Time-dependent density-functional theory calculations: Difference between revisions

Latest revision as of 17:44, 7 February 2024

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 . 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.

Solving Casida equation

The algorithm for solving the Casida equation can be selected by setting ALGO = TDHF. This approach is very similar to BSE but differs in the way the screening of the Coulomb potential is approximated. The TDHF approach uses the exchange-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 only needs to provide the ground-state orbitals (WAVECAR) and the derivatives of the orbitals with respect to $k$ (WAVEDER).

Mind: Unlike BSE, TDHF calculations do not require

W(\omega \to 0)

, i.e., Wxxxx.tmp

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

The TDHF calculations can be performed in two steps:

ground-state calculation
optical absorption calculation

For example, an optical absorption calculation of bulk Si can be performed using a dielectric-dependent hybrid-functional described in Refs.^[1]^[2]^[3].

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. 
LMODELHF  = .TRUE. 
AEXX      = 0.083
HFSCREEN  = 1.22
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

SYSTEM    = Si
ISMEAR    = 0 
SIGMA     = 0.05
NBANDS    = 16     
ALGO      = TDHF
IBSE      = 0
NBANDSO   = 4       ! number of occupied bands
NBANDSV   = 8       ! number of unoccupied bands
LHARTREE  = .TRUE.
LADDER    = .TRUE.
LFXC      = .TRUE.
LMODELHF  = .TRUE. 
AEXX      = 0.083
HFSCREEN  = 1.22

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

The TDDFT calculation using the PBE exchange-correlation kernel can be performed by disabling the ladder diagrams LADDER = .FALSE., i.e., only the PBE exchange-correlation kernel is present in the Hamiltonian.

SYSTEM    = Si
ISMEAR    = 0 
SIGMA     = 0.05
NBANDS    = 16     
ALGO      = TDHF
IBSE      = 0
NBANDSO   = 4       ! determines how many occupied bands are used
NBANDSV   = 8       ! determines how many unoccupied (virtual) bands are used
LFXC      = .TRUE.
LHARTREE  = .TRUE.
LADDER    = .FALSE.

Mind: In TDDFT calculation, where the ladder diagrams are not included (LADDER=.FALSE.) or the fraction of exact exchange in the kernel is zero (AEXX=0), 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 Hamiltonian.

The calculated frequency-dependent dielectric function, transition energies and oscillator strength values are stored in the vasprun.xml file.

Calculations beyond Tamm-Dancoff approximation

Calculations beyond Tamm-Dancoff approximation can be performed in the same manner as in the BSE.

Calculations at finite wavevectors

Calculations at finite wavevectors can be performed in the same manner as in the BSE.

References

[chen2018nonempirical-1] W. Chen, G. Miceli, G.M. Rignanese, and A. Pasquarello, Nonempirical dielectric-dependent hybrid functional with range separation for semiconductors and insulators, Phys. Rev. Mater. 2, 073803 (2018).

[cui2018doubly-2] Z.H. Cui, Y.C. Wang, M.Y. Zhang, X. Xu, and H. Jiang, Doubly Screened Hybrid Functional: An Accurate First-Principles Approach for Both Narrow- and Wide-Gap Semiconductors J. Phys. Chem. Lett., 9, 2338-2345 (2018).

[liu2019assessing-3] P. Liu, C. Franchini, M. Marsman, and G. Kresse, Assessing model-dielectric-dependent hybrid functionals on the antiferromagnetic transition-metal monoxides MnO, FeO, CoO, and NiO, J. Phys.: Condens. Matter 32, 015502 (2020).

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
-VASP offers a powerful module for performing time-dependent density functional theory (TDDFT) or time-dependent Hartree-Fock (TDHF) calculations in the Casida formulation {{cite|albrecht:prl:98}}{{cite|rohlfing:prl:98}}. 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.
+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.
-__TOC__
+== Solving Casida equation ==
+The algorithm for solving the Casida equation can be selected by setting {{TAG|ALGO}} = TDHF. This approach is very similar to BSE but differs in the way the screening of the Coulomb potential is approximated. The TDHF approach uses the exchange-correlation kernel <math>f_{\rm xc}</math>, whereas BSE requires the <math>W(\omega \to 0)</math> from a preceding ''GW '' calculation. Thus, in order to perform a TDHF calculation, one only needs to provide the ground-state orbitals ({{FILE|WAVECAR}}) and the derivatives of the orbitals with respect to <math>k</math> ({{FILE|WAVEDER}}).
-== Solving Casida equations ==
+{{NB|mind|Unlike BSE, TDHF calculations do '''not''' require <math>W(\omega \to 0)</math>, i.e., {{FILE|Wxxxx.tmp}}|}}
-In VASP, the algorithm for constructing and solving the Casida equation can be selected by {{TAG|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 <math>f_{\rm xc}</math>, whereas BSE requires the <math>W(\omega \to 0)</math> from a preceding ''GW '' calculation. Thus, in order to perform a TDHF calculation, one has to provide the ground-state orbitals ({{FILE|WAVECAR}}) and the derivatives of the orbitals with respect to <math>k</math> ({{FILE|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 ==
+== Time-dependent Hartree-Fock ==
-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:
+The TDHF calculations can be performed in two steps:
+# ground-state calculation
+# optical absorption calculation
+For example, an optical absorption calculation of bulk Si can be performed using a dielectric-dependent  hybrid-functional described in Refs.{{cite|chen2018nonempirical}}{{cite|cui2018doubly}}{{cite|liu2019assessing}}.
   {{TAG|SYSTEM}}    = Si
@@ Line 17: / Line 19: @@
   {{TAG|ALGO}}      = D       ! Damped algorithm often required for HF type calculations, {{TAG|ALGO}} = Normal might work as well
   {{TAG|LHFCALC}}   = .TRUE.
-  {{TAG|AEXX}}      = 0.3
+ {{TAG|LMODELHF}}  = .TRUE.
-  {{TAG|HFSCREEN}}  = 0.2
+  {{TAG|AEXX}}      = 0.083
+  {{TAG|HFSCREEN}}  = 1.22
   {{TAG|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
+In the second step, the dielectric function is evaluated by solving the Casida equation
   {{TAG|SYSTEM}}    = Si
@@ Line 28: / Line 31: @@
   {{TAG|NBANDS}}    = 16
   {{TAG|ALGO}}      = TDHF
-  {{TAG|AEXX}}      = 0.3
+ {{TAG|IBSE}}      = 0
-  {{TAG|HFSCREEN}}  = 0.2
+ {{TAG|NBANDSO}}   = 4       ! number of occupied bands
+ {{TAG|NBANDSV}}   = 8       ! number of unoccupied bands
+ {{TAG|LHARTREE}}  = .TRUE.
+ {{TAG|LADDER}}    = .TRUE.
+ {{TAG|LFXC}}      = .TRUE.
+ {{TAG|LMODELHF}}  = .TRUE.
+  {{TAG|AEXX}}      = 0.083
+  {{TAG|HFSCREEN}}  = 1.22
 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 <math>f_{\rm xc}</math> are present
+The TDDFT calculation using the PBE exchange-correlation kernel can be performed by disabling the ladder diagrams {{TAG|LADDER}} = .FALSE., i.e., only the PBE exchange-correlation kernel is present in the Hamiltonian.
   {{TAG|SYSTEM}}    = Si
@@ Line 41: / Line 51: @@
   {{TAG|NBANDS}}    = 16
   {{TAG|ALGO}}      = TDHF
+ {{TAG|IBSE}}      = 0
+ {{TAG|NBANDSO}}   = 4       ! determines how many occupied bands are used
+ {{TAG|NBANDSV}}   = 8       ! determines how many unoccupied (virtual) bands are used
   {{TAG|LFXC}}      = .TRUE.
-  {{TAG|AEXX}}      = 0.0
+  {{TAG|LHARTREE}}  = .TRUE.
+ {{TAG|LADDER}}    = .FALSE.
-Since the ladder diagrams are not included in the TD-DFT calculation, the resulting dielectric function lacks the excitonic effects.
+{{NB|mind|In TDDFT calculation, where the ladder diagrams are not included ({{TAG|LADDER}}{{=}}.FALSE.) or the fraction of exact exchange in the kernel is zero ({{TAG|AEXX}}{{=}}0), the resulting dielectric function lacks the excitonic effects.|}}
+VASP tries to use sensible defaults, but it is highly recommended to check the {{FILE|OUTCAR}} file and make sure that the right bands are included.  The tag {{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 Hamiltonian.
+The calculated frequency-dependent dielectric function, transition energies and oscillator strength values are stored in the {{TAG|vasprun.xml}} file.
-VASP tries to use sensible defaults, but it is highly recommended to check the {{FILE|OUTCAR}} file and make sure that the right bands are included.  The tag {{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 {{TAG|NBANDSV}} and {{TAG|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 {{FILE|WFULLxxxx.tmp}} files and if these files are not found, VASP will read the {{FILE|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 {{TAG|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 {{TAG|vasprun.xml}} file.
 == Calculations beyond Tamm-Dancoff approximation ==
-The TDHF and BSE calculations beyond the Tamm-Dancoff approximation (TDA){{cite|sander:prb:15}} can be performed by setting the {{TAG|ANTIRES}} = 2 in the {{TAG|INCAR}} file
+Calculations beyond Tamm-Dancoff approximation can be performed in the same manner as in the [[BSE calculations#Calculations beyond Tamm-Dancoff approximation|BSE]].
- {{TAG|SYSTEM}}       = Si
- {{TAG|NBANDS}}       = same as in GW calculation
- {{TAG|ISMEAR}}       = 0
- {{TAG|SIGMA}}        = 0.05
- {{TAG|ALGO}}         = BSE
- {{TAG|ANTIRES}}      = 2      ! beyond Tamm-Dancoff
- {{TAG|LORBITALREAL}} = .TRUE.
- {{TAG|NBANDSO}}      = 4
- {{TAG|NBANDSV}}      = 8
-The flag {{TAG|LORBITALREAL}} = .TRUE. forces VASP to make the orbitals <math> \phi({\bf r}) </math> 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 <math>{\bf q}</math>-vector compatible with the k-point grid (finite-momentum excitons).
+Calculations at finite wavevectors can be performed in the same manner as in the [[BSE calculations#Calculations at finite wavevectors|BSE]].
- {{TAG|SYSTEM}}       = Si
- {{TAG|NBANDS}}       = same as in GW calculation
- {{TAG|ISMEAR}}       = 0
- {{TAG|SIGMA}}        = 0.05
- {{TAG|ALGO}}         = BSE
- {{TAG|ANTIRES}}      = 2
- {{TAG|KPOINT_BSE}}   = 3 -1 0 0  ! q-point index,  three integers
- {{TAG|LORBITALREAL}} = .TRUE.
- {{TAG|NBANDSO}}      = 4
- {{TAG|NBANDSV}}      = 8
-The tag {{TAG|KPOINT_BSE}} sets the <math>{\bf q}</math>-point and the shift at which the dielectric function is calculated. The first integer specifies the index of the <math>{\bf q}</math>-point and the other three values shift the provided <math>{\bf q}</math>-point by an arbitrary reciprocal vector <math> \bf G</math>.  The reciprocal lattice vector is supplied by three integer values <math> n_i</math> with <math> {\bf G}= n_1 {\bf G}_1+n_2 {\bf G}_2+n_3 {\bf G}_3</math>.  This feature is only supported as of VASP.6 (in VASP.5 the feature can be enabled, but the results are erroneous).
 == References ==
 <references/>
 ----
-[[Category:Many-body perturbation theory]][[Category:Bethe-Salpeter equations]][[Category:Howto]]
+[[Category:Time-dependent density functional theory]][[Category:Howto]]