Diagonal correction

In the Bom-Oppenheimer picture the nuclei move on a potential energy surface (PES) which is a solution to the electronic Schrodinger equation. The PES is independent of the nuclear masses (i.e. it is the same for isotopic molecules), this is not the case when working in the adiabatic approximation since the diagonal correction (and mass polarization) depends on the nuclear masses. Solution of (3.16) for the nuclear wave function leads to energy levels for molecular vibrations (Section 13.1) and rotations, which in turn are the fundamentals for many forms of spectroscopy, such as IR, Raman, microwave etc. 

This expression constitutes the basis of current interpretations of electron transfer processes in biological systems. From Eq. (9), the functions Hg, (Q) and Hbb (Q) represent potential energy surfaces for the nuclear motion described by Xav and Xbw respectively, if the weak diagonal corrections Taa and T b are neglected. Then, the region Q Q where Xav and Xbw overlap significantly corresponds to the minimum of the intersection hypersurface between Hga (Q) and Hbb (Q)- Referring to definition (5), this implies ... 

Romelt, J. (1983). Prediction and interpretation of collinear reactive scattering resonances by the diagonal corrected vibrational adiabatic hyperspherical model, Chem. Phys. 79, 197-209. 

The (7(R) term is known as the diagonal correction, and is smaller than Ej(R) by a factor roughly equal to the ratio of the electronic and nuclear masses (eq. (3.8)). It is nsnally a slowly varying hmctinn of R, and the shape of the energy surface is therefore... 

To clarify these questions we have studied the quantal and semiclassical theory of reactive resonances. In section II the Diagonal corrected Vibrational Adiabatic Hyperspherical (DIVAH) model 21, ) is reviewed. This theory was the first to provide quantitative predictions (with a typical accuracy of 0.1 kcal/mole) of collinear quantal resonance energies. Furthermore the DIVAH model led us to the new and very surprising phenomenon of vibrationally bonded molecules (29-41). The connection and interrelation with semiclassical RPO theory (25.30.42-47) which predicted and interpreted these results, is presented in section III. Having... 

Note that the excellent agreement can only be obtained after insertion of the diagonal correction. The magnitude of this term is < 0.01 kJ/mol. 

One of the main advantages of dealing with periodic orbits is that they are uniquely determined by the electronic potential energy surface and the masses of the system. This is not true for the quantal adiabatic surfaces. As shown above, the quantal adiabatic approximation became quantitatively reliable only with the advent of DIVAH theory - that is one must incorporate diagonal adiabatic corrections within the adiabatic approximation. By studying the properties of RPO s we have shown that the adiabatic diagonal corrections are basically curvature corrections that are automatically incorporated in the RPO s (A5). 

Handy, N.C., Yamaguchi, Y, Schaefer 111, H.F. The diagonal correction to the Born-Oppenheimer approximation—its effect on the singlet-triplet splitting of CH2 and other molecular effects, J. Chem. Phys. 1986, 84,4481. ... 

From this equation the diagonal corrections to the energies for J= 2>/2,... 

Subsequently, gradient methods were used to evaluate the adiabatic correction or Born-Oppeneimer diagonal correction, ACj j, for SCF wave functions, where... 

As noted in Chapter 2 of this volume, the Born-Oppenheimer diagonal correction produces a mass dependent modification of a mass independent Born-Oppenheimer potential energy surface. This medication can be inferred from experimental measurements of rovibrational levels of a series of isotopomers and can be computed directly, using the methods of this chapter. The state of LiH is considered here. It has been the... 

The Born-Oppenheimer diagonal correction is given in Eq. (2a). In that equation, the gradients refer to space fixed frame (SFF) coordinates. For diatomic molecules, considerable savings result from a transformation to body fixed frame (BFF) coordinates. This transformation is accomplished in two steps. The SFF coordinates are transformed to center of mass fixed frame (CMFF) coordinates and then the CMFF coordinates are transformed to BFF coordinates. The details of the transformation are beyond the scope of this review. Here we sketch the ideas involved. A detailed treatment, based on the pioneering work of Kronig, can be found in Ref. 7. In particular, first the rigorously removable center of mass of the nuclei and... 

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