The hyperspherical method

In this section we will describe the hyperspherical method in more detail. The description of the formalism is based on extractions taken from the work of Launay [2]. 

To see this feature, we consider a one-dimensional (collinear) model in which the wavefunction is expressed in the form of an expansion written in Jacobi coordinates of arrangement a as 

In an attempt to overcome this difficulty, one may use an expansion over eigenstates which are adapted to the true potential and not to the asymptotic potential. One thus defines adiabatic states by solving, for each Ra the 

In the case of three-dimensional reactive scattering, the generalization of polar coordinates yields the hyperspherical coordinates of six-dimensional space. Using the definitions 

This is a mere rotation through the skewing angle in the range tt to 37t/2, given by 

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The hyperspherical method, from a formal viewpoint, is general and thus can be applied to any N-body Coulomb problem. Our analysis of the three body Coulomb problem exploits considerations on the symmetry of the seven-dimensional rotational group. The matrix elements which have to be calculated to set up the secular equation can be very compactly formulated. All intervals can be written in closed form as matrix elements corresponding to coupling, recoupling or transformation coefficients of hyper-angular momenta algebra. 

In the discussion of both the hyperspherical method and the molecular orbital method we assumed that the motion in the Rb or R direction was slow. While in the Born-Oppenheimer approximation the slowness is generally thought to arise from the much higher mass of the nucleii, as pointed out by Feagin and Briggs,22 it is probably the repulsive nature of the internuclear potential which allows the separability. In other words, the success of these methods in treating He is due to the interelectronic repusion of the l/r12 potential. 

Hyperspherical methods have the merit of providing a dynamical picture of double excitation and double escape for which the central field approximation is inappropriate. Initially, very accurate calculations were not achieved in this way, and so the hyperspherical method was mainly used as a framework to understand the results obtained by other methods. This situation was transformed by the work of Tang and Shimamura [330] who have performed the most detailed calculations to date on systems with two electrons. 

In principle, the hyperspherical method corresponds rather well to the strategy advocated by Langmuir (see [308] and section 7.5) namely that one should seek to quantise the motion of more than just one electron in a many-electron system. The coordinates R and a describe the combined motion of an electron pair, and so the quantum numbers which arise in the solution are radial correlation quantum numbers. 

Within the hyperspherical method, new quantum numbers K, T and A are introduced to describe two-electron correlations. Both K and T are angular correlation numbers (omitted here for simplicity, see [333]), while A = 0, 1 is a radial quantum number, often written as 0,+,— because it is related to the + and — classification of Cooper, Fano and Pratts [323] described in section 7.10. Another quantum number which is often used is v = n — 1 — K — T, where n is the principal quantum number. The number v turns out to be the vibrational quantum number of the three-body system, or the number of nodes contained between the position vectors ri and r2 of the two electrons [334]. 

In the final section we briefiy describe some recent developments of the hypersphericeil approach, such as the hyperquantization algorithm as an important computational tool for the solution of dynamical problems and the extensions of the hyperspherical method to treat the dynamics of reactions involving polyatomic systems. 

Being the exact quantum medianical treatment of the dynamics of systems containing more than four atoms presently computationally out of question, only particular cases or approximate methods are nowadays being developed in order to extend the hyperspherical method to complex reactions. Proper formulations of the coordinate systems and relevant hamiltonians have already been referred to [27,28,45], see also [51]. 

The Schrodinger equation for the D-dimensional analogue of hydrogen (equation (88)) can be solved exactly, both in direct space and in reciprocal space and in both cases the solutions involve hyperspheri-cal harmonics. In this section we shall discuss the close relationship between hyperspherical harmonics, harmonic polynomials, and exact D-dimensional hydrogenlike wave functions. We shall also discuss the importance of these functions in dimensional scaling and in the hyperspherical method. 

Here is the generalized Laplace operator, defined by equation (37), while R is the hyperradius (equation (5)). In a later chapter of this book. Professor Fano will discuss the application of the hyperspherical method to nonseparable dynamical problems. Here we shall only note that if mass-weighted coordinates axe used, the Schrodinger equation for any system interacting through Coulomb forces can be written in the form ... 

The hypersphere method, as discussed below, is a very efficient way to sample the configurations of charged or dipolar particles, in particular, because the pair potentials have a closed analytical expression. It was demonstrated to give results identical to those obtained by using the Ewald procedure. The method is certainly an eloquent illustration of the equivalence... 

Finally, we should mention that the hypersphere method can be used for investigating the relaxation path(s) from the FC region. Thus this method can be a tool for locating Cl points whenever there is no barrier or intermediate along the relaxation from the FC region. 

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