Kinetic energy, operator for

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... 

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

The rotational kinetic energy operator for a rigid polyatomic molecule is shown in Appendix G to be... 

Using the feet that the full Hamiltonian H is h plus the kinetic energy operator for nuclear motion T... 

The Schrodinger equation applied to atoms will thus describe the motion of each electron in the electrostatic field created by the positive nucleus and by the other electrons. When the equation is applied to molecules, due to the much larger mass of nuclei, their relative motion is considered negligible as compared to that of the electrons (Bom-Oppenheimer approximation). Accordingly, the electronic distribution in a molecule depends on the position of the nuclei and not on their motion. The kinetic energy operator for the nuclei is considered to be zero. 

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 ... 

As indicated, we shall denote electrons and nuclei with Roman (/) and Greek (a) indices, respectively. In terms of kinetic-energy operators for electrons ( e) and nuclei (7k) and the Coulombic potential-energy interactions of electron-electron (Dee), nuclear-nuclear (UNN), and nuclear-electron (VW) type, we can write the supermolecule Hamiltonian as... 

Here Tn and TE are the kinetic energy operators for the nuclei and electrons respectively, and F(r, R) is the total Coulombic energy of nuclei and electrons, r and R denote the sets of coordinates of the electrons and nuclei respectively. One seeks wave functions of the form... 

LeRoy, J. P., and Wallace, R. (1987), Form of the Quantum Kinetic Energy Operator for Relative Motion of A Group of Particles in A General Non-Inertial Reference Frame, Chem. Phys. 118, 379. 

A detailed discussion of the theoretical evaluation of the adiabatic correction for a molecular system is beyond the scope of this book. The full development involves, among other matters, the investigation of the action of the kinetic energy operators for the nuclei (which involve inverse nuclear masses) on the electronic wave function. Such terms are completely ignored in the Born-Oppenheimer approximation. In order to go beyond the Born-Oppenheimer approximation as a first step one can expand the molecular wave function in terms of a set of Born-Oppenheimer states (designated as lec (S, r ))... 

Tunneling in VTST is handled just like tunneling in TST by multiplying the rate constant by k. The initial tunneling problem in the kinetics was the gas phase reaction H -(- H2 = H2 + H, as well as its isotopic variants with H replaced by D and/or T. For the collinear reaction, the quantum mechanical problem involves the two coordinates x and y introduced in the preceding section. The quantum kinetic energy operator (for a particle with mass fi) is just... 

The Hamiltonian for the electrons in an atom or molecule is the sum of terms in kinetic and potential energies 7Y = T + V. The kinetic energy operator for a particle of mass m is T = — in which the Laplacian operator... 

Kn(Q) and Ke(q) are the kinetic energy operators for m-nuclei and n-electrons. The operator Vc(q,Q) represents the total Coulomb interaction electron-electron (Vee), electron-nuclei (Vcn) and nuclei-nuclei (Vnn)-... 

First of all, the theory presented is based on a few assumptions, which, while valid for the molecular systems considered in the literature so far, need to be care-fidly examined in every specific case. As mentioned in Section 8.3, we assume that the effects of external fields on the kinetic energy operator for the relative motion are negligible and that the interactions with electromagnetic fields are independent of the relative separation of the colliding particles. In addition, we ignore the nonadiabatic interactions that may be induced by external fields and that, at present, cannot be rigorously accounted for in the coupled channel calculations. 

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