%% 28 FEB 97: began TeX-ifying the .asc file ftp'd from Robin's site,
%%            using the .tex version for ideas on some of the coding.
%%            Robin's instructions:
%%
%%            What needs doing is
%%
%%            1) to tidy it all up: in particular, write maths where
%%            the paper has maths and the file has various
%%            incomprehensible runes like 
%%
%%               K <D><$E => 2m<D>e<D>/(M <D><$E +> m<D>e<D>
%%
%%            and if you can spot 'em all, remove fantasies like
%%            <$TVJBelow=267;VJInterLn=0> in the start of BODY 1ST
%%            PARA of the intro 
%%
%%            2) convert obvious things to \section and the like
%%            commands 
%%
%%            3) deal with the diacriticised letters in the
%%            r\'esum\'e, which all appear as 8th-bit set M-garbage to
%%            me (your editor might well be better set up).
%%
%%            4) anything else you can think of that makes sense.
%%
%%    UPDATE: coded the whole file; added some dummy codes to make it
%%            easier to replace them with the appropriate variants;
%%            merged all tables into this file as well; e-mailed it
%%            all off to Robin (Ch.)
%%
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%


%%  \documentclass[fleqn]{article}
\documentclass[type1rest,a4paper]{can-jp}
%\usepackage{mathtime}
\usepackage{mathptm}
\usepackage[figuresright]{rotating}
\usepackage{cite}

%% For now: 
%\setlength{\textwidth}{34pc}
%\setlength{\textheight}{50pc}
%\setlength{\mathindent}{0pc}

\let\mc=\multicolumn

\let\gcite=\cite  %% \gcite  = grouped \cite: e.g., [2], [3] --> [2, 3]
\let\nbcite=\citen %% \nbcite = no brackets around cited ref. number
\def\Hline{\hline*}



%%%%%%%%
\setcounter{page}{11}
\volyear{75}[1862]{1997}
\received{Received June 6, 1996. Accepted September 5, 1996}

\begin{document}

\title{Properties of two-electron systems in an electric field}
%
\author{A. K. Bhatia}
  \address{Laboratory for Astronomy and Solar Physics,
           NASA-Goddard Space Flight Center,
           Greenbelt, MD 20771, U.S.A.}
%
\author[R. J. Drachman]{Richard J. Drachman}
\correspond{\phone{+1 (301) 286-4426};
  \fax{+1 (301) 286-1753};
  \email{drachman@stars.gsfc.nasa.gov}}

\shortauthor{Bhatia and Drachman}

\maketitle
%
\begin{abstract}
   We have extended our previous work on the generalized
   polarizabilities of two-electron atoms and ions to include higher
   order processes and ions of atomic number $Z = 2$ to 6 and 10. The
   pseudostate summation method is used again, and the correct finite
   value of the nuclear mass, $M$, for a variety of isotopes is
   included in the Hamiltonian. The results are tabulated, and
   approximate interpolation formulas in $Z$ and $K = 2m_e/(M + m_e)$
   are given.
\end{abstract}
%
\begin{resume}
   Nous poursuivons notre \'etude sur la polarisabilit\'e des atomes
   et ions \`a deux \'electrons, pour y inclure des effets d'ordre
   plus \'elev\'e et l'\'etendre aux ions de nombre atomique $Z = 2$
   \`a 6 et 10. Nous continuons \`a utiliser la m\'ethode de sommation
   sur les pseudo-\'etats. Les valeurs exactes des masses des
   diff\'erents isotopes, $M$, sont fournies au hamiltonien. Nous
   pr\'esentons les r\'esultats de nos calculs ainsi que des formules
   d'interpolation en $Z$ et en $K = 2m_e/(M + m_e)$.

   [Traduit par la r\'edaction]
\end{resume}


%% 1:
\section{Introduction}

There has been considerable interest recently in the calculation of
multipole polarizabilities of two-electron atomic systems
\cite{PipinBishop,BhatiDrachman}, both for their intrinsic
value and for their application to the computation of Rydberg-state
energies \cite{DrachmanBhatia}. In ref. \nbcite{DrachmanBhatia} we
used the pseudostate summation method to evaluate the eight
generalized polarizabilities and nonadiabatic corrections needed to
construct the effective (or optical) potential acting on an excited
electron in lithium, up to order $1/x^8$, where $x$ is the coordinate
of that outer electron. This involved second-, third-, and
fourth-order perturbation calculations and multipole terms up to
octupole. In the calculation, the finite mass of the lithium nucleus
was accounted for in the Hamiltonian. Using this optical potential we
were able to evaluate the electrostatic fine-structure splitting of
excited (high-$L$) three-electron states analytically, and good
agreement with experiment was obtained in the few cases measured
\cite{Rotheryetal}.

We are now reporting a more complete set of these perturbation sums,
some of which have been reported in previous papers
\cite{BhatiDrachman,DrachmanBhatia}. Not all of them are
needed for the optical-potential method of treating Rydberg states,
but they should be useful anyway. As a convenience, we have made an
analytic fit to the results, enabling one to obtain fairly accurate
values for all of the quantities analytically in terms of the mass of
the particular isotope and its atomic number, at least up to $Z = 10$.


%% p132-01t.asc -- Table 1, p.12, full-page landscape:

\begin{sidewaystable}
\topcaption{Generalized polarizability sums for two-electron systems, $Z
   = 2$ to 6 and 10. The nuclear mass is taken to be infinite ($K =
   0$). The quantity [$\alpha_1$] on the second line is the dipole
   polarizability obtained in ref. 5, shown for comparison.}
%
 \begin{center}
\renewcommand{\arraystretch}{1.75}
\begin{tabular}{@{}lllllll@{}}
\Hline 
             & \mc{1}{c}{He}   
             & \mc{1}{c}{Li$^+$}  
             & \mc{1}{c}{Be$^{2+}$}  
             & \mc{1}{c}{B$^{3+}$}  
             & \mc{1}{c}{C$^{4+}$}  
             & \mc{1}{c}{Ne$^{8+}$} \\
\hline
$\alpha_1$   & \(1.383\,192\,18\)
             & \(0.192\,453\,204\)
             & \(5.226\,876\,68\times 10^{-2}\)
             & \(1.964\,429\,21\times 10^{-2}\) 
             & \(8.963\,931\,65\times 10^{-3}\)
             & \(1.044\,790\,80\times 10^{-3}\)\\ 
\llap{[}$\alpha_1$] 
             & \(1.383\,199\,2  \)
             & \(0.192\,453\,63    \)
             & \(5.226\,881\,4\times 10^{-2}\)
             &               
             & \(8.963\,935\,8\times 10^{-3}\)  
             & \(1.044\,790\,9\times 10^{-3}\) \\ 
$\beta_1$    & \(0.707\,510\,151   \)
             & \(3.528\,083\,04\times 10^{-2}\)
             & \(4.918\,454\,80 \times 10^{-3}\)
             & \(1.124\,263\,60 \times 10^{-3}\)
             & \(3.448\,575\,71 \times 10^{-4}\)
             & \(1.360\,105\,96\times 10^{-5}\)\\ 
$\gamma_1$   & \(0.385\,534\,905 \)
             & \(6.805\,326\,25\times 10^{-3}\)
             & \(4.845\,613\,22 \times 10^{-4}\)
             & \(6.718\,553\,37\times 10^{-5}\)
             & \(1.383\,049\,23 \times 10^{-5}\)
             & \(1.840\,135\,10 \times 10^{-7}\) \\ 
$\delta_1$   & \(0.218\,732\,917 \)
             & \(1.35\,607\,948 \times 10^{-3}\)
             & \(4.915\,642\,08 \times 10^{-5}\)
             & \(4.126\,920\,11 \times 10^{-6}\)
             & \(5.695\,039\,25 \times 10^{-7}\)
             & \(2.550\,907\,81 \times 10^{-9}\) \\ 
$\epsilon_1$ & \(0.127\,536\,701 \)
             & \(2.762\,358\,84 \times 10^{-4}\)
             & \(5.086\,126\,71 \times 10^{-6}\)
             & \(2.582\,330\,59 \times 10^{-7}\)
             & \(2.386\,993\,27 \times 10^{-8}\)
             & \(3.594\,185\,28 \times 10^{-11}\) \\ 
$\alpha_2$   & \(2.445\,083\,33 \)
             & \(0.113\,887\,296 \)
             & \(1.532\,035\,44 \times 10^{-2}\)  
             & \(3.426\,631\,04 \times 10^{-3}\)  
             & \(1.035\,861\,05 \times 10^{-3}\)  
             & \(3.967\,225\,91 \times 10^{-5}\) \\ 
$\beta_2$    & \(1.036\,096\,31\)
             & \(1.667\,862\,59 \times 10^{-2}\)  
             & \(1.131\,688\,59 \times 10^{-3}\)  
             & \(1.523\,658\,62 \times 10^{-4}\)  
             & \(3.075\,204\,46 \times 10^{-5}\)  
             & \(3.932\,082\,22 \times 10^{-7}\) \\ 
$\gamma_2$   & \(0.467\,619\,507 \)
             & \(2.583\,854\,06 \times 10^{-3}\)  
             & \(8.818\,478\,35 \times 10^{-5}\)  
             & \(7.135\,934\,94 \times 10^{-5}\)  
             & \(9.606\,564\,60 \times 10^{-7}\)  
             & \(4.093\,396\,79 \times 10^{-9}\) \\ 
$\alpha_3$   & \(10.620\,344\,9 \)
             & \(0.168\,429\,083 \)
             & \(1.134\,902\,76 \times 10^{-2}\)  
             & \(1.521\,607\,35 \times 10^{-3}\)  
             & \(3.062\,436\,29 \times 10^{-4}\)
             & \(3.893\,400\,36 \times 10^{-6}\) \\ 
$\beta_3$    & \(4.072\,133\,69 \)
             & \(2.199\,545\,00 \times 10^{-2}\)  
             & \(7.429\,394\,46 \times 10^{-4}\)  
             & \(5.975\,072\,33 \times 10^{-5}\)  
             & \(8.010\,882\,86 \times 10^{-6}\)  
             & \(3.385\,421\,93 \times 10^{-8}\) \\ 
$\gamma_3$   & \(1.656\,760\,66 \)
             & \(3.033\,495\,12 \times 10^{-3}\)  
             & \(5.126\,229\,92 \times 10^{-5}\)  
             & \(2.470\,380\,99 \times 10^{-6}\)  
             & \(2.204\,870\,32 \times 10^{-7}\)  
             & \(3.093\,355\,27 \times 10^{-10}\) \\ 
$\delta$     & \(7.323\,852\,32 \)
             & \(0.121\,392\,360 \)
             & \(8.393\,398\,13 \times 10^{-3}\)  
             & \(1.144\,353\,53 \times 10^{-3}\)  
             & \(2.330\,246\,76 \times 10^{-4}\)  
             & \(3.036\,252\,95 \times 10^{-6}\) \\ 
$\epsilon$   & \(4.567\,363\,66 \)
             & \(2.702\,883\,54 \times 10^{-2}\)  
             & \(9.633\,777\,79 \times 10^{-4}\)  
             & \(8.020\,559\,63 \times 10^{-5}\)  
             & \(1.101\,362\,28 \times 10^{-5}\)  
             & \(4.891\,561\,80 \times 10^{-8}\) \\
\Hline
\end{tabular}
 \end{center}
\end{sidewaystable}


%% p132-02t.asc -- Table 2, p.13, top of page, portrait: 

\begin{table}
\topcaption{Parameters of the empirical fitting formula 
   $\sigma (Z) = \sigma_{\mbox{o}} + {A \over Z} 
                                   + {B \over Z^2}
                                   + {C \over Z^4}$, where the
   effective charge $\overline Z = Z - \sigma(Z)$.}
%
\begin{center}
\begin{tabular}{@{}lllllll@{}}
\Hline 
             & \mc{1}{c}{$\sigma_{\mbox{o}}$}
             & \mc{1}{c}{A}  
             & \mc{1}{c}{B}  
             & \mc{1}{c}{C} \\ 
\hline
$\alpha_1$   & 0.359\,403 
             & 0.063\,19 
             & 0.034\,70 
             & 0.051\,16 \\ 
$\beta_1$    & 0.376\,898 
             & 0.071\,79 
             & 0.040\,02 
             & 0.054\,61 \\ 
$\gamma_1$   & 0.390\,222 
             & 0.077\,97 
             & 0.042\,86 
             & 0.054\,90 \\ 
$\delta_1$   & 0.400\,727 
             & 0.082\,57 
             & 0.044\,24 
             & 0.054\,01 \\ 
$\epsilon_1$ & 0.409\,15 
             & 0.087 
             & 0.040 
             & 0.061 \\ 
$\alpha_2$   & 0.450\,589 
             & 0.042\,33 
             & 0.025\,05 
             & 0.052\,55 \\ 
$\beta_2$    & 0.465\,169 
             & 0.046\,77 
             & 0.026\,40 
             & 0.055\,45 \\ 
$\gamma_2$   & 0.476\,311 
             & 0.049\,33 
             & 0.027\,27 
             & 0.057\,22 \\ 
$\alpha_3$   & 0.476\,758 
             & 0.037\,32 
             & 0.031\,36 
             & 0.056\,37 \\ 
$\beta_3$    & 0.489\,311 
             & 0.039\,85 
             & 0.032\,07 
             & 0.058\,51 \\ 
$\gamma_3$   & 0.499\,042 
             & 0.041\,59 
             & 0.032\,25 
             & 0.060\,15 \\ 
$\delta$     & 0.425\,135 
             & 0.079\,86 
             & 0.032\,82 
             & 0.045\,36 \\ 
$\epsilon$   & 0.408\,959 
             & 0.093\,05 
             & 0.051\,49 
             & 0.042\,80 \\
\Hline
\end{tabular}
\end{center}
\end{table}


%% 2:
\section{Definitions and calculation method}

We consider a system consisting of a nucleus of atomic number $Z$ and
mass $M$ (in units of the electron mass) and two electrons, all in the
field of a unit positive point charge fixed at the origin of
coordinates. In addition to the Hamiltonian involving the three
particles written in terms of the two vectors $\rho_1$ and $\rho_2$
and a term involving the motion of the center of mass $x$ in the
external field, there is a perturbation due to the differential
interaction of the external field with the three particles. When
expanded in a multipole series it has the form
%
%% Eq 1:
\begin{equation}
   V \sim \sum_{j = 1}^\infty\ {Q_j \over {x^{j + 1}}}
\end{equation}
%
The first three terms of this series have the following forms:
%
%% Eq 2 (first line not number):
\begin{equation}
   Q_1 = 2w\left[{2 + (Z - 1) K
                  \over 
                  2 + K}\right]P_1 (\widehat w \cdot \widehat x),
%
  \quad
%
   Q_2 = 2\left[\rho_1^2 P_2(\widehat\rho_1 \cdot \widehat x) 
                 + \rho_2^2 P_2(\widehat\rho_2 \cdot \widehat x) 
                 - {4K + ZK^2
                   \over 
                    (2 + K)^2}w^2 P_2(\widehat x \cdot \widehat x)\right] 
\end{equation}
%
%% Just a kludge, to get the eqn. nos. back on track:
\setcounter{equation}{1}
%% Eq 2 (second line numbered):
\begin{equation}
   Q_3 = 2\left\{{2 - K \over 2 + K} 
                 [\rho_1^3 P_3(\widehat\rho_1 \cdot \widehat x)  
                  + \rho_2^3 P_3 (\widehat\rho_2 \cdot \widehat x)] 
                 - {4K - 2K^2 - ZK^3
                    \over 
                    (2 + K)^3}w^3 P_3(\widehat w \cdot \widehat x)\right\}
\end{equation}
%
where $w = \rho_1 + \rho_2$ and $K = 2/(M + 1)$. Reduced rydberg
energy units, $R = (1 - K/2)R_\infty$, are being used. All this is
described more fully in ref. \nbcite{DrachmanBhatia}.


%% p132-a3t.asc -- Table 3 (first part), p.14, full-page landscape:

\newlength\expfive
\settowidth\expfive{$(\times10^5)$}%

\begin{sidewaystable}
%
\tabcolsep2pt
 \begin{center}
\topcaption{Generalized polarizability sums for two-electron systems of
   various masses ($K \ne 0), Z = 2$ to 6 and 10.}
%
\renewcommand{\arraystretch}{1.75}%
\begin{tabular}{@{}llllllllllll@{}}
\Hline 
             && \mc{1}{c}{$^3$He}
             & \mc{1}{c}{$^4$He} 
             && \mc{1}{c}{$^6$Li} 
             & \mc{1}{c}{$^7$Li} 
             && \mc{1}{c}{$^9$Be} 
             && \mc{1}{c}{$^{10}$B} 
             & \mc{1}{c}{$^{11}$B} \\ 
\cline{1-4} \cline{6-7} \cline{9-9} \cline{11-12}
\mc{1}{@{}p{\expfive}}{$K$ \newline
                    $(\times 10^5)$}
             && \mc{1}{c}{36.38427}
             & \mc{1}{c}{27.41493}
             && \mc{1}{c}{18.24336}
             & \mc{1}{c}{15.64041}
             && \mc{1}{c}{12.17642}
             && \mc{1}{c}{10.95983}
             & \mc{1}{c}{9.96774} \\ 
\hline 
$\alpha_1$   && 1.38325698 
             & 1.38324101 
             && 0.192490759 
             & 0.192485401 
             && \(5.22817988 \times 10^{-2}\)  
             && \(1.96508291 \times 10^{-2}\)  
             & \(1.96502375 \times 10^{-2}\)  \\ 
$\beta_1$    && 0.707525195 
             & 0.707521493 
             && \(3.52868769 \times 10^{-2}\)  
             & \(3.52860143 \times 10^{-2}\)  
             && \(4.91958958 \times 10^{-3}\)  
             && \(1.12461743 \times 10^{-3}\)  
             & \(1.12458540 \times 10^{-3}\)  \\ 
$\gamma_1$   && 0.385539496 
             & 0.385538368 
             && \(6.80637618 \times 10^{-3}\)  
             & \(6.80622640 \times 10^{-3}\)  
             && \(4.84666014 \times 10^{-4}\)  
             && \(6.72056877 \times 10^{-5}\)  
             & \(6.72038637 \times 10^{-5}\)  \\ 
$\delta_1$   && 0.218735720 
             & 0.218735026 
             && \(1.35627173 \times 10^{-3}\)  
             & \(1.35624427 \times 10^{-3}\)  
             && \(4.91664578 \times 10^{-5}\)  
             && \(4.12810700 \times 10^{-6}\)  
             & \(4.12799959 \times 10^{-6}\)  \\ 
$\epsilon_1$ && 0.127539290 
             & 0.127538649 
             && \(2.76272482 \times 10^{-4}\)  
             & \(2.76267240 \times 10^{-4}\)  
             && \(5.08711500 \times 10^{-6}\)  
             && \(2.58304573 \times 10^{-7}\)  
             & \(2.58298101 \times 10^{-7}\)  \\ 
$\alpha_2$   && 2.44281327 
             & 2.44337262 
             && 0.113825923 
             & 0.113834677 
             && \(1.53144514 \times 10^{-2}\)  
             && \(2.42539400 \times 10^{-3}\)  
             & \(3.42550584 \times 10^{-3}\)  \\ 
$\beta_2$    && 1.03522605 
             & 1.03544052 
             && \(1.66703257 \times 10^{-2}\)  
             & \(1.66715097 \times 10^{-2}\)  
             && \(1.13128217 \times 10^{-3}\)  
             && \(1.52314312 \times 10^{-4}\)  
             & \(1.52318977 \times 10^{-4}\)  \\ 
$\gamma_2$   && 0.467255468 
             & 0.467345191 
             && \(2.58263815 \times 10^{-3}\)  
             & \(2.58281163 \times 10^{-3}\)  
             && \(8.81546026 \times 10^{-5}\)  
             && \(7.13362419 \times 10^{-6}\)
             & \(7.13383331 \times 10^{-6}\)  \\ 
$\alpha_3$   && 10.6112737 
             & 10.6135092 
             && 0.168351213 
             & 0.168362322 
             && \(1.13493435 \times 10^{-2}\)  
             && \(1.52116309 \times 10^{-3}\)  
             & \(1.52120329 \times 10^{-3}\)  \\ 
$\beta_3$    && 4.06906440 
             & 4.06982082 
             && \(2.19861586 \times 10^{-2}\)  
             & \(2.19874840 \times 10^{-2}\)  
             && \(7.42964591 \times 10^{-4}\)  
             && \(5.97344727 \times 10^{-5}\)  
             & \(5.97359433 \times 10^{-5}\)  \\ 
$\gamma_3$   && 1.65567174 
             & 1.65594012 
             && \(3.03232829 \times 10^{-3}\)  
             & \(3.03249475 \times 10^{-3}\)  
             && \(5.12643356 \times 10^{-5}\)  
             && \(2.46975527 \times 10^{-6}\)  
             & \(2.46981189 \times 10^{-6}\)  \\ 
$\delta$     && 7.31685783 
             & 7.31858218 
             && 0.121337544 
             & 0.121345401 
             && \(8.39141754 \times 10^{-3}\)  
             && \(1.14389541 \times 10^{-3}\)  
             & \(1.14393689 \times 10^{-3}\)  \\ 
$\epsilon$   && 4.56800770 
             & 4.56784901 
             && \(2.70395958 \times 10^{-2}\)  
             & \(2.70380702 \times 10^{-2}\) 
             && \(9.63859236 \times 10^{-4}\)  
             && \(8.02589048 \times 10^{-5}\)  
             & \(8.02540794 \times 10^{-5}\) \\
\Hline
\end{tabular}
 \end{center}
\end{sidewaystable}


%% p132-c3t.asc -- Table 3 (second part), p.15, full-page landscape:

\setcounter{table}{2}  %% forces the digit ``3'' onto this table
\begin{sidewaystable}
\tabcolsep2pt
%
 \begin{center}
\topcaption{(concluded).} 
\renewcommand{\arraystretch}{1.75}
\begin{tabular}{@{}llllllllll@{}}
\Hline 
             & \mc{1}{c}{$^{10}$C} 
             & \mc{1}{c}{$^{11}$C} 
             & \mc{1}{c}{$^{12}$C} 
             & \mc{1}{c}{$^{13}$C} 
             & \mc{1}{c}{$^{14}$C} 
             && \mc{1}{c}{$^{19}$Ne} 
             & \mc{1}{c}{$^{20}$Ne} 
             & \mc{1}{c}{$^{21}$Ne} \\ 
\cline{1-6} \cline{8-10}
\mc{1}{@{}p{\expfive}}{$K$ \newline
                    $(\times 10^5)$}
             & \mc{1}{c}{\(10.95614\)} 
             & \mc{1}{c}{\(9.966312\)} 
             & \mc{1}{c}{\(9.145094\)} 
             & \mc{1}{c}{\(8.439298\)} 
             & \mc{1}{c}{\(7.836581\)} 
             && \mc{1}{c}{\(5.775458\)} 
             & \mc{1}{c}{\(5.489232\)} 
             & \mc{1}{c}{\(5.227334\)} \\ 
\hline
$\alpha_1$   & \(8.96788885 \times 10^{-3}\)  
             & \(8.96753132 \times 10^{-3}\)  
             & \(8.96723470 \times 10^{-3}\)  
             & \(8.96697977 \times 10^{-3}\)  
             & \(8.96676207 \times 10^{-3}\)  
             && \(1.04527428 \times 10^{-3}\)  
             & \(1.04525059 \times 10^{-3}\)  
             & \(1.04522865 \times 10^{-3}\)  \\ 
$\beta_1$    & \(3.45003316 \times 10^{-4}\)  
             & \(3.44990148 \times 10^{-4}\)  
             & \(3.44979224 \times 10^{-4}\)  
             & \(3.44969835 \times 10^{-4}\)  
             & \(3.44961817 \times 10^{-4}\)  
             && \(1.36072119 \times 10^{-5}\)  
             & \(1.36069069 \times 10^{-5}\)  
             & \(1.36066279 \times 10^{-5}\)  \\ 
$\gamma_1$   & \(1.38361204 \times 10^{-5}\)  
             & \(1.38356119 \times 10^{-5}\)  
             & \(1.38351901 \times 10^{-5}\)  
             & \(1.38348275 \times 10^{-5}\)  
             & \(1.38345179 \times 10^{-5}\)  
             && \(1.84095008 \times 10^{-7}\)  
             & \(1.84091014 \times 10^{-7}\)  
             & \(1.84087316 \times 10^{-7}\)  \\ 
$\delta_1$   & \(5.69728028 \times 10^{-7}\)  
             & \(5.69707781 \times 10^{-7}\)  
             & \(5.69690983 \times 10^{-7}\)  
             & \(5.69676547 \times 10^{-7}\)  
             & \(5.69664218 \times 10^{-7}\)  
             && \(2.55201695 \times 10^{-9}\)  
             & \(2.55196259 \times 10^{-9}\)  
             & \(2.55191226 \times 10^{-9}\)  \\ 
$\epsilon_1$ & \(2.38790447 \times 10^{-8}\)  
             & \(2.38782215 \times 10^{-8}\)  
             & \(2.38743099 \times 10^{-8}\)  
             & \(2.38769515 \times 10^{-8}\)  
             & \(2.38764502 \times 10^{-8}\)  
             && \(3.59572248 \times 10^{-11}\)  
             & \(3.59564654 \times 10^{-11}\)  
             & \(3.59557682 \times 10^{-11}\)  \\ 
$\alpha_2$   & \(1.03547724 \times 10^{-3}\)  
             & \(1.03551191 \times 10^{-3}\)  
             & \(1.03554068 \times 10^{-3}\)  
             & \(1.03556540 \times 10^{-3}\)  
             & \(1.03558651 \times 10^{-3}\)  
             && \(3.96641034 \times 10^{-5}\)  
             & \(3.96645031 \times 10^{-5}\)  
             & \(3.96648731 \times 10^{-5}\)  \\ 
$\beta_2$    & \(3.07413270 \times 10^{-5}\)  
             & \(3.07422952 \times 10^{-5}\)  
             & \(3.07430985 \times 10^{-5}\)  
             & \(3.07437889 \times 10^{-5}\)  
             & \(3.07443784 \times 10^{-5}\)  
             && \(3.93131631 \times 10^{-7}\)  
             & \(3.93135384 \times 10^{-7}\)  
             & \(3.93138860 \times 10^{-7}\)  \\ 
$\gamma_2$   & \(9.60335129 \times 10^{-6}\)  
             & \(9.60364158 \times 10^{-6}\)  
             & \(9.60388243 \times 10^{-6}\)  
             & \(9.60408942 \times 10^{-6}\)  
             & \(9.60426618 \times 10^{-6}\)  
             && \(4.09262742 \times 10^{-9}\)  
             & \(4.09266513 \times 10^{-9}\)  
             & \(4.09270004 \times 10^{-9}\)  \\ 
$\alpha_3$   & \(3.06153207 \times 10^{-4}\)  
             & \(3.06161375 \times 10^{-4}\)  
             & \(3.06168152 \times 10^{-4}\)  
             & \(3.06173976 \times 10^{-4}\)  
             & \(3.06178950 \times 10^{-4}\)  
             && \(3.89278110 \times 10^{-6}\)  
             & \(3.89281145 \times 10^{-6}\)  
             & \(3.89283955 \times 10^{-6}\)  \\ 
$\beta_3$    & \(8.00867049 \times 10^{-6}\)  
             & \(8.00887034 \times 10^{-6}\)  
             & \(8.00903615 \times 10^{-6}\)  
             & \(8.00917886 \times 10^{-6}\)  
             & \(8.00930036 \times 10^{-6}\)  
             && \(3.38491443 \times 10^{-8}\)  
             & \(3.38493930 \times 10^{-8}\)  
             & \(3.38496233 \times 10^{-8}\)  \\ 
$\gamma_3$   & \(2.20430058 \times 10^{-7}\)  
             & \(2.20435204 \times 10^{-7}\)  
             & \(2.20439474 \times 10^{-7}\)  
             & \(2.20443144 \times 10^{-7}\)  
             & \(2.20446278 \times 10^{-7}\)  
             && \(3.09291770 \times 10^{-10}\)  
             & \(3.09293915 \times 10^{-10}\)  
             & \(3.09295900 \times 10^{-10}\)  \\ 
$\delta$     & \(2.32754113 \times 10^{-4}\)  
             & \(2.32778574 \times 10^{-4}\)  
             & \(2.32798866 \times 10^{-4}\)  
             & \(2.32816304 \times 10^{-4}\)  
             & \(2.32831194 \times 10^{-4}\)  
             && \(2.99920821 \times 10^{-6}\)  
             & \(3.00102503 \times 10^{-6}\)  
             & \(3.00270689 \times 10^{-6}\)  \\ 
$\epsilon$   & \(1.10233253 \times 10^{-5}\)  
             & \(1.10224486 \times 10^{-5}\)  
             & \(1.10217212 \times 10^{-5}\)  
             & \(1.10210960 \times 10^{-5}\)  
             & \(1.10205622 \times 10^{-5}\)
             && \(4.89607914 \times 10^{-8}\)  
             & \(4.89585768 \times 10^{-8}\)  
             & \(4.89565266 \times 10^{-8}\)  \\
\Hline
\end{tabular}
 \end{center}
\end{sidewaystable}


The quantities we computed are of the following types. The generalized
second-order polarizabilities have the form:
%
%% Eq 3:
\begin{equation}
   S_{k,i} \equiv \sum_{n,m}  
           {\langle 0 | Q_i | n\rangle \langle n | Q_i | 0 \rangle
            \over 
            (E_n - E_0)^k}
\end{equation}
%
(An alternative notation for the second-order sums is $S_{1,i} =
\alpha_i$, $S_{2,i} = \beta_i$, $S_{3,i} = \gamma_i$, $S_{4,i} =
\delta_i$, $S_{5,i} = \epsilon_i)$. 

\vspace{1pc}

\noindent The third-order polarizability has the following form:
%
%% Eq 4:
\begin{equation}
   \delta \equiv \sum_{n,m}
           {\langle 0 | Q_i | n \rangle \langle n | Q_j | m \rangle
               \langle m | Q_k | 0 \rangle
            \over 
            (E_n - E_0) (E_m - E_0)}
\end{equation}
%
(The values that $\{ijk\}$ can take are all the permutations of
$\{112\}$). Finally, the fourth-order
hyper\-polarizability,\footnote{The complete hyperpolarizability is
usually defined as 32($\epsilon - \alpha_1 \beta_1$).} involving only
dipole terms, has the form
%
%% Eq 5:
\begin{equation}
   \epsilon \equiv \sum_{n,m,p}
             {\langle 0 | Q_1 | n \rangle 
                 \langle n | Q_1 | m \rangle 
                 \langle m | Q_1 | p \rangle 
                 \langle p | Q_1 | 0 \rangle
             \over 
              (E_n - E_0) (E_m - E_0) (E_p - E_0)}
\end{equation}

The intermediate states appearing in these sums are the approximate
eigenstates of the appropriate angular momentum, obtained by
diagonalizing the unperturbed Hamiltonian in a generalized Hylleraas
basis, and the corresponding energies are the approximate
eigenvalues. The forms used for the $S$-wave ground state and the $L$
= 1, 2, and 3 intermediate states are shown in previous work of the
authors \cite{BhatiDrachman,DrachmanBhatia}.


%% 3:
\section{Results for infinite nuclear mass}

Our best results for infinite nuclear mass ($K = 0$) appear in Table
1, and in the particular case of $\alpha_1$ recent results of Johnson
and Cheng \cite{JohnsonCheng} are also shown, for comparison. (Earlier
results \cite{BishopRerat} for what we call $\alpha_1$, $\alpha_2$,
$\delta$, and $\epsilon$ are not shown, since no more than five
significant figures are given; to that extent they agree perfectly
with our results.) Since each of these calculations represents a
significant amount of computer time, it would be useful to construct
an analytic expression describing these results. Such a formula would
interpolate between the computed values $Z$ = 6 and 10 and might also
describe the values for $Z > 10$ fairly well. For $Z \rightarrow
\infty$ the product wave function $\Psi = {\bar Z^3 \over \pi}
e^{-\bar Z(r_1 + r_2)}$ is a fairly good representation, since the
two-electron correlation becomes less important as the nuclear charge
increases. In this approximation it is possible, using the method of
Dalgarno and Lewis \cite{DalgarnoLewis}, to evaluate exactly all the
sums of interest here (except $\epsilon$) in terms of the effective
atomic number $\overline Z$. The expressions for all the sums in terms of
$\overline Z$ are as follows:

%% Eq 6:
\begin{equation}
   \alpha_1 = {9 \over \bar Z^4},\quad
   \beta_1 = {43 \over 4 \bar Z^6},\quad
   \gamma_1 = {319 \over 24 \bar Z^8},\quad
   \delta_1 = {9673 \over 576 \bar Z^10},\quad
   \epsilon_ 1 = {297541 \over 13824 \bar Z^12}
\end{equation}


%% Eq 7:
\begin{equation}
   \alpha_2 = {30 \over \bar Z^6},\quad
   \beta_2 = {107 \over 4 \bar Z^8},\quad
   \gamma_2 = {2399 \over 96 \bar Z^10}
\end{equation}


%% Eq 8:
\begin{equation}
   \alpha_3 = {525 \over  2 \bar Z^8},\quad
   \beta_3 = {3265 \over 16 \bar Z^10},\quad
   \gamma_3 = {63901 \over 384 \bar Z^12}
\end{equation}


%% Eq 9:
\begin{equation}
   \delta  = {213 \over \bar Z^8},\quad
   \epsilon = {318.9 \over \bar Z^10}
\end{equation}

%% p132-04t.asc -- Table 4, p.17, top of page, portrait: 

\begin{table}
\topcaption{Parameters of the empirical formula 
         $C (Z) = - A + {B \over Z - v} + \eta Z$ 
   to be used to obtain $K$-dependent (finite nuclear mass) values of
   the polarizability sums. The linear approximation is 
         $f (Z, K) = f (Z, O) [1 + C (Z) K]$, 
   where $f$ represents any one of the 13 quantities calculated here.}
%
\begin{center}
\begin{tabular}{@{}lllllll@{}}
\Hline 
             & \mc{1}{c}{A}
             & \mc{1}{c}{B}  
             & \mc{1}{c}{$\nu$}
             & \mc{1}{c}{$\eta$} \\ 
\hline
$\alpha_1$   & 2.000\,16 
             & 0.152\,1 
             & 0.82 
             & 1 \\ 
$\beta_1$    & 2.202\,5 
             & 0.310\,5 
             & 0.81 
             & 1 \\ 
$\gamma_1$   & 2.382\,9 
             & 0.507\,1 
             & 0.78 
             & 1 \\ 
$\delta_1$   & 2.546\,7 
             & 0.727\,5 
             & 0.75 
             & 1 \\ 
$\epsilon_1$ & 2.698\,2 
             & 0.965 
             & 0.72 
             & 1 \\ 
$\alpha_2$   & 3.840\,6 
             & 2.847 
             & \llap{$-$}0.21 
             & 0 \\ 
$\beta_2$    & 3.678\,5 
             & 3.122 
             & \llap{$-$}0.28 
             & 0 \\ 
$\gamma_2$   & 3.572\,1 
             & 3.250 
             & \llap{$-$}0.27 
             & 0 \\ 
$\alpha_3$   & 2.838\,1 
             & 0.804 
             & 0.36 
             & 0 \\ 
$\beta_3$    & 2.699 
             & 0.992 
             & 0.41 
             & 0 \\ 
$\gamma_3$   & 2.574 
             & 1.189 
             & 0.45 
             & 0 \\ 
$\delta$     & ? 
             & ? 
             & ? 
             & ? \\ 
$\epsilon$   & 4.066\,6
             & 0.550
             & 0.79
             & 2 \\ 
\Hline
\end{tabular}
\end{center}
\end{table}


(The coefficient shown in the formula for $\epsilon$ is not obtained
analytically, but as part of the numerical fitting procedure.) If we
set $\overline Z = Z - \sigma (Z)$, a reasonable guess would be $\sigma(Z)
= 5/16$, which gives the best variational energy in this
approximation, but this agrees poorly with the values in Table 1. In
fact, better values of the effective atomic number have been derived
\cite{DalgarnoStewart} for the dipole and quadrupole
polarizabilities: $\overline Z (\alpha_1) = Z - 23/64, \overline
Z(\alpha_2) = Z - 10381/23040$. These results are exact in the
large-$Z$ limit, and they provide a format for presenting our
empirical formulas. We will write
%
%% Eq 10:
\begin{equation}
   \sigma (Z) = \sigma_{\mbox{o}} + {A \over Z} 
                                  + {B \over Z^2}
                                  + {C \over Z^4}
\end{equation}
%
and will carry out a least-squares fit of the data in Table 1 to the
expressions in (6)--(10); a test of the method is afforded by
comparing the values we obtain for $\sigma_{\mbox{o}}$ with the two
exact cases discussed above. In Table 2 the four parameters
$\sigma_{\mbox{o}}$, $A$, $B$, and $C$ defined in (10) are tabulated
for each of the 13 generalized polarizabilities. The values of
$\sigma_{\mbox{o}}$ obtained in this way can be compared with the two
exact ones from ref. \nbcite{DalgarnoStewart}; for the dipole case
their ratio differs from 1 by only $8 \times 10^{-5}$ and for the
quadrupole case by only $5 \times 10^{-5}$. In addition, our
interpolated values for $Z$ = 7 to 9 agree exactly with the results
given in ref. \nbcite{BishopRerat}, and for various values of $Z > 10$
our values of $\alpha_1$ agree to about seven significant figures with
those in ref. \nbcite{JohnsonCheng}.


%% 4:
\section{Results for finite nuclear mass}

We have included the so-called mass polarization term $-K\overline V_1 
\cdot \overline V_2$ in the Hamiltonian to take into account the
effect of the finite nuclear mass on the wave functions involved in
computing the generalized polarizability sums. In addition, the
$K$-dependent terms in the definitions of the three multipoles
retained here, (2), also have an effect. In Table 3 the 13 quantities
of interest are shown, this time for a variety of nuclear masses, and
for each isotope the corresponding value of the parameter $K$ is
displayed. Except for the third-order calculation\footnote{This
calculation isnot completely correct.} done by Weinhold
\cite{Weinhold} and our earlier work
\cite{BhatiDrachman,DrachmanBhatia} there do not seem to be any
published values of
these quantities for finite mass, although a comparison between theory
and really accurate measurements is not possible without this kind of
calculation. (Because of the use of scaled natural units in the
formulation of the problem, all results are in natural units
appropriate to the particular quantity. For example, $\alpha_1$ is in
units of $a_o^3$ in Table 1 and in scaled units $a_K^3$ in Table 3,
where the scaled Bohr length is $a_K = 2a_o/(2 - K)$.)

Since there is quite a large amount of data in Table 3, it should be
useful to find a simple, analytic expression that can represent the
generalized polarizability sums as functions of $K$. A fairly good
least-squares fit can be made to the formula
%
\begin{equation}
   f (Z, K) = f (Z, O) [1 + C (Z) K]
  \quad \mbox{where}\quad
   C (Z) = - A + {B \over Z - v} + \eta Z
\end{equation}

Here $f$ represents any one of the 13 quantities in Table 3 and a
linear fit in $K$ is assumed. In Table 4 the four parameters $A$, $B$,
$\nu$, and $\eta$ are tabulated for all quantities of interest except
$\delta$; this particular sum has an unusual dependence on $Z$ that is
not easy to model or understand. The linear dependence on $K$ gives a
good description of the data. Note that the parameter $\eta$, which
introduces a linear dependence on $Z$, appears only for cases
involving the dipole potential and comes from the pre-factor in the
definition of $Q_1$ in (2).


%% 5:
\section{Conclusions}

In this paper we have extended our computations
concerning generalized polarizabilities of two-electron atoms and
ions. The three main improvements over previous work are as follows.
%
\begin{enumerate} 
   \item [] First, we believe the present results are more accurate
            than any others, when comparisons are possible.

   \item [] Second, we have taken the effect of finite nuclear mass
            into account in all cases.

   \item [] Third, we have presented a practical way to interpolate
            (and perhaps extrapolate) the data to various values of
            atomic number and nuclear mass.
\end{enumerate}


\section*{Acknowledgments}

These numerical results were obtained with the Cray Y-MP computer of
the NASA Center for Computational Science. Computer costs were funded
under NASA-RTOP Grant No. 432-38-53-14.


%% \section*{References}
\begin{thebibliography}{99}

%% 1:
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   J. Pipin and D.M. Bishop. J. Phys. B: At. Mol. Opt. Phys.
      {\bf 25}, 17 (1992).

%% 2:
\bibitem{BhatiDrachman}
   A.K. Bhatia and R.J. Drachman. Phys. Rev. A, 45, 7752
      (1992); J. Phys. B: At. Mol. Opt. Phys. {\bf 27}, 1299 (1994).

%% 3:
\bibitem{DrachmanBhatia}
   R.J. Drachman and A.K. Bhatia. Phys. Rev. A: At. Mol.
      Opt. Phys. {\bf 51}, 2926 (1995).

%% 4:
\bibitem{Rotheryetal}
   N.E. Rothery, C.H. Storry, and E.A. Hessels. Phys. Rev.
      A: At. Mol. Opt. Phys. {\bf 51}, 2919 (1995).

%% 5:
\bibitem{JohnsonCheng}
   W.R. Johnson and K.T. Cheng. Phys. Rev. A: At. Mol. Opt.
      Phys. {\bf 53}, 1375 (1996).

%% 6:
\bibitem{BishopRerat}
   D.M. Bishop and M.Rerat. J. Chem. Phys. {\bf 91}, 5489 (1989).

%% 7:
\bibitem{DalgarnoLewis}
   A. Dalgarno and J.T. Lewis. Proc. R. Soc. London A, {\bf 233},
      70 (1955).

%% 8:
\bibitem{DalgarnoStewart}
   A. Dalgarno and A.L. Stewart. Proc. R. Soc. London A,
      {\bf 257}, 534 (1960).

%% 9:
\bibitem{Weinhold}
   F. Weinhold. J. Chem. Phys. {\bf 86}, 1111 (1982).
\bibitem{mumble} H. MacMillan and A. Doubtless-Whom. J. Mont. Pyt.
   Stud., \textbf{2446}, 13235 (1963).
\end{thebibliography}


\end{document}

%% END OF FILE

