# 2. Electronic Structure Theories

## 2.1. Density-Functional Theory

Excited state calculations require a reference ground state calculation within density-functional theory. VOTCA-XTP provides both an automated interface to the ORCA package [Neese:2012] and a lightweight internal DFT engine based on atom-centered Gaussian-type orbitals for method developing and testing. It solves the Kohn-Sham Equations for the molecular orbitals \(\phi_n^\textrm{KS}(\mathbf{r})\) with orbital energies \(\varepsilon_n^\textrm{KS}\)

(2.1)\[\left\{ -\frac{\hbar^2}{2m}\nabla^2 + V_\mathrm{ext}(\mathbf{r}) + V_\textrm{H}(\mathbf{r}) +V_\textrm{xc}(\mathbf{r})\right\}\phi_n^\textrm{KS}(\mathbf{r}) =\varepsilon_n^\textrm{KS} \phi_n^\textrm{KS}(\mathbf{r}) ,\]

where \(V_\textrm{ext}\) is the external potential, \(V_\textrm{H}\) the Hartree potential, and \(V_\textrm{xc}\) the exchange-correlation potential. VOTCA-XTP also contains functionality for projector-based-embedding DFT-in-DFT ground state calculations [Manby:2012], in which a chosen active subregion of a molecular system is embedded into an inactive one, reproducing the total energy of the full system ground state exactly.

## 2.2. Many-Body Green’s Functions and the Bethe-Salpeter Equation

Using the ground-state reference, many-body Green’s functions theory with the \(GW\) approximation first calculates single-particle excitations (electron addition or removal) as solutions to the quasiparticle equations

(2.2)\[\left\{ -\frac{\hbar^2}{2m}\nabla^2 + V_\textrm{ext}(\mathbf{r}) + V_\textrm{H}(\mathbf{r})\right\}\phi_n^\textrm{QP}(\mathbf{r}) + \int{\Sigma(\mathbf{r},\mathbf{r}',\varepsilon_n^\textrm{QP})\phi_n^\textrm{QP}(\mathbf{r}')d\mathbf{r}'} = \varepsilon_n^\textrm{QP} \phi_n^\textrm{QP}(\mathbf{r}) .\]

In place of the exchange-correlation potential in Eq.2.1, the energy-dependent self-energy operator \(\Sigma(\mathbf{r},\mathbf{r}',E)\) occurs in the QP equations. This operator is evaluated using the one-body Green’s function in quasi-particle approximation

(2.3)\[G(\mathbf{r},\mathbf{r}',\omega) = \sum_n{\frac{\phi_n(\mathbf{r})\phi_n^*(\mathbf{r}')}{\omega-\varepsilon_n+i0^+\textrm{sgn}(\varepsilon_n -\mu)}}\]

as

(2.4)\[\Sigma(\mathbf{r},\mathbf{r}',E) = \frac{i}{2\pi} \int{e^{-i\omega 0^+}G(\mathbf{r},\mathbf{r}',E-\omega)W(\mathbf{r},\mathbf{r}',\omega)\,d\omega},\]

where \(W\) denotes the dynamically screened Coulomb interaction. Assuming that \(\phi^\textrm{QP}_n\approx \phi^\textrm{KS}_n\), the quasiparticle energies can be evaluated perturbatively according to

(2.5)\[\varepsilon_n^\textrm{QP}= \varepsilon_n^\textrm{KS} + \Delta \varepsilon_n^{GW} = \varepsilon_n^\textrm{KS} + \left\langle\phi^\textrm{KS}_n\left\vert \Sigma(\varepsilon_n^\textrm{QP})-V_\text{xc} \right\vert\phi^\textrm{KS}_n\right\rangle .\]

As the correction \(\Delta \varepsilon_n^{GW}\) itself depends on \(\varepsilon_n^\textrm{QP}\), Eq.2.5 needs to be solved self-consistently.

Neutral excitations with a conserved number of electrons can be obtained from the Bethe-Salpeter Equation (BSE) by expressing coupled electron-hole amplitudes of excitation \(S\) in a product basis of single-particle orbitals, i.e.,

(2.6)\[\chi_S(\mathbf{r}_\textrm{e},\mathbf{r}_\textrm{h})=\sum_{v}^{\mathrm{occ}}\sum_c^{\mathrm{unocc}}A_{vc}^S\phi_{c}(\mathbf{r}_\textrm{e})\phi^*_{v}(\mathbf{r}_\textrm{h})+B_{vc}^S\phi_{v}(\mathbf{r}_\textrm{e})\phi^{*}_{c}(\mathbf{r}_\textrm{h}),\]

where \(\mathbf{r}_\textrm{e}\) (\(\mathbf{r}_\textrm{h}\)) is for the electron (hole) coordinate and \(A_{vc}\) (\(B_{vc}\)) are the expansion coefficients of the excited state wave function in terms of resonant (anti-resonant) transitions between occupied \(v\) and unoccupied \(c\) states, respectively. In this basis, the BSE turns into an effective two-particle Hamiltonian problem of the form

(2.7)\[\begin{split}\begin{pmatrix} \underline{\mathbf{H}}^{\text{res}}&\underline{\mathbf{K}} \\ -\underline{\mathbf{K}} & -\underline{\mathbf{H}}^{\text{res}} \end{pmatrix} \begin{pmatrix} \mathbf{A}^S\\ \mathbf{B}^S \end{pmatrix} =\Omega_S \begin{pmatrix} \mathbf{A}^S\\ \mathbf{B}^S \end{pmatrix}.\end{split}\]

Specifically, the matrix elements of the blocks \(\underline{\mathbf{H}}^{\text{res}}\) and \(\underline{\mathbf{K}}\) are calculated as

(2.8)\[\begin{split}\begin{align} H^{\text{res}}_{vc,v'c'}&=D_{vc,v'c'}+\eta K^\mathrm{x}_{vc,v'c'}+K^\mathrm{d}_{vc,v'c'}\\ K_{cv,v'c'}&=\eta K^\mathrm{x}_{cv,v'c'}+K^\mathrm{d}_{cv,v'c'}\, , \end{align}\end{split}\]

with

(2.9)\[\begin{split}\begin{align} D_{vc,v'c'}&=(\varepsilon_c-\varepsilon_v)\delta_{vv'}\delta_{cc'},\\ K^\text{x}_{vc,v'c'}&=\iint \phi_c^*(\mathbf{r}_\textrm{e})\phi_v(\mathbf{r}_\textrm{e})v_{\mathrm{C}}(\mathbf{r}_\textrm{e},\mathbf{r}_\textrm{h}) \phi_{c'}(\mathbf{r}_\textrm{h})\phi_{v'}^*(\mathbf{r}_\textrm{h}) d^3\mathbf{r}_\textrm{e} d^3\mathbf{r}_\textrm{h}\\ K^\text{d}_{vc,v'c'}&=-\iint \phi_c^*(\mathbf{r}_\textrm{e})\phi_{c'}(\mathbf{r}_\textrm{e})W(\mathbf{r}_\textrm{e},\mathbf{r}_\textrm{h},\omega=0) \phi_v(\mathbf{r}_\textrm{h})\phi_{v'}^*(\mathbf{r}_\textrm{h})d^3\mathbf{r}_\textrm{e} d^3\mathbf{r}_\textrm{h} \, . \end{align}\end{split}\]

and \(\eta=2\) (\(\eta=0\)) for singlet (triplet) excitations. Here, \(K^\text{x}\) is the repulsive exchange interaction originating from the bare Coulomb term \(v_\mathrm{C}\), while the direct interaction \(K^\text{d}\) contains the attractive, but screened, interaction \(W\) between electron and hole, causing the binding of the electron-hole pair. In Eq.2.9 it is assumed that the dynamic properties of \(W(\omega)\) are negligible, and the computationally less demanding static approximation \(\omega=0\) is employed.