mirror of https://github.com/abinit/abinit.git
143 lines
6.1 KiB
Markdown
143 lines
6.1 KiB
Markdown
---
|
|
authors: MG
|
|
---
|
|
|
|
$$
|
|
\newcommand{\aa}{\alpha}
|
|
\newcommand{\PS}{{\text{PS}}}
|
|
\newcommand{\Ylm}{{Y_m^l}}
|
|
\newcommand{\Vloc}{V_{\text{loc}}}
|
|
\newcommand{\vv}{\hat v}
|
|
\newcommand{\vloc}{\vv_{\text{loc}}}
|
|
\newcommand{\Vnl}{V_{\text{nl}}}
|
|
\newcommand{\lm}{{lm}}
|
|
\newcommand{\KK}{{\bf K}}
|
|
\newcommand{\KKp}{{\bf{K'}}}
|
|
\newcommand{\KKhat}{\widehat \KK}
|
|
\newcommand{\KKphat}{\widehat{\KK}'}
|
|
\newcommand{\jl}{j_l}
|
|
\newcommand{\rrhat}{{\widehat\rr}}
|
|
\newcommand{\dd}{{\,\text{d}}}
|
|
$$
|
|
|
|
## Pseudopotentials in the Kleynman Bylander form
|
|
|
|
This page reports the basic definitions and equations needed to evaluate the nonlocal part of the Hamiltonian.
|
|
We mainly focus on norm-conserving pseudopotentials.
|
|
Providing a consistent introduction to the theory of pseudopotentials is beyond the purpose of this section,
|
|
a more complete discussion of the topic can be found in specialized articles available in the literature
|
|
[[cite:Hamann1979]], [[cite:Bachelet1982]], [[cite:Kleinman1982]], [[cite:Hamann1989]], [[cite:Troullier1991]]
|
|
[[cite:Gonze1991]], [[cite:Fuchs1999]].
|
|
For the generalization of norm conservation to multiple projectors (Optimized norm-conserving Vanderbilt pseudopotentials)
|
|
we refer the reader to [[cite:Hamann2013]].
|
|
The generation and validation of ONCVPSP pseudopotentials is described in
|
|
the [PseudoDojo](http://www.pseudo-dojo.org/) paper [[cite:Setten2018]].
|
|
|
|
For our purpose, we can summarize by saying that a pseudopotential is constructed in order to replace
|
|
the atomic all-electron potential such that core states are eliminated and valence electrons are
|
|
described by pseudo wavefunctions whose representation in the Fourier space decays rapidly.
|
|
|
|
Modern norm-conserving pseudopotentials are constructed such that the scattering properties
|
|
of the all-electron atom are reproduced around the reference energy configuration up to first order in energy,
|
|
following the procedure described in [[cite:Hamann1979]], [[cite:Hamann1989]], and [[cite:Troullier1991]].
|
|
In modern ab initio codes, the fully separable KB form, proposed by Kleynman and Bylander, is usually employed
|
|
as it drastically reduces the number of operations required for the application of the nonlocal part of the Hamiltonian
|
|
as well as the memory required to store the operator [[cite:Kleinman1982]].
|
|
In the KB form, the interaction between a valence electron and an ion of type $\aa$ is described by means of the operator:
|
|
|
|
\begin{equation}\label{eq:KBsingleatom}
|
|
\vv_\PS^\aa =
|
|
\vloc^\aa(r) + \sum_\lm |\chi_\lm^\aa\ra\,E_l^\aa\,\la\chi_\lm^\aa|,
|
|
% \vloc_\aa(\rr) + \sum_\lm |\Ylm\,u_{l\aa}^\KB\ra E_{l\aa}^\KB \lau_{l\aa}^\KB\,\Ylm|
|
|
\end{equation}
|
|
|
|
where $\vloc^\aa(r)$ is a purely local potential with a Coulomb tail $\gamma/r$.
|
|
|
|
!!! note
|
|
|
|
For simplicity, the discussion is limited to pseudopotentials with a single projector per angular channel.
|
|
The generalization to multi-projector pseudopotentials requires introducing an additional $n$ index
|
|
in the KB energies i.e. $E_{nl}^\aa$ and in the projectors $\chi_{n\lm}^\aa(\rr)$.
|
|
|
|
The so-called Kleinman-Bylander energies, $E_l^\aa$,
|
|
measure the strength of the nonlocal component with respect to the local part.
|
|
The projectors $\chi_\lm^\aa(\rr)$ are
|
|
short-ranged functions, expressed in terms of a complex spherical harmonic $\Ylm(\theta,\phi)$
|
|
multiplied by an $l$- and atom-dependent radial function:
|
|
|
|
\begin{equation}
|
|
\label{eq:KB_projectors}
|
|
\chi_\lm^\aa(\rr) = \dfrac{1}{r}\,\chi_l^\aa(r)\,\Ylm(\theta,\phi).
|
|
\end{equation}
|
|
|
|
<!--
|
|
where
|
|
|
|
\begin{equation}\label{eq:KBfunctionU}
|
|
u_{l\aa}^\KB (\rr) =
|
|
\frac{\Delta v_{l\aa}(\rr)\,u_{l\aa}^\PS (\rr)}
|
|
{\norm{u_{l\aa}(\rr)\,\Delta_{l\aa}}^{1/2}}
|
|
\end{equation}
|
|
are localized functions defined in terms of the short-ranged ...
|
|
and the pseudo eigenfuncions of the reference atom.
|
|
\begin{equation}\label{eq:KBenergy}
|
|
E_{l\aa}^\KB =
|
|
\frac{\la u_{l\aa}^\PS \Delta v_{l\aa} | \Delta v_{l\aa} u_{l\aa}^\PS \ra }
|
|
{\la u_{l\aa}^\PS | \Delta v_{l\aa} | u_{l\aa}^\PS \ra}
|
|
\end{equation}
|
|
-->
|
|
|
|
The total nonlocal part of the Hamiltonian is obtained by summing the different atom-centered contributions.
|
|
The final expression in real space reads:
|
|
|
|
\begin{equation}
|
|
\Vnl(\rr_1,\rr_2)=
|
|
\sum_{\substack{\RR \\ \aa\tt_\aa}}
|
|
\sum_\lm \la\rr_1-\RR-\tt_\aa|\chi_\lm^\aa\ra E_l^\aa\la\chi_\lm^\aa|\rr_2-\RR-\tt_\aa\ra,
|
|
\end{equation}
|
|
|
|
where $\RR$ runs over the sites of the Bravais lattice, and $\tt_\aa$ over the positions of the atoms with the same type $\aa$ located inside the unit cell.
|
|
Due to the nonlocality of the operator, its Fourier transform depends on two separate indices, $\kk+\GG_1$ and $\kk+\GG_2$, instead of the simple difference $\GG_1-\GG_2$.
|
|
The shorthand notation $\KK = \kk + \GG$ is used in the following to simplify the derivation.
|
|
|
|
Using the Rayleigh expansion of planewaves in terms of spherical harmonics $\Ylm(\theta,\phi)$ and spherical Bessel functions $\jl(Kr)$:
|
|
|
|
\begin{equation}\label{eq:PWinYlmPl}
|
|
e^{i\KK\cdot\rr} =
|
|
4\pi\,\sum_\lm i^l \jl(Kr) \Ylm^\*(\KKhat)\,\Ylm(\rrhat),
|
|
\end{equation}
|
|
|
|
the Fourier representation of the projector, defined by \ref{eq:KB_projectors}, can be expressed as:
|
|
|
|
\begin{equation}
|
|
\la\KK|\chi_\lm^\aa\ra =
|
|
\frac{4\pi}{\Omega^{1/2}}\, (-1)^l\,\Ylm(\KKhat)
|
|
\int_0^\infty\, r\jl(Kr)\,\chi_l^\aa(r)r\,\dd r =
|
|
\frac{4\pi}{\Omega^{1/2}}\, (-1)^l\,\Ylm(\KKhat) F_l^\aa(K).
|
|
\end{equation}
|
|
|
|
<!--
|
|
%where the form factors $F_l^\aa(K)$ related to the atom of type $\aa$ is defined by
|
|
%\begin{equation}\label{wq:defformfactors}
|
|
% F_l^\aa(K) \df
|
|
% \frac{\int_0^\infty r\,j_l (Kr)\,u_{l\aa} \Delta v_{l\aa}\,\dd r}
|
|
% {\norm{u_{l\aa} \Delta v_{l\aa}}^{1/2} }
|
|
%\end{equation}
|
|
-->
|
|
|
|
Finally, the expression for the total nonlocal operator in reciprocal space is:
|
|
|
|
\begin{equation}
|
|
\label{eq:VnlKBmatrixelements}
|
|
\Vnl(\KK,\KKp)= \frac{(4\pi)^2}{\Omega} \sum_{\aa \tt_\aa} \sum_\lm
|
|
\,e^{-i(\KK-\KKp)\cdot\tt_\aa}\,
|
|
\Ylm(\KKhat)\Ylm^\*(\KKphat)\, E_l^\aa F_l^\aa(K) F_l^\aa(K').
|
|
\end{equation}
|
|
|
|
!!! note
|
|
|
|
Abinit employs iterative eigenvalue solvers to solve the KS equations.
|
|
This means that we only need to compute $\Vnl |\Psi\ra$, i.e., we only
|
|
need to **apply** the Hamiltonian onto a set of trial eigenvectors.
|
|
Therefore the full $\Vnl(\KK,\KKp)$ matrix is never constructed explicitly.
|