Kinetic energy matrix elements

Decius JC (1948) A tabulation of general formulas for inverse kinetic energy matrix elements in acyclic molecules. J Chem Phys 16 1025-1034... 

The rotational kinetic energy matrix element is now negative while that of the spin-orbit coupling remains positive. In this case, therefore, y (2) is positive. 

To find the derivative of the kinetic energy matrix element first, note that AkAkiAi may be written as... 

Normally, the kinetic energy matrix elements are expressed in that basis set in terms of the second difference formula... 

One solution to the problem of limited accuracy is to evaluate the kinetic energy matrix elements not in position but in momentum space connected via Fourier transformation [10,45]. In momentum space, the kinetic energy operator is diagonal and the associated matrix elements can be calculated by simple multiplication, followed by a back transformation to position space. 

Recently, an alternative approach has been developed by Zou [71], where within a given basis set size the kinetic matrix elements can be evaluated to a desired order of accuracy using Stirling s interpolation formula. The kinetic energy matrix elements can then be written in terms of the discretized position space as... 

The kinetic energy matrix elements are closely related. Define the general matrix element as... 

The origin of the difficulty with the numerical scale factor k in the Wolfsberg-Helmholtz formula (5) lies chiefly in the variation of kinetic-energy matrix elements with distance apart of the two atoms involved. 

The other generic kinetic-energy matrix element needed here is of the form (Vf( , ln) T ir(P, 1m)>- In the present instance, it specializes to... 

The first term on the right side of Eq.(9.67) represents the kinetic energy matrix element... 

The following equation, which is misprinted in Koehler (1968), Is very useful in evaluating kinetic energy matrix elements ... 

If, however, one intends to use a general quadratic potential function, the applicability of which is as appropriate in any one coordinate system as in any other, the simplicity of the kinetic energy matrix elements in the central force coordinate system may favor the use of these coordinates for some molecules. 

In this appendix, formulas for some of the more frequently used kinetic energy matrix elements will be tabulated. It is evident from Sec. 4-3 that these elements will, in general, depend upon the atomic masses and upon the equilibrium bond lengths and angles of the molecules. Since the masses and bond lengths frequently appear in the denominators, it will be convenient to introduce the symbols for the reciprocal of the mass of the ath atom, and for the reciprocal of the a-/3 interatomic distance. 

A kinetic energy matrix element will be given a double subscript to indicate the general types of the two coordinates involved, as Orr,... 

For the two radial coordinates, we use the radial sinc-DVR given by Colbert and Miller [42]. Considering the scattering coordinate first, a grid of R values is defined by R = iAR where i = 1,2,3,. The point at zero is automatically deleted because of the Jacobian weight at the origin. The radial kinetic energy matrix element is... 

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