Coordinate confocal elliptical

The hydrogen molecule ion is best set up in confocal elliptical coordinates with the two protons at the foci of the ellipse and one electron moving in their combined potential field. Solution follows in mueh the same way as it did for the hydrogen atom but with considerably more algebraic detail (Pauling and Wilson, 1935 Grivet, 2002). The solution is exact for this system (Hanna, 1981). 

D Hooghe, M, and Rahman, A, Physica 23, 26, Approximate diatomic orbitals for H2+. Confocal elliptic coordinates. 

This one-particle equation is sufficiently simple so that it is possible to obtain numerical solutions to any degree of accuracy. As first done by Burrau [84] the equation is transformed (eqn. 1.12) into confocal elliptic coordinates... 

This integral is best evaluated by transformation into confocal elliptical coordinates (33)... 

A useful feature of the molecular orbital approach is that the eigenvalue equation of Eq. (23.22) can be separated in confocal elliptic coordinates,23 and, equally important, these eigenfunctions are apparently somewhat similar to the final atomic eigenfunctions.22 The coordinates are given by22... 

A transformation similar in form to (4) was utilized for the l-c range confocal elliptical coordinates ju used for Hj. The transformations utilized for the remaining coordinates—angular, etc.—are described below. 

For hydrogen, trial wavefunctions identical to those utilized by Goodisman were chosen. These, most conveniently expressed in terms of mixed inter-particle-confocal elliptical coordinates, are cusp corrected. They are composed of terms of the type ... 

Figure 6.3 The confocal elliptical coordinate system used in early calculations to solve Schrodinger s equation for the dihydrogen molecule and its ion.

In Fig. 13.3 the coordinate origin is on the intemuclear axis, midway between the nuclei, with the z axis lying along the intemuclear axis. The H2 electronic Schrodinger equation is not separable in spherical coordinates. However, separation of variables is possible using confocal elliptic coordinates 17, and <. The coordinate is the angle of rotation of the electron about the intemuclear (z) axis, the same as in spherical coordinates. The coordinates f (xi) and 17 (eta) are defined by... 

The electronic Schrddinger equation for H2 can be solved exactly in confocal elliptic coordinates 17, and to give wave functions that are eigenfunctions of L, the... 

In confocal elliptic coordinates, what is the shape of surfaces of constant (a)... 

The Schrodinger equation for H2 can be solved in an alternative coordinate system called confocal elliptical coordinates. Just as with the hydrogen atom, certain wavefunctions result, each of which has a distinct energy. 

Here ra and are the distances of the electron from the two nuclei a and b. Equation (1.3.1) was first solved, with high accuracy, by Burrau (1927), who transformed to confocal elliptic coordinates in which the equation is separable, the solution becoming a product of three factors tlrat may be obtained by solution of three separate differential equations. Solutions in this way is not generally possible, and we pass directly to the construction of simpler approximations. 

Introducing the confocal elliptical coordinates and carrying out the integration first over the azimuthal angle 0 and then over v, we obtain... 

Related to the elliptic integral of the third kind are the Lame functions, which arise in the generalisation of spherical harmonics to confocal ellipsoidal coordinates. Applications of these in molecular electrostatics can be found... 

The electronic Schrodinger equation for the hJ ion can be solved by transforming to a coordinate system that is called confocal polar elliptical coordinates. One coordinate is f = (rA + rB)/rAB, the second coordinate isp = (rA — rB)/rAB, and the third coordinate is the angle , the same angle as in spherical polar coordinates. The solutions to the electronic Schrodinger equation are products of three factors ... 

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