Time-dependent density-functional theory calculations

VASP offers a powerful module for performing time-dependent density functional theory (TDDFT) or time-dependent Hartree-Fock (TDHF) calculations in the Casida formulation . 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 GW approximations.

Solving Casida equations

In VASP, the algorithm for constructing and solving the Casida equation can be selected by ALGO = TDHF. This approach essentially solves the same equations as BSE but differs in the way the screening of the Coulomb potential is approximated. The TDHF approach uses the exact-correlation kernel $f_{\rm xc}$, whereas BSE requires the $W(\omega \to 0)$ from a preceding GW calculation. Thus, in order to perform a TDHF calculation, one has to provide the ground-state orbitals (WAVECAR) and the derivatives of the orbitals with respect to $k$ (WAVEDER).

In summary, both TDHF and BSE approaches require a preceding ground-state calculation, however, the TDHF does not need the preceding GW and can be performed with the DFT or hybrid-functional orbitals and energies.

Time-dependent Hartree-Fock calculation

The TDHF calculations can be performed in two steps: the ground-state calculation and the optical absorption calculation. For example, optical absorption of bulk Si can be performed with a hybrid-functional electronic structure, where the number of bands is increased to include the relevant conduction bands:

SYSTEM    = Si
ISMEAR    = 0 
SIGMA     = 0.05
NBANDS    = 16      ! or any larger desired value
ALGO      = D       ! Damped algorithm often required for HF type calculations, ALGO = Normal might work as well
LHFCALC   = .TRUE. 
AEXX      = 0.3 
HFSCREEN  = 0.2
LOPTICS   = .TRUE.  ! can also be done in an additional intermediate step

In the second step, the dielectric function is evaluated by solving the Casida equation for the

SYSTEM    = Si
ISMEAR    = 0 
SIGMA     = 0.05
NBANDS    = 16     
ALGO      = TDHF
AEXX      = 0.3 
HFSCREEN  = 0.2

THDF calculations can be performed for non-spin-polarized, spin-polarized, and noncollinear cases, as well as the case with spin-orbit coupling. There is, however, one caveat. The local exchange-correlation kernel is approximated by the density-density part only. This makes predictions for spin-polarized systems less accurate than for non-spin-polarized systems.

Time-dependent DFT calculation

Within the TD-DFT approximation, the Fock exchange is not included in the exchange-correlation kernel and the ladder diagrams are not taken into account. Hence, only the local contributions in $f_{\rm xc}$ are present

SYSTEM    = Si
ISMEAR    = 0 
SIGMA     = 0.05
NBANDS    = 16     
ALGO      = TDHF
LFXC      = .TRUE.
AEXX      = 0.0 

Since the ladder diagrams are not included in the TD-DFT calculation, the resulting dielectric function lacks the excitonic effects.

VASP tries to use sensible defaults, but it is highly recommended to check the OUTCAR file and make sure that the right bands are included. The tag OMEGAMAX specifies the maximum excitation energy of included electron-hole pairs and the pairs with the one-electron energy difference beyond this limit are not included in the BSE Hamiltonian. Hint: The convergence with respect to NBANDSV and OMEGAMAX should be thoroughly checked as the real part of the dielectric function, as well as the correlation energy, is usually very sensitive to these values, whereas the imaginary part of the dielectric function converges quickly.

At the beginning of the BSE calculation, VASP will try to read the WFULLxxxx.tmp files and if these files are not found, VASP will read the Wxxxx.tmp files. For small isotropic bulk systems, the diagonal approximation of the dielectric screening may be sufficient and yields results very similar to the calculation with the full dielectric tensor WFULLxxxx.tmp. Nevertheless, for molecules and atoms as well as surfaces, the full-screened Coulomb kernel is strictly required.

Both TDHF and BSE approaches write the calculated frequency-dependent dielectric function as well as the excitonic energies in the vasprun.xml file.

Calculations beyond Tamm-Dancoff approximation

The TDHF and BSE calculations beyond the Tamm-Dancoff approximation (TDA) can be performed by setting the ANTIRES = 2 in the INCAR file

SYSTEM       = Si
NBANDS       = same as in GW calculation
ISMEAR       = 0
SIGMA        = 0.05
ALGO         = BSE  
ANTIRES      = 2      ! beyond Tamm-Dancoff
LORBITALREAL = .TRUE. 
NBANDSO      = 4 
NBANDSV      = 8

The flag LORBITALREAL = .TRUE. forces VASP to make the orbitals $\phi({\bf r})$ real valued at the Gamma point as well as k-points at the edges of the Brillouin zone. This can improve the performance of BSE/TDHF calculations but it should be used consistently with the ground-state calculation.

Calculations at finite wavevectors

VASP can also calculate the dielectric function at a ${\bf q}$-vector compatible with the k-point grid (finite-momentum excitons).

SYSTEM       = Si
NBANDS       = same as in GW calculation
ISMEAR       = 0 
SIGMA        = 0.05
ALGO         = BSE  
ANTIRES      = 2 
KPOINT_BSE   = 3 -1 0 0  ! q-point index,  three integers
LORBITALREAL = .TRUE.
NBANDSO      = 4 
NBANDSV      = 8

The tag KPOINT_BSE sets the ${\bf q}$-point and the shift at which the dielectric function is calculated. The first integer specifies the index of the ${\bf q}$-point and the other three values shift the provided ${\bf q}$-point by an arbitrary reciprocal vector $\bf G$. The reciprocal lattice vector is supplied by three integer values $n_i$ with ${\bf G}= n_1 {\bf G}_1+n_2 {\bf G}_2+n_3 {\bf G}_3$. This feature is only supported as of VASP.6 (in VASP.5 the feature can be enabled, but the results are erroneous).

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