Raman spectra
Raman spectroscopy is used to study the vibrational, rotational, and other low-frequency modes in a system. It is based on the Raman effect of inelastic scattering of monochromatic light. This interaction with vibrations results in the energy of incident photons being shifted up or down. The energy shift is defined by the vibrational frequency and the proportion of the inelastically scattered light is defined by the spatial derivatives of the macroscopic polarization, technical details are described by Porezag and Pederson (1996).
Spatial derivatives of the macroscopic polarization are calculated numerically along eigenvectors of each Raman active phonon mode by calculating the polarization for each displacement using a linear response formalism. Once these derivatives are known, it is straightforward to calculate the Raman cross-section through appropriate space averaging.
Raman activities defined by Porezag and Pederson (1996) characterize phonon mode contributions to the intensity of peaks in Raman spectra. These intensities depend on some other factors such as the temperature and incident light wavelength. It is important to specify these parameters in order to simulate a realistic Raman spectrum that can be compared to experimental results.
The Raman susceptibility tensor from which the Raman intensity can be calculated is defined as:
where:
χ(1) is the first order dielectric susceptibility
ν is a phonon eigenvector (the direction in which atoms, I, at equilibrium positions R, move under excitation of a phonon mode, m in a unit cell with volume V)
Greek subscripts denote Cartesian directions
The derivative of χ(1) inside the sum can be calculated in CASTEP using a finite difference approach, making small atomic displacements corresponding to the eigenvectors of each Raman active mode. However, the derivative can be written in full as:
where:
ϵ is the electric field of a photon in a Raman experiment exciting the phonon
E is the total energy of the system.
We can use the (2n+1) theorem in quantum mechanics to evaluate this in a more direct manner, see Miwa (2011). This theorem states that, under a perturbation, if the perturbed wavefunctions are known to order n then we can evaluate the perturbed energy up to order 2n+1. In particular, in a CASTEP phonon calculation, we need to evaluate the perturbed wavefunctions up to first order and therefore we should have access to the 3rd order energy perturbation, E(3), from which we can compute the susceptibility derivatives.
For Raman intensities, the full expression for this third derivative is given by equations (28)-(31) of Miwa (2011). These expressions are implemented in CASTEP for norm-conserving pseudopotentials.
The finite difference method and the "2n+1" linear response method lead to the same Raman susceptibility tensor, however the mathematical, and hence computational, routes are rather different. In the finite difference method, the phonon eigenvectors are calculated during a phonon task, and are then used to calculate the third order derivative for each mode one at a time. This means that there is a separate Raman intensity calculation for each Raman active mode to obtain its intensity. Computationally, this is usually the most time consuming part of a Raman intensity calculation.
Alternatively, in the linear response method, we first precompute the 9 second order derivatives with respect to electric field: equation 19 of Miwa (2011) and then construct the full set of Raman tensors during the phonon perturbations: equation (28) of Miwa (2011). Computationally, this means that in the linear response Raman intensity method there is an initialization stage and each phonon perturbation calculation has slightly more work to do than in the finite difference method. However, the computational work is then complete, unlike in the finite difference method where additional electric field perturbations are required for every Raman intensity.
For small cells the linear response method will be slower as the initialization stage and the extra cost of each phonon perturbation does not outweigh the effort of calculating the finite difference Raman tensor. For example, a linear response calculation for the primitive cell of boron nitride in the diamond structure takes about twice as long as the finite displacement calculation. However, the linear response method quickly becomes significantly more efficient as the number of degrees of freedom increases. Calculations for molecular crystals containing 50-100 atoms per unit cell can be an order of magnitude faster using the linear response method compared to the finite difference method.
It is important to remember that Raman intensities are effectively third order derivatives, so to obtain reasonable results very accurate calculations are required. Both the energy cutoff and k-point sampling can contribute significantly to a convergence error, so a careful convergence test is necessary to obtain reliable Raman intensities.