Schrodinger equation dissociation energy

The potential energy function U(R) that appears in the nuclear Schrodinger equation is the sum of the electronic energy and the nuclear repulsion. The simplest case is that of a diatomic molecule, which has one internal nuclear coordinate, the separation R of the two nuclei. A typical shape for U(R) is shown in Fig. 19.1. For small separations the nuclear repulsion, which goes like 1 /R, dominates, and liniR >o U(R) = oo. For large separations the molecule dissociates, and U(R) tends towards the sum of the energies of the two separated atoms. For a stable molecule in its electronic ground state U(R) has a minimum at a position Re, the equilibrium separation. 

For energies below the dissociation threshold we can use various coordinate systems to solve the nuclear Schrodinger equation (2.32). If the displacement from equilibrium is small, normal coordinates are most appropriate (Wilson, Decius, and, Cross 1955 ch.2 Weissbluth 1978 ch.27 Daudel et al. 1983 ch.7 Atkins 1983 ch.ll). However, if the vibrational amplitudes increase so-called local coordinates become more advantageous (Child and Halonen 1984 Child 1985 Halonen 1989). Eventually, the molecular vibration becomes unbound and the molecule dissociates. Under such circumstances, Jacobi or so-called scattering coordinates are the most suitable coordinates they facilitate the definition of the boundary conditions of the continuum wavefunctions at infinite distances which we need to determine scattering or dissociation cross sections (Child 1991 ch.l0). Normal coordinates become less and less appropriate if the vibrational amplitudes increase they are completely impractical for the description of unbound motion in the continuum. 

For fixed total energy E, Equation (2.59) defines one possible set of Nopen degenerate solutions I/(.R, r E, n),n = 0,1,2,..., nmax of the full Schrodinger equation. As proven in formal scattering theory they are orthogonal and complete, i.e., they fulfil relations similar to (2.54) and (2.55). Therefore, the (R,r E,n) form an orthogonal basis in the continuum part of the Hilbert space of the nuclear Hamiltonian H(R, r) and any continuum wavefunction can be expanded in terms of them. Since each wavefunction (R, r E, n) describes dissociation into a specific product channel, we call them partial dissociation wavefunctions. 

In Section 2.5 we have constructed the degenerate continuum wavefunc-tions 4/ f(R, r Ef, n), which describe the dissociation of the ABC complex into A+BC(n). They solve the time-independent Schrodinger equation for fixed energy Ef subject to the boundary conditions (2.59). Furthermore, the 4/f(R,r Ef,n) are orthogonal and complete and thus they form a basis in the corresponding Hilbert space, i.e., any function can be represented as a linear combination of them. 

The wavefunction tot Ef) is a solution of the time-independent Schrodinger equation and therefore corresponds to a particular energy Ef. According to the uncertainty relation, it contains all times, i.e., the entire history of the dissociation process (see also Section 16.1). 

Here, De is the dissociation energy and / is a measure of the curvature at the bottom of the potential well. If the Schrodinger equation is solved with this potential, the eigenvalues are (18,19)... 

By comparison, numerical solution of the one-electron Schrodinger equation for clamped nuclei, predicts the correct dissociation energy and interatomic distance for HJ. These calculations show that the electrostatic interaction which stabilizes the molecule arises from quantum-mechanical charge distributions. Classical models, such as the Lewis or van t Hoff models, can therefore be rejected at the outset. Semi-classical models such as LCAO, are of the same kind, in view of the demonstrated classical nature of hybridization. 

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