Quantum kinetic energy operator

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. 

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... 

J. R. Alvarez-Collado, On derivation of curvilinear ro-vibrational quantum kinetic energy operator for polyatomic molecules. J. Mol. Struct. (Theochem.) 433, 69—81 (1998). 

To obtain the quantum kinetic energy operator, we first rewrite the classical expression in terms of momenta conjugate to the coordinates, and then follow the prescription described by Podolsky or Margenau and Murphy.In ot-channel Jacobi coordinates, we obtain... 

The elements of this matrix, transcribed in terms of g only [i.e. in the form ° [x(a)]= (g), which is unrestrictrdly possible because of the translation and rotation invariance of the expressions °g(x)l, are to be used in the exact expression of the quantum kinetic energy operator, obtained in following the so-called gOO-approach, in the general form of Eq.(3), Ref.[32]. They play a role in the part of the Hamiltonian operator that accounts for pure internal deformations, which can be entirely depicted with the help of the 3N-6 internal coordinates. Indeed, the overall exact quantum-mechanical kinetic energy operator can be written as [32] ... 

The L term contains all the dependence on the angular motion of the electron. In classical mechanics, we express the kinetic energy of a particle as wjv /2, and we can break the contributions to the velocity up into a radial speed and an angular speed which allows us to obtain the kinetic energy in a form similar to the quantum kinetic energy operator in Eq. 3.4 ... 

So, for any atom, the orbitals can be labeled by both 1 and m quantum numbers, which play the role that point group labels did for non-linear molecules and X did for linear molecules. Because (i) the kinetic energy operator in the electronic Hamiltonian explicitly contains L2/2mer2, (ii) the Hamiltonian does not contain additional Lz, Lx, or Ly factors. 

Each time step thus involves a calculation of the effect of the Hamilton operator acting on the wave function. In fully quantum methods the wave function is often represented on a grid of points, these being the equivalent of basis functions for an electronic wave function. The effect of the potential energy operator is easy to evaluate, as it just involves a multiplication of the potential at each point with the value of the wave function. The kinetic energy operator, however, involves the derivative of the wave function, and a direct evaluation would require a very dense set of grid points for an accurate representation. 

There are various approximations (7) to the above expression for the absorption rate Rj that offer further insight into the photon absorption process and form a basis for comparison to the non Bom-Oppenheimer rate expression. The most classical (and hence, least quantum) approximation is to ignore the fact that the kinetic energy operator T does not commute with the potentials Vj f and thus to write... 

After the separation of the kinetic energy operator due to the center-of-mass motion from the Hamiltonian, the Hamiltonian describes the internal motions of electrons and nuclei in the system. These in the BO approximation can be separated into the vibrational and rotational motions of the nuclear frame of the molecule and the electronic motion that only parametrically depends on the instantenous positions of the nuclei. When the BO approximation is removed, the electronic and nuclear motions become coupled and the only good quantum numbers, which can be used to quantize the stationary states of the system, are the principle quantum number, the quantum number quantizing the square of the total (nuclear and electronic) squared angular momentum, and the quantum number quantizing the projection of the total angular momentum vector on a selected direction (usually the z axis). The separation of different rotational states is an important feamre that can considerably simplify the calculations. 

This paper considers the hyperspherical harmonics of the four dimensional rotation group 0(4) in the same spirit ofprevious investigations [2,11]), where the possibility is considered of exploiting different parametrizations of the 5" hypersphere to build up alternative Sturmian [12] basis sets. Their symmetry and completeness properties make them in fact adapt to solve quantum mechanical problems where the hyperspherical symmetry of the kinetic energy operator is broken by the interaction potential, but the corresponding perturbation matrix elements can be worked out explicitly, as in the case of Coulomb interactions (see Section 3). A final discussion is given in Section 4. 

As is well known, the vibrational Hamiltonian defined in internal coordinates may be written as the sum of three different terms the kinetic energy operator, the Potential Energy Surface and the V pseudopotential [1-3]. V is a kinetic energy term that arises when the classic vibrational Hamiltonian in non-Cartesian coordinates is transformed into the quantum-mechanical operator using the Podolsky trick [4]. The determination of V is a long process which requires the calculation of the molecular geometry and the derivatives of various structural parameters. 

By using internal coordinates, the quantum-mechanical operator may be obtained by the application of the Podolsky trick [4]. For this purpose, the kinetic energy has to be multiplied by the determinant g of the G matrix. The classical kinetic energy operator is ... 

The problem with these equations is that they correspond to infinite different Hamiltonians so that the solutions for different electronic quantum numbers are incommensurate. To do away with these objections, use instead the complete set of functions rendering the kinetic energy operator Kn diagonal. The set, within normalization factors, is fk(Q) exp(ik Q) k is a vector in nuclear reciprocal space. Including the system in a box of volume V, the reciprocal vectors are discrete, ki, and the functions f (Q) = (1/Vv) exp(iki Q) form an orthonormal set with the completeness relation 8(Q-Q ) = Si fi(Q) fi(Q )- The direct product set ( )j(q)fki(Q) is complete. The matrix elements of eq. (8) reads ... 

The standard quantum chemical model for the molecular hamiltonian Hm contains, besides purely electronic terms, the Coulomb repulsion among the nuclei Vnn and the kinetic energy operator K]. The electronic terms are the electron kinetic energy operator Ke and the electron-electron Coulomb repulsion interaction Vee and interactions of electrons with the nuclei, these latter acting as sources of external (to the electrons) potential designated as Ve]q. The electronic hamiltonian He includes and is defined as... 

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