LFXC

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

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

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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]. 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,

[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.

Related tags and articles

Tags

LADDER, LHARTREE, AEXX, ENCUTGW

Articles

Time-dependent density-functional theory calculations, Time-evolution algorithm

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