Born-Oppenheimer approximation electronic, matrix elements

To deduce whether a transition is allowed between two stationary states, we investigate the matrix element of the electric dipole-moment operator between those states (Section 3.2). We will use the Born-Oppenheimer approximation of writing the stationary-state molecular wave functions as products of electronic and nuclear wave functions ... 

Let us consider the electron-vibrational matrix element. As is usually done, we consider two coordinate systems, the origins of which are located at the center of mass of the molecule. The first coordinate system is fixed in space, while the second system (the rotational one) is fixed to the molecule. For describing the orientation of the rotational system with respect to the fixed frame we use the Eulerian angles 6 = a, / , y. In the Born-Oppenheimer approximation, the motion of nuclei may be decomposed into the vibrations of the nuclei about their equilibrium position and the rotation of the molecule as a whole. Accordingly, the wave function of the nuclei X (R) is presented as a product of the vibrational wave function A V(Q) and the rotational wave function... 

In the Born-Oppenheimer approximation, the relative importance of channels (la) and (lb), together with their dependence on wavelength would depend upon the matrix elements for the transition between the electronic states, the Franck-Condon factors, the Honl-London factors, and upon the probabilities for spontaneous dissociation of the excited state formed. In principle, except for the last one, these are well known quantities whose product is the transition probability for that particular absorption band of Cs. When multiplied by the last quantity, and with an adjustment of numerical constants i becomes the cross section for the photolysis of Cs into Cs + Cs. It is the measurement of this cross section that lies at the focus of this work. 

Fig. 1. The molecular energy level model used to discuss radiationless transitions in polyatomic molecules. 0O, s, and S0,S are vibronic components of the ground, an excited, and a third electronic state, respectively, in the Born-Oppenheimer approximation. 0S and 0 and 0j are assumed to be allowed, while transitions between j0,j and the thermally accessible 00 are assumed to be forbidden. The f 0n are the molecular eigenstates...

It is usually assumed that the electronic coupling matrix element is a constant across the reaction coordinate. Since the electronic wavefunction is a function of both the electronic and nuclear coordinates, even in the Born-Oppenheimer approximation, it is not surprising that in some systems the assumption that the nuclear and electronic coordinates are independent (the Condon approximation) is not appropriate. The most obvious example of the failure of this approximation is for a system in which the matrix element is dominated by superexchange contributions, since the vertical energies, Adb and Eba. vary with the nuclear coordinates. There are other, probably less obvious kinds of such vibronic coupling ... 

Note that the wave functions for the initial and final states and include both donor and acceptor. This equation is usually simplified by making the Born-Oppenheimer approximation for the separation of nuclear and electron wave functions, resulting in equation (12), in which V is the electronic matrix element describing the coupling between the electronic state of the reactants with those of the product, and FC is the Franck-Condon factor. 

The geminal ansatz still requires more effort than the standard one-electron approach of the independent particle model. It is therefore usually restricted to small molecules for feasibility reasons. As an example how the nonlinear optimization problem can be handled we refer to the stochastic variational approach [340]. However, the geminal ansatz as presented above has the useful feature that all elementary particles can be treated on the same footing. This means that we can actually use such an ansatz for total wave functions without employing the Born-Oppenheimer approximation, which exploits the fact that nuclei are much heavier than electrons. Hence, electrons and nuclei can be treated on the same footing [340-342] and even mixed approaches are possible, where protons and electrons are treated in the external field of heavier nuclei [343-346]. The integrals required for the matrix elements are hardly more complicated than those over one-electron Gaussians [338,339,347]. 

George and Ross34 set out to derive symmetry rules for chemical reactions as a set of selection rules on elements of the transition matrix. Each element of this matrix describes the probability of transition from a specified state of the reactants to a specified state of the products. One selection rule on such a matrix is the approximate conservation of total electron spin by making the Born-Oppenheimer approxima-... 

The theory of multi-oscillator electron transitions developed in the works [1, 2, 5-7] is based on the Born-Oppenheimer s adiabatic approach where the electron and nuclear variables are divided. Therefore, the matrix element describing the transition is a product of the electron and oscillator matrix elements. The oscillator matrix element depends only on overlapping of the initial and final vibration wave functions and does not depend on the electron transition type. The basic assumptions of the adiabatic approach and the approximate oscillator terms of the nuclear subsystem are considered in the following section. Then, in the subsequent sections, it will be shown that many vibrations take part in the transition due to relative change of the vibration system in the initial and final states. This change is defined by the following factors the displacement of the equilibrium positions in the... 

If these Born-Oppenheimer product wave functions are to approximate Hamiltonian eigenvectors, we have to minimize all off-diagonal matrix elements K L and u v). To this end, the electronic wave functions are chosen to be eigenvectors of a part of the Hamiltonian operator called the electronic Hamiltonian (adiabatic states) ... 

For vibrational transitions it is not appropriate to use the Born-Oppenheimer (BO) approximation because the ground and excited states in the context of vibrational transitions have the same electronic wavefunction and differ only in the nuclear wavefunctions, a consequence of which is that the electronic contribution to the magnetic dipole transition matrix element vanishes in the BO approximation. In order to include the important electronic contribution to magnetic dipole transition moments, one has to choose either to make further approximations to the magnetic dipole operator yielding effective non-vanishing magnetic transition moments, or to go beyond the BO approximation. Various approximate models and exact a priori methods have resulted in the last 25 years. 

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