mirror of https://github.com/abinit/abinit.git
Add section on pseudos and spherical harmonics to theory notes
This commit is contained in:
Matteo Giantomassi
parent
99938a8066
commit
c0e8cbe9e6
|
|
@ -945,31 +945,31 @@ The bibtex file is available [here](../abiref.bib).
|
|||
url = ""
|
||||
if a.text is None: a.text = fragment
|
||||
else:
|
||||
if name.startswith("lesson_"):
|
||||
self.warn("lesson_NAME is DEPRECATED, use lesson:name. %s in %s is deprecated" % (token, page_rpath))
|
||||
# Handle [[lesson_gw1|text]]
|
||||
url = "/tutorials/%s" % name.replace("lesson_" , " ", 1).strip()
|
||||
if a.text is None: a.text = name
|
||||
html_classes.append("lesson-wikilink")
|
||||
#if name.startswith("lesson_"):
|
||||
# self.warn("lesson_NAME is DEPRECATED, use lesson:name. %s in %s is deprecated" % (token, page_rpath))
|
||||
# # Handle [[lesson_gw1|text]]
|
||||
# url = "/tutorials/%s" % name.replace("lesson_" , " ", 1).strip()
|
||||
# if a.text is None: a.text = name
|
||||
# html_classes.append("lesson-wikilink")
|
||||
|
||||
elif name.startswith("topic_"):
|
||||
# Handle [[topic_SelfEnergy|text]]
|
||||
self.warn("topic_NAME is DEPRECATED, use topic:name. %s in %s is deprecated" % (token, page_rpath))
|
||||
name = name.replace("topic_" , " ", 1).strip()
|
||||
url = "/topics/%s" % name
|
||||
if a.text is None: a.text = "%s topic" % name
|
||||
html_classes.append("topic-wikilink")
|
||||
add_popover(a, content=self.howto_topic[name])
|
||||
#elif name.startswith("topic_"):
|
||||
# # Handle [[topic_SelfEnergy|text]]
|
||||
# self.warn("topic_NAME is DEPRECATED, use topic:name. %s in %s is deprecated" % (token, page_rpath))
|
||||
# name = name.replace("topic_" , " ", 1).strip()
|
||||
# url = "/topics/%s" % name
|
||||
# if a.text is None: a.text = "%s topic" % name
|
||||
# html_classes.append("topic-wikilink")
|
||||
# add_popover(a, content=self.howto_topic[name])
|
||||
|
||||
elif name.startswith("help_"):
|
||||
self.warn("help_NAME is DEPRECATED, use help:name. %s in %s is deprecated" % (token, page_rpath))
|
||||
# Handle [[help_abinit|text]]
|
||||
code = name.replace("help_" , " ", 1).strip()
|
||||
url = "/guide/%s" % code
|
||||
if a.text is None: a.text = "%s help file" % code
|
||||
html_classes.append("user-guide-wikilink")
|
||||
#elif name.startswith("help_"):
|
||||
# self.warn("help_NAME is DEPRECATED, use help:name. %s in %s is deprecated" % (token, page_rpath))
|
||||
# # Handle [[help_abinit|text]]
|
||||
# code = name.replace("help_" , " ", 1).strip()
|
||||
# url = "/guide/%s" % code
|
||||
# if a.text is None: a.text = "%s help file" % code
|
||||
# html_classes.append("user-guide-wikilink")
|
||||
|
||||
elif "@" in name:
|
||||
if "@" in name:
|
||||
# Handle [[dipdip@anaddb|text]]
|
||||
vname, code = name.split("@")
|
||||
var = self.codevars[code][vname]
|
||||
|
|
@ -985,17 +985,17 @@ The bibtex file is available [here](../abiref.bib).
|
|||
if a.text is None:
|
||||
a.text = var.name if not var.is_internal else "%%%s" % var.name
|
||||
|
||||
elif name in self.bib_data.entries:
|
||||
# Handle citation
|
||||
self.warn("DEPRECATED citation without `cite` --> %s in %s is deprecated" % (token, page_rpath))
|
||||
ref = self.bib_data.entries[name]
|
||||
url = "/theory/bibliography#%s" % self.slugify(name)
|
||||
content = ref.fields.get("title", "Unknown")
|
||||
if content == "Unknown":
|
||||
self.warn("Entry for %s does not provide title" % name)
|
||||
add_popover(a, content=content) #+ "\n\n" + ref.authors
|
||||
if a.text is None: a.text = "[%s]" % name
|
||||
html_classes.append("citation-wikilink")
|
||||
#elif name in self.bib_data.entries:
|
||||
# # Handle citation
|
||||
# self.warn("DEPRECATED citation without `cite` --> %s in %s is deprecated" % (token, page_rpath))
|
||||
# ref = self.bib_data.entries[name]
|
||||
# url = "/theory/bibliography#%s" % self.slugify(name)
|
||||
# content = ref.fields.get("title", "Unknown")
|
||||
# if content == "Unknown":
|
||||
# self.warn("Entry for %s does not provide title" % name)
|
||||
# add_popover(a, content=content) #+ "\n\n" + ref.authors
|
||||
# if a.text is None: a.text = "[%s]" % name
|
||||
# html_classes.append("citation-wikilink")
|
||||
|
||||
elif name.startswith("tests/") or name.startswith("~abinit/tests/"):
|
||||
assert fragment is None
|
||||
|
|
@ -1079,9 +1079,9 @@ The bibtex file is available [here](../abiref.bib).
|
|||
if a.text is None: a.text = "%s_%s" % (namespace, name)
|
||||
add_popover(a, content=self.howto_topic[name])
|
||||
|
||||
elif namespace in ("bib", "cite"):
|
||||
if namespace == "bib":
|
||||
self.warn("%s in %s is deprecated" % (token, page_rpath))
|
||||
elif namespace == "cite":
|
||||
#if namespace == "bib":
|
||||
# self.warn("%s in %s is deprecated" % (token, page_rpath))
|
||||
# Handle [[bib:biblio|bibliography]]
|
||||
if name == "biblio":
|
||||
url = "/theory/bibliography/"
|
||||
|
|
@ -1101,9 +1101,9 @@ The bibtex file is available [here](../abiref.bib).
|
|||
(exc.__class__, str(exc), token, page_rpath))
|
||||
url, a.text = "FAKE_URL", "FAKE_URL"
|
||||
|
||||
elif namespace in ("theorydoc", "theory"):
|
||||
if namespace == "theorydoc":
|
||||
self.warn("%s in %s is deprecated" % (token, page_rpath))
|
||||
elif namespace == "theory":
|
||||
#if namespace == "theorydoc":
|
||||
# self.warn("%s in %s is deprecated" % (token, page_rpath))
|
||||
# TODO theorydoc --> theory
|
||||
# Handle [[theorydoc:mbpt|text]]
|
||||
url = "/theory/%s" % name
|
||||
|
|
@ -1140,7 +1140,7 @@ The bibtex file is available [here](../abiref.bib).
|
|||
html_classes.append("abifile-wikilink")
|
||||
|
||||
elif namespace == "ac":
|
||||
# Handle [[ac:abiref_gnu_5.3_debug.ac]]
|
||||
# Handle [[ac:abiref_gnu_5.3_debug.ac]]
|
||||
# The following is incorrect : files in /build/config-examples are generated when */*/makemake is issued.
|
||||
# url = "/build/config-examples/%s" % name
|
||||
# By contrast, the following is a permanent reference
|
||||
|
|
|
|||
|
|
@ -98,6 +98,7 @@
|
|||
"ra": "{\\rangle}",
|
||||
"la": "{\\langle}",
|
||||
// matchcal version
|
||||
mcG: "{\\mathcal{G}}",
|
||||
mcR: "{\\mathcal{R}}",
|
||||
mcS: "{\\mathcal{S}}",
|
||||
mcC: "{\\mathcal{C}}",
|
||||
|
|
|
|||
|
|
@ -3627,3 +3627,84 @@ author = "Philip B. Allen and Božidar Mitrović"
|
|||
year = {1975},
|
||||
month = aug,
|
||||
}
|
||||
@article{Hobbs2000,
|
||||
author = {Hobbs, D. and Kresse, G. and Hafner, J.},
|
||||
doi = {10.1103/physrevb.62.11556},
|
||||
number = {17},
|
||||
pages = {11556-11570},
|
||||
source = {Crossref},
|
||||
url = {http://dx.doi.org/10.1103/physrevb.62.11556},
|
||||
volume = {62},
|
||||
journal = {Phys. Rev. B},
|
||||
publisher = {American Physical Society (APS)},
|
||||
title = {Fully unconstrained noncollinear magnetism within the projector augmented-wave method},
|
||||
issn = {0163-1829, 1095-3795},
|
||||
year = {2000},
|
||||
month = nov,
|
||||
}
|
||||
@article{Perdew1992,
|
||||
author = {Perdew, John P. and Chevary, J. A. and Vosko, S. H. and Jackson, Koblar A. and Pederson, Mark R. and Singh, D. J. and Fiolhais, Carlos},
|
||||
doi = {10.1103/physrevb.46.6671},
|
||||
number = {11},
|
||||
pages = {6671-6687},
|
||||
source = {Crossref},
|
||||
url = {http://dx.doi.org/10.1103/physrevb.46.6671},
|
||||
volume = {46},
|
||||
journal = {Phys. Rev. B},
|
||||
publisher = {American Physical Society (APS)},
|
||||
title = {Atoms, molecules, solids, and surfaces: {Applications} of the generalized gradient approximation for exchange and correlation},
|
||||
issn = {0163-1829, 1095-3795},
|
||||
year = {1992},
|
||||
month = sep,
|
||||
}
|
||||
@article{Blanco1997,
|
||||
title = "Evaluation of the rotation matrices in the basis of real spherical harmonics",
|
||||
journal = "Journal of Molecular Structure: THEOCHEM",
|
||||
volume = "419",
|
||||
number = "1",
|
||||
pages = "19 - 27",
|
||||
year = "1997",
|
||||
issn = "0166-1280",
|
||||
doi = "https://doi.org/10.1016/S0166-1280(97)00185-1",
|
||||
url = "http://www.sciencedirect.com/science/article/pii/S0166128097001851",
|
||||
author = "Miguel A. Blanco and M. Flórez and M. Bermejo",
|
||||
keywords = "Wigner functions, Real spherical harmonics, Electronic structure calculations, Rotation matrices"
|
||||
}
|
||||
|
||||
@book{bassani1975,
|
||||
title={Electronic states and optical transitions in solids},
|
||||
author={Bassani, G.F. and Parravicini, G.P.},
|
||||
isbn={9780080168463},
|
||||
lccn={gb75022672},
|
||||
series={International series of monographs in the science of the solid state},
|
||||
url={https://books.google.be/books?id=cGh5AAAAIAAJ},
|
||||
year={1975},
|
||||
publisher={Pergamon Press}
|
||||
}
|
||||
|
||||
@Article{Bhagavantam1964,
|
||||
author="Bhagavantam, S. and Pantulu, P. V.",
|
||||
title="Crystal symmetry and physical properties: Application of group theory",
|
||||
journal="Proceedings of the Indian Academy of Sciences - Section A",
|
||||
year="1964",
|
||||
month="Jul",
|
||||
day="01",
|
||||
volume="60",
|
||||
number="1",
|
||||
pages="1--10",
|
||||
abstract="A group theoretical method given by one of us in earlier publications for studying the effect of crystal symmetry on physical properties is now extended to cover the galvanomagnetic, thermomagnetic and piezo-galvanomagnetic effects in single crystals. These effects have gained importance in semi-conductor physics and such studies are therefore of current interest and may be expected to yield valuable new information.",
|
||||
issn="0370-0089",
|
||||
doi="10.1007/BF03046362",
|
||||
url="https://doi.org/10.1007/BF03046362"
|
||||
}
|
||||
|
||||
@book{Landau1984,
|
||||
title={Electrodynamics of continuous media: },
|
||||
author={Landau, L.D. and Lifshit͡s, E.M. and Lifshit︠s︡, E.M. and Pitaevski{\u\i}, L.P.},
|
||||
isbn={9780080302751},
|
||||
lccn={83024997},
|
||||
series={Pergamon international library of science, technology, engineering, and social studies},
|
||||
url={https://books.google.be/books?id=j7nvAAAAMAAJ},
|
||||
year={1984},
|
||||
publisher={Pergamon}
|
||||
}
|
||||
|
|
|
|||
Binary file not shown.
|
Before Width: | Height: | Size: 1.3 KiB |
|
|
@ -29,8 +29,8 @@ related to the irreducible polarizability by the following relation
|
|||
|
||||
from which the inverse dielectric function is obtained via matrix inversion.
|
||||
|
||||
The _**macroscopic**_ dielectric function, _є M_LF(ω), can be directly related
|
||||
to the inverse of a single element, the first ( **G** _1_ =0, **G** _2_ =0),
|
||||
The _**macroscopic**_ dielectric function, $\ee_M^{\text{LF}}(\ww)$, can be directly related
|
||||
to the inverse of a single element, the first ($\GG_1 = 0, \GG_2 =0$) element
|
||||
of the inverse of the _**microscopic**_ dielectric matrix by means of:
|
||||
|
||||
\begin{equation}\label{eq:abs_LFE}
|
||||
|
|
@ -39,8 +39,8 @@ of the inverse of the _**microscopic**_ dielectric matrix by means of:
|
|||
|
||||
The microscopic dielectric matrix is the one usually calculated within the RPA in
|
||||
a GW calculation. The optical absorption spectrum is simply given by the
|
||||
imaginary part of _є M_LF(ω). Sometimes, especially when comparing with
|
||||
experimental spectra, the real part is simply called _є 1_ and the imaginary part _є 2_.
|
||||
imaginary part of $\ee_M^{\text{LF}}(\ww)$ . Sometimes, especially when comparing with
|
||||
experimental spectra, the real part is simply called $\ee_1$ and the imaginary part $\ee_2$.
|
||||
|
||||
Note that above equation differs from
|
||||
|
||||
|
|
@ -49,12 +49,15 @@ Note that above equation differs from
|
|||
\end{equation}
|
||||
|
||||
due to the so called local-field effects introduced by the presence of the crystalline environment.
|
||||
The former quantity _with_ local fields (LF) is more accurate than the latter one
|
||||
with _no_ local field (NLF). The reason the two equations are different is
|
||||
because in the first, the expression in the denominator is the first element
|
||||
of the inverse of the _whole_ microscopic dielectric matrix. This element
|
||||
depends on the _entire_ matrix and cannot simply be calculated by taking the
|
||||
inverse of the first element of the non-inverted matrix.
|
||||
The former quantity _with_ local fields (LF) is more accurate than the latter one with _no_ local field (NLF).
|
||||
|
||||
!!! important
|
||||
|
||||
The reason the two equations are different is
|
||||
because in the first, the expression in the denominator is the first element
|
||||
of the inverse of the _whole_ microscopic dielectric matrix. This element
|
||||
depends on the _entire_ matrix and cannot simply be calculated by taking the
|
||||
inverse of the first element of the non-inverted matrix.
|
||||
|
||||
In the [[theory:mbt#RPA_Fourier_space|GW_notes]], we have discussed how
|
||||
to calculate the irreducible polarizability and thus the absorption spectrum
|
||||
|
|
@ -75,8 +78,9 @@ The fundamental reason for the failure of the RPA can be traced back to the
|
|||
neglect of the electron-hole interaction that, in the many-body language,
|
||||
should be included through the vertex function. Replacing the vertex function
|
||||
with a local and instantaneous function is a too crude and unrealistic
|
||||
approximation for the many-body polarizability. In the next section we will
|
||||
discuss how to obtain an improved approximation for the vertex and therefore
|
||||
approximation for the many-body polarizability.
|
||||
|
||||
In the next section we will discuss how to obtain an improved approximation for the vertex and therefore
|
||||
an improved approximation for the polarizability that takes into account many-
|
||||
body effects due to the electron-hole interaction
|
||||
|
||||
|
|
@ -89,12 +93,12 @@ spectra available in ABINIT is given below.
|
|||
|
||||
The Bethe-Salpeter theory is formulated in terms of two-particle propagators.
|
||||
These are four-point functions describing the motion through the system of two
|
||||
particles at once. We will label them _L 0_ and _L_ , where the subscript zero
|
||||
particles at once. We will label them $L^0$ and $L$ , where the subscript zero
|
||||
indicates that the particles are non-interacting. By restricting the starting
|
||||
and ending point of the particles, the two-point contraction of _L 0_ gives
|
||||
and ending point of the particles, the two-point contraction of $L^0$ gives
|
||||
the reducible independent-particle polarizability according to
|
||||
|
||||
$$ \chi^0(12) = L^0(11, 22) $$ while the two-point contraction of _L_ gives
|
||||
$$ \chi^0(12) = L^0(11, 22) $$ while the two-point contraction of $L$ gives
|
||||
the reducible many-body polarizability. $$ \hat\chi(12) = L(11, 22) $$ that
|
||||
should not be confused with the _irreducible_ polarizability of the many-body
|
||||
system (denoted with a tilde in the formulas in the [[theory:mbt#RPA_Fourier_space|GW_notes]]).
|
||||
|
|
@ -104,21 +108,20 @@ _**with**_ local field effects is obtained directly from the reducible
|
|||
polarizability using
|
||||
|
||||
\begin{equation}
|
||||
\ee_M^{\text{LF}}(\ww) = 1 - \lim_{\qq
|
||||
\rightarrow 0} v(\qq)\,\tchi_{00}(\qq;\ww)
|
||||
\ee_M^{\text{LF}}(\ww) = 1 - \lim_{\qq \rightarrow 0} v(\qq)\,\tchi_{00}(\qq;\ww)
|
||||
\end{equation}
|
||||
|
||||
Computing the reducible _L_ directly, if possible, thus allows one to avoid
|
||||
Computing the reducible $L$ directly, if possible, thus allows one to avoid
|
||||
the costly matrix inversion of the dielectric function that should otherwise
|
||||
be performed for each frequency.
|
||||
|
||||
One can show that _L_ satisfies the Dyson-like equation:
|
||||
One can show that $L$ satisfies the Dyson-like equation:
|
||||
|
||||
\begin{equation}
|
||||
L = L^0 + L^0 K L \Longrightarrow L = \bigl [ 1 - L^0 K]^{-1} L^0
|
||||
\end{equation}
|
||||
|
||||
that involves the Bethe-Salpeter kernel _K_ :
|
||||
that involves the Bethe-Salpeter kernel $K$:
|
||||
|
||||
\begin{equation}\label{eq:BSE_kernel_LF}
|
||||
K(1234) = \underbrace{\delta(12)\delta(34)\bar v(13)}_{Exchange} -
|
||||
|
|
@ -126,7 +129,7 @@ K(1234) = \underbrace{\delta(12)\delta(34)\bar v(13)}_{Exchange} -
|
|||
\end{equation}
|
||||
|
||||
where the bar symbol signifies that the Coulomb interaction has to be taken without its
|
||||
long-range Fourier component at **G** =0:
|
||||
long-range Fourier component at $\GG =0$:
|
||||
|
||||
\begin{equation} \begin{cases} {\bar
|
||||
v(\qq)} = v(\qq) \quad {\text{if}}\; \qq \neq 0 \\\ \\\ {\bar v(\qq=0)} = 0
|
||||
|
|
@ -140,11 +143,10 @@ rewriting of the kernel in this way is mostly an algebraic trick which allows
|
|||
one to easily separate the contributions and calculate the optical spectrum
|
||||
both with and without the LF.
|
||||
|
||||
The Dyson-like equation for _L_ becomes a matrix problem once a particular
|
||||
The Dyson-like equation for $L$ becomes a matrix problem once a particular
|
||||
basis set is chosen to expand the four-point functions involved in the
|
||||
problem. In the standard approach, the polarisabilities and the BS kernel are
|
||||
expressed in the so-called _transition_ space (i.e. as products of single-
|
||||
particle orbitals) using:
|
||||
expressed in the so-called _transition_ space (i.e. as products of single-particle orbitals) using:
|
||||
|
||||
\begin{equation} F(1234) =
|
||||
\sum_{ \substack{(n_1 n_2) \\\ (n_3 n_4)} }
|
||||
|
|
@ -158,7 +160,7 @@ and
|
|||
} = \int F(1234) \psi_{n_1}(1) \psi_{n_2}^\*(2) \psi_{n_3}^\*(3)
|
||||
\psi_{n_4}(4) \dd (1234) \end{equation}
|
||||
|
||||
where _n i_ is a shorthand notation to denote band, **k** -point and spin index.
|
||||
where $n_i$ is a shorthand notation to denote band, **k**-point and spin index.
|
||||
The expansion is exact provided that
|
||||
the set of single-particle orbitals from a complete basis set in the Hilbert space.
|
||||
|
||||
|
|
@ -173,14 +175,14 @@ f_{n_1}) } { (\ee_{n_2} - \ee_{n_1} - \ww)} \delta_{n_1 n_3} \delta_{n_2 n_4}
|
|||
thus leading to a considerable simplification in the mathematical formalism.
|
||||
|
||||
After some algebra (see [[cite:Onida2002]] for a more complete derivation) one
|
||||
finds that, in a system with an energy gap, the Dyson-like equation for _L_
|
||||
can be expressed in terms of an effective two-particle Hamiltonian, _H_ , according to
|
||||
finds that, in a system with an energy gap, the Dyson-like equation for $L$
|
||||
can be expressed in terms of an effective two-particle Hamiltonian, $H$ , according to
|
||||
|
||||
\begin{equation}
|
||||
L = \bigl [ H-\ww \bigr]^{-1}\,F
|
||||
\end{equation}
|
||||
|
||||
The explicit form for _H_ and _F_ in the transition space is given by
|
||||
The explicit form for $H$ and $F$ in the transition space is given by
|
||||
|
||||
\begin{equation}
|
||||
H =
|
||||
|
|
@ -204,9 +206,9 @@ F =
|
|||
\right)
|
||||
\end{equation}
|
||||
|
||||
where valence states are indicated by the indices _v_ , _v'_ while _c_ , _c'_
|
||||
are used for conduction bands. The _R_ sub-matrix is Hermitian and is usually
|
||||
referred to as the resonant block while _C_ (the so-called coupling block) is
|
||||
where valence states are indicated by the indices $v$ , $v'$ while $c$ , $c'$
|
||||
are used for conduction bands. The $R$ sub-matrix is Hermitian and is usually
|
||||
referred to as the resonant block while $C$ (the so-called coupling block) is
|
||||
a complex symmetric sub-matrix. Note that, due to the presence of the coupling
|
||||
term, the BS Hamiltonian is non-Hermitian although the operator possesses a real spectrum.
|
||||
|
||||
|
|
@ -222,7 +224,7 @@ W_{(vc\kk)(v'c'\kk')}
|
|||
|
||||
with single particle transition energies on the diagonal.
|
||||
|
||||
The matrix elements of v and _W_ are defined as:
|
||||
The matrix elements of $v$ and $W$ are defined as:
|
||||
|
||||
\begin{equation}
|
||||
{\bar v}_{(n_1 n_2) (n_3 n_4)} = \delta_{\sigma_1 \sigma_2}\,
|
||||
|
|
@ -233,12 +235,12 @@ v(\rr-\rr')} \psi_{n_3}^*(\rr') \psi_{n_4}(\rr') \dd \rr \dd \rr'
|
|||
{W(\rr,\rr',\ww=0)} \psi^*_{n_2}(\rr') \psi_{n_4}(\rr') \dd \rr \dd \rr'
|
||||
\end{equation}
|
||||
|
||||
Note, in particular, that only the static limit ( ω = 0) of _W_ is involved in the last expression.
|
||||
Note, in particular, that only the static limit ($\omega = 0$) of $W$ is involved in the last expression.
|
||||
|
||||
The coupling matrix elements are usually smaller than the resonant ones. This
|
||||
is especially true in bulk systems due to the different spatial behavior of
|
||||
conduction and valence states. In solid state calculations, it is therefore
|
||||
common practice to ignore the _C_ block (the so-called Tamm-Dancoff
|
||||
common practice to ignore the $C$ block (the so-called Tamm-Dancoff
|
||||
approximation). Under this assumption the excitonic Hamiltonian is given by a Hermitian operator.
|
||||
|
||||
|
||||
|
|
@ -246,8 +248,8 @@ approximation). Under this assumption the excitonic Hamiltonian is given by a He
|
|||
The variable [[bs_coupling]] defines whether the coupling block should be included in the calculation.
|
||||
|
||||
The macroscopic dielectric function is obtained by contracting the many-body
|
||||
_L_ and then taking the optical limit of the **G** = **G'** =0 component along
|
||||
a particular direction in **q** -space. The final result reads:
|
||||
$L$ and then taking the optical limit of the $\GG = \GG' = 0$ component along
|
||||
a particular direction in $\qq$-space. The final result reads:
|
||||
|
||||
\begin{equation}
|
||||
\ee_M(\ww) = 1 - \lim_{\qq \rightarrow 0} v(\qq)\,\langle
|
||||
|
|
@ -273,7 +275,7 @@ change the default directions using the variables [[gw_nqlwl]] and [[gw_qlwl]].
|
|||
At this point it is clear that the evaluation of the macroscopic dielectric
|
||||
function within the BS formalism is a two-step process:
|
||||
|
||||
1. Construction of the _H_ matrix in the transition space.
|
||||
1. Construction of the $H$ matrix in the transition space.
|
||||
|
||||
2. Computation of the macroscopic dielectric function using the two equations reported at the end of the previous section.
|
||||
|
||||
|
|
@ -291,7 +293,7 @@ For BS computations, it is common practice to simulate the self-energy
|
|||
corrections by employing the scissors operator whose value can be obtained
|
||||
either from experiments or from ab-initio calculations. The scissors operator
|
||||
allows one to avoid a costly _GW_ calculation that should performed for all
|
||||
the **k** -points and bands included in the transition space (the optional
|
||||
the $\kk$-points and bands included in the transition space (the optional
|
||||
path on the right indicated with yellow arrows that corresponds to [[optdriver]]=4).
|
||||
|
||||
The construction of the Hamiltonian matrix represents a significant portion of
|
||||
|
|
@ -305,7 +307,7 @@ matrix inversion for each frequency. The variable [[bs_algorithm]] is used to
|
|||
select among the different possibilities.
|
||||
|
||||
[[bs_algorithm]]=1 employs standard (Sca)LAPACK routines to obtain the
|
||||
spectral representation of _H_ in terms of eigenvalues and right eigenvectors of _H_:
|
||||
spectral representation of $H$ in terms of eigenvalues and right eigenvectors of $H$:
|
||||
|
||||
\begin{equation}
|
||||
\begin{cases} H |\lambda\rangle = \ee_\lambda |\lambda\rangle
|
||||
|
|
@ -314,7 +316,7 @@ spectral representation of _H_ in terms of eigenvalues and right eigenvectors of
|
|||
\langle\lambda'| \end{cases}
|
||||
\end{equation}
|
||||
|
||||
Then, as discussed in [2], the inverse of [ _H_ -ω] is obtained according to
|
||||
Then, as discussed in [2], the inverse of $H - \ww$ is obtained according to
|
||||
|
||||
\begin{equation} \bigl[ H -\ww
|
||||
\bigr]^{-1} = \sum_{\lambda \lambda'} |\lambda\rangle \dfrac{O_{\lambda
|
||||
|
|
@ -371,7 +373,7 @@ We refer the reader to [[cite:Gruning2009]] for a generalization of the method t
|
|||
the case in which the coupling term cannot be neglected. The main drawback of
|
||||
the method is that it does not give direct access to the excitonic spectrum
|
||||
hence it cannot be used to calculate binding energies or to plot the excitonic
|
||||
wavefunctions. Moreover, for the time being, [[bs_algorithm]]=2 cannot be used
|
||||
wavefunctions. Moreover, for the time being, [[bs_algorithm]] = 2 cannot be used
|
||||
for calculations in which the coupling term is included.
|
||||
|
||||
[[bs_algorithm]]=3 employs the conjugate-gradient method to calculate the
|
||||
|
|
@ -381,9 +383,8 @@ testing and does not support calculations with the coupling term.
|
|||
## 4 Kernel matrix elements in reciprocal space
|
||||
|
||||
Our implementation employs planewaves to expand the periodic part of the Bloch
|
||||
states, _u_ , and the two-point function _W(r,r')_ that becomes a **q**
|
||||
-dependent matrix in reciprocal-space. The conventions used for the transforms
|
||||
are documented in [this section](mbt.md#notations) of the _GW_ notes.
|
||||
states, $u$ , and the two-point function $W(\rr,\rr')$ that becomes a $\qq$-dependent matrix in reciprocal-space.
|
||||
The conventions used for the transforms are documented in [this section](mbt.md#notations) of the $GW$ notes.
|
||||
|
||||
The matrix elements of the exchange term are evaluated in reciprocal space using:
|
||||
|
||||
|
|
@ -414,21 +415,20 @@ calculations without local field effects. The variable [[bs_coulomb_term]]
|
|||
governs the computation of the Coulomb term, the most CPU-demanding part due
|
||||
to the presence of the double sum over **G** -vectors.
|
||||
|
||||
It is also important to stress that, in the two equations above, the **k**
|
||||
-point index runs over the full Brillouin zone hence the size of the
|
||||
It is also important to stress that, in the two equations above, the **k**-point
|
||||
index runs over the full Brillouin zone hence the size of the
|
||||
Hamiltonian is defined by the number of point in the full Brillouin zone and
|
||||
not by the number of points in the irreducible wedge.
|
||||
|
||||
## 5 Matrix elements of the dipole operator
|
||||
|
||||
The accurate calculation of optical properties require the correct treatment
|
||||
of the optical limit ( **G** =0, **q** -> 0) of the oscillator matrix
|
||||
of the optical limit ($\GG = 0, |\qq| \rightarrow 0$) of the oscillator matrix
|
||||
elements. The computation of these terms deserves some clarification, due to
|
||||
the presence of the fully nonlocal pseudopotential term in the Kohn-Sham
|
||||
Hamiltonian.
|
||||
|
||||
A linear expansion up to the first order in **q** of the oscillator matrix
|
||||
element results in:
|
||||
A linear expansion up to the first order in **q** of the oscillator matrix element results in:
|
||||
|
||||
\begin{equation}
|
||||
\langle b_1,\kmq|e^{-i\qq\cdot\rr}|b_2,\kk \rangle
|
||||
|
|
@ -436,9 +436,9 @@ element results in:
|
|||
\rangle + \mcO(q^2)
|
||||
\end{equation}
|
||||
|
||||
where we have assumed _b 1 ≠ b2 _and the difference between the two wave functions
|
||||
at **k** - **q** and **k** has been
|
||||
neglected because it only introduces terms that are quadratic in **q**.
|
||||
where we have assumed $b_1 \neq b_2$ and the difference between the two wave functions
|
||||
at $\kk - \qq$ and $\kk$ has been
|
||||
neglected because it only introduces terms that are quadratic in $\qq$.
|
||||
|
||||
Unfortunately, the above expression cannot be directly used because the matrix
|
||||
elements of the position operator are ill-defined when wavefunctions obey
|
||||
|
|
@ -458,7 +458,7 @@ the pseudopotential is more involved and much more CPU-demanding.
|
|||
|
||||
The role played by this additional term is usually marginal in the case of GW
|
||||
calculations: the QP corrections are obtained by performing an integration in
|
||||
**q** -space and only the **q** -> 0 component of the inverse dielectric
|
||||
$\qq$-space and only the $\qq \rightarrow 0$ component of the inverse dielectric
|
||||
matrix is affected by the commutator of the non-local part of the pseudopotential.
|
||||
|
||||
For this reason it is common practice, especially during the _GW_ convergence
|
||||
|
|
|
|||
|
|
@ -1,17 +1,17 @@
|
|||
---
|
||||
authors: X. Gonze, Y. Suzukawa, M. Mikami
|
||||
authors: M. Giantomassi, X. Gonze, Y. Suzukawa, M. Mikami
|
||||
---
|
||||
|
||||
# Geometric considerations
|
||||
|
||||
## Real space
|
||||
|
||||
The three primitive translation vectors are ${\bf R}_1$, ${\bf R}_2$, ${\bf R}_3$.
|
||||
The three primitive translation vectors are $\RR_1$, $\RR_2$, $\RR_3$.
|
||||
Their representation in Cartesian coordinates (atomic units) is:
|
||||
|
||||
$$ {\bf R}_1 \rightarrow {rprimd(:, 1)} $$
|
||||
$$ {\bf R}_2 \rightarrow {rprimd(:, 2)} $$
|
||||
$$ {\bf R}_3 \rightarrow {rprimd(:, 3)} $$
|
||||
$$ \RR_1 \rightarrow {rprimd(:, 1)} $$
|
||||
$$ \RR_2 \rightarrow {rprimd(:, 2)} $$
|
||||
$$ \RR_3 \rightarrow {rprimd(:, 3)} $$
|
||||
|
||||
Related input variables: [[acell]], [[rprim]], [[angdeg]].
|
||||
|
||||
|
|
@ -33,7 +33,7 @@ Representation in reduced coordinates:
|
|||
|
||||
Related input variables: [[xangst]], [[xcart]], [[xred]]
|
||||
|
||||
The volume of the primitive unit cell (ucvol) is
|
||||
The volume of the primitive unit cell (called *ucvol* in the code) is
|
||||
|
||||
\begin{eqnarray*}
|
||||
\Omega &=& {\bf R}_1 \cdot ({\bf R}_2 \times {\bf R}_3)
|
||||
|
|
@ -77,7 +77,7 @@ $$ {\bf R}^{met}_{ij} \rightarrow {rmet(i,j)} $$
|
|||
## Reciprocal space
|
||||
|
||||
The three primitive translation vectors in reciprocal space are
|
||||
${\bf G}_{1}$,${\bf G}_{2}$,${\bf G}_{3}$
|
||||
$\GG_1$, $\GG_2$,$\GG_3$
|
||||
|
||||
\begin{eqnarray*}
|
||||
{\bf G}_{1}&=&\frac{1}{\Omega}({\bf R}_{2}\times{\bf R}_{3}) \rightarrow {gprimd(:,1)} \\
|
||||
|
|
@ -85,7 +85,7 @@ ${\bf G}_{1}$,${\bf G}_{2}$,${\bf G}_{3}$
|
|||
{\bf G}_{3}&=&\frac{1}{\Omega}({\bf R}_{1}\times{\bf R}_{2}) \rightarrow {gprimd(:,3)}
|
||||
\end{eqnarray*}
|
||||
|
||||
This definition is such that ${\bf G}_{i}\cdot{\bf R}_{j}=\delta_{ij}$
|
||||
This definition is such that $\GG_i \cdot \RR_j = \delta_{ij}$
|
||||
|
||||
!!! warning
|
||||
Often, a factor of $2\pi$ is present in definition of
|
||||
|
|
@ -141,53 +141,33 @@ where ${\bf G}^{met}_{ij}$ is the metric tensor in reciprocal space:
|
|||
|
||||
$$ {\bf G}^{met}_{ij} \rightarrow {gmet(i,j)} $$
|
||||
|
||||
## Symmetries
|
||||
## Fourier series for periodic lattice quantities
|
||||
|
||||
A symmetry operation in real space sends the point ${\bf r}$ to the point
|
||||
${\bf r'}={\bf S_t}\{{\bf r}\}$ whose coordinates are
|
||||
$({\bf r'})_{\alpha}=\sum_{\beta}S_{\alpha \beta}r_{\beta} + t_{\alpha}$ (Cartesian coordinates).
|
||||
Any function with the periodicity of the lattice i.e any function fullfilling the property:
|
||||
|
||||
The symmetry operations that preserves the crystalline structure are
|
||||
those that send every atom location on an atom location with the same atomic type.
|
||||
$$ u(\rr + \RR) = u(\rr) $$
|
||||
|
||||
The application of a symmetry operation to a function of spatial coordinates $f$ is such that:
|
||||
can be represented with the discrete Fourier series:
|
||||
|
||||
$$
|
||||
({\bf S_t}f)({\bf r}) = f(({\bf S_t})^{-1}{\bf r})
|
||||
$$
|
||||
$$ u(\rr)= \sum_\GG u(\GG)e^{i\GG\cdot\rr} $$
|
||||
|
||||
$$
|
||||
({\bf S_t})^{-1}({\bf r}) =
|
||||
\sum_{\beta} S^{-1}_{\alpha \beta}(r_{\beta}-t_{\beta})
|
||||
$$
|
||||
where the Fourier coefficient, $u(\GG)$, is given by:
|
||||
|
||||
For each symmetry operation,$isym=1 \dots nsym$, the $3\times3$
|
||||
${\bf S}^{red}$ matrix is stored in {symrel(:,:,isym)}.
|
||||
$$ u(\GG) = \frac{1}{\Omega} \int_\Omega u(\rr)e^{-i\GG\cdot\rr}\dd\rr $$
|
||||
|
||||
[in reduced coordinates:
|
||||
$r'^{red}_{\alpha}=\sum_{\beta}S^{red}_{\alpha \beta}
|
||||
r^{red}_{\beta}+t^{red}_{\beta}$ ]
|
||||
<!--
|
||||
This appendix reports the conventions used in this work for the Fourier
|
||||
representation in frequency- and momentum-space.
|
||||
The volume of the unit cell is denoted with $\Omega$, while $V$ is the
|
||||
total volume of the crystal simulated employing Born-von K\'arman periodic boundary condition~\cite{Ashcroft1976}.
|
||||
|
||||
and the vector ${\bf t}^{red}$ is stored in {tnons(:,isym)}.
|
||||
\begin{equation}\label{eq:IFT_2points_convention}
|
||||
f(\rr_1,\rr_2)= \frac{1}{V} \sum_{\substack{\qq \\ \GG_1 \GG_2}}
|
||||
e^{i (\qq +\GG_1) \cdot \rr_1}\,f_{\GG_1 \GG_2}(\qq)\,e^{-i (\qq+\GG_2) \cdot \rr_2}
|
||||
\end{equation}
|
||||
|
||||
The conversion between reduced coordinates and Cartesian coordinates is
|
||||
|
||||
$$
|
||||
r'_{\gamma}=\sum_{\alpha \beta}(R_{\alpha p})_{\gamma}
|
||||
[S^{red}_{\alpha \beta} r^{red}_{\beta}+t^{red}_{\alpha}]
|
||||
$$
|
||||
|
||||
with $[{\rm as} \ G_{ip} \cdot R_{jp}=\delta_{ij}]$
|
||||
|
||||
$$
|
||||
r_{\delta}=\sum_{\alpha}(R_{\alpha p})_\delta
|
||||
r^{red}_{\alpha} \rightarrow
|
||||
\sum_{\beta}(G_{\beta p})_{\delta} r_{\delta}=r^{red}_{\beta}
|
||||
$$.
|
||||
|
||||
So
|
||||
|
||||
$$
|
||||
S_{\gamma \delta}=\sum_{\alpha \beta}(R_{\alpha p})_{\gamma}
|
||||
S^{red}_{\alpha \beta}(G_{\beta p})_{\gamma}
|
||||
$$
|
||||
\begin{equation}\label{eq:FT_2points_convention}
|
||||
f_{\GG_1\GG_2}(\qq) = \frac{1}{V} \iint_V
|
||||
e^{-i(\qq+\GG_1) \cdot \rr_1}\,f(\rr_1, \rr_2)\,e^{i (\qq+\GG_2) \cdot \rr_2}\dd\rr_1\dd\rr_2
|
||||
\end{equation}
|
||||
-->
|
||||
|
|
|
|||
|
|
@ -7,7 +7,9 @@ authors: MG, MS
|
|||
The aim of this section is to introduce the Green's function formalism, the
|
||||
concept of self-energy and the set of coupled equations proposed by Hedin
|
||||
whose self-consistent solution, in principle, gives the exact Green's function
|
||||
of the interacting system. We mainly focus on the aspects of the theory that
|
||||
of the interacting system.
|
||||
|
||||
We mainly focus on the aspects of the theory that
|
||||
are important for understanding the different steps of the calculation and the
|
||||
role played by the input variables used to control the run. A much more
|
||||
consistent and rigorous introduction to the physical concept of Green's
|
||||
|
|
@ -47,9 +49,9 @@ E_i^{(N+1)} - E_0^N & \text{if}\,\varepsilon_i > \mu \\\ E_0^N-E_i^{(N-1)} & \te
|
|||
\end{cases}
|
||||
\end{equation}
|
||||
|
||||
where _E 0N_ is the ground state energy of
|
||||
the electron system with _N_ electrons, and _i_ is the set of quantum numbers
|
||||
labeling the excited states with _N ±_1 electrons. Finally, _μ_ is the
|
||||
where $E_0^N$ is the ground state energy of
|
||||
the electron system with $N$ electrons, and $i$ is the set of quantum numbers
|
||||
labeling the excited states with $N \pm 1$ electrons. Finally, $\mu$ is the
|
||||
chemical potential of the system. Other important quantities that will be used
|
||||
in the following are the so-called Lehmann amplitudes defined, in the Schrodinger representation, by
|
||||
|
||||
|
|
@ -67,8 +69,8 @@ G (\rr_1,\rr_2;\ww) = \sum_i \frac{\Psi_i(\rr_1)\Psi_i^\*(\rr_2)} {\ww-\ee_i
|
|||
\end{equation}
|
||||
|
||||
makes it clear that, in the frequency domain, the time-ordered Green's function
|
||||
contains the complete excitation spectrum corresponding to excitations of an (
|
||||
_N_ -1)-particle and an ( _N_ +1)-particle system. Hence, locating the poles
|
||||
contains the complete excitation spectrum corresponding to excitations of an
|
||||
$(N-1)$-particle and an $(N+1)$-particle system. Hence, locating the poles
|
||||
of the Green's function in the complex plane provides the information needed
|
||||
to interpret those processes measured in experiments in which a single
|
||||
electron is inserted to or removed from the system. The figure below gives a
|
||||
|
|
@ -87,14 +89,14 @@ The Dyson equation
|
|||
G(12) = \Go(12) + \int \Go(13)\,\Sigma(34)\,G(42)\dd34.
|
||||
\end{equation}
|
||||
|
||||
establishes a connection between the fully interacting _G_ and the propagator, _G 0_, of an approximate
|
||||
establishes a connection between the fully interacting $G$ and the propagator, $\Go$, of an approximate
|
||||
non-interacting system through a (non-local, non-Hermitian and time dependent)
|
||||
potential called the self-energy, Σ. Since _G 0_ is supposed to be known
|
||||
exactly, the problem of calculating _G_ (12) has now been reduced to the calculation of the self-energy.
|
||||
potential called the self-energy, $\Sigma$. Since $\Go$ is supposed to be known
|
||||
exactly, the problem of calculating $G (12)$ has now been reduced to the calculation of the self-energy.
|
||||
|
||||
The self-energy is not a mere mathematical device used in a roundabout way to
|
||||
obtain _G_ but is has a direct physical meaning. The knowledge of the self-
|
||||
energy operator, allows one to describe the quantum-mechanical state of a
|
||||
obtain $G$ but is has a direct physical meaning. The knowledge of the self-energy operator,
|
||||
allows one to describe the quantum-mechanical state of a
|
||||
renormalized electron in the many-body system by solving the quasiparticle
|
||||
(QP) equation [[cite:Onida2002]]:
|
||||
|
||||
|
|
@ -104,7 +106,7 @@ renormalized electron in the many-body system by solving the quasiparticle
|
|||
\end{equation}
|
||||
|
||||
The QP eigenstates so obtained can be used
|
||||
to construct _G_ according to the Lehmann representation. Note that the QP
|
||||
to construct $G$ according to the Lehmann representation. Note that the QP
|
||||
equation departs from the Kohn Sham equation since the QP eigenvectors and
|
||||
eigenvalues do have a direct physical meaning: they can be used to obtain both
|
||||
the charge density of the interacting system and to describe the properties of charged excitations.
|
||||
|
|
@ -113,7 +115,7 @@ the charge density of the interacting system and to describe the properties of c
|
|||
|
||||
In 1965 Hedin [[cite:Hedin1965]] showed how to derive a set of coupled integro-
|
||||
differential equations whose self-consistent solution, in principle, gives the
|
||||
exact self-energy of the system and therefore the exact _G_.
|
||||
exact self-energy of the system and therefore the exact $G$.
|
||||
|
||||
The fundamental building blocks employed in the formalism are the irreducible polarizability:
|
||||
|
||||
|
|
@ -124,14 +126,14 @@ The fundamental building blocks employed in the formalism are the irreducible po
|
|||
which describes the linear response of the density to changes in the total
|
||||
effective potential (the superposition of the external potential plus the
|
||||
internal classical Hartree potential) and the dynamically screened
|
||||
interaction, _W_ , that is related to the bare Coulomb interaction, _v_ , and
|
||||
interaction, $W$ , that is related to the bare Coulomb interaction, $v$ , and
|
||||
to the inverse dielectric function through:
|
||||
|
||||
\begin{equation} \label{W_def}
|
||||
W(12) \equiv \int \ee^{-1}(13)\,v(32)\dd3.
|
||||
\end{equation}
|
||||
|
||||
The dielectric matrix є(12) is related to the irreducible polarizability _χ_ (12) by the
|
||||
The dielectric matrix $\ee(12)$ is related to the irreducible polarizability $\tchi(12)$ by the
|
||||
following relation:
|
||||
|
||||
\begin{equation}
|
||||
|
|
@ -143,12 +145,12 @@ The pentagon sketched in the figure below shows how the various physical quantit
|
|||
|
||||
|
||||
The polarization function renormalises the bare interaction resulting in the
|
||||
screened interaction _W_ (12). The screened interaction, _W_ (12), the many-
|
||||
body propagator _G_ (12), and the vertex function, Γ(12;3), which describe the
|
||||
screened interaction $W (12)$. The screened interaction, $W (12)$, the many-
|
||||
body propagator $G (12)$, and the vertex function, $\Gamma(12;3)$, which describe the
|
||||
interactions between virtual hole and electron excitations [[cite:Onida2002]], are
|
||||
the essential ingredients for the determination of Σ(12).
|
||||
the essential ingredients for the determination of $\Sigma(12)$.
|
||||
|
||||
The iteration starts by setting _G_ = _G 0_. Then the set of equations should
|
||||
The iteration starts by setting $G = \Go$. Then the set of equations should
|
||||
in principle be iterated until self-consistency in all terms is reached.
|
||||
|
||||
## The GW approximation
|
||||
|
|
@ -160,7 +162,7 @@ challenging. The set of equations can, however, be iterated assuming that only
|
|||
a few iterations are actually needed to obtain physically meaningful results.
|
||||
|
||||
A widely used approach to the approximate solution of Hedin's equations is the
|
||||
so-called _GW_ approximation [[cite:Hedin1965]], which consists in approximating
|
||||
so-called $GW$ approximation [[cite:Hedin1965]], which consists in approximating
|
||||
the vertex function with a local and instantaneous function:
|
||||
|
||||
\begin{equation}
|
||||
|
|
@ -172,7 +174,7 @@ the full set of Hedin's equations, leads to a considerable simplification in the
|
|||
|
||||
|
||||
|
||||
Thanks to the neglect of vertex corrections, the irreducible polarizability _χ_ (12) is now given by
|
||||
Thanks to the neglect of vertex corrections, the irreducible polarizability $\tchi (12)$ is now given by
|
||||
|
||||
\begin{equation}\label{eq:RPA_with_G}
|
||||
\tchi = -i\,G(12)G(21^+).
|
||||
|
|
@ -182,7 +184,7 @@ which, once rewritten in terms of orbitals and energies, reduces to the RPA
|
|||
expression proposed by Adler [[cite:Adler1962]] and Wiser [[cite:Wiser1963]].
|
||||
|
||||
In real space, the self-energy reduces to a simple direct product of the
|
||||
dressed electron propagator, _G_ (12), and the dynamically screened interaction, _W_ (12):
|
||||
dressed electron propagator, $G (12)$, and the dynamically screened interaction, $W (12)$:
|
||||
|
||||
\begin{equation}
|
||||
\Sigma(12) = i G(12)\,W(1^+2).
|
||||
|
|
@ -209,16 +211,16 @@ self-consistent approach would need the inclusion of some kind of vertex
|
|||
correction during the solution of the equations.
|
||||
|
||||
For this reason, the most common approach employed in the _ab initio_
|
||||
community consists of using the best available approximation for _G_ and _W_
|
||||
community consists of using the best available approximation for $G$ and $W$
|
||||
as a starting point and performing only a single-iteration of the
|
||||
parallelogram (the so-called one-shot _GW_ method, or _G 0W0_). In this case
|
||||
parallelogram (the so-called one-shot $GW$ method, or $\Go\Wo$). In this case
|
||||
the self-energy is simply given by:
|
||||
|
||||
\begin{equation}
|
||||
\Sigma(12) = i\Go^\KS(12)\Wo(1^+2)
|
||||
\end{equation}
|
||||
|
||||
where _G0KS_(12) is the independent-particle propagator of the Kohn-Sham (KS)
|
||||
where $\Go^\KS(12)$ is the independent-particle propagator of the Kohn-Sham (KS)
|
||||
Hamiltonian, and the screened interaction is approximated with the RPA
|
||||
calculated with KS energies and wave functions:
|
||||
|
||||
|
|
@ -236,11 +238,11 @@ are usually in qualitative agreement with experiment.
|
|||
|
||||
This observation suggests that a simple, albeit accurate, solution for the QP
|
||||
energies can be obtained using first-order perturbation theory, treating the
|
||||
exchange and correlation potential, _V xc_, as a zeroth-order approximation to
|
||||
exchange and correlation potential, $V_{xc}$, as a zeroth-order approximation to
|
||||
the non-local and energy dependent self-energy [[cite:Hybertsen1985]], [[cite:Hybertsen1986]]
|
||||
|
||||
Under the assumption that the QP wavefunctions equal the KS orbitals, we can
|
||||
expand the self-energy operator around _ε KS_ obtaining a closed expression for _ε QP_:
|
||||
expand the self-energy operator around $\ee^\KS$ obtaining a closed expression for $\ee^\QP$:
|
||||
|
||||
\begin{equation} \label{eq:implicit_QP_energy}
|
||||
\ee^\QP = \ee^\KS + Z \langle\Psi^\KS|\Sigma(\ee^\KS)-\vxc|\Psi^\KS\rangle.
|
||||
|
|
@ -267,9 +269,9 @@ particle polarizability assumes the form:
|
|||
|
||||
|
||||
|
||||
where only the transitions between valence ( _v_ ) and conduction states (_c_) contribute
|
||||
(for simplicity we have assumed a semiconductor with time-
|
||||
reversal invariance, the conventions used for the Fourier transform are
|
||||
where only the transitions between valence ($v$) and conduction states ($c$) contribute
|
||||
(for simplicity we have assumed a semiconductor with time-reversal invariance,
|
||||
the conventions used for the Fourier transform are
|
||||
discussed in the [notation](#notations) section).
|
||||
|
||||
The number of bands used to compute the polarizability is specified by
|
||||
|
|
@ -277,21 +279,22 @@ The number of bands used to compute the polarizability is specified by
|
|||
divergences in the denominators. The frequency mesh is defined by the set of
|
||||
variables [[nfreqre]], [[nfreqim]], [[freqremax]], and [[freqremin]] (a number
|
||||
of more exotic grid choices are available through input variables beginning
|
||||
with gw_... or cd_..., e.g. [[gw_frqim_inzgrid]]).
|
||||
with `gw_...` or `cd_...`, e.g. [[gw_frqim_inzgrid]]).
|
||||
|
||||
_M_ is a shorthand notation to denote the matrix element of a plane wave
|
||||
$M$ is a shorthand notation to denote the matrix element of a plane wave
|
||||
sandwiched between two wavefunctions (i.e. oscillator matrix elements).
|
||||
The number of planewaves (PW) used to describe the wavefunctions is determined by
|
||||
[[ecutwfn]] while the number of _G_ -vectors used to describe the
|
||||
[[ecutwfn]] while the number of $\GG$-vectors used to describe the
|
||||
polarizability (i.e. the number of _G_ vectors in the oscillator matrix
|
||||
elements) is determined by [[ecuteps]].
|
||||
|
||||
The oscillators are ubiquitous in the Many-Body part of ABINIT and their calculation
|
||||
represents one of the most CPU intensive part of the execution. For this reason we
|
||||
devote [[#6|section 6]] to the discussion of some important technical details concerning their computation.
|
||||
|
||||
In principle, the set of **q** -points in the screening matrix is given by all
|
||||
In principle, the set of $\qq$-points in the screening matrix is given by all
|
||||
the possible differences between two crystalline momenta of the wavefunctions
|
||||
stored in the KSS file, so it is controlled by the chosen **k** -point grid.
|
||||
stored in the KSS file, so it is controlled by the chosen $\kk$-point grid.
|
||||
The code, however, exploits the invariance of the two-point function under the
|
||||
action of any symmetry operation of the crystalline space group:
|
||||
|
||||
|
|
@ -300,7 +303,7 @@ action of any symmetry operation of the crystalline space group:
|
|||
\bigl(\Ri(\rr_1-\tt),\Ri(\rr_2-\tt)\bigr)
|
||||
\end{equation}
|
||||
|
||||
so that only the **q** -points in the irreducible Brillouin zone (IBZ) have to be calculated explicitly.
|
||||
so that only the $\qq$-points in the irreducible Brillouin zone (IBZ) have to be calculated explicitly.
|
||||
|
||||
In frequency and reciprocal space, the microscopic dielectric function is
|
||||
related to the irreducible polarizability by the following relation
|
||||
|
|
@ -311,7 +314,7 @@ related to the irreducible polarizability by the following relation
|
|||
\end{equation}
|
||||
|
||||
from which the inverse dielectric function is obtained via matrix inversion. Following Adler
|
||||
[[cite:Adler1962]], the macroscopic dielectric function, єMLF(ω), can be directly
|
||||
[[cite:Adler1962]], the macroscopic dielectric function, $\ee_M^{\text{LF}}(\ww)$, can be directly
|
||||
related to the inverse of the microscopic dielectric matrix by means of:
|
||||
|
||||
\begin{equation} \label{eq:abs_LFE}
|
||||
|
|
@ -319,7 +322,7 @@ related to the inverse of the microscopic dielectric matrix by means of:
|
|||
\end{equation}
|
||||
|
||||
The optical absorption spectrum -- the quantity one can compare with experiments -- is
|
||||
given by the imaginary part: Im[єMLF(ω)].
|
||||
given by the imaginary part.
|
||||
|
||||
Note that the equation above differs from
|
||||
|
||||
|
|
@ -353,13 +356,13 @@ link and use the FFTW3 library in GW calculations (controlled by setting
|
|||
[[fftalg]] 312). The performance of the various FFT libraries for a given type
|
||||
of calculation can be benchmarked with the **fftprof** utility.
|
||||
|
||||
For a given set of indices ( _b 1_, _b 2_, **k**, **q** ), the calculation of
|
||||
For a given set of indices ( $b_1$, $b_2$, $\kk$, $\qq$ ), the calculation of
|
||||
the oscillator is done in four different steps:
|
||||
|
||||
1. The two wavefunctions in the irreducible wedge are FFT transformed from the **G** -space to the real space representation,
|
||||
2. The orbitals are rotated in real space on the FFT mesh to obtain the points **k** and **k-q** in the full Brillouin zone.
|
||||
1. The two wavefunctions in the irreducible wedge are FFT transformed from the $\GG$-space to the real space representation,
|
||||
2. The orbitals are rotated in real space on the FFT mesh to obtain the points $\kk$ and $\kk-\qq$ in the full Brillouin zone.
|
||||
3. Computation of the wavefunction product.
|
||||
4. FFT transform of the product to obtain _M_
|
||||
4. FFT transform of the product to obtain $M$
|
||||
|
||||
Each oscillator thus requires three different FFTs (two transforms to
|
||||
construct the product, one FFT to get M). The number of FFTs can be
|
||||
|
|
@ -371,10 +374,10 @@ time instead. This option is controlled by the second digit of the input variabl
|
|||
|
||||
The third term in the equation defining the oscillators makes it clear that
|
||||
the product of the periodic part of the orbitals has non-zero Fourier
|
||||
components in a sphere whose radius is 2 × _R wfn_ where _R wfn_ is the radius
|
||||
of the **G** -sphere used for the wavefunctions (set by [[ecutwfn]]). To avoid
|
||||
components in a sphere whose radius is $2 × R_{wfn}$ where $R_{wfn}$ is the radius
|
||||
of the $\GG$-sphere used for the wavefunctions (set by [[ecutwfn]]). To avoid
|
||||
aliasing errors in the FFT one should therefore use an FFT box that encloses
|
||||
the sphere with radius 2× _R wfn_, but this leads to a significant increase in
|
||||
the sphere with radius $2 × R_{wfn}$, but this leads to a significant increase in
|
||||
the computing effort as well as in the memory requirements. The input variable
|
||||
[[fftgw]] specifies how to setup the FFT box for the oscillators and should be
|
||||
used to test how the aliasing errors affect the final results. The default
|
||||
|
|
@ -428,7 +431,7 @@ expensive frequency integration (a Hilbert transform):
|
|||
|
||||
The price to be paid, however, is that a large table for the spectral function
|
||||
has to be stored in memory and a Hilbert transform has to be performed for
|
||||
each pair ( **G** 1, **G** 2). Since the computing time required for the
|
||||
each pair ( $\GG_1, \GG_2$). Since the computing time required for the
|
||||
transform scales quadratically with the number of vectors in the
|
||||
polarizability (governed by [[ecuteps]]), the CPU time spent in this part will
|
||||
overcome the computing time of the standard Adler-Wiser formalism for large
|
||||
|
|
@ -450,8 +453,8 @@ W = v + (\ee^{-1} - 1) v
|
|||
|
||||
where matrix notation is used.
|
||||
|
||||
This particular decomposition of _W_ , once inserted in the convolution
|
||||
defining Σ, leads to the split of the self-energy into two different
|
||||
This particular decomposition of $W$ , once inserted in the convolution
|
||||
defining $\Sigma$, leads to the split of the self-energy into two different
|
||||
contributions (exchange and correlation):
|
||||
|
||||
\begin{equation}
|
||||
|
|
@ -467,7 +470,7 @@ quasiparticle amplitudes
|
|||
\sum_\nu^\text{occ} \Psi_{n\kk}(\rr_1){\Psi^\*_{n\kk}}(\rr_2)\,v(\rr_1,\rr_2)
|
||||
\end{equation}
|
||||
|
||||
while the dynamic part Σc(ω) accounts for correlation effects beyond Σx.
|
||||
while the dynamic part $\Sigma_c(\ww)$ accounts for correlation effects beyond $\Sigma_x$.
|
||||
|
||||
It is important to stress that, for computational efficiency, the code does
|
||||
not compute the full self-energy operator by default. Only its matrix elements
|
||||
|
|
@ -489,13 +492,14 @@ converged independently of others if necessary, and given a much larger value
|
|||
in comparison to [[ecut]], [[ecutwfn]] and [[ecuteps]].
|
||||
|
||||
Another point worth noting is the presence in the expression of the Coulomb
|
||||
singularity for | **q** + **G** | -> 0\. From a mathematical point of view,
|
||||
the integral is well-defined since the singularity is integrable in three-
|
||||
dimensional space once the thermodynamical limit, _N_ **q** -> ∞, is reached.
|
||||
On the other hand, only a finite number of **q** -points can be used for
|
||||
singularity for $| \qq + \GG | \rightarrow 0$ .
|
||||
From a mathematical point of view, the integral is well-defined since the singularity is integrable in three-
|
||||
dimensional space once the thermodynamical limit, $N_\qq \rightarrow \infty$, is reached.
|
||||
|
||||
On the other hand, only a finite number of $\qq$-points can be used for
|
||||
practical applications, and a careful numerical treatment is needed to avoid
|
||||
an exceedingly slow convergence with respect to the BZ sampling. To accelerate
|
||||
the convergence in the number of **q** -points, the code implements several
|
||||
the convergence in the number of $\qq$-points, the code implements several
|
||||
techniques proposed in the literature. We refer to the documentation of
|
||||
[[icutcoul]] for a more extensive discussion.
|
||||
|
||||
|
|
@ -512,7 +516,7 @@ approximate way, alternatively, it is possible to use the more sophisticated
|
|||
frequency integration of the contour deformation method [[cite:Lebegue2003]] for
|
||||
accurate QP calculations (see the related variables [[ppmodel]] and [[gwcalctyp]]).
|
||||
|
||||
The double sum over **G** -vectors is performed for all the plane waves
|
||||
The double sum over $\GG$-vectors is performed for all the plane waves
|
||||
contained within a sphere of energy [[ecuteps]] (it cannot be larger than the
|
||||
value used to generate the SCR file). For each state, the correlated matrix
|
||||
elements are evaluated on a linear frequency mesh centered around the initial
|
||||
|
|
@ -520,7 +524,7 @@ KS energy and the derivative needed for the renormalization factor is obtained
|
|||
numerically (see [[nomegasrd]] and [[omegasrdmax]]).
|
||||
|
||||
Note that here, in contrast to the exchange term, the sum over the band index
|
||||
_n_ should extend up to infinity although in practice only a finite number of
|
||||
$n$ should extend up to infinity although in practice only a finite number of
|
||||
states can be used (specified by [[nband]]).
|
||||
|
||||
## Plasmon-pole models
|
||||
|
|
@ -528,14 +532,14 @@ states can be used (specified by [[nband]]).
|
|||
One of the major computational efforts in self-energy calculations is
|
||||
represented by the calculation of the frequency dependence of the screened
|
||||
interaction, which is needed for the evaluation of the convolution. Due to the
|
||||
ragged behavior of _G_ (ω) and _W_ (ω) along the real axis, numerous real
|
||||
ragged behavior of $G(\ww)$ and $W (\ww)$ along the real axis, numerous real
|
||||
frequencies are in principle required to converge the results (note that the
|
||||
maximum frequencies needed are now reported if [[prtvol]] > 9). On the other
|
||||
hand, since the fine details of _W_ (ω) are integrated over, it is reasonable
|
||||
hand, since the fine details of $W(\ww)$ are integrated over, it is reasonable
|
||||
to expect that approximate models, able to capture the main physical features
|
||||
of the screened interaction, should give sufficiently accurate results with a
|
||||
considerably reduction of computational effort. This is the basic idea behind
|
||||
the so-called plasmon-pole models in which the frequency dependence of _W_ (ω)
|
||||
the so-called plasmon-pole models in which the frequency dependence of $W(\ww)$
|
||||
is modelled in terms of analytic expressions depending on coefficients that
|
||||
are derived from first principles, i.e. without any adjustable external parameters.
|
||||
|
||||
|
|
@ -558,7 +562,7 @@ function is modeled according to
|
|||
The two models differ in the approach used to compute the
|
||||
parameters. [[ppmodel]]=1 derives the parameters such that the inverse
|
||||
dielectric matrix is correctly reproduced at two different explicitly
|
||||
calculated frequencies: the static limit (ω = 0) and an additional imaginary
|
||||
calculated frequencies: the static limit ($\ww = 0$) and an additional imaginary
|
||||
point located at [[ppmfrq]]. Unless the user overrides this, the default value
|
||||
is calculated from the average electronic density of the system. The plasmon-
|
||||
pole parameters of [[ppmodel]]=2 are calculated so as to reproduce the static
|
||||
|
|
@ -589,8 +593,8 @@ contributions coming from the poles of the integrand lying inside the contour:
|
|||
\end{equation}
|
||||
|
||||
|
||||
In the above equation, the first sum is restricted to the poles lying inside the path
|
||||
_C_. _W_ c( _z_ ) represents the frequency dependent part of the screened
|
||||
In the above equation, the first sum is restricted to the poles lying inside the path $\mcC$.
|
||||
$W^c(z)$ represents the frequency dependent part of the screened
|
||||
interaction, whose expression in reciprocal space is given by:
|
||||
|
||||
\begin{equation}
|
||||
|
|
@ -619,7 +623,7 @@ v(\rr_1,\rr_2)\,\delta(t_1-t_2) $$
|
|||
|
||||
$$ 1^+ = (\rr_1,t_1 + \eta)_{\eta \rightarrow 0^+} $$
|
||||
|
||||
where _v_ ( **r** 1, **r** 2) represents the bare Coulomb interaction, and η is a positive infinitesimal.
|
||||
where $v(\rr_1, \rr_2)$ represents the bare Coulomb interaction, and $\eta$ is a positive infinitesimal.
|
||||
|
||||
The Fourier transforms for periodic lattice quantities are defined as
|
||||
|
||||
|
|
@ -640,6 +644,6 @@ f_{\GG_1\GG_2}(\qq) =
|
|||
(\qq+\GG_2) \cdot \rr_2}\dd\rr_1\dd\rr_2
|
||||
\end{equation}
|
||||
|
||||
The volume of the unit cell is denoted with Ω, while _V_ is the total volume of the crystal simulated
|
||||
The volume of the unit cell is denoted with $\Omega$, while $V$ is the total volume of the crystal simulated
|
||||
employing Born-von Karman periodic boundary condition. Unless otherwise
|
||||
specified, Hartree atomic units will be used throughout.
|
||||
|
|
|
|||
|
|
@ -19,7 +19,7 @@ $$ \rho(\rr)=\sum_{\alpha} \rho^{\alpha\alpha}(\rr) $$
|
|||
|
||||
and the magnetization density $\vec m(\rr)$ (in units of $\hbar /2$) whose components are:
|
||||
|
||||
$$ m_i(\rr) = \sum_{\alpha\beta} \rho^{\alpha\beta}(\rr) \sigma_i^{\alpha\beta} $$,
|
||||
$$ m_i(\rr) = \sum_{\alpha\beta} \rho^{\alpha\beta}(\rr) \sigma_i^{\alpha\beta}, $$
|
||||
|
||||
where the $\sigma_i$ are the Pauli matrices.
|
||||
|
||||
|
|
@ -99,7 +99,8 @@ Note that only the forurier transform are performed in *mkrho.f*, while the fina
|
|||
$\rho(\rr)$, $\vec m(\rr)$ is performed in *symrhg.f*.
|
||||
|
||||
The computation of $V_{xc}$ is performed in *rhohxc.f*. The only transformation to this routine, is
|
||||
to compute $|\vec m(\rr)|$ as input of the usual (i.e spin polarized) {\tt rhohxc.f} and yield back
|
||||
the four component $V_{xc}$, from the expression of ${\delta E_{xc} \over \delta |m (\rr)| }$.
|
||||
to compute $|\vec m(\rr)|$ and yield back the four component $V_{xc}$, from the expression
|
||||
of ${\delta E_{xc} \over \delta |m (\rr)| }$.
|
||||
|
||||
For more information, see: Hobbs et al., PRB, 62, 11556; Perdew et al. PRB, 46, 6671 (for the xc functional)
|
||||
For more information about noncollinear magnetism see [[cite:Hobbs2000]]
|
||||
and [[cite:Perdew1992]] for the xc functional.
|
||||
|
|
|
|||
|
|
@ -0,0 +1,137 @@
|
|||
---
|
||||
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}}
|
||||
$
|
||||
|
||||
## Norm-conserving Pseudopotentials in the Kleynman Bylander form
|
||||
|
||||
This page reports basic definitions and equations needed
|
||||
to evalute the matrix elements of the commutator between the position operator, $\rr$,
|
||||
and the nonlocal part of the Hamiltonian in the particular case in which norm-conserving pseudopotentials are used.
|
||||
Providing a consistent introduction to the theory of pseudopotentials is beyond the purposes of this section,
|
||||
a more complete discussion of the topic can be found in specialized
|
||||
articles available in the literature ~\cite{Hamann1979,Bachelet1982,Kleinman1982,Hamann1989,Troullier1991,Gonze1991,Fuchs1999}.
|
||||
|
||||
For our purposes, we can summarize by saying that a pseudopotential in constructed in order to
|
||||
replace the atomic all-electron potential such that core states are eliminated
|
||||
and valence electrons are described by nodeless pseudo wavefunctions whose
|
||||
representation in the Fourier space decays rapidly.
|
||||
|
||||
Modern norm-conserving pseudopotentials are constructed following the procedure
|
||||
described in~\cite{Hamann1979,Hamann1989,Troullier1991} such that the scattering properties of the
|
||||
all-electron atom are reproduced around the reference energy configuration, up to first order in energy.
|
||||
In modern ab initio codes, the fully separable 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 requirens 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 =
|
||||
\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}
|
||||
|
|
@ -0,0 +1,222 @@
|
|||
---
|
||||
authors: MG
|
||||
---
|
||||
|
||||
$$
|
||||
\newcommand{\lm}{{lm}}
|
||||
\newcommand{\Ylm}{{Y_m^l}}
|
||||
\newcommand{\rY}{{\mathcal Y}}
|
||||
\newcommand{\rYlm}{{\mathcal Y}_m^l}
|
||||
\newcommand{\rYLM}{{\mathcal Y}_M^L}
|
||||
\newcommand{\lp}{{l'}}
|
||||
\newcommand{\rrhat}{{\widehat\rr}}
|
||||
\newcommand{\rrphat}{{{\widehat\rr}'}}
|
||||
\newcommand{\mcP}{{\mathcal{P}}}
|
||||
\newcommand{\df}{\equiv} %this required mathtools :=
|
||||
\newcommand{\li}{{l_i}}
|
||||
\newcommand{\mi}{{m_i}}
|
||||
\newcommand{\nj}{{n_j}}
|
||||
\newcommand{\lj}{{l_j}}
|
||||
\newcommand{\mj}{{m_j}}
|
||||
\renewcommand{\mp}{{m'}}
|
||||
\newcommand{\mcG}{{\mathcal G}}
|
||||
\newcommand{\omcR}{{\hat\mcR}}
|
||||
\newcommand{\mcR}{{\mathcal{R}}}
|
||||
\newcommand{\ddO}{{\dd\Omega}}
|
||||
\newcommand{\LM}{{LM}}
|
||||
\newcommand{\mcRm}{\mcR^{-1}}
|
||||
\newcommand{\kpGhat}{\widehat{\kpG}}
|
||||
\newcommand{\rrp}{{\bf r'}}
|
||||
\newcommand{\Gaunt}{{\mathcal G}}
|
||||
$$
|
||||
|
||||
# Complex and Real Spherical Harmonics
|
||||
|
||||
## Complex Spherical Harmonics
|
||||
|
||||
Complex spherical harmonics, $\Ylm$, are defined as the eigenfunctions of the orbital
|
||||
angular momentum operators, $\hat L^2$ and $\hat L_z$.
|
||||
They can be labeled by two quantum numbers, $l$ and $m$, related to the eigenvalues of
|
||||
$\hat L^2$ and $\hat L_z$:
|
||||
|
||||
\begin{equation}
|
||||
\hat L^2 \Ylm(\theta,\phi) = l(l+1) \Ylm(\theta,\phi),
|
||||
\end{equation}
|
||||
|
||||
and
|
||||
|
||||
\begin{equation}
|
||||
\hat L_z \Ylm(\theta,\phi) = m \Ylm(\theta,\phi).
|
||||
\end{equation}
|
||||
|
||||
Their explicit expression is:
|
||||
|
||||
\begin{equation}
|
||||
\Ylm(\theta,\phi) = (-1)^m N_\lm P_m^l(\cos\theta) e^{im\phi},
|
||||
\end{equation}
|
||||
|
||||
where $l$ takes non-negative integer values and the possible values for $m$ are
|
||||
integers from $-l$ to $l$.
|
||||
The two symbols $\theta$ and $\phi$ are used to denote angular spherical coordinates.
|
||||
$N_\lm$ is the normalization factor:
|
||||
|
||||
\begin{equation}
|
||||
N_\lm = \sqrt{\frac{2l+1}{4\pi} \dfrac{(l-m)!}{(l+m)!}}
|
||||
\end{equation}
|
||||
|
||||
with $P_m^l$ being the associated Legendre polynomials.
|
||||
Complex spherical harmonics are orthogonal functions on the sphere, i.e.:
|
||||
|
||||
\begin{equation}
|
||||
\la\Ylm|Y_\mp^\lp\ra_\Omega = \delta_{l\lp}\delta_{m\mp},
|
||||
\end{equation}
|
||||
|
||||
where $\la.|.\ra_\Omega$ denotes the integration over the unit sphere.
|
||||
Furthermore, they form a complete basis set since any square-integrable
|
||||
function $f(\theta,\phi)$ can be expanded in series according to:
|
||||
|
||||
\begin{equation}
|
||||
f(\theta,\phi) = \sum_{l=0}^\infty \sum_{m=-l}^l a_{\lm}\Ylm(\theta,\phi).
|
||||
\end{equation}
|
||||
|
||||
Complex spherical harmonics satisfy two useful properties commonly employed in practical applications.
|
||||
The parity properties:
|
||||
|
||||
\begin{equation}\label{eq:parity_property_Ylm}
|
||||
{Y_m^l}^\dagger(\rrhat) = (-1)^m Y_{-m}^l (\rrhat),
|
||||
\end{equation}
|
||||
|
||||
\begin{equation}
|
||||
\Ylm(-\rrhat) = (-1)^l \Ylm(\rrhat),
|
||||
\end{equation}
|
||||
|
||||
and the addition theorem:
|
||||
|
||||
\begin{equation}\label{eq:addition_theorem_CSH}
|
||||
\sum_m \Ylm(\rrhat)\, \Ylm^\*(\rrphat) =
|
||||
\dfrac{2l+1}{4\pi}\,\mcP_l (\rrhat\cdot\rrphat),
|
||||
\end{equation}
|
||||
|
||||
where $\mcP_l(q)$ are Legendre polynomials.
|
||||
Equation \ref{eq:addition_theorem_CSH} is generally employed in the
|
||||
application of the nonlocal part of the Hamiltonian in the case of
|
||||
norm-conserving pseudopotentials when [[useylm]] = 0.
|
||||
|
||||
## Real spherical harmonics <a id="RSH"><a>
|
||||
|
||||
Real spherical harmonics (RSH) are obtained by combining complex conjugate
|
||||
functions associated to opposite values of $m$.
|
||||
RSH are the most adequate basis functions for calculations in which atomic
|
||||
symmetry is important since they can be directly related to the irreducible
|
||||
representations of the subgroups of $D_3$ [[cite:Blanco1997]].
|
||||
Moreover, being real, they have half the memory requirement of complex spherical harmonics.
|
||||
This is clearly an advantage if high angular momenta are needed or several RHS values have to be stored in memory.
|
||||
A possible definition for RHS is [[cite:Blanco1997]]:
|
||||
|
||||
\begin{equation}\label{eq:Definition_real_harmonics}
|
||||
\rYlm \df
|
||||
\begin{cases}
|
||||
\quad \dfrac{(-1)^m}{\sqrt{2}}\,(\Ylm + \Ylm^\*) \quad m > 0
|
||||
\\\\
|
||||
\quad Y_0^l \quad m = 0
|
||||
\\\\
|
||||
\quad \dfrac{(-1)^m}{i\sqrt{2}}\,(Y_{\lvert{m}\rvert}^l - {Y_{\lvert{m}\rvert}^l}^\*) \quad m < 0.
|
||||
\end{cases}
|
||||
\end{equation}
|
||||
|
||||
Equation \ref{eq:Definition_real_harmonics} can be rewritten in matrix notation as
|
||||
|
||||
\begin{equation}\label{eq:Unitaty_transformation_Y_Re2Cmplx}
|
||||
\underline\rY^l = U^l \underline{Y}^l,
|
||||
\end{equation}
|
||||
|
||||
where $\underline\rY^l$ and $\underline{Y}^l$ are column vectors containing real and
|
||||
spherical harmonics ordered by increasing $m$ values.
|
||||
$U^l$ is a unitary matrix whose explicit expression can be found in [[cite:Blanco1997]].
|
||||
|
||||
The basic properties of RSH can be easily derived from
|
||||
the properties of complex spherical harmonics by means of \ref{eq:Definition_real_harmonics}.
|
||||
|
||||
Reality
|
||||
:
|
||||
\begin{equation}
|
||||
{\rYlm}^\*(\rrhat) = \rYlm(\rrhat),
|
||||
\end{equation}
|
||||
|
||||
Parity property
|
||||
:
|
||||
\begin{equation}
|
||||
\rYlm(-\rrhat) = (-1)^l \rYlm (\rrhat),
|
||||
\end{equation}
|
||||
|
||||
Orthonormality
|
||||
:
|
||||
\begin{equation}
|
||||
\la\rYlm|\rY_{\mp}^\lp\ra_\Omega = \delta_{l\lp} \delta_{m\mp}.
|
||||
\end{equation}
|
||||
|
||||
Certain on-site matrix elements include the integration of three real spherical harmonics, which
|
||||
can be evaluated exactly:
|
||||
|
||||
\begin{equation}\label{eq:Gaunt_def}
|
||||
\Gaunt_{\li\mi\lj\mj}^{\LM} \df \int \rY_\mi^\li\,\rY_M^L\,\rY_\mj^\lj\ddO.
|
||||
\end{equation}
|
||||
|
||||
The above integral is known in the literature under the name of Gaunt coefficient.
|
||||
Due to selection rules, the integral in equation \ref{eq:Gaunt_def} is nonzero only if $\mi+\mj = M$
|
||||
and $\lvert{\li+\lj}\rvert \ge (L,L+2,L+4, \dots) \ge \lvert{\li-\lj}\rvert$.
|
||||
|
||||
!!! note
|
||||
|
||||
The expression for $\mcG$ follows the internal conventions
|
||||
used inside the ABINIT code and slightly differs from the one reported in standard text-books.
|
||||
|
||||
## Symmetry transformation of RSH
|
||||
|
||||
The use of symmetry properties is of fundamental importance to simplify the
|
||||
determination of electronic structure as it greatly
|
||||
reduces the number of non-equivalent integrals that have to be computed,
|
||||
as well as the size of any tables stored in memory.
|
||||
|
||||
Frequently one has to evaluate the value of a RSH at a rotated point after
|
||||
the application of a symmetry operation belonging to the space group of the crystal.
|
||||
This usually occurs when $\kk$-dependent matrix elements have to be symmetrized in the full
|
||||
Brillouin zone starting from the knowledge of their symmetrical image in the irreducible wedge.
|
||||
The effect of a proper or improper rotation on a RSH can be deduced from:
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:RSH_rotation}
|
||||
\omcR\,\rYlm(\rrhat) = \rYlm(\mcRm \rrhat) =
|
||||
\sum_\alpha D^l_{\alpha m}(\mcR)\,\rYlm(\rrhat).
|
||||
\end{equation}
|
||||
|
||||
That is, spherical harmonics of given $l$ are transformed into a linear combination
|
||||
of RHS of same $l$ where the coefficients are given by:
|
||||
|
||||
\begin{equation}
|
||||
D^l_{\alpha m}(\mcR) \df \la\rY^l_\alpha|\hat\mcR|\rYlm\ra.
|
||||
\end{equation}
|
||||
|
||||
## Useful expansions in terms of RSH
|
||||
|
||||
By means of the unitary transformation given in \ref{eq:Definition_real_harmonics},
|
||||
it is possible to rewrite the Rayleigh expansion of a plane wave in terms of RSH as
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:Rayleigh_expansion_real_Ylm}
|
||||
e^{i(\kpG)\cdot\rr} =
|
||||
4\pi \sum_\lm i^l j_l(\lvert{\kpG}\rvert r)\,\rYlm(\kpGhat)\,\rYlm(\rrhat)
|
||||
\end{equation}
|
||||
|
||||
with $j_l(x)$ the spherical Bessel function of order $l$.
|
||||
In a similar way, the electrostatic potentials can be expanded in a real spherical harmonics basis set.
|
||||
The final expression is very similar to the one obtained in the case of complex spherical harmonics:
|
||||
|
||||
\begin{equation}\label{eq:Expansion_Coulombian_RSH}
|
||||
\dfrac{1}{\lvert{\rr-\rrp}\rvert} = \sum_{l=0}^\infty
|
||||
\dfrac{4\pi}{2l+1}
|
||||
\sum_{m=-l}^{m=l}
|
||||
\dfrac{r_<^l}{r_>^{l+1}} \rYlm(\rrhat)\rYlm(\rrphat),
|
||||
\end{equation}
|
||||
|
||||
with $r_< \df \min(r,r')$ and $r_> \df \max(r,r')$.
|
||||
File diff suppressed because it is too large
Load Diff
|
|
@ -199,8 +199,10 @@ pages:
|
|||
- tutorial/wannier90.md
|
||||
- Theory:
|
||||
- Geometry: theory/geometry.md
|
||||
- Spherical Harmonics: theory/spherical_harmonics.md
|
||||
- Wavefunctions: theory/wavefunctions.md
|
||||
- Noncollinear: theory/noncollinear.md
|
||||
- Pseudopotentials: theory/pseudopotentials.md
|
||||
- Many-Body Perturbation Theory: theory/mbt.md
|
||||
- Bethe-Salpeter: theory/bse.md
|
||||
- Documents: theory/documents.md
|
||||
|
|
|
|||
Loading…
Reference in New Issue