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  • Frequency Dependent Response TDHF

Last edited by Andrew Wildman Oct 13, 2020
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Frequency Dependent Response TDHF

Frequency dependent response (FDR) using time-dependent Hartree-Fock (or TDDFT) allows one to compute electronic frequency dependent polarizabilties of molecular systems by choosing a perturbing operator, and a specified observable.

Specifically, FDR jobs solve the linear inverse propagator problem for the input perturbations \mathbf{B}, and the output observable \mathbf{A} for a particular frequency to produce the response function. In short, the equation is of the form

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

where \mathbf{H} is the orbital Hessian, \omega is the chosen frequency, and \mathbf{S} is the metric

\mathbf{S} = \begin{bmatrix} \mathbf{I} & 0 \\ 0 & -\mathbf{I} \end{bmatrix} 

In the following example we will compute the FDR across a range of frequencies and perturbing operators for the HCl molecule.

Input

[Molecule]
charge = 0
mult = 1
geom:
 Cl   0.0   0.0   0.0
 H    0.0   0.0   1.27

[QM]
reference = RHF
job = resp

[BASIS]
basis = cc-pvdz

[RESPONSE]
type = fdr
bfreq = range(0.0,10,0.01)
bops  = edl md
dofull = true

For this HCl example, we use a restricted Hartree-Fock reference, set the job to job = resp for response, and use the cc-pvdz basis set. Now, we set the FDR related options in the [RESPONSE] settings.

We set the job type to FDR with the keyword type = fdr. Next, the range of perturbing frequencies we want to solve for is specified by bfreq. In this case, bfreq = range(0.0,10,0.01) meaning that the frequencies will range from 0.0 to 0.09 a.u. for a total of 10 different frequencies to be computed.

Further, the bops keyword corresponds to "\mathbf{B} operators". Specifically it is a space separated list of perturbing operators. In this example, bops = edl md specifies ChronusQ to compute the FDR job with an electric dipole (length gauge) perturbation, edl, and a Magnetic dipole perturbation, md. Additionally, there is an aops keyword, but defaults to using all operators.

Finally, dofull = true specifies to solve for the full \mathbf{H} matrix.

Output

After running the FDR job, we see the following results in the output file.

FREQUENCY DEPENDENT RESPONSE RESULTS


* RESPONSE FUNCTIONS (POLARIZABILITIES)


  Electric Dipole - Electric Dipole (Length) : - Re [<< r_i; r_j >>] (AU)


                         { << X; X >>, << X; Y >>, << X; Z >> }
                         { << Y; X >>, << Y; Y >>, << Y; Z >> }
                         { << Z; X >>, << Z; Y >>, << Z; Z >> }

    W(AU) = 0.0000      6.32053e+00    0.00000e+00    0.00000e+00
                        0.00000e+00    6.32053e+00    0.00000e+00
                        0.00000e+00    0.00000e+00    1.27291e+01

    W(AU) = 0.0100      6.32132e+00    0.00000e+00    0.00000e+00
                        0.00000e+00    6.32132e+00    0.00000e+00
                        0.00000e+00    0.00000e+00    1.27327e+01
.
.
.

In this first polarizability result, we see the dipole-dipole response function tensor for each of the 10 frequencies:

\text{Re} \left[ \langle\langle r_i; r_j \rangle \rangle_\omega \right]

Only the first two frequencies are shown above, but we can see each tensor is ordered by cartesian direction as so:

{ << X; X >>, << X; Y >>, << X; Z >> }
{ << Y; X >>, << Y; Y >>, << Y; Z >> }
{ << Z; X >>, << Z; Y >>, << Z; Z >> }

As expected, we see changes in the polarizability as the frequency of the perturbation is changed.

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Overview and Features

Getting ChronusQ

Running ChronusQ

Input sections

     Overview
     QM and PROTQM
     Molecule
     Basis and DFBasis
     Ints
     DFTInts
     SCF
     RT
     Response
     CC
     MCSCF
     Misc

FAQ

Examples

     HF energy
     Relativistic DFT Energy
     Linear Response TDDFT
     Frequency dependent TDHF
     Model Order Reduction of TDDFT
     Electron dynamics

Keyword Reference

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