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 response¹ 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:

$$\left( H^{(0)} - \epsilon_{i} \right)\left\vert \Phi_{i}^{\l... ...p\omega\left\vert \Phi_{i}^{\left( \pm \right)} \right\rangle,$$ (1)

where $H^{(0)}$ is the ground state (Kohn-Sham) Hamiltonian, $\epsilon_{i}$ are the ground state Kohn-Sham eigenvalues for band index $i$, $\vert \Phi_{i}^{\left( \pm \right)} \rangle$ are the response wavefunctions to a perturbation of frequency $\omega$, $P_{c}$ is the projector on the subspace of unperturbed unoccupied states, $\vert \Phi_{i}^{\left( 0 \right)} \rangle$ are the ground state Kohn Sham orbitals and $V^{(1)}(\pm\omega)$ is the response potential, containing contributions from the Hartree, exchange-correlation, and electric field perterbation terms. In the above we have neglected reference to electronic spin, for brevity. The reponse wavefunctions throughout this report are taken to be orthogonal to the occupied ground state, so there is an implied projector.

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 $\vert \Phi_{i}^{\left( + \right)} \rangle = 0$, so Hutter's equation 17 becomes

$$\left( H^{(0)} - \epsilon_{i} \right)\left\vert \Phi_{i}^{\l... ... = \omega\left\vert \Phi_{i}^{\left( - \right)} \right\rangle,$$ (2)

which is the equation implemented for electric field response in CASTEP (with the right hand side normally zero). A chosen frequency, $\omega$, can be set through the 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 $\omega$ 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 $\omega$ 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 eV² and a cell 21 Å$^3$ 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.