THE GROUND STATE OF TWO-ELECTRON ATOMS AND IONS

A. M. Ermolaev, G. B. Sochilin

(Presented by Academician V. A. Fock on 4 XII 1963)

PHYSICS

1. Variational calculations of the nonrelativistic \(S\)-states of helium and helium-like ions, carried out by many authors in connection with the experimental study of the radiation shift in two-electron atoms, undoubtedly show that solving this problem with the required accuracy is an exceptionally difficult task, unless the properties of the wave equation at singular points are taken into account. A rigorous investigation of the equation for \(S\)-states in the neighborhood of the point of triple collision \(r_1=r_2=0\) was carried out by V. A. Fock [1]. Using the example of the \(1^1S\)-state of He and of some ions, we shall consider a method, based on Fock’s work, for taking singularities into account in the wave equation with the interaction potential of the particles of the system
   \[ V(r_1,r_2,r_{12})=-Z(r_1^{-1}+r_2^{-1})+r_{12}^{-1}. \]

2. First we shall establish the analytic form of a function which, up to certain coefficients, coincides with the function \(\Psi\) satisfying the wave equation at \(R=r_1^2+r_2^2=0\) and \(R\to\infty\). Consider, in the variables \(R,\alpha\), and \(\vartheta\) \(\bigl(\alpha=2\operatorname{arcctg} r_1/r_2,\ \vartheta\) is the angle between \(\mathbf r_1\) and \(\mathbf r_2,\ 0\leq \vartheta\leq \pi\bigr)\), the expression
   \[ \Psi_N(R,\alpha,\vartheta)= \exp\left[-\frac{k}{2}\sigma(\alpha)\sqrt{R}\right] \sum_{n=1,\,3/2,\,2,\ldots} R^{\,n-1} \sum_{p=0}^{[n-1]} g_{np}(\alpha,\vartheta)\ln^p R, \tag{1} \]
   in which \(\sigma(\alpha)=\sin\alpha/2+\cos\alpha/2\), \(k\) is a scale parameter. \(\Psi_N\) has the proper asymptotic behavior as \(R\to\infty\), if \(k=2\sqrt{-E}\) (\(E\) is the total energy of the atom). This function also satisfies the equation near \(R=0\) up to accuracy \(R^N\), if the coefficients \(g_{np}\) entering (1) are defined by the formulas \(g_{np}=(j!)^{-1}\sigma^j\psi_{n-j/2,p}\) (with \(j\) summed from 0 to \(2(n-p-1)\)) through the coefficients \(\psi_{n'p}\) of the expansion of \(\Psi\) in Fock’s series.

As was shown in (1), the functions \(\psi_{np}\) must be determined successively, beginning with \(n=1\), then \(n=3/2,2,\ldots\), and, for a given \(n\), in the order of decreasing second index \(p\), beginning with its maximum value \(p=[n-1]\) down to \(p=0\), from the differential equations
\[ L[\psi_{np}]=\Box^*\psi_{np}+(n^2-1)\psi_{np}=F_{np}, \tag{2} \]
\[ F_{np}=-(p+1)(p+2)\psi_{n,p+2} -2n(p+1)\psi_{n,p+1} +\tfrac12 U\psi_{n-1/2,p} -\tfrac12 E\psi_{n-1,p}. \]
Here \(\Box^*\) is the Laplace operator on the four-dimensional sphere with angles \(\alpha,\vartheta,\varphi\) (the solution does not depend on \(\varphi\), since the potential energy \(U(\alpha,\vartheta)\) on this sphere does not depend on \(\varphi\)). The eigenvalues of the operator \(-\Box^*\) are \(\lambda_n=n^2-1\) for \(n=1,2,3,\ldots\), and its eigenfunctions are the generalized spherical functions \(\Phi_{nl}(\alpha,\vartheta)\). Finite solutions of these equations exist, and all functions \(\psi_{np}\) (under the additional condition \(\psi_{np}(\alpha,\vartheta)=\psi_{np}(\pi-\alpha,\vartheta)\) for singlet states) are determined uniquely, with the exception of \(\psi_{n0}\) with integer \(n\). Arbitrary combinations of \(\Phi_{nl}\) of the corresponding order \(n\) enter into \(\psi_{n0}\) [1]. The functions \(\psi_{n,[n-1]}\) can be written in finite form. In the case of integer \(n\), the equations for \(\psi_{n,n-1}\) are homogeneous, so that \(\psi_{n,n-1}\sim\Phi_{nl}\), and the sum of the indices \(n+l\) must be odd. For half-integer \(n\), when \([n-1]=n-3/2\), the functions \(\psi_{n,n-3/2}\) satisfy equations with singu

regular right-hand side, \({}^{1}/_{2}\) and \(\psi_{n-{}^{1}/_{2},\,n-{}^{3}/_{2}}\). These equations have solutions in the form of polynomials of degree \(2(n-1)\) in \(\sigma\) and in \(\omega=\sqrt{1-\sin\alpha\cos\vartheta}\). For example,

\[ \psi_{{}^{3}/_{2},0}=-Z\sigma+{}^{1}/_{2}\omega;\quad \psi_{{}^{5}/_{2},1}=\frac{Z(\pi-2)}{6\pi} \left[\frac{Z}{2}\sigma(1-\omega^{2})-\frac{1}{2}\omega-\frac{5}{12}\omega^{3}\right] \tag{2} \]

In all other cases, including the most important one, when \(p=0\) (for \(n\geqslant 2\)), the solution of the singular equations (2) is not representable in finite form. However, even in this general case one can separate out from \(\psi_{np}\) the principal part \(\chi_{np}\), which also has the form of a polynomial in \(\sigma\) and in \(\omega\). Its coefficients are determined by the requirement that the function \(\overset{0}{F}_{np}\) and its derivatives be finite at all points of the four-dimensional sphere, including the regions corresponding to double collisions. \(\overset{0}{F}_{np}\) is the free term of the equation \(L[\overset{0}{\psi}_{np}]=\overset{0}{F}_{np}\), where \(\overset{0}{\psi}_{np}=\psi_{np}-\chi_{np}\). After the function \(\chi_{np}\), and together with it also \(\overset{0}{F}_{np}\), have been determined, the difference \(\overset{0}{\psi}_{np}\) can be represented by a rapidly convergent series in generalized spherical functions \(\Phi_{sl}\). The functions \(g_{np}\), which determine \(\Psi_N\), differ only by power factors \(\sigma^j\) from the sum of the functions \(\psi_{n-i/2,p}\). In this connection \(g_{n,[n-1]}\), just as \(\psi_{n,[n-1]}\), can be written in finite form, while for the remaining \(g_{np}\), just as for \(\psi_{np}\) in the general case, a convenient representation of the form

\[ g_{np}(\alpha,\vartheta)=\chi_{np}(\sigma,\omega)+ \sum_{s=1}^{\infty}\sum_{l=0}^{s-1} a^{np}_{sl}\Phi_{sl}, \qquad s+l\ \text{odd}. \tag{3} \]

can be obtained.

The polynomial functions \(\chi_{np}\) take into account the strongest of the singularities of \(g_{np}\), caused by double collisions.

1. The method of numerical calculation was as follows. The series entering into \(\Psi_N\) were truncated. In the expression obtained, \(\widetilde{\Psi}_N\), some of the coefficients in \(\chi_{np}\) and the coefficients of the generalized functions \(\Phi_{sl}\) were regarded as arbitrary (\(\widetilde{\Psi}_N\) is a linear function with respect to these coefficients), and their values, which below we shall denote by \(c_i\), were determined from the variational principle for the total energy of the atom. The condition to which the function \(\Psi_N\) is subject near \(R=0\), in this method, is generally speaking violated, since it makes arbitrary only \({}^{1}/_{2}N(N+1)\) coefficients entering into \(g_{n0}\) with the functions \(\psi_{n0}\). At the same time, the possibility arises of improving the approximation of the wave function at large \(R\). In this case the degree of agreement between the variational and theoretical values of the coefficients entering into the initial terms of the expansions of \(\widetilde{\Psi}_N\) and \(\Psi\) at \(R=0\) may serve as a criterion of the quality of the variational function \(\widetilde{\Psi}_N\) near the nucleus—in the region where the approximation usually proves to be very poor.

We investigated the ground state of \(\mathrm{H}^{-}\), He, \(\mathrm{Li}^{+}\), \(\mathrm{Be}^{+2}\), \(\mathrm{Be}^{+3}\), \(\mathrm{O}^{+6}\), and \(\mathrm{Ne}^{+8}\). The first part of the calculations was carried out with a function \(A\) containing 30 linear parameters \(c_i\) and a nonlinear scale parameter \(k\). The explicit form of this function is determined by formula (1) for \(N=2\), if the following expressions are taken for \(g_{np}\):

\[ \begin{gathered} g_{10}=1;\qquad g_{{}^{3}/_{2},0}=c_1\sigma+c_2\omega;\qquad g_{21}=c_3\Phi_{21};\\ g_{20}=c_4\sigma+c_5\omega+c_6\sigma\omega+c_7\sigma^2+c_8\omega^3+c_9\Phi_{10}+c_{10}\Phi_{21};\\ g_{{}^{5}/_{2},1}=c_{11}\sigma(1-\omega^2)+c_{12}\omega\left(1-\frac{5}{6}\omega^2\right); \tag{4}\\ g_{{}^{5}/_{2},0}=c_{13}\sigma+c_{14}\omega+c_{15}\sigma^2\omega+c_{16}\sigma^3+c_{17}\omega^3+c_{18}\Phi_{10}+c_{19}\Phi_{21};\\ g_{32}=c_{20}\Phi_{30}+c_{21}\Phi_{32};\qquad g_{31}=c_{22}\omega;\\ g_{30}=c_{23}\sigma+c_{24}\omega+c_{25}\sigma^2\omega^2+c_{26}\sigma^4+c_{27}\Phi_{10}+c_{28}\Phi_{21}+c_{29}\Phi_{30}+c_{30}\Phi_{32}. \end{gathered} \]

The values of the nonrelativistic energy \(E_2^A(Z)\) (the motion of the nucleus was not taken into account), calculated with function \(A\), turned out to be as follows:
\(E_2^A(1)=-0.5277318,\ E_2^A(2)=-2.9037233,\ E_2^A(3)=-7.2799118,\ E_2^A(4)=-13.655564,\ E_2^A(5)=-22.030469,\ E_2^A(8)=-59.156592,\ E_2^A(10)=-93.906803\), a.u.* These values are in good agreement with the results of Perkins’s 203-parameter calculations: \(\varepsilon(1)=19.2,\ \varepsilon(2)=1.0,\ \varepsilon(3)=1.4,\ \varepsilon(4)=2.0,\ \varepsilon(5)=\varepsilon(8)=\varepsilon(10)=3.0\) (3).

The deviation near the nucleus of the variational function \(\Psi_N\) from the exact solution is conveniently estimated from the values of the quantities \(\Delta_1=(Z+\alpha_1)\cdot 10^3\) and \(\Delta_2=(1/2-\alpha_2)\cdot 10^3\), where \(\alpha_1\) and \(\alpha_2\) are the coefficients of \(\sigma \sqrt{R}\) and \(\omega \sqrt{R}\) in the expansion of the variational function. For the exact function \(\Psi\) the quantities \(\Delta\) are zero. The largest values, \(\Delta_1=27.2\) and \(\Delta_2=38.4\), as was to be expected, were obtained by us for the negative hydrogen ion. It is remarkable, however, that the order of this deviation is the same as for the 252-parameter helium function, for which it follows from (4): \(\Delta_1=27.2\) and \(\Delta_2=43.6\). The values of \(\Delta\) in our calculations for \(Z\ge 2\) are practically independent of \(Z\) and have the values \(\Delta_1\sim 7\) and \(\Delta_2\sim 15\) (for comparison, we note that the 1078-parameter helium function (4) gives \(\Delta_1=15.6\) and \(\Delta_2=26.1\)). The discrepancy in the ratios for the theoretical coefficients and for the coefficients of function \(A\) increases in the subsequent terms of the expansion to 10–30%. We note that in methods that do not take into account the properties of the solution near \(R=0\), even the initial coefficients of the functions may fail to have stable values.**

Table 1

\(1^1 S\)-state of He, Li\(^+\), and O\({}^{+6}\)

1. To estimate the influence of refining the initial \(g_{np}\) on the convergence of the method, functions \(B\) and \(D\) were considered with a number of parameters \(m\) equal to 31, and \(G\) (\(m=37\)), defined by (1) for \(N=2\). If the \(g_{np}\) from (4) are taken as the initial ones, then \(B\) is obtained by supplementing \(g_{20}\) with the terms \(\Phi_{30}\) and \(\Phi_{32}\), \(g_{5/2\,0}\) with the term \(\Phi_{32}\), and by excluding \(g_{31}\) and \(g_{32}\). In function \(D\) the coefficients in \(g_{3/2\,0}\) were taken to be theoretical, and the terms \(\Phi_{30}\) and \(\Phi_{32}\) were introduced into \(g_{20}\). In function \(G\), \(\Phi_{30}\), \(\Phi_{32}\), and \(\Phi_{41}\) were included in \(g_{20}\), \(\Phi_{30}\) and \(\Phi_{32}\) in \(g_{5/2\,0}\), and a term with \(s\) in \(g_{31}\). To assess the prospects of the subsequent steps in \(N\), function \(C\) (\(m=31\)) was considered, containing four terms proportional to \(R^{5/2}\) and \(R^3\). Finally, function \(F\) was taken as a characteristic example of a function containing no logarithmic terms. The values of \(\varepsilon(Z)\) and \(\Delta\), calculated with these functions for He, Li\(^+\), and O\({}^{+6}\), are given in Table 1.

Table 2

\[ \varepsilon_N(Z) \]

The data of Table 2 show that rapid convergence of \(\varepsilon_N(Z)\) with increasing \(N\) (and with the corresponding increase in \(m\)) occurs for all the ions considered. In contrast to this, for methods in which the features of the wave equation are not taken into account, a quite—

* The values \(E(Z)\) obtained by us in additional calculations, as well as by other authors, will subsequently be given for convenience in the form \(\varepsilon(Z)=[E_2^A(Z)-E(Z)]\cdot 10^6\).

** The 70-parameter helium function of first-order variational perturbation theory has, near the nucleus, deviations \(\Delta_1=81.4\) and \(\Delta_2=-543.3\) (5).

there is weak convergence of \(\varepsilon(Z)\) with increasing \(m\), and not only for helium, but also for ions with \(Z>2\). Thus, the calculations \((^6)\), \((m=20)\), give \(\varepsilon(2)=-5.4\), \(\varepsilon(3)=-6.4\), and \(\varepsilon(8)=-10\), whereas in \((^7)\), with \(m\) equal to 31, for the same atoms the values \(-1.1\), \(-6.3\), and \(-11\) were obtained. To refine the terms corresponding to the transition from \(N=3/2\) to \(N=2\) in Table 2 (where the deviation of the terms from the exact values is reduced to \(2\cdot 10^{-7}\)—\(3\cdot 10^{-8}\)), in these methods it is necessary to introduce into the functions about a hundred additional variational parameters.

1. The mean value of the operator \({\bf p}_1{\bf p}_2\), calculated with the functions \(A\), \(B\), \(D\), and \(G\), which gives the correction to the level due to nuclear motion, as well as the mean values of powers of \(r_1\) and \(r_{12}\), agree to an accuracy of \(10^{-6}\) and better with the known values for He and Li\(^+\) from \((^{4,8})\). At the same time, the quantities \(\langle\delta^{1}_{3}(r_1)\rangle\) and \(\langle\delta^{(3)}(r_{12})\rangle\), which enter into the expression for the relativistic correction and also into the radiation shift and depend strongly on the properties of the variational functions at special points*, show a deviation from the values obtained by Pekeris. Function \(A\) gives, for all ions, values corresponding to \(m=95\); function \(B\), values corresponding to \(m=161\)—203. Functions \(G\) and \(D\) lead to quantities which, for He and Li\(^+\), lie beyond the interpolation values obtained in calculations with \(m=1078\). For helium the latter values are 1.810427 and 0.106345, whereas the values obtained with \(G\) and \(D\) are 1.810462 and 0.106299 and, respectively, 1.810646 and 0.106238. The interpolation values \(\langle\delta\rangle\) from \((^{4,8})\), apparently, are reliable only to an accuracy of \(10^{-5}\)—\(10^{-4}\). This circumstance must be kept in mind in studying the radiation shift, if it proves possible to increase the accuracy of the experimental determination of ionization potentials. An example of another problem in which it would also be important to estimate the influence of allowance for singularities is the study of the hyperfine structure in \(2^3S\) He\(^3\). The discrepancy between the measured value of the splitting and its theoretical value, which also depends on \(\langle\delta^{(3)}(r_1)\rangle\), turns out to be 5—10 times greater than the experimental error \((^{9,4})\), amounting to \(10^{-5}\).

2. The expansion found by V. A. Fock, on which the present method is based, admits generalization to states with nonzero angular momentum and to systems of particles whose number is greater than 3 \((^{10})\). However, the \(S\)-state of a two-electron atom is probably at present an exceptional example of such a problem in which allowance for singularities can actually be carried out.

3. The numerical calculations were performed on the BESM-2 at the Computing Center of the Leningrad Division of the V. A. Steklov Mathematical Institute of the Academy of Sciences of the USSR.

The authors take this opportunity to express their gratitude to Academician V. A. Fock for discussion of the work and valuable comments, and also to Yu. N. Demkov for discussion of a number of questions.

Leningrad State University
named after A. A. Zhdanov

Leningrad Division
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
27 XI 1963

CITED LITERATURE

1. V. A. Fock, Izv. AN SSSR, ser. fiz., 18, 161 (1954).
2. A. M. Ermolaev, Vestn. LGU, No. 16, 19 (1961).
3. C. L. Pekeris, Phys. Rev., 112, 1649 (1958).
4. C. L. Pekeris, Phys. Rev., 115, 1216 (1959); 126, 1470 (1962).
5. R. Knight, C. Scherr, Phys. Rev., 128, 2675 (1962).
6. J. F. Hart, G. Herzberg, Phys. Rev., 106, 79 (1957).
7. H. M. Schwartz, Phys. Rev., 130, 1029 (1963).
8. C. L. Pekeris, Phys. Rev., 126, 143 (1962).
9. J. A. White, L. Y. Chow et al., Phys. Rev. Lett., 3, 428 (1959).
10. A. M. Ermolaev, Vestn. LGU, No. 22, 48 (1958); Yu. N. Demkov, A. M. Ermolaev, ZhETF, 36, 896 (1959).

* The value \(\langle p_1^4\rangle\) also depends strongly on these properties. For the relativistic correction to the ionization potential as a whole, this dependence proves stronger than for the quantities \(\langle p_4^1\rangle\) and \(\langle\delta^{1}_{3}(r_1)\rangle\) entering into it separately.