Category:Time-dependent density functional theory

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 primarily through the Casida-equation formulation (ALGO=TDHF), which builds an excitonic Hamiltonian and solves it using exact diagonalization, time-evolution, or Lanczos algorithms. The detailed theoretical background is given in the theory page.

Casida TDDFT

The Casida formulation of TDDFT (ALGO=TDHF) 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].

Casida TDDFT can be performed using DFT or hybrid-functional orbitals and eigenvalues. VASP provides three algorithms for solving the Casida equation, selected via IBSE:

Exact diagonalization (IBSE=2)

The excitonic Hamiltonian is diagonalized exactly. The excitation energies and oscillator strengths are obtained directly from the eigenvalues and eigenvectors, which makes this approach particularly useful for analyzing individual excitons. The macroscopic dielectric function is obtained from the spectral representation:

[math]\displaystyle{ \varepsilon_M(\omega) = 1 + \frac{4\pi}{\Omega} \sum_\lambda \left|\sum_{cv\mathbf k} \mu_{cv\mathbf k} X_\lambda^{cv\mathbf k}\right|^2 \left[\frac{1}{\omega + \omega_\lambda + \mathrm i\eta} - \frac{1}{\omega - \omega_\lambda + \mathrm i\eta}\right], }[/math]

where [math]\displaystyle{ \mu_{cv\mathbf{k}}^j=\frac{\langle c\mathbf{k}|v_j|v\mathbf{k}\rangle}{\varepsilon_c(\mathbf{k})-\varepsilon_v(\mathbf{k})} }[/math] is the dipole matrix element, [math]\displaystyle{ X_\lambda^{cv\mathbf k} }[/math] are the eigenvector components, and [math]\displaystyle{ \omega_\lambda }[/math] are the excitation energies.

Time-evolution algorithm (IBSE=1)

The Casida equation is solved via real-time propagation. A Dirac delta pulse of the electric field simultaneously excites all valence-to-conduction transitions, and the time-dependent dipole moments are propagated forward in time. The dielectric function is found via a Fourier transform^[1]:

[math]\displaystyle{ \varepsilon_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 \eta) t}, }[/math]

where [math]\displaystyle{ \mu_{cv\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 exact diagonalization 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.

Lanczos algorithm (IBSE=3)

The dielectric function is expressed as a continued fraction

[math]\displaystyle{ \varepsilon_M(\omega) = 1 - \frac{4\pi}{\Omega}\cfrac{|u_0|^2}{(\omega - a_1 + \mathrm i\eta) - \cfrac{b_1^2}{(\omega -a_2 + \mathrm i\eta) - \cfrac{b_2^2}{...}}}, }[/math]

where [math]\displaystyle{ |u_0\rangle }[/math] is an initial guess vector computed from the dipole moments. The [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] coefficients are evaluated iteratively, with the algorithm stopping once the difference between [math]\displaystyle{ \varepsilon(\omega) }[/math] from two consecutive iterations is below a threshold selected by BSEPREC. Because the starting vector is built from dipole moments, the Lanczos algorithm is sensitive only to optically active transitions and can reach convergence faster than other methods for larger matrices.

The following features are currently supported:

• Calculating the dielectric function and eigenvectors (IBSE=2)
• Tamm-Dancoff approximation
• Calculations with hybrid functionals and range-separated hybrids

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 / Lanczos	Exact diagonalization
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]	⋯

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

• Lecture on TDDFT theory and calculations.

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 Casida equation via diagonalization: TDDFT calculations.
• Practical guide for real-time TDDFT calculations: Time-evolution algorithm.

References