An explicitly correlated helium wave function in hyperspherical coordinates

doi:10.1016/j.cplett.2018.02.069

Chemical Physics Letters

Volume 698, 16 April 2018, Pages 120-126

https://doi.org/10.1016/j.cplett.2018.02.069 Get rights and content

Highlights

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Method for calculation of a basis set for solution of 3-Body problem.
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Proposed basis have no problems with crossterms in Hamilton operator.
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Basis B: exact calculations of integrals with Fock logarithmic terms.
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R12-3D integral package: no limitation on power of $r_{12}^{j}$ .

Abstract

Wave functions of a new functional kind has been proposed in this work for helium-like atoms. These functions depend explicitly on interelectronic and hyperspherical coordinates. The best ground state energy for the helium atom $- 2.903724376677 a . u$ . has been calculated using the variational method with a basis including a single exponential parameter. To our knowledge, this is the best result so far using of hyperspherical coordinates. Comparable result has been obtained for the hydrogen anion. For helium atom, our best wave functions matched the Kato cusp conditions within an accuracy below ${6.10}^{- 4}$ . An important feature of proposed wave functions is the inclusion of negative powers of $R = \sqrt{(} r_{1}^{2} + r_{2}^{2})$ in combination with positive powers of $r_{12}$ into the wave function. We showed that this is necessary condition for proposed wave function to be a formal solution of Schrödinger equation.

Graphical abstract

Coordinate system: $R, t, r_{12}$ . Simplified solution - spinless part: $Ψ = \sum_{i, j, k} C_{i, j, k} R^{i} r_{12}^{j} t^{rk} e^{- ζ R}$ . Necessary condition for simplified wave function (which should be a formal solution of 3Body Schrödinger equation): inclusion of the terms $r_{12}^{j} / R^{j^{'} - 2}$ where $j^{'} = 0, \dots, j - 2$ for $j ⩾ 2$ .

Introduction

Since Hylleraas’ work, it is well known [1], [2], that one of the necessary conditions of a rapid convergence towards the exact nonrelativistic ground state helium energy is the inclusion of $r_{12}$ terms into the wave function. Until now many different methods, in which the $r_{12}$ term is used to construct the wave function, have been suggested. These methods could be divided to variational [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], correlated-function hyperspherical-harmonic (hh) methods [13], [14], [15] (a nice overview of hh methods can be found in the paper of Krivec [15]) and ICI method (iterative complement interaction method)[16], [17], [18]. In general, we can state that variational methods converge to globally optimized solutions while hyperspherical-harmonic methods converge pointwise. However it is known (Bartlett et. all [19]), that wave function created from only the Slater functions and powers of $r_{1}, r_{2}$ and $r_{12}$ is not the exact one. Bartlett [20] and Fock [21] were the first, who pointed out that exact wave function must depend on coordinate $R = \sqrt{r_{1}^{2} + r_{2}^{2}}$ . Fock’s wave function depends on the powers of $R^{2}$ multiplied by powers of the logarithmic term $\ln R^{2}$ . Due to the mathematical difficulty of direct implementation of Fock’s approach it was probably firstly implemented in the work of Sochilin and Ermolaev [22]. The next important work was done by Frankowski and Pekeris [5] in 1966. Their results were improved by Freund, Huxtable and Morgan [6], each group used wave functions based on Hylleraas coordinates and powers of the logarithmic term of the coordinate $s = r_{1} + r_{2}$ .

Our effort is concentrated on finding a competitive approach with use of simple functions as coordinate system that could be generalized to more than two electron atoms. We proposed the approach where the wave function depends besides the powers of $r_{12}$ on the hyperradial coordinate $R = \sqrt{r_{1}^{2} + r_{2}^{2}}$ and on the coordinate $t = (r_{2}^{2} - r_{1}^{2}) / (r_{2}^{2} + r_{1}^{2})$ . The function t has only one shortage, it is not well defined at the region where $r_{1} \to 0$ and $r_{2} \to 0$ (it does not have double limit in the point $r_{1} = 0$ and $r_{2} = 0$ , it depends on path that goes to this point), so it has not derivative in this point. But in the rest area it is well defined, so it is not a significant defect. This function is more simple and easier to handle than that Fock’s one. Probably the first attempt to use these coordinates directly in variational approaches was done in [24], but the authors did not used interelectronic coordinates $r_{ij}$ .

As a first test of quality of the basis set all calculations were done with the same exponential scale factor $ζ$ in $\exp (- ζ R)$ . An important feature of this approach is the inclusion of negative powers of R in combination of positive powers of $r_{12}$ . The incorporation of these terms is – similar to the approach of Kinoshita [3] – necessary for the wave function to be a formal solution of Schrödinger equation. In [23] Sochilin and Ermolaev firstly proposed using the powers of the function $w = r_{12} / R$ for the determination of excited $2 S^{3}$ state energy of helium atom. In this approach, the Hylleraas coordinates were used and it is not clear from their article, how they used the proposed function w in actual calculations.

As a first step the variational method has been used. We want to show that this (or similar) proposed basis sets open new possibilities in finding of the proper wave function that satisfies all cusp conditions. We believe that this knowledge will increase a chance to construct general few electron (with number of electrons more than two) atomic (or molecular) wave functions, because of a pairwise character of electron-electron and nuclear-electron interactions.

Section snippets

Hamiltonian transformation and basis set construction

The Hamiltonian for helium atom in S basic state in coordinates $r_{1}, r_{2}$ and $r_{12}$ can be written as $H = - \frac{1}{2} \frac{\partial^{2}}{\partial r_{1}^{2}} - \frac{1}{r_{1}} \frac{\partial}{\partial r_{1}} - \frac{1}{2} \frac{\partial^{2}}{\partial r_{2}^{2}} - \frac{1}{r_{2}} \frac{\partial}{\partial r_{2}} - \frac{\partial^{2}}{\partial r_{12}^{2}} - \frac{2}{r_{12}} \frac{\partial}{\partial r_{12}} - \frac{1}{2} \frac{r_{1}^{2} + r_{12}^{2} - r_{2}^{2}}{r_{1} r_{12}} \frac{\partial^{2}}{\partial r_{1} \partial r_{12}} - \frac{1}{2} \frac{r_{2}^{2} + r_{12}^{2} - r_{1}^{2}}{r_{2} r_{12}} \frac{\partial^{2}}{\partial r_{2} \partial r_{12}} - \frac{Z}{r_{1}} - \frac{Z}{r_{2}} + \frac{1}{r_{12}}$ As it was already mentioned in the introduction, a function constructed with the powers of $r_{1}, r_{2}$ and $r_{12}$ is not an exact eigenfunction of the Hamiltonian (1) due to the cross terms $\frac{r_{2}^{2}}{r_{1} r_{12}} \frac{\partial^{2}}{\partial r_{1} \partial r_{12}}$ and $\frac{r_{1}^{2}}{r_{2} r_{12}} \frac{\partial^{2}}{\partial r_{2} \partial r_{12}}$ . Let us make the following transformation of the

Variational solution of Schrödinger equation

Here we solved the standard matrix equation $Hc = ∊ Sc$ The elements of the matrix $H$ are $〈 Φ_{i^{'}, j^{'}, k^{'}, l^{'}} | H | Φ_{i, j, k, l} 〉$ and those of the overlap matrix $S$ are $〈 Φ_{i^{'}, j^{'}, k^{'}, l^{'}} | Φ_{i, j, k, l} 〉$ . Finally the vectors $c$ consist of the coefficients $C_{i, j, k, l}$ in case of basis A and of $A_{i, j, k}, B_{i, j, k}$ and $C_{i, j, k, l}$ in the case of bases B and C.

Discussion

The basis sets could be further optimised by omitting the contributions with negligible effect on the result. Therefore, it would probably be possible to find different basis sets with the same total number of functions that could result in better energy. We did not follow this way. Let us notice that for each basis for small values of $ζ$ the method does not converge at all. This is most visible for H⁻ anion. The divergence for such values can be explained by non-physically small values of

Conclusions

In this work a new ansatz for the helium-like ion wave function was proposed. To our knowledge it is at present the best energy for He atom obtained with the use of hyperradial coordinates. Our preliminary results show that the wave functions of the type B and C are close to the exact solution of the Schrödinger equation. The remarkable property of the functions B and C is that for properly chosen expansion coefficients functions of the same type occur on both sides of the Schrödinger equation.

Acknowledgement

The author is deeply indebted to Pavel Neogrády and Ján Mašek for many inspirating discussions and to Ilja Martišovits for discussions of mathematical aspect of the problems. Special thanks are due to Štefan Varga for help with calculation of some integrals and for help with preparing of the manuscript and also to Štefan Dobiš for technical support. This work is dedicated to the memory of Ladislav Turi-Nagy the former head of Department of Theoretical Chemistry at the Institute of Inorganic

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View full text

Research paperAn explicitly correlated helium wave function in hyperspherical coordinates

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Hamiltonian transformation and basis set construction

Variational solution of Schrödinger equation

Discussion

Conclusions

Acknowledgement

Chem. Phys. Lett.

Phys. Lett. A

Chem. Phys.

Über den grundzustand des heliumatoms

Z. Phys.

Neue Berechnung der energie des heliums im grundzustande,sowie des terms von ortho-helium

Z. Phys.

Ground state of helium atom

Phys. Rev.

Variational calculations for helium-like ions using generalized Kinoshita-type expansions

Theor. Chem. Acc.

Logarithmic terms in wave functions of the ground state of two-electron atoms

Phys. Rev.

Variational calculations on the helium isoelectronic sequence

Phys. Rev. A

Uncoupling correlated calculations in atomic physics: very high accuracy and easy

Phys. Rev. A

Coulomb three-body bound-state problem: variational calculations of nonrelativistic energies

Phys. Rev. A

Ground state energies for helium, H-, and Ps-

Phys. Rev. A

Int. J. Quantum Chem.

Experiment and theory in computations of the He atom ground state

Int. J. Mod. Phys. E

Hyperspherical-harmonics methods for few body problems

Few Body Syst.

Solving the Schrödinger equation for helium atom and its isoelectronic ions with the free iterative complement interaction (ICI) method

J. Chem. Phys.

Research paper
An explicitly correlated helium wave function in hyperspherical coordinates

Ground state energies for helium, $H^{-}$ , and ${Ps}^{-}$