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\begin_body

\begin_layout Section
Basis states
\begin_inset CommandInset label
LatexCommand label
name "chap:Basis"

\end_inset


\begin_inset Index idx
status collapsed

\begin_layout Plain Layout
basis
\end_layout

\end_inset


\end_layout

\begin_layout Standard

\family typewriter
\lang english
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
hypertarget{sec:basis}{}
\end_layout

\end_inset


\end_layout

\begin_layout Subsection
General definitions
\begin_inset CommandInset label
LatexCommand label
name "subsec:General-definitions"

\end_inset


\end_layout

\begin_layout Subsubsection
Orbital structure
\begin_inset CommandInset label
LatexCommand label
name "subsec:Orbital-structure"

\end_inset


\end_layout

\begin_layout Standard
The 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 basis consists of atom-like orbitals 
\begin_inset Formula $\Phi_{ti}$
\end_inset

 for each Wyckoff position 
\begin_inset Formula $t$
\end_inset

 with atom-like quantom numbers (qns) 
\begin_inset Formula $i$
\end_inset

, which are products of radial and angular functions.
 Depending on the mode of the relativistic treatment we have at atom 
\begin_inset Formula $t$
\end_inset

 orbitals of shape
\begin_inset Formula 
\begin{equation}
\Phi_{ti}=\begin{cases}
\varphi_{tnl}Y_{lm}\chi_{s} & \text{without spin orbit coupling (NREL/SREL)}\\
\left(\begin{array}{c}
g_{tnlj}\chi_{\kappa_{lj}\mu}\\
if_{tnlj}\chi_{-\kappa_{lj}\mu}
\end{array}\right) & \text{with spin orbit coupling (FREL)}
\end{cases}\label{eq:orbital_def}
\end{equation}

\end_inset

where 
\begin_inset Formula $n$
\end_inset

 is the main qn and 
\begin_inset Formula $\chi_{s}$
\end_inset

 are spin-
\begin_inset Formula $\frac{1}{2}$
\end_inset

 basis spinors: 
\begin_inset Formula $\sigma_{z}\chi_{s}=\chi_{s}s$
\end_inset

 (
\begin_inset Formula $s=\pm1$
\end_inset

), 
\begin_inset Formula $Y_{lm}$
\end_inset

 are (real) spherical harmonics with angular momentum qns 
\begin_inset Formula $l$
\end_inset

 and 
\begin_inset Formula $m\in\left[-l,l\right]$
\end_inset

 and 
\begin_inset Formula $\varphi_{tnl}$
\end_inset

 are radial functions (one per 
\begin_inset Formula $nl$
\end_inset

-shell and atom).
 (In scalar relativistic modes there are actually up to four radial functions
 per 
\begin_inset Formula $\varphi_{nl}$
\end_inset

-shell, the details of which are yet to be published.) In FREL mode the orbitals
 are four-spinors with two large 
\begin_inset Formula $g_{tnlj=l\pm\frac{1}{2}}$
\end_inset

 and two small 
\begin_inset Formula $f_{tnlj=l\pm\frac{1}{2}}$
\end_inset

 radial functions per 
\begin_inset Formula $nl$
\end_inset

-shell and 
\begin_inset Formula $\chi_{\kappa_{lj}\mu}$
\end_inset

 are spherical spinors with spin-orbit quantum number 
\begin_inset Formula $\kappa=\left(2j+1\right)\left(l-j\right)$
\end_inset

, total angular momentum qns 
\begin_inset Formula $j=l\pm\frac{1}{2}$
\end_inset

 and 
\begin_inset Formula $\mu=\left[-j,j\right]$
\end_inset

, such that 
\begin_inset Formula $\hat{\kappa}\chi_{\kappa\mu}=-\chi_{\kappa\mu}\kappa$
\end_inset

, 
\begin_inset Formula $\hat{j}\chi_{\kappa\mu}=\chi_{\kappa\mu}j$
\end_inset

, 
\begin_inset Formula $\hat{j}_{z}\chi_{\kappa\mu}=\chi_{\kappa\mu}\mu$
\end_inset

 and in detail 
\begin_inset Formula $\hat{\kappa}=1+\hat{\sigma}\hat{L}$
\end_inset

, 
\begin_inset Formula $\kappa_{lj=l+\frac{1}{2}}=-\left(l+1\right)$
\end_inset

, 
\begin_inset Formula $\kappa_{lj=l-\frac{1}{2}}=l$
\end_inset

.
\end_layout

\begin_layout Standard
Note, that 
\begin_inset Formula $\varphi_{tnl}$
\end_inset

 does not depend on the spin index 
\begin_inset Formula $s$
\end_inset

, so the spin polarization of the final wave functions must be obtained
 via additional orbitals with the same qns (polarization-orbitals).
 More generally, any polarization effect of an imagined effective radial
 function due to some perturbation must be obtained via polarization-orbitals.
 (PO) 
\end_layout

\begin_layout Standard
From an atom point of view polarization-orbitals are orbitals with higher
 angular-momentum- and/or main-qns.
 We use the term PO more loosely than quantum chemists would.
 In a solid the distinction becomes mute, the more orbitals we add, since
 a 
\begin_inset Formula $4p$
\end_inset

 is an 
\begin_inset Formula $l+1$
\end_inset

-PO to the 
\begin_inset Formula $4s$
\end_inset

-orbital but a 
\begin_inset Formula $5s$
\end_inset

-PO (an 
\begin_inset Formula $n+1$
\end_inset

-PO to 
\begin_inset Formula $4s$
\end_inset

) is an 
\begin_inset Formula $l-1$
\end_inset

-PO to the 
\begin_inset Formula $4p$
\end_inset

-PO.
 
\end_layout

\begin_layout Standard
Now, group all orbitals of the same 
\begin_inset Formula $l$
\end_inset

- (
\begin_inset Formula $lj$
\end_inset

-) channel together and we get as possible valence orbitals 
\begin_inset Formula $s$
\end_inset

-, 
\begin_inset Formula $p$
\end_inset

-, 
\begin_inset Formula $d$
\end_inset

- and 
\begin_inset Formula $f$
\end_inset

- orbitals and if polarization-orbitals are included it just determines
 how many orbitals of each 
\begin_inset Formula $l$
\end_inset

-channel there are (see Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:From-atomic-basis-to-fplo-basis"
plural "false"
caps "false"
noprefix "false"

\end_inset

).
 Compare this to an angular momentum expansion of plane waves.
 Of course, also 
\begin_inset Formula $l>3$
\end_inset

 channels exist.
 However, the resulting orbitals are quite high in energy, have a quite
 large Hilbert subspace (
\begin_inset Formula $2l+1$
\end_inset

-dimensional) and rather small effect.
 So, we try to get away with low-
\begin_inset Formula $l$
\end_inset

 polarization-orbitals.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement H
wide false
sideways false
status open

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pic/basisscheme.pdf
	width 90col%
	clip

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:From-atomic-basis-to-fplo-basis"

\end_inset

From atomic basis to 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 basis: the left column shows the essential orbitals of an isolated atom.
 In the second column polarization orbitals are added.
 The third column is how one should thing about the second column, while
 the right column shows the orbital order which results in 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
In contrast to the valence orbitals, which serve to expand wave functions
 of higher and higher energy, the lower lying (fully occupied) semicore
 (core) states are described by a single atom-like orbital (such that the
 isolated atom limit) is correctly described.
 If polarization effects on these rather inert orbitals are important (hyper-fin
d field [
\begin_inset Formula $B^{\mathrm{HF}}$
\end_inset

] calculations) the corresponding polarization orbital needs to live in
 the same energy region as the main orbital and describe the effects of
 the polarization under consideration.
 To be specific if we need good spin polarization of the 
\begin_inset Formula $1s$
\end_inset

-orbitals we could consider the two spin polarized atomic orbitals 
\begin_inset Formula $\varphi_{n=1l=0s}Y_{00}\chi_{s}$
\end_inset

, where 
\begin_inset Formula $\varphi_{n=1l=0s}$
\end_inset

 are the majority 
\begin_inset Formula $s=1$
\end_inset

 and minority 
\begin_inset Formula $s=-1$
\end_inset

 
\begin_inset Formula $1s$
\end_inset

-orbitals of a spin polarized atom.
 Equivalent/similar sets of orbitals are a possibility, e.g.
\begin_inset space ~
\end_inset

one spin-unpolarized orbital (and say the majority orbital).
 These two functions are quit similar, but after orthogonalization we get
 a proper 
\begin_inset Formula $1s$
\end_inset

-basis of dimension 2 which spans a subspace of spin-polarized 
\begin_inset Formula $1s$
\end_inset

-orbitals.
 What is important here is that we did not add a 
\begin_inset Formula $2s$
\end_inset

 PO (as we would do for the open valence section) but a 
\begin_inset Formula $1s^{\prime}$
\end_inset

-orbital with slight perturbation.
 Afterall, the 
\begin_inset Formula $2s$
\end_inset

-orbital also exists in the basis.
 Since the core has a predetermined occupation pattern we need some bookkeeping
 to separate the first set of core orbitals from the polarization-orbitals
 by sorting the polarization-orbitals towards the end of the core-section
 and only occupy the first set.
 For the semicore section possible polarization-orbitals need to be sorted
 to the very end of the valence section for various reasons (see Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:From-atomic-basis-to-fplo-basis"
plural "false"
caps "false"
noprefix "false"

\end_inset

).
\end_layout

\begin_layout Subsubsection
Orbital classification
\end_layout

\begin_layout Standard
The orbitals at each atom are classified into sections
\end_layout

\begin_layout Description
core: orbitals which can be treated separately from all other orbitals since
 they 
\end_layout

\begin_layout Itemize
are fully occupied 
\end_layout

\begin_layout Itemize
have (essentially) 
\emph on
no 
\emph default
inter-site overlap/dispersion.
\end_layout

\begin_layout Itemize
have energies well below the chemical spectrum (valence band structure).
\end_layout

\begin_layout Description
semicore: orbitals which need to be included in the non-core (valence) section
 since they 
\end_layout

\begin_layout Itemize
have small inter-site overlap/dispersion
\end_layout

\begin_layout Itemize
although they are fully occupied
\end_layout

\begin_layout Itemize
and have energies below the chemical spectrum (valence band structure).
\end_layout

\begin_layout Description
valence: orbitals which 
\end_layout

\begin_layout Itemize
have large inter-site overlap/dispersion and form the valence and conduction
 bands
\end_layout

\begin_layout Itemize
are significantly partially occupied (chemical valence+
\begin_inset Formula $l$
\end_inset

-polarization)
\end_layout

\begin_layout Itemize
are essentially un-occupied and form high-lying bands (polarization orbitals)
\end_layout

\begin_layout Standard
The differentiation into semi-core and valence is used in automatic Wannier
 basis determination and in the order of on-site orthogonalization: first
 the semi-cores are orthogonalized among themselves, then the valence orbitals
 are made orthogonal to the semicore section and then among themselves.
 Exception occur if multi-semicore orbitals are defined, in which case the
 first orbital of the multi-orbital is in the semi-core section and fully
 occupied, while the additional (nearly empty) orbitals are sorted towards
 the end of the valence section.
 Otherwise, the orthogonalization process would deform the valence orbitals
 of the same 
\begin_inset Formula $nl$
\end_inset

-qns.
 But we want to keep the shape of the valence orbitals as close to an atomic
 orbital with corresponding qns as possible.
 Also adding a semicore polarization orbital should not change the already
 existing valence orbitals.
\end_layout

\begin_layout Subsubsection
Radial functions 
\begin_inset CommandInset label
LatexCommand label
name "par:Radial-functions"

\end_inset


\end_layout

\begin_layout Standard
The radial functions 
\begin_inset Formula $\varphi_{tnl}$
\end_inset

 or 
\begin_inset Formula $g_{tlnj}$
\end_inset

/
\begin_inset Formula $f_{tlnj}$
\end_inset

 in Eq.
\begin_inset space ~
\end_inset

(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:orbital_def"
plural "false"
caps "false"
noprefix "false"

\end_inset

) are solutions to a Schr
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash

\begin_inset Quotes erd
\end_inset

o}
\end_layout

\end_inset

\SpecialChar softhyphen
din\SpecialChar softhyphen
ger/Dirac equation with a modified atom potential 
\begin_inset Formula $V_{tnl}=V_{t}^{\mathrm{at}}+V_{t}^{\infty}+V_{tnl}^{\mathrm{conf}}+V_{tQS}$
\end_inset

, where 
\begin_inset Formula $t$
\end_inset

 denotes the Wyckoff position (sort) and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $V^{\mathrm{at}}$
\end_inset

 is the self consistent atom potential, obtained by solving a DFT atom with
 confining potential 
\begin_inset Formula $V_{t}^{\infty}+V_{tnl}^{\mathrm{conf}}$
\end_inset

.
\end_layout

\begin_layout Itemize
\begin_inset Formula $V^{\infty}$
\end_inset

 is an infinite potential well outside the compact support radius 
\begin_inset Formula $\rho$
\end_inset

, which is individual for each atom and depends on the atom environment
 in a complicated manner.
 For FREL mode only the large component feels this well, which gives leaking
 small component at 
\begin_inset Formula $r=\rho$
\end_inset

, which however is a very tiny error.
\end_layout

\begin_layout Itemize
\begin_inset Formula $V_{nl}^{\mathrm{conf}}=\left(\frac{r}{\left(PP_{nl}\rho\right)}\right)^{N}$
\end_inset

 is a confining potential with atom-wide compression parameter 
\begin_inset Formula $P$
\end_inset

, orbital-specific compression parameter 
\begin_inset Formula $P_{nl}$
\end_inset

, which is made to be close to 
\begin_inset Formula $1$
\end_inset

 (default value), and with compression power 
\begin_inset Formula $N$
\end_inset

 (default: 
\begin_inset Formula $N=14$
\end_inset

).
\end_layout

\begin_layout Itemize
\begin_inset Formula $V_{QS}$
\end_inset

 is added to the self consistent potential, when calculating the radial
 functions, and is obtained after self consistency by recreating the atom
 potential (for each orbital separately) with modified atomic occupation
 numbers such that the resulting atom is missing charge of amount 
\begin_inset Formula $Q_{nl}$
\end_inset

 and/or has total magnetic moment 
\begin_inset Formula $S_{nl}$
\end_inset

.
 If 
\begin_inset Formula $Q$
\end_inset

 is larger than the total atom charge (light atoms) a suitable exponentially
 localized potential is used.
 In FREL mode both 
\begin_inset Formula $j$
\end_inset

-sub-shells use the same 
\begin_inset Formula $Q$
\end_inset

-, 
\begin_inset Formula $S$
\end_inset

-parameters.
 The 
\begin_inset Formula $S_{nl}$
\end_inset

 option is experimental and was used in the context of the unofficial hyper-fine
 field (
\begin_inset Formula $B^{\mathrm{HF}}$
\end_inset

) module, where core and semi-core 
\begin_inset Formula $s$
\end_inset

-orbitals need to describe the spin polarization at the nucleus extremely
 well.
 If 
\begin_inset Formula $S>0$
\end_inset

 the majority orbital for the corresponding qns is used, and if 
\begin_inset Formula $S<0$
\end_inset

 the minority orbital (which has a different shape) is used.
\end_layout

\begin_layout Subsubsection
Multi orbitals
\begin_inset CommandInset label
LatexCommand label
name "subsec:Multi-orbitals"

\end_inset


\end_layout

\begin_layout Standard
To increase variational flexibility, polarization orbitals can be added,
 which results in the concept of multi-orbitals (MuO) (see Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:From-atomic-basis-to-fplo-basis"
plural "false"
caps "false"
noprefix "false"

\end_inset

).
 These are denoted by prepending a multiplicity character to the orbital
 name.
 Currently, multiplicities up to six are implemented with character: 
\emph on
SDTQUX
\emph default
 which stand for 
\emph on
S
\emph default
ingle, 
\emph on
D
\emph default
ouble, 
\emph on
T
\emph default
riple, 
\emph on
Q
\emph default
uadruple, q
\emph on
U
\emph default
intuple, si
\emph on
X
\emph default
tuple.
\end_layout

\begin_layout Standard
A multi orbital 
\begin_inset Formula $T3d$
\end_inset

 then means that besides the first 
\begin_inset Formula $3d$
\end_inset

-orbital two more 
\begin_inset Formula $d$
\end_inset

-orbitals are added to the basis, which of course will have their own 
\begin_inset Formula $Q$
\end_inset

-, 
\begin_inset Formula $P$
\end_inset

- and 
\begin_inset Formula $S$
\end_inset

-parameters.
 Which qns the orbitals carry depends on the orbital classification (see
 below and Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:From-atomic-basis-to-fplo-basis"
plural "false"
caps "false"
noprefix "false"

\end_inset

).
\end_layout

\begin_layout Standard
All orbitals of a multi-
\series bold
core
\series default
 and multi-
\series bold
semicore
\series default
 orbital will have the 
\series bold
same
\series default
 qns as the first orbital, since we want to add slightly distorted orbitals
 with the same qns to add flexibility in the resulting effective radial
 functions (implicit result of diagonalization).
 So a double core ore semicore orbital expands as
\begin_inset Formula 
\[
D2p\to2p_{1}~2p_{2}
\]

\end_inset


\end_layout

\begin_layout Standard
The orbitals of a multi-
\series bold
valence
\series default
 orbital will have 
\series bold
increasing main
\series default
 qns, such that the resulting set of orbitals looks like a set of atomic
 orbitals for ever higher energies:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
T3d\to3d~4d~5d
\]

\end_inset


\end_layout

\begin_layout Subsubsection
Default basis
\end_layout

\begin_layout Standard
Using our classification we can define a save core section in the sense
 that the inter-site overlap is negligible even for close packed solids
 under pressure, say up to 200-300GPa.
\end_layout

\begin_layout Standard
The valence orbitals contain the 
\begin_inset Quotes eld
\end_inset

chemical
\begin_inset Quotes erd
\end_inset

 valence orbitals and one or two polarization orbitals (POs) of the lowest
 unoccupied shells (in an atom) with different angular momentum.
 Chemical valence is roughly defined as the HOMO+LUMO or HOMO-1+HOMO of
 the atom, (exceptions apply).
 Additionally the 
\begin_inset Quotes eld
\end_inset

chemical
\begin_inset Quotes erd
\end_inset

 orbitals are doubled to become double-orbitals (MuO with multiplicity 2),
 thus adding variational flexibility to the corresponding 
\begin_inset Formula $nl$
\end_inset

-channel.
 
\end_layout

\begin_layout Standard
Examples:
\end_layout

\begin_layout Itemize
H has chemical valence 
\begin_inset Formula $1s$
\end_inset

, to which we add the lowest lying 
\begin_inset Formula $l\ne0$
\end_inset

 PO 
\begin_inset Formula $2p$
\end_inset

.
 Then we double the chemical valence by adding 
\begin_inset Formula $2s$
\end_inset

 resulting in 
\begin_inset Formula $D1s$
\end_inset

, 
\begin_inset Formula $S2p$
\end_inset

.
\end_layout

\begin_layout Itemize
C has chemical valence 
\begin_inset Formula $2s$
\end_inset

, 
\begin_inset Formula $2p$
\end_inset

 (HOMO-1+HOMO), to which we add the lowest lying 
\begin_inset Formula $l\ne0,1$
\end_inset

 PO 
\begin_inset Formula $3d$
\end_inset

.
 After doubleing the chemical valence we get 
\begin_inset Formula $D2s$
\end_inset

, 
\begin_inset Formula $D2p$
\end_inset

, 
\begin_inset Formula $S3d$
\end_inset

.
\end_layout

\begin_layout Itemize
Na has chemical valence 
\begin_inset Formula $3s$
\end_inset

, 
\begin_inset Formula $3p$
\end_inset

 (HOMO+LUMO) to which we add the lowest lying 
\begin_inset Formula $l\ne0,1$
\end_inset

 PO 
\begin_inset Formula $3d$
\end_inset

.
 Then we double 
\begin_inset Formula $3s$
\end_inset

 and 
\begin_inset Formula $3p$
\end_inset

 to obtain the valence section 
\begin_inset Formula $D3s$
\end_inset

, 
\begin_inset Formula $D3p$
\end_inset

, 
\begin_inset Formula $S3d$
\end_inset

.
\end_layout

\begin_layout Itemize
Fe has chemical valence 
\begin_inset Formula $4s$
\end_inset

, 
\begin_inset Formula $3d$
\end_inset

 and the lowest lying PO with different 
\begin_inset Formula $l$
\end_inset

 is 
\begin_inset Formula $4p$
\end_inset

.
 We double 
\begin_inset Formula $4s$
\end_inset

 and 
\begin_inset Formula $3d$
\end_inset

 to obtain the valence section 
\begin_inset Formula $D4s$
\end_inset

, 
\begin_inset Formula $D3d$
\end_inset

, 
\begin_inset Formula $S4p$
\end_inset

.
\end_layout

\begin_layout Itemize
Eu has chemical valence 
\begin_inset Formula $6s$
\end_inset

, 
\begin_inset Formula $5d$
\end_inset

, 
\begin_inset Formula $4f$
\end_inset

 and lowest lying polarization orbital with different 
\begin_inset Formula $l$
\end_inset

 is 
\begin_inset Formula $6p$
\end_inset

.
 We double 
\begin_inset Formula $6s$
\end_inset

, 
\begin_inset Formula $5d$
\end_inset

 and 
\begin_inset Formula $4f$
\end_inset

 to obtain the valence section 
\begin_inset Formula $D6s$
\end_inset

, 
\begin_inset Formula $D5d$
\end_inset

, 
\begin_inset Formula $S6p$
\end_inset

, 
\begin_inset Formula $D4f$
\end_inset

.
\end_layout

\begin_layout Itemize
La has chemical valence 
\begin_inset Formula $6s$
\end_inset

, 
\begin_inset Formula $5d$
\end_inset

, 
\begin_inset Formula $4f$
\end_inset

 and lowest lying polarization orbital with different 
\begin_inset Formula $l$
\end_inset

 is 
\begin_inset Formula $6p$
\end_inset

.
 We double 
\begin_inset Formula $6s$
\end_inset

, 
\begin_inset Formula $5d$
\end_inset

 to obtain the valence section 
\begin_inset Formula $D6s$
\end_inset

, 
\begin_inset Formula $D5d$
\end_inset

, 
\begin_inset Formula $S6p$
\end_inset

, 
\begin_inset Formula $S4f$
\end_inset

.
\begin_inset Newline newline
\end_inset

La is a bit of an exception since 
\begin_inset Formula $4f$
\end_inset

 is not doubled.
 The reason is that 
\begin_inset Formula $4f$
\end_inset

 should be essentially empty (which it is not in LDA).
 This might be less optimal and be changed in the future.
 It can also be changed via modifications.
\end_layout

\begin_layout Standard
According to these definitions the semicore section is formed by the remaining
 orbitals between the lowest valence and the highest core orbital.
\end_layout

\begin_layout Standard
In the default basis all core and semicore orbitals are single.
\end_layout

\begin_layout Subsubsection
Parameters
\begin_inset CommandInset label
LatexCommand label
name "subsec:Parameters"

\end_inset


\end_layout

\begin_layout Standard
The parameters which determine the basis are the atom specific compact support
 radius 
\begin_inset Formula $\rho$
\end_inset

, the atom-specific compression parameter 
\begin_inset Formula $P$
\end_inset

 and compression power 
\begin_inset Formula $N$
\end_inset

.
 
\begin_inset Formula $\rho$
\end_inset

 and 
\begin_inset Formula $P$
\end_inset

 are determined internally (based on optimizations performed once and for
 all) while 
\begin_inset Formula $N$
\end_inset

 is fixed to it's default 
\begin_inset Formula $N=14$
\end_inset

 .
\end_layout

\begin_layout Standard
The orbital specific charges 
\begin_inset Formula $Q_{tnl}$
\end_inset

 (and/or spin moments 
\begin_inset Formula $S_{tnl}$
\end_inset

, currently experimental) and compression modifiers 
\begin_inset Formula $P_{nl}$
\end_inset

 (see Par.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "par:Radial-functions"
plural "false"
caps "false"
noprefix "false"

\end_inset

) were also determined by optimizations and are tabulated internally.
 However, they can be changed if circumstances make this seem reasonable.
 They also have to be chosen, when the basis is modified (extended).
\end_layout

\begin_layout Standard
The default parameter values are 
\begin_inset Formula $Q_{nl}=S_{nl}=0$
\end_inset

 and 
\begin_inset Formula $P_{nl}=1$
\end_inset

.
 
\begin_inset Formula $S$
\end_inset

 is not currently used; it is an option for hyper-fine (
\begin_inset Formula $B^{\mathrm{HF}}$
\end_inset

) field calculations (experimental), in which case the spin polarization
 of 
\begin_inset Formula $s$
\end_inset

-orbitals at the nucleus must be very precise.
 Making core and semicore MuOs, with the added orbitals having 
\begin_inset Formula $S\ne0$
\end_inset

, will improve the resulting spin polarization at the nucleus.
\end_layout

\begin_layout Standard
It seems to be the right strategy to let the essential orbitals have 
\begin_inset Formula $Q=S=0$
\end_inset

 in order to get atomic (yet compressed) orbitals with energies such that
 the single-atom limit is basically correct.
 Essential means: first core/semicore and first chemical valence orbitals,
 basically the orbitals which are occupied in the atom.
 Imagine calculating a free standing atom, then compression goes to zero
 (
\begin_inset Formula $\rho$
\end_inset

 becomes large) and these orbitals need to be solutions to the free atom,
 hence 
\begin_inset Formula $Q=S=0$
\end_inset

.
\end_layout

\begin_layout Standard
For orientation:
\end_layout

\begin_layout Itemize
the first core orbitals (usually core is single) have default values
\end_layout

\begin_layout Itemize
the first semicore orbitals (usually semicore is single) have default 
\begin_inset Formula $Q$
\end_inset

 and 
\begin_inset Formula $S$
\end_inset

 but may have 
\begin_inset Formula $P\ne1$
\end_inset

 (from optimization in solids), especially for heavier atoms under pressure.
\end_layout

\begin_layout Itemize
The chemical valence orbitals (first of MuOs) usually have default 
\begin_inset Formula $Q=S=0$
\end_inset

 but some 
\begin_inset Formula $P\ne1$
\end_inset

.
\end_layout

\begin_layout Itemize
The additional orbitals of a MuO and the true polarization orbitals usually
 need 
\begin_inset Formula $Q\ne0$
\end_inset

 (determined by optimization and tabulated).
\end_layout

\begin_layout Itemize

\emph on
Experimental
\emph default
: additional orbitals for core and semicore MuOs (for 
\begin_inset Formula $B^{\mathrm{HF}}$
\end_inset

) should use 
\begin_inset Formula $Q_{2\ldots}=0,2,4,\ldots$
\end_inset

 and 
\begin_inset Formula $S_{2,\ldots}=5,-5,10,-10$
\end_inset

 (remember 
\begin_inset Formula $Q_{0}=S_{0}=0$
\end_inset

), where 
\begin_inset Formula $Q_{i}=0$
\end_inset

 seems to be good for ligther atoms and 
\begin_inset Formula $Q=2$
\end_inset

 for heavier.
 If problems occur, they are visible in the population analysis, where the
 net occupations become significantly larger than the gross occupations.
 Example: 
\begin_inset Formula $D2s$
\end_inset

 
\begin_inset Formula $Q=(0,0)$
\end_inset

, 
\begin_inset Formula $S=\left(0,5\right)$
\end_inset

 or 
\begin_inset Formula $D2s$
\end_inset

 
\begin_inset Formula $Q=(0,2)$
\end_inset

, 
\begin_inset Formula $S=\left(0,5\right)$
\end_inset

.
 This business is prone to instabilities.
\end_layout

\begin_layout Subsubsection
Modifications
\end_layout

\begin_layout Standard
The default basis can be modified in various ways: using 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fedit{}
\end_layout

\end_inset

, using 
\family typewriter
pyfplo.fploio.Basis
\family default
 (
\begin_inset Flex URL
status collapsed

\begin_layout Plain Layout

../pyfplo/pyfplo.pdf
\end_layout

\end_inset

) or editing 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
filei{=.basdef}
\end_layout

\end_inset

 by hand (see Sec.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Basis-modifications"
plural "false"
caps "false"
noprefix "false"

\end_inset

).
 The following are the most important modifications
\end_layout

\begin_layout Itemize
Extend the basis 
\end_layout

\begin_deeper
\begin_layout Itemize
add more polarization orbitals to existing orbitals (increase the multiplicity
 of multi-orbitals)
\end_layout

\begin_layout Itemize
add a 
\begin_inset Formula $d$
\end_inset

-orbital if none exists (H and He)
\end_layout

\begin_layout Itemize
add an 
\begin_inset Formula $f$
\end_inset

-orbital if none exist
\end_layout

\end_deeper
\begin_layout Itemize
Core treatment of 
\begin_inset Formula $4f$
\end_inset

 (for crazy people who know what they do)
\end_layout

\begin_deeper
\begin_layout Itemize
move 
\begin_inset Formula $4f$
\end_inset

 into core
\end_layout

\begin_layout Itemize
move 
\begin_inset Formula $4f$
\end_inset

 into core and remove all remaining 
\begin_inset Formula $f$
\end_inset

-POs from the valence altogether.
\end_layout

\end_deeper
\begin_layout Itemize
Experimental: Improve core/semicore for 
\begin_inset Formula $B^{\mathrm{HF}}$
\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
double/triple the core
\end_layout

\begin_layout Itemize
double/triple the semicore
\end_layout

\end_deeper
\begin_layout Itemize
Modeling, testing \SpecialChar ldots
 (not accurate)
\end_layout

\begin_deeper
\begin_layout Itemize
make all valence orbitals single (not a standard option, but possible using
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
pyfplo{}
\end_layout

\end_inset

)
\end_layout

\begin_layout Itemize
remove orbitals (possible using 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
pyfplo{}
\end_layout

\end_inset

)
\end_layout

\end_deeper
\begin_layout Subsection
Basis setup
\begin_inset CommandInset label
LatexCommand label
name "subsec:Basis-setup"

\end_inset


\end_layout

\begin_layout Standard
As explained in Sec.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:General-definitions"
plural "false"
caps "false"
noprefix "false"

\end_inset

, the default basis is out of the box and ready to go.
 However, it turns out that in some contexts a modified (especially extended)
 basis gives better results.
 To understand how such modifications are achieved we need to explain the
 basis setup of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 in detail.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement H
wide false
sideways false
status open

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pic/basissetup.pdf
	width 50col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Flow of basis setup on 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 run.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
When 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 is running 
\end_layout

\begin_layout Enumerate
for each Wyckoff position the default basis is extracted from internal tables
 according to a user selected basis-version (currently there is still only
 one version available).
 
\end_layout

\begin_layout Enumerate
\begin_inset space ~
\end_inset


\end_layout

\begin_deeper
\begin_layout Enumerate
If standard modifications are requested by the user these are applied.
\end_layout

\begin_layout Enumerate
If the file 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
filei{=.basdef}
\end_layout

\end_inset

 is found in the working directory it will be read and it's content will
 overwrite the default basis and sidestep the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fedit{}
\end_layout

\end_inset

-defined modifications!
\end_layout

\end_deeper
\begin_layout Enumerate
The basis obtained thus far is written in 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
filei{=.basdef}
\end_layout

\end_inset

 format to the output in a section marked like this:
\end_layout

\begin_deeper
\begin_layout LyX-Code
------------------------------------------------------------------------
 
\end_layout

\begin_layout LyX-Code
Start: content of =.basdef
\end_layout

\begin_layout LyX-Code
------------------------------------------------------------------------
\end_layout

\begin_layout LyX-Code
\SpecialChar ldots

\end_layout

\begin_layout LyX-Code
\SpecialChar ldots

\end_layout

\begin_layout LyX-Code
\SpecialChar ldots

\end_layout

\begin_layout LyX-Code
------------------------------------------------------------------------
\end_layout

\begin_layout LyX-Code
End  : content of =.basdef
\end_layout

\begin_layout LyX-Code
------------------------------------------------------------------------
\end_layout

\end_deeper
\begin_layout Enumerate
If the virtual crystal option was chosen (VCA) the basis obtained thus far
 will be merged with a default basis of appropriate atomic number to interpolate
d the basis parameters (as good as possible) between the two bases belonging
 to the interval boundaries of the atomic number interval which contains
 the VCA-nuclear charge.
 This process cannot be made absolutely consistent, since neighboring atoms
 can have differing basis sets, e.g.
 Zn and Ga.
 So keep in mind that VCA interpolation is applied after the basis definition/re
ading of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
filei{=.basdef}
\end_layout

\end_inset

! This also means that VCA will not be completely symmetric in the sense
 that Zn
\begin_inset Formula $_{x}$
\end_inset

Ga
\begin_inset Formula $_{1-x}$
\end_inset

 with Zn as nominal (input) element will be slightly different from Zn
\begin_inset Formula $_{x}$
\end_inset

Ga
\begin_inset Formula $_{1-x}$
\end_inset

 with Ga as nominal (input) element.
 Plan your investigations accordingly.
 VCA is a crude approximation after all.
\end_layout

\begin_layout Standard
Note, that there are two mutually exclusive routes to modifying the basis
\end_layout

\begin_layout Itemize
use-specified modification via 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fedit{}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
whatever is written in 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
filei{=.basdef}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Also note, that starting from 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 version >21.00, the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
filei{=.basdef}
\end_layout

\end_inset

 content in the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 output is no longer the preferred starting point to created a user-defined
 basis.
 Instead use 
\family typewriter
pyfplo.fploio.Basis
\family default
 (and various examples) explained in 
\begin_inset Flex URL
status collapsed

\begin_layout Plain Layout

../pyfplo/pyfplo.pdf
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
Some users used 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
filei{=.basdef}
\end_layout

\end_inset

 in the past as the only way to modify the basis.
 However, it is advised to use the newer mechanisms for basis manipulation
 in future projects.
 In almost all cases the modifications provided by 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fedit{}
\end_layout

\end_inset

/
\family typewriter
pyfplo.fedit.Fedit
\family default
 should be enough.
\end_layout

\begin_layout Subsection
=.basdef
\begin_inset CommandInset label
LatexCommand label
name "subsec:=.basdef"

\end_inset


\end_layout

\begin_layout Standard
Although 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
filei{=.basdef}
\end_layout

\end_inset

 is not the preferred path to modifying the basis it will be explained first,
 since it is a useful representation of the basis to discuss extensions
 in detail.
 (It actually is the internal representation.) 
\end_layout

\begin_layout Standard
The file 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
filei{=.basdef}
\end_layout

\end_inset

 contains one line for each Wyckoff position (sort), which looks like:
\end_layout

\begin_layout LyX-Code

\size footnotesize
3 P=(0.67); N=(14); 1s2s2p : 3s S3p P=(0.79); / D4s Q=(0,3.1) P=(0.92,1.01) ;
 D3d ...
 S4p Q=(4.2) P=(1);
\end_layout

\begin_layout Standard
and has the structure:
\end_layout

\begin_layout LyX-Code

\size footnotesize
sort P=(); N=(); core : semicore / valence
\end_layout

\begin_layout Itemize
The first number is the sort number.
 Hence, if 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
filei{=.basdef}
\end_layout

\end_inset

 does not contain certain sorts, the basis definitions of these missing
 sorts will retain their default settings.
\end_layout

\begin_layout Itemize
The leading 
\family typewriter
P=()
\family default
 defines the atom-wide compression parameter 
\begin_inset Formula $P_{t}$
\end_inset

 (see Secs.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "par:Radial-functions"
plural "false"
caps "false"
noprefix "false"

\end_inset

,
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Parameters"
plural "false"
caps "false"
noprefix "false"

\end_inset

).
\end_layout

\begin_layout Itemize
The leading 
\family typewriter
N=()
\family default
 defines the compression power 
\begin_inset Formula $N_{t}$
\end_inset

.
\end_layout

\begin_layout Itemize
Between the power and '
\family typewriter
:
\family default
' follows the core section
\end_layout

\begin_layout Itemize
Between '
\family typewriter
:
\family default
' and '
\family typewriter
/
\family default
' follows the semicore section
\end_layout

\begin_layout Itemize
After '
\family typewriter
/
\family default
' follows the valence section
\end_layout

\begin_layout Itemize
An orbital with default parameters 
\begin_inset Formula $Q_{nl}=S_{nl}=0$
\end_inset

 (and 
\begin_inset Formula $P_{nl}=1$
\end_inset

 for non-core orbitals) can simply be written with the usual notation 
\family typewriter
2p
\family default
.
\end_layout

\begin_layout Itemize
A single orbital with non-default parameters must be written 
\family typewriter
S3p P=()
\family default
; , where only the non-default parameters need to be explicitly specified.
 Of course all parameters can be specified, which for this example would
 read 
\family typewriter
S3p Q=() S=() P=();
\family default
.
\end_layout

\begin_layout Itemize
A multi-orbital starts with a multi-orbital character (Sec.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Multi-orbitals"
plural "false"
caps "false"
noprefix "false"

\end_inset

) followed by the orbital name and the parameter list, which can consist
 of 
\begin_inset Formula $Q$
\end_inset

, 
\begin_inset Formula $S$
\end_inset

 and 
\begin_inset Formula $P$
\end_inset

 (if not core).
 The number of values in the parameter list must match the multiplicity
 of the multi-orbtial, e.g.
\begin_inset space ~
\end_inset


\family typewriter
T3d
\family default
 requires 
\family typewriter
Q=(Q1,Q2,Q3)
\family default
.
 
\end_layout

\begin_layout Itemize
For the core a condensed notation like 
\family typewriter
D{1s2s2p} Q=(Q1,Q1) S=(S1,S2);
\family default
 can be used to group several core orbitals together, having the same parameters.
\end_layout

\begin_layout LyX-Code

\end_layout

\begin_layout Subsection
Basis modifications
\begin_inset CommandInset label
LatexCommand label
name "subsec:Basis-modifications"

\end_inset


\end_layout

\begin_layout Subsubsection
In 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fedit{}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Some standard basis modifications are implemented in the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fedit{}
\end_layout

\end_inset

 menu 
\family typewriter
\series bold
B
\series default
asis
\family default
 since 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 version 22.
 The modifications are applied to the default basis in the order as they
 appear on the menu screen! This together with the explanations below determines
 the resulting basis.
 Naturally, all calculations of one (sub-) project should use the same basis!
 So, using simple python scripts (
\family typewriter
pyfplo.fedit.Fedit
\family default
) to create all inputs seems advisable.
\end_layout

\begin_layout Standard
Modifications:
\end_layout

\begin_layout Description
basis-version identifies a complete set of default basis definitions for
 all (implemented) atoms.
 As of version 22, only one basis version is implemented, but others might
 follow.
 
\end_layout

\begin_layout Description
extension-level determines how many valence polarization orbitals will be
 added to the default basis.
 The default level is 
\begin_inset Formula $1$
\end_inset

 and yields the default basis with certain predefined valence MuOs.
 Using the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
filei{=.basdef}
\end_layout

\end_inset

 notation (Sec.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:=.basdef"
plural "false"
caps "false"
noprefix "false"

\end_inset

) a valence section example could look like this:
\end_layout

\begin_deeper
\begin_layout LyX-Code

\size footnotesize
\SpecialChar ldots
 / D4s Q=(Q4s,Q5s) P=(P4s,P5s); D3d Q=(Q3d,Q4d) P=(P3d,P4d); S4p Q=(Q4p)
 P=(P4p)
\end_layout

\begin_layout Standard
which expands to the valence orbitals
\end_layout

\begin_layout LyX-Code
 4s5s3d4d4p
\end_layout

\begin_layout Standard
We used place holders for the parameter values.
 Note, that we name the values Q4s, Q5s, \SpecialChar ldots
 according to the rules of Sec.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Multi-orbitals"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
An extension level 
\begin_inset Formula $2$
\end_inset

 would increase the multiplicity of all MuOs by one (
\family typewriter
S
\family default

\begin_inset Formula $\to$
\end_inset


\family typewriter
D
\family default
, 
\family typewriter
D
\family default

\begin_inset Formula $\to$
\end_inset


\family typewriter
T
\family default
, \SpecialChar ldots
), level 3 by two and so on, such that our example for extension level
 
\begin_inset Formula $2$
\end_inset

 would yield
\end_layout

\begin_layout LyX-Code

\size footnotesize
\SpecialChar ldots
 / T4s Q=(Q4s,Q5s,Q6s) P=(P4s,P5s,P6s); T3d Q=(Q3d,Q4d,Q5d) P=(P3d,P4d,P5d);
 D4p Q=(Q4p,Q5p)
\backslash

\begin_inset Newline newline
\end_inset

       P=(P4p,P5p)
\end_layout

\begin_layout Standard
which expands to the valence orbitals
\end_layout

\begin_layout LyX-Code
 
\size footnotesize
4s5s6s3d4d5d4p5p
\end_layout

\begin_layout Standard
The increased multiplicity requires that all parameter lists get a new value
 appended.
 Since only the default basis parameters were optimized and since the optimizati
on becomes less stable the more parameters are added, we need a rule to
 chose additional parameters, which is 
\begin_inset Formula 
\begin{align}
Q_{n+1,l} & =Q_{n,l}+2\nonumber \\
P_{n+1,l} & =\max\left(0.85,\min\left(P_{nl},1\right)\right)\label{eq:Pnplus1}
\end{align}

\end_inset


\end_layout

\end_deeper
\begin_layout Description
add-3d The default basis for H and He does not contain a 
\begin_inset Formula $d$
\end_inset

-orbital.
 This options adds a single 
\family typewriter
S3d Q=(5) P=(1);
\family default
 orbital to the basis, if the atom does not contain any 
\begin_inset Formula $d$
\end_inset

-orbital in any section.
\end_layout

\begin_layout Description
add-f If the valence section of an atom does not contain an 
\begin_inset Formula $f$
\end_inset

-orbital this option adds a single 
\family typewriter
Snf Q=(5) P=(1);
\family default
 , where 
\family typewriter
n
\family default
 is the lowest 
\begin_inset Formula $f$
\end_inset

-main qn, not contained in any section.
 So, if there is no 
\begin_inset Formula $f$
\end_inset

-orbital at all this adds a 
\family typewriter
S4f
\family default
 and if the core contains a 
\begin_inset Formula $4f$
\end_inset

-orbital this adds a 
\family typewriter
S5f
\family default
.
 If the valence contains an 
\begin_inset Formula $f$
\end_inset

-orbital none is added.
 
\end_layout

\begin_layout Description
Put-4f-into-core uses a list of elements and/or a list of sorts as input.
 For some reasons it might be desirable to put the 
\begin_inset Formula $4f$
\end_inset

-valence orbital into the core and give it a particular (non-full) occupation
 (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fedit{}
\end_layout

\end_inset

 submenu 
\family typewriter
Core-
\series bold
O
\series default
ccupation
\family default
).
 The union of the sorts resulting from the two input lists defines for which
 atoms this happens.
 This option moves an existing 
\begin_inset Formula $4f$
\end_inset

-orbital to the core and leaves the remaining 
\begin_inset Formula $f$
\end_inset

-POs in the valence.
 Such a calculation, even if it does not have a bad population analysis
 (net- much larger than gross-occupation), will have unreliable total energies!
 So, only use this option if you need to get rid of the 
\begin_inset Formula $f$
\end_inset

-states from the Fermi energy and if you know what you are doing.
 I would use LDA+U instead! Example: Eu has default basis
\end_layout

\begin_deeper
\begin_layout LyX-Code

\size footnotesize
core  : semicore / non-f-valence D4f Q=(Q4f,Q5f) P=(P4f,P5f); 
\end_layout

\begin_layout Standard
which applying this option becomes
\end_layout

\begin_layout LyX-Code

\size footnotesize
core 4f : semicore / non-f-valence S5f Q=(Q5f) P=(P5f); 
\end_layout

\end_deeper
\begin_layout Description
Put-4f-into-core-and-remove-f-POs uses a list of elements and/or a list
 of sorts as input.
 This option puts any existing 
\begin_inset Formula $4f$
\end_inset

-valence orbital into the core and removes the remaining 
\begin_inset Formula $4f$
\end_inset

-POs of the corresponding MuO altogether.
 This option might be more stable then the previous one, since a (physical
 valence) 
\begin_inset Formula $4f$
\end_inset

-orbital in the core violates the zero inter-site overlap condition and
 the additional 
\begin_inset Formula $f$
\end_inset

-valence POs of the corresponding 
\begin_inset Formula $f$
\end_inset

-MuO will lead to wrong matrix elements between the core-
\begin_inset Formula $4f$
\end_inset

 and the remaining 
\begin_inset Formula $f$
\end_inset

-valence POs.
 By removing the POs this error is removed (other similar errors remain
 though).
 The closing remarks of the previous option apply.
 Eu has default basis
\end_layout

\begin_deeper
\begin_layout LyX-Code

\size footnotesize
core  : semicore / non-f-valence D4f Q=(Q4f,Q5f) P=(P4f,P5f); 
\end_layout

\begin_layout Standard
which applying this option becomes
\end_layout

\begin_layout LyX-Code

\size footnotesize
core 4f : semicore / non-f-valence 
\end_layout

\end_deeper
\begin_layout Description
Multi-core-orbitals uses a multiplicity and a 
\begin_inset Formula $Q$
\end_inset

- and 
\begin_inset Formula $S$
\end_inset

-list of said multiplicity as input.
 The default are empty lists, which indicate 
\begin_inset Quotes eld
\end_inset

no change
\begin_inset Quotes erd
\end_inset

.
 This option will turn all core orbitals into corresponding multi-core orbitals
 each having the specified 
\begin_inset Formula $Q$
\end_inset

- and 
\begin_inset Formula $S$
\end_inset

-parameter values (experimental for 
\begin_inset Formula $B^{\mathrm{HF}}$
\end_inset

 calculations).
 Example: multiplicity 2 with two 
\begin_inset Formula $Q$
\end_inset

- and 
\begin_inset Formula $S$
\end_inset

-values turns a core section
\end_layout

\begin_deeper
\begin_layout LyX-Code

\size footnotesize
1s2s2p  / \SpecialChar ldots

\end_layout

\begin_layout Standard
into
\end_layout

\begin_layout LyX-Code

\size footnotesize
 D{1s2s2p} Q=(Q1,Q2) S=(S1,S2); /\SpecialChar ldots

\end_layout

\end_deeper
\begin_layout Description
Multi-semicore-orbitals uses a multiplicity and a 
\begin_inset Formula $Q$
\end_inset

- and 
\begin_inset Formula $S$
\end_inset

-list of said multiplicity as input.
 The default are empty lists, which indicate 
\begin_inset Quotes eld
\end_inset

no change
\begin_inset Quotes erd
\end_inset

.
 This option will turn all semicore orbitals into corresponding multi-core
 orbitals each having the specified 
\begin_inset Formula $Q$
\end_inset

- and 
\begin_inset Formula $S$
\end_inset

-parameter values and 
\begin_inset Formula $P$
\end_inset

-values copied from the 
\begin_inset Formula $P$
\end_inset

-value of the existing single orbital (experimental for 
\begin_inset Formula $B^{\mathrm{HF}}$
\end_inset

 calculations).
 Example: multiplicity 2 with two 
\begin_inset Formula $Q$
\end_inset

- and 
\begin_inset Formula $S$
\end_inset

-values turns a semicore section
\end_layout

\begin_deeper
\begin_layout LyX-Code

\size footnotesize
core : 3s S3p P=(0.79); / valence
\end_layout

\begin_layout Standard
into
\end_layout

\begin_layout LyX-Code

\size footnotesize
core : D3s Q=(Q1,Q2) S=(S1,S2); D3p Q=(Q1,Q2) S=(S1,S2) P=(0.79,0.79); / valence
\end_layout

\begin_layout Standard
which in turn gets internally expanded according to the rules of Sec.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Orbital-structure"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\end_deeper
\begin_layout Standard
If you want to check which basis actually resulted from the modifications
 have a look at the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
filei{=.basdef}
\end_layout

\end_inset

 section in the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 output which was explained in Sec.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Basis-setup"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
In practical applications we have used extension level 2, add-3d and add-f
 together as a kind of 
\begin_inset Quotes eld
\end_inset

standard extended basis
\begin_inset Quotes erd
\end_inset

, if such a basis seemed to be indicated by enhanced accuracy requirements
 or for testing that the default accuracy was indeed sufficient.
\end_layout

\begin_layout Standard
In publications one should report the use of basis modifications.
 Assuming a 
\begin_inset Quotes eld
\end_inset

standard extended basis
\begin_inset Quotes erd
\end_inset

 was used we suggest a formulation like:
\end_layout

\begin_layout Quotation
\SpecialChar ldots
 we used the default basis with extension level 2 and compulsory valence
 
\begin_inset Formula $f$
\end_inset

 and 
\begin_inset Formula $d$
\end_inset

-orbitals \SpecialChar ldots

\end_layout

\begin_layout Standard
or if more than one basis versions exist in the future 
\end_layout

\begin_layout Quotation
\SpecialChar ldots
 we used the FPLO9 basis with extension level 2 and compulsory valence 
\begin_inset Formula $f$
\end_inset

 and 
\begin_inset Formula $d$
\end_inset

-orbitals \SpecialChar ldots

\end_layout

\begin_layout Standard
where FPLO9 stands for the basis version descriptor as found in the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fedit{}
\end_layout

\end_inset

 basis menu.
\end_layout

\begin_layout Standard
When in doubt list the orbitals explicitly.
\end_layout

\begin_layout Subsubsection
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
pyfplo{}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
As an advanced option the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
filei{=.basdef}
\end_layout

\end_inset

-method of modifying the basis, using class 
\family typewriter
pyfplo.fploio.Basis
\family default
, is outlined in the following.
 
\end_layout

\begin_layout Standard
You will need the basis-version identifier (as for 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fedit{}
\end_layout

\end_inset

) on which to base the modification and a list of the elements or atomic
 numbers of all Wyckoff positions (explained in 
\begin_inset Flex URL
status collapsed

\begin_layout Plain Layout

../pyfplo/pyfplo.pdf
\end_layout

\end_inset

).
 With this you can extract the default basis directly from 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
pyfplo{}
\end_layout

\end_inset

 (without running 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 first, as in the past).
 Then you can modify the basis for each sort at will (as provided by the
 python interface) and save it into 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
filei{=.basdef}
\end_layout

\end_inset

.
 This way one can setup the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 input (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
file{=.in}
\end_layout

\end_inset

) together with the basis (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
filei{=.basdef}
\end_layout

\end_inset

).
\end_layout

\begin_layout Standard
Let us stress again that the preferred way to achieve a non-default basis
 is to use 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fedit{}
\end_layout

\end_inset

 or in scripts 
\family typewriter
pyfplo.fploio.Fedit
\family default
 (if it provides what the user needs)!
\end_layout

\begin_layout Standard
Of course one can also create a default 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
filei{=.basdef}
\end_layout

\end_inset

 via 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
pyfplo{}
\end_layout

\end_inset

 (or extract the basis from an output file via 
\family typewriter
fout2in -b
\family default
) and then edit it by hand as was done in the past.
\end_layout

\end_body
\end_document
