Abstract
In this chapter, the focus lies on the approximate evaluation of excited states in the many-electron system. The discussion is limited to neutral excitations in which the many-electron system couples to light by absorbing photons of specific wavelengths.
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Notes
- 1.
Here, we limit the discussion to vertical excitations of closed-shell systems. DFT can be used to yield excitation energies in cases where different spin configurations yield different ground state energies. The energy difference between two different configurations can then be interpreted as the excitation energy of the system going from one symmetry to the other.
- 2.
- 3.
In principle, requiring the electron density to vanish at infinity means that the proof of the Runge–Gross theorem presented here is not valid for infinite systems. This limitation is avoided in TDCDFT, as the mapping between the current density and external potential can be proven without relying on any quantity vanishing at infinity (see Eq. (3.9)).
- 4.
The quantity in brackets linking \(\rho ^{\{1\}}_{ij}(\omega )\) and \(\delta V^\mathrm{{{pert}}}_{ij}(\omega )\) actually corresponds to the inverse of \(\chi (\omega )\) in matrix form.
- 5.
Here, the factor of 2 originates from an implied summation over spin indices. The spin structure of the equation is discussed in some more detail in Sect. 3.2.5.
- 6.
Note that such a measure of basis set quality requires the calculation of all possible excitations and thus a full diagonalisation of the entire 2-particle Hamiltonian and is thus not practical for larger systems.
- 7.
In order to perform the inverse Fourier transform, one has to make use of Cauchy’s theorem and consider \(G(\omega )\) as a function of the complex variable \(\omega \) in the lower half of the complex plane. The contour chosen for the integration is a semicircle in the lower half plane, where the factor \(\mathrm {i}\delta \) ensures that the pole is inside the contour region.
- 8.
It should be noted that there is no full formal justification why the perturbed density matrix should show the same sparsity properties as the ground state density matrix. A more detailed discussion on the question of sparsity of response density matrices is delivered in Sect. 5.2.
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Zuehlsdorff, T.J. (2015). Approximations to Excited States. In: Computing the Optical Properties of Large Systems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-19770-8_3
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