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\begin{document}

\title{A momentum-space Argonne V18 interaction}

\author{S. Veerasamy}
\affiliation{
Department of Physics and Astronomy, The University of Iowa, Iowa City, IA
52242}

\author{W. N. Polyzou}
\affiliation{
Department of Physics and Astronomy, The University of Iowa, Iowa City, IA
52242}

\vspace{10mm}
\date{\today}

\begin{abstract}

  This paper gives two representations of the Argonne V18 potential in
  momentum space.  One is as an expansion in terms elementary
  functions and other as an expansion in terms of Chebyshev
  polynomials.  Both provide practical and efficient representations
  for computing the momentum-space potential that does not require
  integration or interpolation.  Programs based on both expansions are
  available as supplementary material.
 
\end{abstract}

\vspace{10mm}

\pacs{21.45.Bc ,21.30.Cb}
\maketitle

%\newpage

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%\narrowtext

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\section{Introduction}

The Argonne V18 potential \cite{v18} is one of a number of realistic
nucleon-nucleon interactions \cite{reid}\cite{v18}\cite{cdbonn}. It is
distinguished from other realistic interactions because it is
expressed as an operator expansion with local configuration-space
coefficient functions.  This representation has advantages when used
in variational Monte Carlo calculations.  On the other hand there are
a number of calculations that require a realistic interaction that are
more naturally performed in momentum space.  These include some
Faddeev calculations, relativistic few-body calculations, and
electromagnetic calculations.  In the momentum representation the
variable conjugate to the relative coordinate is the difference
between the final and initial momenta.  In calculations, both momenta
appear, which requires a separate Fourier transform for each pair of
momenta.  While the direct Fourier transforms of the V18 potential
have been used in some applications, this is not the most efficient
method to calculate the Fourier transform of the potential.  The
purpose of this paper is to provide useful, tested, easily
reproducible analytic forms of the Fourier transform of the Argonne
V18 potential for use in momentum-space calculations.  The analytic
forms allow for a direct calculation of the momentum-space interaction
for any pair of initial and final momenta.  In keeping with the
traditional Argonne form, the momentum-space potential is given in
operator form.  The resulting momentum space potential has 24 terms.
This is because the $\mathbf{L}^2 V(r) $ and $(\mathbf{L}\cdot
\mathbf{S})^2 V(r)$ operators become linear combinations of two
momentum-space operators with different coefficient functions.  In
this work the Fourier transform is applied to the strong part of the
Argonne V18 potential, without the electromagnetic terms.
The Argonne V18 potential has the form 
\beq 
V = \sum_{n=1}^{18} V_n (r) O_n 
\label{a.1}
\eeq 
where $V_n(r)$ are rotationally invariant coefficient functions of the
relative coordinate of the nucleons and the $O_n$ are the following
set of eighteen spin-isospin operators:
\beq 
O_1 = I, \qquad O_2 =
(\btau_1 \cdot \btau_2), \qquad O_3 = (\pmb{\sigma}_1 \cdot
\pmb{\sigma}_2), 
\label{a.2}
\eeq 
\beq 
O_4 = (\mathbf{s}_1 \cdot
\mathbf{s}_2)(\btau_1 \cdot \btau_2), \qquad O_5 =
S_{12}=3(\pmb{\sigma}_1 \cdot \hat{\mathbf{r}})( \pmb{\sigma}_2 \cdot
\hat{\mathbf{r}}) - \pmb{\sigma}_1 \cdot \pmb{\sigma}_2, \qquad O_6
=S_{12} (\btau_1 \cdot \btau_2), 
\label{a.3}
\eeq 
\beq 
O_7 = (\mathbf{L}\cdot
\mathbf{S}), \qquad O_8 = (\mathbf{L}\cdot \mathbf{S})(\btau_1 \cdot
\btau_2) \qquad O_9 = (\mathbf{L}\cdot \mathbf{L}) 
\label{a.4}
\eeq 
\beq 
O_{10}
=(\mathbf{L}\cdot \mathbf{L})(\btau_1 \cdot \btau_2), \qquad
O_{11}=(\mathbf{L}\cdot \mathbf{L})(\pmb{\sigma}_1 \cdot \pmb{\sigma}_2),
\qquad O_{12}= (\mathbf{L}\cdot \mathbf{L})(\pmb{\sigma}_1 \cdot
\pmb{\sigma}_2) (\btau_1 \cdot \btau_2), 
\label{a.5}
\eeq 
\beq 
O_{13}=
(\mathbf{L}\cdot \mathbf{S})^2, \qquad O_{14}= (\mathbf{L}\cdot
\mathbf{S})^2 (\btau_1 \cdot \btau_2) , \qquad O_{15}= T_{12} = ( 3
\tau_{1z}\tau_{2z} - \btau \cdot \btau), 
\label{a.6}
\eeq 
\beq 
O_{16}= 
(\pmb{\sigma}_1 \cdot \pmb{\sigma}_2)T_{12}, \qquad O_{17} = S_{12} 
T_{12}, \qquad O_{18} = (\tau_{1z} + \tau_{2z}).  
\label{a.7}
\eeq  
where $T_{12}$ is the isotensor operator $T_{12}:= 3
\tau_{1z}\tau_{2z}- \pmb{\tau}_1 \cdot \pmb{\tau}_2 $.  While the
isospin operators, $\pmb{\tau}_i$, factor out of the Fourier
transforms, the operators $L^2$, $\mathbf{L}\cdot \mathbf{S}$,
$(\mathbf{L}\cdot \mathbf{S})^2$ and the tensor operator $S_{12}$
contribute to the Fourier transform.

For the eighteen operators there are four types of integrals that must
be computed to calculate the Fourier transforms.  The potential matrix
element $\langle \mathbf{k}'\vert V \vert \mathbf{k} \rangle$, with
$\mathbf{q}:=\mathbf{k} - \mathbf{k}'$, has the following
contributions:

\begin{itemize}

\item [1.] Identity
\beq
{1 \over (2 \pi)^3} \int e^{-i (\mathbf{k}'-\mathbf{k})\cdot \mathbf{r}}
V_j(r) d\mathbf{r} =
{1 \over 2 \pi^2} \int_0^\infty j_0 (qr) V_j(r) r^2 dr 
\label{a.8}
\eeq

\item [2.] $\mathbf{L} \cdot \mathbf{S}$

\beq
{1 \over (2 \pi)^3} \int e^{-i \mathbf{k}'\cdot \mathbf{r}}
V_j(r) \mathbf{L} \cdot \mathbf{S}
e^{i \mathbf{k}\cdot \mathbf{r}} d\mathbf{r} =
i \mathbf{k} \times \mathbf{k}' 
{1 \over 2 \pi^2 q} \int_0^\infty j_1 (qr) V_j(r) r^3 dr 
\label{a.9}
\eeq

\item [3.] $\mathbf{L} \cdot \mathbf{L}$
 
\[
{1 \over (2 \pi)^3} \int e^{-i \mathbf{k}'\cdot \mathbf{r}}
V_j(r) \mathbf{L} \cdot \mathbf{L}
e^{i \mathbf{k}\cdot \mathbf{r}} d\mathbf{r} =
\]
\beq
- (\mathbf{k}'\times \mathbf{k}) \cdot 
(\mathbf{k}'\times \mathbf{k})
{1 \over 2 \pi^2 q^2}\int_0^\infty j_2 (qr) V_j(r) r^4 dr
+
2 (\mathbf{k}'\cdot \mathbf{k}) 
{1 \over 2 \pi^2 q}\int_0^\infty j_1 (qr) V_j(r) r^3 dr 
\label{a.10}
\eeq

\item [4] $(\mathbf{L} \cdot \mathbf{S})^2$

\[
{1 \over (2 \pi)^3} \int e^{-i \mathbf{k}'\cdot \mathbf{r}}
V_j(r) (\mathbf{L} \cdot \mathbf{S})^2
e^{i \mathbf{k}\cdot \mathbf{r}} d\mathbf{r} =
\]

\beq
- (\mathbf{S} \cdot ( \mathbf{k}\times \mathbf{k}'))^2
 {1 \over 2 \pi^2 q^2 } \int_0^\infty j_2 (qr) V_j(r) r^4 dr
+ 
( \mathbf{k}'\times \mathbf{S})\cdot
( \mathbf{k}\times \mathbf{S})  
{1 \over 2 \pi^2 q} \int_0^\infty j_1 (qr) V_j(r) r^3 dr
\label{a.11}
\eeq

\item [5.] $S_{12}= 3(\hat{\mathbf{r}} \cdot \pmb{\sigma}_1)
(\hat{\mathbf{r}} \cdot \pmb{\sigma}_2) - \pmb{\sigma}_1\cdot \pmb{\sigma}_2$
\[
{1 \over (2 \pi)^3} \int e^{-i \mathbf{k}'\cdot \mathbf{r}}
V(r) \left (3 (\hat{\mathbf{r}} \cdot \pmb{\sigma}_1)
(\hat{\mathbf{r}} \cdot \pmb{\sigma}_2) 
-  \pmb{\sigma}_1\cdot \pmb{\sigma}_2 \right )
e^{i \mathbf{k}\cdot \mathbf{r}} d\mathbf{r} =
\]
\beq
-
\left (3 (\mathbf{q} \cdot \pmb{\sigma}_1)( \mathbf{q} \cdot \pmb{\sigma}_2)
- q^2 \pmb{\sigma}_1 \cdot \pmb{\sigma}_2 \right )
{1 \over 2 \pi^2 q^2} \int_0^\infty j_2 (qr) V(r) r^2 dr 
\label{a.12}
\eeq

\end{itemize}

These expressions are used to represent the momentum-space
interaction as a sum of scalar functions of $q:=\vert \mathbf{q}\vert$ 
multiplied by spin-isospin operators.  The scalar functions of momentum transfer
that multiply the spin-isospin operators are:
\beq
\tilde{V}_{m}(q) =  {1 \over 2 \pi^2 }\int_0^\infty j_0 (qr) V_m(r) r^2 dr
\qquad m \in \{1,2,3,4,15,16,18 \}
\label{a.13}
\eeq
\beq
\tilde{V}_{m}(q) = {1 \over 2 \pi^2 q }\int_0^\infty j_1 (qr) V_m(r) r^3 dr
\qquad m \in \{7,8,9b,10b,11b,12b,13b,14b \}
\label{a.14}
\eeq
\beq
\tilde{V}_{m}(q) =  {1 \over 2 \pi^2 q^2}\int_0^\infty j_2 (qr) V_m(r) r^4 dr
\qquad m \in \{9a,10a,11a,12a,13a,14a \}
\label{a.15}
\eeq
\beq
\tilde{V}_{m}(q) =  {1 \over 2 \pi^2 q^2}\int_0^\infty j_2 (qr) V_m(r) r^2 dr
\qquad m \in \{5,6,17 \}
\label{a.16}
\eeq
where $V_m(r)$ is the $m^{th}$ potential in
the expansion (\ref{a.1}) and $\tilde{V}_{ma} (q)$ and $\tilde{V}_{mb}
(q)$ are the two different functions that appear in (\ref{a.10}) and
(\ref{a.11}).  These functions have a finite limit as $q\to 0$ in
spite of the $1/q^l$ coefficient because the Bessel function $j_l(qr)$
vanishes like $q^l$ as $q\to 0$.

The resulting momentum-space potential can be expressed as
\beq
\langle \mathbf{k}' \vert V \vert \mathbf{k} \rangle = 
\sum_{m\in S}  \tilde{V}_{m} (q) \tilde{O}_m 
\label{a.17}
\eeq
where $S=\{1,2,3,4,5,6,7,8,9a,9b,10a,10b,11a,11b,12a,12b,13a,13b,14a,14b,15,16,17$,$18\}$.  The 24 operators $\tilde{O}_m $ are 
\beq
\tilde{O}_{1} = I
\qquad
\tilde{O}_{2} = (\btau_1 \cdot \btau_2)
\qquad
\tilde{O}_{3} = (\pmb{\sigma}_1 \cdot \pmb{\sigma}_2)
\qquad
\tilde{O}_{4} =(\pmb{\sigma}_1 \cdot \pmb{\sigma}_2)(\btau_1 \cdot \btau_2) 
\label{a.18}
\eeq
\beq
\tilde{O}_{5} = - \left (3 (\mathbf{q} \cdot \pmb{\sigma}_1)( \mathbf{q} \cdot \pmb{\sigma}_2)
- {q^2 } \pmb{\sigma}_1 \cdot \pmb{\sigma}_2 \right )
\qquad
\tilde{O}_{6} =
- \left (3 (\mathbf{q} \cdot \pmb{\sigma}_1)( \mathbf{q} \cdot \pmb{\sigma}_2)
- {q^2 } \pmb{\sigma}_1 \cdot \pmb{\sigma}_2 \right )
(\btau_1 \cdot \btau_2)
\label{a.19}
\eeq
\beq
\tilde{O}_{7} =
i \mathbf{k} \times \mathbf{k}'   
\qquad
\tilde{O}_{8} =
i \mathbf{k} \times \mathbf{k}' (\btau_1 \cdot \btau_2)
\qquad
\tilde{O}_{9a} =
-(\mathbf{k}'\times \mathbf{k}) \cdot 
(\mathbf{k}'\times \mathbf{k})  
\qquad
\tilde{O}_{9b}=
2 (\mathbf{k}'\cdot \mathbf{k}) 
\label{a.20}
\eeq
\beq
\tilde{O}_{10a} =
-(\mathbf{k}'\times \mathbf{k}) \cdot 
(\mathbf{k}'\times \mathbf{k})  (\btau_1 \cdot \btau_2)
\qquad
\tilde{O}_{10b} =
2 (\mathbf{k}'\cdot \mathbf{k})(\btau_1 \cdot \btau_2) 
\eeq
\beq
\tilde{O}_{11a}=
-(\mathbf{k}'\times \mathbf{k}) \cdot 
(\mathbf{k}'\times \mathbf{k}) (\pmb{\sigma}_1 \cdot \pmb{\sigma}_2) 
\label{a.21}
\qquad
\tilde{O}_{11b} =
2(\mathbf{k}'\cdot \mathbf{k})(\pmb{\sigma}_1 \cdot \pmb{\sigma}_2)  
\eeq
\beq
\tilde{O}_{12a}=
-(\mathbf{k}'\times \mathbf{k}) \cdot 
(\mathbf{k}'\times \mathbf{k})  (\pmb{\sigma}_1 \cdot
\pmb{\sigma}_2) (\btau_1 \cdot \btau_2) 
\qquad
\tilde{O}_{12b} =
2 (\mathbf{k}'\cdot \mathbf{k})(\pmb{\sigma}_1 \cdot
\pmb{\sigma}_2) (\btau_1 \cdot \btau_2) 
\label{a.22}
\eeq
\beq
\tilde{O}_{13a} =
- (\mathbf{S} \cdot ( \mathbf{k}\times \mathbf{k}'))^2
\qquad
\tilde{O}_{13b}=
( \mathbf{k}'\times \mathbf{S})\cdot
( \mathbf{k}\times \mathbf{S}) 
\qquad
\tilde{O}_{14a} =
- (\mathbf{S} \cdot ( \mathbf{k}\times \mathbf{k}'))^2
 (\btau_1 \cdot \btau_2) 
\label{a.23}
\eeq
\beq
\tilde{O}_{14b} =
( \mathbf{k}'\times \mathbf{S})\cdot
( \mathbf{k}\times \mathbf{S}) (\btau_1 \cdot \btau_2)
\qquad
\tilde{O}_{15} =
T_{12} 
\qquad
\tilde{O}_{16}=
(\pmb{\sigma}_1 \cdot \pmb{\sigma}_2) T_{12} 
\label{a.24}
\eeq
\beq
\tilde{O}_{17} =
- \left (3 (\mathbf{q} \cdot \pmb{\sigma}_1)( \mathbf{q} \cdot \pmb{\sigma}_2)
- {q^2 } \pmb{\sigma}_1 \cdot \pmb{\sigma}_2 \right )T_{12}
\qquad
\tilde{O}_{18} =
(\tau_{1z} + \tau_{2z}). 
\label{a.25}
\eeq
In the approximate potentials the coefficient functions $\tilde{V}_{m}
(q)$ are replaced by expansions in known functions.

\section{Expansions}

Two approaches are used to compute the scalar functions
(\ref{a.13}-\ref{a.16}).  Both approaches involve approximating these
functions by a finite linear combination of elementary functions.  The
first method approximates these functions by linear combinations of
Chebyshev polynomials on three distinct intervals of momenta, for
momenta up to 100$fm^{-1}$.  The second approach approximates the
integrand by a finite linear combination of orthogonal functions that
have analytic Fourier transforms.  The configuration-space basis
functions are associated Laguerre polynomials multiplied by decaying
exponentials which have analytic Fourier transforms that can be
expressed in terms of Jacobi polynomials \cite{keister}.  In both
approaches the coefficients of the expansion are stored and the
potentials can then be efficiently computed for any values of momenta
by summing a finite series of elementary functions.  In both cases 
the required expansion functions at any point can be determined
recursively, leading to efficient and accurate approximations to the
momentum space potential.

This section discusses the Chebyshev basis.  Because the configuration
space potential falls off asymptotically like $e^{-m_{\pi} r}$, the
radial integrals are evaluated with a finite cutoff at 20 $fm$.  The
Chebyshev expansion is used for $q<100\,fm^{-1}$.  With these cutoffs
the maximum value of $x:=qr$ that can appear in the argument of the
spherical Bessel functions in the integrals (\ref{a.13}-\ref{a.16}) is
$x_{max}=2000$.  To evaluate these integrals  
the zeros of the spherical Bessel functions $j_0(x)$, $j_1(x)$, and $j_2(x)$
for $0\leq x \leq 2000$ are computed for each fixed value of
$q$.  For each value of $q$ the integrals are
expressed as sums integrals between successive zeros of the spherical
Bessel function that appear in the integral.  If $q$ is such that $qr$
is never a zero of $j_l(qr)$ for $0<r<20 fm$ then the integral over $r$ is
performed using a 100 point Gauss-Legendre quadrature on the interval
$[0,20fm]$.  If $q$ is such that $qr$ has zeros of $j_l(qr)$ for $0<r<20fm$,
then the integrals between zeros $[qr_i , qr_{i+1}]$ are computed
using 20 Gauss-Legendre points when $r_{i+1} \leq 5\, fm$, 
40 Gauss-Legendre points when $5\,fm< r_{i+1} \leq 10\,fm$ and 80 Gauss-Legendre
points when $10\,fm< r_{i+1} \leq 20\,fm$.

The functions $\tilde{V}_m(q)$ are replaced by a Chebyshev polynomial 
approximation on the interval $q \in [a,b]$ using \cite{broucke}
\beq
\tilde{V}_m(q) \approx 
c_0/2 +\sum_{n=1}^{100} c_n T_n (-{a+b\over b-a}+ {2 \over b-a}q)
\label{aa.1}
\eeq
where 
\beq 
T_{n}(x) = \cos(n \cos^{-1} (x)))
\label{aa.2}
\eeq 
are Chebyshev polynomials and the coefficients $c_n$ are
computed using a Clenshaw-Curtiss quadrature\cite{broucke} :
\beq
c_n = {2 \over N} 
[
{1 \over 2}\tilde{V}_m(b)+ 
\sum_{j=1}^{N-1} 
\tilde{V}_m({a+b\over 2} + {b-a \over 2} \cos (\pi j /N))  \cos (nj \pi /N)+
(-)^n {1 \over 2} \tilde{V}_m(a)
]
\label{aa.3}
\eeq
with $N=101$.
The functions $\tilde{V}_m (q)$ are evaluated at the quadrature
points $q_j:={a+b\over 2} + {b-a \over 2} \cos (\pi j /N)$ using the
methods discussed above.  This is repeated for $q$ in each of three
intervals, $[a,b]= [0,10], [10,50], [50,100]$ and the 101 expansion
coefficients associated with each of these three intervals are
stored.  The Chebyshev polynomials are computed using the 
recurrence relations
\beq
T_{n+1} (x) = 2x T_n(x) - T_{n-1}(x), \qquad T_0 (x)=1, \, T_1 (x) = x .
\label{aa.4}
\eeq
For $q$ larger than 100 $fm^{-1}$ $\tilde{V}_m(q)$ is approximated by 0.

This method provides an accurate and efficient representation 
for computing a momentum space $V18$ interaction.  While it requires a 
substantial effort to compute the integrals at the required 
quadrature points, this only has to be done once.

\section{Basis Functions} 

While the method of the previous section gives accurate results,
a more straightforward approach is to represent the potential directly as an
expansion in basis functions that have analytic Fourier transforms.
The plots in figures 1-24 show the configuration-space
functions that must be
Fourier-Bessel transformed for each of the 24 integrals listed in
(\ref{a.13}-\ref{a.16}).  These graphs actually show both the exact
radial functions and the basis function expansions of these functions,
which are discussed below.  The curves are indistinguishable on these plots.
The plots show that the radial functions are simple smooth functions.

In order to represent the potential, each of the scalar potentials
$\tilde{V}_m (q)$, is approximated by an expansion in known basis
functions.  A method to compute both the expansion
coefficients and a recursion formula to compute basis functions
are given below.


The functions $V_m(r)$, $rV_m(r)$, and $r^2V_m(r)$
that appear in the integrals (\ref{a.13}-\ref{a.16})
are expanded using an
orthonormal set of radial functions that have analytic Fourier-Bessel
transforms \cite{keister}.  These functions are associated
Laguerre polynomials multiplied by decaying exponentials in
configuration space.  Their Fourier-Bessel transforms have power-law
fall of in momentum space.  In addition, they vanish at the origin in
a manner that can be used to explicitly cancel the factors $1/q$ and
$1/q^2$ that appear in the definitions (\ref{a.12}-\ref{a.15}) of
$\tilde{V}_{m}$.  Both sets of basis functions can be generated
efficiently using recursion relations.  The cancellation of the
factors $1/q$ and $1/q^2$ can be directly incorporated into the
recursion that generates the momentum-space basis functions so the
final expression for the potential does not require a special
treatment for $q$ near $0$.

The radial basis functions for different values of $l$ are given below.
The dimensionless parameter $x := \Lambda r$ is used in the basis 
functions,  where 
$\Lambda$ is a scale parameter that can be chosen to improve convergence.  
The parameterization of the Argonne V18 interaction 
uses the value $\Lambda = 7(fm)^{-1}$.  
The configuration space basis functions are  
\beq
\phi_{nl}(r) = {1 \over \sqrt{N_{nl}}}x^l L_n^{2l+2}(2x) e^{-x} 
\label{b.1}
\eeq
where 
\beq
L_n^\alpha = \sum_{m=0}^n (-)^m 
\left (
\begin{array}{c}
n+\alpha \\
n-m \\
\end{array}
\right )
{x^m \over m!}
\label{b.2}
\eeq
and the normalization coefficient is 
\beq
N_{nl} = \Lambda^{-3} ({1 \over 2})^{2l+3} {\Gamma (n+\alpha+1 ) \over n!}.
\label{b.3}
\eeq
These functions satisfy the orthogonality relations
\beq
\int_0^\infty 
\phi_{nl}(r) \phi_{ml}(r) r^2 dr = \delta_{mn} .
\label{b.4}
\eeq
They have analytic Fourier-Bessel transforms defined by 
\beq
\tilde{\phi}_{nl}(q) = \sqrt{2 \over \pi} \int_0^{\infty} 
j_l(qr) \phi_{nl} (r) r^2 dr .
\label{b.5}
\eeq
For $y=q / \Lambda$ the $\tilde{\phi}_{nl}(q)$ can be expressed in terms
of Jacobi polynomials:
\beq
\tilde{\phi}_{nl}(q) =
{1 \over \sqrt{\tilde{N}_{nl}}}{y^l \over (y^2 +1)^{l+2}}
P_n^{l+{3 \over 2},l+{1 \over 2}}({y^2-1 \over y^2+1})
\label{b.6}
\eeq
with normalization coefficient
\beq
\tilde{N}_{nl} = {\Lambda^3 \over 2 (2n+2l+3)}
{\Gamma(n+l+{5 \over 2})\Gamma(n+l+{3 \over 2}) \over
n! \Gamma (n+2l+3)}
\label{b.7}
\eeq
and
\beq
P^{(\alpha,\beta)}_n (z) = 
{\Gamma (\alpha+n+1) \over n! \Gamma (\alpha +\beta +n +1)}
\sum_{m=0}^n \left ( 
\begin{array}{c}
n\\
m
\end{array}
\right ) 
{\Gamma (\alpha+\beta +n +m+1) \over 2^m \Gamma (\alpha+m +1)}
(z-1)^m .
\label{b.8}
\eeq
These functions satisfy the orthogonality relations
\beq
\int_0^\infty 
\tilde{\phi}_{nl}(q) \tilde{\phi}_{ml}(q) q^2 dq = \delta_{mn} .
\label{b.9}
\eeq

These basis functions can be generated by using the following recursion
formulas for the associated Laguerre functions and Jacobi
polynomials 
\beq
(n+1) L^\alpha_{n+1} (x) = 
(2n+\alpha +1 -x)  L^\alpha_{n+} (x)
- (n+\alpha)  L^\alpha_{n-1} (x)
\label{b.12}
\eeq
and 
\[
2(n+1)(n+\alpha+\beta +1) (2n+\alpha + \beta) 
P_{n+1} ^{(\alpha,\beta)} (x) = 
\]
\[
[(2n+\alpha +\beta +1)(\alpha^2 -\beta^2)
+ x((2n+\alpha+\beta)(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)]
P_{n} ^{(\alpha,\beta)} (x)
\]
\beq 
- 
2(n+\alpha)(n+\beta)(2n+\alpha+\beta+2) 
P_{n-1} ^{(\alpha,\beta)} (x) .
\label{b.13}
\eeq
These recursion relations can be modified to incorporate the 
normalization constants (\ref{b.3}) and ({\ref{b.7}) directly 
into the recursion.  The recursion for the 
normalized radial basis functions with $(x=\Lambda r)$ is given by: 
\beq
\phi_{0l} (r) = {1 \over \sqrt{(2*l+1)!}}{1 \over \sqrt{2^{2l+3}}}
\Lambda^{3/2} x^l e^{-x} 
\label{b.14}
\eeq
\beq
\phi_{1l} (r)  =  {2l+3-2x \over \sqrt{2l+3}} \phi_{0l} (r) 
\label{b.15}
\eeq
\beq
\phi_{nl} (r) = {2n+1+2l-2x \over \sqrt{n+1+2l}\sqrt{n}} \phi_{n-1,l} (r)
- 
\sqrt{{(n-1)(n+1+2l) \over n (n+2+2l)}} \phi_{n-2,l} (r)
\label{b.16}
\eeq
Similarly, the normalized momentum-space basis functions with 
$(y=q/\Lambda)$ are generated by the recursion:
\[
\tilde{\phi}_{0l} (q) =
\]
\beq
{1 \over \sqrt{(2l+3)!}}{1 \over \sqrt{\Lambda^3}}
{1 \over \sqrt{{1 \over 2} \cdots {2l+3 \over 2}}}
{1 \over \sqrt{{1 \over 2} \cdots {2l+1 \over 2}}} y^l {1 \over (y^2 +1)^{2l+2}}
\label{b.17}
\eeq
\beq
\tilde{\phi}_{1l} (q) = 
({1 \over 2} + (l+2){y^2-1 \over y^2+1} )
\sqrt{{2l+5 \over (l+2 +{1 \over 2})(l+1 +{1 \over 2})}}
\tilde{\phi}_{0l} (q) 
\label{b.18}
\eeq 
\[
\tilde{\phi}_{nl} (q) =
\] 
\[
\sqrt{{(2n + 2l +3)n (n+2l+2)\over 
(2n+2l +1)(n+l +{3 \over 2})(n+l +{1 \over 2}) }}
\]
\[
\times {(2n+2l+1)(2l+2)(y^2+1) + (2n+2l+1)(2n+2l)(2n+2l+2)(y^2-1)  
\over 
2n(n+2l+2)(2n+2l)(y^2-1)
}\tilde{\phi}_{n-1l} (q)
\]
\[
-
\sqrt{{(2n + 2l +3)(n-1)n(n+1+2l)(n+2l+2)\over 
(2n+2l -1)(n+l +{3 \over 2})(n+l +{1 \over 2})
(n+l +{1 \over 2})(n+l -{1 \over 2})
}}
\]
\beq
\times
{(n+l+{1 \over 2}) (n+l-{1 \over 2}) (2n+2l+2) \over
(n)(n+2l+2)(2n+2l)
}\tilde{\phi}_{n-1l} (q).
\label{b.19}
\eeq

Replacing $\tilde{\phi}_{0l} (q)$ by
$\hat{\phi}_{0l} (q):= \tilde{\phi}_{0l} (q)/q^l$ given by
\[
\hat{\phi}_{0l} (q) =
\]
\beq
{1 \over \sqrt{(2l+3)!}}{1 \over \sqrt{\Lambda^3}}
{1 \over \sqrt{{1 \over 2} \cdots {2l+3 \over 2}}}
{1 \over \sqrt{{1 \over 2} \cdots {2l+1 \over 2}}} \Lambda^{-l} 
{1 \over (y^2 +1)^{2l+2}}
\label{b.20}
\eeq
to start the recursion
in equations 
(\ref{b.18})-(\ref{b.19})  generates
$\hat{\phi}_{nl} (q):= \tilde{\phi}_{nl} (q)/q^l$, which are
well-behaved as $q \to 0$.  
Seventy expansion
coefficients are used 
to construct the momentum-space potential for each value of $m$
\beq
c_{nm}=  {1 \over 2 \pi^2 }\int_0^\infty \phi_{n0} (r) V_m(r) r^2 dr
\qquad m \in \{1,2,3,4,15,16,18 \}
\label{b.21}
\eeq
\beq
c_{nm} = {1 \over 2 \pi^2  }\int_0^\infty \phi_{n1} (r) V_m(r) r^3 dr
\qquad m \in \{7,8,9b,10b,11b,12b,13b,14b \}
\label{b.22}
\eeq
\beq
c_{nm} =  {1 \over 2 \pi^2 }\int_0^\infty \phi_{n2} (r) V_m(r) r^4 dr
\qquad m \in \{9a,10a,11a,12a,13a,14a \}
\label{b.23}
\eeq
\beq
c_{nm} =  {1 \over 2 \pi^2 }\int_0^\infty \phi_{n2} (r) V_m(r) r^2 dr
\qquad m \in \{5,6,17 \}
\label{b.24}
\eeq
The integrals are approximated using an 80 point Gauss Legendre
quadrature between 0 and 10$fm$.  The basis functions $\phi_{nl}(r)$ are
generated using (\ref{b.14}-\ref{b.16}).  The scale parameter in the
recursion for $\phi_{nl}(r)$ is taken as $\Lambda =7fm^{-1} $.

The 70x24 expansion coefficients $c_{nm}$ are stored.  The momentum
space potential functions are then given by 
\beq
\tilde{V}_{m}(q) = \sum_{n=1}^{70} c_{nm} \hat{\phi}_{nl} (q) 
\eeq
where the reduced expansion functions $\hat{\phi}_{nl}(q) := 
\tilde{\phi}_{nl}(q)/q^l$ are generated recursively using 
(\ref{b.17}-\ref{b.19}).

The full momentum-space potential in operator form is given by
\beq
V = \sum_{m\in S} \tilde{V}_{m}(q) \tilde{O}_{m} 
\eeq
where $\tilde{O}_{m}$ are the 24 operators (\ref{a.17}-\ref{a.24})
and $q=\sqrt{k^2 + k^{\prime 2} - 2 \mathbf{k}' \cdot \mathbf{k}}$.

\section{Tests} 

Three tests are performed on the two potentials.  First, the exact
radial functions that appear on the right side of the integrals
(\ref{a.13}-\ref{a.16}) are compared with their configuration space
expansions in the basis (\ref{b.1}).  Second, the momentum space
coefficient functions, $\tilde{V}_m(q)$ are compared to the more
precise calculation of the Fourier-Bessel transforms based on direct
Fourier-Bessel transforms with a Chebyshev polynomial interpolation.
Finally, both potentials are used to compute the deuteron 
binding energy and wave functions.  These results are compared 
to a direct calculation of these quantities using the partial wave
expansion of the original configuration space potential. 
 
Figures 1-24 compare the structure of the 24 configuration space
functions (the integrands in (\ref{a.13}-\ref{a.16}) for $r<5 fm$
computed from the 70 term expansion (dashed lines) to the exact
expressions (solid lines).  The difference between the exact and
approximate radial functions is too small to be seen in these plots.
The quality of the approximations is not surprising given that exact
functions are smooth with minimal structure.
 
To test the Fourier transforms the Fourier-Bessel transforms
(\ref{a.13}-\ref{a.16}) based on the Chebyshev expansion, discussed in
the previous section, are compared to the Fourier-Bessel transforms
obtained by analytically Fourier transforming the basis functions.
Figures 25-48 show both plots.  The solid lines are the series
expansions while the dashed lines are the Chebyshev expansion.  The
figures for the potentials associated with the
 
Again differences are not visible on these plots.  A better
comparison is given in tables 1-4, which list values of the 
Fourier-Bessel transforms of the 24 radial functions using both methods for
momenta of 1,5,15 and 25 $fm^{-1}$.

\begin{center}
Table 1. - Fourier transforms at 1 fm$^{-1}$
\end{center}
\begin{center}
\begin{tabular}{lll}
%\begin{table}{lll}
\hline
n &  series  & Chebyshev expansion \\
\hline
1    &   6.789973e-01  &     6.789990e-01 \\
2    &  -4.019392e-01  &    -4.019400e-01 \\
3    &  -1.692090e-01  &    -1.692090e-01 \\
4    &   2.358519e-01  &     2.356720e-01 \\
5    &   7.216739e-03  &     7.218230e-03 \\
6    &   2.857732e-01  &     2.860470e-01 \\
7    &  -5.511547e-01  &    -5.511560e-01 \\
8    &  -1.678888e-01  &    -1.678890e-01 \\
9    &   1.741415e-01  &     1.741420e-01 \\
10   &  -3.272988e-02  &    -3.272990e-02 \\
11   &   1.999136e-02  &     1.999140e-02 \\
12   &  -7.414060e-03  &    -7.414070e-03 \\
13   &   9.084422e-02  &     9.084440e-02 \\
14   &   1.245017e-01  &     1.245020e-01 \\
15   &   1.122388e-02  &     1.122390e-02 \\
16   &  -1.214926e-02  &    -1.216020e-02 \\
17   &   2.403290e-03  &     2.420790e-03 \\
18   &   6.124964e-03  &     6.124970e-03 \\
19   &   1.304278e-02  &     1.304280e-02 \\
20   &  -1.702409e-02  &    -1.702400e-02 \\
21   &  -7.227244e-03  &    -7.227270e-03 \\
22   &  -7.849686e-03  &    -7.849720e-03 \\
23   &   4.518193e-02  &     4.518270e-02 \\
24   &   3.980251e-02  &     3.980280e-02 \\
\hline
\end{tabular}
\end{center} 
                         

\begin{center}
Table 2. - Fourier transforms at 5 fm$^{-1}$
\end{center}
\begin{center}
\begin{tabular}{lll}
%\begin{table}{lll}
\hline
n &  series  & Chebyshev expansion \\
\hline
1  &  1.160699e+00   &    1.160700e+00  \\
2  & -1.360382e-02   &   -1.360380e-02  \\
3  & -1.148807e-01   &   -1.148810e-01  \\
4  & -1.065288e-01   &   -1.065200e-01  \\
5  &  4.489757e-03   &    4.489760e-03  \\
6  &  4.405849e-03   &    4.405360e-03  \\
7  &  -4.623736e-02  &   -4.623740e-02  \\
8  &  -1.871380e-02  &   -1.871380e-02  \\
9  &   2.471311e-02  &    2.471320e-02  \\
10 &   1.480758e-03  &    1.480760e-03  \\
11 &   6.027203e-03  &    6.027210e-03  \\
12 &   1.465070e-03  &    1.465070e-03  \\
13 &   5.222260e-03  &    5.222270e-03  \\
14 &   8.233502e-03  &    8.233520e-03  \\
15 &   4.828280e-03  &    4.828290e-03  \\
16 &  -4.815794e-03  &   -4.815310e-03  \\
17 &   1.656921e-06  &    1.627536e-06  \\
18 &   4.274306e-04  &    4.274310e-04  \\
19 &   4.273833e-03  &    4.273840e-03  \\
20 &   1.791462e-04  &    1.791460e-04  \\
21 &   9.672551e-04  &    9.672570e-04  \\
22 &   1.814761e-04  &    1.814760e-04  \\
23 &   1.620319e-03  &    1.620320e-03  \\
24 &   1.790086e-03  &    1.790090e-03  \\
%\end{table}
\hline
\end{tabular}
\end{center} 

\begin{center}
Table 3. - Fourier transforms at 15 fm$^{-1}$
\end{center}
\begin{center}
\begin{tabular}{lll}
%\begin{table}{lll}
\hline
n &  series  & Chebyshev expansion \\
\hline
1    &    9.321365e-04  &     9.321050e-04   \\
2    &    4.123439e-05  &     4.123390e-05   \\
3    &   -1.924812e-05  &    -1.924670e-05   \\
4    &   -6.648375e-05  &    -6.643920e-05   \\
5    &   -9.010902e-06  &    -9.010533e-06   \\
6    &    1.026393e-05  &     1.026324e-05   \\
7    &    5.541260e-06  &     5.540870e-06   \\
8    &    2.632043e-06  &     2.631910e-06   \\
9    &   -1.962835e-06  &    -1.962590e-06   \\
10   &   -9.304609e-07  &    -9.304860e-07   \\
11   &   -6.015901e-07  &    -6.015370e-07   \\
12   &   -1.047669e-07  &    -1.047530e-07   \\
13   &   -4.725022e-06  &    -4.725160e-06   \\
14   &   -1.527634e-06  &    -1.527590e-06   \\
15   &    2.942747e-06  &     2.942630e-06   \\
16   &   -2.895027e-06  &    -2.892440e-06   \\
17   &   -2.865458e-10  &    -3.050827e-10   \\
18   &    9.986465e-08  &     9.985120e-08   \\
19   &   -2.604487e-07  &    -2.604660e-07   \\
20   &   -6.335039e-08  &    -6.334960e-08   \\
21   &   -7.055132e-08  &    -7.055530e-08   \\
22   &   -1.454468e-08  &    -1.454570e-08   \\
23   &   -3.115148e-07  &    -3.115090e-07   \\
24   &   -1.394089e-07  &    -1.394130e-07   \\
%\end{table}
\hline
\end{tabular}
\end{center} 


\begin{center}
Table 4. - Fourier transforms at 25 fm$^{-1}$
\end{center}
\begin{center}
\begin{tabular}{lll}
%\begin{table}{lll}
\hline
n &  series  & Chebyshev expansion \\
\hline
1   &  -1.386301e-05  &    -1.383430e-05    \\
2   &  -6.108349e-08  &    -6.010020e-08    \\
3   &   8.598072e-07  &     8.595170e-07    \\
4   &   1.014189e-06  &     1.003920e-06    \\
5   &  -4.600082e-07  &    -4.599216e-07    \\
6   &   4.739733e-07  &     4.738720e-07    \\
7   &   2.443040e-08  &     2.442090e-08    \\
8   &   9.428095e-09  &     9.412980e-09    \\
9   &  -1.534834e-08  &    -1.533920e-08    \\
10  &   3.457372e-10  &     3.607580e-10    \\
11  &  -3.619628e-09  &    -3.613210e-09    \\
12  &  -1.005784e-09  &    -1.003140e-09    \\
13  &   4.666338e-09  &     4.709400e-09    \\
14  &  -3.274714e-09  &    -3.270330e-09    \\
15  &  -5.425469e-08  &    -5.415690e-08    \\
16  &   5.452722e-08  &     5.399650e-08    \\
17  &  -2.888773e-12  &    -4.234000e-12    \\
18  &  -5.852151e-09  &    -5.841450e-09    \\
19  &  -2.512190e-10  &    -2.555620e-10    \\
20  &   7.827015e-12  &     7.411800e-12    \\
21  &  -5.864134e-11  &    -5.980770e-11    \\
22  &  -1.617550e-11  &    -1.652710e-11    \\
23  &   8.297311e-11  &     8.271960e-11    \\
24  &  -5.322500e-11  &    -5.424550e-11    \\
%\end{table}
\hline
\end{tabular}
\end{center} 

\begin{center}
Table 5 s and d wave functions using Chebyshev, series and partial waves
\end{center}
\begin{center}
\begin{tabular}{lllllll}
%\begin{table}{lll}
\hline
k$fm^{-1}$ & s-Cheb.  & s-pw  & s-series & d-Cheb, & d-pw & d-series\\
\hline
0.0 &   1.2695e+01 &  1.2638e+01 &  1.2695e+01  &   0.00000e+00  &   0.00000e+00 &  0.00000e+00 \\ 
0.5 &   1.9609e+00 &  1.9606e+00 &  1.9609e+00  &  -2.19827e-01  &  -2.19752e-01 & -2.19811e-01 \\
1.0 &   3.7684e-01 &  3.7687e-01 &  3.7685e-01  &  -1.72164e-01  &  -1.72174e-01 & -1.7216e-01  \\
1.5 &   8.2472e-02 &  8.2472e-02 &  8.2473e-02  &  -1.1243e-01   &  -1.12418e-01 & -1.1243e-01  \\
2.0 &   6.0810e-03 &  6.1039e-03 &  6.0810e-03  &  -7.10859e-02  &  -7.10846e-02 & -7.10868e-02 \\
2.5 &  -1.3615e-02 & -1.3601e-02 & -1.3616e-02  &  -4.4543e-02   &  -4.45444e-02 & -4.45435e-02 \\	
3.0 &  -1.6153e-02 & -1.6154e-02 & -1.6153e-02  &  -2.76854e-02  &  -2.76819e-02 & -2.76856e-01 \\	 
3.5 &  -1.3648e-02 & -1.3648e-02 & -1.3648e-02  &  -1.69881e-02  &  -1.69856e-02 & -1.69882e-02 \\ 
4.0 &  -1.0153e-02 & -1.0146e-02 & -1.0153e-02  &  -1.02234e-02  &  -1.02219e-02 & -1.02235e-02 \\
4.5 &  -6.9954e-03 & -6.9994e-03 & -6.9955e-03  &  -5.98474e-03  &  -5.98642e-03 & -5.9848e-03  \\
5.0 &  -4.5270e-03 & -4.5234e-03 & -4.5271e-03  &  -3.37042e-03  &  -3.36900e-03 & -3.37046e-03 \\
\hline   
\end{tabular}
\end{center} 
                                                     
As a final test the deuteron binding energy and wave functions
using the two different momentum space potentials are compared
to each other and to the same quantities using a direct calculation
based on the configuration space potential in partial waves.

We solve for the deuteron binding energy and the $s$ and $d$ wave
functions using direct integration of the vector variables.  The
method of solution, which is discussed in \cite{charlotte}, uses the
expansion (\ref{a.17}) directly without using partial waves.  Calculations
are performed for both momentum space potentials.  

For comparison, the wave functions are represented by an expansion in
70 configuration space basis using the configuration space basis
functions (\ref{b.1}).  Matrix elements of the partial wave projection of
the Hamiltonian in this basis are directly computed and the eigenvalue
problem is solved.  The solution of the eigenvalue problem gives an
independent evaluation of both the binding energy and wave functions
that do not require Fourier transforms of the potential.
 
The Deuteron binding energy obtained by both Fourier transform methods
was $e=-2.24225MeV$ compared with $e=-2.24221MeV$ using a partial wave
calculation that directly uses the configuration space version of the
Argonne V18 potential.  These eigenvalues differ slightly from the
eigenvalues obtained including the electromagnetic corrections of
$+17.6$ keV \cite{wiringa}.  If this correction is added to the
numbers above the binding energy $e=-2.2246MeV$, is obtained.

The $s$ and $d$ wave functions for all three calculations are shown in
figures 49 and 50.  The solid lines represent the wave functions
computed using the Chebyshev expansion of the Fourier transformed
potential, the long dashes represent the partial wave wave-functions
and the short dashes represent the wave functions computed using the
series expansion of the Fourier transformed potential.  Explicit
values of both wave functions in all three calculations are compared
table 5. The wave functions differ in the fourth significant figure
for momenta less that 5 $fm^{-1}$, while binding energies of all three
calculations differ in the seventh significant figure.  This level of
accuracy should be adequate for most applications.

The programs to compute the potentials $\tilde{V}_m(q)$ are freely
available as supplementary material to the electronic version of this
article.

This work supported by the U.S. Department of Energy,  
contract \# DE-FG02-86ER40286.

\section{Appendix} In this appendix we calculate the Fourier transform 
of the operators that appear (\ref{a.9}-\ref{a.12}) in V18.

{\bf \noindent $\mathbf{L} \cdot \mathbf{S}$:} 
Let $\mathbf{q}= \mathbf{k}-\mathbf{k}'$ 

\[
{1 \over (2 \pi)^3} \int e^{-i \mathbf{k}'\cdot \mathbf{r}}
V_j(r) \mathbf{L} \cdot \mathbf{S}
e^{i \mathbf{k}\cdot \mathbf{r}} d\mathbf{r} =
{1 \over (2 \pi)^3} \int e^{-i \mathbf{k}'\cdot \mathbf{r}}
V_j(r) \mathbf{S} \cdot (\mathbf{r}\times \mathbf{p}) 
e^{i \mathbf{k}\cdot \mathbf{r}} d\mathbf{r} =
\]
\[
{1 \over (2 \pi)^3} \int e^{-i \mathbf{k}'\cdot \mathbf{r}}
V_j(r) \mathbf{S} \cdot (\mathbf{r}\times \mathbf{k}) 
e^{i \mathbf{k}\cdot \mathbf{r}} d\mathbf{r} =
{1 \over (2 \pi)^3} \int e^{i \mathbf{q}\cdot \mathbf{r}}
V_j(r) \mathbf{S} \cdot (\mathbf{r}\times \mathbf{k}) 
d\mathbf{r} =
\]
\beq
{4 \pi \over (2 \pi)^3} \int
\sum_{l=0}^\infty \sum_{m=-l}^l
i^l j_l (qr) Y_{lm} (\hat{\mathbf{q}})Y^*_{lm} (\hat{\mathbf{r}}) 
V_j(r) \mathbf{S} \cdot (\mathbf{r}\times \mathbf{k}) 
d\mathbf{r}.
\label{ap.1}
\eeq
Since $\mathbf{r}$ can be expanded as a linear combination of
$Y_{1m}(\hat{\mathbf{r}})$ the only terms that survive are the $l=1$
terms.  The integral over angles and the spherical harmonics simply
replace $\hat{\mathbf{r}}$ by $\hat{\mathbf{q}}$, giving
\[
= {4\pi i \over (2 \pi)^3}
\int_{0}^\infty
j_1 (qr)  
V_j(r) \mathbf{S} \cdot (\mathbf{q}\times \mathbf{k}) 
r^3 dr =
\mathbf{S} \cdot (\mathbf{k}\times \mathbf{k}')
\times [
{4\pi i \over q(2 \pi)^3}
\int_{0}^\infty
j_1 (qr)  
V_j(r)  
r^3 dr ]=
\]
\beq
\mathbf{S} \cdot (\mathbf{k}\times \mathbf{k}')
\times [
{ i \over 2 \pi^2 q}
\int_{0}^\infty
j_1 (qr)  
V_j(r)  
r^3 dr ] 
\label{ap.2}
\eeq
Thus we get
\beq
{1 \over (2 \pi)^3} \int e^{-i \mathbf{k}'\cdot \mathbf{r}}
V_j(r) \mathbf{L} \cdot \mathbf{S}
e^{i \mathbf{k}\cdot \mathbf{r}} d\mathbf{r} =
i \mathbf{S} \cdot (\mathbf{k}\times \mathbf{k}') 
I_1(q)
\label{ap.3}
\eeq
where 
\beq
I_1 (q) := { 1 \over 2 \pi^2 q}
\int_{0}^\infty j_1 (qr) V_j(r)  
r^3 dr  
\label{ap.4}
\eeq

The following relations are useful for treating the remaining three
operators:

\beq
\pmb{\nabla}_q f(q) = f'(q) {\mathbf{q} \over q}
\eeq
\beq
\nabla_q^2 f(q)  =
f''(q) 
+ f'(q) {3 \over q}
- f'(q) {1 \over q} =
f''(q) + {2 \over q} f'(q) 
\eeq
\beq
(\mathbf{a} \cdot \pmb{\nabla}_q)
(\mathbf{b} \cdot \pmb{\nabla}_q) f(q) = 
f''(q) 
{\mathbf{a} \cdot \mathbf{q} \over q}{\mathbf{b} \cdot \mathbf{q} \over q}
+ f'(q) {\mathbf{a} \cdot \mathbf{b} \over q}
- f'(q) {(\mathbf{a} \cdot \mathbf{q}) (\mathbf{b} \cdot \mathbf{q}) \over q^3}=
\eeq
\beq
{\mathbf{a} \cdot \mathbf{q} \over q}{\mathbf{b} \cdot \mathbf{q} \over q}(
(f''(q) - {f'(q) \over q} ) + {\mathbf{a} \cdot \mathbf{b} \over q} f'(q)  
\eeq

{\bf \noindent $\mathbf{L} \cdot \mathbf{L}$:} Let $\mathbf{q}= \mathbf{k}-\mathbf{k}'$

\[
{1 \over (2 \pi)^3} \int e^{-i \mathbf{k}'\cdot \mathbf{r}}
V_j(r) (\mathbf{r}\times \mathbf{p}) \cdot (\mathbf{r}\times \mathbf{p})
e^{i \mathbf{k}\cdot \mathbf{r}} d\mathbf{r} =
{1 \over (2 \pi)^3} \int e^{-i \mathbf{k}'\cdot \mathbf{r}}
V_j(r) (\mathbf{r}\times \mathbf{k}') \cdot (\mathbf{r}\times \mathbf{k})
e^{i \mathbf{k}\cdot \mathbf{r}} d\mathbf{r} =
\]
\beq
(-i\pmb{\nabla}_{q}\times \mathbf{k}') 
\cdot (-i\pmb{\nabla}_q\times \mathbf{k})
{1 \over (2 \pi)^3} \int V_j(r) 
e^{i \mathbf{q}\cdot \mathbf{r}} d\mathbf{r} =
-(\pmb{\nabla}_{q}\times \mathbf{k}') 
\cdot (\pmb{\nabla}_{q}\times \mathbf{k})
{4\pi \over (2 \pi)^3} \int_0^\infty V_j(r) 
j_0 (qr) r^2 dr .
\eeq
To compute the derivatives
note
\[
(\pmb{\nabla}_{q}\times \mathbf{k}') 
\cdot (\pmb{\nabla}_{q}\times \mathbf{k}) =
\]
\[
(\mathbf{k} \cdot \mathbf{k}')(\pmb{\nabla}_{q}\cdot \pmb{\nabla}_{q})
-
(\mathbf{k} \cdot \pmb{\nabla}_{q} )
(\mathbf{k} \cdot \pmb{\nabla}_{q} )
\]

Using this in the above gives 
\beq
-(\pmb{\nabla}_{q}\times \mathbf{k}') 
\cdot (\pmb{\nabla}_{q}\times \mathbf{k})
{4\pi \over (2 \pi)^3} \int_0^\infty V_j(r) 
j_0 (qr) r^2 dr = 
- \left ((\mathbf{k}'\cdot \mathbf{k})\nabla^2_q
- (\mathbf{k}'\cdot \pmb{\nabla}_{q})(
\mathbf{k}\cdot \pmb{\nabla}_{q}) \right )I_0 (q)
\eeq
where
\beq
I_0(q)= {4\pi \over (2 \pi)^3} \int_0^\infty V_j(r) 
j_0 (qr) r^2 dr =
{1 \over 2 \pi^2} \int_0^\infty V_j(r) j_0 (qr) r^2 dr
\eeq
Evaluating this gives
\[
- \left ((\mathbf{k}'\cdot \mathbf{k})\nabla^2_q
- (\mathbf{k}'\cdot \pmb{\nabla}_{q})(
\mathbf{k}\cdot \pmb{\nabla}_{q}) \right )I_0 (q)
=
\]
\[ 
- (\mathbf{k}'\cdot \mathbf{k})(I_0''(q) + {2 \over q}I_0'(q)) 
+ 
I_0'(q) ({\mathbf{k}'\cdot \mathbf{k}\over q}-
{(\mathbf{k}'\cdot \mathbf{q})
 (\mathbf{k} \cdot \mathbf{q})\over q^3})
+ I_0''(q) 
{(\mathbf{k}'\cdot \mathbf{q})
 (\mathbf{k} \cdot \mathbf{q})\over q^2} = 
\]
\beq
-(\mathbf{k}'\cdot \mathbf{k})(I_0''(q) + {1 \over q}I_0'(q)) +
{(\mathbf{k}'\cdot \mathbf{q})(\mathbf{k}\cdot \mathbf{q})\over q^2}
(I_0''(q)- {1 \over q}I_0'(q))
\eeq

To eliminate the derivatives use 
\[
I_0''(q)-  {1 \over q}I_0'(q) =
{1 \over 2 \pi^2} \int_0^{\infty} V(r) (j_0'' (qr) r^2 - j_0' (qr) {r \over q} ) r^2 dr =
\]
\[
{1 \over 2 \pi^2} \int_0^{\infty} V(r) (j_0'' (qr) - j_0' (qr) {1\over qr} ) r^4 dr =
{1 \over 2 \pi^2} \int_0^{\infty} V(r) j_2 (qr) r^4 dr = I_2(q) 
\]
\[
I_0''(q)+  {1 \over q}I_0'(q) = 
{1 \over 2 \pi^2} \int_0^{\infty} V(r) (j_0'' (qr) - j_0' (qr) {1\over qr}
+ 2 j_0' (qr) {1\over qr} 
 ) r^4 dr =
\]
\[
{1 \over 2 \pi^2} \int_0^{\infty} V(r) j_2'' (qr) r^4 dr
- {1 \over 2 \pi^2} {2\over q}  \int_0^{\infty} V(r) j_1 (qr) r^3 dr 
=
I_2(q) - {2\over q}I_1(q)
\]

This gives
\beq
{1 \over (2 \pi)^3} \int e^{-i \mathbf{k}'\cdot \mathbf{r}}
V_j(r) (\mathbf{r}\times \mathbf{p}) \cdot (\mathbf{r}\times \mathbf{p})
e^{i \mathbf{k}\cdot \mathbf{r}} d\mathbf{r} =
-(\mathbf{k}'\cdot \mathbf{k})(I_2(q) - {2\over q}I_1(q)) +
{(\mathbf{k}'\cdot \mathbf{q})( \mathbf{k}\cdot \mathbf{q} )\over q^2}
I_2(q)
\eeq
which can be reexpressed in terms of cross products
\beq
{1 \over (2 \pi)^3} \int e^{-i \mathbf{k}'\cdot \mathbf{r}}
V_j(r) (\mathbf{r}\times \mathbf{p}) \cdot (\mathbf{r}\times \mathbf{p})
e^{i \mathbf{k}\cdot \mathbf{r}} d\mathbf{r} =
- I_2(q) {(\mathbf{k}'\times \mathbf{k}) \cdot 
(\mathbf{k}'\times \mathbf{k}) \over q^2} + {2\over q}
(\mathbf{k}'\cdot \mathbf{k}) I_1(q) 
\eeq


{\noindent \bf $(\mathbf{L} \cdot \mathbf{S})^2$:} 
\[
{1\over (2\pi)^3} 
\int e^{-i(\mathbf{k}'-\mathbf{k})\cdot \mathbf{r}} V_j (r) 
(\mathbf{L} \cdot \mathbf{S})^2 d \mathbf{r} =
{1\over (2\pi)^3} 
\int e^{-i(\mathbf{k}'-\mathbf{k})\cdot \mathbf{r}} V_j (r) 
(\mathbf{S} \cdot (\mathbf{r} \times \mathbf{p}))^2 d \mathbf{r} =
\]
\[
-{4\pi\over (2\pi)^3} 
(( \mathbf{k}'\times \mathbf{S})
\cdot \pmb{\nabla}_q)
(( \mathbf{k}\times \mathbf{S})
\cdot \pmb{\nabla}_q)
\int j_0(qr) V_j (r) r^2 dr =
\]
\[
- (( \mathbf{k}'\times \mathbf{S})
\cdot \pmb{\nabla}_q)
(( \mathbf{k}\times \mathbf{S})
\cdot \pmb{\nabla}_q) I_0 (q) =
\]
\[
- (( \mathbf{k}'\times \mathbf{S})\cdot \mathbf{q}
(( \mathbf{k}\times \mathbf{S})\cdot \mathbf{q} ))
{1 \over q^2} (I_0'' (q) - {1 \over q} I'_0 (q))
- 
( \mathbf{k}'\times \mathbf{S})\cdot
( \mathbf{k}\times \mathbf{S}){1 \over q} I'_0 (q) =
\]

\beq
- (( \mathbf{k}'\times \mathbf{S})\cdot \mathbf{q}
(( \mathbf{k}\times \mathbf{S})\cdot \mathbf{q} ))=
{1 \over q^2} I_2 (q)
+ 
( \mathbf{k}'\times \mathbf{S})\cdot
( \mathbf{k}\times \mathbf{S}){1 \over q}  I_1 (q)
\eeq
which gives

\beq
- ((\mathbf{S} \cdot ( \mathbf{k}\times \mathbf{k}'))^2 
{1 \over q^2} I_2 (q)
+ 
( \mathbf{k}'\times \mathbf{S})\cdot
( \mathbf{k}\times \mathbf{S}){1 \over q}  I_1 (q)
\eeq

or
\beq
{1\over (2\pi)^2} 
\int e^{-i(\mathbf{k}'-\mathbf{k})} V_j (r) 
(\mathbf{L} \cdot \mathbf{S})^2 d \mathbf{r} =
- ((\mathbf{S} \cdot ( \mathbf{k}\times \mathbf{k}'))^2
{1 \over q^2} I_2 (q)
+ 
( \mathbf{k}'\times \mathbf{S})\cdot
( \mathbf{k}\times \mathbf{S}){1 \over q}  I_1 (q)
\eeq

{\noindent \bf $ \left (\hat{\mathbf{r}} \cdot \pmb{\sigma}_1)
(\hat{\mathbf{r}} \cdot \pmb{\sigma}_2) 
- {1 \over 3} \pmb{\sigma}_1\cdot \pmb{\sigma}_2 \right )$:}

\[
{1 \over (2 \pi)^3} \int e^{-i \mathbf{k}'\cdot \mathbf{r}}
V(r) \left ( (\hat{\mathbf{r}} \cdot \pmb{\sigma}_1)
(\hat{\mathbf{r}} \cdot \pmb{\sigma}_2) 
- {1 \over 3} \pmb{\sigma}_1\cdot \pmb{\sigma}_2 \right )
e^{i \mathbf{k}\cdot \mathbf{r}} d\mathbf{r} =
\]
\[
-\left ( (\pmb{\nabla}_q \cdot \pmb{\sigma}_1)
(\pmb{\nabla}_q \cdot \pmb{\sigma}_2) 
- {1 \over 3} \pmb{\sigma}_1\cdot \pmb{\sigma}_2 \pmb{\nabla}_q^2 \right )
{4\pi \over (2 \pi)^3} \int
V(r){r^2 \over r^2} j_0(qr)  dr =
\]
\[
-{\pmb{\sigma}_1 \cdot \mathbf{q} \over q}
{\pmb{\sigma}_2 \cdot \mathbf{q} \over q}(
(I_{0-}''(q) - {I_{0-}'(q) \over q} ) - 
{\mathbf{s}_1 \cdot \mathbf{s}_2 \over q} I_0'(q) =  
\]
\beq
+{1 \over 3} \pmb{\sigma}_1\cdot \pmb{\sigma}_2 
(I''_{0-}(q) + {2 \over q} I_{0-}'(q))
\eeq
Thus,
\beq
{1 \over (2 \pi)^3} \int e^{-i \mathbf{k}'\cdot \mathbf{r}}
V(r) \left ( (\hat{\mathbf{r}} \cdot \pmb{\sigma}_1)
(\hat{\mathbf{r}} \cdot \pmb{\sigma}_2) 
- {1 \over 3} \pmb{\sigma}_1\cdot \pmb{\sigma}_2 \right )
e^{i \mathbf{k}\cdot \mathbf{r}} d\mathbf{r} =
\left ( 
-{\pmb{\sigma}_1 \cdot \mathbf{q} \over q}
{\pmb{\sigma}_2 \cdot \mathbf{q} \over q} 
+ 
{\pmb{\sigma}_1 \cdot \pmb{\sigma}_2 \over 3}
\right ) 
I_{2-}(q)
\eeq
where 
\beq
I_{0-}(q) = {4\pi \over (2 \pi)^3} \int
V(r) j_2(qr) r^2 dr 
\eeq

\vfill\eject

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig1.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 1]{\label{labelFig1} 
Calculated $V_1(r)$ compared to exact.
}
\end{center}
\label{fig.1}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig2.eps}
\caption[Short caption for figure 2]{\label{labelFig2}
Calculated $V_2(r)$ compared to exact.
}
\end{center}
\end{minipage}
\label{fig.2}
\end{figure}

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig3.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 3]{\label{labelFig3} 
Calculated $V_3(r)$ compared to exact.
}
\end{center}
\label{fig.3}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig4.eps}
\caption[Short caption for figure 4]{\label{labelFig4}
Calculated $V_4(r)$ compared to exact.
}
\end{center}
\end{minipage}
\label{fig.4}
\end{figure}

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig5.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 5]{\label{labelFig5} 
Calculated $V_5(r)$ compared to exact.
}
\end{center}
\label{fig.5}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig6.eps}
\caption[Short caption for figure 6]{\label{labelFig6}
Calculated $V_6(r)$ compared to exact.
}
\end{center}
\end{minipage}
\label{fig.6}
\end{figure}

\vfill\eject

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig7.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 7]{\label{labelFig7} 
Calculated $V_7(r)$ compared to exact.
}
\end{center}
\label{fig.7}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig8.eps}
\caption[Short caption for figure 8]{\label{labelFig8}
Calculated $V_8(r)$ compared to exact.
}
\end{center}
\end{minipage}
\label{fig.8}
\end{figure}

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig9.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 9]{\label{labelFig9} 
Calculated $V_9(r)$ compared to exact.
}
\end{center}
\label{fig.9}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig10.eps}
\caption[Short caption for figure 10]{\label{labelFig10}
Calculated $V_{10}(r)$ compared to exact.
}
\end{center}
\end{minipage}
\label{fig.10}
\end{figure}

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig11.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 11]{\label{labelFig11} 
Calculated $V_{11}(r)$ compared to exact.
}
\end{center}
\label{fig.11}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig12.eps}
\caption[Short caption for figure 12]{\label{labelFig12}
Calculated $V_{12}(r)$ compared to exact.
}
\end{center}
\end{minipage}
\label{fig.12}
\end{figure}

\vfill\eject

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig13.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 13]{\label{labelFig13} 
Calculated $V_{13}(r)$ compared to exact.
}
\end{center}
\label{fig.13}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig14.eps}
\caption[Short caption for figure 14]{\label{labelFig14}
Calculated $V_{14}(r)$ compared to exact.
}
\end{center}
\end{minipage}
\label{fig.14}
\end{figure}

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig15.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 15]{\label{labelFig15} 
Calculated $V_{15}(r)$ compared to exact.
}
\end{center}
\label{fig.15}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig16.eps}
\caption[Short caption for figure 16]{\label{labelFig16}
Calculated $V_{16}(r)$ compared to exact.
}
\end{center}
\end{minipage}
\label{fig.16}
\end{figure}

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig17.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 18]{\label{labelFig17} 
Calculated $V_{18}(r)$ compared to exact.
}
\end{center}
\label{fig.17}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig18.eps}
\caption[Short caption for figure 18]{\label{labelFig18}
Calculated $V_{18}(r)$ compared to exact.
}
\end{center}
\end{minipage}
\label{fig.18}
\end{figure}

\vfill\eject

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig19.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 19]{\label{labelFig19} 
Calculated $r^2 V_{9}(r)$ compared to exact.
}
\end{center}
\label{fig.19}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig20.eps}
\caption[Short caption for figure 20]{\label{labelFig20}
Calculated $r^2V_{10}(r)$ compared to exact.
}
\end{center}
\end{minipage}
\label{fig.20}
\end{figure}


\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig21.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 21]{\label{labelFig21} 
Calculated $r^2 V_{11}(r)$ compared to exact.
}
\end{center}
\label{fig.21}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig22.eps}
\caption[Short caption for figure 22]{\label{labelFig22}
Calculated $r^2V_{12}(r)$ compared to exact.
}
\end{center}
\end{minipage}
\label{fig.22}
\end{figure}

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig23.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 23]{\label{labelFig23} 
Calculated $r^2 V_{13}(r)$ compared to exact.
}
\end{center}
\label{fig.23}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig24.eps}
\caption[Short caption for figure 24]{\label{labelFig24}
Calculated $r^2V_{14}(r)$ compared to exact.
}
\end{center}
\end{minipage}
\label{fig.24}
\end{figure}

\vfill\eject

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig25.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 25]{\label{labelFig25} 
Calculated $V_1(k)$ compared to Chebyshev.
}
\end{center}
\label{fig.25}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig26.eps}
\caption[Short caption for figure 26]{\label{labelFig26}
Calculated $V_2(k)$ compared to Chebyshev.
}
\end{center}
\end{minipage}
\label{fig.26}
\end{figure}

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig27.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 27]{\label{labelFig27} 
Calculated $V_3(k)$ compared to Chebyshev.
}
\end{center}
\label{fig.27}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig28.eps}
\caption[Short caption for figure 28]{\label{labelFig28}
Calculated $V_4(k)$ compared to Chebyshev.
}
\end{center}
\end{minipage}
\label{fig.28}
\end{figure}

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig29.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 29]{\label{labelFig29} 
Calculated $V_5(k)$ compared to Chebyshev.
}
\end{center}
\label{fig.29}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig30.eps}
\caption[Short caption for figure 30]{\label{labelFig30}
Calculated $V_6(k)$ compared to Chebyshev.
}
\end{center}
\end{minipage}
\label{fig.30}
\end{figure}

\vfill\eject

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig31.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 31]{\label{labelFig31} 
Calculated $V_7(k)$ compared to Chebyshev.
}
\end{center}
\label{fig.31}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig32.eps}
\caption[Short caption for figure 32]{\label{labelFig32}
Calculated $V_8(k)$ compared to Chebyshev.
}
\end{center}
\end{minipage}
\label{fig.32}
\end{figure}

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig33.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 33]{\label{labelFig33} 
Calculated $V_9(k)$ compared to Chebyshev.
}
\end{center}
\label{fig.33}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig34.eps}
\caption[Short caption for figure 34]{\label{labelFig34}
Calculated $V_{10}(k)$ compared to Chebyshev.
}
\end{center}
\end{minipage}
\label{fig.34}
\end{figure}

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig35.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 35]{\label{labelFig35} 
Calculated $V_{11}(k)$ compared to Chebyshev.
}
\end{center}
\label{fig.35}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig36.eps}
\caption[Short caption for figure 36]{\label{labelFig36}
Calculated $V_{12}(k)$ compared to Chebyshev.
}
\end{center}
\end{minipage}
\label{fig.36}
\end{figure}

\vfill\eject

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig37.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 37]{\label{labelFig37} 
Calculated $V_{13}(k)$ compared to Chebyshev.
}
\end{center}
\label{fig.37}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig38.eps}
\caption[Short caption for figure 38]{\label{labelFig38}
Calculated $V_{14}(k)$ compared to Chebyshev.
}
\end{center}
\end{minipage}
\label{fig.38}
\end{figure}

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig39.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 39]{\label{labelFig39} 
Calculated $V_{15}(k)$ compared to Chebyshev.
}
\end{center}
\label{fig.39}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig40.eps}
\caption[Short caption for figure 40]{\label{labelFig40}
Calculated $V_{16}(k)$ compared to Chebyshev.
}
\end{center}
\end{minipage}
\label{fig.40}
\end{figure}

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig41.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 41]{\label{labelFig41} 
Calculated $V_{18}(k)$ compared to Chebyshev.
}
\end{center}
\label{fig.41}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig42.eps}
\caption[Short caption for figure 42]{\label{labelFig42}
Calculated $V_{18}(k)$ compared to Chebyshev.
}
\end{center}
\end{minipage}
\label{fig.42}
\end{figure}

\vfill\eject

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig43.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 43]{\label{labelFig43} 
Calculated $V_{9b}(k)$ compared to Chebyshev.
}
\end{center}
\label{fig.43}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig44.eps}
\caption[Short caption for figure 44]{\label{labelFig44}
Calculated $V_{10b}(k)$ compared to Chebyshev.
}
\end{center}
\end{minipage}
\label{fig.44}
\end{figure}


\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig45.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 45]{\label{labelFig45} 
Calculated $V_{11b}(k)$ compared to Chebyshev.
}
\end{center}
\label{fig.45}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig46.eps}
\caption[Short caption for figure 46]{\label{labelFig46}
Calculated $V_{12b}(k)$ compared to Chebyshev.
}
\end{center}
\end{minipage}
\label{fig.46}
\end{figure}

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig47.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 47]{\label{labelFig47} 
Calculated $V_{13b}(k)$ compared to Chebyshev.
}
\end{center}
\label{fig.47}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig48.eps}
\caption[Short caption for figure 48]{\label{labelFig48}
Calculated $V_{14b}(k)$ compared to Chebyshev.
}
\end{center}
\end{minipage}
\label{fig.48}
\end{figure}

\vfill\eject

\begin{figure}
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig49.eps}
%\includegraphics[width=6cm,clip,bb=000 250 614 1044]{fig23.eps}
\caption[Short caption for figure 49]{\label{labelFig49} 
Deuteron $d$-state wave function compared to partial wave calculation.
}
\end{center}
\label{fig.49}
\end{minipage}
\hfill
\begin{minipage}[t]{7.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{fig50.eps}
\caption[Short caption for figure 50]{\label{labelFig50}
Deuteron $d$-state wave function compared to partial wave calculation.
}
\end{center}
\end{minipage}
\label{fig.50}
\end{figure}


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\end{thebibliography}

\end{document}
