Symmetry-breaking and Tunneling in

Department of Chemistry Harvard University 12 Oxford Street Cambridge MA 02138, USA 

Our aim is to develop and test practical, numerically stable means of treating nonsepaxable potentials in which tuimeling occurs in two or more degrees of freedom. The system is particularly suitable for this purpose. Exact numerical calculations [4] are available for comparison over a wide range of R. Also, in spheroidal coordinates the double-minimum potential is separable and timneling occurs in only one coordinate [5] whereas in cylindrical coordinates [6] the potential is nonseparable and tunneling occurs in two coordinates. This offers an opportunity to compare approximation methods for separable and nonseparable versions of the same system. 

Our results agree well with numerical calculation. It is remarkable that use of the effective potential for large-dimension, which is exactly calculable from classical electrostatics, yields quantitative results for electronic tunneling, an intrinsically quanta phenomenon. 

We consider H with clamped nuclei, corresponding to the Bom-Oppenheimer approximation, for which very accurate numerical so- 

Although the H2 problem for D-dimensions is separable in spheroidal coordinates, just as for D = 3, since we want to examine the nonseparable situation, we employ cylindrical coordinates. In these coordinates the nuclei are located on the z-axis at —i /2 and -t-iJ/2, respectively, and the electron is at p,z). Dimensional scaling is introduced by using units of jZ bohr radii for distance and hartrees for energy, with Z the nuclear charge and k = D — l)/2. The scaled Schrodinger equation for H then takes a simple form. 

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