Octopus

ArnoldiOrthogonalization

InteractionTiming

IonsConstantVelocity

IonsTimeDependentDisplacements

MoveIons

RecalculateGSDuringEvolution

TDDynamics

TDEnergyUpdateIter

TDExponentialMethod

Name TDExponentialMethod
Section Time-Dependent::Propagation
Type integer
Default taylor

Method used to numerically calculate the exponential of the Hamiltonian, a core part of the full algorithm used to approximate the evolution operator, specified through the variable TDPropagator. In the case of using the Magnus method, described below, the action of the exponential of the Magnus operator is also calculated through the algorithm specified by this variable.

Options:

lanczos:
Allows for larger time-steps. However, the larger the time-step, the longer the computational time per time-step. In certain cases, if the time-step is too large, the code will emit a warning whenever it considers that the evolution may not be properly proceeding -- the Lanczos process did not converge. The method consists in a Krylov subspace approximation of the action of the exponential (see M. Hochbruck and C. Lubich, SIAM J. Numer. Anal. 34, 1911 (1997) for details). Two more variables control the performance of the method: the maximum dimension of this subspace (controlled by variable TDExpOrder), and the stopping criterion (controlled by variable TDLanczosTol). The smaller the stopping criterion, the more precisely the exponential is calculated, but also the larger the dimension of the Arnoldi subspace. If the maximum dimension allowed by TDExpOrder is not enough to meet the criterion, the above-mentioned warning is emitted.
taylor:
This method amounts to a straightforward application of the definition of the exponential of an operator, in terms of its Taylor expansion.
$\exp_{\rm STD} (-i\delta t H) = \sum_{i=0}^{k} {(-i\delta t)^i\over{i!}} H^i.$

The order k is determined by variable TDExpOrder. Some numerical considerations from Jeff Giansiracusa and George F. Bertsch suggest the 4th order as especially suitable and stable.
chebyshev:
In principle, the Chebyshev expansion of the exponential represents it more accurately than the canonical or standard expansion. As in the latter case, TDExpOrder determines the order of the expansion.
There exists a closed analytic form for the coefficients of the exponential in terms of Chebyshev polynomials:

$\exp_{\rm CHEB} \left( -i\delta t H \right) = \sum_{k=0}^{\infty} (2-\delta_{k0})(-i)^{k}J_k(\delta t) T_k(H),$

where $J_k$ are the Bessel functions of the first kind, and H has to be previously scaled to $[-1,1]$. See H. Tal-Ezer and R. Kosloff, J. Chem. Phys. 81, 3967 (1984); R. Kosloff, Annu. Rev. Phys. Chem. 45, 145 (1994); C. W. Clenshaw, MTAC 9, 118 (1955).

TDExpOrder

TDIonicTimeScale

TDLanczosTol

TDMaxSteps

TDPhotonicTimeScale

TDPropagationTime

TDPropagator

Name TDPropagator
Section Time-Dependent::Propagation
Type integer
Default etrs

This variable determines which algorithm will be used to approximate the evolution operator $U(t+\delta t, t)$. That is, given $\psi(\tau)$ and $H(\tau)$ for $\tau \le t$, calculate $t+\delta t$. Note that in general the Hamiltonian is not known at times in the interior of the interval $[t,t+\delta t]$. This is due to the self-consistent nature of the time-dependent Kohn-Sham problem: the Hamiltonian at a given time $\tau$ is built from the "solution" wavefunctions at that time.

Some methods, however, do require the knowledge of the Hamiltonian at some point of the interval $[t,t+\delta t]$. This problem is solved by making use of extrapolation: given a number $l$ of time steps previous to time $t$, this information is used to build the Hamiltonian at arbitrary times within $[t,t+\delta t]$. To be fully precise, one should then proceed self-consistently: the obtained Hamiltonian at time $t+\delta t$ may then be used to interpolate the Hamiltonian, and repeat the evolution algorithm with this new information. Whenever iterating the procedure does not change the solution wavefunctions, the cycle is stopped. In practice, in Octopus we perform a second-order extrapolation without a self-consistency check, except for the first two iterations, where obviously the extrapolation is not reliable.

The proliferation of methods is certainly excessive. The reason for it is that the propagation algorithm is currently a topic of active development. We hope that in the future the optimal schemes are clearly identified. In the mean time, if you do not feel like testing, use the default choices and make sure the time step is small enough.

Options:

etrs:
The idea is to make use of time-reversal symmetry from the beginning:
$ \exp \left(-i\delta t H_{n} / 2 \right)\psi_n = \exp \left(i\delta t H_{n+1} / 2 \right)\psi_{n+1}, $

and then invert to obtain:

$ \psi_{n+1} = \exp \left(-i\delta t H_{n+1} / 2 \right) \exp \left(-i\delta t H_{n} / 2 \right)\psi_{n}. $

But we need to know $H_{n+1}$, which can only be known exactly through the solution $\psi_{n+1}$. What we do is to estimate it by performing a single exponential: $\psi^{*}_{n+1}=\exp \left( -i\delta t H_{n} \right) \psi_n$, and then $H_{n+1} = H[\psi^{*}_{n+1}]$. Thus no extrapolation is performed in this case.
aetrs:
Approximated Enforced Time-Reversal Symmetry (AETRS). A modification of previous method to make it faster. It is based on extrapolation of the time-dependent potentials. It is faster by about 40%. The only difference is the procedure to estimate $H_{n+1}$: in this case it is extrapolated via a second-order polynomial by making use of the Hamiltonian at time $t-2\delta t$, $t-\delta t$ and $t$.
caetrs:
(experimental) Corrected Approximated Enforced Time-Reversal Symmetry (AETRS), this is the previous propagator but including a correction step to the exponential.
exp_mid:
Exponential Midpoint Rule (EM). This is maybe the simplest method, but it is very well grounded theoretically: it is unitary (if the exponential is performed correctly) and preserves time-reversal symmetry (if the self-consistency problem is dealt with correctly). It is defined as: $ U_{\rm EM}(t+\delta t, t) = \exp \left( -i\delta t H_{t+\delta t/2}\right)\,. $
crank_nicholson:
crank_nicolson:
Classical Crank-Nicolson propagator. $ (1 + i\delta t H_{n+1/2} / 2) \psi_{n+1} = (1 - i\delta t H_{n+1/2} / 2) \psi_{n} $
crank_nicholson_sparskit:
crank_nicolson_sparskit:
Classical Crank-Nicolson propagator. Requires the SPARSKIT library. $ (1 + i\delta t H_{n+1/2} / 2) \psi_{n+1} = (1 - i\delta t H_{n+1/2} / 2) \psi_{n} $
magnus:
Magnus Expansion (M4). This is the most sophisticated approach. It is a fourth-order scheme (a feature which it shares with the ST scheme; the other schemes are in principle second-order). It is tailored for making use of very large time steps, or equivalently, dealing with problem with very high-frequency time-dependence. It is still in a experimental state; we are not yet sure of when it is advantageous.
qoct_tddft_propagator:
WARNING: EXPERIMENTAL
runge_kutta4:
WARNING: EXPERIMENTAL. Implicit Gauss-Legendre 4th order Runge-Kutta.
runge_kutta2:
WARNING: EXPERIMENTAL. Implicit 2nd order Runge-Kutta (trapezoidal rule). Similar, but not identical, to Crank-Nicolson method.
expl_runge_kutta4:
WARNING: EXPERIMENTAL. Explicit RK4 method.
cfmagnus4:
WARNING EXPERIMENTAL

TDSCFThreshold

TDStepsWithSelfConsistency

TDSystemPropagator

TDTimeStep

TemperatureFunction

Thermostat

ThermostatMass