Nuclear kinetic energy

Note the stnicPiral similarity between equation (A1.6.72) and equation (Al.6.41). witii and E being replaced by and the BO Hamiltonians governing the quanPim mechanical evolution in electronic states a and b, respectively. These Hamiltonians consist of a nuclear kinetic energy part and a potential energy part which derives from nuclear-electron attraction and nuclear-nuclear repulsion, which differs in the two electronic states. 

The effective nuclear kinetic energy operator due to the vector potential is formulated by multiplying the adiabatic eigenfunction of the system, t t(/ , r) with the HLH phase exp(i/2ai ctan(r/R)), and operating with T R,r), as defined in Eq. fl), on the product function and after little algebraic simplification, one can obtain the following effective kinetic energy operator. 

The general form of the effective nuclear kinetic energy operator (T) can be written as... 

The coordinates p,Tx are called the principal axes of inertia symmetrized hyperspherical coordinates. The nuclear kinetic energy operator in these coordinates is given by... 

Here, t is the nuclear kinetic energy operator, and so all terms describing the electronic kinetic energy, electron-electron and electron-nuclear interactions, as well as the nuclear-nuclear interaction potential function, are collected together. This sum of terms is often called the clamped nuclei Hamiltonian as it describes the electrons moving around the nuclei at a particular configrrration R. 

In this picture, the nuclei are moving over a PES provided by the function V(R), driven by the nuclear kinetic energy operator, 7. More details on the derivation of this equation and its validity are given in Appendix A. The potential function is provided by the solutions to the electronic Schrddinger equation. 

Let US consider the simplified Hamiltonian in which the nuclear kinetic energy term is neglected. This also implies that the nuclei are fixed at a certain configuration, and the Hamiltonian describes only the electronic degrees of freedom. This electronic Hamiltonian is... 

Next, we shall consider how the nuclear kinetic energy is taken into consideration perturbatively. The natural perturbation index k is chosen to be... 

Consider a diatomic molecule as shown in Figure 1. The nuclear kinetic energy is expressed as... 

We assume that the nuclei are so slow moving relative to electrons that we may regard them as fixed masses. This amounts to separation of the Schroedinger equation into two parts, one for nuclei and one for electrons. We then drop the nuclear kinetic energy operator, but we retain the intemuclear repulsion terms, which we know from the nuclear charges and the intemuclear distances. We retain all terms that involve electrons, including the potential energy terms due to attractive forces between nuclei and electrons and those due to repulsive forces... 

The nuclear kinetic energy is essentially a differential operator, and we may write it as ... 

The Hamiltonian for this system should include the kinetic and potential energy of the electron and both of the nuclei. However, since the electron mass is more than a thousand times smaller than that of the lightest nucleus, one can consider the nuclei to be effectively motionless relative to the quickly moving electron. This assumption, which is basically the Born-Oppenheimer approximation, allows one to write the Schroedinger equation neglecting the nuclear kinetic energy. For the Hj ion the Born-Oppenheimer Hamiltonian is... 

Therefore, the simplest classical treatment in which the propagator exp(it (T+V) ) is approximated in the product form exp(it (T) ) exp(it (V)/fc) and die nuclear kinetic energy T is conserved during the transition produces a nonsensical approximation to the non BO rate. This should not be surprising because (a) In the photon absorption case, the photon induces a transition in the electronic degrees of freedom which subsequently cause changes in the vibration-rotation energy, while (b) in the non BO case, the electronic and vibration-... 

The derivative (nonadiabatic) coupling, ffy, is the term neglected in the Bom-Oppenheimer approximation that is responsible for nonadiabatic transitions between different states I and. /. It originates from the nuclear kinetic energy operator operating on the electronic wavefunctions ijf] and is given by... 

Both of these matrix elements are readily computed analytically (the subscript R denotes integration over the nuclear coordinates and by definition Su and S/y vanish for / / J). In Eq. (2.11), H/y is the full Hamiltonian matrix including both electronic and nuclear terms. Each matrix element of H is written as the sum of the nuclear kinetic energy (7r) and the electronic Hamiltonian (He)... 

The obstacle to simultaneous quantum chemistry and quantum nuclear dynamics is apparent in Eqs. (2.16a)-(2.16c). At each time step, the propagation of the complex coefficients, Eq. (2.11), requires the calculation of diagonal and off-diagonal matrix elements of the Hamiltonian. These matrix elements are to be calculated for each pair of nuclear basis functions. In the case of ab initio quantum dynamics, the potential energy surfaces are known only locally, and therefore the calculation of these matrix elements (even for a single pair of basis functions) poses a numerical difficulty, and severe approximations have to be made. These approximations are discussed in detail in Section II.D. In the case of analytic PESs it is sometimes possible to evaluate these multidimensional integrals analytically. In either case (analytic or ab initio) the matrix elements of the nuclear kinetic energy... 

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