Born-Huang expansion

In order to discuss as accmately as possible these complex theories, we will now briefly review the traditional and canonical way of describing a molecular system. Within the nonrelativistic scheme, the quantum-mechanical molecular Hamiltonian is generally written as 

Here and in what follows, r and R denote the electronic and nuclear coordinates, respectively, and pj and Pk are the operators of their conjugate momenta of the jth and fcth component of r (denoted as rg) and R Rk), respectively. 14(1 R) represents collectively the Coulombic interactions among electrons and nuclei, which is =.2 

This is often referred to as the Born-Huang expansion [58]. We assume that the electronic basis set [ /), either adiabatic or diabatic, are orthonormal at each nuclear configuration as 

In what follows, the bra-ket inner product represents integration over the electronic coordinates only. 

Inserting this form of the total wavefunction into the time-dependent Schrodinger equation and using the above orthonormal property, one obtains the standard coupled equations of motion for nuclear wavefunctions. .d 

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II. n-ELECTRONIC STATE ADIABATIC REPRESENTATION A. Born-Huang Expansion... 

Born-Huang expansion, 286—289 first-derivative coupling matrix, 290—291 nuclear motion Schrodinger equation, 289-290... 

In this equation, the gradient term U(q>)(R/jU(q>) /R) x i(R),) = W,Ik (R/J Vr x Rl) still appears and, as mentioned before, introduces numerical inefficiencies in its solution. Even though a truncated Born-Huang expansion was used to obtain Eq. (53), W VR).), although no longer zero, has no poles at conical intersection geometries [as opposed to the full W bad(R j matrix]. 

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... 

Born-Huang Expansion of the Total Wave Function... 

There are two points to emphasize here. When a conical intersection exists, the wave functions c (X) that satisfy Eq. (15a) are not suitable as a basis for the Born-Huang expansion, Eq. (1), as they are doublevalued, taking on either the value c (X). To correct this deficiency, either a similarly double-valued vibrational basis must be used in Eq. (1) or a phase factor must be included to account for this deficiency, that... 

Vr is the gradient with respect to all nuclear degrees of freedom which locate the nuclei in a center-of-mass frame, and the semicolon indicates the parametric dependence of the electronic wave functions on R. Since the latter form a complete orthonormal set, the total wave function may be written as the Born-Huang ° expansion... 

To be a little more precise about the notion of quantum entanglement, it should be noted that even a single term in the Born-Huang expansion such as x(R) R) in Eq- (5-1) has already, at least to some extent, taken into account quantum entanglement through the functional form of the electronic wavefunction. This is made clearer if we formally expand it into independent basis functions as... 

The corresponding wavefunctions are usually represented in the Born-Huang expansion as... 

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