The nuclear kinetic energy operator

3C(/a) is included in the electronic Hamiltonian since, as we shall see, its most important effects arise from interactions involving electronic motions. The interactions which arise from electron spin, Si), will be derived later from relativistic quantum mechanics for the moment electron spin is introduced in a purely phenomenological manner. The electron-electron and electron-nuclear potential energies are included in equation (2.36) and the purely nuclear electrostatic repulsion is in equation (2.37). The double prime superscripts have been dropped for the sake of simplicity. We remind ourselves that jx in equation (2.37) is the reduced nuclear mass, M M.2l M -h Mi). 

We wish to divide Xx into a part describing the nuclear motion and a part describing the electronic motion in a fixed nuclear configuration, as far as possible. Equations (2.36) and (2.37) do not themselves represent such a separation because 3Cei is still a function of R,(p and 0 and cannot therefore commute with 3Qiuci which, as we shall see, involves partial differential operators with respect to these coordinates. The obvious way to remove the effects of nuclear motion from 3Q1 is by transforming from space-fixed axes to molecule-fixed axes gyrating with the nuclei. 

In the Born-Oppenheimer approximation the basis set for 3Cei would consist of products of electronic space and spin functions. Transformation to the gyrating axis system may involve transformation of both space and spin variables, leading to a Hamiltonian in which the spin is quantised in the molecule-fixed axis system (as, for example, in a Hund s case (a) coupling scheme) or transformation of spatial variables only, in which case spatially quantised spin is implied (for example, Hund s case (b)). We will deal in detail with the former transformation and subsequently summarise the results appropriate to spatially quantised spin. 

The rotational and vibrational kinetic energies of the nuclei are represented by the term —(ti /2n)Vg in equation (2.37) we now seek its explicit form and the relation between the momentum operators Pr and in equation (2.6). If we take components of Pr 

These rotations are performed sequentially and a rotation which takes one along an axis in the sense of a right-handed screw is defined as being positive. The nuclei are labelled so that the molecule-fixed z axis points fi om nucleus 1 to nucleus 2. It must be appreciated that this rotating coordinate system is a completely new one it was not mentioned in section 2.3 where all the various coordinate systems have a fixed orientation in laboratory space. 

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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 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 ... 

Initially we hold all nuclei fixed so that the nuclear kinetic energy operator (T ) is zero. If there is no interaction... 

Here Q represents a vector of normal nuclear coordinates Qu Q2>- , Tn is the nuclear kinetic energy operator, and HSF(Q) is the electronic spin-free Hamiltonian... 

To consider the nature of this approximation one should notice that the nuclear kinetic energy operator acts both on the electronic and the nuclear parts of the BO wavefunction. Hence, the deviations from the adiabatic approximation will be measured by the matrix elements of the nuclear kinetic energy, T(Q), and of the nuclear momentum. The approximate adiabatic wavefunctions have the following off-diagonal matrix elements between different vibronic states ... 

It can be safely assumed that the contribution of the term involving the nuclear kinetic energy operator is negligibly small. The coupling matrix element then assumes the form... 

For a recent discussion of the nuclear kinetic energy operator and of its application to nuclear... 

We now note that the nuclear kinetic energy operator in equations (2.35) and (2.37) is... 

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