Category:Time-dependent density functional theory: Difference between revisions

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 ^[1][2]. 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 perturbation:

[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) }[/math] is:

[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} }[/math]:

[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{ \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) }[/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 ^[3]

[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{ 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. }[/math]

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

[math]\displaystyle{ \begin{aligned} 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 \\ &\quad- \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{aligned} }[/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.
• Neglecting only the [math]\displaystyle{ f_{\rm xc} }[/math] gives the dielectric function in the Random Phase Approximation (RPA)
• If [math]\displaystyle{ f_{\rm xc} }[/math] is local, i.e., LDA or PBE, no excitonic effects are included
• If [math]\displaystyle{ f_{\rm xc} }[/math] includes the non-local exact exchange terms, for example PBE0, the dielectric function is described within TDPBE0 approximation and takes into account the electron-hole interaction, i.e., excitonic effects, with approximate screening described by the fraction of the exact exchange, i.e, 0.25.

Additional resources

Lectures

• Lecture on TDDFT.

Tutorials

• Tutorial calculating optical absorption of C diamond using TDDDH.
• Tutorial on the efficient Brillouin zone sampling using TDDDH and exciton analysis using TDDDH.

How to

• Practical guide for solving the BSE via diagonalization BSE calculations

References