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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.