Computing the Optical Properties of Large Systems

Linear-scaling time-dependent density-functional theory beyond the Tamm-Dancoff approximation: Obtaining efficiency and accuracy with in situ optimised local orbitals

“…Here, the elements of S ϕφ are given by S ϕφ αβ = ϕ α |φ β . While using the joint representation for the unoccupied subspace instead of {χ α } roughly doubles the computational cost compared to using {χ α }, it yields consistently good TDDFT excitation energies [31,32] and will be used throughout in Sec. IV.…”

“…While it is possible to iteratively apply the projection at the end of each conjugate gradient step until some measure of the violation of the invariance constraint is kept below a certain threshold [20,32], in practice this can lead to convergence problems as it destroys the variational nature of the algorithm presented here. This problem is overcome by introducing a set of auxiliary density matrices L {q} and L {q} [32] in the spirit of ground state linear-scaling approaches [38]. The auxiliary matrices can be arbitrarily truncated and are used to define the real density matrices P {q} and P {q} used in the algorithm via…”

“…at every step of the calculation. While this scheme comes at a computational cost as P {p} is less sparse than L {p} , it guarantees that every point in the algorithm, P {q} and P {q} fulfil their respective invariance constraints to the degree that P {v} and P {c} fulfil their respective idempotency constraint [32]. Since linear-scaling DFT calculations employ a number of techniques that ensure that for sensible truncation schemes, P {v} and P {c} retain nearidempotency [38], the above scheme yields a robust convergence of TDDFT calculations when truncations are applied to the respective response density matrices.…”

Predicting solvatochromic shifts and colours of a solvated organic dye: The example of nile red

“…As discussed in previous work [11,51], the representation of the TDDFT response density matrix in terms of localised atom-centered orbitals {χ β } and {φ α } allows us remove certain unwanted excitations from the subspace of solutions to the TDDFT eigenvalue problem by placing appropriate constraints on the sparsity pattern of the response density matrix.…”

Oxygen K-edge X-ray Absorption Spectra

“…Liang et al reported the oxygen K edge spectra of several transition metal oxides obtained through the BSE procedure and they showed the accuracy of the method in predicting the excitonic character of some peaks. The reproduction of the correct excitonic features of the spectra, is insured in BSE by the screened Coulomb interaction W, present in the interaction kernel, that acts on the electron hole matrices and introduces a long range coupling between the electron and the hole wave functions [46].…”