Category:Wannier functions: Difference between revisions
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Here, <math>U_{mn\mathbf{k}}</math> is a unitary matrix which can be generated using different approaches discussed below, <math>m</math> is an index enumerating Wannier functions with position <math>\mathbf{R}</math>, <math>n</math> is the band index, and <math>\mathbf{k}</math> is the Bloch vector. | Here, <math>U_{mn\mathbf{k}}</math> is a unitary matrix which can be generated using different approaches discussed below, <math>m</math> is an index enumerating Wannier functions with position <math>\mathbf{R}</math>, <math>n</math> is the band index, and <math>\mathbf{k}</math> is the Bloch vector. | ||
Generally, one starts with an initial guess for <math>U_{mn\mathbf{k}}</math> that is build from <math>A_{mn\mathbf{k}}</math>. The latter can be | Generally, one starts with an initial guess for <math>U_{mn\mathbf{k}}</math> that is build from <math>A_{mn\mathbf{k}}</math>. The latter can be built from projections onto some localized-orbital basis. | ||
== One-shot singular-value decomposition (SVD)== | == One-shot singular-value decomposition (SVD)== | ||
Revision as of 11:47, 17 May 2024
Wannier functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle |w_{m\mathbf{R}}\rangle} are constructed by a linear combination of Bloch states Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle |\psi_{n\mathbf{k}}\rangle} , i.e., the computed Kohn-Sham (KS) orbitals, as follows:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle |w_{m\mathbf{R}}\rangle = \sum_{n\mathbf{k}} e^{-i\mathbf{k}\cdot\mathbf{R}} U_{mn\mathbf{k}} |\psi_{n\mathbf{k}}\rangle. }
Here, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle U_{mn\mathbf{k}}} is a unitary matrix which can be generated using different approaches discussed below, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): m is an index enumerating Wannier functions with position Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\mathbf {R}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): n is the band index, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\mathbf {k}} is the Bloch vector. Generally, one starts with an initial guess for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle U_{mn\mathbf{k}}} that is build from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle A_{mn\mathbf{k}}} . The latter can be built from projections onto some localized-orbital basis.
One-shot singular-value decomposition (SVD)
In one-shot SVD, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle A_{mn\mathbf{k}}} is computed by projecting the KS orbitals onto localized orbitals basis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \phi_{m\mathbf{k}}} that is specified by the LOCPROJ tag:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle A_{mn\mathbf{k}} = \langle \psi_{n\mathbf{k}} | S |\phi_{m\mathbf{k}}\rangle, }
where
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \phi_{i\mathbf{k}}(\mathbf{r}) = e^{\mathrm{i}\mathbf{k}\cdot\mathbf{r}} Y_{lm}(\hat{r})R_n(r). }
Note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): i encodes the quantum numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): n , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): l , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): m . Thus, in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle A_{mn\mathbf{k}}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): m is not the magnetic quantum number.
Then, VASP performs one-shot SVD for each k point
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle A_{mn\mathbf{k}} = [D \Sigma V^*]_{mn\mathbf{k}} }
to obtain the unitary matrix
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle U_{mn\mathbf{k}} = [DV^*]_{mn\mathbf{k}}. }
Selected columns of the density matrix (SCDM)
The SCDM method [1] is switched on using LSCDM. It has the advantage that the specification of a local basis in terms of atomic quantum numbers is omitted.
Maximally localized Wannier functions using Wannier90
The interface of VASP with the Wannier90 code is mainly controlled by LWANNIER90 and LWANNIER90_RUN. First, the initial guess for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle A_{mn\mathbf{k}}} can be created by providing the projections block in the wannier90.win file (also see WANNIER90_WIN) and setting LWANNIER90=True.
In order to obtain maximally localized Wannier functions, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle U_{mn\mathbf{k}}} is constructed in a second step. For this, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle A_{mn\mathbf{k}}} could be created using any projection method in the first step, i.e., single-shot SVD method (LOCPROJ), SCDM method (LSCDM), or Wannier90 (LWANNIER90). Then, Wannier90 can be executed directly or through VASP with the LWANNIER90_RUN tag.
References
Pages in category "Wannier functions"
The following 19 pages are in this category, out of 19 total.