Coordinates elliptic

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). 

You will find the detailed solution of the electronic Schrddinger equation for H2" in any rigorous and old-fashioned quantum mechanics text (such as EWK), together with the potential energy curve. If you are particularly interested in the method of solution, the key reference is Bates, Lodsham and Stewart (1953). Even for such a simple molecule, solution of the electronic Schrddinger equation is far from easy and the problem has to be solved numerically. Burrau (1927) introduced the so-called elliptic coordinates... 

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

Evaluating the energy e for different values of R gives the effective potential for the nuclei in the presence of the electron. This function is called the Born-Oppenheimer potential surface or just the potential surface. In order to evaluate e(R) we have to determine HAA, HAB, and SAB. These quantities, which can be evaluated using elliptical coordinates, are given by... 

The Hartree-Fock orbitals are expanded in an infinite series of known basis functions. For instance, in diatomic molecules, certain two-center functions of elliptic coordinates are employed. In practice, a limited number of appropriate atomic orbitals (AO) is adopted as the basis. Such an approach has been developed by Roothaan 10>. In this case the Hartree-Fock differential equations are replaced by a set of nonlinear simultaneous equations in which the limited number of AO coefficients in the linear combinations are unknown variables. The orbital energies and the AO coefficients are obtained by solving the Fock-Roothaan secular equations by an iterative method. This is the procedure of the Roothaan LCAO (linear-combination-of-atomic-orbitals) SCF (self-consistent-field) method. 

Transferred to the observation that the reflections in moderately anisotropic scattering images are found on ellipses3, it appears reasonable to parameterize such images in elliptical coordinates ( , v). The transformation relations are [266]... 

Parameterization of Reflections in Elliptical Coordinates. The authors of the technique define the intensity... 

Another curvilinear coordinate system of importance in two-centre problems, such as the diatomic molecule, derives from the more general system of confo-cal elliptical coordinates. The general discussion as represented, for instance by Margenau and Murphy [5], will not be repeated here. Of special interest is the case of prolate spheroidal coordinates. In this system each point lies at the intersection of an ellipsoid, a hyperboloid and and a cylinder, such that... 

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)... 

Because of the interelectronic repulsion term l/ri2, the electronic Hamiltonian is not separable and only approximate solution of the wave equation can be considered. The obvious strategy would be to use Hj wave functions in a variation analysis. Unfortunately, these are not known in functional form and are available only as tables. A successful parameterization, first proposed by James and Coolidge [89] and still the most successful procedure, consists of expressing the Hamiltonian operator in terms of the four elliptical coordinates 1j2 and 771 >2 of the two electrons and the variable p = 2ri2/rab. The elliptical coordinates 4> 1 and 2, as in the case of Hj, do not enter into the ground-state wave function. The starting wave function for the lowest state was therefore taken in the power-series form... 

The classical potential energy term is just a sum of the Coulomb interaction terms (Equation 2.1) that depend on the various inter-particle distances. The potential energy term in the quantum mechanical operator is exactly the same as in classical mechanics. The operator Hop has now been obtained in terms of second derivatives with respect to Cartesian coordinates and inter-particle distances. If one desires to use other coordinates (e.g., spherical polar coordinates, elliptical coordinates, etc.), a transformation presents no difficulties in principle. The solution of a differential equation, known as the Schrodinger equation, gives the energy levels Emoi of the molecular system... 

Being the lowest stable excited state, the electronic structure of the B state of H2 has been of considerable interest. The calculation of Kolos and Wolniewicz using the variational method with elliptic coordinates [57] showed that the wavefunction is well represented by a mixture of three configurations ionic,... 

H[ is the operator of the relative kinetic energy of the nuclei, H is a correction to the kinetic energy of the electrons, is a mass polarisation correction and // denotes the reduced mass of the nuclei. The explicit expression for H R) in terms of elliptic coordinates is given in Ref(Kolos and Wolniewicz, 1964). In the BO approximation the term is neglected. 

The function F(l,2) is in fact the space part of the total wave function, since a non-relativistic two-electron wave function can always be represented by a product of the spin and space parts, both having opposite symmetries with respect to the electrons permutations. Thus, one may skip the spin function and use only the space part of the wave function. The only trace that spin leaves is the definite per-mutational symmetry and sign in Eq.(14) refers to singlet as "+" and to triplet as Xi and yi denote cartesian coordinates of the ith electron. A is commonly known angular projection quantum number and A is equal to 0, 1, and 2 for L, II and A symmetry of the electronic state respectively. The linear variational coefficients c, are found by solving the secular equations. The basis functions i(l,2) which possess 2 symmetry are expressed in elliptic coordinates as ... 

The power series expansion of the generalised James-Coolidge firnction, given in elliptic coordinates, has been, therefore, developed specially for two-electron systems and, moreover, cannot be used for nonlinear molecules. 

The operator makes the wave function antisymmetric to the exchange of any two electrons, x is the appropriate spin function and v takes value 0 or 1. N denotes the number of electrons in the molecule (3 or 4). QmfrnJ are non-orthogonal orbitals given in the elliptic coordinates as ... 

What I would do is the following (a) For low l take the quantum defect matrix elements as they result from the fittings, (b) For high l evaluate them in elliptic coordinates assuming no penetration (the effects of the dipole field are then fully included) in this way a full calculation with an arbitrary number of 1 components can be carried out. 

Show that = HBB, as well as HAb = H, using elliptical coordinates as in Problem 5. With the overlap integral from Problem 5, obtain the expression for the lowest orbital in H2+ as a function of R and Z. Putting Z = 1, sketch the curve W = f(R). 

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. Dupays, B. Lepetit, J.A. Beswick, C. Rizzo, D. Bakalov, Nonzero total-angular-momentum three-body dynamics using hyperspherical elliptic coordinates Application to muon transfer from muonic hydrogen to atomic oxygen and neon, Phys. Rev. A 69 (2004) 062501. 

The only model available for direct quantum-mechanical study of interatomic interaction is the hydrogen molecular ion Hj. If the two protons are considered clamped in position at a fixed distance apart, the single electron is represented by a Schrodinger equation, which can be separated in confo-cal elliptic coordinates. On varying the interproton distance for a series of calculations a complete mapping of the interaction for all possible configurations is presumably achieved. This is not the case. Despite its reasonable appearance the model is by no means unbiased. 

Michael et al. [394] used the mapping function used earlier by Saito [490], transforming to elliptic coordinates [404],... 

The first matrix element in (6.82) arises in the treatment of the hydrogen atom, and the second can be evaluated by transforming to elliptical coordinates [18], giving... 

A large number of variation calculations have been performed for H2, some of the most accurate employing elliptical coordinates with the nuclei as foci to describe electronic positions relative to the nuclei. The electron coordinates may be written... 

Figure 6.29. Coordinate system for the description of the Ht ion, showing the two-dimensional transformation from cartesian to elliptical coordinates.

Combining equations (6.436) and (6.430) we obtain the Schrodinger equation for the II2 molecular ion in elliptical coordinates ... 

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