Nuclear kinetic operator

In an electronic adiabatic representation, however, the electronic Hamiltonian becomes diagonal,i.e. ( a 77e C/3) = da,0Va, where the adiabatic Va potentials for initial (A,B,B ) and final (X) electronic states were described in Ref.[31]. The couplings between different electronic states arises from the matrix elements of the nuclear kinetic operator Tn, giving rise to the so-called non-adiabatic coupling matrix elements (NACME) and are due to the dependence of the electronic functions on the nuclear coordinates. The actual form of these matrix elements depends on the choice of the coordinates. 

The selection rules for radiationless transitions are just the opposite of those for radiative transitions. The nuclear kinetic operator is symmetric. The symmetric aromatic molecules normally have symmetrical ground state and antisymmetrical excited state. Therefore, allowed transitions are ... 

It is well known, that the full electrostatic Hamiltonian operator may be written, in terms of an electronic Hamiltonian operator and a nuclear kinetic operator ... 

In cases where the Born-Oppenheimer perturbation theory of Eq. (2.36) is not valid, mainly as a result of the degenerate situation among the static electronic states, quantum mechanical mixing among them through the nuclear kinetic operator becomes significant. Consequently nonadiabatic... 

The approximation inherent in Equation [5b] means that the nuclear kinetic operator, -hyi) d ldQ ), has no influence on the electronic wavefunction, namely, the terms, -h d g/dQ)ei[S/dQi) and (-fi2/2)(a2 aQ2), which operate on the electronic part of the wavefunction, are omitted. This implies that infinitesimal changes in the nuclear configuration do not afiect the electronic wavefunction, g. This approximation is, however, not good enough in evaluations of the magnetic dipole transition moments. The disturbance to the electronic state caused by changes in the nuclear configuration has to be taken... 

Let us start with the Born-Handy ansatz [6] for the groundstate electron wave-function y/oC ) where represents nuclear coordinates. The adiabatic correction A o to the groundstate is expressed as a mean value of the nuclear kinetic operator Tn [22],... 

Here the vector r = (ri,r2,..., r ) where ri = (xi,yi,Zi) denotes collectively all electronic coordinates and the coordinates of the nuclei are specified by q = (qi, q2,..., qn), where N = 3K—6. In the following, we shall adopt the convention that the components of the vector q are labeled by Greek indices if thqr range from 1 to N, and the Latin ones denote the components of the electronic coordinates. The electronic kinetic energy operator Te(r) and the nuclear kinetic operator TN(q) are presented in a diagonal form ... 

Non-adiabatic transitions are induced by off-diagonal matrix elements of die nuclear kinetic operator on the electronic wavefunctions. In the case of OCS, a rotational (Coriolis) coupling is essential. If we assume the total angular momentum J to be zero, a rotational coupling term is expressed as follows ... 

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. 

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 ... 

Here ho is the kinetic energy and nuclear attraction operator while and 1C are the coulomb and exchange operators, respectively. The coefficients X and Y are solutions of the RPA equations, which for the / singlet transition with excitation energy can be written as... 

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... 

Pj and p2 represent the displacement vectors of the nuclei A and D (the corresponding polar coordinates are p1 cji, and p2, < )2, respectively) p, and pc are the displacement vectors and pT, r and pc, <[)f the corresponding polar coordinates of the terminal nuclei at the (collective) trans-bending and cis-bending vibrations, respectively. As a consequence of the use of these symmetry coordinates the nuclear kinetic energy operator for small-amplitude bending vibrations represents the kinetic energy of two uncoupled 2D harmonic oscillators ... 

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