[Pw_forum] Re: projwfc.f90

[Andrea Floris]

Many thanks Paolo for your suggestions, very useful.
cheers
Andrea

> Message: 7
> From: Paolo Giannozzi <giannozz at nest.sns.it>
> Organization: Scuola Normale Superiore di Pisa
> To: pw_forum at pwscf.org
> Subject: Re: [Pw_forum] projwfc.f90
> Date: Tue, 11 Oct 2005 17:33:46 +0200
> Reply-To: pw_forum at pwscf.org
>
> On Tuesday 11 October 2005 16:52, Andrea Floris wrote:
>
> > does it exists some documentation (papers, some notes or whatever)
> > on which the program projwfc.f90 (and the routines that it calls) is
> > based?
>
> some time ago I wrote some notes (latex file below) about Loewdin
> charges. Hope this helps
>
> Paolo
> --
> Paolo Giannozzi             e-mail:  giannozz at nest.sns.it
> Scuola Normale Superiore    Phone:   +39/050-509876, Fax:-563513
> Piazza dei Cavalieri 7      I-56126 Pisa, Italy
> --
> \documentclass[a4paper,12pt]{article}
>
> \begin{document}
>
> \section{L\"owdin and Mulliken population analysis}
>
> We have $| \phi_\mu\rangle$ atomic (or atomic-like)
> orbitals, not necessarily orthonormal, and we want
> to write KS orbitals for our system:
> $|\psi_\alpha\rangle$, as sums over said
> atomic orbitals:
> $$
> |\psi_\alpha\rangle = \sum_\mu c_\mu^{(\alpha)} | \phi_\mu \rangle.
> $$
> The generalized orthonormality relations for KS orbitals
> is written as:
> $$
> \langle\psi_\alpha| \hat S |\psi_\beta\rangle = \delta_{\alpha\beta}.
> $$
> where $\hat S$ is an operator defined in the US PP framework.
> The charge density $\rho({\bf r})$ is given by
> $$
> \rho({\bf r}) = \sum_{\mu,\nu} P_{\mu\nu}
> \left (\phi^*_\nu \hat S \phi_\mu\right) ({\bf r})
> $$
> where
> $$
> P_{\mu\nu} = \sum_\alpha  c_\mu^{(\alpha)} c_\nu^{(\alpha)*}
> $$
> define an operator $\hat P$, and
> $$
> \left (\phi^*_\nu \hat S \phi_\mu\right) ({\bf r}) =
> \phi^*_\nu({\bf r}) \phi_\mu({\bf r}) +
> \sum_{lm} \langle\phi_\nu|\beta_l\rangle q_{lm}({\bf r})
>           \langle\beta_m|\phi_\mu\rangle
> $$
> where the $\beta$'s and $q$'s are components of the US PP.
>
> \subsection{Mulliken population analysis}
>
> We write the total number of electrons $N$ as:
> $$
> N = \int \rho({\bf r}) d{\bf r}
>   = \mbox{Tr} \hat P\hat S = \sum_\mu (\sum_\nu P_{\mu\nu} S_{\nu\mu})
>          \equiv \sum_\mu q_\mu
> $$
> where $q_\mu$ is the {\em Mulliken charge} associated to state $\mu$ :
> $$
> q_\mu = \sum_\nu \sum_\alpha c_\mu^{(\alpha)} c_\nu^{(\alpha)*} S_{\nu\mu}
> $$
> and
> $$
> S_{\nu\mu} = \langle\phi_\nu|\hat S|\phi_\mu\rangle
>            = \int \phi^*_\nu({\bf r}) \phi_\mu({\bf r}) d{\bf r} +
>              \sum_{lm} \langle\phi_\nu|\beta_l\rangle Q_{lm}
>              \langle\beta_m|\phi_\mu\rangle
> $$
> with
> $$
> Q_{lm}=\int q_{lm}({\bf r}) d{\bf r}.
> $$
> In general, this matrix is not diagonal, even with NC PP.
> The coefficients $c_\mu^{(\alpha)}$ are obtained by inverting the linear
> system:
> $$
> \langle \phi_\nu | \hat S | \psi_\alpha\rangle =
>   \sum_\mu c_\mu^{(\alpha)} S_{\nu\mu}
> $$
> that is
> $$
>   c_\mu^{(\alpha)} = (\hat S^{-1})_{\mu\nu}
>   \langle \phi_\nu | \hat S | \psi_\alpha\rangle
> $$
> and finally
> $$
> q_\mu  = \sum_{\alpha\nu}
>          \langle \psi_\alpha | \hat S | \phi_\mu\rangle
>          (\hat S^{-1})_{\mu\nu}
>          \langle \phi_\nu | \hat S | \psi_\alpha\rangle.
> $$
>
> \subsection{L\"owdin population analysis}
>
> The total number of electrons $N$ can be alternatively written as
> $$
> N = \mbox{Tr} \left[\hat S^{1/2}\hat P \hat S^{1/2}\right]
>   = \sum_\mu \tilde q_\mu.
> $$
> where $\tilde q_\mu$ is called {\em L\"owdin charge} associated to state
> $\mu$. Let us introduce an auxiliary set of atomic orbitals
> $\tilde\phi$ via :
> $$
>  |\tilde\phi_\mu\rangle = \sum_\nu (\hat S^{-1/2})_{\nu\mu}| \phi_\nu\rangle,
> \qquad
>  | \phi_\mu\rangle = \sum_\nu (\hat S^{1/2})_{\nu\mu}|\tilde\phi_\nu\rangle,
> $$
> for which the generalized orthonormality relation
> $$
>  \langle\tilde\phi_\mu|\hat S| \tilde\phi_\nu\rangle = \delta_{\mu\nu}
> $$
> holds. The KS orbitals for our systems can be rewritten as
> $$
> |\psi_\alpha\rangle = \sum_\mu c_\mu^{(\alpha)} | \phi_\mu \rangle
>                     = \sum_\mu \tilde c_\mu^{(\alpha)} | \tilde\phi_\mu
> \rangle
> $$
> where
> $$
> \tilde c_\mu^{(\alpha)} = \sum_\nu (\hat S^{1/2})_{\nu\mu} c_\nu^{(\alpha)}.
> $$
> By comparison with the above expression of $\tilde q_\mu$ we get
> $$
> \tilde q_\mu = \sum_\alpha |\tilde c_\mu^{(\alpha)}|^2
>        = |\langle \tilde\phi_\mu | \hat S | \psi_\alpha\rangle|^2
> $$
>
> Reference: {\em Modern Quantum Chemistry}, A. Szabo and N. Ostlund (Dover, NY
> 1996),
> p. 153
> \end{document}
>
>