Quantum mechanics three-body problem

The H7+ molecule-ion, which consists of two protons and one electron, represents an even simpler case of a covalent bond, in which only one electron is shared between the two nuclei. Even so, it represents a quantum mechanical three-body problem, which means that solutions of the wave equation must be obtained by iterative methods. The molecular orbitals derived from the combination of two Is atomic orbitals serve to describe the electronic configurations of the four species H2+, H2, He2+ and He2. 

W. Zickendraht, Construction of a complete orthogonal system for the quantum-mechanical three-body problem. Ann. Phys., 35 18 41,1965. 

E. W. Schmid and H. Ziegelmann, The Quantum Mechanical Three-Body Problem (Pergamon, Oxford, 1974). 

According to modern science, all various kinds of matter consist essentially of a few types of elementary particles combined together in different ways. Since these particles do not obey the laws of classical physics but the laws of modern wave mechanics, the problem of the constitution of matter is a quantum-mechanical many-particle problem of a much higher degree of complexity than even the famous classical three-body problem. 

The transition from a macroscopic description to the microscopic level is always a complicated mathematical problem (the so-called many-particle problem) having no universal solution. To illustrate this point, we recommend to consider first the motion of a single particle and then the interaction of two particles, etc. The problem is well summarized in the following remark from a book by Mattuck [18] given here in a shortened form. For the Newtonian mechanics of the 18th century the three-body problem was unsolvable. The general theory of relativity and quantum electrodynamics created unsolvable two-body and single-body problems. Finally, for the modem quantum field... 

Once electron repulsion is taken into account, this separation of a many-electron wavefunction into a product of one-electron wavefunctions (orbitals) is no longer possible. This is not a failing of quantum mechanics scientists and engineers reach similar conclusions whenever they have to deal with problems involving three or more mutually interacting particles. We speak of the three-body problem. 

This problem is called the three-body problem by the people who study such things (the theoreticians of quantum mechanics). When you have two particles in motion that attract each other, you can describe the situation with an equation. But when you have three particles, and there are attractions and repulsions, and all these particles are in motion, there are too many things going on for one neat equation. The problem is one of clouds a cloud exists and we can point to it and measure it, but to predict in advance just where it will be and what form it will take is not possible. There are too many factors, too many variables, many of which are unknown or unknowable. This problem is at the heart of the probability approach to atomic structure. 

In this paper we consider some coordinate sets used for the treatment in classical and quantum mechanics of the motion of three particles in space. The alternative sets of coordinate systems for the three-body problem have been studied extensively [1-5] and a good choice of the coordinate systems is of crucial importance. Key references for the basic theory are 11 1,6], where also history is sketched and credits are given. 

V. Aquilanti and S. Tonzani, Three-body problem in quantum mechanics Hyperspherical elliptic coordinates and harmonic basis sets. J. Chem. Phys., 120(9) 4073, 2004. 

For a prototypical three body problem, the Helium-like atom, the procedure is well known since the early days of quantum mechanics. More recently, Fano, Macek and Klar [18-23] identified a near separable variable p = (rf -I- rD, where rj and T2 are the two Jacobi vectors of the system, named hyperradius , corresponding to the radius of a six-dimensional hypersphere pareuneterized by five hyperangles . However, note that the hyperradius is independent of the numbering of particles and is therefore very useful for rearrangement problems. 

Very extensive computations have been carried out on the dynamics of reactions of the type A+BC. They are either founded on classical or on quantum mechanics, are either to be considered exact or involving more or less drastic approximations and have been based either in the real three dimensional world or in somewhat artificial spaces of lower dimensionality. These computations are thus attempts to solve the three-body problem more or less accurately. Other Chapters in this book extensively review this subject [41]. The papers, presented at a meeting celebrating "Fifty Years of Chemical Dynamics", held in Berlin in 1982 and published as an issue of the Berichte der Bunsen Gesellschaft in 1982, should be consulted, also for providing a historical perspective [42],... 

Finally, these adiabatic approaches have shown to be able to predict the formation of stable molecules trapped on a repulsive potential energy surface [54] (the stability of these molecules has been confirmed also by three dimensional calculations). This is a purely quantum mechanical effect, since classically such systems would dissociate this prediction is a big success of the adiabatic approach to the three body problems, which therefore is being shown useful for a unified view of bound states and collisions. 

Conversely, for slow collisions the combined system of incoming electron and target molecule has to be considered, leading in the exit channel to a full three-body problem. Quantum-mechanical (approximate) calculations are difficult and have been carried out only for a few selected examples. Therefore, other methods have been developed with the goal of obtaining reasonably accurate cross sections using either classical or semiclassical theories and by devising semiempirical formulae. Some of these concepts are based on the Born-Bethe formula [22] and on the observation that the ejection of an atomic electron with quantum numbers (n,J) is approximately proportional to the mean-square radius of the electron shell (n,J). This leads also to proposed correlations of the ionization cross section with polarizability, dia-... 

The number of problems that can be solved exactly in mechanics is not large. Once we have to treat three interacting bodies, life becomes very difficult indeed. This comment applies to classical mechanics just as to quantum mechanics. What we often do is to look for a simple, idealized problem that we can solve exactly, and then treat the real problem in hand as some kind of perturbation on the idealized one. 

A simple estimate of the computational difficulties involved with the customary quantum mechanical approach to the many-electron problem illustrates vividly the point [255]. Consider a real-space representation of ( ii 2, , at) on a mesh in which each coordinate is discretized by using 20 mesh points (which is not very much). For N electrons, becomes a variable of 3N coordinates (ignoring spin), and 20 values are required to describe on the mesh. The density n(r) is a function of three coordinates and requires only 20 values on the same mesh. Cl and the Kohn-Sham formulation of DFT (see below) additionally employ sets of single-particle orbitals. N such orbitals, used to build the density, require 20 values on the same mesh. (A Cl calculation employs in addition unoccupied orbitals and requires more values.) For = 10 electrons, the many-body wave function thus requires 20 °/20 10 times more storage space than the density and sets of single-particle orbitals 20 °/10x 20 10 times more. Clever use of symmetries can reduce these ratios, but the full many-body wave function remains inaccessible for real systems with more than a few electrons. 

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