Model Order Reduction or MOR is a general linear algebra technique that allows for a reduction in dimension of a linear system solver while retaining high accuracy, ubiquitous in many scientific domains.
For our purposes, we will can use MOR in ChronusQ to compute the oneelectron absorption cross section over an arbitrary energy range for the TDHF or TDDFT response matrices. The frequency (\omega
) dependent absorption cross section^{1} is defined as
{\boldsymbol \sigma}(\omega) = \omega \text{Im}\left[ \text{Tr} \left( {\boldsymbol \alpha}(\tilde{\omega}) \right) \right]
where {\boldsymbol \alpha}(\tilde{\omega})
is the dynamic polarizabilty tensor, and \tilde{\omega} = \omega + i \eta
, where \eta
is a small positive damping factor to numerically converge resonant excitations.
Luckily, we can use a frequency dependent response (FDR) calculation to compute {\boldsymbol \alpha}(\tilde{\omega})
, which is detailed in the example: Frequency Dependent TDHF.
Input
The MOR absorption cross section job only requires a small modification to the input file in the Frequency dependent TDHF example.
[Molecule]
charge = 0
mult = 1
geom:
Cl 0.0 0.0 0.0
H 0.0 0.0 1.27
[QM]
reference = RB3LYP
job = resp
[BASIS]
basis = ccpvdz
[RESPONSE]
type = mor
damp = 0.01
bfreq = range(0.15,300,0.01)
bops = edl md
dofull = true
[MOR]
refine = true
The main difference from the FDR job input file is in the [RESPONSE]
section and the additional [MOR]
section. First, we specify that the additional MOR absorption cross section is to be computed with type = mor
in the [RESPONSE]
section. Additionally, we sample 300 different frequencies bfreq = range(0.15,300,0.01)
to have good resolution for the output MOR spectrum.
Next, the damp
parameter is required. This is a user choice that is equivalent to dressing a typical TDDFT stick spectra with a lorentzian where the width is chosen by said damp parameter. In this case damp = 0.01
Hartrees.
Finally there's an optional parameter refine
in the optional section [MOR]
. This method is turns on an adaptive scheme for the MOR method,^{2} reducing computational overhead, and is recommended for large and dense spectra.
Output
For the output file of an MOR spectrum TDDFT job, we see the typical polarizabilties that are produced by the FDR portion.
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) = 0.1500 6.68923e+00 0.00000e+00 0.00000e+00
0.00000e+00 6.68923e+00 0.00000e+00
0.00000e+00 0.00000e+00 1.38417e+01
W(AU) = 0.1600 6.72890e+00 0.00000e+00 0.00000e+00
0.00000e+00 6.72890e+00 0.00000e+00
0.00000e+00 0.00000e+00 1.39914e+01
.
.
.
However, below all the polarizabilities, the absorption cross section {\boldsymbol \sigma}(\omega)
at each frequency is reported in the following format:
* OBSERVABLES
ONEPHOTON ABSORPTION CROSSSECTION (EDA)
* SIGMA(W) = 4 * PI * W / C * IM[ALPHA(W,W)]
* ALPHA(W,W) = TR[ << r; r >>(W) ]
W SIGMA(W) (AU)
1.5000000000e01 3.0037708606e03
1.6000000000e01 3.5462240120e03
1.7000000000e01 4.1741142323e03
1.8000000000e01 4.9075595328e03
1.9000000000e01 5.7753132370e03
.
.
.
where W
is the frequency and SIGMA(W)
is the absorption cross section. Using this data, the absorption spectrum can be easily plotted with your favorite plotting software. For this example, the spectrum looks like the following:
References

Oddershede, J., & Jørgensen, P. & Yeager, D. L. (1984). Polarization Propagator Methods in Atomic and Molecular Calculations. Computer Physics Reports, 3392.
↩ 
Van Beeumen, R., & WilliamsYoung, D. B., & Kasper, J. M., & Yang, C., & Ng, E. G., & Li, X. (2017). Model Order Reduction Algorithm for Estimating the Absorption Spectrum. Journal of Chemical Theory and Computation, 13(10), 49504961.
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