Effective kinetic energy operator

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. 

Another approach to separation of the large- and small-amplitude modes is applicable when the kinetic and potential energy coupling terms between these modes are small. In such cases, a Van Vleck transformation may be used23. The effective kinetic energy operator for the large-amplitude modes then becomes... 

Letting the kinetic part of H act on j,k), multiplying from the left by (/, A I and integrating over all angular coordinates yields an effective kinetic energy operator (J=0) which works only on functions of the radial variables ... 

The second term can be thought of as an effective kinetic energy operator that goes to the non-relativistic one when V 0. Proper renormalization gives the Infinite Order Regular Approximation (lORA) [17], often approximated by scaled ZORA [16], which improves on ZORA. 

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

This matrix represents an effective operator that still has to act on the bending functions/ (p),/ (p). A generalization of (24) to the case when the kinetic energy operator (i.e., the coefficients 7 and A) has a different form in the... 

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... 

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. 

The effective one-electron operator indicated in brackets includes the kinetic energy operator —and an effective potential energy V ri) taken as an averaged function of ri—the distance of electron 1 from the nucleus. In this approximation, electron 1... 

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. 

Here f is the electron kinetic energy operator, VTe and Vpe are the potential operators for the interactions (coulombic or effective) between the electron and, respectively, the target and the projectile centers. 

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