关注：

1） HSE引入所要解决的问题

2） HSE的具体物理内涵

3） HSE03与HSE06的区别

    the Heyd-Scuseria-Ernzerhof (HSE) screened hybrid functional

    The parameter  a=1/4 is the HF mixing constant derived from perturbation theory.    

   

 

is the parameter that defines the  range-separation, and is related to a characteristic distance, (),  at which the short-range interactions become negligible.

Note: It has been shown [93] that the optimum ,  controlling the range separation is approximately Å. To conform with the HSE06 functional you need to select (HFSCREEN=0.2) [93,94,95].

Using the decomposed Coulomb kernel and Equ. (6.59), one straightforwardly obtains:

 

 

 

 

参考：

http://scitation.aip.org/content/aip/journal/jcp/125/22/10.1063/1.2404663

http://cms.mpi.univie.ac.at/wiki/index.php/Specific_hybrid_functionals

http://scitation.aip.org/content/aip/journal/jcp/124/21/10.1063/1.2204597

 

 

Applications of KohnSham density functional theory to solids mostly employ semilocal approximations to the exchange-correlation energy, 

     such as the functional of Perdew, Burke, and Ernzerhof (PBE).    ¹ 

where the screening parameter  w  defines the separation range.

 

However, semi-local functionals have some important shortcomings.【半局域函数】 

 

For instance,

(1)  they systematically underestimate band gaps  with respect to experiment.                         ^2–4 

(2)  Also, semilocal functionals overestimate electron delocalization effects 【高估电子局域作用】and therefore fail for many                          d - and                          f -element compounds.    

 

 

Recent studies  have shown that hybrid density functionals, which include a portion of Hartree-Fock (HF) exchange, partially solve these   problems.

 

However, HF exchange drastically increases the computational demand for calculations in periodic systems【HF非常耗时】.

                   

An efficient alternative to conventional hybrids is the screened hybrid functional【屏蔽杂化泛函】 of Heyd, Scuseria, and Ernzerhof (HSE).                        ^12–16

 

 

In HSE, the spatial decay of the HF exchange interaction 【HF交换项空间衰变】is accelerated by substitution of the full                          1/r  Coulomb potential with a screened one【全库伦势】.

 

 

This enables a substantial lowering of the computational cost for calculations in  extended systems【与Roald的观点一致，extended systems】.

 

The HSE functional partitions 【分开/分割】the Coulomb potential for exchange into short-range (SR) and long-range (LR)  components:                          

where the screening parameter  w  defines the separation range.

 

The exchange-correlation energy is then calculated as

 

                       

where xx is the short-range HF exchange【短程作用HF交换项】,                          E PBE,SR x   and xxx and xxx                         E PBE,LR x   are the short-range and long-range components of the PBE exchange functional obtained by integration of the model PBE exchange            hole,                         ^13,17 and              xxxx            E PBE c   is the PBE correlation energy.

 

 

The parameter    a=1/4   is the HF mixing constant derived from perturbation theory.    

 

                    ¹⁸ Note that HSE may be viewed as an adiabatic【绝热的】 connection functional,                         ¹⁸ but only in the short-range portion of the potential.

 

There is no long-range HF exchange in HSE, only short-range HF.

 

 

For w=0 HSE reduces to the hybrid functional PBEh [also known as PBE0 (Ref.                          19) or PBE1PBE (Ref.                          20) in the literature], and

for     w--- $\infty$                     ω→∞ , HSE becomes identical with PBE.  

 

                     ²¹ HSE with a finite value of    w   can be regarded as an interpolation between these two limits  [PBEh和PBE]. 

 

A comparative study of HSE and PBEh performance for solids            was recently published.                         ²²

 

The value of the screening parameter was originally selected based on molecular tests                         ¹² and yielded excellent band gaps for solids.

 

                       ^15,16 However, the screening parameter                          ω=0.15bohr −1   quoted in our previous publications                         ^11–16 is not the one used in the code.

 

Because of interpretation errors,  was used instead of   ω  in the Hartree-Fock part of the code, whereas                          2 1/3 ω  was used in the PBE part both for short and long ranges (see recent Erratum to Ref.                          12).

There is no   a priori reason to use different screening parameters for PBE and HF exchange. Moreover, HSE is exact in the uniform electron gas   limit only when  W_HF=W_PBE                        ω HF =ω PBE  .                      

In this work, we reexamine how the choice of the screening parameter                          ω  affects the HSE performance when both screening parameters are the same.

 

 

Note that in previous publications                         ^12–16 only the influence of screening parameter on enthalpies of formation was discussed.                         ¹² To avoid confusion, we will here refer to the previous HSE version with   as HSE03. The current version with  

W_HF=W_PBE        will here be denoted HSE06.

 

 

摘录：HFSCREEN and LTHOMAS

http://cms.mpi.univie.ac.at/vasp/vasp/HFSCREEN_LTHOMAS.html

 

HFSCREEN= [real]

Default: none

HFSCREEN determines the range separation parameter in range separated hybrid functionals.

 

In combination with PBE potentials, attributing a value to HFSCREEN will switch from the PBE0 functional (in case LHFCALC=.TRUE.) to the closely related HSE03 or HSE06 functional [93,94,95].

 

The HSE03 and HSE06 functional replaces the slowly decaying long-ranged part of the  Fock exchange, by the corresponding density functional counterpart.

 

The resulting expression for the exchange-correlation energy is given by:

 

 

As can be seen above, the separation of the electron-electron interaction into a short- and long-ranged part, labeled SR and LR respectively, is realized only in the exchange interactions.

Electronic correlation is represented by the corresponding part of the PBE density functional.

 

The decomposition of the Coulomb kernel is obtained using the  following construction (HFSCREEN):

 

 

 

where , and is the parameter that defines the  range-separation, and is related to a characteristic distance, (),  at which the short-range interactions become negligible.

Note: It has been shown [93] that the optimum ,  controlling the range separation is approximately Å. To conform with the HSE06 functional you need to select (HFSCREEN=0.2) [93,94,95].

Using the decomposed Coulomb kernel and Equ. (6.59), one straightforwardly obtains:

 

 

The representation of the corresponding short-ranged Fock potential in reciprocal space is given by

Clearly, the only difference to the reciprocal space representation of the  complete (undecomposed) Fock exchange potential, given by Equ. (6.63), is the second factor in the summand in Equ. (6.68), representing the  complementary error function in reciprocal space.

The short-ranged PBE exchange energy and potential, and their long-ranged counterparts, are arrived at using the same decomposition  [Equ. (6.66)], in accordance with Heyd et al. [93] It is easily seen from Equ. (6.66) that the long-range term becomes zero for , and the short-range contribution then equals the full Coulomb operator, whereas for it is the other way around. Consequently, the two limiting cases of the HSE03/HSE06 functional [see Equ. (6.65)] are a true PBE0 functional for , and a pure PBE calculation for .

Note: A comprehensive study of the performance of the HSE03/HSE06 functional compared to the PBE and PBE0 functionals can be found in Ref. [99].  The B3LYP functional was investigated in Ref. [100]. Further applications of hybrid functionals to selected materials can be found in the following references: Ceria (Ref. [101]), lead chalcogenides (Ref. [102]),  CO adsorption on metals (Refs. [103,104]), defects in ZnO  (Ref. [105]),  excitonic properties (Ref. [106]),  SrTiO and BaTiO (Ref.  [107]).

LTHOMAS= .TRUE. | .FALSE.

Default: LTHOMAS=.FALSE.

If the flag LTHOMAS is set, a similar decomposition of the exchange functional into a long range and a short range part is used. This time, it is more convenient to write the decomposition in reciprocal space:

 

 

where is the Thomas-Fermi screening length. HFSCREEN is used to specify the parameter . For typical semi-conductors, the Thomas-Fermi screening length is about 1.8 Å, and setting  HFSCREEN to this value yields reasonable band gaps for most materials. In principle, however, the  Thomas-Fermi screening length depends on the valence electron density;  VASP determines this parameter from the number of valence electrons (POTCAR) and the volume and writes the corresponding value to the OUTCAR file: Thomas-Fermi vector in A = 2.00000 Since, VASP counts the semi-core states and -states as valence electrons, although these states  do not contribute to the screening, the values reported by VASP are, however, often incorrect. Details can be found in literature [96,97,98]. Another important detail concerns that implementation of the density functional part in  the screened exchange case. Literature suggests that a global enhancement factor (see Equ. (3.15) in Ref. [98])  should be used, whereas VASP implements a local density dependent enhancement factor , where is the Fermi wave vector corresponding to the local density (and not the average density as suggested in Ref. [98]). The VASP implementation is in the spirit of the local density approximation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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