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Response

Table of contents

Details

The RESP section handles job parameters for response calculations in ChronusQ. It is strongly recommended to use a stable reference wave function for any 2-component response calculation. The RESP section is required for response jobs (QM.JOB = RESP) and takes the following optional keywords

Keywords

Keyword Type Description Default
TYPE String The response job type RESIDUE
PROPAGATOR String The quantum propagator used in the response function PARTICLEHOLE
DOFULL Boolean Whether or not to do the "full" problem, i.e. with (Sca)LAPACK. Implies FULLMAT if true TRUE
FULLMAT Boolean Whether or not to build the full propagator in memory TRUE
NROOTS Integer Number of desired roots for a RESIDUE job (see RESP.TYPE keyword). Only used if DOFULL=FALSE 3
DEMIN Double precision float Minimum energy (in a.u.) for a RESIDUE job (see RESP.TYPE keyword). Only used if DOFULL=FALSE 0.0
MAXITER Integer Maximum number of iterations for iterative response job 500
AOPS String Output observable operators ALLOPS
BOPS String Input perturbation operators EDL
BFREQ String Frequencies for applied BOPs 0.0
DAMP Double precision float Damping parameter for FDR jobs (see RESP.TYPE keyword) 0.0
CONV Double precision float Residual convergence critera for iterative response job 1e^{-7}
GPLHR_M Integer Internal GPLHR subspace expansion parameter 3
GPLHR_SIGMA Double precision float Internal GPLHR energy shift parameter 0.0
DISTMATFROMROOT Boolean Whether to form matrix on root, then distribute (MPI only) FALSE
FORMMATDIST Boolean Whether to form the matrix distributed (MPI only) FALSE

The TYPE Keyword

ChronusQ supports Three types of response jobs:1 RESIDUE, FDR (Frequency Dependent Response), and MOR. (Model Order Reduction) RESIDUE calculations solve for the eigenvectors and eigenvalues from the "traditional" response matrix. FDR calculations solve the linear inverse propagator problem for specific input perturbations (type and frequency) and output observables. MOR calculations solve for frequency dependent response over a range of frequencies in a more efficient manner. In general, if you want a "traditional" TDHF/TDDFT calculation, run a RESIDUE calculation.

RESIDUE calculations solve:

\mathbf{H} \mathbf{X} = \Omega\mathbf{S}\mathbf{X}

where \mathbf{H} is the orbital hessian, \mathbf{X} is the transition density, \mathbf{S} is the metric, and \Omega are the eigenvalues.

FDR calculations solve:

\langle\langle \mathcal{A}; \mathcal{B} \rangle \rangle_\omega = \mathbf{A}^* (\mathbf{H} - (\omega + i\gamma)\mathbf{S})^{-1}\mathbf{B}

where \mathbf{B} is the perturbing operator, \mathbf{A} is the output observable, \omega is the perturbing frequency, and \gamma is a damping parameter in order to ensure convergence even at the poles.

The PROPAGATOR Keyword

ChronusQ supports both the polarization (particle-hole) propagator (PARTICLEHOLE) and the particle-particle propagator. (PARTICLEPARTICLE) The polarization propagator gives rise to single excitations, and the particle-particle propagator gives rise to effective "double" excitations by adding two electrons. Note that this does not correspond to double excitations from a neutral reference. Particle-particle propagators are currently considered an expert feature and are controlled by undocumented keywords. If this application is important for your research, please open an issue!

Iterative Keywords

ChronusQ supports both full and iterative diagonalization of the response problems. If the size of iterative problem requested would exceed the size of full diagonalization, ChronusQ falls back to full diagonalization.

Full Diagonalization

Full diagonalization is controlled by the DOFULL keyword. If you have enough memory for this, it is typically the fastest and most robust method for getting more than a few roots from the response problem.

Iterative Diagonalization

Iterative diagonalization in ChronusQ is performed using the Generalized Preconditioned Locally Harmonic Residual (GPLHR) method.2 Like other iterative methods, you can set a minimum energy for the roots to be solved for (DEMIN) and the number of roots to solve for (NROOTS) as well as the maximum number of iterations. (MAXITER)

Frequency Dependent Response / Model Order Reduction

Frequency dependent response calculations solve:

\langle\langle \mathcal{A}; \mathcal{B} \rangle \rangle_\omega = \mathbf{A}^* (\mathbf{H} - (\omega + i\gamma)\mathbf{S})^{-1}\mathbf{B}

where \mathbf{H} is the orbital hessian, \mathbf{X} is the transition density, \mathbf{S} is the metric, \mathbf{B} is the perturbing operator, \mathbf{A} is the output observable, \omega is the perturbing frequency, and \gamma is a damping parameter in order to ensure convergence even at the poles.

Response Operators

Both AOPS and BOPS may be:

Keyword Operator
EDL Electric Dipole (Length gauge)
EQL Electric Quadrupole (Length gauge)
EOL Electric Octupole (Length gauge)
EDV Electric Dipole (Velocity gauge)
EQV Electric Quadrupole (Velocity gauge)
EOV Electric Octupole (Velocity gauge)
MD Magnetic Dipole
MQ Magnetic Quadrupole
ALLOPS All of the above

The DAMP Keyword

Without a small imaginary shift, frequency dependent response diverges at the poles of the propagator. Physically, this corresponds to finite excited state lifetimes. The DAMP keyword controls the magnitude of this damping.

The BFREQ Keyword

The BFREQ keyword determines the frequencies at which to solve the FDR/MOR problem. This can take two forms:

  1. One or more frequencies delimited with space, tab, comma, or semicolon.
  2. A string of the format Range(<start>,<count>,<step>) to generate a set of uniformly spaced frequencies covering the interval [\mathrm{start}, \mathrm{count}\times\mathrm{step} + \mathrm{start}) with a stepsize of <step>.

For example, a set of 10 uniformly distributed frequencies over the interval [0,5) could be specified as:

BFreq = 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Or:

BFreq = Range(0.0,10,0.5)

Fine Control of Diagonalization

For most cases, the defaults for this section are sufficient. However, some users may desire more fine control over the diagonalization for difficult or large cases.

The CONV keyword controls the convergence criterion for iterative diagonalization. (Compared to norm of residual vectors)

GPLHR Parameters

GPLHR has two parameters that primarily control the method: GPLHR_M and GPLHR_SIGMA.2

  • GPLHR_M is a parameter that determines how much the subspace is increased internally with every iteration. It typically ranges from 2 to 7, with a larger M increasing the work done in each step significantly. Consequently, choosing a different m value may allow users to converge very hard cases at the cost of significantly more work done per step.
  • GPLHR_SIGMA controls the shift of the subspace. GPLHR is a harmonic method, so it will converge roots that are close to \sigma. (on either side) Using DEMIN will dynamically choose \sigma so that the roots selected are above the given energy, so this option is typically less useful to users than DEMIN.

MPI Options

When ChronusQ is compiled with MPI enabled, it can form the propagator matrix either completely on the root process or distributed among all processes. The DISTMATFROMROOT and FORMMATDIST control these choices.

Examples

Residue Response

[Response]
Type = RESIDUE

Damped FDR (All observables to electric dipole perturbation, iterative)

[Response]
Type = FDR
Damp = 0.01
BFreq = 0.5
DoFull = False
FullMat = False

Damped MOR (Electric dipole response to electric dipole perturbations)

[Response]
Type = MOR
Damp = 0.01
BFreq = Range(0.0,300,0.01)

References

  1. Williams‐Young, D. B., Petrone, A., Sun, S., Stetina, T. F., Lestrange, P., Hoyer, C. E., ... & Li, X. (2020). The Chronus Quantum software package. Wiley Interdisciplinary Reviews: Computational Molecular Science, 10(2), e1436. ↩

  2. Vecharynski, E., Yang, C., & Xue, F. (2016). Generalized preconditioned locally harmonic residual method for non-Hermitian eigenproblems. SIAM Journal on Scientific Computing, 38(1), A500-A527. ↩ ↩2