Research paperAn explicitly correlated helium wave function in hyperspherical coordinates
Graphical abstract
Coordinate system: . Simplified solution - spinless part: . Necessary condition for simplified wave function (which should be a formal solution of 3Body Schrödinger equation): inclusion of the terms where for .
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 terms into the wave function. Until now many different methods, in which the 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 and is not the exact one. Bartlett [20] and Fock [21] were the first, who pointed out that exact wave function must depend on coordinate . Fock’s wave function depends on the powers of multiplied by powers of the logarithmic term . 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 .
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 on the hyperradial coordinate and on the coordinate . The function t has only one shortage, it is not well defined at the region where and (it does not have double limit in the point and , 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 .
As a first test of quality of the basis set all calculations were done with the same exponential scale factor in . An important feature of this approach is the inclusion of negative powers of R in combination of positive powers of . 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 for the determination of excited 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 and can be written asAs it was already mentioned in the introduction, a function constructed with the powers of and is not an exact eigenfunction of the Hamiltonian (1) due to the cross terms and . Let us make the following transformation of the
Variational solution of Schrödinger equation
Here we solved the standard matrix equationThe elements of the matrix are and those of the overlap matrix are . Finally the vectors consist of the coefficients in case of basis A and of and 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 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
References (32)
- et al.
Variational eigenvalues for the S states of helium
Chem. Phys. Lett.
(1994) - et al.
A fast convergent hyperspherical expansion for the helium ground state
Phys. Lett. A
(1987) - et al.
The formalism and matrix elements of a complete potential-harmonic scheme for directly solving the Schrödinger equation of the helium atom
Chem. Phys.
(1996) Über den grundzustand des heliumatoms
Z. Phys.
(1928)Neue Berechnung der energie des heliums im grundzustande,sowie des terms von ortho-helium
Z. Phys.
(1929)Ground state of helium atom
Phys. Rev.
(1957)- et al.
Variational calculations for helium-like ions using generalized Kinoshita-type expansions
Theor. Chem. Acc.
(2003) - et al.
Logarithmic terms in wave functions of the ground state of two-electron atoms
Phys. Rev.
(1966) - et al.
Variational calculations on the helium isoelectronic sequence
Phys. Rev. A
(1984) Uncoupling correlated calculations in atomic physics: very high accuracy and easy
Phys. Rev. A
(1998)
Coulomb three-body bound-state problem: variational calculations of nonrelativistic energies
Phys. Rev. A
Ground state energies for helium, , and
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.
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