Category:Time-dependent density functional theory

The formalism of the Bethe-Salpeter equation (BSE) allows for calculating the polarizability with the electron-hole interaction and constitutes the state of the art for calculating absorption spectra in solids.

Theory

Time-dependent density-functional theory (TDDFT) is an extension of DFT to address excited-state properties, dynamics, and spectroscopy. In principle, TDDFT is an exact theory for neutral electronic excitations, however, similarly to DFT, the exchange-correlation functional is unknown and needs to be approximated.

In the linear response approximation, we split the external potential into a static term and a time-dependent perturbation [math]\displaystyle{ v(r,t) =v(r)+\delta v(r,t), }[/math] where the perturbation term is much smaller than the static potential [math]\displaystyle{ \delta v(r,t) \ll v(r) }[/math]. In this case the Hohenberg–Kohn and Runge–Gross theorems state the correspondence [math]\displaystyle{ \delta \rho(r,t) \Leftrightarrow \delta v(r,t) }[/math]. A TDDFT calculation is a two-step procedure: first, we perform an ordinary DFT calculation with a static external potential [math]\displaystyle{ v(r) }[/math] and then we perform a TDDFT calculation of the density variation [math]\displaystyle{ \delta \rho(r,t) }[/math] corresponding to the external time-dependent perturbation [math]\displaystyle{ \delta v(r,t) }[/math]. From [math]\displaystyle{ \delta \rho(r,t) }[/math] we can calculate the polarizability of the system [math]\displaystyle{ \chi }[/math] using [math]\displaystyle{ \delta \rho(r_1,t_1)= \int dr_2dt_2 \chi(r_1,t_1,r_2,t_2)\delta v(r_2,t_2). }[/math]

Following a Kohn-Sham (KS) scheme we assume that the density response of KS system is equivalent to that of the real system, i.e., [math]\displaystyle{ \delta \rho = \delta \rho^{\rm KS} }[/math], in response to an effective KS petrubation

[math]\displaystyle{ \delta v^{\mathrm{KS}}(x)=\delta v(x)+\delta v_{\mathrm{H}}(x)+\delta v_{\mathrm{xc}}(x). }[/math] Here, [math]\displaystyle{ \delta v(r_1,t_1) }[/math] is the real external perturbation, the Hartree term [math]\displaystyle{ \delta v_H(x) }[/math] is [math]\displaystyle{ \delta v_{\mathrm{H}}(x)=\int dr_2dt_2 V(r_1,t_1,r_2,t_2)\delta \rho(r_2,t_2) }[/math] and the exchange-correlation term [math]\displaystyle{ \delta v_{\mathrm{xc}}(x)=\int dr_2dt_2 f_{\rm xc}[\rho](r_1,t_1,r_2,t_2)\delta \rho(r_2,t_2). }[/math] The main challenge lays in finding an accurate approximation for the exchange-correlation kernel [math]\displaystyle{ f_{\rm xc}[\rho](r_1,t_1,r_2,t_2)=\frac{\delta v_{\rm xc}[\rho](r_1,t_1)}{\delta \rho(r_2,t_2)} }[/math]

The response of the non-interacting KS particles is then [math]\displaystyle{ \delta \rho(r_1,t_1)= \int dr_2dt_2 \chi^{\rm KS}(r_1,t_1,r_2,t_2)\delta v^{\rm KS}(r_2,t_2). }[/math]

Then, writing the Adler-Wiser expression in reciprocal space and frequency domain we can find the response function of the KS system

[math]\displaystyle{ \begin{gathered} \chi_{\mathbf{G}, \mathbf{G}^{\prime}}^{K S}(\mathbf{q}, \omega) =-\frac{1}{V} \sum_{n \mathbf{k}} \sum_{m \mathbf{k}^{\prime}} 2 f_{n \mathbf{k}}\left(1-f_{m \mathbf{k}^{\prime}}\right)\left(\frac{\left\langle m \mathbf{k}^{\prime}\left|e^{i(\mathbf{q}+\mathbf{G}) \mathbf{r}}\right| n \mathbf{k}\right\rangle\left\langle n \mathbf{k}\left|e^{-i\left(\mathbf{q}+\mathbf{G}^{\prime}\right) \mathbf{r}^{\prime}}\right| m \mathbf{k}^{\prime}\right\rangle}{\epsilon_{m \mathbf{k}^{\prime}}-\epsilon_{n \mathbf{k}}-\bar{\omega}}+\right. \\ \left.+\frac{\left\langle n \mathbf{k}\left|e^{i(\mathbf{q}+\mathbf{G}) \mathbf{r}}\right| m \mathbf{k}^{\prime}\right\rangle\left\langle m \mathbf{k}^{\prime}\left|e^{-i\left(\mathbf{q}+\mathbf{G}^{\prime}\right) \mathbf{r}^{\prime}}\right| n \mathbf{k}\right\rangle}{\epsilon_{m \mathbf{k}^{\prime}}-\epsilon_{n \mathbf{k}}+\bar{\omega}}\right) \\ \end{gathered} }[/math]

and using the Dyson equation for the polarizability we find the polarizability of the real system

[math]\displaystyle{ \chi = \chi^{\rm KS}+ \chi^{\rm KS}(v+f_{\rm xc})\chi. }[/math]

The exciation frequencies of the system can be extracted from the analytic structure of polarizability [math]\displaystyle{ \chi }[/math].

Finally, the dielectric function is found [math]\displaystyle{ \varepsilon^{-1}_{\mathbf{G}, \mathbf{G}^{\prime}}(\mathbf{q}, \omega)= \delta_{\mathbf{G}, \mathbf{G}^{\prime}}+\frac{4 \pi e^2}{|\mathbf{G}+\mathbf{q}|\left|\mathbf{G}^{\prime}+\mathbf{q}\right|} \chi_{\mathbf{G}, \mathbf{G}^{\prime}}(\mathbf{q}, \omega) }[/math]

Casida equation Alternatively, the excitation energies [math]\displaystyle{ \omega_\lambda }[/math] of the real system can be found by mapping Eq. (1) onto an eigenvalue problem

[math]\displaystyle{ \left(\begin{array}{cc} A & B \\ B^* & A^* \end{array}\right)\left(\begin{array}{l} X_\lambda \\ Y_\lambda \end{array}\right)=\omega_\lambda\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)\left(\begin{array}{l} X_\lambda \\ Y_\lambda \end{array}\right) }[/math]

The structure of the Casida equation is very similar to that of the Bethe-Salpeter equation. And similarly to BSE, the standard way to solve the Casida equation is to to neglect the coupling terms [math]\displaystyle{ B }[/math] and [math]\displaystyle{ B^* }[/math], i.e., the Tamm-Dancoff approximation

[math]\displaystyle{ AX_\lambda=\omega_\lambda X_\lambda~. }[/math]

The interaction between in TDDFT is described by the bare Coulomb [math]\displaystyle{ V_\mathbf{G} }[/math] and the exchange-correlation kernel

[math]\displaystyle{ \begin{gathered} A_{vc\mathbf{k}}^{v'c'\mathbf{k}'} = (\varepsilon_v-\varepsilon_c)\delta_{vv'}\delta_{cc'}\delta_{\mathbf{kk}'}+ \frac{2}{\Omega}\sum_{\mathbf{G}\neq0}\bar{V}_\mathbf{G}(\mathbf{q})\langle c\mathbf{k}|e^{i\mathbf{Gr}}|v\mathbf{k}\rangle\langle v'\mathbf{k}'|e^{-i\mathbf{Gr}}|c'\mathbf{k}'\rangle\\ + \frac{2}{\Omega}\sum_{\mathbf{G}\neq0}f^{\rm xc}_{\mathbf{G,G'}}\langle c\mathbf{k}|e^{i\mathbf{Gr}}|v\mathbf{k}\rangle\langle v'\mathbf{k}'|e^{-i\mathbf{Gr}}|c'\mathbf{k}'\rangle. \end{gathered} }[/math]

If xc potential includes the non-local exact exchange contribution, an additional term will appear

[math]\displaystyle{ \begin{gathered} A_{vc\mathbf{k}}^{v'c'\mathbf{k}'} = (\varepsilon_v-\varepsilon_c)\delta_{vv'}\delta_{cc'}\delta_{\mathbf{kk}'}+ \frac{2}{\Omega}\sum_{\mathbf{G}\neq0}\bar{V}_\mathbf{G}(\mathbf{q})\langle c\mathbf{k}|e^{i\mathbf{Gr}}|v\mathbf{k}\rangle\langle v'\mathbf{k}'|e^{-i\mathbf{Gr}}|c'\mathbf{k}'\rangle\\ + \frac{2}{\Omega}\sum_{\mathbf{G}\neq0}f^{\rm xc,loc}_{\mathbf{G,G'}}\langle c\mathbf{k}|e^{i\mathbf{Gr}}|v\mathbf{k}\rangle\langle v'\mathbf{k}'|e^{-i\mathbf{Gr}}|c'\mathbf{k}'\rangle \\-\frac{2}{\Omega}\sum_{\mathbf{G,G}'}c_{\rm x}(\mathbf{q+G})V_{\mathbf{G}}(\mathbf{q})\delta_{\mathbf{q,k-k}'} \langle c\mathbf{k}|e^{i(\mathbf{q+G})}|c'\mathbf{k}'\rangle \langle v'\mathbf{k}'|e^{-i(\mathbf{q+G})}|v\mathbf{k}\rangle. \end{gathered} }[/math]

Here, [math]\displaystyle{ c_x(\mathbf{q+G}) }[/math] is the range-dependent fraction of the exact exchange potential.

Common approximations Neglecting both interaction terms, i.e., [math]\displaystyle{ v }[/math] and [math]\displaystyle{ f_{\rm xc} }[/math] yields the independent particle approximation.

Bethe-Salpeter equation

In the BSE, the excitation energies correspond to the eigenvalues [math]\displaystyle{ \omega_\lambda }[/math] of the following linear problem

[math]\displaystyle{ \left(\begin{array}{cc} \mathbf{A} & \mathbf{B} \\ \mathbf{B}^* & \mathbf{A}^* \end{array}\right)\left(\begin{array}{l} \mathbf{X}_\lambda \\ \mathbf{Y}_\lambda \end{array}\right)=\omega_\lambda\left(\begin{array}{cc} \mathbf{1} & \mathbf{0} \\ \mathbf{0} & -\mathbf{1} \end{array}\right)\left(\begin{array}{l} \mathbf{X}_\lambda \\ \mathbf{Y}_\lambda \end{array}\right)~. }[/math]

The matrices [math]\displaystyle{ A }[/math] and [math]\displaystyle{ A^* }[/math] describe the resonant and anti-resonant transitions between the occupied [math]\displaystyle{ v,v' }[/math] and unoccupied [math]\displaystyle{ c,c' }[/math] states

[math]\displaystyle{ A_{vc}^{v'c'} = (\varepsilon_v-\varepsilon_c)\delta_{vv'}\delta_{cc'} + \langle cv'|V|vc'\rangle - \langle cv'|W|c'v\rangle. }[/math]

The energies and orbitals of these states are usually obtained in a [math]\displaystyle{ G_0W_0 }[/math] calculation, but DFT and Hybrid functional calculations can be used as well. The electron-electron interaction and electron-hole interaction are described via the bare Coulomb [math]\displaystyle{ V }[/math] and the screened potential [math]\displaystyle{ W }[/math].

The coupling between resonant and anti-resonant terms is described via terms [math]\displaystyle{ B }[/math] and [math]\displaystyle{ B^* }[/math]

[math]\displaystyle{ B_{vc}^{v'c'} = \langle vv'|V|cc'\rangle - \langle vv'|W|c'c\rangle. }[/math]

Due to the presence of this coupling, the Bethe-Salpeter Hamiltonian is non-Hermitian.

Tamm-Dancoff approximation

A common approximation to the BSE is the Tamm-Dancoff approximation (TDA), which neglects the coupling between resonant and anti-resonant terms, i.e., [math]\displaystyle{ B }[/math] and [math]\displaystyle{ B^* }[/math]. Hence, the TDA reduces the BSE to a Hermitian problem

[math]\displaystyle{ AX_\lambda=\omega_\lambda X_\lambda~. }[/math]

In reciprocal space, the matrix [math]\displaystyle{ A }[/math] is written as

[math]\displaystyle{ A_{vc\mathbf{k}}^{v'c'\mathbf{k}'} = (\varepsilon_v-\varepsilon_c)\delta_{vv'}\delta_{cc'}\delta_{\mathbf{kk}'}+ \frac{2}{\Omega}\sum_{\mathbf{G}\neq0}\bar{V}_\mathbf{G}(\mathbf{q})\langle c\mathbf{k}|e^{i\mathbf{Gr}}|v\mathbf{k}\rangle\langle v'\mathbf{k}'|e^{-i\mathbf{Gr}}|c'\mathbf{k}'\rangle -\frac{2}{\Omega}\sum_{\mathbf{G,G}'}W_{\mathbf{G,G}'}(\mathbf{q},\omega)\delta_{\mathbf{q,k-k}'} \langle c\mathbf{k}|e^{i(\mathbf{q+G})}|c'\mathbf{k}'\rangle \langle v'\mathbf{k}'|e^{-i(\mathbf{q+G})}|v\mathbf{k}\rangle, }[/math]

where [math]\displaystyle{ \Omega }[/math] is the cell volume, [math]\displaystyle{ \bar{V} }[/math] is the bare Coulomb potential without the long-range part

[math]\displaystyle{ \bar{V}_{\mathbf{G}}(\mathbf{q})=\begin{cases} 0 & \text { if } G=0 \\ V_{\mathbf{G}}(\mathbf{q})=\frac{4 \pi}{|\mathbf{q}+\mathbf{G}|^2} & \text { else } \end{cases}~, }[/math]

and the screened Coulomb potential [math]\displaystyle{ W_{\mathbf{G}, \mathbf{G}^{\prime}}(\mathbf{q}, \omega)=\frac{4 \pi \epsilon_{\mathbf{G}, \mathbf{G}^{\prime}}^{-1}(\mathbf{q}, \omega)}{|\mathbf{q}+\mathbf{G}|\left|\mathbf{q}+\mathbf{G}^{\prime}\right|}. }[/math]

Here, the dielectric function [math]\displaystyle{ \epsilon_\mathbf{G,G'}(\mathbf{q}) }[/math] describes the screening in [math]\displaystyle{ W }[/math] within the random-phase approximation (RPA)

[math]\displaystyle{ \epsilon_{\mathbf{G}, \mathbf{G}^{\prime}}^{-1}(\mathbf{q}, \omega)=\delta_{\mathbf{G}, \mathbf{G}^{\prime}}+\frac{4 \pi}{|\mathbf{q}+\mathbf{G}|^2} \chi_{\mathbf{G}, \mathbf{G}^{\prime}}^{\mathrm{RPA}}(\mathbf{q}, \omega). }[/math]

Although the dielectric function is frequency-dependent, the static approximation [math]\displaystyle{ W_{\mathbf{G}, \mathbf{G}^{\prime}}(\mathbf{q}, \omega=0) }[/math] is considered a standard for practical BSE calculations.

Macroscopic dielectric function

The macroscopic dielectric which accounts for the excitonic effects is found via eigenvalues [math]\displaystyle{ \omega_\lambda }[/math] and eigenvectors [math]\displaystyle{ X_\lambda }[/math] of the BSE

[math]\displaystyle{ \epsilon_M(\mathbf{q}=0,\omega)= 1-\lim_{\mathbf{q}\rightarrow 0}v(q)\sum_{\lambda} \left|\sum_{c,v,k}\langle c\mathbf{k}|e^{i\mathbf{qr}}|v\mathbf{k}\rangle X_\lambda^{cv\mathbf{k}}\right|^2 \times \left(\frac{1}{\omega - \omega_\lambda + i\delta}\right)~. }[/math]

Scaling

The scaling of the BSE equation strongly limits its application for large systems. The main limiting factor is the diagonalization of the BSE Hamiltonian. The rank of the Hamiltonian is

[math]\displaystyle{ N_{\rm rank} = N_k\times N_c\times N_v }[/math],

where [math]\displaystyle{ N_k }[/math] is the number of k-points in the Brillouin zone and [math]\displaystyle{ N_c }[/math] and [math]\displaystyle{ N_v }[/math] are the number of conduction and valence bands, respectively. The diagonalization of the matrix scales cubically with the matrix rank, i.e., [math]\displaystyle{ N_{\rm rank}^3 }[/math].

Despite the fact that this matrix diagonalization is usually the bottleneck for bigger systems, the construction of the BSE Hamiltonian also scales unfavorably and can play a dominant role in big systems, i.e.,

[math]\displaystyle{ N_k\times N_q\times (N_v\times N_v\times N_G\times N_c\times N_c) }[/math],

where [math]\displaystyle{ N_q }[/math] is the number of q-points and [math]\displaystyle{ N_G }[/math] number of G-vectors.

Solution of BSE

Diagonalization

The exact diagonalization of the BSE Hamiltonian can be perform using various eigensolvers provided in ScaLAPACK, ELPA, and cuSolver libraries. The advantage of this approach is that the eigenvectors can be directly obtained and used for the analysis of the excitons.

The following features are currently supported:

Time evolution

The alternative approach is to formulate the BSE as the initial-value problem for the macroscopic polarizability. This approach converges to the same solution as the exact diagonalization and can be used for obtaining the absorption spectrum, but does not yield the eigenvectors, which can be limiting for the analysis of the excitons. The advantage of this approach is the quadratic scaling with the size of the BSE Hamiltonian [math]\displaystyle{ N_{rank}^2 }[/math].

The following features are currently supported:

Obtaining the spectra
Calculations beyond Tamm-Dancoff approximation
Calculations of [math]\displaystyle{ \varepsilon(\mathbf{q},\omega) }[/math] for [math]\displaystyle{ \mathbf{q}\neq0 }[/math]

How to

Practical guide for solving the BSE via diagonalization BSE calculations

References

Pages in category "Time-dependent density functional theory"

The following 3 pages are in this category, out of 3 total.

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