperature and magnetic field dependent multiplicative constant (chi(B,T)) is evaluated whenever temperature-dependent MCD parameters are considered. This constant includes all magnetic field and temperature dependence of the temperature-dependent MCD. Thus chi(B,T)*C can be used in place of B*C/kT in equation 1 of [278] when MCD spectra are to be simulated. Note that, since the g-factor for all states is here approximated by 2.0, chi applies to all transitions.

Applications of the Excitation feature in ADF¶

It may be useful to consult the following (early) applications of the Excitation feature in ADF:

• For excitation energies based on exact XC potentials: [82]
• Calculations on Free Base Porphin: [83]; calculations on metal-porphyrins: a series of papers by Rosa, Ricciardi, Baerends, e.g. [84, 85].
• Calculations on MnO₄ -, Ni(CO)₄ and Mn₂ (CO)₁₀ : [86]
• Calculations on M(CO)₅ (M=Cr, Mo, W), using the scalar ZORA relativistic approach: [87]
• Excitation energies of open-shell molecules: [154, 157]
• Calculations on [PtCl₄ ]^2- , [PtBr₄ ]^2- , and [Pt(CN)₄ ]^2- , using the ZORA relativistic approach including spin-orbit coupling: [183]
• For details regarding the (near linear scaling and parallelized) implementation, please check Refs.[71, 88]