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Table of contents

• Table of contents
• Details
• Keywords
  • The INTALG Keyword
  • The FIELD Keyword
  • The RESTARTSTEP and IRSTRT Keywords
  • The RESTART Keyword
• Examples
  • Propagation after a "delta kick"
  • Propagation after SCF in a field
• References

Details

The RT input section includes all specifications for real-time electron dynamics calculations. This section is required for all calculations where QM.JOB is RT. The defaults are suitable for most applications, although the time step and simulation length must be specified by the user.

Keywords

The INTALG Keyword

The integration algorithm used to propagate the density matrix. The currently available methods are both 2^nd order integrators based on the Magnus expansion,¹ specifically MMUT and MAGNUS2. The default algorithm is MMUT due to its lower cost (single Fock formation and diagonalization per step) and superior performance for oscillatory solutions.

MMUT: The Modified Midpoint Unitary Transform (MMUT) propagator²

MMUT is a "leapfrog" integrator in which the density from the previous time step is propagated to the next time step based on the Fock matrix of the current time step.

\mathbf{P}(t_{k+1}) = \mathbf{U}(t_k) \mathbf{P}(t_{k-1}) \mathbf{U}^\dag(t_k)

The propagator, \mathbf{U} is an exponential based on the Fock matrix at the current time.

\mathbf{U}(t_k) = e^{2i\Delta t \mathbf{F}(t_k)}

Currently, this expression is evaluated through explicit diagonalization of the time dependent Fock matrix. Other methods for evaluating the matrix exponential are possible and are being implemented.

MAGNUS2: A second order, trapezoidal approximation to the Magnus expansion³

The 2nd order, explicit Magnus propagator is a single step integrator, propagating the density from the current step to the next step.

\mathbf{P}(t_{k+1}) = \mathbf{U}(t_k) \mathbf{P}(t_k) \mathbf{U}^\dag(t_k)

Where the propagator is based on the Fock matrix at the current time step and the next time step.

\mathbf{U}(t_k) = e^{\frac{i\Delta t}{2} (\mathbf{F}(t_k) + \mathbf{F}(t_{k+1}))}

Currently, the Fock matrix at the next time step is obtained through a forward Euler propagation of the density.

The FIELD Keyword

The FIELD keyword specifies a time-dependent electromagnetic perturbation for the real-time electron dynamics simulation. The specification of the RT field is very similar to that of the SCF.FIELD, with a few caveats:

• One must specify a time-dependent envelope and
• Multiple fields may be specified.

Thus, the total field at a given time is given by

    E(t) = \sum_k e_k(t) a_{kj} O_j

Where e_k is the envelope for the k-th field, O_j is some multipole operator and a_{kj} is the amplitude of O_j for the k-th field.

In general, this may be specified as

Field:
  <ENVELOPE> <FIELD TYPE> <AMPLITUDES>
  <ENVELOPE> <FIELD TYPE> <AMPLITUDES>
  <ENVELOPE> <FIELD TYPE> <AMPLITUDES>

The <FIELD TYPE> and <AMPLITUDES> follow the same rules as SCF.FIELD. The <ENVELOPE> specification may be given (in general) as,

<ENVELOPE NAME>(<ENVELOPE PARAM>)

Where <ENVELOPE PARAM> is a comma separated list of envelope parameters (see table). As of 10/2/2020, ChronusQ supports the following envelopes:

Note: Currently, ChronusQ only supports Electric Dipole fields in real-time electron dynamics.

The RESTARTSTEP and IRSTRT Keywords

When using the MMUT algorithm to propagate the density, single step methods must be used to start the algorithm. Furthermore, if allowed to propagate for long time frames, the branches of the MMUT algorithm with even and odd steps can diverge, leading to a highly oscillatory solution. This can be remedied by using a single step method to periodically restart the MMUT algorithm. IRSTRT controls the number of MMUT steps between the single step propagator.

For RESTARTSTEP, both MAGNUS2 and FORWARDEULER are accepted, but the FORWARDEULER restart step may introduce significant error and should not be used. It is included for the sake of backward compatibility.

The RESTART Keyword

RESTART will trigger an attempt to restart from the last recorded checkpoint on the binary ChronusQ file. It will load the last time dependent density saved and propagate to the final time. RESTART cannot currently be used to extend the simulation time specified in the original input file. If this is important for your work, please open an issue!

Examples

Propagation after a "delta kick"

The absorption spectrum of a system can be extracted by analyzing the dipole oscillations after an instantaneous "kick" from an external field.¹ One of the required simulations for this process can be achieved in ChronusQ with the following input.

[RT]
TMAX   = 620.15
DELTAT = 0.005
FIELD:
 StepField(0.,0.00001) Electric 0. 0.001 0.

Propagation after SCF in a field

Real time electron dynamics can also be used to study molecular plasmons.⁴ In this case, one performs an optimization of the molecular wavefunction in a static electric field, but then removes the electric field for the propagation. This may be achieved (combined with the required SCF field) with the following RT input.

[RT]
TMax     = 1000.
DeltaT   = 0.005
SCFField = False

References