Kinetic energy operators

When a molecule is isolated from external fields, the Hamiltonian contains only kinetic energy operators for all of the electrons and nuclei as well as temis that account for repulsion and attraction between all distinct pairs of like and unlike charges, respectively. In such a case, the Hamiltonian is constant in time. Wlien this condition is satisfied, the representation of the time-dependent wavefiinction as a superposition of Hamiltonian eigenfiinctions can be used to detemiine the time dependence of the expansion coefficients. If equation (Al.1.39) is substituted into the tune-dependent Sclirodinger equation... 

This expression, in combination with (A3.11.122), detemiines the action of the kinetic energy operator on the wavefiinction at each grid point. The action of Vis just at each grid point. 

When relaxation of the internal motion during the collision is fast compared with the slow collision speed v, or when the relaxation time is short compared with the collision time, the kinetic energy operator... 

Flere we distinguish between nuclear coordinates R and electronic coordinates r is the single-particle kinetic energy operator, and Vp is the total pseudopotential operator for the interaction between the valence electrons and the combined nucleus + frozen core electrons. The electron-electron and micleus-micleus Coulomb interactions are easily recognized, and the remaining tenu electronic exchange and correlation... 

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. 

Similarly, the expression for the effective kinetic energy operator in polar coordinates will be. 

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

In hyperspherical coordinates, the wave function changes sign when <]) is increased by 2k. Thus, the cotTect phase beatment of the (]) coordinate can be obtained using a special technique [44 8] when the kinetic energy operators are evaluated The wave function/((])) is multiplied with exp(—i(j)/2), and after the forward EFT [69] the coefficients are multiplied with slightly different frequencies. Finally, after the backward FFT, the wave function is multiplied with exp(r[Pg.60]

The kinetic energy operator evaluation and then the propagation of the 0, <]) degrees of freedom have been performed using the FFT [69] method followed... 

We will omit the kinetic energy operator Tg of the center of mass <5, since no external fields act on the system and consider only its internal kinetic energy operator 7 " given by [26]... 

In Eqs. (5) and (6), M is the total mass of the nuclei and is the mass of one electron. By using Eq. (2), the system s internal kinetic energy operator is given in terms of the mass-scaled Jacobi vectors by... 

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. 

The Hamiltonian again has the basic form of Eq. (63). The system is described by the nuclear coordinates, Q, which are relative to a suitable nuclear configuration Q. In conbast to Section in.C, this may be any point in configmation space. As a diabatic representation has been assumed, the kinetic energy operator matrix, T, is diagonal with elements... 

Finally, we shall look briefly at the form of the non-adiabatic operators. Taking the kinetic energy operator in Cartesian form, and using mass-scaled coordinates where Ma is the nuclear mass associated with the ath... 

The f operators are the usual kinetic energy operators, and the potential energy V(r,R) includes all of the Coulomb interactions ... 

Note that the electronic kinetic energy operator does not depend on the nuclear configuration explicitly. Therefore, we can conclude that... 

Although all of the nuclear coordinates participate in this kinetic energy operator, and in our previous discussions, all of the nuclear coordinates are expanded, with respect to an equivalent position, in power series of the parameter K, here in the specific case of a diatomic molecule, we found that only the R coordinate seems to have an equilibrium position in the molecular fixed coordinates. This means that actually we only have to, or we can only, expand the R coordinate, but not the other coordinates, in the way that... 

Explicit forms of the coefficients Tt and A depend on the coordinate system employed, the level of approximation applied, and so on. They can be chosen, for example, such that a part of the coupling with other degrees of freedom (typically stretching vibrations) is accounted for. In the space-fixed coordinate system at the infinitesimal bending vibrations, Tt + 7 reduces to the kinetic energy operator of a two-dimensional (2D) isotropic haiinonic oscillator. 

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