# Resonant and non-resonant Raman spectra¶

Note: Raman Calculations with SIESTA and PYSCF-NAO are possible also.

Raman spectra can be calculated in various approximations 1. While the examples below are using GPAW explicitly, the modules are intended to work with other calculators also. The strategy is to calculate vibrational properties first and obtain the spectra from these later.

## 1. Finite difference calculations¶

### 1a. Forces¶

It is recommended to do a vibrational analysis first by using
the `Vibrations`

or `Infrared`

modules. In the example of molecular hydrogen this is

```
from ase.build import molecule
from ase import optimize
from ase.vibrations.infrared import InfraRed
from gpaw.cluster import Cluster
from gpaw import GPAW, FermiDirac
h = 0.22
atoms = Cluster(molecule('H2'))
atoms.minimal_box(3.5, h=h)
# relax the molecule
calc = GPAW(h=h, occupations=FermiDirac(width=0.1))
atoms.calc = calc
dyn = optimize.FIRE(atoms)
dyn.run(fmax=0.05)
atoms.write('relaxed.traj')
# finite displacement for vibrations
ir = InfraRed(atoms)
ir.run()
```

This produces a calculation with rather accurate forces in order to get the Hessian and thus the vibrational frequencies as well as Eigenstates correctly.

### 1b. Excitations¶

In the next step we perform a finite difference optical calculation with less accuracy, where the optical spectra are evaluated using TDDFT

```
from ase.vibrations.resonant_raman import ResonantRaman
from gpaw.cluster import Cluster
from gpaw import GPAW, FermiDirac
from gpaw.lrtddft import LrTDDFT
h = 0.25
atoms = Cluster('relaxed.traj')
atoms.minimal_box(3.5, h=h)
# relax the molecule
calc = GPAW(h=h, occupations=FermiDirac(width=0.1),
eigensolver='cg', symmetry={'point_group': False},
nbands=10, convergence={'eigenstates':1.e-5,
'bands':4})
atoms.calc = calc
# use only the 4 converged states for linear response calculation
rr = ResonantRaman(atoms, LrTDDFT, exkwargs={'jend':3})
rr.run()
```

Albrecht B+C terms need wave function overlaps at equilibrium and displaced structures. These are assumed to be calculated in the form

where \(\phi_j^{{\rm eq}}\) is an orbital at equilibrium position
and \(\phi_i^{\rm disp}\) is an orbital at displaced position.
This is implemented in `Overlap`

in GPAW
(approximated by pseudo-wavefunction overlaps) and can be triggered
in `ResonantRaman`

by:

```
from gpaw.analyse.overlap import Overlap
rr = ResonantRaman(atoms, LrTDDFT, exkwargs={'jend':3}
overlap=lambda x, y: Overlap(x).pseudo(y),
)
```

### 2. Analysis of the results¶

We assume that the steps above were performed and are able to analyse the results in different approximations.

In order to do the full Albrecht analysis later we We save the standard names:

```
# standard name for Vibrations
gsname='vib'
# standard name for Infrared
gsname='ir'
```

#### Placzek¶

The most popular form is the Placzeck approximation that is present in two implementations. The simplest is the direct evaluation from derivatives of the frequency dependent polarizability:

```
from ase.vibrations.placzek import Placzek
photonenergy = 7.5 # eV
pz = Placzek()
x, y = pz.get_spectrum(photonenergy, start=0, end=2000, method='frederiksen', type='Lorentzian')
```

The second implementation evaluates the derivatives differently allowing for more analysis:

```
from ase.vibrations.placzek import Profeta
photonenergy = 7.5 # eV
pr = Profeta(approximation='Placzek')
x, y = pr.get_spectrum(photonenergy, start=0, end=2000, method='frederiksen', type='Lorentzian')
```

Both should lead to the same spectrum.

#### Albrecht¶

`ResonantRaman`

calls the displaced excited state objects’ function
`overlap`

with the matrix \(o_{ij}\) and expects the function to
return the corresponding overlap matrix for the transition dipoles.
In case of Kohn-Sham transitions with \(i,j\) for occupied
and \(\alpha,\beta\) for empty orbitals, this is

Example:

```
from ase.vibrations.albrecht import Albrecht
al = Albrecht()
```

- 1
“Ab-initio wave-length dependent Raman spectra: Placzek approximation and beyond” Michael Walter, Michael Moseler arXiv:1806.03840 [physics.chem-ph]