LFXC

LFXC = .TRUE. | .FALSE.
Default: LFXC = .FALSE.

Description: LFXC enables the (semi-)local exchange-correlation kernel in Casida and time-evolution TDDFT calculations.

In linear-response TDDFT, the density-density response function [math]\displaystyle{ \chi }[/math] obeys the Dyson equation

[math]\displaystyle{ \chi(\mathbf r, \mathbf r'; \omega) = \chi_\mathrm{KS}(\mathbf r, \mathbf r'; \omega) + \int \mathrm d\mathbf r_1 \mathrm d\mathbf r_2 \, \chi_\mathrm{KS}(\mathbf r, \mathbf r_1; \omega) \left[ v(\mathbf r_1, \mathbf r_2) + f_\mathrm{xc}(\mathbf r_1, \mathbf r_2; \omega) \right] \chi(\mathbf r_2, \mathbf r'; \omega), }[/math]

where [math]\displaystyle{ \chi_\mathrm{KS} }[/math] is the non-interacting Kohn-Sham response function, [math]\displaystyle{ v }[/math] is the bare Coulomb interaction, and [math]\displaystyle{ f_\mathrm{xc} }[/math] is the exchange-correlation kernel. VASP uses the adiabatic approximation, [math]\displaystyle{ f_\mathrm{xc}(\mathbf r, \mathbf r'; \omega) \approx f_\mathrm{xc}(\mathbf r, \mathbf r') }[/math].

Setting LFXC=.TRUE. includes the (semi-)local part of [math]\displaystyle{ f_\mathrm{xc} }[/math] in both the Casida eigenvalue problem (ALGO=TDHF) and the time-evolution TDDFT (or real-time TDDFT) (ALGO=TIMEEV).

(Semi-)local exchange-correlation kernel

The exchange-correlation kernel is computed very differently in the Casida and time-evolution TDDFT approaches. It is defined as the second functional derivative of the exchange-correlation energy density with respect to the charge density,

[math]\displaystyle{ f_\mathrm{xc}(\mathbf r, \mathbf r') = \frac{\partial^2 \varepsilon_\mathrm{xc}}{\partial n(\mathbf r) \, \partial n(\mathbf r')}\delta(\mathbf r - \mathbf r'). }[/math]

The Casida approach requires the derivative to be evaluated explicitly and therefore implemented for each functional. The time-evolution TDDFT does not require an explicit kernel: its contribution is included implicitly through the propagation of the charge density and the exchange-correlation potential.

Casida equation

For an LDA functional,

[math]\displaystyle{ f_\mathrm{xc}^\mathrm{LDA}(\mathbf r, \mathbf r') = \frac{\partial^2 \varepsilon_\mathrm{xc}}{\partial n^2} \, \delta(\mathbf r - \mathbf r'). }[/math]

For a GGA functional, gradient terms appear,

[math]\displaystyle{ f_\mathrm{xc}^\mathrm{GGA}(\mathbf r, \mathbf r') = \frac{\partial^2 \varepsilon_\mathrm{xc}}{\partial n^2}(\mathbf r) \, \delta(\mathbf r - \mathbf r') - \left[\nabla \frac{\partial^2 \varepsilon_\mathrm{xc}}{\partial n \, \partial \nabla n}(\mathbf r)\right] \delta(\mathbf r - \mathbf r') - \nabla_i \frac{\partial^2 \varepsilon_\mathrm{xc}}{\partial \nabla_i n \, \partial \nabla_j n}(\mathbf r) \, \nabla_j \delta(\mathbf r - \mathbf r'), }[/math]

where [math]\displaystyle{ i, j }[/math] are summed Cartesian indices. In the Casida approach these gradient terms are dropped and only the density derivatives are kept. Meta-GGA kernels are not supported.

Time-evolution TDDFT (Real-time TDDFT)

The real-time propagation applies [math]\displaystyle{ f_\mathrm{xc} }[/math] directly to the time-dependent density, so LDA and GGA kernels are used in full, including the gradient terms.

For meta-GGA functionals, the dependence of [math]\displaystyle{ \varepsilon_\mathrm{xc} }[/math] on the kinetic-energy density [math]\displaystyle{ \tau(\mathbf r) }[/math] makes [math]\displaystyle{ \delta v_\mathrm{xc}/\delta n }[/math] non-local through the orbital dependence of [math]\displaystyle{ \tau }[/math]^[1]. These non-local contributions are not implemented in VASP, so the [math]\displaystyle{ 1/q^2 }[/math] long-range component of [math]\displaystyle{ f_\mathrm{xc} }[/math] responsible for excitonic effects is missing.

Hybrid functionals

For a hybrid functional, a fraction [math]\displaystyle{ c_\mathrm{x} }[/math] of the (semi-)local exchange is replaced by exact (Fock) exchange in both solvers^[2],

[math]\displaystyle{ f_\mathrm{xc}(\mathbf r, \mathbf r') = \left(1-c_\mathrm{x}\right) \frac{\partial^2 \varepsilon_\mathrm{x}}{\partial n(\mathbf r) \, \partial n(\mathbf r')} + \frac{\partial^2 \varepsilon_\mathrm{c}}{\partial n(\mathbf r) \, \partial n(\mathbf r')} + c_\mathrm{x} \frac{\partial^2 \varepsilon_\mathrm{x}^\mathrm{Exact}}{\partial^2 n(\mathbf r, \mathbf r')}, }[/math]

where [math]\displaystyle{ c_\mathrm{x} }[/math] is set by AEXX and [math]\displaystyle{ n(\mathbf r, \mathbf r') }[/math] is the one-particle density matrix. LFXC=.TRUE. enables the first two terms only; the Fock contribution is enabled separately by LADDER=.TRUE..

Compare Casida and time-evolution TDDFT results

The Casida and time-evolution approaches produce very similar results for LDA exchange-correlation. Small differences typically remain because one-center terms in the PAW method are treated differently in the two approaches. To bring the Casida results into closer agreement, increase ENCUTGW beyond its default value.

References

[nazarov:vignale:2011-1] V. U. Nazarov, G. Vignale, Optics of semiconductors from meta-generalized-gradient-approximation-based time-dependent density-functional theory, Phys. Rev. Lett. 107, 216402 (2011).

[sander:prb:15-2] T. Sander, E. Maggio, and G. Kresse, Beyond the Tamm-Dancoff approximation for extended systems using exact diagonalization, Phys. Rev. B 92, 045209 (2015).

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