Testbed implementation of TDDFT

Milestone 1 - *Implement a straightforward scheme based on the existing DFPT module code in CASTEP to compute the electronic response to an external electric field of a set frequency. This will provide a reference calculation against which the later, more sophisticated calculations may be benchmarked. Test results on small molecular systems will be compared against previous calculations published in scientific literature. This work comprises stages 1 and 2 of the work plan.*

The above refers to equation 17 of Hutter's paper [6], which describes the first-order response^{1} of electrons to an electric field at a particular frequency. The density functional perturbation theory implementation in CASTEP[9] already had much of the functionality required for this. Hutter's equation 17 is:

In the Tamm-Dancoff approximation[4], occupied-virtual contributions to equation (1) are disregarded but the virtual-ocupied ones are kept, under the assumption that the contribution from the former is small. This amounts to setting
, so Hutter's equation 17 becomes

(2) |

`excited_state_scissors`

parameter.
In the `secondd`

module, we have two different methods available for calculating the response. Namely, a variational solver, and a Green's function solver. The variational solver is more stable than the Green's function solver when approaching the first excitation energy. However, the variational solver cannot converge for values of above the first excitation energy, whether the polarisability is negative or otherwise.

We chose an isolated methane molecule for our test system. To get the energy of the excited states, we calculate the polarisability at a number of values, and extrapolate for the divergence. Extensive tests were performed to investigate convergence of the first excitation energy with respect to plane wave cutoff energy and size of supercell. To get the excitation energy to two decimal places, a cutoff energy of 750 eV^{2} and a cell 21 Å was required.

We found the first excitation energy to be 9.10 eV, using the LDA. This compares well with a literature value of 9.053 eV [8], where they used GAUSSIAN 98. As we are using the Tamm-Dancoff approximation, this value for the first excitation energy can be compared directly to that obtained by a direct calculation of the poles, covered in the next section.