Raman intensities

Calculating Raman intensities is by no means a trivial task, and because of the computational cost of such calculations, there are therefore few theoretical investigations of basis set requirements and correlation effects on calculated Raman intensities. The Raman intensities calculated are the ones obtained within the Placzek approximation [41], and the implementation is described in Ref. [42].

The Raman intensity is the differentiated frequency-dependent polarizability with respect to nuclear displacements. As it is a third derivative depending on the nuclear positions through the basis set, numerical differentiation of the polarizability with respect to nuclear coordinates is necessary.

The input looks very similar to the input needed for the calculation of Raman optical activity described in Section 11.4

**DALTON INPUT
.WALK
*WALK
.NUMERI
**WAVE FUNCTIONS
.HF
*SCF INPUT
.THRESH
1.0D-8
**START
.RAMAN
*ABALNR
.THRESH
1.0D-7
.FREQUE
     2
0.0 0.09321471
**EACH STEP
.RAMAN
*ABALNR
.THRESH
1.0D-7
.FREQUE
     2
0.0 0.09321471
**PROPERTIES
.RAMAN
.VIBANA
*RESPONSE
.THRESH
1.0D-6
*ABALNR
.THRESH
1.0D-7
.FREQUE
     2
0.0 0.09321471
*VIBANA
.PRINT
 1
.ISOTOP
   1   5
 1 1 1 2 3
**END OF DALTON INPUT

The keyword .RAMAN in the general input module indicates that a frequency-dependent polarizability calculation is to be done. The keyword .RAMAN indicates that we are only interested in the Raman intensities and depolarization ratios. Note that these parameters are also obtainable by using the keyword .VROA. In this calculation we calculate the Raman intensities for two frequencies, the static case and a frequency of the incident light corresponding to a laser of wavelength 488.8 nm.

Due to the numerical differentiation that is done, the threshold for the iterative solution of the response equations are by default 10$^{-7}$, in order to get Raman intensities that are numerically stable to one decimal digit.

In the *WALK input module we have specified that the walk is a numerical differentiation. This will automatically turn off the calculation of the geometric Hessian, putting limitations on what kind of properties that may be calculated at the same time as Raman intensities. Because the Hessian is not calculated, there will not be any prediction of the energy at the new point.

It should also be noted that in a numerical differentiation, the program will step plus and minus one displacement unit along each Cartesian coordinate of all nuclei, as well as calculating the property at the reference geometry. Thus, for a molecule with $N$ atoms the properties will need to be calculated in a total of 2*3*$N$ + 1 points, which for a molecule with five atoms will amount to 31 points. The default maximum number of steps in DALTON is 20. However, in numerical differentiation calculations, the number of iterations will always be reset (if there are more than 20 steps that need to be taken) to 6$N$+1, as it is assumed that the user always wants the calculation to complete correctly. The maximum number of allowed iterations can be manually set by adding the keyword .MAX IT in the **DALTON input module.

The default step length in the numerical differentiation is $1.0\cdot 10^{-4}$ a.u., and this step length may be adjusted by the keyword .DISPLA in the *WALK input module. The steps are taken in the Cartesian directions and not along normal modes. This enables us to study the Raman intensities of a large number of isotopically substituted molecules at once. This is done in the **PROPERTIES input section, where we have requested one isotopically substituted species in addition to the isotopic species containing the most abundant isotope of each element.