ROKS |  $\omega$B97X-V | 687.1 | 288.5 |  | $-0.3$ | $0.5$
Experiment |  | 687.4 | 288.0 |  | — | —

  * •

aData are from Ref. 408, aug-cc-pCVTZ basis set. bUsing approximate spin
projection (ASP), Eq. (66). cWith respect to experiment.

For second-row atoms, errors in both core-level IEs343, 416, 418 and also
core-level excitation energies343, 408, 348 are comparatively small when using
the $\Delta$SCF approach, although even for these elements Hartree-Fock theory
is competitive with DFT,418 suggesting that orbital relaxation is much more
important than correlation. These rather small errors should be contrasted
with much larger ones encountered when LR-TDDFT is applied to the same states
using the frozen-valence approximation. Table 2 shows results for two
different molecules (HF and CH4) using two different functionals (SCAN and
$\omega$B97X-V) that both perform well in $\Delta$SCF benchmarks. In contrast
to the sub-eV errors obtained using the $\Delta$SCF approach, LR-TDDFT
calculations exhibit errors in excess of 10 eV for the carbon K-edge
transition and $\sim 20$ eV for the fluorine K-edge transition. For K-edge
transitions of Mn(II) at $\hbar\omega\sim 6540$ eV, LR-TDDFT errors of
$\approx 32$ eV are obtained using B3LYP, and errors using the GGA functional
BP86 are $\approx 62$ eV.419 Errors are even larger for heavier elements.420
Notably, the sign of the errors in Table 2 points to underestimation of the
excitation energy, consistent with too-soft asymptotic decay of the potential
for a transition with CT character from a very compact 1s orbital to a
radially-diffuse LUMO. LRC functionals perform much better in this
capacity.421, 422

That said, the precision of core excitation energies computed using LR-TDDFT
is rather good even if the accuracy is not,422 meaning that chemical shifts
can be obtained even if absolute excitation energies must be shifted to match
experiment. It is the magnitude of the required shifts that is somewhat
unnerving. This inspired work on RSH functionals that partition $r_{12}^{-1}$
into short-, middle-, and long-range components,211, 212, 213, 214 with the
intention to use a larger fraction of exact exchange
($\alpha_{\text{hfx}}\approx 0.87$) at a length scale $\mu^{-1}\approx 0.24$
Å.423 These “short-range corrected” (SRC) functionals work rather well for LR-
TDDFT x-ray calculations,213 although they are empirically parameterized
specifically for that purpose and may not be good functionals for other
applications such as ground-state thermochemistry or valence excitation
energies. Furthermore, the SRC functionals are not LRC functionals in the
sense of Eq. (48), because they do not go to a limit of 100% HFX as
$r_{12}\rightarrow\infty$. Proper LRC functionals have been developed to
describe core-level excitations.211

Figure 14:  (a) Pictorial view of TP-type methods based on Kohn-Sham
eigenvalues, classified into “explicit” methods that require a separate SCF
calculation for each excited state, versus “implicit” methods that do not. (b)
Percent error for each method (versus experiment), applying the PBE functional
to a data set of K-edge excitation energies. Box plots extend from the 25th to
the 75th error percentile with the median value indicated, while the whiskers
show the largest outliers. Adapted from Ref. 377; copyright 2019 American
Institute of Physics.

As illustrated by the porphyrin spectrum in Fig. 13 and other examples,338
accurate electronic absorption spectra can be computed via $\Delta$SCF
calculations that are carried out in a state-by-state manner. However, this
does present something of a nuisance as compared to automatic generation of
numerous excitation energies in a LR-TDDFT calculation. Some of the TP or
occupancy-constrained $\Delta$SCF approaches that were described in Section
3.1.3 bypass this annoyance by using orbital eigenvalues from one or two
fractional-occupancy SCF calculations to obtain the entire spectrum of
excitation energies. These methods are assessed (versus experiment) in Fig.
14, for a data set of K-edge transitions.377 Following the notation of Ref.
377, these methods are characterized as either “explicit” or “implicit”, with
explicit methods involving occupancy constraints that are applied state-by-
state, as suggested by the original Slater method, whereas the implicit
methods place no electron at all into the virtual space, which makes for a
very simple computational scheme. The “neutral implicit” methods follow the
paradigm of the XCH approach,388 placing an electron (or a fraction of an
electron) into the LUMO only; it is often possible to optimize a core-to-LUMO
SCF solution without specialized algorithms. Finally, the “ground state”
results in Fig. 14 represent a control experiment in which the ground-state
eigenvalue difference ${\varepsilon}_{\text{LUMO}}-\varepsilon_{\text{1s}}$ is
used to approximate the K-edge transition energy. That method fares
surprisingly well, or perhaps the other approaches should be said to fare
surprisingly poorly. If the user desires to avoid state-by-state optimization
of an occupancy-constrained determinant, then the XCH approach would seem to
be the best option.

## 4 Time-Dependent Kohn-Sham Theory: “Real-Time” TDDFT

The cost of the $\Delta$SCF approaches described in Section 3 is typically no
more than a few times the cost of ground-state DFT, depending somewhat on the
algorithm that is used to converge the non-aufbau SCF solution. With the
exception of the $\sigma$-SCF method, the algorithms described in that section
each possess the same $\mathcal{O}(n_{\text{basis}}^{3})$ scaling as the
ground-state calculation. However, these methods must be applied in a state-
by-state manner, constructing a different initial guess for each state of
interest, which limits their applicability to problems where only a small
number of states is desired or required. In contrast, using LR-TDDFT it is
possible to obtain a large number of states in an automated fashion, at least
for medium-sized molecules. For large molecules, the cost of computing a very
large number of excited states can become prohibitively expensive, especially
in systems where the density of states is large. The appearance of spurious CT
states in the spectrum exacerbates the cost, even if their affect on the
overall spectral envelope is nil.286, 424 In situations such as these, where
proliferation of states (whether real or spurious) makes the iterative
diagonalization cost-prohibitive, the TDKS approach may be advantageous. Using
this method, a broadband spectrum can be computed from the oscillating dipole
moment function obtained from time-dependent electron dynamics.

### 4.1 Theory

The TDKS approach is also known as “real-time” (RT-)TDDFT,51, 52 to
distinguish it from LR-TDDFT. Starting from the ground-state Kohn-Sham
determinant, an external electric field $\bm{\mathcal{E}}(\mathbf{r},t)$ is
turned on at $t=0$, either as an impulse or as a continuous wave, and the
resulting perturbation creates a time-evolving superposition state whose
Fourier components encode the excitation energies. This is analogous to
propagation of a non-stationary wave packet according to the time-dependent
Schrödinger equation. Similar to that situation, the time-dependent MOs
$\psi_{k\sigma}(\mathbf{r},t)$ are complex-valued for $t>0$. Unlike the many-
electron time-dependent Schrödinger equation, the effective Hamiltonian
$\hat{F}$ in Kohn-Sham theory depends on its own time-evolving eigenfunctions,
and $\hat{F}(t)$ does not commute with $\hat{F}(t^{\prime})$ for $t\neq
t^{\prime}$. Therefore the time evolution operator

$\hat{U}(t_{2},t_{1})=\exp\left[\frac{-\text{i}(t_{2}-t_{1})}{\hbar}\hat{H}\right]$
(77)

for time-independent Hamiltonian $\hat{H}$ must be generalized to

$\hat{U}(t_{2},t_{1})=\hat{\cal
T}\exp\left[-\text{i}\int_{t_{1}}^{t_{2}}\hat{F}(t)\;dt\right]$ (78)

for TDKS calculations, where $\hat{\cal T}$ is a time-ordering operator. This
leads to so-called Magnus expansion that generalizes the time-independent
Baker-Campbell-Hausdorff expansion.425, 426, 427

A variety of time-propagation algorithms have been developed for TDKS
simulations,428, 429, 430, 431, 427, 52 the simplest of which is a “modified-
midpoint” algorithm,429 a type of explicit Euler integration scheme that
requires only one Fock matrix construction per time step. Denoting the Fock
and density matrices in the orthonormal MO basis as $\mathbf{F}$ and
$\mathbf{P}$, respectively, the Fock matrix $\mathbf{F}$ is first propagated
forward in time by a half step, $(\Delta t)/2$. At each subsequent instant in
time, $t_{n}=n\Delta t$, the matrix representation of the propagator is
constructed according to

$\mathbf{U}_{n}=\exp[-\text{i}(\Delta t)\mathbf{F}_{n+1/2}]\;,$ (79)

again in the MO basis.427 The MO density matrix is then propagated from
$t_{n-1/2}$ to $t_{n+1/2}$ according to

$\mathbf{P}_{n+1/2}=\mathbf{U}_{n}\;\mathbf{P}_{n-1/2}\;\mathbf{U}_{n}^{\dagger}\;.$
(80)

(Spin indices are omitted as these equations are valid for either spin.)
Construction of $\mathbf{U}_{n}$ requires diagonalization of the Fock matrix,
which is not a problem in Gaussian basis sets but is not feasible in plane-
wave basis sets or on a grid. See Ref. 428 for a discussion of alternatives
when $\mathbf{F}$ is too large to diagonalize. For Gaussian basis
calculations, self-consistent propagators based on predictor–corrector
algorithms have also been developed.427, 432 These may require more than one
Fock build per time step but allow for the use of somewhat larger time steps
as well as for automatic detection of time steps that are too long, which is
not always obvious from the usual criterion of checking to make sure that
fluctuations in the total energy are bounded.427

For the modified-midpoint algorithm that is encapsulated by Eqs. (79) and
(80), the cost of a single time step is comparable to the cost of a single SCF
cycle of the ground-state calculation. The storage requirement is also modest,
amounting to a few complex-valued matrices of dimension
$n_{\text{basis}}\times n_{\text{basis}}$. This should be contrasted with the
storage requirement for iterative solution of the LR-TDDFT pseudo-eigenvalue
problem, which is $\mathcal{O}(n_{\text{roots}}n_{\text{occ}}n_{\text{vir}})$
with a prefactor that reflects the number of iterations and therefore the size
of the iterative subspace. The modified-midpoint approach works well provided
that the time step $\Delta t$ is sufficiently small; values ranging from
0.01–0.50 a.u. are typical, where $\text{1~{}a.u.}\approx 2.42\times 10^{-17}$
s = 24.2 attoseconds. (The time for one orbit in the Bohr model of the
hydrogen atom is $2\pi$ times this value, which establishes a timescale for
electronic motion.) The maximum acceptable value of $\Delta t$ is limited not
only by stability of the time integration but also by the excitation energies
that one desires to access, as discussed below.

Unlike LR-TDDFT, which operates by definition in the limit of a vanishingly
weak external field, the TDKS approach is non-perturbative and in principle
can be used to simulate electron dynamics in strong laser fields, e.g., to
simulate nonlinear optical properties of materials,433 or to make contact with
emerging attosecond spectroscopies57 that create electronic wave packets that
are out of equilibrium with the nuclei and thus outside of the Born-
Oppenheimer approximation.54, 53, 55, 56 In practice, there are various issues
related to the use of the adiabatic approximation (Section 2.1.2),434, 435,
290 meaning the use of ground-state functionals with no memory, such that the
time dependence is carried solely by the time-evolving density,
$E_{\text{xc}}[\rho(\mathbf{r},t)]$. On the other hand, within the adiabatic
approximation the initial-state dependence vanishes since the XC kernel is
fully specified in terms of the instantaneous time-evolving density.436 The
topic of strong-field electron dynamics and how it can be described using TDKS
calculations is not considered here, except to note that there have been
successful TDKS simulations of strong-field photoionization,437, 438, 439,
440, 441, 442, 443, 444, 445, 446, 447, 448 and also of high harmonic
generation,449, 450, 451, 452, 453, 454, 455, 456, 457 both in Gaussian-
orbital representations of the density. Unlike the grid-based treatments that
are common in atomic physics, Gaussian-based methods are scalable to
molecules. However, self-interaction error is known to significantly suppress
strong-field ionization rates,458 therefore much of the aforementioned work
has been performed at the TD-CIS level, where self-interaction is not a
concern and the exchange potential has the correct asymptotic form.

This section focuses on the use of TDKS simulations to obtain broadband
spectra. Within the electric dipole approximation (which is also invoked in
LR-TDDFT insofar as oscillator strengths are proportional to transition dipole
matrix elements), the absorption spectrum corresponds to the dipole strength
function459, 163

$S(\omega)=\left(\frac{4\pi\omega}{3c}\right)\mathfrak{Im}\big{[}\alpha_{xx}(\omega)+\alpha_{yy}(\omega)+\alpha_{zz}(\omega)\big{]}$
(81)

where for example

$\alpha_{xy}(\omega)=\frac{\partial\mu_{x}(\omega)}{\partial\mathcal{E}_{y}(\omega)}$
(82)

is an element of the dynamic polarizability tensor, $\bm{\alpha}(\omega)$. The
quantities $\mu_{x}(\omega)$ and $\mathcal{E}_{y}(\omega)$ are the Fourier
transforms of the time-dependent dipole moment and the external electric
field, respectively, although for an impulsive $\delta$-function pulse the
denominator in Eq. (82) can be replaced by the field amplitude while the
numerator is replaced by

$\mu(\omega)=\int_{0}^{\infty}w(t)\;\mu(t)\;e^{-\text{i}\omega t}\;dt\;.$ (83)

Here, $w(t)$ is a windowing or padding function, as in standard signal
processing.52 To obtain a linear absorption spectrum the external field
$\bm{\mathcal{E}}(\mathbf{r},t)$ should be weak, impulsive, and off-resonant,
and should contain components in all three Cartesian directions in order to
excite states of all symmetries, effectively averaging over molecular
orientations. With an appropriately chosen integration time step, this
procedure reproduces the same spectrum that is obtained using LR-TDDFT, if all
of the LR-TDDFT excitations within the energy window of interest are
included.459, 427 The TDKS approach affords the entire broadband spectrum from
a single Fourier transform [Eq. (83)] following sufficient time propagation,
but it is not straightforward to assign the features in the TDKS spectrum to
transitions between specific MOs. Techniques to do so have been developed,
based on identifying individual Fourier modes in the dipole moment matrix
(expressed in the MO basis) at a specific transition frequency.460, 461, 462,
463, 464 This does require some post-processing and some insight regarding the
important MOs. Time-dependent generalization of the NTO basis (Section 2.2.4)
have also been proposed.465

The cost of the time steps needed to propagate the Kohn-Sham MOs in time, and
thus to obtain the time-evolving density and dipole moment function, is
comparable to the cost of a single ground-state SCF cycle but many time steps
are required. A typical simulation time might be 30 fs to obtain a fully-
converged spectrum,427 but with $\Delta t=0.1~{}\text{a.u.}=2.4$ as (1 as =
$10^{-18}$ s), this represents $>10^{5}$ time steps. Recently, Padé
approximant techniques have been introduced in order to obtain $\mu(\omega)$
based on a short time series of input data $\mu(t)$.462, 52 Using this
approach, spectra that are well-converged (with respect to LR-TDDFT results)
can be obtained in $<10$ fs of time propagation and rough spectra can be
obtained with as little as 3–5 fs.163 The time step $\Delta t$ dictates the
spectral window that can be accessed, via the usual time-energy uncertainty
relationship $(\Delta E)(\Delta t)\gtrsim\hbar$ that comes from the Fourier
transform. In practice, the spectrum is only reliably converged up the Nyquist
frequency $f_{\text{Ny}}=\pi/(\Delta t)$, and perhaps only a fraction of that
value.427, 163 This implies that especially small time steps are required for
x-ray applications. A time step $\Delta t=0.1$ a.u., for example, corresponds
to a Nyquist frequency $\hbar f_{\text{Ny}}=854$ eV, which is well above the K
edge for second-row elements (C, N, O, etc.) but not for third-row elements.
The K edge for elements Al–Cl lies above 1,500 eV.

### 4.2 Examples

Figure 15:  Experimental x-ray absorption spectra of $\alpha$-quartz at the
L2,3 edge,466 along with LR- and RT-TDDFT calculations using a cluster model
Si5O16H12 of the bulk material (as shown), with modified hydrogen charges to
enforce charge neutrality.467 The spectrum labeled “RT-TDDFT +
$\text{i}\Gamma$” includes phenomenological lifetime parameters for the
virtual orbitals. All calculations were performed with an optimally-tuned
version of LRC-$\omega$PBEh. The dashed line labeled $E_{0}$ is the
experimental ionization energy of Si(2p). Reproduced from Ref. 467; copyright
2015 American Chemical Society.

As an example of a broadband spectrum of interest in materials science, Fig.
15 presents x-ray spectra at the L2,3 edge of $\alpha$-quartz, computed using
several different TDDFT methods.467 Starting from the lowest valence
excitations, a LR-TDDFT calculation with $n_{\text{roots}}=300$ reproduces the
first two features in the experimental spectrum (labeled “A” and “B”) but is
unable to resolve the higher-energy features. (The calculations do not include
spin-orbit coupling and thus do not reproduce the doublet for peak A, which
arises due to the 0.6 eV splitting of the 2p3/2 and 2p1/2 levels.466) Although
the most intense feature (peak B) is evident in the RT-TDDFT spectrum, at a
peak position that precisely matches the corresponding LR-TDDFT spectrum, the
RT-TDDFT spectrum is quite noisy and other features in that spectrum are
obscured by this noise.

That noise is actually a basis-set artifact arising from the absence of proper
continuum states (or the inability to describe ionization within a finite-
basis approximation), which has the effect of artificially trapping metastable
excitations that lie above the ionization threshold.468, 467 A solution to
this problem is to incorporate phenomenological lifetimes for the unbound MOs,
meaning those with $\varepsilon_{p\sigma}>0$. This procedure is described in
Ref. 468 and corresponds to a modification
$\varepsilon_{p\sigma}\rightarrow\varepsilon_{p\sigma}+\text{i}\Gamma$ in the
MO basis, where $\Gamma^{-1}$ is a phenomenological lifetime that is modeled
as a function of energy, decreasing exponentially above the vacuum level. As
seen in Fig. 15, this “$\text{i}\Gamma$” modification removes the noise from
the RT-TDDFT spectrum, such that most of the experimental features become
evident even at energies far above what can feasibly be reached with LR-TDDFT.
This does require some phenomenological modeling, however.

Figure 16:  Absorption spectrum of 4-aminophenol at the oxygen K edge
(PBE0/def2-TZVP level), illustrating the appearance of pre-edge intruder peaks
in the RT-TDDFT spectrum that are not present in the LR-TDDFT spectrum. These
can be suppressed by filtering the dipole moment function. Reproduced from
Ref. 469; copyright 2022 American Chemical Society.

When computing K-edge x-ray spectra using high-quality basis sets, similar
artifacts can manifest as spurious pre-edge features that are not seen in
compact basis sets where the density of levels $\varepsilon_{p\sigma}$ is more
sparse. In a molecule that contains both nitrogen and oxygen, for example,
excitations from N(1s) core orbitals to the highest-energy virtual MOs can
manifest as spurious pre-edge features at the oxygen K-edge,163, 469 as shown
in Fig. 16. These artifacts appear despite the fact that the nitrogen K-edge
lies $>100$ eV below the oxygen K-edge! These intruder peaks could potentially
be mitigated via heuristic lifetime models for the unbound states, as
described above, although a simpler fix is to modify the time-dependent dipole
moment matrix in the MO basis. That matrix is

$D_{jk\sigma,x}(t)=-e\langle\psi_{j\sigma}(t)|\hat{x}|\psi_{k\sigma}(t)\rangle$
(84)

for the $x$ component. The time-dependent dipole moment function that is
needed in Eq. (83) is then

$\mu_{x}(t)=\mbox{tr}(\mathbf{P}_{\\!\alpha}\mathbf{D}_{\alpha,x}+\mathbf{P}_{\\!\beta}\mathbf{D}_{\beta,x})\;.$
(85)

By eliminating the rows and columns of $\mathbf{D}_{\sigma,x}$ that correspond
to occupied MOs other than the ones of interest, prior to computing the
Fourier transform in Eq. (83), the undesired resonances can be removed from
the spectrum.469 For the oxygen K edge, this means retaining only those rows
and columns where $j$ and $k$ refer to the O(1s) orbitals. This is precisely
analogous to the frozen-valence truncation of the LR-TDDFT excitation manifold
that is used to obtain core-level spectra (Section 2.2.1), and may have
similar limitations for L- and M-edge spectra. For the oxygen K edge, Fig. 16
shows that this procedure affords a spectrum in good agreement with LR-TDDFT,
free of contamination by N(1s) excitations. This filtering procedure does
require the user to decide in advance which edges are of interest, so that
multiple edges can no longer be computed in a single calculation unless the
entire trajectory of dipole moment matrices is stored.

The TDKS approach has been extended to compute excited-state absorption
spectra,436, 470 using an excited-state density prepared via LR-TDDFT as the
initial density at $t=0$. Because the initial state is non-stationary, this
requires that the “field on” simulation be referenced to a time-evolving
“field-off” simulation.436 This approach has recently been applied to simulate
emerging transient x-ray experiments,471, 472, 473, 474 carried out at free-
electron laser facilities using x-ray pulses with femtosecond time resolution.
As an example, it is possible to follow metal-to-metal CT dynamics in the
mixed-valence $\rm[(CN)_{5}Fe^{II}CNRu^{III}(NH_{3})_{5}]^{-}$ compound, which
occur on a $\sim 60$ fs timescale following excitation at 800 nm, using time-
resolved x-ray emission spectroscopy at the iron K edge ($\rm
2p_{3/2}\rightarrow 1s$ transition at 7,114 eV).474

## Acknowledgements

The author’s work on excited-state DFT has been supported for many years by
the U.S. National Science Foundation, at present under grant no. CHE-1955282.

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