Kinetic energy operator expansion

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... 

In the original mathematical treatment of nuclear and electronic motion, M. Bom and J. R. Oppenheimer (1927) applied perturbation theory to equation (10.5) using the kinetic energy operator Tq for the nuclei as the perturbation. The proper choice for the expansion parameter is A = (me/M) /", where M is the mean nuclear mass... 

Note that the variables defined in Eq. (3.18) are not normal modes so that the kinetic energy operator is not diagonal. As in the case of a single variable, x, discussed in Chapter 2, the expansion (3.19) has convergence problems. A better expansion is... 

The kinetic energy operator in the Schrodinger equation corresponds to the quadratic term in this nonrelativistic expansion, and thus the Schrodinger equation describes only the leading nonrelativistic approximation to the hydrogen energy levels. 

The Fourier method is best suited to cartesian coordinates because the expansion functions QtkR/LR etlr Lr are just the eigenfunctions of the kinetic energy operator. For problems including the rotational degree of freedom other propagation methods have been developed (Mowrey, Sun, and Kouri 1989 Le Quere and Leforestier 1990 Dateo, Engel, Almeida, and Metiu 1991 Dateo and Metiu 1991). 

This expansion is valid to second order with respect to St. This is a convenient and practical method for computing the propagation of a wave packet. The computation consists of multiplying X t)) by three exponential operators. In the first step, the wave packet at time t in the coordinate representation is simply multiplied by the first exponential operator, because this operator is also expressed in coordinate space. In the second step, the wave packet is transformed into momentum space by a fast Fourier transform. The result is then multiplied by the middle exponential function containing the kinetic energy operator. In the third step, the wave packet is transformed back into coordinate space and multiplied by the remaining exponential operator, which again contains the potential. 

The difference between H(t) and /70(/,) stems from the kinetic energy operator in the adiabatic Hamiltonian H0, which can be treated as a perturbation. Using the Cambell-Baker-Hausdorff expansion, to the first order we have... 

Within the real-space method, the kinetic energy operator is expressed by the finite-difference scheme. Here, we derive the matrix elements for the kinetic energy operator of one dimension in the first-order finite difference. By the Taylor expansion of a wavefunction i/r (/) at the grid point Z we obtain the equations,... 

Recently, the dynamical formulations in these two coordinate systems were compared (159) for the nine-mode CD3H molecule. This study focused upon the validity of the kinetic approximation in curvilinear coordinates the Hamiltonian contained the kinetic energy operator, which could be truncated, and only harmonic potential terms. The kinetic energy operator was expanded to high order, with up to 140 terms. Low-order expansions, particularly first or second order, gave a poor description of both the spectroscopic and dynamic features. It thus appears, from a computational point of view, that the curvilinear description may not give a simpler approach than one based upon rectilinear coordinates. 

Again, this expression is not unique, and alternatives continue to be explored [39,65,77-80]. An expansion of the x"> as linear combinations of momentum states, which are themselves also eigenkets of the kinetic-energy operator, leads to Q = limP-,x g°PI with [62]... 

For the general case, we remark that the kinetic energy operator normally has the required form (23), but the potential energy operator often does not. It then may be fitted to the product form. A convenient, systematic, and efficient approach to obtain an optimal product representation is described in Refs. 6 and 19. We finally note that there are other methods which evaluate the Hamiltonian matrix elements efficiently without relying on a product expansion (23). Most notable here is the CDVR method of Manthe. ... 

These are the representations of the overlap, the potential energy, and the kinetic energy operators in the expansion basis. Note that = (77 )l, so that... 

In the Hamiltonian (O Eq. 2.4), the nuclear variables are free and not constant and there are no nuclear kinetic energy operators to dominate the potential operators involving these free nuclear variables. The Hamiltonian thus specified cannot be self-adjoint in the Kato sense. The Hamiltonian can be made self-adjoint by clamping the nuclei because the electronic kinetic energy operators can dominate the potential operators which involve only electronic variables. The Hamiltonian (O Eq. 2.2) is thus a proper one and the solutions O Eq. 2.3 are a complete set. But since the Hamiltonian (O Eq. 2.4) is not self-adjoint it is not at all clear that the hoped for eigensolutions of O Eq. 2.5 form a complete set suitable for the expansion (O Eq. 2.7). 

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