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小木虫论坛-学术科研互动平台 » 计算模拟区 » 第一性原理 » Vasp&MedeA » 请问VASP的LDA+U ，LDAUTYPE=1是什么方法？
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jugengfans

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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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1楼2016-05-06 13:30:44
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jugengfans

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2楼: Originally posted by monkeylove at 2016-05-06 19:22:54
对于有些特别的材料需要+U、、我+U是LDAUTYPE用的默认值2、、通过U，J可以调变你要加的Ueff值、、

看有的方法是 全各向异性 计算
其中U和J是矩阵，J等于0时U就只有对角元。J不等于0时U还有非对角元。这个貌似用的不多不过和effective U应该不同吧。问题在于哪一个更靠谱？

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3楼2016-05-06 21:19:18
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monkeylove

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对于有些特别的材料需要+U、、我+U是LDAUTYPE用的默认值2、、通过U，J可以调变你要加的Ueff值、、
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2楼2016-05-06 19:22:54
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半尾残狐

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我记着这个参数代表着不同的加U方法的，2的话只需要考虑U-J的差值

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4楼2016-05-07 07:24:45
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jugengfans

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引用回帖:
4楼: Originally posted by 半尾残狐 at 2016-05-07 07:24:45
我记着这个参数代表着不同的加U方法的，2的话只需要考虑U-J的差值

谢谢回复，这个是vasp默认的值。
对于这种方法U-J相同的时候，无论U和J的绝对数值怎么变
给出来的电子结构都一模一样。

但是似乎LDAUTYPE=1的方法考虑了U的各向异性，这种各向异性是和J有关的。U是一个矩阵。这个矩阵的矩阵元和U、J都有关系，在J=0时，矩阵变成一个只有对角元的矩阵，此时U才没有各向异性，变成一个同样的值。

我在想vasp里面的第一种方法是不是就是所谓的 full- anisotropy 的U方法？
它的精确性又如何呢？如果更精确的话为什么没有广泛应用？有谁用过吗？
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5楼2016-05-07 10:14:04
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