\documentclass[11pt]{article}
\usepackage{geometry}                % See geometry.pdf to learn the layout options. There are lots.
\geometry{a4paper}                   % ... or a4paper or a5paper or ...
%\geometry{landscape}                % Activate for for rotated page geometry
%\usepackage[parfill]{parskip}    % Activate to begin paragraphs with an empty line rather than an indent
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{lipsum}
\usepackage{authblk}
%\usepackage{amsaddr}
\usepackage{epstopdf}
\usepackage{booktabs}
\usepackage{xcolor}
\usepackage{fancyhdr}
\usepackage[yyyymmdd,hhmmss]{datetime}
\pagestyle{fancy}
\rfoot{Compiled on \today\ at \currenttime}
\cfoot{}
\lfoot{Page \thepage}
\RequirePackage[colorinlistoftodos,prependcaption,textsize=tiny]{todonotes} % look for '\todo'
\definecolor{darkred}{rgb}{0.4,0.0,0.0}
\definecolor{darkgreen}{rgb}{0.0,0.4,0.0}
\definecolor{darkblue}{rgb}{0.0,0.0,0.4}
\usepackage[bookmarks,linktocpage,colorlinks,
    linkcolor = darkred,
    urlcolor  = darkblue,
    citecolor = darkgreen]{hyperref}

%\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}

\renewcommand{\headrulewidth}{0pt}
\fancyhead[L]{
%\includegraphics[width=4cm]{/Users/tomluu/Research/talks/fzjTemplate/uniBonn_logo.jpg}
}
\fancyhead[R]{
\includegraphics[width=4cm]{figs/fzj_logo.jpg}
}
\pagestyle{plain}

\title{Tower of operators}
\author[1,2]{Thomas Luu}
\affil[1]{Institute for Advanced Simulation 4\\
Forschungszentrum J\"ulich, Germany}
\affil[2]{Rheinische Friedrich-Williams-Universit\"at Bonn, Germany}

%\email{t.luu@fz-juelich.de}
\date{}                                           % Activate to display a given date or no date

\begin{document}
\maketitle
\begin{center}
email: \href{mailto:t.luu@fz-juelich.de}{t.luu@fz-juelich.de}
\end{center}
\abstract{
Here I try to determine the scattering amplitude for an interaction of the type
\begin{displaymath}
V(p',p)=C_0+\frac{C_2}{2}\left(p'^2+p^2\right)
\end{displaymath}
and I assume a hard-cutoff regulator.
}

\thispagestyle{fancy}

\clearpage{}
%\tableofcontents
%\newpage

\section{Interaction}
I assume an interaction of the form
\begin{equation}\label{eqn:potential}
V(p',p)=\left(C_0+C_2\frac{\left(p'^2+p^2\right)}{2}\right)f_\Lambda(p')f_\Lambda(p)
\end{equation}
with an implicit hard-cutoff regulator at some scale $\Lambda$,
\begin{equation}
f_\Lambda(p)=\Theta\left(|p|-\Lambda\right)\ .
\end{equation}
 Note that this interaction is a sum of separable terms, and there is NO delta function (we are in momentum space).  The coefficients $C_n$ are bare parameters and must be ``renormalized"

 \section{T-matrix}
 The T-matrix is defined as
 \begin{equation}
 T(p',p;E)\equiv V(p',p)+\int \frac{d^3q}{(2\pi)^3} V(p',q) G_0(q;E) T(q,p;E)\ ,
 \end{equation}
 where
 \begin{equation}
 G_0(q;E)=\frac{1}{E-q^2/m+i\epsilon}\ .
 \end{equation}
 Another relation, which is actually more useful when deriving the T-matrix, is the expression for the fully interacting propagator, which I write in operator form
 \begin{equation}\label{eqn:full G}
 \hat G(E)\equiv \frac{1}{E-\hat H_0-\hat V+i\epsilon}=\hat G_0(E)+\hat G_0(E)\hat T(E)\hat G_0(E)\ .
 \end{equation}

 \subsection{Summing the T-matrix}
I can sum the T-matrix to all orders in $V(p',p)$ since $V(p',p)$ is a sum of separable terms.   I will present my derivations (based off eq.~\eqref{eqn:full G}) for the full T-matrix in later sections, but only express the result here
\begin{multline}\label{eqn:T-matrix 1}
T(p',p;E)=\\
\frac{\mathbb{C}_0+\mathbb{C}_2\frac{\left(p'^2+p^2\right)}{2}+\mathbb{X}_4\  p'^2p^2}
{1-C_0I_0(E)-C_2I_2(E)-\frac{1}{4}C^2_2\left[I_0(E)I_4(E)-I_2(E)^2\right]}f_\Lambda(p')f_\Lambda(p)\ ,
\end{multline}
where
\begin{align}
\mathbb{C}_0&=C_0+\frac{1}{4}C^2_2I_4(E)\\
\mathbb{C}_2&=C_2-\frac{1}{2}C^2_2I_2(E)\\
\mathbb{X}_4&=\frac{1}{4}C^2_2I_0(E)\ ,
\end{align}
and\footnote{I assume we are in the infinite-volume, continuum limit, and therefore perform the integrals in a spherical basis.}
\begin{equation}
I_n(E)\equiv \frac{1}{2\pi^2}\int_0^\Lambda dq\frac{q^{2+n}}{E-q^2/m+i\epsilon}\ .
\end{equation}
Note that the solution to the T-matrix has an \emph{induced} higher-order term $\mathbb{X}_4p'^2p^2$.

We can also relate all terms $I_{n>0}(E)$ to $I_0(E)$,
\begin{align}
I_2(E)&=-\frac{m}{6\pi^2}\Lambda^3+[mE]\  I_0(E)\\
I_4(E)&=-\frac{m}{10\pi^2}\Lambda^5-[mE]\ \frac{m}{6\pi^2}\Lambda^3+[mE]^2\ I_0(E)\ .
\end{align}
And finally, we have that
\begin{equation}
I_0(E)=-\frac{m}{2\pi^2}\Lambda -i \frac{m}{4\pi}\sqrt{mE}+\frac{m}{2\pi^2\Lambda}mE+\mathcal{O}(\Lambda^{-3})\ .
\end{equation}
With these relations, the \emph{denominator} in eq.~\eqref{eqn:T-matrix 1} becomes
\begin{multline}\label{eqn:denominator}
1-C_0 I_0(E)-C_2 \left([mE]\ I_0(E) -\frac{\Lambda ^3 m}{6 \pi ^2}\right)\\
+C_2^2 \left(I_0(E) \left(\frac{\Lambda ^5 m}{40 \pi
   ^2}-\frac{\Lambda ^3 m\ [mE]}{24 \pi ^2}\right)+\frac{\Lambda ^6 m^2}{144 \pi
   ^4}\right)
   \end{multline}

\section{Scattering amplitude near threshold}
Near threshold I set everything on-shell, $p'=p=\sqrt{mE}$.  In this limit, the \emph{numerator} of eq.~\eqref{eqn:T-matrix 1} becomes
\begin{equation}\label{eqn:numerator}
C_0+C_2 p^2+C_2^2 \left(\frac{\Lambda ^3 m p^2}{24 \pi ^2}-\frac{\Lambda ^5 m}{40 \pi
   ^2}\right)\ .
   \end{equation}
% and one has that
%\begin{equation}
%T(p'=p; E=p^2/m)\equiv T(p)=\frac{4\pi/m}{p\cot \delta -ip}
%\end{equation}
%or equivalently
%\begin{equation}
%p\cot \delta -ip=\frac{4\pi}{m\ T(p)}\ .
%\end{equation}
%Plugging everything in, I get
%\begin{multline}\label{eqn:matching 1}
%p\cot \delta -ip=\\
%\frac{4\pi}{m}\frac{1-(C_0 +C_2 p^2)I_0(E)+C_2 \frac{\Lambda ^3 m}{6 \pi ^2}
%+C_2^2 \left(I_0(E) \left(\frac{\Lambda ^5 m}{40 \pi
%   ^2}-\frac{\Lambda ^3 mp^2}{24 \pi ^2}\right)+\frac{\Lambda ^6 m^2}{144 \pi
%   ^4}\right)}{C_0+C_2 p^2+C_2^2 \left(\frac{\Lambda ^3 m p^2}{24 \pi ^2}-\frac{\Lambda ^5 m}{40 \pi
%   ^2}\right)}
%   \end{multline}
%
Now I define the renormalized coefficients,
\begin{align}
C_0(\Lambda)&=C_0-C_2^2\frac{\Lambda^5 m}{40 \pi^2}\\
C_2(\Lambda)&=C_2\left(1+C_2\frac{\Lambda^3 m}{24\pi^2}\right)\ ,
\end{align}
which means eq.~\eqref{eqn:numerator} becomes
\begin{equation}
\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}\ .
\end{equation}
The denominator, eq.~\eqref{eqn:denominator}, becomes
\begin{multline}
1-I_0(E)\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}+\frac{\Lambda ^3 m}{6 \pi ^2}C_2 \left(1+
C_2 \frac{\Lambda ^3 m}{24 \pi
   ^2}\right)\\
   =1-I_0(E)\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}+C_2(\Lambda) \frac{\Lambda ^3 m}{6 \pi ^2}\ .
\end{multline}
So the full on-shell T-matrix near threshold is
\begin{equation}
T(E=p^2/m)=-\frac{4\pi/m}{p\cot \delta -ip}=\frac{\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}}{1-I_0(E)\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}+C_2(\Lambda) \frac{\Lambda ^3 m}{6 \pi ^2}}\ .
\end{equation}
This expression is \emph{almost} like eq.(1) of Beane et al. (hep-lat0312004) ``Two-Nucleons on a Lattice",
\begin{displaymath}
\mathcal{A}=\frac{\sum C_{2 n}(\mu) p^{2 n}}{1-I_{0} \sum C_{2 n}(\mu) p^{2 n}}\ ,
\end{displaymath}
except for the extra \emph{induced} term $C_2(\Lambda) \frac{\Lambda ^3 m}{6 \pi ^2}$ in the denominator.

\section{So do we have a ``Tower of Interactions"?}
I \emph{suspect} what happens (this is by no means a proof) is that as we include more terms in our potential, eq.~\eqref{eqn:potential}, the induced term gets pushed to higher and higher order.  One must be consistent in adding more terms into the potential, however (i.e. derivative expansion w/ Galilean invariance, etc. . .--essentially pionless EFT).

For example, if we include a $C_4$ term into our potential, then I suspect we would get for the scattering amplitude
 \begin{equation}
 \frac{\sum_{n=0,1,2}C_{2n}(\Lambda)p^{2n}}{1-I_0(E)\sum_{n=0,1,2}C_{2n}(\Lambda)p^{2n}+C_4(\Lambda) \mathcal{C}\Lambda^5}\ ,
 \end{equation}
 where $\mathcal{C}$ is some coefficient.  If this is true (and I'm not going to even attempt to prove this), then only in the limit of an ``infinite tower of interactions" do we recover Beane et al.'s expression,
\begin{equation}
\frac{\sum C_{2 n}(\Lambda) p^{2 n}}{1-I_{0}(E) \sum C_{2 n}(\Lambda) p^{2 n}}\ ,
\end{equation}

\section{Summing the T-matrix}
Later. . .

%Then eq.~\eqref{eqn:matching 1} becomes
%\begin{align}\label{eqn:matching 2}
%p\cot \delta -ip&=
%\frac{4\pi}{m}\frac{1-I_0(E)\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}+C_2 \frac{\Lambda ^3 m}{6 \pi ^2}\left(1+
%C_2 \frac{\Lambda ^3 m}{24 \pi
%   ^2}\right)}{\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}}\\\
%   &=\frac{4\pi}{m}\left(\frac{1}{\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}}-I_0(E)+\frac{C_2 \frac{\Lambda ^3 m}{6 \pi ^2}\left(1+
%C_2 \frac{\Lambda ^3 m}{24 \pi
%   ^2}\right)}{\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}}\right)
%\end{align}
%\subsection{}

%\newpage
%\appendix

%\clearpage
%\bibliography{references}

\end{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Kind Inference for Datatypes}
\label{chap:kindinference}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In recent years, languages like Haskell have seen a dramatic surge of new
features that significantly extends the expressive power of their type systems.
With these features, the challenge of \emph{kind inference} for datatype
declarations has presented itself and become a worthy research problem on its
own.

In this chapter, we apply promotion to kind inference for datatypes. Inspired by
previous research on type-inference, we offer declarative specifications for
what datatype declarations should be accepted, both for \hne and for a more
advanced system we call \tit, based on the extensions in modern Haskell,
including a limited form of dependent types. We believe these formulations to be
novel and without precedent, even for \hne. These specifications are
complemented with implementable algorithmic versions. We study \emph{soundness},
\emph{completeness} and the existence of \emph{principal kinds} in these
systems, proving the properties where they hold. This work can serve as a guide
both to language designers who wish to formalize their datatype declarations and
also to implementors keen to have principled inference of principal types.

\input{Gen/KindInference/introduction.lhstex}
\input{Gen/KindInference/haskell98.lhstex}
\input{Gen/KindInference/haskell98_algo.lhstex}
\input{Gen/KindInference/haskell98_tparm.lhstex}
\input{Gen/KindInference/typeintype.lhstex}
\input{Gen/KindInference/typeintype_algo.lhstex}
\input{Gen/KindInference/extension.lhstex}

%%% Local Variables:
%%% mode: latex
%%% TeX-master: "../Thesis"
%%% org-ref-default-bibliography: "../Thesis.bib"
%%% End:


\section{Performance Measurement}

In this section, we will outline some basics of several of the more useful
profilers available in \thispackage.

\subsection{flips and flops}

\thispackage offers two high-level utilities for measuring floating point
performance data: \code{system.flips()} and \code{system.flops()}.
The former captures floating point instruction measurements; a \textit{flip} is
the rate of floating point instructions (flpins), or the number of flpins per
second.  Perhaps the more well-known measurement is the rate of floating point
\emph{operations}.  Like its cousin \code{system.flips()}, the \thispackage
function \code{system.flops()} will measure both the number of floating point
operations as well as their rate --- the number of floating point operations
per second, or flops.

Generally, reports of flops (or flips) are not given, but Mega-flops (Mflops);
as the name implies, 1 Mflop is 1,000,000 flops.

\paragraph{Theoretical flops}

A processor has a theoretical peak number of flops, which is typically much
higher than what is found experimentally.  Still, understanding the peak flops
of a system can be useful in understanding ``good'' flops performance of a
program (just understand that it will always be lower in practice than the
theoretical peak).

The equation below demonstrates how to compute the peak Mflops of a processor:
\begin{align*}
\text{Mflops} = \text{(\# cores)} * \text{(Speed in Mhz)} *
\text{(\# of SSE units per core)} *  \text{(\# SSE operations per cycle)}
\end{align*}

single precision Mflops (divide by 2 for double precision).  So for this Intel
Sandy Bridge Core i7 as a reference, the theoretical peak is:
\begin{align*}
\text{Mflops} &= (4) * (2800 Mhz) * 2 * 2
\end{align*}

which is roughly 45 single precision Gflops, or 22.5 double precision Gflops
--- 22,500,000,000 floating point operations per second!

See where your computer stacks up against the fastest supercomputers in the
world at \href{http://www.top500.org/}{top500.org}.  For instance, again using
this laptop as a reference, and using theoretical flops as a proxy for running
the Linpack benchmark (which is not really fair, we grant), we would crush
every supercomputer on the list from June 1997, but wouldn't make the cut for
June 1999.

\subsection{Cache Misses and Cache Hits}

\paragraph{Memory and Cache}

Computers operate at \emph{billions} of cycles per second.  Of course, those
operations occur on data.  A useful abstraction we use in thinking about
processing data is you load the stuff up into RAM and then the processor does
things to it.  This is usually fine, or at least convenient, but it's not
accurate, as you are probably aware.

\begin{figure}[ht]
  \centering
  \includegraphics[scale=.54]{./include/pics/memory}
  \caption{Computer Memory Hierarchy}
  \label{fig:mem}
\end{figure}
Another more accurate abstraction is that shown in Figure~\ref{fig:mem}.  In a
sense, the magic really happens when things get into the CPU registers.  But
something that's in RAM that you want to operate on, as it's headed to the CPU,
gets cached into various levels of (comparatively) fast access storage along
the way.  Understanding this behavior, and writing your code to take advantage
of it, can have \emph{tremendous} impacts on performance.

\begin{figure}[ht]
  \centering
  \includegraphics[scale=.54]{./include/pics/cache}
  \caption{Detailed Computer Memory Hierarchy}
  \label{fig:mem2}
\end{figure}
A more detailed presentation of Figure~\ref{fig:mem} is that found in
Figure~\ref{fig:mem2}.  Here we see a more accurate presentation; for example,
L3 cache (if it exists) is shared by all cores on a moder CPU, with L2 private,
and L1 split into data and instruction caches.  The lookup costs in terms of
processor cycles are provided for each type of memory.  These ``times'' are
presented in terms of the CPU cycle for mostly historical reasons, though there
is benefit to thinking in these terms (for example, when trying to minimize
wait cycles).  Though note that a faster processor could actually
\emph{increase} these costs, specifically RAM to L3 lookups.

If you are unfamiliar with the importance (or existence) of cache, I would
strongly encourage you to experiment with this great
\href{http://www.overbyte.com.au/misc/Lesson3/CacheFun.html}%
{interactive visualization}
showing (relative) speeds of cache misses. It too involves some simplifications
of how modern hardware actually works, so if this is at all confusing, let us
all take a moment to pity the tragic life of the computer engineer.  Another
great resource is this
\href{http://www.eecs.berkeley.edu/~rcs/research/interactive_latency.html}%
{interactive visualization} showing memory and cache latency numbers by year.

\paragraph{Cache Misses} Fundamentally, a cache miss occurs when the cache
needs some piece of data to pass along to registers, but it isn't immediately
available and the computer has to go digging through RAM (or god help you,
disk) to get it.  Cache misses are bad and reduce performance.  You can't get
rid of them completely, unless your entire problem --- copies and all ---
comfortably fits into cache, but you can eliminate \emph{unnecessary} cache
misses being aware of how your data and algorithms interact with cache.  See
the demo \code{cache_access.r} in \thispackage for an example of good versus
bad cache access.


\section{{\Large$\langle j_0\rangle$ }Form factors for 3d transition elements and their ions}
\begin{table}[H]
\caption{\noindent $\langle j_0\rangle$ Form factors for 3d transition elements and their ions\hfill}
\label{3dj0}
\vspace{2mm}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Sc0 &$0.2512$ &$90.0296$ &$0.3290$ &$39.4021$ &$0.4235$ &$14.3222$ &$-0.0043$ \\
Sc1 &$0.4889$ &$51.1603$ &$0.5203$ &$14.0764$ &$-0.0286$ &$0.1792$ &$0.0185$ \\
Sc2 &$0.5048$ &$31.4035$ &$0.5186$ &$10.9897$ &$-0.0241$ &$1.1831$ &$0.0000$ \\
Ti0 &$0.4657$ &$33.5898$ &$0.5490$ &$9.8791$ &$-0.0291$ &$0.3232$ &$0.0123$ \\
Ti1 &$0.5093$ &$36.7033$ &$0.5032$ &$10.3713$ &$-0.0263$ &$0.3106$ &$0.0116$ \\
Ti2 &$0.5091$ &$24.9763$ &$0.5162$ &$8.7569$ &$-0.0281$ &$0.9160$ &$0.0015$ \\
Ti3 &$0.3571$ &$22.8413$ &$0.6688$ &$8.9306$ &$-0.0354$ &$0.4833$ &$0.0099$ \\
V0 &$0.4086$ &$28.8109$ &$0.6077$ &$8.5437$ &$-0.0295$ &$0.2768$ &$0.0123$ \\
V1 &$0.4444$ &$32.6479$ &$0.5683$ &$9.0971$ &$-0.2285$ &$0.0218$ &$0.2150$ \\
V2 &$0.4085$ &$23.8526$ &$0.6091$ &$8.2456$ &$-0.1676$ &$0.0415$ &$0.1496$ \\
V3 &$0.3598$ &$19.3364$ &$0.6632$ &$7.6172$ &$-0.3064$ &$0.0296$ &$0.2835$ \\
V4 &$0.3106$ &$16.8160$ &$0.7198$ &$7.0487$ &$-0.0521$ &$0.3020$ &$0.0221$ \\
Cr0 &$0.1135$ &$45.1990$ &$0.3481$ &$19.4931$ &$0.5477$ &$7.3542$ &$-0.0092$ \\
Cr1 &$-0.0977$ &$0.0470$ &$0.4544$ &$26.0054$ &$0.5579$ &$7.4892$ &$0.0831$ \\
Cr2 &$1.2024$ &$-0.0055$ &$0.4158$ &$20.5475$ &$0.6032$ &$6.9560$ &$-1.2218$ \\
Cr3 &$-0.3094$ &$0.0274$ &$0.3680$ &$17.0355$ &$0.6559$ &$6.5236$ &$0.2856$ \\
Cr4 &$-0.2320$ &$0.0433$ &$0.3101$ &$14.9518$ &$0.7182$ &$6.1726$ &$0.2042$ \\
Mn0 &$0.2438$ &$24.9629$ &$0.1472$ &$15.6728$ &$0.6189$ &$6.5403$ &$-0.0105$ \\
Mn1 &$-0.0138$ &$0.4213$ &$0.4231$ &$24.6680$ &$0.5905$ &$6.6545$ &$-0.0010$ \\
Mn2 &$0.4220$ &$17.6840$ &$0.5948$ &$6.0050$ &$0.0043$ &$-0.6090$ &$-0.0219$ \\
Mn3 &$0.4198$ &$14.2829$ &$0.6054$ &$5.4689$ &$0.9241$ &$-0.0088$ &$-0.9498$ \\
Mn4 &$0.3760$ &$12.5661$ &$0.6602$ &$5.1329$ &$-0.0372$ &$0.5630$ &$0.0011$ \\
Fe0 &$0.0706$ &$35.0085$ &$0.3589$ &$15.3583$ &$0.5819$ &$5.5606$ &$-0.0114$ \\
Fe1 &$0.1251$ &$34.9633$ &$0.3629$ &$15.5144$ &$0.5223$ &$5.5914$ &$-0.0105$ \\
Fe2 &$0.0263$ &$34.9597$ &$0.3668$ &$15.9435$ &$0.6188$ &$5.5935$ &$-0.0119$ \\
Fe3 &$0.3972$ &$13.2442$ &$0.6295$ &$4.9034$ &$-0.0314$ &$0.3496$ &$0.0044$ \\
Fe4 &$0.3782$ &$11.3800$ &$0.6556$ &$4.5920$ &$-0.0346$ &$0.4833$ &$0.0005$ \\
Co0 &$0.4139$ &$16.1616$ &$0.6013$ &$4.7805$ &$-0.1518$ &$0.0210$ &$0.1345$ \\
Co1 &$0.0990$ &$33.1252$ &$0.3645$ &$15.1768$ &$0.5470$ &$5.0081$ &$-0.0109$ \\
Co2 &$0.4332$ &$14.3553$ &$0.5857$ &$4.6077$ &$-0.0382$ &$0.1338$ &$0.0179$ \\
Co3 &$0.3902$ &$12.5078$ &$0.6324$ &$4.4574$ &$-0.1500$ &$0.0343$ &$0.1272$ \\
Co4 &$0.3515$ &$10.7785$ &$0.6778$ &$4.2343$ &$-0.0389$ &$0.2409$ &$0.0098$ \\
Ni0 &$-0.0172$ &$35.7392$ &$0.3174$ &$14.2689$ &$0.7136$ &$4.5661$ &$-0.0143$ \\
Ni1 &$0.0705$ &$35.8561$ &$0.3984$ &$13.8042$ &$0.5427$ &$4.3965$ &$-0.0118$ \\
Ni2 &$0.0163$ &$35.8826$ &$0.3916$ &$13.2233$ &$0.6052$ &$4.3388$ &$-0.0133$ \\
Ni3 &$0.0012$ &$34.9998$ &$0.3468$ &$11.9874$ &$0.6667$ &$4.2518$ &$-0.0148 $ \\
Ni4 &$-0.0090$ &$35.8614$ &$0.2776$ &$11.7904$ &$0.7474$ &$4.2011$ &$-0.0163$ \\
Cu0 &$0.0909$ &$34.9838$ &$0.4088$ &$11.4432$ &$0.5128$ &$3.8248$ &$-0.0124$ \\
Cu1 &$0.0749$ &$34.9656$ &$0.4147$ &$11.7642$ &$0.5238$ &$3.8497$ &$-0.0127$ \\
Cu2 &$0.0232$ &$34.9686$ &$0.4023$ &$11.5640$ &$0.5882$ &$3.8428$ &$-0.0137$ \\
Cu3 &$0.0031$ &$34.9074$ &$0.3582$ &$10.9138$ &$0.6531$ &$3.8279$ &$-0.0147$ \\
Cu4 &$-0.0132$ &$30.6817$ &$0.2801$ &$11.1626$ &$0.7490$ &$3.8172$ &$-0.0165$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large$\langle j_0\rangle$ }Form factors for 4d transition elements and their ions}
\begin{table}[H]
\caption{
$\langle j_0\rangle$ form factors for 4d atoms and ions}\vspace{2mm}
\label{4dj0}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Y0 &$0.5915$ &$67.6081$ &$1.5123$ &$17.9004$ &$-1.1130$ &$14.1359$ &$0.0080$ \\
Zr0 &$0.4106$ &$59.9961$ &$1.0543$ &$18.6476$ &$-0.4751$ &$10.5400$ &$0.0106$ \\
Zr1 &$0.4532$ &$59.5948$ &$0.7834$ &$21.4357$ &$-0.2451$ &$9.0360$ &$0.0098$ \\
Nb0 &$0.3946$ &$49.2297$ &$1.3197$ &$14.8216$ &$-0.7269$ &$9.6156$ &$0.0129$ \\
Nb1 &$0.4572$ &$49.9182$ &$1.0274$ &$15.7256$ &$-0.4962$ &$9.1573$ &$0.0118$ \\
Mo0 &$0.1806$ &$49.0568$ &$1.2306$ &$14.7859$ &$-0.4268$ &$6.9866$ &$0.0171$ \\
Mo1 &$0.3500$ &$48.0354$ &$1.0305$ &$15.0604$ &$-0.3929$ &$7.4790$ &$0.0139$ \\
Tc0 &$0.1298$ &$49.6611$ &$1.1656$ &$14.1307$ &$-0.3134$ &$5.5129$ &$0.0195$ \\
Tc1 &$0.2674$ &$48.9566$ &$0.9569$ &$15.1413$ &$-0.2387$ &$5.4578$ &$0.0160$ \\
Ru0 &$0.1069$ &$49.4238$ &$1.1912$ &$12.7417$ &$-0.3176$ &$4.9125$ &$0.0213$ \\
Ru1 &$0.4410$ &$33.3086$ &$1.4775$ &$9.5531$ &$-0.9361$ &$6.7220$ &$0.0176$ \\
Rh0 &$0.0976$ &$49.8825$ &$1.1601$ &$11.8307$ &$-0.2789$ &$4.1266$ &$0.0234$ \\
Rh1 &$0.3342$ &$29.7564$ &$1.2209$ &$9.4384$ &$-0.5755$ &$5.3320$ &$0.0210$ \\
Pd0 &$0.2003$ &$29.3633$ &$1.1446$ &$9.5993$ &$-0.3689$ &$4.0423$ &$0.0251$ \\
Pd1 &$0.5033$ &$24.5037$ &$1.9982$ &$6.9082$ &$-1.5240$ &$5.5133$ &$0.0213$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large$\langle j_0\rangle$ }Form factors for 5d transition elements and their ions}
\begin{table}[H]
 \caption{$\langle j_0\rangle$ form factors for the 5d electrons of transition atoms and ions from Hf to Re.\cite{kob:11}}
\label{5dj0} \vspace{1ex}
{\tablesize
\begin{tabular}{llrrrrrrrrr}
\hline
\multicolumn{1}{c}{ Ion}&\multicolumn{1}{c}{ Config}&\multicolumn{1}{c}{ A }&\multicolumn{1}{c}{  a }&\multicolumn{1}{c}{B }&\multicolumn{1}{c}{ b }&\multicolumn{1}{c}{ C }&\multicolumn{1}{c}{ c }&\multicolumn{1}{c}{ D }&\multicolumn{1}{c}{ d }&\multicolumn{1}{c}{E }\\
\hline
Hf2 & 6s05d2 &$0.4229$ &$50.465$ &$0.7333$ &$23.865$ &$-0.3798$ &$4.051$ &$0.2252$ &$2.497$ &$-0.0018$ \\
Hf3 & 6s05d1 &$0.3555$ &$40.954$ &$0.8483$ &$21.726$ &$-0.4116$ &$4.305$ &$0.2101$ &$2.349$ &$-0.0023$ \\
Ta2 & 6s05d3 &$0.3976$ &$45.095$ &$0.7746$ &$21.028$ &$-0.6098$ &$3.471$ &$0.4395$ &$2.570$ &$-0.0020$ \\
Ta3 & 6s05d2 &$0.3611$ &$36.921$ &$0.8579$ &$19.195$ &$-0.4945$ &$3.857$ &$0.2781$ &$2.303$ &$-0.0026$ \\
Ta4 & 6s05d1 &$0.3065$ &$31.817$ &$0.9611$ &$17.749$ &$-0.5463$ &$3.979$ &$0.2816$ &$2.232$ &$-0.0030$ \\
W0 & 6s05d6 &$0.3990$ &$73.810$ &$0.7138$ &$22.815$ &$-2.0436$ &$2.710$ &$1.9319$ &$2.559$ &$-0.0023$ \\
W0 & 6s15d5 &$0.3811$ &$62.707$ &$0.7523$ &$21.434$ &$-12.5449$ &$2.702$ &$12.4130$ &$2.674$ &$-0.0023$ \\
W0 & 6s25d4 &$0.3653$ &$53.965$ &$0.7926$ &$20.078$ &$-0.8142$ &$3.030$ &$0.6581$ &$2.476$ &$-0.0023$ \\
W0 & 6s05d5 &$0.4077$ &$51.367$ &$0.7436$ &$20.256$ &$-9.8283$ &$2.780$ &$9.6788$ &$2.740$ &$-0.0021$ \\
W0 & 6s15d4 &$0.3834$ &$46.233$ &$0.7890$ &$19.278$ &$-1.4650$ &$2.947$ &$1.2945$ &$2.628$ &$-0.0022$ \\
W2 & 6s05d4 &$0.3876$ &$40.340$ &$0.8008$ &$18.621$ &$-1.3911$ &$2.995$ &$1.2048$ &$2.627$ &$-0.0023$ \\
W3 & 6s05d3 &$0.3610$ &$33.519$ &$0.8717$ &$17.176$ &$-0.6183$ &$3.445$ &$0.3883$ &$2.276$ &$-0.0028$ \\
W4 & 6s05d2 &$0.3221$ &$29.047$ &$0.9574$ &$15.979$ &$-0.6287$ &$3.597$ &$0.3525$ &$2.174$ &$-0.0033$ \\
W5 & 6s05d1 &$0.2725$ &$25.966$ &$1.0558$ &$14.954$ &$-0.6990$ &$3.643$ &$0.3745$ &$2.145$ &$-0.0037$ \\
Re0 & 6s05d7 &$0.3807$ &$63.042$ &$0.7497$ &$19.967$ &$-6.5300$ &$2.501$ &$6.4013$ &$2.451$ &$-0.0028$ \\
Re0 & 6s15d6 &$0.3691$ &$53.934$ &$0.7837$ &$18.790$ &$-9.1491$ &$2.558$ &$8.9983$ &$2.517$ &$-0.0027$ \\
Re0 & 6s25d5 &$0.3548$ &$47.108$ &$0.8210$ &$17.769$ &$-9.8674$ &$2.599$ &$9.6938$ &$2.556$ &$-0.0027$ \\
Re0 & 6s05d6 &$0.3944$ &$45.427$ &$0.7742$ &$17.948$ &$-3.1692$ &$2.653$ &$3.0028$ &$2.521$ &$-0.0026$ \\
Re0 & 6s15d5 &$0.3736$ &$41.151$ &$0.8160$ &$17.158$ &$-7.0396$ &$2.642$ &$6.8523$ &$2.577$ &$-0.0026$ \\
Re2 & 6s05d5 &$0.3825$ &$36.336$ &$0.8218$ &$16.636$ &$-8.7220$ &$2.657$ &$8.5201$ &$2.601$ &$-0.0026$ \\
Re3 & 6s05d4 &$0.3585$ &$30.671$ &$0.8863$ &$15.527$ &$-0.8682$ &$3.047$ &$0.6263$ &$2.280$ &$-0.0030$ \\
Re4 & 6s05d3 &$0.2974$ &$27.372$ &$0.9826$ &$14.807$ &$-1.8869$ &$2.840$ &$1.6100$ &$2.476$ &$-0.0031$ \\
Re5 & 6s05d2 &$0.3143$ &$23.522$ &$1.0276$ &$13.505$ &$-0.7438$ &$3.393$ &$0.4059$ &$2.030$ &$-0.0041$ \\
Re6 & 6s05d1 &$0.2146$ &$22.496$ &$1.1616$ &$13.064$ &$-1.0455$ &$3.162$ &$0.6734$ &$2.196$ &$-0.0041$ \\
\hline
\end{tabular}
}
\end{table}
\begin{table}[H]
 \caption{$\langle j_0\rangle$ form factors for the 5d electrons of transition atoms and ions from Os to Au.\cite{kob:11}}
 \label{5dj0b}
 \vspace{1ex}
{ \tablesize
\begin{tabular}{llrrrrrrrrr}
\hline
\multicolumn{1}{c}{ Ion}&\multicolumn{1}{c}{ Config}&\multicolumn{1}{c}{ A }&\multicolumn{1}{c}{  a }&\multicolumn{1}{c}{B }&\multicolumn{1}{c}{ b }&\multicolumn{1}{c}{ C }&\multicolumn{1}{c}{ c }&\multicolumn{1}{c}{ D }&\multicolumn{1}{c}{ d }&\multicolumn{1}{c}{E}\\
\hline
Os0 & 6s05d8 &$0.3676$ &$54.835$ &$0.7793$ &$17.716$ &$-2.0669$ &$2.418$ &$1.9224$ &$2.247$ &$-0.0034$ \\
Os0 & 6s15d7 &$0.3571$ &$47.458$ &$0.8123$ &$16.770$ &$-1.2072$ &$2.556$ &$1.0404$ &$2.211$ &$-0.0033$ \\
Os0 & 6s25d6 &$0.3467$ &$41.778$ &$0.8458$ &$15.918$ &$-5.6370$ &$2.459$ &$5.4472$ &$2.381$ &$-0.0032$ \\
Os0 & 6s05d7 &$0.3837$ &$40.665$ &$0.8006$ &$16.096$ &$-3.5305$ &$2.487$ &$3.3488$ &$2.366$ &$-0.0030$ \\
Os0 & 6s15d6 &$0.3666$ &$36.997$ &$0.8390$ &$15.425$ &$-2.6944$ &$2.537$ &$2.4916$ &$2.360$ &$-0.0031$ \\
Os2 & 6s05d6 &$0.3786$ &$33.005$ &$0.8412$ &$14.990$ &$-7.0632$ &$2.503$ &$6.8462$ &$2.433$ &$-0.0030$ \\
Os3 & 6s05d5 &$0.3557$ &$28.222$ &$0.9002$ &$14.140$ &$-2.5972$ &$2.601$ &$2.3444$ &$2.376$ &$-0.0032$ \\
Os4 & 6s05d4 &$0.3337$ &$24.723$ &$0.9655$ &$13.288$ &$-0.9653$ &$2.906$ &$0.6698$ &$2.117$ &$-0.0037$ \\
Os5 & 6s05d3 &$0.3055$ &$22.152$ &$1.0395$ &$12.529$ &$-0.9158$ &$3.016$ &$0.5750$ &$2.032$ &$-0.0042$ \\
Os6 & 6s05d2 &$0.2714$ &$20.218$ &$1.1211$ &$11.851$ &$-0.9773$ &$3.050$ &$0.5894$ &$2.005$ &$-0.0046$ \\
Os7 & 6s05d1 &$0.2101$ &$19.108$ &$1.2240$ &$11.347$ &$-1.2543$ &$2.933$ &$0.8250$ &$2.088$ &$-0.0048$ \\
Ir0 & 6s05d9 &$0.3564$ &$48.464$ &$0.8049$ &$15.923$ &$-2.5258$ &$2.265$ &$2.3675$ &$2.121$ &$-0.0040$ \\
Ir0 & 6s15d8 &$0.3492$ &$42.195$ &$0.8350$ &$15.113$ &$-5.1496$ &$2.279$ &$4.9686$ &$2.201$ &$-0.0038$ \\
Ir0 & 6s25d7 &$0.3400$ &$37.499$ &$0.8675$ &$14.402$ &$-2.3703$ &$2.370$ &$2.1661$ &$2.177$ &$-0.0037$ \\
Ir0 & 6s05d8 &$0.3744$ &$36.764$ &$0.8240$ &$14.576$ &$-8.8616$ &$2.303$ &$8.6664$ &$2.255$ &$-0.0035$ \\
Ir0 & 6s15d7 &$0.3604$ &$33.570$ &$0.8597$ &$13.993$ &$-2.1686$ &$2.412$ &$1.9518$ &$2.188$ &$-0.0036$ \\
Ir2 & 6s05d7 &$0.3802$ &$30.032$ &$0.8550$ &$13.567$ &$-1.6185$ &$2.488$ &$1.3866$ &$2.162$ &$-0.0035$ \\
Ir3 & 6s05d6 &$0.3678$ &$25.828$ &$0.9065$ &$12.788$ &$-0.8587$ &$2.745$ &$0.5883$ &$1.960$ &$-0.0040$ \\
Ir4 & 6s05d5 &$0.3969$ &$22.050$ &$0.9310$ &$11.768$ &$-0.7090$ &$3.017$ &$0.3857$ &$1.778$ &$-0.0047$ \\
\hline
\end{tabular}
}
\end{table}
\section{{\large$\langle j_0\rangle$ }Form factors for Rare earth ions}
\begin{table}[H]
\caption{$\langle j_0\rangle$ form factors for the 4f electrons of rare earth ions}
\label{re4fj0}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Ce2 &$0.2953$ &$17.6846$ &$0.2923$ &$6.7329$ &$0.4313$ &$5.3827$ &$-0.0194$ \\
Nd2 &$0.1645$ &$25.0453$ &$0.2522$ &$11.9782$ &$0.6012$ &$4.9461$ &$-0.0180$ \\
Nd3 &$0.0540$ &$25.0293$ &$0.3101$ &$12.1020$ &$0.6575$ &$4.7223$ &$-0.0216$ \\
Sm2 &$0.0909$ &$25.2032$ &$0.3037$ &$11.8562$ &$0.6250$ &$4.2366$ &$-0.0200$ \\
Sm3 &$0.0288$ &$25.2068$ &$0.2973$ &$11.8311$ &$0.6954$ &$4.2117$ &$-0.0213$ \\
Eu2 &$0.0755$ &$25.2960$ &$0.3001$ &$11.5993$ &$0.6438$ &$4.0252$ &$-0.0196$ \\
Eu3 &$0.0204$ &$25.3078$ &$0.3010$ &$11.4744$ &$0.7005$ &$3.9420$ &$-0.0220$ \\
Gd2 &$0.0636$ &$25.3823$ &$0.3033$ &$11.2125$ &$0.6528$ &$3.7877$ &$-0.0199$ \\
Gd3 &$0.0186$ &$25.3867$ &$0.2895$ &$11.1421$ &$0.7135$ &$3.7520$ &$-0.0217$ \\
Tb2 &$0.0547$ &$25.5086$ &$0.3171$ &$10.5911$ &$0.6490$ &$3.5171$ &$-0.0212$ \\
Tb3 &$0.0177$ &$25.5095$ &$0.2921$ &$10.5769$ &$0.7133$ &$3.5122$ &$-0.0231$ \\
Dy2 &$0.1308$ &$18.3155$ &$0.3118$ &$7.6645$ &$0.5795$ &$3.1469$ &$-0.0226$ \\
Dy3 &$0.1157$ &$15.0732$ &$0.3270$ &$6.7991$ &$0.5821$ &$3.0202$ &$-0.0249$ \\
Ho2 &$0.0995$ &$18.1761$ &$0.3305$ &$7.8556$ &$0.5921$ &$2.9799$ &$-0.0230$ \\
Ho3 &$0.0566$ &$18.3176$ &$0.3365$ &$7.6880$ &$0.6317$ &$2.9427$ &$-0.0248$ \\
Er2 &$0.1122$ &$18.1223$ &$0.3462$ &$6.9106$ &$0.5649$ &$2.7614$ &$-0.0235$ \\
Er3 &$0.0586$ &$17.9802$ &$0.3540$ &$7.0964$ &$0.6126$ &$2.7482$ &$-0.0251$ \\
Tm2 &$0.0983$ &$18.3236$ &$0.3380$ &$6.9178$ &$0.5875$ &$2.6622$ &$-0.0241$ \\
Tm3 &$0.0581$ &$15.0922$ &$0.2787$ &$7.8015$ &$0.6854$ &$2.7931$ &$-0.0224$ \\
Yb2 &$0.0855$ &$18.5123$ &$0.2943$ &$7.3734$ &$0.6412$ &$2.6777$ &$-0.0213$ \\
Yb3 &$0.0416$ &$16.0949$ &$0.2849$ &$7.8341$ &$0.6961$ &$2.6725$ &$-0.0229$ \\
Pr3 &$0.0504$ &$24.9989$ &$0.2572$ &$12.0377$ &$0.7142$ &$5.0039$ &$-0.0219$ \\
\hline\\[-2ex]
\end{tabular}
}
\vspace{2mm}
\end{table}
\begin{table}[H]
 \caption{$\langle j_0\rangle$ form factors for the 5d electrons of rare earth ions.\cite{kob:12}}
 \vspace{1ex}
 \label{re5dj0}
{\tablesize
\begin{tabular}{llrrrrrrrrr}
\hline
\multicolumn{1}{c}{ Ion}&\multicolumn{1}{c}{ Config}&\multicolumn{1}{c}{ A }&\multicolumn{1}{c}{  a }&\multicolumn{1}{c}{B }&\multicolumn{1}{c}{ b }&\multicolumn{1}{c}{ C }&\multicolumn{1}{c}{ c }&\multicolumn{1}{c}{ D }&\multicolumn{1}{c}{ d }&\multicolumn{1}{c}{E}\\
\hline
La2 & 4f05d1 &$0.5488$ &$63.822$ &$0.7238$ &$34.429$ &$-6.0375$ &$7.092$ &$5.7655$ &$6.839$ &$-0.0008$ \\
Ce2 & 4f15d1 &$0.4959$ &$63.797$ &$0.7571$ &$34.334$ &$-5.9903$ &$6.595$ &$5.7381$ &$6.370$ &$-0.0009$ \\
Ce3 & 4f05d1 &$1.2395$ &$35.447$ &$-1.6420$ &$4.939$ &$1.9467$ &$3.715$ &$-0.5481$ &$2.671$ &$0.0020$ \\
Pr2 & 4f25d1 &$0.4568$ &$63.765$ &$0.7795$ &$34.094$ &$-5.1096$ &$6.186$ &$4.8743$ &$5.950$ &$-0.0011$ \\
Pr3 & 4f15d1 &$0.7735$ &$40.670$ &$0.6284$ &$21.775$ &$-7.2962$ &$6.114$ &$6.8955$ &$5.866$ &$-0.0015$ \\
Pr4 & 4f05d1 &$1.3076$ &$27.762$ &$-3.2363$ &$4.278$ &$4.3424$ &$3.471$ &$-1.4168$ &$2.758$ &$0.0018$ \\
Nd2 & 4f35d1 &$1.2972$ &$27.071$ &$-2.4336$ &$4.208$ &$3.2415$ &$3.271$ &$-1.1081$ &$2.568$ &$0.0016$ \\
Nd3 & 4f25d1 &$0.6179$ &$41.924$ &$0.7344$ &$23.743$ &$-5.2140$ &$5.726$ &$4.8631$ &$5.427$ &$-0.0016$ \\
Nd4 & 4f15d1 &$1.2972$ &$27.071$ &$-2.4336$ &$4.208$ &$3.2415$ &$3.271$ &$-1.1081$ &$2.568$ &$0.0016$ \\
Pm3 & 4f35d1 &$0.5149$ &$42.825$ &$0.8077$ &$24.559$ &$-3.6851$ &$5.432$ &$3.3642$ &$5.053$ &$-0.0018$ \\
Sm2 & 4f55d1 &$0.3908$ &$63.383$ &$0.8062$ &$32.974$ &$-5.5271$ &$5.174$ &$5.3313$ &$5.015$ &$-0.0013$ \\
Sm3 & 4f45d1 &$0.4725$ &$42.826$ &$0.8301$ &$24.562$ &$-3.0608$ &$5.170$ &$2.7600$ &$4.755$ &$-0.0019$ \\
Eu2 & 4f65d1 &$0.3827$ &$63.023$ &$0.8044$ &$32.481$ &$-1.8540$ &$5.066$ &$1.6682$ &$4.618$ &$-0.0014$ \\
Eu3 & 4f55d1 &$0.3784$ &$44.430$ &$0.9025$ &$25.126$ &$-3.1521$ &$4.871$ &$2.8731$ &$4.511$ &$-0.0020$ \\
Gd0 & 4f85d1 &$0.4982$ &$104.440$ &$0.5928$ &$44.451$ &$-14.5061$ &$4.692$ &$14.4155$ &$4.664$ &$-0.0008$ \\
Gd0 & 6s14f75d1 &$0.4132$ &$75.211$ &$0.7389$ &$33.606$ &$-1.8951$ &$4.755$ &$1.7441$ &$4.411$ &$-0.0013$ \\
Gd2 & 4f75d1 &$0.3745$ &$62.755$ &$0.8026$ &$32.071$ &$-8.6327$ &$4.644$ &$8.4569$ &$4.560$ &$-0.0014$ \\
Gd3 & 4f65d1 &$0.3587$ &$44.300$ &$0.9082$ &$24.875$ &$-2.6769$ &$4.664$ &$2.4120$ &$4.274$ &$-0.0020$ \\
Gd4 & 4f55d1 &$0.4512$ &$32.712$ &$0.9183$ &$19.602$ &$-2.8665$ &$4.569$ &$2.4997$ &$4.098$ &$-0.0027$ \\
Tb0 & 4f95d1 &$0.5057$ &$104.335$ &$0.5796$ &$43.869$ &$-1.6061$ &$4.572$ &$1.5212$ &$4.339$ &$-0.0008$ \\
Tb0 & 6s14f95d1 &$0.4131$ &$74.925$ &$0.7311$ &$33.131$ &$-1.2648$ &$4.612$ &$1.1216$ &$4.133$ &$-0.0013$ \\
Tb2 & 4f85d1 &$0.3734$ &$62.276$ &$0.7955$ &$31.562$ &$-3.4780$ &$4.492$ &$3.3105$ &$4.298$ &$-0.0014$ \\
Tb3 & 4f75d1 &$0.3223$ &$44.892$ &$0.9303$ &$24.826$ &$-3.9948$ &$4.375$ &$3.7443$ &$4.140$ &$-0.0021$ \\
Tb4 & 4f65d1 &$0.3614$ &$33.730$ &$0.9844$ &$19.945$ &$-2.6536$ &$4.355$ &$2.3105$ &$3.892$ &$-0.0028$ \\
Dy2 & 4f95d1 &$0.3754$ &$61.715$ &$0.7859$ &$31.049$ &$-1.7452$ &$4.388$ &$1.5853$ &$4.024$ &$-0.0014$ \\
Dy3 & 4f85d1 &$0.3014$ &$45.110$ &$0.9392$ &$24.622$ &$-3.2549$ &$4.203$ &$3.0163$ &$3.936$ &$-0.0021$ \\
Ho3 & 4f95d1 &$0.2974$ &$44.755$ &$0.9335$ &$24.262$ &$-2.3286$ &$4.079$ &$2.0998$ &$3.728$ &$-0.0021$ \\
Er3 & 4f105d1 &$0.2916$ &$44.550$ &$0.9304$ &$23.933$ &$-1.3854$ &$4.035$ &$1.1655$ &$3.468$ &$-0.0021$ \\
Tm2 & 4f125d1 &$0.4067$ &$59.305$ &$0.7386$ &$29.168$ &$-0.3795$ &$4.593$ &$0.2356$ &$2.956$ &$-0.0015$ \\
Tm3 & 4f115d1 &$0.3302$ &$42.750$ &$0.8905$ &$23.057$ &$-0.5894$ &$4.397$ &$0.3708$ &$2.923$ &$-0.0022$ \\
Tm4 & 4f105d1 &$0.2406$ &$35.005$ &$1.0453$ &$19.560$ &$-4.5727$ &$3.572$ &$4.2895$ &$3.383$ &$-0.0027$ \\
Yb2 & 4f135d1 &$0.4170$ &$58.627$ &$0.7233$ &$28.615$ &$-0.3308$ &$4.553$ &$0.1919$ &$2.742$ &$-0.0015$ \\
Yb3 & 4f125d1 &$0.3498$ &$41.749$ &$0.8666$ &$22.455$ &$-0.4923$ &$4.434$ &$0.2780$ &$2.671$ &$-0.0022$ \\
Lu2 & 4f145d1 &$0.4216$ &$58.262$ &$0.7131$ &$28.218$ &$-0.3048$ &$4.446$ &$0.1715$ &$2.593$ &$-0.0015$ \\
Lu3 & 4f135d1 &$0.3217$ &$42.404$ &$0.8845$ &$22.448$ &$-0.4726$ &$4.258$ &$0.2686$ &$2.566$ &$-0.0022$ \\
\hline
\end{tabular}
}
\end{table}
\section{{\large$\langle j_0\rangle$ }Form factors for Actinide ions}
\begin{table}[H]
\caption{$\langle j_0\rangle$ Form factors for 5f electrons of actinide ions}\vspace{2mm}
\label{acj0}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
U3 &$0.5058$ &$23.2882$ &$1.3464$ &$7.0028$ &$-0.8724$ &$4.8683$ &$0.0192$ \\
U4 &$0.3291$ &$23.5475$ &$1.0836$ &$8.4540$ &$-0.4340$ &$4.1196$ &$0.0214$ \\
U5 &$0.3650$ &$19.8038$ &$3.2199$ &$6.2818$ &$-2.6077$ &$5.3010$ &$0.0233$ \\
Np3 &$0.5157$ &$20.8654$ &$2.2784$ &$5.8930$ &$-1.8163$ &$4.8457$ &$0.0211$ \\
Np4 &$0.4206$ &$19.8046$ &$2.8004$ &$5.9783$ &$-2.2436$ &$4.9848$ &$0.0228$ \\
Np5 &$0.3692$ &$18.1900$ &$3.1510$ &$5.8500$ &$-2.5446$ &$4.9164$ &$0.0248$ \\
Np6 &$0.2929$ &$17.5611$ &$3.4866$ &$5.7847$ &$-2.8066$ &$4.8707$ &$0.0267$ \\
Pu3 &$0.3840$ &$16.6793$ &$3.1049$ &$5.4210$ &$-2.5148$ &$4.5512$ &$0.0263$ \\
Pu4 &$0.4934$ &$16.8355$ &$1.6394$ &$5.6384$ &$-1.1581$ &$4.1399$ &$0.0248$ \\
Pu5 &$0.3888$ &$16.5592$ &$2.0362$ &$5.6567$ &$-1.4515$ &$4.2552$ &$0.0267$ \\
Pu6 &$0.3172$ &$16.0507$ &$3.4654$ &$5.3507$ &$-2.8102$ &$4.5133$ &$0.0281$ \\
Am2 &$0.4743$ &$21.7761$ &$1.5800$ &$5.6902$ &$-1.0779$ &$4.1451$ &$0.0218$ \\
Am3 &$0.4239$ &$19.5739$ &$1.4573$ &$5.8722$ &$-0.9052$ &$3.9682$ &$0.0238$ \\
Am4 &$0.3737$ &$17.8625$ &$1.3521$ &$6.0426$ &$-0.7514$ &$3.7199$ &$0.0258$ \\
Am5 &$0.2956$ &$17.3725$ &$1.4525$ &$6.0734$ &$-0.7755$ &$3.6619$ &$0.0277$ \\
Am6 &$0.2302$ &$16.9533$ &$1.4864$ &$6.1159$ &$-0.7457$ &$3.5426$ &$0.0294$ \\
Am7 &$0.3601$ &$12.7299$ &$1.9640$ &$5.1203$ &$-1.3560$ &$3.7142$ &$0.0316$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large $\langle j_2\rangle$} Form factors for 3d transition elements and their ions}
\label{3dj2}
\begin{table}[H]
\caption{
$\langle j_2\rangle$ form factors for 3d transition elements and their ions}
 \vspace{2mm}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Sc0 &$10.8172$ &$54.3270$ &$4.7353$ &$14.8471$ &$0.6071$ &$4.2180$ &$0.0011$ \\
Sc1 &$8.5021$ &$34.2851$ &$3.2116$ &$10.9940$ &$0.4244$ &$3.6055$ &$0.0009$ \\
Sc2 &$4.3683$ &$28.6544$ &$3.7231$ &$10.8233$ &$0.6074$ &$3.6678$ &$0.0014$ \\
Ti0 &$4.3583$ &$36.0556$ &$3.8230$ &$11.1328$ &$0.6855$ &$3.4692$ &$0.0020$ \\
Ti1 &$6.1567$ &$27.2754$ &$2.6833$ &$8.9827$ &$0.4070$ &$3.0524$ &$0.0011$ \\
Ti2 &$4.3107$ &$18.3484$ &$2.0960$ &$6.7970$ &$0.2984$ &$2.5476$ &$0.0007$ \\
Ti3 &$3.3717$ &$14.4441$ &$1.8258$ &$5.7126$ &$0.2470$ &$2.2654$ &$0.0005$ \\
V0 &$3.8099$ &$21.8313$ &$2.4026$ &$7.5458$ &$0.4464$ &$2.6628$ &$0.0017$ \\
V1 &$4.7474$ &$23.3226$ &$2.3609$ &$7.8082$ &$0.4105$ &$2.7063$ &$0.0014$ \\
V2 &$3.4386$ &$16.5303$ &$1.9638$ &$6.1415$ &$0.2997$ &$2.2669$ &$0.0009$ \\
V3 &$2.3005$ &$14.6821$ &$2.0364$ &$6.1304$ &$0.4099$ &$2.3815$ &$0.0014$ \\
V4 &$1.8377$ &$12.2668$ &$1.8247$ &$5.4578$ &$0.3979$ &$2.2483$ &$0.0012$ \\
Cr0 &$3.7600$ &$20.1267$ &$2.1006$ &$6.8020$ &$0.4266$ &$2.3941$ &$0.0019$ \\
Cr1 &$3.7768$ &$20.3456$ &$2.1028$ &$6.8926$ &$0.4010$ &$2.4114$ &$0.0017$ \\
Cr2 &$2.6422$ &$16.0598$ &$1.9198$ &$6.2531$ &$0.4446$ &$2.3715$ &$0.0020$ \\
Cr3 &$1.6262$ &$15.0656$ &$2.0618$ &$6.2842$ &$0.5281$ &$2.3680$ &$0.0023$ \\
Cr4 &$1.0293$ &$13.9498$ &$1.9933$ &$6.0593$ &$0.5974$ &$2.3457$ &$0.0027$ \\
Mn0 &$2.6681$ &$16.0601$ &$1.7561$ &$5.6396$ &$0.3675$ &$2.0488$ &$0.0017$ \\
Mn1 &$3.2953$ &$18.6950$ &$1.8792$ &$6.2403$ &$0.3927$ &$2.2006$ &$0.0022$ \\
Mn2 &$2.0515$ &$15.5561$ &$1.8841$ &$6.0625$ &$0.4787$ &$2.2323$ &$0.0027$ \\
Mn3 &$1.2427$ &$14.9966$ &$1.9567$ &$6.1181$ &$0.5732$ &$2.2577$ &$0.0031$ \\
Mn4 &$0.7879$ &$13.8857$ &$1.8717$ &$5.7433$ &$0.5981$ &$2.1818$ &$0.0034$ \\
Fe0 &$1.9405$ &$18.4733$ &$1.9566$ &$6.3234$ &$0.5166$ &$2.1607$ &$0.0036$ \\
Fe1 &$2.6290$ &$18.6598$ &$1.8704$ &$6.3313$ &$0.4690$ &$2.1628$ &$0.0031$ \\
Fe2 &$1.6490$ &$16.5593$ &$1.9064$ &$6.1325$ &$0.5206$ &$2.1370$ &$0.0035$ \\
Fe3 &$1.3602$ &$11.9976$ &$1.5188$ &$5.0025$ &$0.4705$ &$1.9914$ &$0.0038$ \\
Fe4 &$1.5582$ &$8.2750$ &$1.1863$ &$3.2794$ &$0.1366$ &$1.1068$ &$-0.0022$ \\
Co0 &$1.9678$ &$14.1699$ &$1.4911$ &$4.9475$ &$0.3844$ &$1.7973$ &$0.0027$ \\
Co1 &$2.4097$ &$16.1608$ &$1.5780$ &$5.4604$ &$0.4095$ &$1.9141$ &$0.0031$ \\
Co2 &$1.9049$ &$11.6444$ &$1.3159$ &$4.3574$ &$0.3146$ &$1.6453$ &$0.0017$ \\
Co3 &$1.7058$ &$8.8595$ &$1.1409$ &$3.3086$ &$0.1474$ &$1.0899$ &$-0.0025$ \\
Co4 &$1.3110$ &$8.0252$ &$1.1551$ &$3.1792$ &$0.1608$ &$1.1301$ &$-0.0011$ \\
Ni0 &$1.0302$ &$12.2521$ &$1.4669$ &$4.7453$ &$0.4521$ &$1.7437$ &$0.0036$ \\
Ni1 &$2.1040$ &$14.8655$ &$1.4302$ &$5.0714$ &$0.4031$ &$1.7784$ &$0.0034$ \\
Ni2 &$1.7080$ &$11.0160$ &$1.2147$ &$4.1031$ &$0.3150$ &$1.5334$ &$0.0018$ \\
Ni3 &$1.4683$ &$8.6713$ &$1.1068$ &$3.2574$ &$0.1794$ &$1.1058$ &$-0.0023$ \\
Ni4 &$1.1612$ &$7.7000$ &$1.0027$ &$3.2628$ &$0.2719$ &$1.3780$ &$0.0025$ \\
Cu0 &$1.9182$ &$14.4904$ &$1.3329$ &$4.7301$ &$0.3842$ &$1.6394$ &$0.0035$ \\
Cu1 &$1.8814$ &$13.4333$ &$1.2809$ &$4.5446$ &$0.3646$ &$1.6022$ &$0.0033$ \\
Cu2 &$1.5189$ &$10.4779$ &$1.1512$ &$3.8132$ &$0.2918$ &$1.3979$ &$0.0017$ \\
Cu3 &$1.2797$ &$8.4502$ &$1.0315$ &$3.2796$ &$0.2401$ &$1.2498$ &$0.0015$ \\
Cu4 &$0.9568$ &$7.4481$ &$0.9099$ &$3.3964$ &$0.3729$ &$1.4936$ &$0.0049$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large $\langle j_2\rangle$} Form factors for 4d transition elements}
\begin{table}[H]
\caption{$\langle j_2\rangle$ form factors for 4d atoms and ions}\vspace{2mm}
\label{4dj2}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Y0 &$14.4084$ &$44.6577$ &$5.1045$ &$14.9043$ &$-0.0535$ &$3.3189$ &$0.0028$ \\
Zr0 &$10.1378$ &$35.3372$ &$4.7734$ &$12.5453$ &$-0.0489$ &$2.6721$ &$0.0036$ \\
Zr1 &$11.8722$ &$34.9200$ &$4.0502$ &$12.1266$ &$-0.0632$ &$2.8278$ &$0.0034$ \\
Nb0 &$7.4796$ &$33.1789$ &$5.0884$ &$11.5708$ &$-0.0281$ &$1.5635$ &$0.0047$ \\
Nb1 &$8.7735$ &$33.2848$ &$4.6556$ &$11.6046$ &$-0.0268$ &$1.5389$ &$0.0044$ \\
Mo0 &$5.1180$ &$23.4217$ &$4.1809$ &$9.2080$ &$-0.0505$ &$1.7434$ &$0.0053$ \\
Mo1 &$7.2367$ &$28.1282$ &$4.0705$ &$9.9228$ &$-0.0317$ &$1.4552$ &$0.0049$ \\
Tc0 &$4.2441$ &$21.3974$ &$3.9439$ &$8.3753$ &$-0.0371$ &$1.1870$ &$0.0066$ \\
Tc1 &$6.4056$ &$24.8243$ &$3.5400$ &$8.6112$ &$-0.0366$ &$1.4846$ &$0.0044$ \\
Ru0 &$3.7445$ &$18.6128$ &$3.4749$ &$7.4201$ &$-0.0363$ &$1.0068$ &$0.0073$ \\
Ru1 &$5.2826$ &$23.6832$ &$3.5813$ &$8.1521$ &$-0.0257$ &$0.4255$ &$0.0131$ \\
Rh0 &$3.3651$ &$17.3444$ &$3.2121$ &$6.8041$ &$-0.0350$ &$0.5031$ &$0.0146$ \\
Rh1 &$4.0260$ &$18.9497$ &$3.1663$ &$6.9998$ &$-0.0296$ &$0.4862$ &$0.0127$ \\
Pd0 &$3.3105$ &$14.7265$ &$2.6332$ &$5.8618$ &$-0.0437$ &$1.1303$ &$0.0053$ \\
Pd1 &$4.2749$ &$17.9002$ &$2.7021$ &$6.3541$ &$-0.0258$ &$0.6999$ &$0.0071$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large $\langle j_2\rangle$} Form factors for 5d Transition elements}
\begin{table}[H]
 \caption{$\langle j_2\rangle$ form factors for the 5d electrons of transition atoms and ions from Hf to Re.\cite{kob:11}}
 \label{5dj2}
 \vspace{1ex}
 {\tablesize
\begin{tabular}{llrrrrrrrrr}
\hline
\multicolumn{1}{c}{ Ion}&\multicolumn{1}{c}{ Config}&\multicolumn{1}{c}{ A }&\multicolumn{1}{c}{  a }&\multicolumn{1}{c}{B }&\multicolumn{1}{c}{ b }&\multicolumn{1}{c}{ C }&\multicolumn{1}{c}{ c }&\multicolumn{1}{c}{ D }&\multicolumn{1}{c}{ d }&\multicolumn{1}{c}{E}\\
\hline
Hf2 & 6s05d2 &$9.6670$ &$33.435$ &$5.2429$ &$13.529$ &$-0.5533$ &$1.402$ &$0.4934$ &$1.254$ &$-0.0033$ \\
Hf3 & 6s05d1 &$7.5646$ &$27.367$ &$5.0743$ &$12.402$ &$-0.4133$ &$1.742$ &$0.3163$ &$1.437$ &$-0.0012$ \\
Ta2 & 6s05d3 &$8.1746$ &$29.871$ &$4.9405$ &$12.188$ &$-1.1294$ &$1.254$ &$1.0658$ &$1.181$ &$-0.0046$ \\
Ta3 & 6s05d2 &$6.5998$ &$25.026$ &$4.7913$ &$11.303$ &$-0.7801$ &$1.529$ &$0.6820$ &$1.382$ &$-0.0019$ \\
Ta4 & 6s05d1 &$5.5048$ &$21.620$ &$4.6734$ &$10.556$ &$-0.2181$ &$2.100$ &$0.0810$ &$1.049$ &$-0.0016$ \\
W0 & 6s05d6 &$8.4216$ &$64.652$ &$7.6983$ &$23.124$ &$2.3605$ &$8.583$ &$-0.3876$ &$4.620$ &$0.0033$ \\
W0 & 6s15d5 &$6.9289$ &$53.425$ &$7.2157$ &$20.016$ &$5.7754$ &$6.543$ &$-4.2114$ &$5.854$ &$0.0036$ \\
W0 & 6s25d4 &$5.4017$ &$47.110$ &$6.9113$ &$18.760$ &$7.0586$ &$6.363$ &$-5.4525$ &$5.802$ &$0.0040$ \\
W0 & 6s05d5 &$5.6854$ &$44.743$ &$6.8903$ &$18.753$ &$7.8523$ &$6.308$ &$-6.3447$ &$5.842$ &$0.0038$ \\
W0 & 6s15d4 &$4.5118$ &$41.281$ &$6.7136$ &$17.964$ &$7.6331$ &$6.269$ &$-6.0365$ &$5.758$ &$0.0042$ \\
W2 & 6s05d4 &$7.0301$ &$26.990$ &$4.6604$ &$11.083$ &$-0.8220$ &$1.145$ &$0.7581$ &$1.042$ &$-0.0068$ \\
W3 & 6s05d3 &$5.8036$ &$22.969$ &$4.5243$ &$10.361$ &$-0.7897$ &$1.388$ &$0.6927$ &$1.248$ &$-0.0031$ \\
W4 & 6s05d2 &$4.9006$ &$20.117$ &$4.4360$ &$9.765$ &$-2.0009$ &$1.497$ &$1.8714$ &$1.431$ &$-0.0014$ \\
W5 & 6s05d1 &$4.1973$ &$17.967$ &$4.3791$ &$9.255$ &$-1.8830$ &$1.603$ &$1.7205$ &$1.518$ &$-0.0004$ \\
Re0 & 6s05d7 &$6.7574$ &$55.529$ &$6.7931$ &$20.125$ &$2.3113$ &$7.529$ &$-0.5004$ &$4.412$ &$0.0037$ \\
Re0 & 6s15d6 &$5.5830$ &$46.852$ &$6.4516$ &$17.855$ &$5.0609$ &$6.037$ &$-3.5427$ &$5.312$ &$0.0040$ \\
Re0 & 6s25d5 &$4.4322$ &$41.798$ &$6.2063$ &$16.844$ &$9.8763$ &$5.751$ &$-8.3294$ &$5.406$ &$0.0043$ \\
Re0 & 6s05d6 &$4.7231$ &$40.130$ &$6.2139$ &$16.902$ &$8.8240$ &$5.766$ &$-7.3608$ &$5.396$ &$0.0042$ \\
Re0 & 6s15d5 &$3.7875$ &$37.254$ &$6.0587$ &$16.264$ &$6.9896$ &$5.786$ &$-5.4457$ &$5.279$ &$0.0045$ \\
Re2 & 6s05d5 &$3.0708$ &$33.896$ &$5.9941$ &$15.817$ &$6.7816$ &$5.768$ &$-5.2056$ &$5.231$ &$0.0047$ \\
Re3 & 6s05d4 &$1.7870$ &$31.248$ &$5.9068$ &$15.160$ &$17.4262$ &$5.551$ &$-15.6856$ &$5.339$ &$0.0053$ \\
Re4 & 6s05d3 &$4.4033$ &$18.733$ &$4.1971$ &$9.047$ &$-2.3106$ &$1.352$ &$2.1866$ &$1.299$ &$-0.0025$ \\
Re5 & 6s05d2 &$3.8162$ &$16.843$ &$4.1400$ &$8.605$ &$-4.0882$ &$1.443$ &$3.9333$ &$1.408$ &$-0.0012$ \\
Re6 & 6s05d1 &$3.3349$ &$15.341$ &$4.1015$ &$8.213$ &$-2.3511$ &$1.531$ &$2.1651$ &$1.459$ &$-0.0003$ \\
\hline
\end{tabular}
}
\end{table}
\begin{table}[H]
 \caption{$\langle j_\rangle$ form factors for the 5d electrons of transition atoms and ions from Os to Au.\cite{kob:11}}
\label{5dj2b} \vspace{1ex}
{\tablesize
\begin{tabular}{llrrrrrrrrr}
\hline
\multicolumn{1}{c}{ Ion}&\multicolumn{1}{c}{ Config}&\multicolumn{1}{c}{ A }&\multicolumn{1}{c}{  a }&\multicolumn{1}{c}{B }&\multicolumn{1}{c}{ b }&\multicolumn{1}{c}{ C }&\multicolumn{1}{c}{ c }&\multicolumn{1}{c}{ D }&\multicolumn{1}{c}{ d }&\multicolumn{1}{c}{E }\\
\hline
Os0 & 6s05d8 &$5.5418$ &$48.893$ &$6.0803$ &$17.984$ &$2.2542$ &$6.853$ &$-0.5285$ &$4.095$ &$0.0040$ \\
Os0 & 6s15d7 &$4.6511$ &$41.610$ &$5.8194$ &$16.062$ &$6.7172$ &$5.447$ &$-5.2689$ &$4.986$ &$0.0043$ \\
Os0 & 6s25d6 &$3.7421$ &$37.491$ &$5.6137$ &$15.256$ &$10.0488$ &$5.293$ &$-8.5683$ &$4.991$ &$0.0046$ \\
Os0 & 6s05d7 &$4.0236$ &$36.272$ &$5.6349$ &$15.338$ &$8.6568$ &$5.318$ &$-7.2505$ &$4.980$ &$0.0044$ \\
Os0 & 6s15d6 &$3.2611$ &$33.818$ &$5.4945$ &$14.808$ &$5.7610$ &$5.391$ &$-4.2811$ &$4.827$ &$0.0048$ \\
Os2 & 6s05d6 &$2.7289$ &$30.900$ &$5.4357$ &$14.396$ &$18.6546$ &$5.151$ &$-17.1708$ &$4.993$ &$0.0050$ \\
Os3 & 6s05d5 &$4.5913$ &$19.692$ &$4.0615$ &$8.862$ &$-0.9950$ &$1.086$ &$0.9081$ &$0.986$ &$-0.0077$ \\
Os4 & 6s05d4 &$3.9724$ &$17.514$ &$3.9817$ &$8.421$ &$-0.7719$ &$1.264$ &$0.6552$ &$1.107$ &$-0.0042$ \\
Os5 & 6s05d3 &$3.4764$ &$15.826$ &$3.9241$ &$8.032$ &$-0.6854$ &$1.397$ &$0.5394$ &$1.184$ &$-0.0024$ \\
Os6 & 6s05d2 &$3.0642$ &$14.479$ &$3.8847$ &$7.686$ &$-4.7235$ &$1.384$ &$4.5485$ &$1.352$ &$-0.0012$ \\
Os7 & 6s05d1 &$2.7164$ &$13.366$ &$3.8554$ &$7.370$ &$-5.0211$ &$1.439$ &$4.8166$ &$1.405$ &$-0.0004$ \\
Ir0 & 6s05d9 &$4.6102$ &$43.878$ &$5.4892$ &$16.394$ &$2.1707$ &$6.423$ &$-0.4761$ &$3.722$ &$0.0043$ \\
Ir0 & 6s15d8 &$3.9372$ &$37.508$ &$5.2846$ &$14.661$ &$3.7267$ &$5.263$ &$-2.3158$ &$4.416$ &$0.0045$ \\
Ir0 & 6s25d7 &$3.2263$ &$33.922$ &$5.1086$ &$13.921$ &$6.5993$ &$4.978$ &$-5.1841$ &$4.549$ &$0.0048$ \\
Ir0 & 6s05d8 &$3.4956$ &$32.991$ &$5.1369$ &$13.998$ &$8.3991$ &$4.924$ &$-7.0561$ &$4.612$ &$0.0047$ \\
Ir0 & 6s15d7 &$2.8732$ &$30.809$ &$5.0094$ &$13.522$ &$6.8656$ &$4.933$ &$-5.4669$ &$4.526$ &$0.0050$ \\
Ir2 & 6s05d7 &$2.4419$ &$28.356$ &$4.9470$ &$13.222$ &$4.7478$ &$5.029$ &$-3.3259$ &$4.393$ &$0.0052$ \\
Ir3 & 6s05d6 &$1.5883$ &$25.969$ &$4.8472$ &$12.711$ &$5.6507$ &$4.949$ &$-4.1190$ &$4.388$ &$0.0056$ \\
Ir4 & 6s05d5 &$3.5964$ &$16.439$ &$3.7872$ &$7.873$ &$-1.2303$ &$1.092$ &$1.1232$ &$1.003$ &$-0.0068$ \\
\hline
\end{tabular}
}
\end{table}
\section{{\large $\langle j_2\rangle$} Form factors for 4f electrons of rare earth ions}
\begin{table}[H]
\caption{$\langle j_2\rangle$ form factors for 4f electrons of rare-earth ions}
\vspace{2mm}
\label{re4fj2}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Ce2 &$0.9809$ &$18.0630$ &$1.8413$ &$7.7688$ &$0.9905$ &$2.8452$ &$0.0120$ \\
Nd2 &$1.4530$ &$18.3398$ &$1.6196$ &$7.2854$ &$0.8752$ &$2.6224$ &$0.0126$ \\
Nd3 &$0.6751$ &$18.3421$ &$1.6272$ &$7.2600$ &$0.9644$ &$2.6016$ &$0.0150$ \\
Sm2 &$1.0360$ &$18.4249$ &$1.4769$ &$7.0321$ &$0.8810$ &$2.4367$ &$0.0152$ \\
Sm3 &$0.4707$ &$18.4301$ &$1.4261$ &$7.0336$ &$0.9574$ &$2.4387$ &$0.0182$ \\
Eu2 &$0.8970$ &$18.4429$ &$1.3769$ &$7.0054$ &$0.9060$ &$2.4213$ &$0.0190$ \\
Eu3 &$0.3985$ &$18.4514$ &$1.3307$ &$6.9556$ &$0.9603$ &$2.3780$ &$0.0197$ \\
Gd2 &$0.7756$ &$18.4695$ &$1.3124$ &$6.8990$ &$0.8956$ &$2.3383$ &$0.0199$ \\
Gd3 &$0.3347$ &$18.4758$ &$1.2465$ &$6.8767$ &$0.9537$ &$2.3184$ &$0.0217$ \\
Tb2 &$0.6688$ &$18.4909$ &$1.2487$ &$6.8219$ &$0.8888$ &$2.2751$ &$0.0215$ \\
Tb3 &$0.2892$ &$18.4973$ &$1.1678$ &$6.7972$ &$0.9437$ &$2.2573$ &$0.0232$ \\
Dy2 &$0.5917$ &$18.5114$ &$1.1828$ &$6.7465$ &$0.8801$ &$2.2141$ &$0.0229$ \\
Dy3 &$0.2523$ &$18.5172$ &$1.0914$ &$6.7362$ &$0.9345$ &$2.2082$ &$0.0250$ \\
Ho2 &$0.5094$ &$18.5155$ &$1.1234$ &$6.7060$ &$0.8727$ &$2.1589$ &$0.0242$ \\
Ho3 &$0.2188$ &$18.5157$ &$1.0240$ &$6.7070$ &$0.9251$ &$2.1614$ &$0.0268$ \\
Er2 &$0.4693$ &$18.5278$ &$1.0545$ &$6.6493$ &$0.8679$ &$2.1201$ &$0.0261$ \\
Er3 &$0.1710$ &$18.5337$ &$0.9879$ &$6.6246$ &$0.9044$ &$2.1004$ &$0.0278$ \\
Tm2 &$0.4198$ &$18.5417$ &$0.9959$ &$6.6002$ &$0.8593$ &$2.0818$ &$0.0284$ \\
Tm3 &$0.1760$ &$18.5417$ &$0.9105$ &$6.5787$ &$0.8970$ &$2.0622$ &$0.0294$ \\
Yb2 &$0.3852$ &$18.5497$ &$0.9415$ &$6.5507$ &$0.8492$ &$2.0425$ &$0.0301$ \\
Yb3 &$0.1570$ &$18.5553$ &$0.8484$ &$6.5403$ &$0.8880$ &$2.0367$ &$0.0318$ \\
Pr3 &$0.8734$ &$18.9876$ &$1.5594$ &$6.0872$ &$0.8142$ &$2.4150$ &$0.0111$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
%
\section{{\large $\langle j_2\rangle$} Form factors for 5d electrons of rare earth elements}
\begin{table}[H]
 \caption{$\langle j_2\rangle$ form factors of the 5d electrons of rare earth ions.\cite{kob:12}}
 \vspace{1ex}
 \label{re5dj2}
{\tablesize
\begin{tabular}{llrrrrrrrrr}
\hline
\multicolumn{1}{c}{ Ion}&\multicolumn{1}{c}{ Config}&\multicolumn{1}{c}{ A }&\multicolumn{1}{c}{  a }&\multicolumn{1}{c}{B }&\multicolumn{1}{c}{ b }&\multicolumn{1}{c}{ C }&\multicolumn{1}{c}{ c }&\multicolumn{1}{c}{ D }&\multicolumn{1}{c}{ d }&\multicolumn{1}{c}{E}\\
\hline
La2 & 4f05d1 &$12.9681$ &$48.694$ &$9.8018$ &$22.831$ &$-0.5443$ &$4.354$ &$0.2584$ &$2.743$ &$-0.0001$ \\
Ce2 & 4f15d1 &$12.6550$ &$47.222$ &$9.3025$ &$21.878$ &$-0.6896$ &$3.762$ &$0.4405$ &$2.849$ &$0.0000$ \\
Ce3 & 4f05d1 &$8.7013$ &$35.824$ &$8.5894$ &$18.242$ &$-0.6110$ &$4.306$ &$0.2181$ &$2.336$ &$-0.0003$ \\
Pr2 & 4f25d1 &$12.4487$ &$45.891$ &$8.8239$ &$21.010$ &$-17.0630$ &$3.120$ &$16.8417$ &$3.095$ &$0.0001$ \\
Pr3 & 4f15d1 &$8.5203$ &$34.877$ &$8.2069$ &$17.602$ &$-0.5841$ &$3.958$ &$0.2347$ &$2.302$ &$-0.0002$ \\
Pr4 & 4f05d1 &$6.3910$ &$28.309$ &$7.6746$ &$15.204$ &$-0.6973$ &$4.010$ &$0.2304$ &$2.119$ &$-0.0004$ \\
Nd2 & 4f35d1 &$6.2640$ &$27.628$ &$7.3661$ &$14.731$ &$-0.6560$ &$3.743$ &$0.2354$ &$2.073$ &$-0.0003$ \\
Nd3 & 4f25d1 &$8.3541$ &$34.036$ &$7.8675$ &$17.031$ &$-0.5998$ &$3.584$ &$0.2896$ &$2.326$ &$0.0000$ \\
Nd4 & 4f15d1 &$6.2640$ &$27.628$ &$7.3661$ &$14.731$ &$-0.6560$ &$3.743$ &$0.2354$ &$2.073$ &$-0.0003$ \\
Pm3 & 4f35d1 &$8.2028$ &$33.286$ &$7.5637$ &$16.518$ &$-0.8756$ &$3.105$ &$0.6007$ &$2.486$ &$0.0002$ \\
Sm2 & 4f55d1 &$12.0956$ &$42.678$ &$7.6076$ &$18.857$ &$-1.4355$ &$2.749$ &$1.2720$ &$2.547$ &$0.0001$ \\
Sm3 & 4f45d1 &$8.1212$ &$32.502$ &$7.2340$ &$15.990$ &$-5.8639$ &$2.698$ &$5.6142$ &$2.630$ &$0.0003$ \\
Eu2 & 4f65d1 &$12.0033$ &$41.860$ &$7.2896$ &$18.289$ &$-1.4882$ &$2.592$ &$1.3411$ &$2.425$ &$0.0000$ \\
Eu3 & 4f55d1 &$8.0653$ &$31.753$ &$6.9151$ &$15.484$ &$-5.5701$ &$2.578$ &$5.3406$ &$2.515$ &$0.0003$ \\
Gd0 & 4f85d1 &$22.2510$ &$65.913$ &$7.5925$ &$23.045$ &$-3.5803$ &$1.952$ &$3.5355$ &$1.930$ &$-0.0010$ \\
Gd0 & 6s14f75d1 &$14.0295$ &$48.586$ &$7.3596$ &$18.927$ &$-5.1053$ &$2.242$ &$5.0055$ &$2.211$ &$-0.0004$ \\
Gd2 & 4f75d1 &$11.9202$ &$41.129$ &$7.0021$ &$17.772$ &$-5.3982$ &$2.392$ &$5.2663$ &$2.354$ &$-0.0001$ \\
Gd3 & 4f65d1 &$8.0082$ &$31.074$ &$6.6281$ &$15.022$ &$-1.9932$ &$2.515$ &$1.7825$ &$2.354$ &$0.0003$ \\
Gd4 & 4f55d1 &$5.8839$ &$25.328$ &$6.3297$ &$13.143$ &$-1.1650$ &$2.597$ &$0.8829$ &$2.219$ &$0.0004$ \\
Tb0 & 4f95d1 &$22.4050$ &$65.581$ &$7.3158$ &$22.565$ &$-4.9513$ &$1.745$ &$4.9149$ &$1.732$ &$-0.0014$ \\
Tb0 & 6s14f85d1 &$14.0011$ &$48.000$ &$7.0994$ &$18.468$ &$-0.6304$ &$2.188$ &$0.5431$ &$1.960$ &$-0.0007$ \\
Tb2 & 4f85d1 &$11.8504$ &$40.480$ &$6.7412$ &$17.303$ &$-11.6794$ &$2.248$ &$11.5615$ &$2.233$ &$-0.0002$ \\
Tb3 & 4f75d1 &$7.9534$ &$30.456$ &$6.3665$ &$14.596$ &$-0.8418$ &$2.508$ &$0.6482$ &$2.145$ &$0.0002$ \\
Tb4 & 4f65d1 &$5.8406$ &$24.788$ &$6.0831$ &$12.775$ &$-2.0431$ &$2.400$ &$1.7828$ &$2.221$ &$0.0004$ \\
Dy2 & 4f95d1 &$11.7905$ &$39.898$ &$6.5024$ &$16.874$ &$-11.5993$ &$2.115$ &$11.4943$ &$2.102$ &$-0.0003$ \\
Dy3 & 4f85d1 &$7.8866$ &$29.902$ &$6.1361$ &$14.212$ &$-12.5093$ &$2.229$ &$12.3329$ &$2.210$ &$0.0002$ \\
Ho3 & 4f95d1 &$7.8279$ &$29.392$ &$5.9224$ &$13.854$ &$-3.0002$ &$2.151$ &$2.8393$ &$2.080$ &$0.0001$ \\
Er3 & 4f105d1 &$7.7700$ &$28.922$ &$5.7263$ &$13.523$ &$-4.1104$ &$2.035$ &$3.9640$ &$1.990$ &$0.0000$ \\
Tm2 & 4f125d1 &$11.6672$ &$38.468$ &$5.8888$ &$15.774$ &$-4.8013$ &$1.717$ &$4.7296$ &$1.697$ &$-0.0012$ \\
Tm3 & 4f115d1 &$7.7142$ &$28.488$ &$5.5450$ &$13.214$ &$-5.7987$ &$1.923$ &$5.6660$ &$1.895$ &$-0.0002$ \\
Tm4 & 4f105d1 &$5.6664$ &$23.000$ &$5.2790$ &$11.534$ &$-3.2389$ &$1.988$ &$3.0496$ &$1.919$ &$0.0002$ \\
Yb2 & 4f135d1 &$11.6437$ &$38.074$ &$5.7103$ &$15.455$ &$-2.9992$ &$1.582$ &$2.9369$ &$1.554$ &$-0.0017$ \\
Yb3 & 4f125d1 &$7.6609$ &$28.088$ &$5.3775$ &$12.926$ &$-5.9455$ &$1.817$ &$5.8255$ &$1.793$ &$-0.0004$ \\
Lu2 & 4f145d1 &$11.6308$ &$37.717$ &$5.5427$ &$15.158$ &$-6.0682$ &$1.432$ &$6.0147$ &$1.420$ &$-0.0024$ \\
Lu3 & 4f135d1 &$7.6106$ &$27.715$ &$5.2212$ &$12.656$ &$-2.1969$ &$1.729$ &$2.0887$ &$1.671$ &$-0.0007$ \\
\hline
\end{tabular}
}
\end{table}
\section{{\large $\langle j_2\rangle$} Form factors for 5f electrons  of actinide ions}
\begin{table}[H]
\caption{$\langle j_2\rangle$ Form factors for actinide ions}\vspace{2mm}
\label{acj2}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
U3 &$4.1582$ &$16.5336$ &$2.4675$ &$5.9516$ &$-0.0252$ &$0.7646$ &$0.0057$ \\
U4 &$3.7449$ &$13.8944$ &$2.6453$ &$4.8634$ &$-0.5218$ &$3.1919$ &$0.0009$ \\
U5 &$3.0724$ &$12.5460$ &$2.3076$ &$5.2314$ &$-0.0644$ &$1.4738$ &$0.0035$ \\
Np3 &$3.7170$ &$15.1333$ &$2.3216$ &$5.5025$ &$-0.0275$ &$0.7996$ &$0.0052$ \\
Np4 &$2.9203$ &$14.6463$ &$2.5979$ &$5.5592$ &$-0.0301$ &$0.3669$ &$0.0141$ \\
Np5 &$2.3308$ &$13.6540$ &$2.7219$ &$5.4935$ &$-0.1357$ &$0.0493$ &$0.1224$ \\
Np6 &$1.8245$ &$13.1803$ &$2.8508$ &$5.4068$ &$-0.1579$ &$0.0444$ &$0.1438$ \\
Pu3 &$2.0885$ &$12.8712$ &$2.5961$ &$5.1896$ &$-0.1465$ &$0.0393$ &$0.1343$ \\
Pu4 &$2.7244$ &$12.9262$ &$2.3387$ &$5.1633$ &$-0.1300$ &$0.0457$ &$0.1177$ \\
Pu5 &$2.1409$ &$12.8319$ &$2.5664$ &$5.1522$ &$-0.1338$ &$0.0457$ &$0.1210$ \\
Pu6 &$1.7262$ &$12.3240$ &$2.6652$ &$5.0662$ &$-0.1695$ &$0.0406$ &$0.1550$ \\
Am2 &$3.5237$ &$15.9545$ &$2.2855$ &$5.1946$ &$-0.0142$ &$0.5853$ &$0.0033$ \\
Am3 &$2.8622$ &$14.7328$ &$2.4099$ &$5.1439$ &$-0.1326$ &$0.0309$ &$0.1233$ \\
Am4 &$2.4141$ &$12.9478$ &$2.3687$ &$4.9447$ &$-0.2490$ &$0.0215$ &$0.2371$ \\
Am5 &$2.0109$ &$12.0534$ &$2.4155$ &$4.8358$ &$-0.2264$ &$0.0275$ &$0.2128$ \\
Am6 &$1.6778$ &$11.3372$ &$2.4531$ &$4.7247$ &$-0.2043$ &$0.0337$ &$0.1892$ \\
Am7 &$1.8845$ &$9.1606$ &$2.0746$ &$4.0422$ &$-0.1318$ &$1.7227$ &$0.0020$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large$\langle j_4\rangle$} Form factors for 3d atoms and ions}
\begin{table}[H]
\caption{$\langle j_4\rangle$ form factors for 3d atoms and ions}\vspace{2mm}
\label{3dj4}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Sc0 &$1.3420$ &$10.2000$ &$0.3837$ &$3.0786$ &$0.0468$ &$0.1178$ &$-0.0328$ \\
Sc1 &$7.1167$ &$15.4872$ &$-6.6671$ &$18.2692$ &$0.4900$ &$2.9917$ &$0.0047$ \\
Sc2 &$-1.6684$ &$15.6475$ &$1.7742$ &$9.0624$ &$0.4075$ &$2.4116$ &$0.0042$ \\
Ti0 &$-2.1515$ &$11.2705$ &$2.5149$ &$8.8590$ &$0.3555$ &$2.1491$ &$0.0045$ \\
Ti1 &$-1.0383$ &$16.1899$ &$1.4699$ &$8.9239$ &$0.3631$ &$2.2834$ &$0.0044$ \\
Ti2 &$-1.3242$ &$15.3096$ &$1.2042$ &$7.8994$ &$0.3976$ &$2.1562$ &$0.0051$ \\
Ti3 &$-1.1117$ &$14.6349$ &$0.7689$ &$6.9267$ &$0.4385$ &$2.0886$ &$0.0060$ \\
V0 &$-0.9633$ &$15.2729$ &$0.9274$ &$7.7315$ &$0.3891$ &$2.0530$ &$0.0063$ \\
V1 &$-0.9606$ &$15.5451$ &$1.1278$ &$8.1182$ &$0.3653$ &$2.0973$ &$0.0056$ \\
V2 &$-1.1729$ &$14.9732$ &$0.9092$ &$7.6131$ &$0.4105$ &$2.0391$ &$0.0067$ \\
V3 &$-0.9417$ &$14.2045$ &$0.5284$ &$6.6071$ &$0.4411$ &$1.9672$ &$0.0076$ \\
V4 &$-0.7654$ &$13.0970$ &$0.3071$ &$5.6739$ &$0.4476$ &$1.8707$ &$0.0081$ \\
Cr0 &$-0.6670$ &$19.6128$ &$0.5342$ &$6.4779$ &$0.3641$ &$1.9045$ &$0.0073$ \\
Cr1 &$-0.8309$ &$18.0428$ &$0.7252$ &$7.5313$ &$0.3828$ &$2.0032$ &$0.0073$ \\
Cr2 &$-0.8930$ &$15.6641$ &$0.5590$ &$7.0333$ &$0.4093$ &$1.9237$ &$0.0081$ \\
Cr3 &$-0.7327$ &$14.0727$ &$0.3268$ &$5.6741$ &$0.4114$ &$1.8101$ &$0.0085$ \\
Cr4 &$-0.6748$ &$12.9462$ &$0.1805$ &$6.7527$ &$0.4526$ &$1.7999$ &$0.0098$ \\
Mn0 &$-0.5452$ &$15.4713$ &$0.4406$ &$4.9024$ &$0.2884$ &$1.5430$ &$0.0059$ \\
Mn1 &$-0.7947$ &$17.8673$ &$0.6078$ &$7.7044$ &$0.3798$ &$1.9045$ &$0.0087$ \\
Mn2 &$-0.7416$ &$15.2555$ &$0.3831$ &$6.4693$ &$0.3935$ &$1.7997$ &$0.0093$ \\
Mn3 &$-0.6603$ &$13.6066$ &$0.2322$ &$6.2175$ &$0.4104$ &$1.7404$ &$0.0101$ \\
Mn4 &$-0.5127$ &$13.4613$ &$0.0313$ &$7.7631$ &$0.4282$ &$1.7006$ &$0.0113$ \\
Fe0 &$-0.5029$ &$19.6768$ &$0.2999$ &$3.7762$ &$0.2576$ &$1.4241$ &$0.0071$ \\
Fe1 &$-0.5109$ &$19.2501$ &$0.3896$ &$4.8913$ &$0.2810$ &$1.5265$ &$0.0069$ \\
Fe2 &$-0.5401$ &$17.2268$ &$0.2865$ &$3.7422$ &$0.2658$ &$1.4238$ &$0.0076$ \\
Fe3 &$-0.5507$ &$11.4929$ &$0.2153$ &$4.9063$ &$0.3468$ &$1.5230$ &$0.0095$ \\
Fe4 &$-0.5352$ &$9.5068$ &$0.1783$ &$5.1750$ &$0.3584$ &$1.4689$ &$0.0097$ \\
Co0 &$-0.4221$ &$14.1952$ &$0.2900$ &$3.9786$ &$0.2469$ &$1.2859$ &$0.0063$ \\
Co1 &$-0.4115$ &$14.5615$ &$0.3580$ &$4.7170$ &$0.2644$ &$1.4183$ &$0.0074$ \\
Co2 &$-0.4759$ &$14.0462$ &$0.2747$ &$3.7306$ &$0.2458$ &$1.2504$ &$0.0057$ \\
Co3 &$-0.4466$ &$13.3912$ &$0.1419$ &$3.0110$ &$0.2773$ &$1.3351$ &$0.0093$ \\
Co4 &$-0.4091$ &$13.1937$ &$-0.0194$ &$3.4169$ &$0.3534$ &$1.4214$ &$0.0112$ \\
Ni0 &$-0.4428$ &$14.4850$ &$0.0870$ &$3.2345$ &$0.2932$ &$1.3305$ &$0.0096$ \\
Ni1 &$-0.3836$ &$13.4246$ &$0.3116$ &$4.4619$ &$0.2471$ &$1.3088$ &$0.0079$ \\
Ni2 &$-0.3803$ &$10.4033$ &$0.2838$ &$3.3780$ &$0.2108$ &$1.1036$ &$0.0050$ \\
Ni3 &$-0.4014$ &$ 9.0462$ &$0.2314$ &$3.0753$ &$0.2192$ &$1.0838$ &$0.0060$ \\
Ni4 &$-0.3509$ &$8.1572$ &$0.2220$ &$2.1063$ &$0.1567$ &$0.9253$ &$0.0065$ \\
Cu0 &$-0.3204$ &$15.1324$ &$0.2335$ &$4.0205$ &$0.2312$ &$1.1957$ &$0.0068$ \\
Cu1 &$-0.3572$ &$15.1251$ &$0.2336$ &$3.9662$ &$0.2315$ &$1.1967$ &$0.0070$ \\
Cu2 &$-0.3914$ &$14.7400$ &$0.1275$ &$3.3840$ &$0.2548$ &$1.2552$ &$0.0103$ \\
Cu3 &$-0.3671$ &$14.0816$ &$-0.0078$ &$3.3149$ &$0.3154$ &$1.3767$ &$0.0132$ \\
Cu4 &$-0.2915$ &$14.1243$ &$-0.1065$ &$4.2008$ &$0.3247$ &$1.3516$ &$0.0148$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large$\langle j_4\rangle$} Form factors 4d atoms and ions}
\begin{table}[H]
\noindent\caption{$\langle j_4\rangle$ form factors for 4d atoms and ions}\
\label{4dj4}
 \vspace{1ex}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Y0 &$-8.0767$ &$32.2014$ &$7.9197$ &$25.1563$ &$1.4067$ &$6.8268$ &$-0.0001$ \\
Zr0 &$-5.2697$ &$32.8680$ &$4.1930$ &$24.1833$ &$1.5202$ &$6.0481$ &$-0.0002$ \\
Zr1 &$-5.6384$ &$33.6071$ &$4.6729$ &$22.3383$ &$1.3258$ &$5.9245$ &$-0.0003$ \\
Nb0 &$-3.1377$ &$25.5948$ &$2.3411$ &$16.5686$ &$1.2304$ &$4.9903$ &$-0.0005$ \\
Nb1 &$-3.3598$ &$25.8202$ &$2.8297$ &$16.4273$ &$1.1203$ &$4.9824$ &$-0.0005$ \\
Mo0 &$-2.8860$ &$20.5717$ &$1.8130$ &$14.6281$ &$1.1899$ &$4.2638$ &$-0.0008$ \\
Mo1 &$-3.2618$ &$25.4862$ &$2.3596$ &$16.4622$ &$1.1164$ &$4.4913$ &$-0.0007$ \\
Tc0 &$-2.7975$ &$20.1589$ &$1.6520$ &$16.2609$ &$1.1726$ &$3.9427$ &$-0.0008$ \\
Tc1 &$-2.0470$ &$19.6830$ &$1.6306$ &$11.5925$ &$0.8698$ &$3.7689$ &$-0.0010$ \\
Ru0 &$-1.5042$ &$17.9489$ &$0.6027$ &$9.9608$ &$0.9700$ &$3.3927$ &$-0.0010$ \\
Ru1 &$-1.6278$ &$18.5063$ &$1.1828$ &$10.1886$ &$0.8138$ &$3.4180$ &$-0.0009$ \\
Rh0 &$-1.3492$ &$17.5766$ &$0.4527$ &$10.5066$ &$0.9285$ &$3.1555$ &$-0.0009$ \\
Rh1 &$-1.4673$ &$17.9572$ &$0.7381$ &$9.9444$ &$0.8485$ &$3.1263$ &$-0.0012$ \\
Pd0 &$-1.1955$ &$17.6282$ &$0.3183$ &$11.3094$ &$0.8696$ &$2.9089$ &$-0.0006$ \\
Pd1 &$-1.4098$ &$17.7650$ &$0.7927$ &$9.9991$ &$0.7710$ &$2.9297$ &$-0.0006$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large$\langle j_4\rangle$} Form factors for 5d atoms and ions}
\begin{table}[H]
 \caption{$\langle j_4\rangle$ form factors for the 5d electrons of transition atoms and ions from Hf to Re.\cite{kob:11}}
\label{5dj4}
 \vspace{1ex}
 {\tablesize
\begin{tabular}{llrrrrrrrrr}
\hline
\multicolumn{1}{c}{ Ion}&\multicolumn{1}{c}{ Config}&\multicolumn{1}{c}{ A }&\multicolumn{1}{c}{  a }&\multicolumn{1}{c}{B }&\multicolumn{1}{c}{ b }&\multicolumn{1}{c}{ C }&\multicolumn{1}{c}{ c }&\multicolumn{1}{c}{ D }&\multicolumn{1}{c}{ d }&\multicolumn{1}{c}{E }\\
\hline
Hf2 & 6s05d2 &$-2.5342$ &$43.826$ &$1.8466$ &$10.393$ &$0.7761$ &$4.888$ &$-0.0327$ &$1.589$ &$0.0017$ \\
Hf3 & 6s05d1 &$-2.3574$ &$32.651$ &$1.8717$ &$8.476$ &$0.6367$ &$3.953$ &$-0.1133$ &$2.169$ &$0.0017$ \\
Ta2 & 6s05d3 &$-2.1974$ &$38.294$ &$1.6220$ &$8.838$ &$0.6836$ &$4.212$ &$-0.0539$ &$1.746$ &$0.0016$ \\
Ta3 & 6s05d2 &$-2.0884$ &$29.531$ &$1.7145$ &$7.385$ &$1.1809$ &$2.994$ &$-0.7705$ &$2.577$ &$0.0016$ \\
Ta4 & 6s05d1 &$-2.0226$ &$24.035$ &$1.7084$ &$6.594$ &$1.1799$ &$2.775$ &$-0.8384$ &$2.430$ &$0.0020$ \\
W0 & 6s05d6 &$-2.1307$ &$79.955$ &$1.4055$ &$11.876$ &$0.8808$ &$4.969$ &$-0.0120$ &$0.419$ &$0.0051$ \\
W0 & 6s15d5 &$-1.9667$ &$60.069$ &$1.3446$ &$9.992$ &$0.7861$ &$4.554$ &$-0.0197$ &$1.087$ &$0.0021$ \\
W0 & 6s25d4 &$-1.8575$ &$47.554$ &$1.3868$ &$8.437$ &$0.6537$ &$3.939$ &$-0.0570$ &$1.722$ &$0.0015$ \\
W0 & 6s05d5 &$-2.0231$ &$46.962$ &$1.3937$ &$9.263$ &$0.7420$ &$4.332$ &$-0.0288$ &$1.327$ &$0.0018$ \\
W0 & 6s15d4 &$-1.9122$ &$39.952$ &$1.4385$ &$8.019$ &$0.6336$ &$3.725$ &$-0.0819$ &$1.868$ &$0.0015$ \\
W2 & 6s05d4 &$-1.9355$ &$33.935$ &$1.5020$ &$7.541$ &$0.6453$ &$3.375$ &$-0.1601$ &$2.110$ &$0.0015$ \\
W3 & 6s05d3 &$-1.8752$ &$26.706$ &$1.5440$ &$6.585$ &$5.9785$ &$2.575$ &$-5.6111$ &$2.516$ &$0.0018$ \\
W4 & 6s05d2 &$-1.8309$ &$22.142$ &$1.5605$ &$5.935$ &$1.4898$ &$2.482$ &$-1.1930$ &$2.266$ &$0.0022$ \\
W5 & 6s05d1 &$-1.7958$ &$18.987$ &$1.5913$ &$5.419$ &$2.0498$ &$2.278$ &$-1.8262$ &$2.156$ &$0.0026$ \\
Re0 & 6s05d7 &$-1.8013$ &$63.944$ &$1.1773$ &$9.808$ &$0.7912$ &$4.380$ &$-0.0155$ &$0.795$ &$0.0027$ \\
Re0 & 6s15d6 &$-1.7056$ &$49.628$ &$1.2209$ &$8.231$ &$0.6637$ &$3.836$ &$-0.0443$ &$1.514$ &$0.0015$ \\
Re0 & 6s25d5 &$-1.6402$ &$40.319$ &$1.3192$ &$7.064$ &$0.8659$ &$2.901$ &$-0.4299$ &$2.306$ &$0.0013$ \\
Re0 & 6s05d6 &$-1.7723$ &$40.683$ &$1.2795$ &$7.798$ &$0.6385$ &$3.630$ &$-0.0656$ &$1.686$ &$0.0015$ \\
Re0 & 6s15d5 &$-1.6968$ &$34.939$ &$1.3535$ &$6.851$ &$1.7292$ &$2.681$ &$-1.3118$ &$2.445$ &$0.0014$ \\
Re2 & 6s05d5 &$-1.7305$ &$30.305$ &$1.3808$ &$6.606$ &$1.5787$ &$2.630$ &$-1.1785$ &$2.378$ &$0.0015$ \\
Re3 & 6s05d4 &$-1.6969$ &$24.325$ &$1.4088$ &$5.901$ &$1.5067$ &$2.424$ &$-1.1834$ &$2.209$ &$0.0019$ \\
Re4 & 6s05d3 &$-1.6679$ &$20.454$ &$1.4439$ &$5.357$ &$0.8175$ &$2.321$ &$-0.5685$ &$1.976$ &$0.0024$ \\
Re5 & 6s05d2 &$-1.6427$ &$17.722$ &$1.4880$ &$4.916$ &$0.5419$ &$2.189$ &$-0.3694$ &$1.781$ &$0.0029$ \\
Re6 & 6s05d1 &$-1.6211$ &$15.673$ &$1.5419$ &$4.548$ &$1.9205$ &$1.833$ &$-1.8287$ &$1.766$ &$0.0034$ \\
\hline
\end{tabular}
}
\end{table}
\begin{table}[H]
 \caption{$\langle j_4\rangle$ form factors for the 5d electrons of transition atoms and ions from Os to Au.\cite{kob:11}}
\label{5dj4b} \vspace{1ex}
{\tablesize
\begin{tabular}{llrrrrrrrrr}
\hline
\multicolumn{1}{c}{ Ion}&\multicolumn{1}{c}{ Config}&\multicolumn{1}{c}{ A }&\multicolumn{1}{c}{  a }&\multicolumn{1}{c}{B }&\multicolumn{1}{c}{ b }&\multicolumn{1}{c}{ C }&\multicolumn{1}{c}{ c }&\multicolumn{1}{c}{ D }&\multicolumn{1}{c}{ d }&\multicolumn{1}{c}{E }\\
\hline
Os0 & 6s05d8 &$-1.5677$ &$53.075$ &$1.0631$ &$8.143$ &$0.6808$ &$3.771$ &$-0.0308$ &$1.243$ &$0.0017$ \\
Os0 & 6s15d7 &$-1.5109$ &$42.193$ &$1.1910$ &$6.850$ &$2.4597$ &$2.602$ &$-2.0163$ &$2.444$ &$0.0011$ \\
Os0 & 6s25d6 &$-1.4734$ &$34.814$ &$1.2105$ &$6.165$ &$0.9468$ &$2.534$ &$-0.5773$ &$2.131$ &$0.0015$ \\
Os0 & 6s05d7 &$-1.5777$ &$35.746$ &$1.2222$ &$6.650$ &$1.7260$ &$2.590$ &$-1.2942$ &$2.360$ &$0.0012$ \\
Os0 & 6s15d6 &$-1.5274$ &$30.891$ &$1.2364$ &$6.044$ &$1.4100$ &$2.431$ &$-1.0480$ &$2.188$ &$0.0016$ \\
Os2 & 6s05d6 &$-1.5637$ &$27.292$ &$1.2627$ &$5.880$ &$1.3526$ &$2.393$ &$-1.0026$ &$2.146$ &$0.0017$ \\
Os3 & 6s05d5 &$-1.5453$ &$22.300$ &$1.3058$ &$5.301$ &$0.7208$ &$2.294$ &$-0.4480$ &$1.881$ &$0.0022$ \\
Os4 & 6s05d4 &$-1.5267$ &$18.972$ &$1.3619$ &$4.834$ &$3.0121$ &$1.938$ &$-2.8240$ &$1.877$ &$0.0027$ \\
Os5 & 6s05d3 &$-1.5094$ &$16.573$ &$1.4158$ &$4.458$ &$0.8448$ &$1.797$ &$-0.7349$ &$1.638$ &$0.0033$ \\
Os6 & 6s05d2 &$-1.4938$ &$14.751$ &$1.4678$ &$4.149$ &$0.8091$ &$1.535$ &$-0.7720$ &$1.442$ &$0.0041$ \\
Os7 & 6s05d1 &$-0.0341$ &$37.994$ &$-1.4680$ &$13.159$ &$1.5216$ &$3.898$ &$-0.0308$ &$0.550$ &$0.0083$ \\
Ir0 & 6s05d9 &$-1.3913$ &$45.243$ &$1.0627$ &$6.722$ &$2.5141$ &$2.534$ &$-2.0510$ &$2.383$ &$0.0009$ \\
Ir0 & 6s15d8 &$-1.3605$ &$36.399$ &$1.0953$ &$5.990$ &$1.7223$ &$2.353$ &$-1.3416$ &$2.162$ &$0.0014$ \\
Ir0 & 6s25d7 &$-1.3382$ &$30.628$ &$1.1376$ &$5.420$ &$1.4261$ &$2.160$ &$-1.1282$ &$1.974$ &$0.0019$ \\
Ir0 & 6s05d8 &$-1.4233$ &$31.680$ &$1.1221$ &$5.872$ &$2.2721$ &$2.301$ &$-1.8973$ &$2.163$ &$0.0015$ \\
Ir0 & 6s15d7 &$-1.3875$ &$27.660$ &$1.1508$ &$5.362$ &$0.6586$ &$2.302$ &$-0.3554$ &$1.810$ &$0.0020$ \\
Ir2 & 6s05d7 &$-1.4233$ &$24.796$ &$1.1799$ &$5.246$ &$3.2548$ &$2.072$ &$-2.9649$ &$1.997$ &$0.0020$ \\
Ir3 & 6s05d6 &$-1.4149$ &$20.563$ &$1.2388$ &$4.761$ &$1.1780$ &$1.928$ &$-0.9708$ &$1.761$ &$0.0026$ \\
Ir4 & 6s05d5 &$-1.4039$ &$17.664$ &$1.2993$ &$4.371$ &$3.5599$ &$1.661$ &$-3.4340$ &$1.625$ &$0.0033$ \\
\hline
\end{tabular}
}
\end{table}
\section{{\large$\langle j_4\rangle$} Form factors for 4f electrons of rare earth ions}
\begin{table}[H]
\caption{$\langle j_4\rangle$ form factors for rare earth ions}\vspace{2mm}
\label{rej4}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Ce2 &$-0.6468$ &$10.5331$ &$0.4052$ &$5.6243$ &$0.3412$ &$1.5346$ &$0.0080$ \\
Nd2 &$-0.5744$ &$10.9304$ &$0.4210$ &$6.1052$ &$0.3124$ &$1.4654$ &$0.0081$ \\
Nd2 &$-0.5416$ &$12.2043$ &$0.3571$ &$6.1695$ &$0.3154$ &$1.4847$ &$0.0098$ \\
Nd3 &$-0.4053$ &$14.0141$ &$0.0329$ &$7.0046$ &$0.3759$ &$1.7074$ &$0.0209$ \\
Sm2 &$-0.4150$ &$14.0570$ &$0.1368$ &$7.0317$ &$0.3272$ &$1.5825$ &$0.0192$ \\
Sm3 &$-0.4288$ &$10.0525$ &$0.1782$ &$5.0191$ &$0.2833$ &$1.2364$ &$0.0088$ \\
Eu2 &$-0.4145$ &$10.1930$ &$0.2447$ &$5.1644$ &$0.2661$ &$1.2054$ &$0.0065$ \\
Eu3 &$-0.4095$ &$10.2113$ &$0.1485$ &$5.1755$ &$0.2720$ &$1.2374$ &$0.0131$ \\
Gd2 &$-0.3824$ &$10.3436$ &$0.1955$ &$5.3057$ &$0.2622$ &$1.2032$ &$0.0097$ \\
Gd3 &$-0.3621$ &$10.3531$ &$0.1016$ &$5.3104$ &$0.2649$ &$1.2185$ &$0.0147$ \\
Tb2 &$-0.3443$ &$10.4686$ &$0.1481$ &$5.4156$ &$0.2575$ &$1.1824$ &$0.0104$ \\
Tb3 &$-0.3228$ &$10.4763$ &$0.0638$ &$5.4189$ &$0.2566$ &$1.1962$ &$0.0159$ \\
Dy2 &$-0.3206$ &$12.0714$ &$0.0904$ &$8.0264$ &$0.2616$ &$1.2296$ &$0.0143$ \\
Dy3 &$-0.2829$ &$9.5247$ &$0.0565$ &$4.4292$ &$0.2437$ &$1.0665$ &$0.0092$ \\
Ho2 &$-0.2976$ &$9.7190$ &$0.1224$ &$4.6345$ &$0.2279$ &$1.0052$ &$0.0063$ \\
Ho3 &$-0.2717$ &$9.7313$ &$0.0474$ &$4.6378$ &$0.2292$ &$1.0473$ &$0.0124$ \\
Er2 &$-0.2975$ &$9.8294$ &$0.1189$ &$4.7406$ &$0.2116$ &$1.0039$ &$0.0117$ \\
Er3 &$-0.2568$ &$9.8339$ &$0.0356$ &$4.7415$ &$0.2172$ &$1.0281$ &$0.0148$ \\
Tm2 &$-0.2677$ &$9.8883$ &$0.0925$ &$4.7838$ &$0.2056$ &$0.9896$ &$0.0124$ \\
Tm3 &$-0.2292$ &$9.8948$ &$0.0124$ &$4.7850$ &$0.2108$ &$1.0071$ &$0.0151$ \\
Yb2 &$-0.2393$ &$9.9469$ &$0.0663$ &$4.8231$ &$0.2009$ &$0.9651$ &$0.0122$ \\
Yb3 &$-0.2121$ &$8.1967$ &$0.0325$ &$3.1533$ &$0.1975$ &$0.8842$ &$0.0093$ \\
Pr3 &$-0.3970$ &$10.9919$ &$0.0818$ &$5.9897$ &$0.3656$ &$1.5021$ &$0.0110$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large$\langle j_4\rangle$} Form factors for 4f electrons of actinide ions}
\begin{table}[H]
\caption{$\langle j_4\rangle$ Form factors for actinide ions}\vspace{2mm}
\label{acj4}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
U3 &$-0.9859$ &$16.6010$ &$0.6116$ &$6.5147$ &$0.6020$ &$2.5970$ &$-0.0010$ \\
U4 &$-1.0540$ &$16.6055$ &$0.4339$ &$6.5119$ &$0.6746$ &$2.5993$ &$-0.0011$ \\
U5 &$-0.9588$ &$16.4851$ &$0.1576$ &$6.4397$ &$0.7785$ &$2.6402$ &$-0.0010$ \\
Np3 &$-0.9029$ &$16.5858$ &$0.4006$ &$6.4699$ &$0.6545$ &$2.5631$ &$-0.0004$ \\
Np4 &$-0.9887$ &$12.4415$ &$0.5918$ &$5.2941$ &$0.5306$ &$2.2625$ &$-0.0021$ \\
Np5 &$-0.8146$ &$16.5809$ &$-0.0055$ &$6.4751$ &$0.7956$ &$2.5623$ &$-0.0004$ \\
Np6 &$-0.6738$ &$16.5531$ &$-0.2297$ &$6.5055$ &$0.8513$ &$2.5528$ &$-0.0003$ \\
Pu3 &$-0.7014$ &$16.3687$ &$-0.1162$ &$6.6971$ &$0.7778$ &$2.4502$ &$0.0000$ \\
Pu4 &$-0.9160$ &$12.2027$ &$0.4891$ &$5.1274$ &$0.5290$ &$2.1487$ &$-0.0022$ \\
Pu5 &$-0.7035$ &$16.3601$ &$-0.0979$ &$6.7057$ &$0.7726$ &$2.4475$ &$0.0000$ \\
Pu6 &$-0.5560$ &$16.3215$ &$-0.3046$ &$6.7685$ &$0.8146$ &$2.4259$ &$0.0001$ \\
Am2 &$-0.7433$ &$16.4163$ &$0.3481$ &$6.7884$ &$0.6014$ &$2.3465$ &$0.0000$ \\
Am3 &$-0.8092$ &$12.8542$ &$0.4161$ &$5.4592$ &$0.5476$ &$2.1721$ &$-0.0011$ \\
Am4 &$-0.8548$ &$12.2257$ &$0.3037$ &$5.9087$ &$0.6173$ &$2.1881$ &$-0.0016$ \\
Am5 &$-0.6538$ &$15.4625$ &$-0.0948$ &$5.9971$ &$0.7295$ &$2.2968$ &$0.0000$ \\
Am6 &$-0.5390$ &$15.4491$ &$-0.2689$ &$6.0169$ &$0.7711$ &$2.2970$ &$0.0002$ \\
Am7 &$-0.4688$ &$12.0193$ &$-0.2692$ &$7.0415$ &$0.7297$ &$2.1638$ &$-0.0011$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large $\langle j_6\rangle$} Form factors for 4f electrons of rare earth ions}
\begin{table}[H]
\caption{$\langle j_6\rangle$ Form factors for rare earth ions}\vspace{2mm}
\label{rej6}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Ce2 &$-0.1212$ &$7.9940$ &$-0.0639$ &$4.0244$ &$0.1519$ &$1.0957$ &$0.0078$ \\
Nd2 &$-0.1600$ &$8.0086$ &$0.0272$ &$4.0284$ &$0.1104$ &$1.0682$ &$0.0139$ \\
Nd3 &$-0.0416$ &$8.0136$ &$-0.1261$ &$4.0399$ &$0.1400$ &$1.0873$ &$0.0102$ \\
Sm2 &$-0.1428$ &$6.0407$ &$0.0723$ &$2.0329$ &$0.0550$ &$0.5134$ &$0.0081$ \\
Sm3 &$-0.0944$ &$6.0299$ &$-0.0498$ &$2.0743$ &$0.1372$ &$0.6451$ &$-0.0132$ \\
Eu2 &$-0.1252$ &$6.0485$ &$0.0507$ &$2.0852$ &$0.0572$ &$0.6460$ &$0.0132$ \\
Eu3 &$-0.0817$ &$6.0389$ &$-0.0596$ &$2.1198$ &$0.1243$ &$0.7639$ &$-0.0001$ \\
Gd2 &$-0.1351$ &$5.0298$ &$0.0828$ &$2.0248$ &$0.0315$ &$0.5034$ &$0.0187$ \\
Gd3 &$-0.0662$ &$6.0308$ &$-0.0850$ &$2.1542$ &$0.1323$ &$0.8910$ &$0.0048$ \\
Tb2 &$-0.0758$ &$6.0319$ &$-0.0540$ &$2.1583$ &$0.1199$ &$0.8895$ &$0.0051$ \\
Tb3 &$-0.0559$ &$6.0311$ &$-0.1020$ &$2.2365$ &$0.1264$ &$1.1066$ &$0.0167$ \\
Dy2 &$-0.0568$ &$6.0324$ &$-0.1003$ &$2.2396$ &$0.1401$ &$1.1062$ &$0.0109$ \\
Dy3 &$-0.0423$ &$6.0376$ &$-0.1248$ &$2.2437$ &$0.1359$ &$1.2002$ &$0.0188$ \\
Ho2 &$-0.0725$ &$6.0453$ &$-0.0318$ &$2.2428$ &$0.0738$ &$1.2018$ &$0.0252$ \\
Ho3 &$-0.0289$ &$6.0504$ &$-0.1545$ &$2.2305$ &$0.1550$ &$1.2605$ &$0.0177$ \\
Er2 &$-0.0648$ &$6.0559$ &$-0.0515$ &$2.2303$ &$0.0825$ &$1.2638$ &$0.0250$ \\
Er3 &$-0.0110$ &$6.0609$ &$-0.1954$ &$2.2242$ &$0.1818$ &$1.2958$ &$0.0149$ \\
Tm2 &$-0.0842$ &$4.0699$ &$0.0807$ &$0.8492$ &$-0.2087$ &$0.0386$ &$0.2095$ \\
Tm3 &$-0.0727$ &$4.0730$ &$0.0243$ &$0.6888$ &$3.9459$ &$0.0023$ &$-3.9076$ \\
Yb2 &$-0.0739$ &$5.0306$ &$0.0140$ &$2.0300$ &$0.0351$ &$0.5080$ &$0.0174$ \\
Yb3 &$-0.0345$ &$5.0073$ &$-0.0677$ &$2.0198$ &$0.0985$ &$0.5485$ &$-0.0076$ \\
Pr3 &$-0.0224$ &$7.9931$ &$-0.1202$ &$3.9406$ &$0.1299$ &$0.8938$ &$0.0051$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large$\langle j_6\rangle$} Form factors for 4f electrons of actinide ions}
\begin{table}[H]
\caption{$\langle j_6\rangle$ Form factors for actinide ions}\vspace{2mm}
\label{acj6}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
U3 &$-0.3797$ &$9.9525$ &$0.0459$ &$5.0379$ &$0.2748$ &$1.6072$ &$0.0016$ \\
U4 &$-0.1793$ &$11.8961$ &$-0.2269$ &$5.4280$ &$0.3291$ &$1.7008$ &$0.0030$ \\
U5 &$-0.0399$ &$11.8909$ &$-0.3458$ &$5.5803$ &$0.3340$ &$1.6448$ &$0.0029$ \\
Np3 &$-0.2427$ &$11.8444$ &$-0.1129$ &$5.3774$ &$0.2848$ &$1.5676$ &$0.0022$ \\
Np4 &$-0.2436$ &$9.5988$ &$-0.1317$ &$4.1014$ &$0.3029$ &$1.5447$ &$0.0019$ \\
Np5 &$-0.1157$ &$9.5649$ &$-0.2654$ &$4.2599$ &$0.3298$ &$1.5494$ &$0.0025$ \\
Np6 &$-0.0128$ &$9.5692$ &$-0.3611$ &$4.3035$ &$0.3419$ &$1.5406$ &$0.0032$ \\
Pu3 &$-0.0364$ &$9.5721$ &$-0.3181$ &$4.3424$ &$0.3210$ &$1.5233$ &$0.0041$ \\
Pu4 &$-0.2394$ &$7.8367$ &$-0.0785$ &$4.0243$ &$0.2643$ &$1.3776$ &$0.0012$ \\
Pu5 &$-0.1090$ &$7.8188$ &$-0.2243$ &$4.1000$ &$0.2947$ &$1.4040$ &$0.0015$ \\
Pu6 &$-0.0001$ &$7.8196$ &$-0.3354$ &$4.1439$ &$0.3097$ &$1.4027$ &$0.0020$ \\
Am2 &$-0.3176$ &$7.8635$ &$0.0771$ &$4.1611$ &$0.2194$ &$1.3387$ &$0.0018$ \\
Am3 &$-0.3159$ &$6.9821$ &$0.0682$ &$3.9948$ &$0.2141$ &$1.1875$ &$-0.0015$ \\
Am4 &$-0.1787$ &$7.8805$ &$-0.1274$ &$4.0898$ &$0.2565$ &$1.3152$ &$0.0017$ \\
Am5 &$-0.0927$ &$6.0727$ &$-0.2227$ &$3.7840$ &$0.2916$ &$1.3723$ &$0.0026$ \\
Am6 &$0.0152$ &$6.0788$ &$-0.3549$ &$3.8610$ &$0.3125$ &$1.4031$ &$0.0036$ \\
Am7 &$0.1292$ &$6.0816$ &$-0.4689$ &$3.8791$ &$0.3234$ &$1.3934$ &$0.0042$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}


\documentclass[main_zanardi.tex]{subfiles}

\begin{document}

% Zanardi è uno iettatore come Pauli

\paragraph{Questions}

How do we implement the POVM we described last time? That is left to the experimentalists.

\paragraph{Proof of the theorem}

This is useful because it shows us technicques, tools.

We have these \( E _{i}  \) from the POVM.

We wish to prove

\begin{equation}
  \cos ^{-1} \abs{\braket{\phi}{\psi}}
  = \cos ^{-1} \qty(\sqrt{p_i ^{\mathbb{E}} (\phi)} \sqrt{p_i(\psi)^{\mathbb{E}}})
\end{equation}

\begin{align}
  \abs{\braket{\phi}{\psi}} &= \abs{\sum_i \bra{\phi} E_i \ket{\psi}}  \\
  &\leq \sum_i \abs{\bra{\phi} \sqrt{E _{i} } \sqrt{E _{i}} \ket{\psi}}  \\
  &= \sum _{i} \abs{\braket{\hat{\phi_i}}{\hat{\psi_i}}}  \\
  &\leq \sum_i \norm{\sqrt{E_i} \ket{\phi}}\norm{\sqrt{E_i} \ket{\psi}}  \\
  &= \sum_i \sqrt{\ev{E_i}{\phi}} \sqrt{\ev{E_i}{\psi}}  \\
  &= \sum_i \sqrt{p_i ^{\mathbb{E}} (\phi)} \sqrt{p_i(\psi)^{\mathbb{E}}}
\end{align}

so we just apply the \( \cos ^{-1} \) to both sides.

\begin{equation}
  \mathbb{E} = \qty{\ketbra{\phi} | , \qty{\ketbra{\phi_i}} \text{span } \ketbra{\phi}^{\perp}}
\end{equation}

so the first term just gives us the upper bound, the other terms in the sum are 0: the inequality is saturated. We can just measure the projector associated with the state we want to know about.

\paragraph{Infinitesimal distance}

What is \( \dd _{B} \qty(\vec{p} , \vec{p} + \dd{\vec{p}}) \)?

\begin{align}
  \cos ^{-1} \qty{\sum_i \sqrt{p_i(p_i + \dd{p}_i)}}
  &= \cos ^{-1} \qty{\sum_i p_i \sqrt{1 + \frac{\dd{p}_i }{p_i}}}  \\
  &\sim \cos ^{-1} \qty(\sum_i p_i \qty(1 + \underbrace{\cancelto{0}{\frac{1}{2} \frac{\dd{p}_i}{p_i}}}_{\substack{\text{The probabilities}\\ \text{are normalized}}}     - \frac{1}{8} \qty(\frac{\dd{p_i}}{p_i})^{2} ))
\end{align}

but the term \( \sum_i \dd{p}_i \) vanishes, so we have

\begin{equation}
  \cos \dd{s} = 1 - \frac{1}{8}\frac{\dd{p}_i ^{2} }{p_i}
\end{equation}

but \( \cos \dd{s} \sim 1 - 1/2 \dd{s}^{2} _{B}  \), so

\begin{equation} \label{eq:fisher-metric}
  \dd{s}^{2}_{B}  = \frac{1}{4} \sum_i  \frac{\qty(\dd{p}_i)^{2} }{p_i}
\end{equation}

This is the \emph{Fisher metric}. More differential-geometry-like: take some \( \lambda \in \mathcal{M} = \qty{\text{manifold of control parameters of dimension N}} \), such that \( p_i = p_i(\lambda) \) and  \( \dd{p}_i = \sum_\mu (\partial_\mu p_i) \dd{\lambda_\mu} \).

Then

\begin{equation}
  g _{\mu \nu} = \frac{1}{4} \sum_i \frac{(\partial_\mu p_i)(\partial_\nu p_i)}{p_i}
\end{equation}

\begin{equation}
  \dd{s}_B ^2 = g _{\mu \nu} \dd{\lambda}_\mu \dd{\lambda}_\nu
\end{equation}

\section{Parameter estimation}

Take a random variable \( x \) distributed according to \(p_\theta (x)\) (a one-parameter distribution, with \(\theta\) as the parameter), and let \(\Theta\) be an estimator for the parameter \(\theta\): then \( \expval{\Theta} = \theta = \int  p_\theta(x) \Theta(x) \dd{x}  \).

We prove a super-famous bound. Differentiate the previous equation wrt \( \theta \).

\begin{subequations}\label{eq:classical-crao-start}
\begin{align}
  1 &= \int  p' _\theta (x) \Theta (x) \dd{x}   \\
  &= \braket{\frac{p'_\theta}{p_\theta}}{\Theta}_p \label{eq:classical-crao-start-subeq}
\end{align}
\end{subequations}

\textbf{Claim}: the notation \( \braket{f}{g}_p = \int  p_\theta f g \dd{x}  \) defines a scalar product.

\begin{greenbox}
  \textbf{Proof: }
  The axioms for a real scalar product are symmetry, linearity, and positive definiteness. The first two come from the properties of the regular product \(fg\); the interesting one to prove is the third. It is true since \(p_ \theta \geq 0\), and then \(\forall f: \int  p_ \theta f^2  \dd{x} \geq 0 \), also (modulo equality almost everywhere) \(\braket{f}{f}=0 \iff f=0\).
\end{greenbox}

We can subtract \( \theta \) from \(\Theta \rightarrow \overline{\Theta} = \Theta - \theta \) leaving the result unchanged, since it holds for any estimator: we restrict ourselves to \emph{unbiased} ones.

% When differentiting in  \eqref{eq:classical-crao-start} the additional terms are
%
% \begin{equation}
%   -\partial _ \theta \int  p_ \theta \theta \dd{x} = - \theta \int  p' _ \theta \dd{x}
%   - \int p_ \theta \dd{x} = - 1
% \end{equation}

\begin{align}
  1 &\leq \norm{\frac{p_\theta '}{p_\theta}}^{2}_p \norm{\Theta}^2_p  \\
  &= \qty(\int  p_\theta \frac{(p_\theta ')^2}{p_\theta^2} \dd{x} )
  \qty(\int p_\theta (\Theta(x) - \theta)^2 \dd{x} )  \\
  &= F \var{\Theta}
\end{align}

Where \( F \) is just the Fischer metric. (To check: \( \bar{\Theta} = \Theta - \theta \) )

But this means \(\var{\Theta} \geq F ^{-1} \): this is the \emph{Cramer-Rao} inequality.

\subsection{Quantize it!}

Unbiased estimator \( \Tr (\rho_\theta ' \hat{\Theta}) = 1 \).

We can do \( \Tr (\rho_\theta \rho_\theta ^{-1} \rho_\theta ' \hat{\Theta}) = 1 \)

But this is noncommutative! We can do

\begin{equation} \label{eq:SLD-def}
  L_\rho (x)  = \frac{1}{2} \qty(\rho X + X \rho) = \rho'
\end{equation}

This is the Symmetric Logarithmic Derivative, SLD.
(recall the logarithmic derivative \( \rho' / \rho = \dv*{\log \rho}{x} \) ).

Now

\begin{equation}
  \frac{1}{2} \Tr \qty[\qty(\rho x + x \rho) \bar{\Theta}] = \Re \Tr (\rho X \Theta)
\end{equation}

and we can take this equation in absolute value. Then,

\begin{equation}
  1 \leq \abs{\Tr (\rho X \bar{\Theta})}^{2} = \abs{\braket{X}{\bar{\Theta}}}^{2}_{\rho}
\end{equation}

And like before

\begin{equation}
  1 \leq \norm{X}^{2} _{\rho}  \norm{\bar{\Theta}}^{2} _{\rho}
  = \Tr (\rho X^2 ) Tr(\rho (\Theta - \theta)^2)
\end{equation}

so then \( \var{\Theta} \geq 1/F_Q \).

\textbf{Claim}: take \( \rho = \sum_i \ketbra{i} \), then \( X _{ij} = 2\bra{i}\rho' \ket{j}/\qty(p_i + p_j) \).

So we can compute

\begin{align}
  F_Q &= \Tr(\rho X^2) = \sum_i p_i \ev{X^2}{i}  \\
  &= \sum _{ij} p_i \bra{i} X \ketbra{j} X \ket{i}  \\
  &= \sum _{ij} p_i \frac{2}{p_i + p_j} \bra{i} \rho' \ketbra{j} \rho' \ket{i} \frac{2}{p_i + p_j}  \\
  &= 2 \sum _{ij} \frac{\abs{\bra{i} \rho' \ket{j}}^2 }{p_i + p_j}
\end{align}

This is Quantum Fischer. It really is the result of \emph{classical} optimization.

The denominator diverges! but the numerator goes to zero quadratically (?).

\paragraph{Fischer metric for pure states}

Take \( \rho_\theta = \ketbra{\psi_\theta} \) gound-eigenstate of a many-body system.

\begin{equation}
  F_Q ^{\text{pure}} \sim \braket{\psi_\theta'}{\psi_\theta'} - \abs{\braket{\psi_\theta'}{\psi_\theta}}^2
\end{equation}

where \( \psi' = \partial_\theta \psi \). This is a way to solve the Lyapunov equation \( L_\theta (\rho) = \rho' \).

\( \Pi = \ketbra{\psi} \). Then \(\Pi ^2 = \Pi\), we differentiate it.
So \( \Pi' \Pi + \Pi \Pi' = \Pi' \): in the pure state case, the solution is then just \( \Pi ' = X \).

\textbf{Claim}: we can calculate  \( F_Q =  \Tr \qty(\rho \qty(\rho')^2)\)...

\begin{greenbox}
  \textbf{Proof}
  The derivative of \( \rho \) is \( \rho' = \ketbra{\psi}{\psi'} + \ketbra{\psi'}{\psi} \).
  We also know, by differentiating \(1 = \braket{\psi}{\psi} \), that \(\braket{\psi}{\psi'} = - \braket{\psi'}{\psi} \). Therefore \(\braket{\psi}{\psi'} = s\) is purely imaginary, \(s = -s^*\).

  Then computing \( \Tr(\rho \rho' \rho')\) (in a basis consisting of \(\ket{\psi } \) and other perpendicular vectors, whose contribution is null because of the first term \(\rho\)) we get:

\begin{subequations}
\begin{align}
   \Tr(\rho \rho' \rho')
   &= \bra{\psi} \qty(\ketbra{\psi}{\psi'} + \ketbra{\psi'}{\psi})^2 \ket{\psi}   \\
   &= \cancelto{1}{\braket{\psi}{\psi}}\braket{\psi'}{\psi}\braket{\psi'}{\psi}
   + \cancelto{1}{\braket{\psi}{\psi}}\braket{\psi'}{\psi'}\cancelto{1}{\braket{\psi}{\psi}} \nonumber \\
   &+ \braket{\psi}{\psi'}\cancelto{1}{\braket{\psi}{\psi}}\braket{\psi'}{\psi}
   + \braket{\psi}{\psi'}\braket{\psi}{\psi'}\cancelto{1}{\braket{\psi}{\psi}} \\
   &= \braket{\psi'}{\psi}^2
   + \braket{\psi'}{\psi'} + \braket{\psi}{\psi'}\braket{\psi'}{\psi}
   + \braket{\psi}{\psi'}^2  \\
   &= \braket{\psi'}{\psi'} + s^2 + (-s)^2 + s(-s)  \\
   &= \braket{\psi'}{\psi'} + s^2  \\
   &= \braket{\psi'}{\psi'} - \abs{\braket{\psi}{\psi'}}^2
\end{align}
\end{subequations}

since \(s^2 \) must be real and negative.

\end{greenbox}
\end{document}