Born-Huang approximation

Additional insight into the significance of the quantities introduced above can be obtained from the nuclear Schrodinger equation in the Born-Huang approximation, which we briefly discuss below. 

In the two-electronic-state Born-Huang expansion, the full-Hilbert space of adiabatic electronic states is approximated by the lowest two states and furnishes for the corresponding electronic wave functions the approximate closure relation... 

The familiar BO approximation is obtained by ignoring the operators A completely. This results in the picture of the nuclei moving over the PES provided by the electrons, which are moving so as to instantaneously follow the nuclear motion. Another common level of approximation is to exclude the off-diagonal elements of this operator matrix. This is known as the Born-Huang, or simply the adiabatic, approximation (see [250] for further details of the possible approximations and nomenclature associated with the nuclear Schrodinger equation). 

Equation (6.130), which we will call the complete non-relativistic Hamiltonian, contains terms which couple the electronic and nuclear motions, making it impossible to obtain exact eigenfunctions and eigenvalues. This is where the Born-Oppenheimer approximation enters, in a method suggested by Bom and Huang [46]. We choose to expand the complete molecular wave function as the series... 

The approximation with (36) and (37) is called a Born-Huang (BH) approximation. In this approximation, the molecular wavefunction is written as... 

This sort of difficulty is a general one and obviously not confined simply to one-electron diatomic molecules. It would clearly be unwise to attempt to approximate solutions for molecules at energies close to their dissociation limits in terms of electronic coordinates with the origin at the center-of-nuclear mass. A trial function for the general case of the Born-Huang form... 

However that maybe, it was observed more than 30 years ago (Woolley and Sutcliffe 1977) that the arguments for an expansion (O Eq. 2.7) are quite formal because the Coulomb Hamiltonian has a completely continuous spectrum arising from the possibility of uniform translational motion and so its solutions cannot be properly approximated by a sum of this kind. This means too that the arguments of Born and Oppenheimer (1927), and of Born and Huang (1955) for their later approach to representations of this kind, are also quite formal. 

The corresponding wavefunction must then include both electronic and nuclear coordinates. The validity of the fixed-nucleus model discussed so far was first established by Bom and Oppenheimer (1927) (see also Born and Huang, 1954), who expanded the total molecular wavefunction in terms of products of electronic and nuclear wavefunctions, and showed that in good approximation a single product was usually appropriate the electronic wavefunction is then a solution of (1.1.1), while the nuclear wavefunction is derived from a nuclear eigenvalue equation in which obtained in (1.1.10), as a function of nuclear positions (via solution of (1.1.1)), is used as a potential function. It is because of the validity of this separation, which depends on the large ratio between electronic and nuclear masses, that we may confine our attention initially to a purely electronic problem. 

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