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(June 30, 2021)

Following a perturbation to the ground-state MOs ${\varphi}_{k}(\mathbf{r},0)$ at $t=0$, these MOs evolve in time according to the time-dependent Schrödinger equation. For an SCF level of theory, this is a one-electron equation

$$i\mathrm{\hslash}\frac{d{\varphi}_{k}(\mathbf{r},t)}{dt}=\widehat{F}(t){\varphi}_{k}(\mathbf{r},t).$$ | (7.30) |

This time evolution can equivalently be expressed in terms of the Liouville-von Neumann equation for the time evolution of the density $\rho (\mathbf{r},t)$:

$$i\mathrm{\hslash}\frac{d\rho (\mathbf{r},t)}{dt}=[\widehat{F}(t),\rho (\mathbf{r},t)].$$ | (7.31) |

In addition to obtaining broadband spectra,
real-time SCF methods can be used to simulate attosecond
dynamics of electrons, perhaps in the presence of strong fields. Note that the dynamics that is simulated by
integrating either Eq. (7.30) or Eq. (7.31) is *electron* dynamics, the fundamental
timescale of which is attoseconds, as can be estimated by the magnitude of the atomic unit of time
($\mathrm{\hslash}/{E}_{h}\approx 2.4\times {10}^{-17}$ s). The finite integration time step $\mathrm{\Delta}t$
must be small compared to this value, and the default in Q-Chem is set to $\mathrm{\Delta}t=0.02\text{a.u.}=4.8\times {10}^{-4}$ fs.
The maximum timescale that can therefore reasonably
be simulated is likely only picoseconds, and at present this time propagation is available only within the clamped-nuclei
approximation, *i.e.*, it is not possible to simulate the couple electron–nuclear dynamics.

Because the Fock operator $\widehat{F}$ depends on its own (time-evolving) eigenfunctions ${\varphi}_{i}(\mathbf{r},t)$,
the operator $\widehat{F}(t)$ that governs the time evolution in Eq. (7.30) or Eq. (7.31) is time-dependent,
which complicates the integration of these equations.
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J. Chem. Phys.

(2018),
148,
pp. 044117.
Link
The simplest possible algorithm to integrate
these equations (over a finite time step $\mathrm{\Delta}t$ is the *modified midpoint unitary transformation* (MMUT)
procedure,
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663
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Phys. Chem. Chem. Phys.

(2005),
7,
pp. 233.
Link
which approximates the operator $\widehat{F}(t+\mathrm{\Delta}t/2)$. When the MMUT algorithm is used,
the cost of a single electron dynamics time step is comparable to the cost of a single SCF cycle of a ground-state
SCF calculation, *i.e.*, it requires a single construction and diagonalization of the Fock matrix. The memory footprint
is about twice that of the ground state, because the time-dependent MOs are complex-valued, but this is usually
considerably smaller than the memory footprint for linear-response (LR-)TDDFT, especially of the number of roots
requested in the LR-TDDFT calculation is large (as required for broadband spectra), or if the density of states is
high (as in models of semiconductors).
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771
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J. Phys. Chem. Lett.

(2015),
6,
pp. 4390.
Link

As compared to the first-order MMUT algorithm, higher-order predictor/corrector algorithms to integrate
the dynamics are also available.
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J. Chem. Phys.

(2018),
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pp. 044117.
Link
These algorithms enable the use of larger time steps $\mathrm{\Delta}t$ at a cost of a few Fock builds per time step.
Perhaps more importantly, the predictor/corrector algorithms iterate the Fock operators $\widehat{F}(t)$ and
$\widehat{F}(t+\mathrm{\Delta}t)$ to self-consistency over each time step, which guarantees stable time propagation (assuming
that the self-consistent procedure converges). Stable dynamics is *not* guaranteed by the MMUT algorithm,
and total energy conservation turns out to be a necessary but not sufficient criterion to ensure that the trajectory has
been integrated accurately. Using MMUT, examples can be found where energy is conserved yet spectra are still shifted
(with respect to benchmarks results obtained using very small time steps) due to the use of a too-large value of $\mathrm{\Delta}t$
that is undetected and undiagnosed by non-self-consistent MMUT algorithm.
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1256
}
J. Chem. Phys.

(2018),
148,
pp. 044117.
Link

Propagation of the electron dynamics requires only ground-state computational machinery (albeit with complex-valued orbitals), and thus is available at any SCF level of theory, including Hartree-Fock theory or DFT. The name “TDKS” (in contrast to the cumbersome “RT-TDDFT”) emphasizes that Eq. (7.30) is the Kohn-Sham analogue of the time-dependent Schrödinger equation. The cost per time step for a TDKS calculation should be no larger than a few times the cost of a ground-state SCF cycle. Q-Chem’s implementation exploits shared-memory parallelism and the use of at least 8 (but possibly more) processor cores is highly recommended, since the number of required time steps (and thus the number of Fock builds) is likely to be quite large. (The use of multiple cores is requested using the -nt flag.)

A TDKS calculation is requested by setting TDKS = TRUE in the *$rem* input section, and other
job control options are discussed in Section 7.4.2.

TDKS

Job control keyword to turn on TDKS calculation

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

TRUE
Perform a TDKS calculation following a ground-state SCF calculation
FALSE
Do not perform a TDKS calculation

RECOMMENDATION:

Also need to set PURECART = 2222.