Hyperspherical symmetry

This paper considers the hyperspherical harmonics of the four dimensional rotation group 0(4) in the same spirit ofprevious investigations [2,11]), where the possibility is considered of exploiting different parametrizations of the 5" hypersphere to build up alternative Sturmian [12] basis sets. Their symmetry and completeness properties make them in fact adapt to solve quantum mechanical problems where the hyperspherical symmetry of the kinetic energy operator is broken by the interaction potential, but the corresponding perturbation matrix elements can be worked out explicitly, as in the case of Coulomb interactions (see Section 3). A final discussion is given in Section 4. 

Hyperspherical harmonics are now explicitly considered as expansion basis sets for atomic and molecular orbitals. In this treatment the key role is played by a generalization of the famous Fock projection [5] for hydrogen atom in momentum space, leading to the connection between hydrogenic orbitals and four-dimensional harmonics, as we have seen in the previous section. It is well known that the hyperspherical harmonics are a basis for the irreducible representations of the rotational group on the four-dimensional hypersphere from this viewpoint hydrogenoid orbitals can be looked at as representations of the four-dimensional hyperspherical symmetry [14]. 

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 presence of reflection symmetry with respect to the diagonal of the potential-energy surface, as in symmetric molecules or in the four-disk scatterer, Burghardt and Gaspard have shown that a further symmetry reduction can be performed in which the symbolic dynamics still contains three symbols A = 0, +, - [10]. The orbit 0 is the symmetric-stretch periodic orbit as before. The orbit + is one of the off-diagonal orbits 1 or 2 while - represents a half-period of the asymmetric-stretch orbit 12. Note that the latter has also been denoted the hyperspherical periodic orbit in the literature. 

Figure 4.13 Adiabatic hyperspherical potentials, Eq. (101), in a.u. without the adiabatic correction term for He of symmetries 1,3SC and 1,3P° converging to the asymptotic limit e + He+(n = 2), plotted against the hyperradius R in a.u. Each potential supports an infinite number of Rydberg states of Feshbach resonance, of which the lowest level is indicated by a horizontal line. Figure from Ref. [90], Note the difference in notation.

Clary, D.C. (1994) Four-atom reaction dynamics, J. Phys. Chem. 98, 10678-10688. Pack, R.T. and Parker, G.A. (1987) Quantum reactive scattering in three dimensions using hyperspherical (APH) coordinates. Theory, J. Chem. Phys. 87, 3888-3921. Truhlar, D.G., Mead, C.A. and Brandt, 5I.A. (1975) Time-Reversal Invariance, Representations for Scattering Wavcfunctions, Symmetry of the Scattering Matrix, and Differential Cross-Sections, Adv. Chem. Phys. 33, 295-344. 

We can calculate the right-hand side of (22) using the definition of Hna (q) together with (17). In the vicinity of the intersection of interest it is appropriate to retain in (22) only the lowest-order terms in each of the Qk (k = 1, 2,.. ., 3N — 6). In addition, in order to explore the point group symmetry properties of ij/ (rel q0) and v /j(rel q0) it is desirable to choose for the Qk coordinates which display simple transformation properties under the operations of that group, such as normal mode coordinates [13,15] or, in some circumstances, symmetrized hyperspherical coordinates [12,16,17]. Such choices lead to simple point group symmetries for the / (rel q0) and (rel q0) and permit the identification of which. (q0) and q0) vanish due to symmetry [18]. 

For maximal symmetry, i.e. all points except for the centre point, lie on a hypersphere of radius a, the appropriate value for a is given by the equation Eq. 13... 

In Section 2 we define the symmetrized hyperspherical coordinates for the electron-hydrogen atom system and express the hamiltonian in these coordinates. In Section 3 symmetry is discussed. The appropriate symmetry wave functions are introduced in Section 4, the local surface eigenfunctions and energy eigenvalues in Section 5, and the scattering equations and asymptotic analysis in Section 6. Finally, some representative results are given and discussed in Section 7 and a summary of the conclusions is presented in Section 8. 

---