3.3 Electronic structure: Exchange, correlation (incl. DFT+U), and excited states

This keyword applies only to the calculation of the MP2 correlation energy (if requested). It does not imply a frozen-core treatment anywhere else.

In a nutshell, this is a simple way to exclude the large contribution from certain core electrons to the MP2 correlation energy. This contribution is mostly systematic, and therefore tends to cancel in energy differences. However, it is also the hardest to compute unless specialized basis sets are invoked that “know” about core correlation; it may thus be the source of a large systematic error that also cancels if excluded from the beginning. For consistency between different calculations, the number of excluded “core” orbitals must be readjusted between calculations with different numbers of atoms.

Compared to the keyword frozen_core, frozen_core_postscf is more friendly, especially for large systems, as you don’t need to count by hand which is the first valence orbital in the frozen-core algorithm.

For example, valence_shell_number=2 means that at most two outer shells are taken as valence shells in the frozen-core calculations:

If (and only if) a hybrid functional is specified using the xc keyword, hybrid_xc_coeff allows to change the Hartree-Fock mixing parameter to a different, given value. For example, the mixing parameter in pbe0 could be specified away from its literature value, $\alpha$=0.25. No effect for xc functionals that do not have any Hartree-Fock exchange admixed in the first place.

Obviously, this option is only useful for test purposes and does change the definition of any functional away from its literature value. Handle with care.

The hse06 functional comes with a screening parameter $\omega$ which must be specified explicitly (see the xc keyword for a detailed explanation). Unfortunately, different codes and authors appear to have adopted different conventions for $\omega$ – either Å^-1 or [bohr radius]^-1. To avoid any possible confusion when using HSE06 in FHI-aims, we therefore only run hse06 if the unit has been explicitly specified, using the above keyword. We apologize for the inconvenience, but the risk of an innocent misunderstanding is rather high in the present case.

There are two common schemes for dealing with the double counting problem in DFT+U: The AMF method assumes that the effect of the DFT+U term on the actual occupations remains small, so that the occupations can be assumed to be equal within each shell for the purpose of the double counting correction. The FLL method, on the other hand, assums a maximal effect of the DFT+U term on the occupation numbers, handling double counting correctly in the case that all orbitals with in the shell are either fully occupied or empty. The self consistent mixing of both limits improves the handling of the intermediate range (see Ref. [248]).

Note that quasiparticle corrections ($GW$, MP2) are currently possible only for cluster geometries (no periodic boundary conditions).

After the normal self-consistency cycle for a given exchange-correlation functional (set using the xc keyword) is complete, qpe_calc can be used to specify a perturbative quasiparticle correction to be applied as a post-processing step. Valid self-energy options selfenergy-type are:

• •

  gw : Perturbative $G_0{W}_0$-type self-energy, where both the Green’s function $G_0$ and the screened Coulomb interaction $W_0$ are computed only once, based on the self-consistent DFT or Hartree-Fock ground state eigenvalues and eigenfunctions.

• •

  ev_scgw Perturbative $G_0{W}_0$-type self-energy, where self-energy is evaluated with partial self-consistency in the eigenvalues. The eigenvalues are iterated in $G$ and $W$. Molecular orbitals are kept unchanged from the preliminary calculation. This scheme is often abbreviated as ev$GW$ in literature. For true self-consistent $GW$, see the sc_self_energy further below.

• •

  ev_scgw0 Perturbative $G_0{W}_0$-type self-energy, where self-energy is evaluated with partial self-consistency in the eigenvalues. The eigenvalues are iterated only in $G$, but not in $W$. Molecular orbitals are kept unchanged from the preliminary calculation as in ev_scgw. This scheme is often abbreviated as ev$G{W}_0$ in literature. For true self-consistent $GW$, see the sc_self_energy further below.

• •

  mp2 : Perturbative MP2-type self-energy, based on the self-consistent DFT or Hartree-Fock ground state eigenvalues and eigenfunctions.

For more details on different $GW$ flavors ($G_0{W}_0$, partially and fully-selfconsistent schemes) see review article [107].

Note that self-consistent $GW$ calculation (sc-$GW$, sc-$G{W}_0$) are currently possible only for cluster geometries (no periodic boundary conditions).

After the normal self-consistency cycle for a given exchange-correlation functional (set using the xc keyword) is complete, sc_self_energy can be used to specify a self-consistent scheme for the calculation of the $GW$ self-energy. The output consists of the total energy calculated from the Galitskii-Migdal formula, an output file (spectrum_sc.dat for spin unpolarized, spectrum_sc_up.dat and spectrum_sc_do.dat for spin up and down respectively in the case of spin polarized calculation) containing the spectral function calculated from the self-consistent Green’s function. At the end of the calculation, the output include the dipole moment evaluated from the self-consistent density.

Currently implemented self-consistent methods are:

• •

  scgw : Calculate the Green’s function by solving until full self-consistency the Dyson’s equation by using a self-energy in the $GW$ approximation.

• •

  scgw0 : Solve self-consistently the Dyson’s equation with the self-energy in the $G{W}_0$ approximation. Differently from fully self-consistent $GW$, in this case the screened Coulomb interaction is kept fixed at the RPA level.

The standard RPA method is a adiabatic-connection advanced DFT method, which integrates the contribution along the adiabatic-connection path analytically. This keyword rpa_along_ac_path allows you to unpack the adiabatic-connection path in the RPA approximation in detail.

This keyword printout_dft_components is repeatable in the contril.in allowing to inspect several DFT methods in one task.

The MP2 correlation energy (total_energy_method mp2 or xc mp2) can be separated into a sum of triplet (spin-up-spin-up) and singlet (spin-up-spin-down) two-electron terms:

$$E_{\text{corr,MP2}}=E_T+E_S.$$

(3.4)

Grimme [115] pointed out that empirical scaling factors $p_T$ and $p_S$ can be introduced and fitted to improve the accuracy of MP2 results compared to quantum-chemical benchmark methods:

$$E_{\text{SCS,MP2}}=p_T{E}_T+p_S{E}_S.$$

(3.5)

For example, $p_T$=1/3 and $p_S$=6/5 are employed to obtain the reaction energies of Table I in Ref. [115].

After the regular scf cycle is complete for a given exchange-correlation method as given by the xc tag, the resulting Kohn-Sham orbitals and eigenvalues are used to recalculate only the exchange-correlation energy, and only once (i.e., perturbative post-processing).

All LDA, GGA and meta-GGA DFT functionals listed under xc can be used with total_energy_method, including those implemented through LibXC and dfauto; See the relevant section of xc options for more detail. Hybrid-DFT functionals are also available, with the exception of range-separated or long-range corrected approaches.

Other valid post-processing options type are:

• •

  C6_coef : Molecular C₆ dispersion coefficients at the MP2 / RPA level will be calculated after the s.c.f. calculation. (This functionality is somewhat experimental, be sure to check for consistency.)

• •

  hf or HF: Calculate Hartree-Fock exchange on the given orbitals.

• •

  ll_vdwdf : The nonlocal part of correlation energy is calculated using the van der Waals density functional proposed by M. Dion et al. [76] and the total correlation energy will be re-evaluated as proposed in their paper. For details about additional Tags needed for the calculation, please visit Sec. 3.24. Note that an alternative implementation by the Helsinki group is available as well, the present keyword is not your only option.

• •

  mp2 : The correlation energy is calculated in second-order Møller-Plesset perturbation theory (MP2), with Hartree-Fock added for the exchange part. Note added in March 2016: A periodic implementation of MP2 is available but, at the time of writing, computationally extremely expensive. If you decide to use it, please keep in mind that the periodic version is included here as a matter of protocol but is not yet optimized to be fully usable in production calculations.

• •

  pbe_vdw, revpbe_vdw, or nlcorr : Evaluates the non-local van der Waals density functional (vdW-DF) proposed by M. Dion et al. [76] with the methodology of Sec. 3.24.2. (pbe_vdw uses PBE exchange, revpbe_vdw uses revpbe exchange, nlcorr calculates only the nonlocal correlation term.) Crucially, this is not the Tkatchenko-Scheffler correction [295] or any other simpler dispersion correction. If you are looking for Tkatchenko-Scheffler, please use the keyword vdw_correction_hirshfeld instead. In contrast, the present, non-local vdW-DF implementation will be very slow compared to any of the other dispersion corrections implemented in FHI-aims.

• •

  ppRPA : The full particle-particle RPA correlation energy will be calculated.

• •

  ppRPA_dir : The direct particle-particle RPA correlation energy will be calculated.

• •

  dppRPA+SOSEX : The SOSEX approximation to the direct particle-particle RPA correlation energy will be calculated. In case of any difficulty please contact "bundesha@gmail.com".

• •

  rpa : The RPA total energy as defined in Eq. (3.2) will be calculated. When this option is specified, the SE and rSE corrections to RPA are also evaluated. The total enegies computed with the RPA, RPA+SE, and RPA+rSE schemes are listed in items ‘‘RPA total energy’’, ‘‘PRA+SE total energy’’, and ‘‘RPA+rSE (full) total energy’’ respectively in the output file.

• •

  rpa+2ox : Just RPA plus second-order exchange (not screened). Likely only useful for testing / benchmarking, use rpt2 for completeness.

• •

  rpa+sosex : Just RPA plus second-order screened exchange. Likely only useful for testing / benchmarking, use rpt2 for completeness.

• •

  rpt2 : The rPT2 total energy as defined in Eq. (3.3) will be calculated. When this option is specified, the “RPA+SOSEX” total energy without the rSE correction will also be printed out in the output file.

• •

  xyg3 : “XYG3” double-hybrid functional[330], which is defined only for a self-consistent B3LYP reference, i.e., xc b3lyp is mandatory. Note that double-hybrid functionals include MP2 components. When using the tier basis sets, you must use a counterpoise correction of energy differences to get reliable results.

• •

  xdh-pbe0 : “xDH-PBE0” double-hybrid functional[326], which is defined only for a self-consistent PBE0 reference, i.e., xc pbe0 is mandatory. Note that double-hybrid functionals include MP2 components. When using the tier basis sets, you must use a counterpoise correction of energy differences to get reliable results.

• •

  lh_b86bpbe0_erfcor : Local hybrid functional that is under development. Do not use at this time. It does not yet produce converged numbers with standard basis sets.
  Syntax:
  xc lh_b86bpbe0_erfcor $c$
  

  $$E_{\mathrm{XC}}=E_{\mathrm{X}}^{\mathrm{GGA}}+\sum _{\sigma }\int f({\rho }_{\sigma },\nabla {\rho }_{\sigma })\left[{\epsilon }_{\mathrm{X},\sigma }^{\mathrm{HF}}(𝐫)-{\epsilon }_{\mathrm{X},\sigma }^{\mathrm{GGA}}(𝐫)\right]𝑑𝐫+E_{\mathrm{C}}^{\mathrm{GGA}}$$

  (3.6)

  where:

  $$f({\rho }_{\sigma },\nabla {\rho }_{\sigma })=\mathrm{erf}\left(c\frac{{\rho }_{\sigma }}{{\epsilon }_{\mathrm{X},\sigma }^{\mathrm{GGA}}}\right)$$

  (3.7)

  • –

    GGA is b86bpbe0

  • –

    $c$ is a variable that can be fitted (usually 0.2)

• •

  lh_b86bpbe0_erfs : Local hybrid functional that is under development. Do not use at this time. It does not yet produce converged numbers with standard basis sets.
  Syntax:
  xc lh_b86bpbe0_erfs $c$
  

  $$E_{\mathrm{XC}}=E_{\mathrm{X}}^{\mathrm{GGA}}+\sum _{\sigma }\int f({\rho }_{\sigma },\nabla {\rho }_{\sigma })\left[{\epsilon }_{\mathrm{X},\sigma }^{\mathrm{HF}}(𝐫)-{\epsilon }_{\mathrm{X},\sigma }^{\mathrm{GGA}}(𝐫)\right]𝑑𝐫+E_{\mathrm{C}}^{\mathrm{GGA}}$$

  (3.8)

  where:

  $$f({\rho }_{\sigma },\nabla {\rho }_{\sigma })=\mathrm{erf}\left(c\frac{|\nabla {\rho }_{\sigma }|}{{\rho }_{\sigma }^{\frac{4}{3}}}\right)$$

  (3.9)

  • –

    GGA is b86bpbe0

  • –

    $c$ is a variable that can be fitted (usually 0.3)

Note that some of the correlation methods available here are only supported for cluster geometries at this time. Note also that when advanced correlation methods (e.g. rpa, rpt2, xyg3, xdh-pbe0 and mp2) are used for binding energy calculations, a counterpoise correction should always be performed with the default NAO basis sets in FHI-aims to get reliable results, since the basis set superposition error (BSSE) for these correlation methods is significant. For these advanced correlation methods, the sequence of NAO valence-correlation consistent basis sets (NAO-VCC-nZ[327]) is a better choice, which reduces the basis set incompleteness error, including BSSE, with increasing the basis size, and especially enables to approach the completeness-basis-set limit with the aid of extrapolation scheme.

FHI-aims provides a wide range of current exchange-correlation options, ranging from local-density and generalized-gradient approximations (LDAs and GGAs) via hybrid functionals and Hartree-Fock to two-electron treatments of the correlated many-body system, such as second-order Møller-Plesset (MP2) theory and the random-phase approximation (RPA).

Implementations of exchange-correlation functionals are either available via the LibXC library [194] or, otherwise, as functionals that are implemented internally in FHI-aims in some form. Both options are covered below.

The following choices for the xc-type option are currently available:

• •

  A broad range of functionals is available through the LibXC library [194], of which v6.1.0 is distributed with FHI-aims at the time of writing. The implementation covers spin paired and polarised calculations for LDA, GGA and meta-GGA approaches (excluding those meta-GGAs on the laplacian of the density), as well as hybrid DFT, in equivalence to functionality with canonical functionals (energy, forces and stress).

  • –

    The general syntax is
    xc libxc <name>
    where e<name> is using the LibXC nomenclature (available at https://libxc.gitlab.io/functionals/), and exchange and correlation density functionals can be combined using a "+" sign.

  • –

    As an example, the LibXC call for PBE would be
    xc libxc GGA_C_PBE+GGA_X_PBE.

• •

  Local-density approximation (different parameterizations):

  • –

    pw-lda : Homogeneous electron gas based on Ceperley and Alder [54] as parameterized by Perdew and Wang 1992 [242]. Recommended LDA parameterization.

  • –

    pz-lda : Homogeneous electron gas based on Ceperley and Alder [54], as parameterized by Perdew and Zunger 1981 [243].

  • –

    vwn : LDA of Vosko, Wilk, and Nusair 1980 [308].

  • –

    vwn-gauss : LDA of Vosko, Wilk, and Nusair 1980, but based on the random phase approximation [273]. Do not use this LDA unless for one specific reason: In the B3LYP implementation of the Gaussian code, this functional is allegedly used instead of the correct VWN functional. It is therefore now present in many reference results in the literature, and also available here for comparison.

• •

  Generalized-gradient approximations:

  • –

    am05 : GGA functional designed to include surface effects in self-consistent density functional theory, according to Armiento and Mattsson [14]

  • –

    blyp : The BLYP functional: Becke (1988) exchange [27] and Lee-Yang-Parr correlation [190].

  • –

    pbe : GGA of Perdew, Burke and Ernzerhof 1997 [240].

  • –

    pbeint : PBEint functional of Ref. [88]

  • –

    pbesol : Modified PBE GGA according to Ref. [246].

  • –

    rpbe : The RPBE modified PBE functional according to Ref. [125].

  • –

    revpbe : The revPBE modified PBE GGA suggested in Ref. [329].

  • –

    r48pbe : The mixed functional containing 0.52*pbe and 0.48*rpbe according to Ref. [228]

  • –

    srp : The mixed functional containing a mix of RPBE and PBE according to Ref. [314] In this case, the additional option value is needed, representing the mixing ratio, mixing of RPBE. For example, a mixing of 0.48 is equivalent to R48PBE.

  • –

    pw91_gga : GGA according to Perdew and Wang, usually referred to as “Perdew-Wang 1991 GGA”. This GGA is most accessibly described in References 26 and 27 of Ref. [241]. Note that the often mis-quoted reference [242] does not(!) describe the Perdew-Wang GGA but instead only the correlation part of the local-density approximation described above.

  • –

    b86bbpe : Becke’s B86b exchange[28] plus PBE correlation[240].

• •

  Meta-generalized gradient approximations:

  • –

    m06-l : Truhlar’s optimized meta-GGA of the “M06” suite of functionals. [331]

  • –

    m11-l : Truhlar’s optimized range-separated local meta-GGA of the “M11” suite of functionals. [250]

  • –

    revtpss : Meta-GGA revTPSS functional of Ref. [244, 245].

  • –

    tpss : Meta-GGA TPSS functional of Ref. [293]

  • –

    tpssloc : Meta-GGA TPSSloc functional, thanks to E. Fabiano and F. Della Sala. L.A. Constantin, E. Fabiano, F.Della Sala, Ref. [63].

  • –

    scan or SCAN: “Strongly Constrained and Appropriately Normed Semilocal Density Functional,” i.e., the SCAN meta-GGA functional by Sun, Ruzsinszky, and Perdew [288]. Note that the in-house implementation of SCAN functional has been deprecated and the dfauto scan functional is currently provided in the FHI-aims. However, the old in-house SCAN functional can also be used by the keyword scan-deprecated.

• •

  Hartree-Fock and hybrid functionals (including non-local exchange): Please also see Secs. 3.25 and 3.26 for related keywords and technical hints.

  • –

    b3lyp : “B3LYP” hybrid functional as allegedly implemented in the Gaussian code (i.e., using the RPA version of the Vosk-Wilk-Nusair local-density approximation, see Refs. [308, 273] for details). Note that this is therefore not exactly the same B3LYP as originally described by Becke in 1993.

  • –

    hf : Hartree-Fock exchange only.

  • –

    hse03 : Hybrid functional as used in Heyd, Scuseria and Ernzerhof [139, 140]. In this functional, 25 % of the exchange energy is split into a short-ranged, screened Hartree-Fock part, and a PBE GGA-like functional for the long-range part of exchange. The remaining 75 % exchange and full correlation energy are treated as in PBE. As clarified in Refs. [180, 140], two different screening parameters were used in the short-range exchange part and long-range exchange part of the original HSE functional, respectively:
    Screened Hartree-Fock exchange: ${\omega }_{\text{HF}}=0.15/\sqrt{2}$
    Screened PBE-like exchange: ${\omega }_{\text{PBE}}=0.15\times 2^{1/3}$
    Following the notation of Ref. [180], the ’hse03’ functional in FHI-aims reproduces these original values exactly.

  • –

    hse06 : Hybrid functional according to Heyd, Scuseria and Ernzerhof [139], following the naming convention suggested in Ref. [180]. In this case, the additional option value is needed, representing the single real, positive screening parameter omega ($\omega$) as clarified in Ref. [180]. In this functional, 25 % of the exchange energy is split into a short-ranged, screened Hartree-Fock part, and a PBE GGA-like functional for the long-range part of exchange. The remaining 75 % exchange and full correlation energy are treated as in PBE.
    In the literature, the unit for $\mathrm{\omega }$ is either Å^-1 or (bohr radius)^-1, depending on the code, authors, and their favorite convention. To avoid any confusion, a separate keyword hse_unit must be specified in control.in, specifying either Å^-1 (’A’) or bohr^-1 (’b’). The code will no longer run without this explicit clarification. A correct calling syntax example is therefore:
    xc hse06 0.11
    hse_unit bohr-1¹¹ 1 The hse_unit flag reads only the first character. Thus this is equivalent to hse_unit b (case insensitive).
    or similar.
    A few comments on typical choices for $\omega$ in the earlier literature:
    The original value of 0.15 bohr^-1 by Heyd, Scuseria and Ernzerhoff 2003 [139] was never true - see their 2006 erratum. In FHI-aims, the ’hse03’ functional implements their actual choice.
    Krukau, Vydrov, Izmaylov and Scuseria 2006 [180] clarify the distinction between ’hse03’ and ’hse06’ (in addition to the Erratum mentioned above). Their conclusion is that omega=0.11 bohr^-1 is a reasonable choice.
    Vydrov, Heyd, Krukau and Scuseria in 2006 [236] appear to favor omega=0.25 bohr^-1, but with a mixing parameter (keyword hybrid_xc_coeff) of 0.5 for the short-range exchange. (The default for hybrid_xc_coeff in FHI-aims is 0.25, i.e., only a quarter of HF-like exchange.)
    You get the idea. As much as we would like to, we can not specify a single omega parameter for hse06 by default – the choice is up to you. Apologies for the inconvenience.

  • –

    pbe0 : PBE0 hybrid functional [3], mixing 75 % GGA exchange with 25 % Hartree-Fock exchange.

  • –

    pbe0_prime : PBE0 hybrid functional [3], mixing 75 % GGA exchange with 25 % Hartree-Fock exchange, and using the extended range-separation function with $\omega =0.11$ Bohr^-1 [174]. For the use of the extended range-separation function for other hybrid functionals, please use the keyword exx_lr_approximation.

  • –

    pbesol0 : Hybrid functional in analogy to PBE0 [3], except that the PBEsol [246] GGA functionals are used, mixing 75 % GGA exchange with 25 % Hartree-Fock exchange.

  • –

    b86bpbe-25 : B86bPBE hybrid functional [28, 240], mixing 75 % GGA exchange with 25 % Hartree-Fock exchange.

  • –

    b86bpbe-50 : B86bPBE hybrid functional [28, 240], mixing 50 % GGA exchange with 50 % Hartree-Fock exchange.

  • –

    b86bpbe0 : B86bPBE hybrid functional [28, 240], defaults to mixing 75 % GGA exchange with 25 % Hartree-Fock exchange, but is intended to be set with hybrid_xc_coeff

  • –

    lc_wpbeh : Range separated hybrid functional LC-$\omega$PBEh using 100 % Hartree-Fock exchange in the long-range part and $\omega$PBE [236] in the short-range part. The full correlation energy is treated as in PBE. The hse_unit must be specified as in hse06!
    Syntax:
    xc lc_wpbeh $\omega$ $\alpha$
    

    $$E_{xc}^{\text{LC-}\omega \text{PBEh}}=\alpha E_{xx}^{\text{SR}}+(1-\alpha )E_{x_{\omega \text{PBE}}}^{\text{SR}}+\left(\frac{1}{ϵ}\right)E_{xx}^{\text{LR}}+\left(1-\frac{1}{ϵ}\right)E_{\omega \text{PBE}}^{\text{LR}}+E_{c_{\text{PBE}}}$$

    (3.10)

    • *

      $ϵ$ can be the dielectric constant. The default value is 1. One might change this parameter with the keyword lc_dielectric_constant

    • *

      If $\alpha =0$ the functional is also known as LC-$\omega$PBE [102]

    • *

      $\alpha =1$ would correspond to a PBE0 calcuation with 100 % Hartree-Fock exchange

  • –

    kc_xc: Koopmans-complliant screend exchange [204]. For more details and example input for control.in, see 3.26.

• •

  Hybrid Meta-generalized gradient functionals (including non-local exchange): Please also see Secs. 3.25 and 3.26 for related keywords and technical hints. Currently the non-local exchange contribution is fixed in all implementations due to the parameterised nature of these density functionals.

  • –

    m06 : Truhlar’s optimized hybrid meta-GGA of the “M06” suite of functionals; with 27% exact exchange. [333]

  • –

    m06-2x : Truhlar’s optimized hybrid meta-GGA of the “M06” suite of functionals, with double contribution (54%) from the hartree-fock exact exchange. [333]

  • –

    m06-hf : Truhlar’s optimized hybrid meta-GGA of the “M06” suite of functionals, with 100% exact exchage contribution. [332]

  • –

    m08-hx : Truhlar’s optimized hybrid meta-GGA of the “M08” suite of functionals, with 52.23% contribution from the hartree-fock exact exchange. [334]

  • –

    m08-so : Truhlar’s optimized hybrid meta-GGA of the “M08” suite of functionals, with 56.79% contribution from the hartree-fock exact exchange. [334]

  • –

    m11 : Truhlar’s optimized range-separated local meta-GGA of the “M11” suite of functionals [249]. The range-separation variable is also hardcoded in this implementation with $\omega =0.25$ bohr^-1.

• •

  Alternative implementations of some XC functionals via the dfauto program [286]. These implementations are generated automatically from Maple definitions that are located in xc_dfauto/. The general syntax is xc dfauto <name> where <name> can be one of (case-insensitive):

  • –

    dfauto pw-lda|pbe|pbe0|tpss : This is practically identical to specifying directly xc <name>, and essentially provides alternative implementations of those functionals for testing purposes.

  • –

    dfauto scan : This is the meta-GGA functional SCAN [288].

  • –

    dfauto rscan : This is a Revision of the meta-GGA functional SCAN (rSCAN) [23].

  • –

    dfauto scan0 : This is the meta-GGA hybrid functional SCAN0 [149], which mixes SCAN with 25% of exact exchange.

• •

  Double-hybrid functionals (including non-local exchange and correlation): Double-hybrid functionals are emerging quickly in the last decade. “double-hybrid” here means that the exchange functional mixes LDA(and/or GGA) exchange with “Hartree-Fock like exact exchange”. Meanwhile, the correlation functional is composed of both conventional LDA(and/or GGA) correlation and second-order perturbation energy. Doubly-hybrid functionals are “semi-empirical”, generally including several empirical parameters determined by optimizing against one or several well-chosen databases. Double-hybrid functionals show a remarkable improvement over conventional (hybrid-)GGAs in the description of heats of formation, bond dissociation enthalpies, reaction barrier heights and weak interactions of the main group elements. Doulbe-hybrid functionals have become new leading actors in the field of computational chemistry.

  • –

    xyg3 : Double-hybrid functional XYG3, containing 80.33% Hartree-Fock exchange and 32.11% second-order perturbation energy [328].

  • –

    xdh-pbe0 : Double-hybrid functional xDH-PBE0, containing 83.51% Hartree-Fock exchange and 52.42% opposite-spin second-order perturbation correlation [326].

• •

  Some specific correlated methods: Only a subset. For many correlated methods that can be used as non-selfconsistent perturbative post-processing methods after an initial s.c.f. calculation, see the total_energy_method keyword. Most of these are not available for periodic geometries, or, if at all, in a very experimental state.

  • –

    mp2 : Self-consistent Hartree-Fock, followed by a second-order Møller-Plesset perturbative addition of the correlation energy. Note that the frozen_core keyword can be used to specify if and which low-lying states should be excluded from the correlation energy. For spin-component scaled MP2 [115], see keyword scs_mp2_parameters.
    Note that when mp2 is used for binding energy calculations, a counterpoise correction should always be performed to get reliable results, since the basis set superposition error (BSSE) for these correlation methods is significant. Note added in March 2016: A periodic implementation of MP2 is available but, at the time of writing, computationally extremely expensive. If you decide to use it, please keep in mind that the periodic version is included here as a matter of protocol but is not yet optimized to be fully usable in production calculations.

  • –

    screx : experimental! Self-consistent, screened Hartree-Fock exchange only. The Coulomb operator is screened as:

    $$\frac{1}{r-r^{\prime }}\to \frac{1}{\epsilon (r,r^{\prime })}\cdot \frac{1}{r-r^{\prime }}$$

    (3.11)

    $\epsilon (r,r^{\prime })$ is the non-local microscopic dielectric function, obtained in the $\omega \to$0 frequency limit of the random-phase approximation (RPA). See Ref. [130] for details.

  • –

    cohsex : experimental! Self-consistent screened exchange plus Coulomb-hole (COH) correlation. See Ref. [130] for details.

• •

  Method of non-local correlation using the “van der Waals density functional” (vdw-DF) as presented by Dion and coworkers in Ref. [76] or the second version of the vdw-DF functional (vdw-DF2) as presented by Lee and coworkers in Ref. [191]. As implemented here, the vdw-DF is provided for reference purposes but it is much more expensive than other dispersion corrections in FHI-aims. The following options are available – as mentioned, very slow:

  • –

    pbe_vdw : the vdw-DF functional with pbe exchange

  • –

    revpbe_vdw : the vdW-DF functional with revpbe exchange

  • –

    vdw-df2 : the vdW-DF2 functional with PW86 exchange

  • –

    beef-vdw : the generalized gradient BEEFvdW functional by Wellendorff and coworkers [313]. This implementation uses exchange and correlation from libxc.

  • –

    beef2-vdw2 : the generalized gradient BEEFvdW functional by Wellendorff and coworkers [313]. This implementation uses only exchange from libxc.

  Note that these keywords are not the correction due to Tkatchenko and Scheffler 2009 [295]. To activate the Tkatchenko-Scheffler correction instead, use the vdw_correction_hirshfeld keyword. The functional by Dion et al. is a very different functional. To use the functional by Dion et al., please review the numerical options described in Sec. 3.24.2.

  A benchmark of options vdw-df2, beef-vdw, beef2-vdw2 (implemented and available), and mbeef2_vdw2 (implemented but disabled due to non-negligible errors in testing) is available in Ref. [155]

Note that our version of the Coulomb operator (which is the basis for Hartree-Fock exchange also in hybrid functionals, as well as MP2 theory) is based on an auxiliary basis in what is known as resolution of the identity (Refs. [41, 6, 302, 86] and others). While our default settings should be safe, you may wish to consult Sec. 3.25 for particulars regarding this auxiliary basis.

Note also that some different perturbative exchange-correlation treatments for post-processing (after a self-consistent DFT or HF calculation is complete) may be invoked using the tag total_energy_method. Likewise, perturbative postprocessing for single-quasiparticle energies through a self-energy formalism (e.g., $GW$) is reached by specifying the qpe_calc tag and its options.

Right now, the correlated beyond-hybrid and beyond-meta methods are not implemented on top of the HSE03 or HSE06 functionals.

Calculating the GGA functional explicitly brings the lowest eigenvalues for the heavy atoms (Pb, Bi) calculated on light grids on par with their counterparts calculated on the logarithmic grid in the atomic solver, thus bringing the calculated analytical stress on the sparse grids closer to the numerical value.

The explicit calculation of potential makes use of the xc energy derivatives readily available in libxc, so it’s dependent on the libxc representation of the potential. The formula for the potential is given by:

$${𝐕}_{\mathrm{𝐱𝐜}}^{\alpha }=\frac{\partial f}{\partial {\rho }^{\alpha }}-2\left[\frac{\partial f}{\partial {\sigma }^{\alpha }}\mathrm{\Delta }{\rho }^{\alpha }+\sum _{\beta }\left(\frac{{\partial }^{2}f}{\partial {\sigma }^{\alpha }\partial {\rho }^{\beta }}\nabla {\rho }^{\beta }+\frac{{\partial }^{2}f}{\partial {\sigma }^{\alpha }\partial {\sigma }^{\beta }}\nabla {\sigma }^{\beta }\right)\cdot \nabla {\rho }^{\alpha }\right],$$

(3.12)

where $\alpha$ and $\beta$ are the spin indices, ${\sigma }^{\alpha }={|\nabla {\rho }^{\alpha }|}^2$, and $f$ is the energy density of the GGA functional. Here we rely on the functional not depending on the cross-spin term $\nabla {\rho }^{\uparrow }\cdot \nabla {\rho }^{\downarrow }$.