Two-Electron Atomic Systems—A Simple Method for Calculating the Ground State near the Nucleus: Some Applications

Abstract

A simple method of non-relativistic variational calculations of the electronic structure of a two-electron atom/ion, primarily near the nucleus, is proposed. The method as a whole consists of a standard solution of a generalized matrix eigenvalue equation, all matrix elements of which are reduced to a numerical calculation of one-dimensional integrals. The distinctive features of the method are as follows: The use of the hyperspherical coordinate system. The inclusion of logarithms of the hyperspherical radius R in the basis functions, similar to the Fock expansion. Using a special basis function including the leading angular Fock coefficients to provide the correct behavior of the wave function near the nucleus. The main numerical parameters characterizing the properties of the helium atom and a number of helium-like ions near the nucleus are calculated and presented in tables. Among others, the specific coefficients, $a_{21}$, of the Fock expansion, which can only be calculated using a wave function with the correct behavior near the nucleus, are presented in table and graphs.

1. Introduction

The trivial statement is that all the main characteristics of a two-electron atomic system in the S state, having an infinitely massive nucleus with charge, Z, and non-relativistic energy, E, are determined by the corresponding wave function (WF) $\Phi (r_1,r_2,r_{12})$, where $r_1$ and $r_2$ are the electron-nucleus distances and $r_{12}$ is the distance between the electrons. It is also well known that the behavior of the ground state WF in the vicinity of the nucleus located at the origin is determined by the Fock expansion (FE) [1,2],

$$\Phi (r_1,r_2,r_{12})\equiv \Psi (R,\alpha ,\theta )=\sum _{k=0}^{\infty }{R}^k\sum _{p=0}^{[k/2]}{\psi }_{k,p}(\alpha ,\theta ){\ln }^{p}R,$$

(1)

where the hyperspherical coordinates (HSC) $R,\alpha$ and $\theta$ are defined by the relations

$$R=\sqrt{r_1^2+r_2^2},\alpha =2arctan\left({\displaystyle \frac{r_2}{r_1}}\right),\theta =arccos\left({\displaystyle \frac{r_1^2+r_2^2-r_{12}^2}{2r_1{r}_2}}\right),$$

(2)

whereas ${\psi }_{k,p}(\alpha ,\theta )$ are the angular Fock coefficients (AFC), many of which have been calculated previously (see, e.g., [3,4,5] and references therein). It is worth noting that the exact representation of the AFCs ${\psi }_{k,p}$ with even k and maximum values $p=k/2$ was recently presented in Ref. [6]. These AFCs will be used in what follows. Recall that the convergence of expansion (1) has been proven in Ref. [7].

There are a large amount of methods for calculating the electronic structure of the two-electron atomic systems. An excellent review on this topic can be found in Refs. [3,8,9,10,11]. However, we know only one technique that correctly represents the WF near the nucleus. It is the so-called correlation function hyperspherical harmonic method (CFHHM) [12,13,14]. This method uses an expansion similar to the FE to represent the WF near the nucleus and uses an expansion in hyperspherical harmonics (HHs) to represent the AFCs. However, the HH-expansion is known to converge very slowly. Although this method makes it possible to increase the convergence of the HH-expansion, a sufficiently good accuracy requires a large HH basis size (thousands of basis functions). Moreover, instead of the widely used variational method for solving the problem, an algebraic method for solving coupled systems of ordinary differential equations is used. All of the above creates great computational difficulties in implementing the method.

It follows that it would be extremely useful to develop a much simpler method for calculating the WF with correct behavior near the nucleus. The main objective of this article is to present such a method, as well as some of its non-trivial applications.

2. General Characteristics of the Method

The simplicity of the proposed method primarily implies the use of a standard variational method, which involves solving the generalized matrix eigenvalue equation

$$\widehat{\mathcal{H}}=E\widehat{\mathcal{S}},$$

(3)

where $\widehat{\mathcal{H}}\equiv \widehat{\mathcal{V}}+\widehat{\mathcal{T}}$ represents the matrix of the non-relativistic Hamiltonian, whereas $\widehat{\mathcal{S}}$, $\widehat{\mathcal{V}}$ and $\widehat{\mathcal{T}}$ are the overlap, potential and kinetic energy matrices, respectively, the elements of which are defined as follows:

$${\mathcal{S}}_{j,j^{\prime }}=\int f_j{f}_{j^{\prime }}dv,{\mathcal{V}}_{j,j^{\prime }}=\int f_{j}V{f}_{j^{\prime }}dv,{\mathcal{T}}_{j,j^{\prime }}=\int f_{j}T{f}_{j^{\prime }}dv.$$

(4)

The potential, V, and kinetic energy, T, operators will be defined later. The number of basis functions (BFs), $f_j$, is limited by the basis size, $N_b$$(1\le j\le N_b)$. The resulting WF can then be constructed in the form

$$\Psi =\sum _{j=1}^{N_b}{C}_j{f}_j,$$

(5)

where $C_j$ is the j-th component of the eigenvector corresponding to solution of Equation (3) with eigenvalue E. Note that all matrix elements (MEs) will be calculated using the HSC defined by Equation (2), and the atomic units system. This choice of coordinate system will help us to construct a WF, $\Psi$, with the behavior near the nucleus represented by the FE (1). Thus, for the volume element in the HSC we have [3]:

$$dv=R^{5}dRd\Omega ,d\Omega ={\pi }^2{sin}^2\alpha d\alpha sin\theta d\theta ,R\in [0,\infty ),\alpha \in [0,\pi ],\theta \in [0,\pi ]$$

(6)

The kinetic energy operator, T, is equal to $-(1/2)\Delta$, where the Laplacian is:

$$\Delta ={\displaystyle \frac{{\partial }^2}{\partial R^2}}+{\displaystyle \frac{5}{R}}{\displaystyle \frac{\partial }{\partial R}}-{\displaystyle \frac{{\Lambda }^2}{R^2}}.$$

(7)

The hyperspherical angular momentum operator ${\Lambda }^2\equiv {\Lambda }^2(\alpha ,\theta )$ is defined as follows (see, e.g., [3,4]):

$${\Lambda }^2=-4\left[{\displaystyle \frac{{\partial }^2}{\partial {\alpha }^2}}+2cot\alpha {\displaystyle \frac{\partial }{\partial \alpha }}+{\displaystyle \frac{1}{{sin}^2\alpha }}\left({\displaystyle \frac{{\partial }^2}{\partial {\theta }^2}}+cot\theta {\displaystyle \frac{\partial }{\partial \theta }}\right)\right].$$

(8)

It is reasonable to discuss the form of potential operator, V, for a two-electron atomic system later. At this stage, we will begin to introduce a set of BFs, which is the main feature of the proposed method. In particular, we propose to divide the entire set of BFs into two unequal parts. The first part will consist of only one, but rather complex, special basis function (SBF). The second part will be represented by a large set of simple BFs.

To proceed to introducing SBF, let us first recall the form of AFC for the linear in the R term of the FE:

$${\psi }_{1,0}(\alpha ,\theta )=-{\displaystyle \frac{Z(r_1+r_2)}{R}}+{\displaystyle \frac{r_{12}}{2R}}=-Z\eta +{\displaystyle \frac{1}{2}}\xi ,$$

(9)

where

$$\eta =\sqrt{1+sin\alpha },\xi =\sqrt{1-sin\alpha cos\theta }.$$

(10)

Secondly, it was already mentioned in the “Introduction” that the AFC ${\psi }_{k,k/2}\equiv {\psi }_{k,k/2}(\alpha ,\theta )$ with even k have recently been presented in the following closed analytical form:

$${\psi }_{k,k/2}=Z^{{\displaystyle \frac{k}{2}}}\sum _{l=k/2,(-2)}^0{\mathrm{c}}_{k,l}{Y}_{k,l}(\alpha ,\theta ),$$

(11)

where $Y_{k,l}(\alpha ,\theta )$ are HHs, and the closed form of the coefficients ${\mathrm{c}}_{k,l}$ for $k\le 10$ can be found in Ref. [6]. It is seen that both the AFCs (11) and the linear AFC (9) depend on the nucleus charge, Z, and do not depend on the energy, E. However, more importantly, the right-hand side (RHS) of the representation (11) can be reduced to the polynomial in variables $\eta$ and $\xi$ defined by Equation (10). In particular, we obtain:

$${\psi }_{k,k/2}={\left({\displaystyle \frac{Z}{\pi }}\right)}^{{\displaystyle \frac{k}{2}}}{\tilde{b}}_{{\displaystyle \frac{k}{2}}}\sum _{n,m}{b}_{{\displaystyle \frac{k}{2}},n,m}{\eta }^n{\xi }^m,$$

(12)

where the coefficients ${\tilde{b}}_{{\displaystyle \frac{k}{2}}}$ and $b_{{\displaystyle \frac{k}{2}},n,m}$ are represented in

Table 1. Factor ${\tilde{b}}_{k^{\prime }}$ and components of coefficients $b_{k^{\prime },n,m}={\tilde{q}}_{k^{\prime },n,m}{\sum }_{j=0}^{J_{k^{\prime }}}{q}_{k^{\prime },n,m}^{\left(j\right)}{\pi }^j$ ($0\le k^{\prime }\equiv k/2\le 5$), representing the polynomials in $\eta$ and $\xi$ according to Equation (12). The upper limit of summation is defined as $J_{k^{\prime }}=k^{\prime }-2$ for $k^{\prime }\ge 2$ and $J_{k^{\prime }}=0$ for $k^{\prime }<2$.

${\mathit{k}}^{\prime }$	${\tilde{\mathit{b}}}_{{\mathit{k}}^{\prime }}$	${\mathit{N}}^{\underline{\mathit{o}}}$	n	m	${\tilde{\mathit{q}}}_{{\mathit{k}}^{\prime },\mathit{n},\mathit{m}}$	${\mathit{q}}_{{\mathit{k}}^{\prime },\mathit{n},\mathit{m}}^{\left(0\right)}$	${\mathit{q}}_{{\mathit{k}}^{\prime },\mathit{n},\mathit{m}}^{\left(1\right)}$	${\mathit{q}}_{{\mathit{k}}^{\prime },\mathit{n},\mathit{m}}^{\left(2\right)}$	${\mathit{q}}_{{\mathit{k}}^{\prime },\mathit{n},\mathit{m}}^{\left(3\right)}$
5	$(2-\pi )(5\pi -14)/$	1	0	0	$1$	$-1089387008240$	$1047586705720$	$-335763343914$	$35868493935$
	5012817232500000	2	0	2	$15$	$4286509281040$	$-4129722511880$	$1326134522774$	$-141939797505$
		3	0	4	$60$	$-5335564976800$	$5141563833740$	$-1651435721371$	$176799081450$
		4	0	6	$40$	$12056179167440$	$-11618550815440$	$3732041258007$	$-399570465060$
		5	0	8	$-15$	$18776793358080$	$-18095537797140$	$5812646794643$	$-622341848670$
		6	0	10	$3$	$18776793358080$	$-18095537797140$	$5812646794643$	$-622341848670$
		7	8	0	$5$	$-31900530538240$	$30758808723020$	$-9885507900549$	$1058977460610$
		8	8	2	$-5$	$-31900530538240$	$30758808723020$	$-9885507900549$	$1058977460610$
		9	6	0	$-20$	$-31900530538240$	$30758808723020$	$-9885507900549$	$1058977460610$
		10	6	2	$20$	$-31900530538240$	$30758808723020$	$-9885507900549$	$1058977460610$
		11	2	0	$160$	$-967277327240$	$932324864590$	$-299527103049$	$32074335000$
		12	2	2	$120$	$9531787759840$	$-9188929121180$	$2952640515759$	$-316236217230$
		13	2	4	$60$	$-24726253970560$	$23837487905180$	$-7659813135081$	$820411311690$
		14	2	6	$-20$	$-24726253970560$	$23837487905180$	$-7659813135081$	$820411311690$
		15	4	0	$20$	$-28031421229280$	$2702950926466$	$-8687399488353$	$930680120610$
		16	4	2	$160$	$413145907340$	$-399002669935$	$128448294159$	$-13783601115$
		17	4	4	$30$	$24726253970560$	$-23837487905180$	$7659813135081$	$-820411311690$
		18	4	6	$-10$	$24726253970560$	$-23837487905180$	$7659813135081$	$-820411311690$
4	$(\pi -2)(5\pi -14)/$	1	0	0	$12$	$483896$	$-308420$	$49095$
	15431472000	2	2	0	$288$	$1201904$	$-775505$	$125073$
		3	4	0	$4$	$48803648$	$-31601320$	$5117187$
		4	6	0	$-4$	$92072192$	$-59519500$	$9619815$
		5	8	0	$1$	$92072192$	$-59519500$	$9619815$
		6	0	2	$-96$	$1685800$	$-1083925$	$174168$
		7	0	4	$12$	$37545376$	$-24164696$	$3886989$
		8	0	6	$-12$	$30802176$	$-19828996$	$3190317$
		9	0	8	$3$	$30802176$	$-19828996$	$3190317$
		10	2	2	$-24$	$46119680$	$-29773780$	$4804833$
		11	2	4	$12$	$46119680$	$-29773780$	$4804833$
		12	4	2	$12$	$46119680$	$-29773780$	$4804833$
		13	4	4	$-6$	$46119680$	$-29773780$	$4804833$
3	$(2-\pi )(5\pi -14)/$	1	0	0	$-2$	$106$	$-33$
	170100	2	2	0	$-2$	2064	$-675$
		3	4	0	$1$	2064	$-675$
		4	0	2	$4$	$609$	$-195$
		5	0	4	$-3$	1112	$-357$
		6	0	6	$1$	1112	$-357$
		7	2	2	$2$	2064	$-675$
		8	4	2	$-1$	2064	$-675$
2	$(\pi -2)(5\pi -14)/$	1	0	0	$1$	$1$
	180	2	2	0	$1$	$4$
		3	4	0	$1$	$-2$
		4	0	2	$-1$	$4$
		5	0	4	$-1$	$-2$
1	$-(\pi -2)/3$	1	0	0	$1$	$1$
		2	0	2	$1$	$-1$
0	1	1	0	0	$1$	$1$

.

Taking into account Equations (9) and (12), and also the fact that the products ${\left(\ln R\right)}^{k/2}{R}^k{\psi }_{k,k/2}$ represent the leading terms of the logarithmic series [6] of the FE, it is reasonable to introduce the SBF in the form:

$$f_0\equiv f_0(R,\alpha ,\theta )=exp\left[-R\left(Z\eta -{\displaystyle \frac{1}{2}}\xi \right)\right]\sum _{k^{\prime }=0\left(2\right)}^K{R}^{k^{\prime }}{\left(\ln R\right)}^{k^{\prime }/2}{\psi }_{k^{\prime },k^{\prime }/2}(\alpha ,\theta ),$$

(13)

where the AFCs are represented by Equation (12) and K is the parameter of the problem under consideration. It should be emphasized that the SBF (13) is designed to serve two purposes. The first one is to provide the correct representation of the FE terms with even k and maximum power $p=k/2$ of $\ln R$. And the second one is to guarantee the correct form of the AFC ${\psi }_{1,0}(\alpha ,\theta )$ or (which is equivalent) to guarantee that the WF will satisfy the Kato’s cusp conditions [15,16].

The main part of the complete basis set can be taken as:

$$f_{kp,nm}\equiv f_{kp,nm}(R,\alpha ,\theta )=exp\left[-R\left(\lambda \eta -\gamma \xi \right)\right]R^k{\left(\ln R\right)}^p{\eta }^n{\xi }^m,(k>1)$$

(14)

where, in general, $\lambda$ and $\gamma$ are variational parameters. It is clear that the basis set (14) is bounded by $k\ge 2$, since the constant ($k=0$) and linear ($k=1$) terms in R are described by the SBF (13).

The Coulomb potential operator in the HSC can now be introduced as follows:

$$V\equiv {\displaystyle \frac{1}{r_{12}}}-Z\left({\displaystyle \frac{1}{r_1}}+{\displaystyle \frac{1}{r_2}}\right)={\displaystyle \frac{1}{R}}\left({\displaystyle \frac{1}{\xi }}-{\displaystyle \frac{2Z\eta }{sin\alpha }}\right).$$

(15)

Note that, according to our building of the complete basis set, the MEs (4) of each of the three matrices $\widehat{\mathcal{T}}$, $\widehat{\mathcal{V}}$ and $\widehat{\mathcal{S}}$ can be divided into three groups. The first group includes MEs between BFs of the form (14), the second group between BFs (14) and (13) and the third group between BFs (13).

It is also important to note that the constant ${\pi }^2$ included in the angular space element $d\Omega$ (see definitions (6)) is a factor for all terms of the matrix Equation (3), and will therefore be omitted in what follows.

3. Overlap and Potential Energy Matrices

According to definition (4), the overlap MEs calculated between BFs of the form (14) are:

$$\begin{array}{c}{\mathcal{S}}_{kp,nm;k^{\prime }{p}^{\prime },n^{\prime }{m}^{\prime }}\equiv {\pi }^{-2}\int f_{kp,nm}{f}_{k^{\prime }{p}^{\prime },n^{\prime }{m}^{\prime }}dv=\hfill \\ {\int }_0^{\infty }{R}^{5}dR{\int }_0^{\pi }{sin}^2\alpha d\alpha {\int }_0^{\pi }{R}^{k+k^{\prime }}{\left(\ln R\right)}^{p+p^{\prime }}exp\left[-2R(\lambda \eta -\gamma \xi )\right]{\eta }^{n+n^{\prime }}{\xi }^{m+m^{\prime }}sin\theta d\theta =\hfill \\ {\mathcal{K}}_2(k+k^{\prime },p+p^{\prime },n+n^{\prime },m+m^{\prime };2\lambda ,2\gamma ),\hfill \end{array}$$

(16)

where the specific calculations of the general 3D (three-dimensional) integral

$$\begin{array}{c}{\mathcal{K}}_L(k,p,n,m;a,b)={\int }_0^{\pi }{\eta }^n{sin}^L\alpha d\alpha {\int }_0^{\pi }{\xi }^{m}sin\theta d\theta {\int }_0^{\infty }{R}^{k+5}{\left(\ln R\right)}^{p}exp\left[-R(a\eta -b\xi )\right]dR\hfill \end{array}$$

(17)

will be discussed in Appendix A.

The overlap MEs calculated between BFs (14) and SBF (13) can be represented in terms of the $\mathcal{K}$-integrals (17) as follows:

$$\begin{array}{c}{\mathcal{S}}_{kp,nm;0}\equiv {\pi }^{-2}\int f_{kp,nm}{f}_{0}dv=\hfill \\ =\sum _{k^{\prime }=0,\left(2\right)}^K{\left({\displaystyle \frac{Z}{\pi }}\right)}^{{\displaystyle \frac{k^{\prime }}{2}}}{\tilde{b}}_{{\displaystyle \frac{k^{\prime }}{2}}}\sum _{n^{\prime },m^{\prime }}{b}_{{\displaystyle \frac{k^{\prime }}{2}},n^{\prime },m^{\prime }}{\mathcal{K}}_2\left(k_1,p_2,n_1,m_1;\lambda +Z,\gamma +{\displaystyle \frac{1}{2}}\right),\hfill \end{array}$$

(18)

where we introduced notations:

$$k_1=k+k^{\prime },p_1=p+p^{\prime },p_2=p+k^{\prime }/2,n_1=n+n^{\prime },m_1=m+m^{\prime },$$

(19)

that remain relevant in what follows.

It is worth noting that the integer parameter K is determined by the number of the FE terms $R^{k^{\prime }}{\left(\ln R\right)}^{p^{\prime }}{\psi }_{k^{\prime },p^{\prime }}(\alpha ,\theta )$, with the maximum value $p^{\prime }=k^{\prime }/2$ included into the SBF (13) for this method option.

The third group of MEs, consisting of only one overlap ME calculated between SBFs (13), can be determined as:

$$\begin{array}{c}{\mathcal{S}}_{0;0}\equiv {\pi }^{-2}\int f_0^{2}dv=\hfill \\ \sum _{k=0,\left(2\right)}^K{\tilde{b}}_{{\displaystyle \frac{k}{2}}}\sum _{n,m}{b}_{{\displaystyle \frac{k}{2}},n,m}\sum _{k^{\prime }=0,\left(2\right)}^K{\left({\displaystyle \frac{Z}{\pi }}\right)}^{{\displaystyle \frac{k+k^{\prime }}{2}}}{\tilde{b}}_{{\displaystyle \frac{k^{\prime }}{2}}}\sum _{n^{\prime },m^{\prime }}{b}_{{\displaystyle \frac{k^{\prime }}{2}},n^{\prime },m^{\prime }}{\mathcal{K}}_2\left(k_1,{\displaystyle \frac{k_1}{2}},n_1,m_1;2Z,1\right).\hfill \end{array}$$

(20)

Following the previously proposed scheme for calculating the overlap matrix, it is easy to derive the corresponding formulas for the elements of the potential energy matrix. Thus, using the notations (19), we obtain:

$$\begin{array}{c}{\mathcal{V}}_{kp,nm;k^{\prime }{p}^{\prime },n^{\prime }{m}^{\prime }}\equiv {\pi }^{-2}\int f_{kp,nm}{\displaystyle \frac{1}{R}}\left({\displaystyle \frac{1}{\xi }}-{\displaystyle \frac{2Z\eta }{sin\alpha }}\right)f_{k^{\prime }{p}^{\prime },n^{\prime }{m}^{\prime }}dv=\hfill \\ {\mathcal{K}}_2(k_1-1,p_1,n_1,m_1-1;2\lambda ,2\gamma )-2Z{\mathcal{K}}_1(k_1-1,p_1,n_1+1,m_1;2\lambda ,2\gamma ),\hfill \end{array}$$

(21)

$$\begin{array}{c}{\mathcal{V}}_{kp,nm;0}\equiv {\pi }^{-2}\int f_{kp,nm}{\displaystyle \frac{1}{R}}\left({\displaystyle \frac{1}{\xi }}-{\displaystyle \frac{2Z\eta }{sin\alpha }}\right)f_{0}dv=\sum _{k^{\prime }=0,\left(2\right)}^K{\left({\displaystyle \frac{Z}{\pi }}\right)}^{{\displaystyle \frac{k^{\prime }}{2}}}{\tilde{b}}_{{\displaystyle \frac{k^{\prime }}{2}}}\sum _{n^{\prime },m^{\prime }}{b}_{{\displaystyle \frac{k^{\prime }}{2}},n^{\prime },m^{\prime }}\times \hfill \\ \left[{\mathcal{K}}_2\left(k_1-1,p_2,n_1,m_1-1;\lambda +Z,\gamma +{\displaystyle \frac{1}{2}}\right)-2Z{\mathcal{K}}_1\left(k_1-1,p_2,n_1+1,m_1;\lambda +Z,\gamma +{\displaystyle \frac{1}{2}}\right)\right],\hfill \end{array}$$

(22)

$$\begin{array}{c}{\mathcal{V}}_{0;0}\equiv {\pi }^{-2}\int f_0^2{\displaystyle \frac{1}{R}}\left({\displaystyle \frac{1}{\xi }}-{\displaystyle \frac{2Z\eta }{sin\alpha }}\right)dv=\sum _{k=0,\left(2\right)}^K{\tilde{b}}_{{\displaystyle \frac{k}{2}}}\sum _{n,m}{b}_{{\displaystyle \frac{k}{2}},n,m}\sum _{k^{\prime }=0,\left(2\right)}^K{\left({\displaystyle \frac{Z}{\pi }}\right)}^{{\displaystyle \frac{k+k^{\prime }}{2}}}{\tilde{b}}_{{\displaystyle \frac{k^{\prime }}{2}}}\sum _{n^{\prime },m^{\prime }}{b}_{{\displaystyle \frac{k^{\prime }}{2}},n^{\prime },m^{\prime }}\times \hfill \\ \left[{\mathcal{K}}_2\left(k_1-1,{\displaystyle \frac{k_1}{2}},n_1,m_1-1;2Z,1\right)-2Z{\mathcal{K}}_1\left(k_1-1,{\displaystyle \frac{k_1}{2}},n_1+1,m_1;2Z,1\right)\right].\hfill \end{array}$$

(23)

4. Kinetic Energy Matrix

To simplify the derivation of calculation formulas for the kinetic energy MEs defined by Equations (4) and (7), we divide them into two parts as follows:

$${\mathcal{T}}_{j,j^{\prime }}=-{\displaystyle \frac{1}{2}}\left[{\mathcal{T}}_{j,j^{\prime }}^{\left(1\right)}-{\mathcal{T}}_{j,j^{\prime }}^{\left(2\right)}\right],$$

(24)

where

$${\mathcal{T}}_{j,j^{\prime }}^{\left(1\right)}={\pi }^{-2}\int f_{j^{\prime }}\left({\displaystyle \frac{{\partial }^2}{\partial R^2}}+{\displaystyle \frac{5}{R}}{\displaystyle \frac{\partial }{\partial R}}\right)f_{j}dv,{\mathcal{T}}_{j,j^{\prime }}^{\left(2\right)}={\pi }^{-2}\int f_{j^{\prime }}{\displaystyle \frac{{\Lambda }^2}{R^2}}{f}_{j}dv.$$

(25)

To proceed, we use the results of the action of the Laplace operator components on the constituents of our complete basis set defined by Equations (13) and (14). In particular, we obtain:

$$\begin{array}{c}\left({\displaystyle \frac{{\partial }^2}{\partial R^2}}+{\displaystyle \frac{5}{R}}{\displaystyle \frac{\partial }{\partial R}}\right)exp\left[-R\left(a\eta -b\xi \right)\right]R^k{\left(\ln R\right)}^p=exp\left[-R\left(a\eta -b\xi \right)\right]R^k{\left(\ln R\right)}^p\times \hfill \\ \left[{\displaystyle \frac{p(p-1)}{R^2{\ln }^{2}R}}+{\displaystyle \frac{2p(k+2)}{R^2\ln R}}+{\displaystyle \frac{k(k+4)}{R^2}}+{(a\eta -b\xi )}^2-\left({\displaystyle \frac{2p}{R\ln R}}+{\displaystyle \frac{2k+5}{R}}\right)(a\eta -b\xi )\right],\hfill \end{array}$$

(26)

$$\begin{array}{c}{\Lambda }^{2}exp\left[-R\left(a\eta -b\xi \right)\right]{\eta }^n{\xi }^m=exp\left[-R\left(a\eta -b\xi \right)\right]{\eta }^n{\xi }^m\times \hfill \\ \left\{(m+n)(m+n+4)-2(a^2+b^2)R^2-{\displaystyle \frac{2n(n-1)}{{\eta }^2}}+{\displaystyle \frac{4naR}{\eta }}+a^2{\eta }^2{R}^2-{\displaystyle \frac{2m(m+n+1)}{{\xi }^2}}\right.\hfill \\ -{\displaystyle \frac{2(2m+n+2)bR}{\xi }}+b(2m+2n+5)\xi R+b^2{\xi }^2{R}^2-a(2m+2n+1)\eta R+{\displaystyle \frac{2ma\eta R}{{\xi }^2}}\hfill \\ +2ab\eta \left({\displaystyle \frac{1}{\xi }}-\xi \right)R^2+{\displaystyle \frac{2}{sin\alpha }}\left[-n(m+2)+{\displaystyle \frac{mn}{{\xi }^2}}+nb\left({\displaystyle \frac{1}{\xi }}-\xi \right)R\right.\hfill \\ \left.\left.+a(m+4)\eta R-2a{\eta }^{3}R-{\displaystyle \frac{ma\eta R}{{\xi }^2}}-ab\eta \left({\displaystyle \frac{1}{\xi }}-\xi \right)R^2\right]\right\}.\hfill \end{array}$$

(27)

It can be seen that the angular parts of both results represented by the RHSs of the last two equations are expressed in terms of the angular variables $\eta$ and $\xi$, and the factor ${sin}^{-1}\alpha$. This enables us to represent the kinetic energy MEs through the integrals of the form (17).

Thus, using Equation (25), as well as Equations (26) and (27) with $a=\lambda$ and $b=\gamma$, we obtain for the kinetic energy MEs calculated between BFs of the form (14):

$$\begin{array}{c}{\mathcal{T}}_{k^{\prime }{p}^{\prime },n^{\prime }{m}^{\prime };kp,nm}^{\left(1\right)}=p(p-1){\mathcal{K}}_2(k_1-2,p_1-2,n_1,m_1)+2p(k+2){\mathcal{K}}_2(k_1-2,p_1-1,n_1,m_1)\hfill \\ +k(k+4){\mathcal{K}}_2(k_1-2,p_1,n_1,m_1)+{\lambda }^2{\mathcal{K}}_2(k_1,p_1,n_1+2,m_1)-2\lambda \gamma {\mathcal{K}}_2(k_1,p_1,n_1+1,m_1+1)\hfill \\ +{\gamma }^2{\mathcal{K}}_2(k_1,p_1,n_1,m_1+2)-2p\left[\lambda {\mathcal{K}}_2(k_1-1,p_1-1,n_1+1,m_1)-\gamma {\mathcal{K}}_2(k_1-1,p_1-1,n_1,m_1+1)\right]\hfill \\ -(2k+5)\left[\lambda {\mathcal{K}}_2(k_1-1,p_1,n_1+1,m_1)-\gamma {\mathcal{K}}_2(k_1-1,p_1,n_1,m_1+1)\right],\hfill \end{array}$$

(28)

$$\begin{array}{c}{\mathcal{T}}_{k^{\prime }{p}^{\prime },n^{\prime }{m}^{\prime };kp,nm}^{\left(2\right)}=(m+n)(m+n+4){\mathcal{K}}_2(k_1-2,p_1,n_1,m_1)-2({\lambda }^2+{\gamma }^2){\mathcal{K}}_2(k_1,p_1,n_1,m_1)\hfill \\ -2n(n-1){\mathcal{K}}_2(k_1-2,p_1,n_1-2,m_1)+4n\lambda {\mathcal{K}}_2(k_1-1,p_1,n_1-1,m_1)\hfill \\ -2m(m+n+1){\mathcal{K}}_2(k_1-2,p_1,n_1,m_1-2)-2(2m+n+2)\gamma {\mathcal{K}}_2(k_1-1,p_1,n_1,m_1-1)\hfill \\ +\gamma (2m+2n+5){\mathcal{K}}_2(k_1-1,p_1,n_1,m_1+1)+{\gamma }^2{\mathcal{K}}_2(k_1,p_1,n_1,m_1+2)\hfill \\ -\lambda (2m+2n+1){\mathcal{K}}_2(k_1-1,p_1,n_1+1,m_1)+2m\lambda {\mathcal{K}}_2(k_1-1,p_1,n_1+1,m_1-2)\hfill \\ +2\lambda \gamma \left[{\mathcal{K}}_2(k_1,p_1,n_1+1,m_1-1)-{\mathcal{K}}_2(k_1,p_1,n_1+1,m_1+1)\right]\hfill \\ +{\lambda }^2{\mathcal{K}}_2(k_1,p_1,n_1+2,m_1)-2n(m+2){\mathcal{K}}_1(k_1-2,p_1,n_1,m_1)\hfill \\ +2n\gamma \left[{\mathcal{K}}_1(k_1-1,p_1,n_1,m_1-1)-{\mathcal{K}}_1(k_1-1,p_1,n_1,m_1+1)\right]\hfill \\ +2mn{\mathcal{K}}_1(k_1-2,p_1,n_1,m_1-2)+2\lambda (m+4){\mathcal{K}}_1(k_1-1,p_1,n_1+1,m_1)\hfill \\ -4\lambda {\mathcal{K}}_1(k_1-1,p_1,n_1+3,m_1)-2m\lambda {\mathcal{K}}_1(k_1-1,p_1,n_1+1,m_1-2)\hfill \\ -2\lambda \gamma \left[{\mathcal{K}}_1(k_1,p_1,n_1+1,m_1-1)-{\mathcal{K}}_1(k_1,p_1,n_1+1,m_1+1)\right].\hfill \end{array}$$

(29)

For simplicity, we replaced the designations ${\mathcal{K}}_L(k,p,n,m;2\lambda ,2\gamma )$ with ${\mathcal{K}}_L(k,p,n,m)$ for all $\mathcal{K}$-integrals in Equations (28) and (29), that is, we simply omitted the last two common parameters. We also used the notations introduced in Equation (19).

Similarly, for the kinetic energy MEs calculated between BFs (13) and SBF (14), we obtain:

$${\mathcal{T}}_{0;kp,nm}^{\left(j\right)}=\sum _{k^{\prime }=0,\left(2\right)}^K{\left({\displaystyle \frac{Z}{\pi }}\right)}^{{\displaystyle \frac{k^{\prime }}{2}}}{\tilde{b}}_{{\displaystyle \frac{k^{\prime }}{2}}}\sum _{n^{\prime },m^{\prime }}{b}_{{\displaystyle \frac{k^{\prime }}{2}},n^{\prime },m^{\prime }}{\widehat{\mathcal{T}}}_{k^{\prime }{p}^{\prime },n^{\prime }{m}^{\prime };kp,nm}^{\left(j\right)},(j=1,2)$$

(30)

where ${\widehat{\mathcal{T}}}_{k^{\prime }{p}^{\prime },n^{\prime }{m}^{\prime };kp,nm}^{\left(1\right)}$ and ${\widehat{\mathcal{T}}}_{k^{\prime }{p}^{\prime },n^{\prime }{m}^{\prime };kp,nm}^{\left(2\right)}$ represent the RHSs of Equations (28) and (29), respectively, but in which $p_1$ should be replaced by $p_2$ (see Equation (19)), and the last two common parameters in the $\mathcal{K}$-integrals should be taken successively as $\lambda +Z$ and $\gamma +1/2$, similar to Equations (18) and (22).

Accordingly, the kinetic energy MEs calculated between SBFs can be represented as:

$${\mathcal{T}}_{0;0}^{\left(j\right)}=\sum _{k=0,\left(2\right)}^K{\tilde{b}}_{{\displaystyle \frac{k}{2}}}\sum _{n,m}{b}_{{\displaystyle \frac{k}{2}},n,m}\sum _{k^{\prime }=0,\left(2\right)}^K{\left({\displaystyle \frac{Z}{\pi }}\right)}^{{\displaystyle \frac{k+k^{\prime }}{2}}}{\tilde{b}}_{{\displaystyle \frac{k^{\prime }}{2}}}\sum _{n^{\prime },m^{\prime }}{b}_{{\displaystyle \frac{k^{\prime }}{2}},n^{\prime },m^{\prime }}{\widehat{T}}_{k^{\prime }{p}^{\prime },n^{\prime }{m}^{\prime };kp,nm}^{\left(j\right)},(j=1,2)$$

(31)

where ${\widehat{T}}_{k^{\prime }{p}^{\prime },n^{\prime }{m}^{\prime };kp,nm}^{\left(1\right)}$ and ${\widehat{T}}_{k^{\prime }{p}^{\prime },n^{\prime }{m}^{\prime };kp,nm}^{\left(2\right)}$ represent the RHSs of Equations (28) and (29), respectively, but in which the parameters $p_1,\lambda$ and $\gamma$ should be replaced by $k_1/2,Z$ and $1/2$, respectively, and the last two common parameters in the corresponding $\mathcal{K}$-integrals should be taken successively as $2Z$ and 1.

5. Numerical Results and Discussion

For convenience, we will call the method described above as the variational method of near the nucleus calculations (VMNN). We applied VMNN to calculate the ground state energies, E, and the corresponding WFs, $\Psi$, for the helium atom and a number of two-electron ions.

The results corresponding to the simplest version of the method with $\lambda =Z$ and $\gamma =1/2$ are presented in

Table 2. Absolute values of the ground state energies, E, and the corresponding FE coefficients $a_{21}$ calculated by VMNN with basis size $N_b$. The sixth column presents an estimate of the deviation from the virial theorem. The two relevant matrix elements are presented in the last two columns.

Atom/Ion	Z	${\mathit{a}}_{21}$	${\mathit{N}}_{\mathit{b}}$	$\left|\mathit{E}\right|$	$\langle \mathit{V}\rangle /\langle \mathit{T}\rangle +2$	$\langle \mathit{\delta }\left({\mathbf{r}}_1\right)\rangle$	$\langle \mathit{\delta }\left({\mathbf{r}}_1\right)\mathit{\delta }\left({\mathbf{r}}_2\right)\rangle$
	$1.5$	0.38238	516	1.465 279 051 825 740	4.6 (−13)	0.682115722	0.20291
He	2	0.47675	516	2.903 724 377 034 119	8.7 (−15)	1.8104293184	1.86874
Li+	3	0.62285	516	7.279 913 412 669 306	4.1 (−17)	6.852009437	33.32118
Be2+	4	0.72787	516	13.655 566 238 423 587	1.4 (−16)	17.198172547	231.0389
B3+	5	0.80232	516	22.030 971 580 242 781	−1.4 (−16)	34.75874366	996.0099
C4+	6	0.85240	516	32.406 246 601 898 530	−1.3 (−16)	61.44357805	3221.850
N5+	7	0.88221	441	44.781 445 148 772 703	5.3 (−16)	99.1625345	8596.074
O6+	8	0.89466	441	59.156 595 122 757 924	3.6 (−16)	149.8254726	19,974.31
F7+	9	0.89193	441	75.531 712 363 959 490	2.6 (−16)	215.3422520	41,827.47
Ne8+	10	0.87572	441	93.906 806 515 037 548	2.0 (−16)	297.6227308	80,761.89
Na9+	11	0.84738	441	114.281 883 776 072 721	1.5 (−16)	398.5767736	146,112.4
Mg10+	12	0.80801	441	136.656 948 312 646 929	1.2 (−16)	520.1142308	250,608.2
Al11+	13	0.75854	441	161.032 003 026 058 359	1.0 (−16)	664.1449752	411,112.2
Si12+	14	0.69974	441	187.407 049 998 662 925	8.5 (−17)	832.5788588	649,432.6
P13+	15	0.63229	441	215.782 090 763 537 159	7.2 (−17)	1027.325721	993,207.9
S14+	16	0.55675	441	246.157 126 474 254 738	6.2 (−17)	1250.295442	147,686.5 (+1)
Cl15+	17	0.47364	441	278.532 158 015 400 094	5.3 (−17)	1503.397895	214,264.8 (+1)
Ar16+	18	0.38340	441	312.907 186 076 611 148	4.7 (−17)	1788.543432	304,172.8 (+1)
K17+	19	0.28642	441	349.282 211 203 453 166	4.1 (−17)	2107.640384	423,536.9 (+1)
Ca18+	20	0.18307	441	387.657 233 833 158 555	3.7 (−17)	2462.600113	579,618.9 (+1)
Sc19+	21	0.07366	441	428.032 254 320 234 690	3.3 (−17)	2855.332029	780,947.6 (+1)
Ti20+	22	−0.04152	441	470.407 272 955 138 383	2.9 (−17)	3287.745943	103,745.9 (+2)
V21+	23	−0.16221	441	514.782 289 978 111 773	2.7 (−17)	3761.751772	136,064.1 (+2)
Cr22+	24	−0.28816	441	561.157 305 589 581 271	2.4 (−17)	4279.259269	176,369.1 (+2)
Mn23+	25	−0.41916	441	609.532 319 958 075 745	2.2 (−17)	4842.178389	226,167.5 (+2)
Fe24+	26	−0.55501	441	659.907 333 226 327 804	2.0 (−17)	5452.418955	287,169.7 (+2)
Co25+	27	−0.69552	441	712.282 345 516 026 550	1.8 (−17)	6111.890859	361,307.4 (+2)
Ni26+	28	−0.84052	441	766.657 356 931 557 099	1.7 (−17)	6822.503872	450,752.1 (+2)
Zn28+	30	−1.14334	441	881.407 377 488 360 605	1.5 (−17)	8404.792902	685,562.3 (+2)
Zr38+	40	−2.88133	441	1575.157 449 525 559 44	7.8 (−18)	20,034.060	392,478.5 (+3)
Sn48+	50	−4.92537	441	2468.907 492 812 711 76	4.9 (−18)	39,260.262	151,406.5 (+4)
Nd58+	60	−7.21331	441	3562.657 521 697 319 28	3.3 (−18)	67,993.257	455,483.1 (+4)
Yb68+	70	−9.70406	441	4856.407 542 342 006 26	2.4 (−18)	108,142.90	115,468.8 (+5)
Hg78+	80	−12.3685	441	6350.157 557 832 474 81	1.8 (−18)	161,619.06	258,314.5 (+5)
Th88+	90	−15.1846	441	8043.907 569 884 711 27	1.4 (−18)	230,331.59	525,303.7 (+5)
Fm98+	100	−18.1357	441	9937.657 579 529 067 14	1.1 (−18)	316,190.36	990,905.1 (+5)

. Note that we call this variant “simplest” because the mentioned choice of parameters implies that both the SBF and the set of BFs (14) contain the same exponential. In the sixth column we present one of the most important characteristics of the obtained state, namely, the corresponding deviation from the virial theorem for the Coulomb interactions. The ${10}^{-M}$ factor has been replaced with $(-M)$. To confirm the main purpose of VMNN, in the last two columns we present two matrix elements of the relevant operators, which represent the main properties of WF near the nucleus and can be calculated in the following simple way:

$$\langle \delta \left({\mathbf{r}}_1\right)\rangle ={\displaystyle \frac{4\pi }{N}}{\int }_0^{\infty }{\Psi }^2(R,0,0)R^{2}dR,\langle \delta \left({\mathbf{r}}_1\right)\delta \left({\mathbf{r}}_2\right)\rangle ={\displaystyle \frac{1}{N}}{\Psi }^2(0,0,0),$$

(32)

where $\delta \left(\mathbf{r}\right)$ denotes the Dirac delta function, and the normalization coefficient is:

$$N=\int {\Psi }^2(R,\alpha ,\theta )dv.$$

(33)

The last significant digit in energy and matrix elements (32) presented in Table 2 is the first one that differs from the more accurate calculations (see, e.g., [11,17,18,19]) for $2\le Z\le 28$, or from the Pekeris-like method [20,21] for non-integer values of Z. The case $Z=28$ is the maximum nucleus charge for which we found numerical results in the relevant scientific literature. Note that fairly accurate values of the ground state energy, especially for large Z, can be calculated using the so called $1/Z$ expansion (see, e.g., [22] and references therein). In particular, all 14 digits after the decimal point for the energy values presented in Table 2 for $Z\ge 40$ are in complete agreement with the corresponding results obtained using the $1/Z$ expansion. In addition, the energies for any Z obtained using this approximation can be used to solve the eigenvalue Equation (3) by a very efficient “Arnoldi” method, which is also known as the “Lanczos” method.

It is well-known that the negative ion of hydrogen is the only ion in the two-electron atomic sequence that has only one bound (ground) state. In other words, there are no excited bound states in this ion. At the same time, its WF is rather diffuse. These features cause special computational difficulties when using not only variational methods, including the one presented here, or the Pekeris-like method [20,21], but also CFHHM [12,13,14]. To solve this problem, both a significantly higher calculation accuracy and a significantly longer basis size are required. It should be emphasized that the given problem concerns only a two-electron negative ion with a nuclear charge, Z, equal to exactly 1. Note that there is no mathematical (computational) method of calculating the electronic structure for a two-electron system with non-integer Z. In particular, the unrealistic case $Z=1.5$, presented in Table 2, was calculated without any specific problems. These results will be useful in solving the problem of calculating the function $a_{21}\left(Z\right)$, which will be discussed below.

It is seen from representation (5) that the WF is defined, first of all, by the set of BFs, which in turn are determined by the sets of integers $\left\{k,p,n,m\right\}$ for the BFs (14), and by integer number K for the SBF (13). To construct the large set (14) we used two characteristic numbers, $k_{max}$ and ${\omega }_{max}$, such that

$$1<k\le k_{max},0\le n\le {\omega }_{max},0\le m\le {\omega }_{max},$$

(34)

but under the additional condition

$$k+p+n+m\le max(k_{max},{\omega }_{max}).$$

(35)

The power, p, of logarithm can vary as follows: $0\le p\le [k/2]$, where $\left[x\right]$ denotes (as in the FE) the integer part of x. It is clear that all cases of $p=k/2$ for even k must be excluded from the set (14), since these cases are represented by the SBF (13) with upper limit $K=2[k_{max}/2]$. To reduce the length of the set (14), one can use the well-known closed form of the AFC [3,4]:

$${\psi }_{3,1}(\alpha ,\theta )={\displaystyle \frac{Z(\pi -2)}{36\pi }}\left[6Z\eta (1-{\xi }^2)+\xi (5{\xi }^2-6)\right].$$

(36)

The representation (36), together with the fact that the series expansion of the SBF (13) gives the correct coefficient $Z(2-\pi )/\left(6\pi \right)$ for $\xi$ in the angular part for $R^3\ln R$, enables us to restrict BFs (14) with $\left\{k,p\right\}=\left\{3,1\right\}$ to angular parts defined by only three sets of integers: $\left\{n,m\right\}=\left\{1,0\right\},\left\{1,2\right\},\left\{0,3\right\}$. Thus, using all the above conditions, we obtain basis sets, in particular those presented in Table 2, of length 516 and 441 for $\left\{k_{max},{\omega }_{max}\right\}$ equal to $\left\{7,13\right\}$ and $\left\{8,12\right\}$, respectively.

The following comments are needed regarding the FE parameter $a_{21}$, calculated by VMNN and presented in Table 2.

The AFCs ${\psi }_{k,p}$ satisfy the Fock recurrence relation,

$$\left[{\Lambda }^2-k(k+4)\right]{\psi }_{k,p}(\alpha ,\theta )=h_{k,p}(\alpha ,\theta ),$$

(37)

where the operator ${\Lambda }^2$ is defined by Equation (8). The specific form of the RHS $h_{k,p}(\alpha ,\theta )$, which is not important for our consideration, can be found, for example, in Ref. [3]. What really matters is that the general solution of the inhomogeneous differential Equation (37) can be expressed as the sum of a particular solution, ${\tilde{\psi }}_{k,p}$, of that equation and the general solution ${\varphi }_{k,p}$ of the associated homogeneous equation

$$\left[{\Lambda }^2-k(k+4)\right]{\varphi }_{k,p}(\alpha ,\theta )=0,$$

(38)

which can be represented as (see, e.g., [3,4])

$${\varphi }_{k,p}(\alpha ,\theta )=\sum _{l=0}^{[k/2]}{a}_{kl}^{\left(p\right)}{Y}_{kl}(\alpha ,\theta ),$$

(39)

where $Y_{kl}$ are the unnormalized HHs. A particular solution, ${\tilde{\psi }}_{k,p}$, can be obtained by directly solving Equation (37) under suitable boundary conditions. The contribution of ${\varphi }_{k,p}$ is determined by the coefficients $a_{k,l}^{\left(p\right)}$, which can only be found using the WF, $\Psi (R,\alpha ,\theta )$, calculated, e.g., by VMNN.

We are interested in the case $k=2,p=0$ which has been discussed, e.g., in Ref. [16]. For this case, Equation (39) becomes

$${\varphi }_{2,0}(\alpha ,\theta )=a_{20}^{\left(0\right)}{Y}_{20}(\alpha ,\theta )+a_{21}^{\left(0\right)}{Y}_{21}(\alpha ,\theta ).$$

(40)

For singlet S states (including the ground state) the coefficient $a_{20}^{\left(0\right)}$ is identically zero, since the unnormalized HH, $Y_{20}(\alpha ,\theta )\equiv 2cos\alpha$ does not preserve the sign under the transformation $\alpha \rightleftharpoons \pi -\alpha$, which essentially corresponds to a permutation of electrons. Recall that the WF of a two-electron atomic system must preserve its parity with such a permutation. To calculate the only remaining non-zero coefficient, $a_{21}\equiv a_{21}^{\left(0\right)}$, we must first obtain the angular function ${\tilde{\varphi }}_{20}(\alpha ,\theta )\simeq {\varphi }_{20}(\alpha ,\theta )$, which is a factor for $R^2$ in the series expansion of the WF, $\Psi (R,\alpha ,\theta )$ near the the nucleus $(R\to 0)$, calculated by VMNN. These functions can then be used to calculate the required coefficient $a_{21}$ as follows:

$$a_{21}={\pi }^2{N}_{21}^2{\int }_0^{\pi }{\int }_0^{\pi }{\tilde{\varphi }}_{20}(\alpha ,\theta )Y_{21}(\alpha ,\theta ){sin}^2\alpha sin\theta d\alpha d\theta ,$$

(41)

where $N_{21}=2{\pi }^{-3/2}$ is the normalization coefficient for the HH, $Y_{21}(\alpha ,\theta )=sin\alpha cos\theta$. Note that the last relation represents a particular case for calculating the expansion coefficients in HHs for the general function of angles $\alpha$ and $\theta$ (see, e.g., [3]). It is the coefficients $a_{21}$, computed according to Equation (41), that are presented in Table 2.

It is worth noting the following. The term ${\varphi }_{20}(\alpha ,\theta )=a_{21}sin\alpha cos\theta$ represents the total contribution of the HH, $Y_{21}(\alpha ,\theta )$ into the AFC ${\psi }_{2,0}(\alpha ,\theta )={\widehat{\psi }}_{2,0}(\alpha ,\theta )+{\varphi }_{20}(\alpha ,\theta )$. On the other hand, in practice, all obtained particular solutions ${\widehat{\psi }}_{2,0}(\alpha ,\theta )$ of the corresponding Fock recurrence relation also contain an admixture of $Y_{21}(\alpha ,\theta )$. Therefore, before adding the components ${\varphi }_{20}(\alpha ,\theta )$ and ${\widehat{\psi }}_{2,0}(\alpha ,\theta )$, it is necessary to get rid of the admixture $C_{21}{Y}_{21}(\alpha ,\theta )$ in the latter component to obtain the correct AFC, ${\psi }_{20}(\alpha ,\theta )$. In particular, the authors of Ref. [23] were the first to obtain a particular solution ${\widehat{\psi }}_{2,0}(\alpha ,\theta )$ in a closed analytic form (see also, [4,10]) for the helium-like sequence. The corresponding admixture coefficient (at least for singlet S states) was obtained [24] in the form:

$$C_{21}={\displaystyle \frac{Z\left(62+17\pi -48G\right)}{72\pi }},$$

(42)

where G is Catalan’s constant.

The results for $a_{21}$ as a function of the nucleus charge, Z, are graphically presented in Figure 1. The curves corresponding to the full range, $Z\in [1.5,100]$, and the partial range, $Z\in [1.5,22]$, are shown in the left and right columns, respectively. It is seen that the function $a_{21}\left(Z\right)$ has two characteristic points. These are the maximum point, $Z_{max}\simeq 8.30613237$, and the zero point, $Z_0\simeq 21.645108$, of the function. Note that, in order to obtain more accurate results, we calculated $a_{21}$ for a set of additional half-integer points, in particular for $Z=1.5,2.5,3.5,7.5,8.5,9.5$. It was found that a simple analytic function of the form

$$g\left(Z\right)={\displaystyle \frac{c_{-1}}{Z}}+c_0+c_{1}Z(c_2+\ln Z),$$

(43)

with properly chosen parameters $c_{-1},c_0,c_1$ and $c_2$ provides an excellent interpolation of the curve calculated using VMNN. Direct numerical interpolation gives a good hint for the following choice of two of the four parameters:

$$c_0=-{\displaystyle \frac{1}{9}},c_1={\displaystyle \frac{2-\pi }{3\pi }}.$$

(44)

It is easy to verify that the second parameter in the last equation represents the factor for the AFC ${\psi }_{21}$. The two remaining parameters can be found by solving the system of two equations, $g^{\prime }\left(Z_{max}\right)=0$ and $g\left(Z_0\right)=0$, which, given the results (44), yields

$$c_{-1}=0.0012024376,c_2=-3.117138.$$

(45)

The accuracy of the resulting function, $g\left(Z\right)$, was estimated using the function ${log}_{10}\left|1-a_{21}/g\left(Z\right)\right|$, which characterizes the relative difference between the functions under consideration. This evaluation function is shown in the bottom row of Figure 1. It can be seen that the difference is less than a hundredth of a percent. It is important to note that all the coefficients $a_{21}$ that were used to construct Figure 1, most of which are presented in Table 2, coincide (at least to 5 significant digits) with those calculated using the AFCs computed by CFHHM.

It should be emphasized that although the above results confirm the correct behavior of the WF near the nucleus, this does not mean that the WF behaves incorrectly at sufficiently large R. On the contrary, the sufficiently high accuracy of the calculations of the energy and parameters characterizing the virial theorem tells us the opposite.

6. Conclusions

The results presented here are devoted to the development of a simple method of non-relativistic variational calculations of the electronic structure of a two-electron atom/ion in the ground state. Although we show numerical results (see Table 2), including for positive ions with a nuclear charge $Z\ge 30$, which suggest the presence of a significant relativistic component, it should be emphasized that our method is intended only for calculating energies and wave functions corresponding to the non-relativistic Hamiltonian. The results for large Z are intended only to demonstrate the accuracy of the presented method, which is fully consistent with correlation function hyperspherical harmonic [12,13,14] and Pekeris-like [20,21] methods, as well as with the $1/Z$ expansion [22], which is intended solely for calculating energies. Unlike the $1/Z$ expansion and some other techniques, the method described in Section 2, Section 3 and Section 4 is designed to calculate both the non-relativistic energy and the wave function, representing with high accuracy the ground state of a two-electron atom/ion, especially near the nucleus.

The method as a whole is variational, consisting of a standard solution of a generalized matrix eigenvalue equation, all matrix elements of which are reduced to the numerical calculation of one-dimensional integrals (see Appendix A).

High accuracy of the wave function near the nucleus is achieved primarily due to the following features of the calculation method.

The complete basis set is divided into two unequal parts, each of which is constructed using hyperspherical coordinates $\alpha ,\theta$ and R.

Unlike other methods that use hyperspherical harmonics $Y_{k,l}(\alpha ,\theta )$, our method instead uses powers of specific functions $\eta \equiv \sqrt{1+sin\alpha }=(r_1+r_2)/R$ and $\xi \equiv \sqrt{1-sin\alpha cos\theta }=r_{12}/R$ (see Equation (2)) as constructive elements of the angular parts of the basis functions. This enables us to significantly reduce the basis size, which in turn simplifies and speeds up computations.

The main (in quantity, but not in importance) part of the complete basis set is represented by the basis functions of the form (14), including logarithms of the hyperspherical radius R, similar to the Fock expansion (1).

It should be emphasized that the most characteristic feature of our method is the additional inclusion of a special basis function of the form (13) in the complete basis set. It is this basis function, involving the leading angular Fock coefficients, that ensures the correct behavior of the wave function near the nucleus, including the guarantee of satisfying the Kato’s cusp conditions [15].

To test the method under consideration, we calculated the four most characteristic parameters presented in Table 2 for various charges Z of the nuclei. These are the non-relativistic energy, E, and the deviation from the virial theorem, $\langle V\rangle /\langle T\rangle +2$, demonstrating the high accuracy of the method, as well as the matrix elements of the operators $\delta \left({\mathbf{r}}_1\right)$ and $\delta \left({\mathbf{r}}_1\right)\delta \left({\mathbf{r}}_2\right)$, demonstrating the high accuracy of the wave function near the nucleus in particular.

The correct behavior of the wave function at all points of the configuration space enabled us to calculate the Fock expansion coefficient $a_{21}$ (see Equation (41)) as a function of the nucleus charge Z. The corresponding results presented in Table 2 are in full agreement with the correlation function hyperspherical harmonic method [12,13,14], giving the most accurate results related to the behavior of the wave function near the nucleus. A plot of the function $a_{21}\left(Z\right)$, together with a very simple interpolating function, $g\left(Z\right)$ (see Equations (43)–(45)), are shown in Figure 1.