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[交流] 请问VASP的LDA+U ，LDAUTYPE=1是什么方法？已有3人参与

看了文献，说vasp的U可以采用各向异性的方法，U和J都是矩阵
这个是不是就对应LDAUTYPE=1?
默认的是2,有人用1吗？
付手册内容。

LDAUTYPE = 1 | 2 | 4
Default: LDAUTYPE = 2

Description: LDAUTYPE specifies which type of L(S)DA+U approach will be used.

The L(S)DA often fails to describe systems with localized (strongly correlated) d and f-electrons (this manifests itself primarily in the form of unrealistic one-electron energies). In some cases this can be remedied by introducing a strong intra-atomic interaction in a (screened) Hartree-Fock like manner, as an on-site replacement of the L(S)DA. This approach is commonly known as the L(S)DA+U method. Setting LDAU=.TRUE. in the INCAR file switches on the L(S)DA+U.

LDAUTYPE=1: The rotationally invariant LSDA+U introduced by Liechtenstein et al.[1]

This particular flavour of LSDA+U is of the form

      E_{{{\rm {HF}}}}={\frac {1}{2}}\sum _{{\{\gamma \}}}(U_{{\gamma _{1}\gamma _{3}\gamma _{2}\gamma _{4}}}-U_{{\gamma _{1}\gamma _{3}\gamma _{4}\gamma _{2}}}){{\hat n}}_{{\gamma _{1}\gamma _{2}}}{{\hat n}}_{{\gamma _{3}\gamma _{4}}}

and is determined by the PAW on-site occupancies

      {{\hat n}}_{{\gamma _{1}\gamma _{2}}}=\langle \Psi ^{{s_{2}}}\mid m_{2}\rangle \langle m_{1}\mid \Psi ^{{s_{1}}}\rangle

and the (unscreened) on-site electron-electron interaction

      U_{{\gamma _{1}\gamma _{3}\gamma _{2}\gamma _{4}}}=\langle m_{1}m_{3}\mid {\frac {1}{|{\mathbf {r}}-{\mathbf {r}}^{\prime }|}}\mid m_{2}m_{4}\rangle \delta _{{s_{1}s_{2}}}\delta _{{s_{3}s_{4}}}

where |m〉 are real spherical harmonics of angular momentum L=LDAUL.

The unscreened e-e interaction Uγ1γ3γ2γ4 can be written in terms of the Slater integrals F^{0}, F^{2}, F^{4}, and F^{6} (f-electrons). Using values for the Slater integrals calculated from atomic orbitals, however, would lead to a large overestimation of the true e-e interaction, since in solids the Coulomb interaction is screened (especially F^{0}).

In practice these integrals are therefore often treated as parameters, i.e., adjusted to reach agreement with experiment in some sense: equilibrium volume, magnetic moment, band gap, structure. They are normally specified in terms of the effective on-site Coulomb- and exchange parameters, U and J (LDAUU and LDAUJ, respectively). U and J are sometimes extracted from constrained-LSDA calculations.

These translate into values for the Slater integrals in the following way (as implemented in VASP at the moment):

      L\; F^{0}\; F^{2}\; F^{4}\; F^{6}\;
      1\; U\; 5J\; - -
      2\; U\; {\frac {14}{1+0.625}}J 0.625F^{2}\; -
      3\; U\; {\frac {6435}{286+195\cdot 0.668+250\cdot 0.494}}J 0.668F^{2}\; 0.494F^{2}\;

The essence of the LSDA+U method consists of the assumption that one may now write the total energy as:

      E_{{{\mathrm {tot}}}}(n,{\hat n})=E_{{{\mathrm {DFT}}}}(n)+E_{{{\mathrm {HF}}}}({\hat n})-E_{{{\mathrm {dc}}}}({\hat n})

where the Hartree-Fock like interaction replaces the LSDA on site due to the fact that one subtracts a double counting energy E_{{{\mathrm {dc}}}}, which supposedly equals the on-site LSDA contribution to the total energy,

      E_{{{\mathrm {dc}}}}({\hat n})={\frac {U}{2}}{{\hat n}}_{{{\mathrm {tot}}}}({{\hat n}}_{{{\mathrm {tot}}}}-1)-{\frac {J}{2}}\sum _{\sigma }{{\hat n}}_{{{\mathrm {tot}}}}^{\sigma }({{\hat n}}_{{{\mathrm {tot}}}}^{\sigma }-1).

LDAUTYPE=2: The simplified (rotationally invariant) approach to the LSDA+U, introduced by Dudarev et al.[2]

This flavour of LSDA+U is of the following form:

      E_{{{\mathrm {LSDA+U}}}}=E_{{{\mathrm {LSDA}}}}+{\frac {(U-J)}{2}}\sum _{\sigma }\left[\left(\sum _{{m_{1}}}n_{{m_{1},m_{1}}}^{{\sigma }}\right)-\left(\sum _{{m_{1},m_{2}}}{\hat n}_{{m_{1},m_{2}}}^{{\sigma }}{\hat n}_{{m_{2},m_{1}}}^{{\sigma }}\right)\right].

This can be understood as adding a penalty functional to the LSDA total energy expression that forces the on-site occupancy matrix in the direction of idempotency,

      {\hat n}^{{\sigma }}={\hat n}^{{\sigma }}{\hat n}^{{\sigma }}.

Real matrices are only idempotent when their eigenvalues are either 1 or 0, which for an occupancy matrix translates to either fully occupied or fully unoccupied levels.

Note: in Dudarev's approach the parameters U and J do not enter seperately, only the difference (U-J) is meaningful.

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