%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% INSTITUTE OF PHYSICS PUBLISHING %%% %%% %%% %%% `Preparing an article for publication in an Institute of Physics %%% %%% Publishing journal using LaTeX' %%% %%% %%% %%% LaTeX source code `ioplau.tex' used to generate `author %%% %%% guidelines', the documentation explaining and demonstrating use %%% %%% of the Institute of Physics Publishing LaTeX preprint macro file %%% %%% `ioplppt.sty'. %%% %%% %%% %%% `ioplau.tex' itself uses LaTeX with `ioplppt.sty' %%% %%% an optional file iopfts.sty can also be loaded to allow the %%% %%% use of the AMS extension fonts msam and msbm with the IOP %%% %%% preprint style. %%% %%% %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% %%% %%% First we have a character check %%% %%% ! exclamation mark " double quote %%% # hash ` opening quote (grave) %%% & ampersand ' closing quote (acute) %%% $ dollar % percent %%% ( open parenthesis ) close paren. %%% - hyphen = equals sign %%% | vertical bar ~ tilde %%% @ at sign _ underscore %%% { open curly brace } close curly %%% [ open square ] close square bracket %%% + plus sign ; semi-colon %%% * asterisk : colon %%% < open angle bracket > close angle %%% , comma . full stop %%% ? question mark / forward slash %%% \ backslash ^ circumflex %%% %%% ABCDEFGHIJKLMNOPQRSTUVWXYZ %%% abcdefghijklmnopqrstuvwxyz %%% 1234567890 %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% %\documentstyle{ioplppt} % use this for journal style \documentstyle[12pt]{ioplppt} % use this for preprint style \def\cqo#1#2{\cos[q\Omega^{\rm #1}_S(b_f,R^s_{#2})]} % \begin{document} \title{Variational principle for critical parameters of quantum systems}[Variational principle for critical parameters] \author{A V Sergeev and S Kais} %\ftnote{3}{To %whom correspondence should be addressed.}} \address{Purdue University, Department of Chemistry, 1393 Brown Building, West Lafayette, IN 47907} \begin{abstract} Variational principle for eigenvalue problems with a non-identity weight operator is used to establish upper or lower bounds on critical parameters of quantum systems. Three problems from atomic physics are considered as examples. Critical screening parameters for the exponentially screened Coulomb potential are found using a trial function with one non-linear variational parameter. The critical charge for the helium isoelectronic series is found using a Hylleraas-type trial function. Finally, critical charges for the same system subject to a magnetic field are found using a product of two hydrogen-like basis sets. \end{abstract} % % Uncomment out if preprint format required % \pacs{03.65.Ge, 03.65.Db} %% http://www.th.physik.uni-frankfurt.de/~cbest/pacs.numbers.00 %% 03.65.-w Quantum theory; quantum mechanics %% 03.65.Db Functional analytical methods %% 03.65.Ge Solutions of wave equations: bound states %\maketitle \section{Introduction} The variational principle for eigenvalue problems is a well-known method and widely used in quantum calculations since the foundation of quantum mechanics. For the eigenvalue equation of the form \begin{equation} \label{eig}L(\psi)=\lambda M(\psi), \end{equation} where $L$ and $M$ are self-adjoint operators, the variational principle reads (Morse and Feshbach 1953): \begin{equation} \label{var}\delta[\lambda]=\delta\left[\int \psi M(\psi)dV/\int \psi L(\psi)dV\right]=0. \end{equation} For a time-independent Schr\"{o}dinger equation, $M$ and $\lambda$ are usually considered as the identity operator and the energy, in this case \Eref{var} represents the variational principle for the energy. Here, we point out the usefulness of the variational principle with a non-identity "weight" operator $M$ for calculations of critical parameters of quantum-mechanical systems. We consider a Hamiltonian that depends on some continuous parameter $\gamma$. We call the parameter $\gamma=\gamma_c$ {\it critical} if the energy of the system reaches the ionization border $E_I$, \begin{equation} \label{cri} H(\gamma_c)\psi=E_I(\gamma_c)\psi. \end{equation} If the Hamiltonian and the border of ionization depend on the parameter $\gamma$ linearly, i.e. $H(\gamma)=H_0 +H_1 \gamma$, $E_I(\gamma)=E_0 +E_1 \gamma$ then the Schr\"{o}dinger equation for the critical parameter, \Eref{cri} takes the form of \Eref{eig} where $L=H_0-E_0$, $M=E_1-H_1$, and $\lambda=\gamma_c$, in which case the variational principle can be used immediately. Often, a linear dependence on $\gamma$ should be achieved by an appropriate scaling transformation before using the variational principle. A similar approach was used earlier for critical screening parameters of the exponentially screened (Hulth\'{e}n and Laurikainen, 1951) and of the cut-off (Dutt, Singh, and Varshni 1985) Coulomb potentials. Here we present a further demonstration of the power of the general variational principle, \Eref{var} to obtain very accurate critical parameters for both simple one degree of freedom problems such as the Yukawa potential and for problems of several degrees of freedom such as the two-electron atoms in a magnetic field. To the best of our knowledge, this general variational principle is used here for the first time to calculate critical parameters for several degrees of freedom. Let us consider several examples to illustrate this approach. \section{Yukawa potential} The radial Schr\"{o}dinger equation for one particle in a Yukawa potential, $v(r) = -\exp(-\delta r)/r$, after the scaling transformation $r\rightarrow r/\delta$, takes the form \begin{equation} \label{yuk} \left[-\frac{1}{2}\frac{d^2}{dr^2} + \frac{l(l+1)}{2r^2} - \delta^{-1}\frac{\exp(-r)}{r} - \delta^{-2} E\right]P(r)=0, \end{equation} where $l$ is the azimuthal quantum number, $\delta$ is the screening parameter, $E$ is the energy, and $P(r)$ is the radial wave function multiplied by $r$ (we use atomic units $\hbar=e=m=1$). For a sufficiently large critical screening parameter $\delta=\delta_c$ the energy reaches the ionization border $E_I =0$. The equation for the $\delta_c$ has a form of a general eigenvalue equation \ref{eig} in which $L = -\frac{1}{2}\frac{d^2}{dr^2} + \frac{l(l+1)}{2r^2}$, $M = \exp(-r)/r$, $\psi=P$, and $\lambda = \delta_c^{-1}$. We are looking for an extremum of the functional \begin{equation} \label{fun} W=\int \psi M(\psi)dV/\int \psi L(\psi)dV \end{equation} using a trial function with only one variational parameter $a$: \begin{equation} \label{tri1} \widetilde{P}(r)=r^{-l}\left(1-e^{(-ar)}\sum_{n=0}^{2l}\frac{a^n}{n!}r^n\right). \end{equation} The function, \Eref{tri1}, behaves like $\widetilde{P}(r) \sim r^{l+1}$ at $r\rightarrow 0$ and $\widetilde{P}(r)\sim r^{-l}$ at $r\rightarrow \infty$ which is consistent with the behavior of the general solution of the radial Schr\"{o}dinger equation at zero energy. The functional, \Eref{fun}, appears to have a minimum at a certain $a=a_{\min}$, these values are listed in Table \ref{res1} for different values of $l$. \begin{table} \noindent \caption{Results of minimization of the functional, \Eref{fun}, for the Yukawa potential with one-parameter trial function, \Eref{tri1}. The exact critical parameters are given in the last column for comparison.} \begin{indented} \label{res1} \item[] \begin{tabular}{@{}cccc} \br $l$&$a_{\min}$&$\widetilde{\delta}_c$&$\delta_c$\\ \mr 0&$1.535$&$1.190\,213$&$1.190\,612$\\ 1&$2.534$&$0.219\,800$&$0.220\,217$\\ 2&$3.525$&$0.091\,085$&$0.091\,345$\\ 3&$4.517$&$0.049\,670$&$0.049\,831$\\ 4&$5.513$&$0.031\,240$&$0.031\,344$\\ 5&$6.510$&$0.021\,455$&$0.021\,525$\\ 6&$7.507$&$0.015\,642$&$0.015\,691$\\ 7&$8.506$&$0.011\,909$&$0.011\,945$\\ \br \end{tabular} \end{indented} \end{table} Its minimum gives an approximation for the eigenvalue $\widetilde{\lambda}$. The corresponding approximations for the critical screening parameter $\widetilde{\delta}_c=\widetilde{\lambda}^{-1}$ together with the exact critical screening parameters, which have been found numerically by integration of the Schr\"{o}dinger equation, are listed in the last two columns of Table \ref{res1}. The variational method yields excellent lower bounds for the critical screening parameters of the lowest states in each $l$ subspace. Hulth\'{e}n and Laurikainen (1951) used, for the ground state ($l=0$), a different trial function in the form of the expansion $\left(1-\e^{-x}\right)\sum_{\nu=0}^n h_{\nu} \e^{-\nu x}$. Since it has several variational variables, their results are more accurate. \section{Two-electron atoms} Calculation of the critical nuclear charge $Z_c$ for two-electron atoms has long history (Br\"{a}ndas and Goscinski 1972, Stillinger 1966, Stillinger and Stillinger 1974, Stillinger and Weber 1974, Reinhardt 1977) with controversial results of whether or not the value of $1/Z_c$ is the same as the radius of convergence of the perturbation series in $1/Z$, $1/Z_*$. Baker \etal (1990) have performed a 400-order perturbation calculation to resolve this controversy and found that $1/Z_*=1/Z_c$ where numerically $Z_*^{-1}\approx 1.097\,66$. Using Euler transformation of the series that accelerates its convergence, Ivanov (1995) estimated the value of the radius of convergence as $Z_*^{-1}\approx 1.097\,660\,79$. The Schr\"{o}dinger equation for a two-electron atom, after the scaling transformation $r\rightarrow r/Z$, takes the form \begin{equation} \label{two} \left[-\frac{1}{2}\nabla_1^2 -\frac{1}{2}\nabla_2^2 -\frac{1}{r_1} -\frac{1}{r_2} +Z^{-1}\frac{1}{r_{12}}- Z^{-2} E\right]\psi=0, \end{equation} where $Z$ is the charge of the nucleus and $E$ is the energy (in atomic units). For a sufficiently small nuclear charge, at $Z=Z_c$ the energy reaches the ionization border $E_I =-Z^2/2$, which is the energy of one-electron atom. The equation for the $Z_c$ has a form of a general eigenvalue equation \ref{eig} in which $L = -\frac{1}{2}\nabla_1^2 -\frac{1}{2}\nabla_2^2 -1/r_1 -1/r_2 +1/2$, $M = 1/r_{12}$, and $\lambda = -Z_c^{-1}$. We are looking for an extremum of the functional, \Eref{fun}, using a Hylleraas-type trial function of the form \begin{equation} \label{tri2} \begin{array}{c} \psi_N= \sum_{i+j^2+k^2\leq N}^{} C_{i,j,k} \left[r_1^i r_2^j \exp(-ar_1-br_2)+r_2^i r_1^j \exp(-ar_2-br_1)\right] \\ \exp(-cr_{12}) r_{12}^k. \end{array} \end{equation} The restriction on the summation indexes $i+j^2+k^2\leq N$ is used instead of the more common restriction $i+j+k\leq N$ in order to decrease the number of terms in the sum from $\sim \frac{1}{6}N^3$ to $\sim \frac{\pi}{8}N^2$ . Here, we suppose that correlation terms with higher degrees of $r_{12}$ are relatively unimportant and we suppress them by rising $k$ to $k^2$. We also assume that expanding over $r_2$ is less important that expanding over $r_1$ because we are ordering the parameters $a$ and $b$ so that $a