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VASP offers a powerful module for performing time-dependent density-functional theory (TDDFT) or time-dependent Hartree-Fock (TDHF) calculations by solving the Casida equation {{cite|casida:jomst:2009}}. This approach can be used for obtaining the frequency-dependent dielectric function with the excitonic effects and can be based on the ground-state electronic structure in the DFT, hybrid-functional or even ''GW'' approximations. You can watch a lecture covering {{Video|bse:alexey:2026|TDDFT theory and calculations}} on our YouTube channel.
Time-dependent density-functional theory (TDDFT) extends [[density-functional theory]] to time-varying external potentials, enabling the computation of neutral electronic excitations and frequency-dependent response functions. VASP implements TDDFT in two complementary forms: a Casida-equation formulation based on the diagonalization of an excitonic Hamiltonian, and a real-time (RT-TDDFT) formulation based on the propagation of the Kohn-Sham orbitals after a delta-pulse perturbation. The detailed theoretical background is given in the [[Time-dependent density-functional theory|theory page]].


== Theory ==
== Theory ==


[[Time-dependent_density-functional_theory_calculations |Time-dependent density-functional theory]] (TDDFT) is an extension of DFT to address excited-state properties, dynamics, and spectroscopy {{Cite|gross:kohn:1990}}{{Cite|marques:gross:2004}}. 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.
The Casida formulation of TDDFT recasts the linear-response problem as a non-Hermitian eigenvalue problem
::<math>
\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>
with the same mathematical structure as the [[Bethe-Salpeter equation]] (BSE). The eigenvalues <math>\omega_\lambda</math> are the excitation energies, and the eigenvectors <math>\mathbf{X}_\lambda, \mathbf{Y}_\lambda</math> determine the oscillator strengths and the dielectric function with excitonic effects. The difference with respect to BSE is that the beyond-RPA part of the kernel is described by the exchange-correlation kernel <math>f_\mathrm{xc}</math> instead of the screened Coulomb interaction <math>W</math>.


In the linear response approximation, we split the external potential into a static term and a time-dependent perturbation <math display="block">v(r,t) =v(r)+\delta v(r,t),</math> where the perturbation term is much smaller than the static potential <math display="inline">\delta
== Scaling ==
v(r,t) \ll v(r)</math>. In this case the Hohenberg–Kohn and Runge–Gross theorems state the correspondence <math display="inline">\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 display="inline">v(r)</math> and then we perform a TDDFT calculation of the density variation <math display="inline">\delta \rho(r,t)</math> corresponding to the external time-dependent perturbation <math display="inline">\delta v(r,t)</math>. From <math display="inline">\delta \rho(r,t)</math> we can calculate the polarizability of the system <math display="inline">\chi</math> using <math display="block">\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 display="inline">\delta \rho = \delta \rho^{\rm
The size of the excitonic Hamiltonian is
KS}</math>, in response to an effective KS perturbation:
::<math>N_{\rm rank} = N_k \times N_c \times N_v,</math>
where <math>N_k</math>, <math>N_c</math>, and <math>N_v</math> are the number of '''k''' points, conduction bands, and valence bands. Building the Hamiltonian scales as <math>N^4</math>--<math>N^5</math> with the system size. Solving the resulting eigenvalue problem by exact diagonalization scales as <math>N_{\rm rank}^3</math>, or as <math>N^6</math> with the system size. The real-time propagation alternative avoids diagonalization entirely and scales as <math>N_{\rm rank}^2</math>, or as <math>N^4</math> with the system size, making it the method of choice for large systems with many bands or '''k''' points.


<math display="block">\delta v^{\mathrm{KS}}(x)=\delta v(x)+\delta v_{\mathrm{H}}(x)+\delta
{| class="wikitable"
v_{\mathrm{xc}}(x).</math>  
|-
! !! Time evolution !! Casida
|-
| Memory || <math>N_{\mathbf{k}} \times (N_v + N_c) \times N_G</math> || <math>(N_{\mathbf{k}} \times N_v \times N_c)^2</math>
|-
| Compute time || <math>N_{\mathbf{k}} \times N_v \times N_G</math> || <math>(N_{\mathbf{k}} \times N_v \times N_c)^3</math>
|-
| || <math>+ N_{\mathbf{k}} \times N_v \times N_c \times N_G</math> ||
|-
| Nonlocal exchange || <math>+ N_{\mathbf{k}}^2 \times N_v^2 \times N_G</math> || ⋯
|}


Here, <math display="inline">\delta v(r_1,t_1)</math> is the real external perturbation, the Hartree term <math display="inline">\delta v_H(x)</math> is:
== Casida TDDFT ==


<math display="block">\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>  
The Casida equation is solved in VASP by setting {{TAG|ALGO}}{{=}}TDHF. The excitation energies and oscillator strengths are obtained directly from the eigenvalues and eigenvectors of the excitonic Hamiltonian, which makes this approach particularly useful for analyzing individual excitons. Casida TDDFT can be performed using DFT or [[hybrid functional|hybrid-functional]] orbitals and eigenvalues. The macroscopic dielectric function then reads
::<math>
\epsilon_M(\mathbf q, \omega) = 1 + 2 \lim_{\mathbf q \to 0} v(\mathbf q) \sum_\lambda \left| \sum_{c v \mathbf k} \langle c \mathbf k | e^{\mathrm i \mathbf{qr}} | v \mathbf k\rangle X_\lambda^{c v \mathbf k}\right|^2 \left(\frac{1}{\omega_\lambda - \omega - \mathrm i \delta}\right).
</math>


and the exchange-correlation term <math display="block">\delta v_{\mathrm{xc}}(x)</math> is:
The following features are currently supported:
* [[TDDFT calculations|Calculating the dielectric function and eigenvectors]]
* [[TDDFT calculations#Tamm-Dancoff approximation|Tamm-Dancoff approximation]]
* Calculations with [[hybrid functional|hybrid functionals]] and range-separated hybrids


<math display="block">\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>
== Time-evolution TDDFT (Real-time TDDFT) ==


The main challenge lays in finding an accurate approximation for the exchange-correlation kernel <math display="block">f_{\rm xc}</math>:
Real-time TDDFT (RT-TDDFT) is selected with {{TAG|ALGO}}{{=}}TIMEEV. The ground-state Kohn-Sham orbitals are perturbed by a Dirac delta pulse of the electric field
 
::<math>
<math display="block">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>
v_\mathrm{ext}(\mathbf r, t) = \lambda \, \mathbf r \cdot \mathbf D \, \delta(t),
 
</math>
The response of the non-interacting KS particles is then:
which simultaneously excites all valence-to-conduction transitions. The time-dependent dipole moments are then propagated and Fourier-transformed to yield the macroscopic dielectric function{{cite|sander:jcp:2017}}
 
::<math>
<math display="block">\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>
\epsilon_M(\omega)=1-\frac{4\pi}{\Omega}\int_0^{\infty} \mathrm{d} t \sum_{c,v,\mathbf{k}}\left(\langle\mu_{cv\mathbf{k}}| \xi_{cv\mathbf{k}}(t)\rangle + \mathrm{c.c.}\right) e^{-\mathrm i(\omega-\mathrm i \delta) t},
 
Then, writing the Adler-Wiser expression in reciprocal space and frequency domain we can find the response function of the KS system
 
<math>
\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>
</math>
where <math>\mu_{c v \mathbf k}</math> are the dipole moments and <math>|\xi_{cv\mathbf k}(t)\rangle</math> is the time-evolved dipole vector. The solution is strictly equivalent to that of the Casida equation for the dielectric function, but does not yield eigenvectors and so cannot be used directly for exciton analysis. Its main advantage is the quadratic scaling with <math>N_{\rm rank}</math>.


and using the Dyson equation for the polarizability we find the polarizability of the real system
The required number of propagation steps is controlled by the broadening {{TAG|CSHIFT}} and the maximum energy {{TAG|OMEGAMAX}}, and does not depend on the size of the Hamiltonian.
 
<math display="block">\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 display="inline">\chi</math>. Finally, the dielectric function is found:


<math display="block">\varepsilon^{-1}_{\mathbf{G}, \mathbf{G}^{\prime}}(\mathbf{q}, \omega)=
The following features are currently supported:
\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>
* Calculating the dielectric function in the [[independent-particle approximation]], [[random phase approximation|RPA]], and full TDDFT
* Calculations with local kernels, [[hybrid functional|hybrid functionals]], and range-separated hybrids
* [[Improving the dielectric function#Model-BSE|Model-screened ladder diagrams]] via {{TAG|LADDER}}


== Casida equation ==
== Exchange-correlation kernel ==
Alternatively, the excitation energies <math display="inline">\omega_\lambda</math> of the real system can be found by mapping Eq. (1) onto an eigenvalue problem {{cite|sander:jcp:2017}}


<math display="block">\left(\begin{array}{cc}
The exchange-correlation kernel <math>f_\mathrm{xc}</math> determines how electron-hole interactions beyond the [[random phase approximation]] are described in TDDFT. The choice of kernel is tied to the exchange-correlation functional used in the ground-state calculation and is controlled by the tags {{TAG|LHARTREE}}, {{TAG|LADDER}}, and {{TAG|LFXC}}.
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 display="inline">B</math> and <math display="inline">B^*</math>, i.e., the Tamm-Dancoff approximation
* '''Local and semilocal kernels (ALDA, APBE)''': obtained as the second functional derivative of an LDA or PBE exchange-correlation energy. These kernels are computationally cheap and work well for plasmons and metallic systems, but lack the long-range <math>-1/q^2</math> behavior and therefore fail to describe bound excitons in semiconductors and insulators.


<math display="block">AX_\lambda=\omega_\lambda X_\lambda~.</math>
* '''Hybrid-functional kernels''': when a fraction of exact exchange is included in the ground-state functional (e.g., PBE0 or HSE), the corresponding TDDFT kernel inherits a long-range non-local exchange contribution. This restores the <math>-1/q^2</math> behavior and allows for an approximate description of excitonic effects. The fraction of exact exchange is controlled by {{TAG|AEXX}}, and the range-separation parameter by {{TAG|HFSCREEN}}. When a hybrid functional is used, {{TAG|LADDER}} must be set to .TRUE. so that the non-local exchange contribution of the kernel (the ladder diagrams) is actually included in the time propagation; otherwise the calculation only contains the local part of <math>f_\mathrm{xc}</math> and the excitonic effects from the hybrid are lost.


The interaction between in TDDFT is described by the bare Coulomb <math display="inline">V_\mathbf{G}</math> and the exchange-correlation kernel
* '''Screened exchange (ladder diagrams)''': enabling {{TAG|LADDER}} adds the screened Coulomb interaction <math>W</math>, yielding the proper long-range electron-hole attraction. In the time-evolution implementation, <math>W</math> is currently obtained from a [[Improving the dielectric function#Model-BSE|model dielectric function]] controlled by {{TAG|LMODELHF}}, {{TAG|AEXX}}, and {{TAG|HFSCREEN}}.


<math>
The combination of these three switches selects the approximation level: setting all three to .FALSE. yields the [[independent-particle approximation]] (equivalent to {{TAG|LOPTICS}}); enabling only {{TAG|LHARTREE}} gives the [[random phase approximation|RPA]]; further enabling {{TAG|LFXC}} or {{TAG|LADDER}} adds the beyond-RPA contributions.
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
== Additional resources ==
 
<math display="block">
\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 display="inline">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 display="inline">v</math> and <math display="inline">f_{\rm xc}</math> yields the independent particle approximation.
* Neglecting only the <math display="inline">f_{\rm xc}</math> gives the dielectric function in the Random Phase Approximation (RPA)
* If <math>f_{\rm xc}</math> is local, i.e., LDA or PBE, no excitonic effects are included
* If <math>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 ===
=== Lectures ===
*Lecture on [https://youtu.be/arTPHW28qNM TDDFT].
* Lecture on {{Video|bse:alexey:2026|TDDFT theory and calculations}}.


=== Tutorials ===
=== Tutorials ===
*Tutorial calculating optical absorption of C diamond using [https://www.vasp.at/tutorials/latest/bse/part1/#BSE-e03 TDDDH].
* Tutorial calculating optical absorption of C diamond using {{Tutorial|bse:e03|TDDDH}}.
*Tutorial on the [https://www.vasp.at/tutorials/latest/bse/part3/#BSE-e09 efficient Brillouin zone sampling] using TDDDH and [https://www.vasp.at/tutorials/latest/bse/part3/#BSE-e10 exciton analysis] using TDDDH.
* Tutorial on the {{Tutorial|bse:e09|efficient Brillouin zone sampling}} using TDDDH and {{Tutorial|bse:e10|exciton analysis}} using TDDDH.


=== How to ===
=== How to ===
* Practical guide for solving the BSE via diagonalization [[BSE calculations]]
* Practical guide for solving the Casida equation via diagonalization: [[TDDFT calculations]].
* [[Time-dependent density-functional theory calculations]]
* Practical guide for real-time TDDFT calculations: [[Time-evolution algorithm]].


== References ==
== References ==


<!-- [[Category:VASP|Bethe-Salpeter equation]][[Category:Many-body perturbation theory]] -->
<references/>
 
[[Category:VASP|TDDFT]][[Category:Many-body perturbation theory]][[Category:Linear response]]

Revision as of 16:05, 5 June 2026

Time-dependent density-functional theory (TDDFT) extends density-functional theory to time-varying external potentials, enabling the computation of neutral electronic excitations and frequency-dependent response functions. VASP implements TDDFT in two complementary forms: a Casida-equation formulation based on the diagonalization of an excitonic Hamiltonian, and a real-time (RT-TDDFT) formulation based on the propagation of the Kohn-Sham orbitals after a delta-pulse perturbation. The detailed theoretical background is given in the theory page.

Theory

The Casida formulation of TDDFT recasts the linear-response problem as a non-Hermitian eigenvalue 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]

with the same mathematical structure as the Bethe-Salpeter equation (BSE). The eigenvalues [math]\displaystyle{ \omega_\lambda }[/math] are the excitation energies, and the eigenvectors [math]\displaystyle{ \mathbf{X}_\lambda, \mathbf{Y}_\lambda }[/math] determine the oscillator strengths and the dielectric function with excitonic effects. The difference with respect to BSE is that the beyond-RPA part of the kernel is described by the exchange-correlation kernel [math]\displaystyle{ f_\mathrm{xc} }[/math] instead of the screened Coulomb interaction [math]\displaystyle{ W }[/math].

Scaling

The size of the excitonic Hamiltonian is

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

where [math]\displaystyle{ N_k }[/math], [math]\displaystyle{ N_c }[/math], and [math]\displaystyle{ N_v }[/math] are the number of k points, conduction bands, and valence bands. Building the Hamiltonian scales as [math]\displaystyle{ N^4 }[/math]--[math]\displaystyle{ N^5 }[/math] with the system size. Solving the resulting eigenvalue problem by exact diagonalization scales as [math]\displaystyle{ N_{\rm rank}^3 }[/math], or as [math]\displaystyle{ N^6 }[/math] with the system size. The real-time propagation alternative avoids diagonalization entirely and scales as [math]\displaystyle{ N_{\rm rank}^2 }[/math], or as [math]\displaystyle{ N^4 }[/math] with the system size, making it the method of choice for large systems with many bands or k points.

Time evolution Casida
Memory [math]\displaystyle{ N_{\mathbf{k}} \times (N_v + N_c) \times N_G }[/math] [math]\displaystyle{ (N_{\mathbf{k}} \times N_v \times N_c)^2 }[/math]
Compute time [math]\displaystyle{ N_{\mathbf{k}} \times N_v \times N_G }[/math] [math]\displaystyle{ (N_{\mathbf{k}} \times N_v \times N_c)^3 }[/math]
[math]\displaystyle{ + N_{\mathbf{k}} \times N_v \times N_c \times N_G }[/math]
Nonlocal exchange [math]\displaystyle{ + N_{\mathbf{k}}^2 \times N_v^2 \times N_G }[/math] ⋯

Casida TDDFT

The Casida equation is solved in VASP by setting ALGO=TDHF. The excitation energies and oscillator strengths are obtained directly from the eigenvalues and eigenvectors of the excitonic Hamiltonian, which makes this approach particularly useful for analyzing individual excitons. Casida TDDFT can be performed using DFT or hybrid-functional orbitals and eigenvalues. The macroscopic dielectric function then reads

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

The following features are currently supported:

Time-evolution TDDFT (Real-time TDDFT)

Real-time TDDFT (RT-TDDFT) is selected with ALGO=TIMEEV. The ground-state Kohn-Sham orbitals are perturbed by a Dirac delta pulse of the electric field

[math]\displaystyle{ v_\mathrm{ext}(\mathbf r, t) = \lambda \, \mathbf r \cdot \mathbf D \, \delta(t), }[/math]

which simultaneously excites all valence-to-conduction transitions. The time-dependent dipole moments are then propagated and Fourier-transformed to yield the macroscopic dielectric function[1]

[math]\displaystyle{ \epsilon_M(\omega)=1-\frac{4\pi}{\Omega}\int_0^{\infty} \mathrm{d} t \sum_{c,v,\mathbf{k}}\left(\langle\mu_{cv\mathbf{k}}| \xi_{cv\mathbf{k}}(t)\rangle + \mathrm{c.c.}\right) e^{-\mathrm i(\omega-\mathrm i \delta) t}, }[/math]

where [math]\displaystyle{ \mu_{c v \mathbf k} }[/math] are the dipole moments and [math]\displaystyle{ |\xi_{cv\mathbf k}(t)\rangle }[/math] is the time-evolved dipole vector. The solution is strictly equivalent to that of the Casida equation for the dielectric function, but does not yield eigenvectors and so cannot be used directly for exciton analysis. Its main advantage is the quadratic scaling with [math]\displaystyle{ N_{\rm rank} }[/math].

The required number of propagation steps is controlled by the broadening CSHIFT and the maximum energy OMEGAMAX, and does not depend on the size of the Hamiltonian.

The following features are currently supported:

Exchange-correlation kernel

The exchange-correlation kernel [math]\displaystyle{ f_\mathrm{xc} }[/math] determines how electron-hole interactions beyond the random phase approximation are described in TDDFT. The choice of kernel is tied to the exchange-correlation functional used in the ground-state calculation and is controlled by the tags LHARTREE, LADDER, and LFXC.

  • Local and semilocal kernels (ALDA, APBE): obtained as the second functional derivative of an LDA or PBE exchange-correlation energy. These kernels are computationally cheap and work well for plasmons and metallic systems, but lack the long-range [math]\displaystyle{ -1/q^2 }[/math] behavior and therefore fail to describe bound excitons in semiconductors and insulators.
  • Hybrid-functional kernels: when a fraction of exact exchange is included in the ground-state functional (e.g., PBE0 or HSE), the corresponding TDDFT kernel inherits a long-range non-local exchange contribution. This restores the [math]\displaystyle{ -1/q^2 }[/math] behavior and allows for an approximate description of excitonic effects. The fraction of exact exchange is controlled by AEXX, and the range-separation parameter by HFSCREEN. When a hybrid functional is used, LADDER must be set to .TRUE. so that the non-local exchange contribution of the kernel (the ladder diagrams) is actually included in the time propagation; otherwise the calculation only contains the local part of [math]\displaystyle{ f_\mathrm{xc} }[/math] and the excitonic effects from the hybrid are lost.
  • Screened exchange (ladder diagrams): enabling LADDER adds the screened Coulomb interaction [math]\displaystyle{ W }[/math], yielding the proper long-range electron-hole attraction. In the time-evolution implementation, [math]\displaystyle{ W }[/math] is currently obtained from a model dielectric function controlled by LMODELHF, AEXX, and HFSCREEN.

The combination of these three switches selects the approximation level: setting all three to .FALSE. yields the independent-particle approximation (equivalent to LOPTICS); enabling only LHARTREE gives the RPA; further enabling LFXC or LADDER adds the beyond-RPA contributions.

Additional resources

Lectures

Tutorials

How to

References

  1. ↑ T. Sander, G. Kresse, Macroscopic dielectric function within time-dependent density functional theory—Real time evolution versus the Casida approach , J. Chem. Phys. 146, 064110 (2017)

Pages in category "Time-dependent density functional theory"

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

T

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