CIS, TDHF, TDDFT¶

Overview¶

NWChem supports a spectrum of single excitation theories for vertical excitation energy calculations, namely, configuration interaction singles (CIS)1, time-dependent Hartree-Fock (TDHF or also known as random-phase approximation RPA), time-dependent density functional theory (TDDFT)2, and Tamm-Dancoff approximation3 to TDDFT. These methods are implemented in a single framework that invokes Davidson’s trial vector algorithm (or its modification for a non-Hermitian eigenvalue problem). The capabilities of the module are summarized as follows:

• Vertical excitation energies (valence and core),
• Spin-restricted singlet and triplet excited states for closed-shell systems,
• Spin-unrestricted doublet, etc., excited states for open-shell systems,
• Tamm-Dancoff and full time-dependent linear response theories,
• Davidson’s trial vector algorithm,
• Symmetry (irreducible representation) characterization and specification,
• Spin multiplicity characterization and specification,
• Transition moments and oscillator strengths,
• Analytical first derivatives of vertical excitation energies with a selected set of exchange-correlation functionals (see TDDFT gradients documentation for further information),
• Numerical second derivatives of vertical excitation energies,
• Disk-based and fully incore algorithms,
• Multiple and single trial-vector processing algorithms,
• Frozen core and virtual approximation,
• Asymptotically correct exchange-correlation potential by van Leeuwen and Baerends4,
• Asymptotic correction by Casida and Salahub5,
• Asymptotic correction by Hirata, Zhan, Aprà, Windus, and Dixon6.

These are very effective way to rectify the shortcomings of TDDFT when applied to Rydberg excited states (see below).

Performance of CIS, TDHF, and TDDFT methods¶

The accuracy of CIS and TDHF for excitation energies of closed-shell systems are comparable to each other, and are normally considered a zeroth-order description of the excitation process. These methods are particularly well balanced in describing Rydberg excited states, in contrast to TDDFT. However, for open-shell systems, the errors in the CIS and TDHF excitation energies are often excessive, primarily due to the multi-determinantal character of the ground and excited state wave functions of open-shell systems in a HF reference. The scaling of the computational cost of a CIS or TDHF calculation per state with respect to the system size is the same as that for a HF calculation for the ground state, since the critical step of the both methods are the Fock build, namely, the contraction of two-electron integrals with density matrices. It is usually necessary to include two sets of diffuse exponents in the basis set to properly account for the diffuse Rydberg excited states of neutral species.

The accuracy of TDDFT may vary depending on the exchange-correlation functional. In general, the exchange-correlation functionals that are widely used today and are implemented in NWChem work well for low-lying valence excited states. However, for high-lying diffuse excited states and Rydberg excited states in particular, TDDFT employing these conventional functionals breaks down and the excitation energies are substantially underestimated. This is because of the fact that the exchange-correlation potentials generated from these functionals decay too rapidly (exponentially) as opposed to the slow -1/r asymptotic decay of the true potential. A rough but useful index is the negative of the highest occupied KS orbital energy; when the calculated excitation energies become close to this threshold, these numbers are most likely underestimated relative to experimental results. It appears that TDDFT provides a better-balanced description of radical excited states. This may be traced to the fact that, in DFT, the ground state wave function is represented well as a single KS determinant, with less multi-determinantal character and less spin contamination, and hence the excitation thereof is described well as a simple one electron transition. The computational cost per state of TDDFT calculations scales as the same as the ground state DFT calculations, although the prefactor of the scaling may be much greater in the former.

A very simple and effecive way to rectify the TDDFT’s failure for Rydberg excited states has been proposed by Tozer and Handy7 and by Casida and Salahub5. They proposed to splice a -1/r asymptotic tail to an exchange-correlation potential that does not have the correct asymptotic behavior. Because the approximate exchange-correlation potentials are too shallow everywhere, a negative constant must be added to them before they can be spliced to the -1/r tail seamlessly in a region that is not sensitive to chemical effects or to the long-range behavior. The negative constant or the shift is usually taken to be the difference of the HOMO energy from the true ionization potential, which can be obtained either from experiment or from a ΔSCF calculation. Recently, we proposed a new, expedient, and self-contained asymptotic correction that does not require an ionization potential (or shift) as an external parameter from a separate calculation. In this scheme, the shift is computed by a semi-empirical formula proposed by Zhan, Nichols, and Dixon6. Both Casida-Salahub scheme and this new asymptotic correction scheme give considerably improved (Koopmans type) ionization potentials and Rydberg excitation energies. The latter, however, supply the shift by itself unlike to former.

Input syntax¶

The module is called TDDFT as time-dependent density functional theory employing a hybrid HF-DFT functional encompasses all of the above-mentioned methods implemented. To use this module, one needs to specify TDDFT on the task directive, e.g.,

     TASK TDDFT ENERGY


for a single-point excitation energy calculation, and

     TASK TDDFT OPTIMIZE


for an excited-state geometry optimization (and perhaps an adiabatic excitation energy calculation), and

     TASK TDDFT FREQUENCIES


for an excited-state vibrational frequency calculation. The TDDFT module first invokes DFT module for a ground-state calculation (regardless of whether the calculations uses a HF reference as in CIS or TDHF or a DFT functional), and hence there is no need to perform a separate ground-state DFT calculation prior to calling a TDDFT task. When no second argument of the task directive is given, a single-point excitation energy calculation will be assumed. For geometry optimizations, it is usually necessary to specify the target excited state and its irreducible representation it belongs to. See the subsections TARGET and TARGETSYM for more detail.

Individual parameters and keywords may be supplied in the TDDFT input block. The syntax is:

 TDDFT
[(CIS||RPA) default RPA]
[NROOTS <integer nroots default 1>]
[MAXVECS <integer maxvecs default 1000>]
[(SINGLET||NOSINGLET) default SINGLET]
[(TRIPLET||NOTRIPLET) default TRIPLET]
[THRESH <double thresh default 1e-4>]
[MAXITER <integer maxiter default 100>]
[TARGET <integer target default 1>]
[TARGETSYM <character targetsym default 'none'>]
[SYMMETRY]
[ECUT] <-cutoff energy>
[EWIN] <-lower cutoff energy>  <-higher cutoff energy>
[ALPHA] <integer lower orbital>  <integer upper orbital>
[BETA] <integer lower orbital>  <integer upper orbital>
[CIVECS]
[CDSPECTRUM]
[GIAO]
[VELOCITY]
[SIMPLESO]
[ALGORITHM <integer algorithm default 0>]
[FREEZE [[core] (atomic || <integer nfzc default 0>)] \
[virtual <integer nfzv default 0>]]
[PRINT (none||low||medium||high||debug)
<string list_of_names ...>]
END


The user can also specify the reference wave function in the DFT input block (even when CIS and TDHF calculations are requested). See the section of Sample input and output for more details.

Since each keyword has a default value, a minimal input file will be

 GEOMETRY
Be 0.0 0.0 0.0
END
BASIS
Be library 6-31G**
END


Note that the keyword for the asymptotic correction must be given in the DFT input block, since all the effects of the correction (and also changes in the computer program) occur in the SCF calculation stage. See DFT (keywords CS00 and LB94) for details.

Keywords of TDDFT input block¶

CIS and RPA: the Tamm-Dancoff approximation¶

These keywords toggle the Tamm-Dancoff approximation. CIS means that the Tamm-Dancoff approximation is used and the CIS or Tamm-Dancoff TDDFT calculation is requested. RPA, which is the default, requests TDHF (RPA) or TDDFT calculation.

The performance of CIS (Tamm-Dancoff TDDFT) and RPA (TDDFT) are comparable in accuracy. However, the computational cost is slightly greater in the latter due to the fact that the latter involves a non-Hermitian eigenvalue problem and requires left and right eigenvectors while the former needs just one set of eigenvectors of a Hermitian eigenvalue problem. The latter has much greater chance of aborting the calculation due to triplet near instability or other instability problems.

NROOTS: the number of excited states¶

One can specify the number of excited state roots to be determined. The default value for NROOTS is 1. It is advised that the users request several more roots than actually needed, since owing to the nature of the trial vector algorithm, some low-lying roots can be missed when they do not have sufficient overlap with the initial guess vectors.

MAXVECS: the subspace size¶

The MAXVECS keyword limits the subspace size of Davidson’s algorithm; in other words, it is the maximum number of trial vectors that the calculation is allowed to hold. Typically, 10 to 20 trial vectors are needed for each excited state root to be converged. However, it need not exceed the product of the number of occupied orbitals and the number of virtual orbitals. The default value is 1000.

SINGLET and NOSINGLET: singlet excited states¶

SINGLET || NOSINGLET requests (suppresses) the calculation of singlet excited states when the reference wave function is closed shell. The default is SINGLET.

TRIPLET and NOTRIPLET: triplet excited states¶

TRIPLET || NOTRIPLET requests (suppresses) the calculation of triplet excited states when the reference wave function is closed shell. The default is TRIPLET.

THRESH: the convergence threshold of Davidson iteration¶

The THRESH keyword specifies the convergence threshold of Davidson’s iterative algorithm to solve a matrix eigenvalue problem. The threshold refers to the norm of residual, namely, the difference between the left-hand side and right-hand side of the matrix eigenvalue equation with the current solution vector. With the default value of 1e-4, the excitation energies are usually converged to 1e-5 hartree.

MAXITER: the maximum number of Davidson iteration¶

It typically takes 10-30 iterations for the Davidson algorithm to get converged results. The default value for MAXITER is 100.

TARGET and TARGETSYM: the target root and its symmetry¶

At the moment, excited-state first geometry derivatives can be calculated analytically for a set of functionals, while excited-state second geometry derivatives are obtained by numerical differentiation. These keywords may be used to specify which excited state root is being used for the geometrical derivative calculation. For instance, when TARGET 3 and TARGETSYM a1g are included in the input block, the total energy (ground state energy plus excitation energy) of the third lowest excited state root (excluding the ground state) transforming as the irreducible representation a1g will be passed to the module which performs the derivative calculations. The default values for TARGET and TARGETSYM are 1 and none, respectively.

The keyword TARGETSYM is essential in excited state geometry optimization, since it is very common that the order of excited states changes due to the geometry changes in the course of optimization. Without specifying the TARGETSYM, the optimizer could (and would likely) be optimizing the geometry of an excited state that is different from the one the user had intended to optimize at the starting geometry. On the other hand, in the frequency calculations, TARGETSYM must be none, since the finite displacements given in the course of frequency calculations will lift the spatial symmetry of the equilibrium geometry. When these finite displacements can alter the order of excited states including the target state, the frequency calculation is not be feasible.

SYMMETRY: restricting the excited state symmetry¶

By adding the SYMMETRY keyword to the input block, the user can request the module to generate the initial guess vectors transforming as the same irreducible representation as TARGETSYM. This causes the final excited state roots be (exclusively) dominated by those with the specified irreducible representation. This may be useful, when the user is interested in just the optically allowed transitions, or in the geometry optimization of an excited state root with a particular irreducible representation. By default, this option is not set. TARGETSYM must be specified when SYMMETRY is invoked.

ECUT: energy cutoff¶

The ECUT keyword enables restricted excitation window TDDFT (REW-TDDFT)8. This is an approach best suited for core excitations. By specifying this keyword only excitations from occupied states below the energy cutoff will be considered.

EWIN: energy window¶

The EWIN keyword enables a restricted energy window between a lower energy cutoff and a higher energy cutoff. For example, ewin -20.0 -10.0 will only consider excitations from occupied orbitals within the specified energy window

Alpha, Beta: alpha, beta orbital windows¶

Orbital windows can be specified using the following keywords:

  alpha 1 4
beta 2 5


Here alpha excitations will be considered from orbitals 1 through 4 depending on the number of roots requested and beta excitations will be considered from orbitals 2 through 5 depending on the number of roots requested.

CIVECS: CI vectors¶

The CIVECS keyword will result in the CI vectors being written out. By default this is off. Please note this can be a very large file, so avoid turning on this keyword if you are calculating a very large number of roots. CI vectors are needed for excited-state gradient and transition density calculations.

Analytical TDDFT gradients can be calculated by specifying a grad block within the main TDDFT block

For example, the following will perform a TDDFT optimization on the first singlet excited state (S1). Note that the civecs keyword must be specified. To perform a single TDDFT gradient, replace the optimize keyword with gradient in the task line. A complete TDDFT optimization input example is given the Sample Inputs section. A TDDFT gradients calculation can be used to calculate the density of a specific excited state. The excited stated density is written to a file with the .dmat suffix.

tddft
nroots 2
algorithm 1
notriplet
target 1
targetsym a
civecs
root 1
end
end


At the moment the following exchange-correlation functionals are supported with TDDFT gradients

LDA, BP86, PBE, BLYP, B3LYP, PBE0, BHLYP, CAM-B3LYP, LC-PBE, LC-PBE0, BNL, LC-wPBE, LC-wPBEh, LC-BLYP


CDSpectrum: optical rotation calculations¶

Perform optical rotation calculations. We recommend to use the GIAO keyword

VELOCITY: velocity gauge¶

Perform CD spectrum calculations with the velocity gauge.

SIMPLESO: simplified Spin-Orbit coupling¶

Perform excited states calculations with a simplied Spin-Orbit coupling that uses eigenvalues from a spin-orbit calculation, instead of a standard dft calculation.
Here is a snippet of an input example (please notice the use of molecular orbitals).

start au2
geometry
au 0 0 1
au 0 0 -1
symmetry d2h
end
#basis sets, ecp and so-ecp skipped  for simplicity
...
dft
odft
vectors output au2_noso.mos
end
dft
vectors input au2_noso.mos output au2_so.mos
end
dft
odft
vectors input u2_noso.mos
end

tddft
simpleso au2.evals
nroots 1
notriplet
end



ALGORITHM: algorithms for tensor contractions¶

There are four distinct algorithms to choose from, and the default value of 0 (optimal) means that the program makes an optimal choice from the four algorithms on the basis of available memory. In the order of decreasing memory requirement, the four algorithms are:

• ALGORITHM 1 : Incore, multiple tensor contraction,
• ALGORITHM 2 : Incore, single tensor contraction,
• ALGORITHM 3 : Disk-based, multiple tensor contraction,
• ALGORITHM 4 : Disk-based, single tensor contraction.

The incore algorithm stores all the trial and product vectors in memory across different nodes with the GA, and often decreases the MAXITER value to accommodate them. The disk-based algorithm stores the vectors on disks across different nodes with the DRA, and retrieves each vector one at a time when it is needed. The multiple and single tensor contraction refers to whether just one or more than one trial vectors are contracted with integrals. The multiple tensor contraction algorithm is particularly effective (in terms of speed) for CIS and TDHF, since the number of the direct evaluations of two-electron integrals is diminished substantially.

FREEZE: the frozen core/virtual approximation¶

Some of the lowest-lying core orbitals and/or some of the highest-lying virtual orbitals may be excluded in the CIS, TDHF, and TDDFT calculations by the FREEZE keyword (this does not affect the ground state HF or DFT calculation). No orbitals are frozen by default. To exclude the atom-like core regions altogether, one may request

 FREEZE atomic


To specify the number of lowest-lying occupied orbitals be excluded, one may use

 FREEZE 10


which causes 10 lowest-lying occupied orbitals excluded. This is equivalent to writing

 FREEZE core 10


To freeze the highest virtual orbitals, use the virtual keyword. For instance, to freeze the top 5 virtuals

 FREEZE virtual 5


TRIALS: restart¶

Setting the keyword trials restart the calculation from the trials vector of a previous run.

trials


The PRINT keyword changes the level of output verbosity. One may also request some particular items in the table below.

Item Print Level Description
“timings” high CPU and wall times spent in each step
“trial vectors” high Trial CI vectors
“initial guess” debug Initial guess CI vectors
“general information” default General information
“xc information” default HF/DFT information
“memory information” default Memory information
“convergence” debug Convergence
“subspace” debug Subspace representation of CI matrices
“transform” debug MO to AO and AO to MO transformation of CI vectors
“diagonalization” debug Diagonalization of CI matrices
“iteration” default Davidson iteration update
“contract” debug Integral transition density contraction
“ground state” default Final result for ground state
“excited state” low Final result for target excited state

Printable items in the TDDFT modules and their default print levels.

Sample input¶

The following is a sample input for a spin-restricted TDDFT calculation of singlet excitation energies for the water molecule at the B3LYP/6-31G*.

START h2o
TITLE "B3LYP/6-31G* H2O"
GEOMETRY
O     0.00000000     0.00000000     0.12982363
H     0.75933475     0.00000000    -0.46621158
H    -0.75933475     0.00000000    -0.46621158
END
BASIS
* library 6-31G*
END
DFT
XC B3LYP
END
TDDFT
RPA
NROOTS 20
END


To perform a spin-unrestricted TDHF/aug-cc-pVDZ calculation for the CO+ radical,

START co
title "TDHF/aug-cc-pVDZ CO+"
charge 1
geometry
c  0.0  0.0  0.0
o  0.0  0.0  1.5
symmetry c2v # enforcing abelian symmetry
end
basis
* library aug-cc-pvdz
end
dft
xc hfexch
mult 2
end
tddft
rpa
nroots 5
end


A geometry optimization followed by a frequency calculation for an excited state is carried out for BF at the CIS/6-31G* level in the following sample input.

start bf
title "CIS/6-31G* BF optimization frequencies"
geometry
b 0.0 0.0 0.0
f 0.0 0.0 1.2
symmetry c2v # enforcing abelian symmetry
end
basis
* library 6-31g*
end
dft
xc hfexch
end
tddft
cis
nroots 3
notriplet
target 1
civecs
root 1
end
end


TDDFT with an asymptotically corrected SVWN exchange-correlation potential. Casida-Salahub scheme has been used with the shift value of 0.1837 a.u. supplied as an input parameter.

START tddft_ac_co
GEOMETRY
O 0.0 0.0  0.0000
C 0.0 0.0  1.1283
symmetry c2v # enforcing abelian symmetry
END
BASIS SPHERICAL
C library aug-cc-pVDZ
O library aug-cc-pVDZ
END
DFT
XC Slater VWN_5
CS00 0.1837
END
TDDFT
NROOTS 12
END


TDDFT with an asymptotically corrected B3LYP exchange-correlation potential. Hirata-Zhan-Apra-Windus-Dixon scheme has been used (this is only meaningful with B3LYP functional).

START tddft_ac_co
GEOMETRY
O 0.0 0.0  0.0000
C 0.0 0.0  1.1283
symmetry c2v # enforcing abelian symmetry
END
BASIS SPHERICAL
C library aug-cc-pVDZ
O library aug-cc-pVDZ
END
DFT
XC B3LYP
CS00
END
TDDFT
NROOTS 12
END


TDDFT for core states. The following example illustrates the usage of an energy cutoff and energy and orbital windows.8

echo
start h2o_core
memory 1000 mb
geometry units au noautosym noautoz
O 0.00000000     0.00000000     0.22170860
H 0.00000000     1.43758081    -0.88575430
H 0.00000000    -1.43758081    -0.88575430
end
basis
O library 6-31g*
H library 6-31g*
end
dft
xc beckehandh
print "final vector analysis"
end
tddft
ecut -10
nroots 5
notriplet
thresh 1d-03
end
tddft
ewin -20.0 -10.0
cis
nroots 5
notriplet
thresh 1d-03
end
dft
odft
mult 1
xc beckehandh
print "final vector analysis"
end
tddft
alpha 1 1
beta 1 1
cis
nroots 10
notriplet
thresh 1d-03
end


TDDFT optimization with LDA of Pyridine with the 6-31G basis9

echo
title "TDDFT/LDA geometry optimization of Pyridine with 6-31G"
geometry nocenter
N     0.00000000    0.00000000    1.41599295
C     0.00000000   -1.15372936    0.72067272
C     0.00000000    1.15372936    0.72067272
C     0.00000000   -1.20168790   -0.67391011
C     0.00000000    1.20168790   -0.67391011
C     0.00000000    0.00000000   -1.38406147
H     0.00000000   -2.07614628    1.31521089
H     0.00000000    2.07614628    1.31521089
H     0.00000000    2.16719803   -1.19243296
H     0.00000000   -2.16719803   -1.19243296
H     0.00000000    0.00000000   -2.48042299
symmetry c1
end
basis spherical
* library "6-31G"
end
driver
clear
maxiter 100
end
dft
iterations 500
grid xfine
end
tddft
nroots 2
algorithm 1
notriplet
target 1
targetsym a
civecs
root 1
end
end


TDDFT calculation followed by a calculation of the transition density for a specific excited state using the DPLOT block

echo
start h2o-td
title h2o-td

charge 0
geometry units au
symmetry group c1
O    0.00000000000000      0.00000000000000      0.00000000000000
H    0.47043554760291      1.35028113274600      1.06035416576826
H   -1.74335410533480     -0.23369304784300      0.27360785442967
end
basis "ao basis"
* library "Ahlrichs pVDZ"
end
dft
xc bhlyp
grid fine
direct
convergence energy 1d-5
end
tddft
rpa
nroots 5
thresh 1d-5
singlet
notriplet
civecs
end
dplot
civecs h2o-td.civecs_singlet
root 2
LimitXYZ
-3.74335 2.47044 50
-2.23369 3.35028 50
-2 3.06035 50
gaussian
output root-2.cube
end


TDDFT protocol for calculating the valence-to-core (1s) X-ray emission spectrum 10

1. Calculate the neutral ground state.
2. Calculate a full core hole (FCH) ionized state self-consistently, where the
1s core orbital of the absorbing center is swapped with a virtual orbital. Use the
maximum overlap constraint to prevent core hole collapse during the FCH calculation.
3. Perform a LR-TDDFT calculation within the TDA is performed with the FCH ionized
state as reference.
4. Final spectra is produced by taking the absolute value of the negative eigenvalues.

Spectrum parser¶

A Python script is available for parsing NWChem output for TDDFT/vspec excitation energies, and optionally Lorentzian broadenening the spectra . The nw_spectrum.py file can be found at https://raw.githubusercontent.com/nwchemgit/nwchem/master/contrib/parsers/nw_spectrum.py

Usage: nw_spectrum.py [options]

Reads NWChem output from stdin, parses for the linear response TDDFT or DFT
vspec excitations, and prints the absorption spectrum to stdout.  It will
optionally broaden peaks using a Lorentzian with FWHM of at least two
energy/wavelength spacings.  By default, it will automatically determine data
format (tddft or vspec) and generate a broadened spectrum in eV.

Example:

nw_spectrum -b0.3 -p5000 -wnm < water.nwo > spectrum.dat

Create absorption spectrum in nm named "spectrum.dat" from the NWChem output
file "water.nwo" named spectrum.dat with peaks broadened by 0.3 eV and 5000
points in the spectrum.

Options:
-h, --help            show this help message and exit
-f FMT, --format=FMT  data file format: auto (default), tddft, vspec, dos
-b WID, --broad=WID   broaden peaks (FWHM) by WID eV (default 0.1 eV)
-n NUM, --nbin=NUM    number of eigenvalue bins for DOS calc (default 20)
-p NUM, --points=NUM  create a spectrum with NUM points (default 2000)
-w UNT, --units=UNT   units for frequency:  eV (default), au, nm
-d STR, --delim=STR   use STR as output separator (four spaces default)
-x, --extract         extract unbroadened roots; do not make spectrum
-c CHA, --comment=CHA
comment character for output ('#' default)
-v, --verbose         echo warnings and progress to stderr


References¶

1. J. B. Foreman, M. Head-Gordon, J. A. Pople, and M. J. Frisch, J. Phys. Chem. 96, 135 (1992), DOI:10.1021/j100180a030

2. C. Jamorski, M. E. Casida, and D. R. Salahub, J. Chem. Phys. 104, 5134 (1996), DOI:10.1063/1.471140; R. Bauernschmitt and R. Ahlrichs, Chem. Phys. Lett. 256, 454 (1996), DOI:10.1016/0009-2614(96)00440-X; R. Bauernschmitt, M. Häser, O. Treutler, and R. Ahlrichs, Chem. Phys. Lett. 264, 573 (1997), DOI:10.1016/S0009-2614(96)01343-7

3. S. Hirata and M. Head-Gordon, Chem. Phys. Lett. 314, 291 (1999). DOI:10.1016/S0009-2614(99)01149-5

4. R. van Leeuwen and E. J. Baerends, Phys. Rev. A 49, 2421 (1994), DOI:10.1103/PhysRevA.49.2421

5. M. E. Casida, C. Jamorski, K. C. Casida, and D. R. Salahub, J. Chem. Phys. 108, 4439 (1998), DOI:10.1063/1.475855

6. S. Hirata, C.-G. Zhan, E. Aprà, T. L. Windus, and D. A. Dixon, J. Phys. Chem. A 107, 10154 (2003). DOI:10.1021/jp035667x

7. D. J. Tozer and N. C. Handy, J. Chem. Phys. 109, 10180 (1998), DOI:10.1063/1.477711

8. K. Lopata, B. E. Van Kuiken, M. Khalil, N. Govind, “Linear-Response and Real-Time Time-Dependent Density Functional Theory Studies of Core-Level Near-Edge X-Ray Absorption”, J. Chem. Theory Comput., 2012, 8 (9), pp 3284–3292, DOI:10.1021/ct3005613

9. D. W. Silverstein, N. Govind, H. J. J. van Dam, L. Jensen, “Simulating One-Photon Absorption and Resonance Raman Scattering Spectra Using Analytical Excited State Energy Gradients within Time-Dependent Density Functional Theory” J. Chem. Theory Comput., 2013, 9 (12), pp 5490–5503, DOI:10.1021/ct4007772

10. Y. Zhang, S. Mukamel, M. Khalil, N. Govind, “Simulating Valence-to-Core X-ray Emission Spectroscopy of Transition Metal”, J. Chem. Theory Comput., 2015, 11 (12), pp 5804–5809, DOI:10.1021/acs.jctc.5b00763