Interaction potential, space-fixed

These simple relations motivate a more formal approximation in which we first re-expand the interaction potential in a space-fixed ("laboratory-frame") coordinate system as... 

For a description of the induced transitions between rotovibrational levels vj) — v f) of H2, radial matrix elements, Eq. 4.17, are needed. Since the H-H interaction potential is well known, the vibrational wave-functions, (pVj(r), which depend on the rotational excitation j, can be computed with the help of digital computers [217], At fixed intermolecu-lar separations R, the AXt(r,R) are known at three vibrational spacings,... 

We have derived the total Hamiltonian expressed in a space-fixed (i.e. non-rotating) coordinate system in (2.36), (2.37) and (2.75). We can now simplify the electronic Hamiltonian 3Q,i by transforming the electronic coordinates to the molecule-fixed axis system defined by (2.40) because the Coulombic potential term, when expressed as a function of these new coordinates, is independent of 0, ip and x From a physical standpoint it is obviously sensible to transform the electronic coordinates in this way because under the influence of the electrostatic interactions, the electrons rotate in space with the nuclei. We shall take the opportunity to refer the electron spins to the molecule-fixed axis system in this section also, and leave discussion of the alternative scheme of space quantisation to a later section. Since we assume the electron spin wave function to be completely separable from the spatial (i.e. orbital) wave function,... 

We start by considering the hydrogen atom, the simplest possible system, in which one electron interacts with a nucleus of unit positive charge. Only two terms are required from the master equation (3.161) in chapter 3, namely, those describing the kinetic energy of the electron and the electron-nuclear Coulomb potential energy. In the space-fixed axes system and SI units these terms are... 

Recently, sapt and dynamical calculations for the He-CO complex have been reported (69). Although the anisotropy of the interaction potential around the van der Waals minimum is not particularly weak, we followed Ref. (67) and described the nuclear dynamics in the space-fixed coordinates. The computed infrared spectra for the 4He-CO and 3He-CO complexes are presented in Fig. 3. 

In these wave packet simulations, the molecular axis of the FHF system is assumed to be aligned along the space-fixed axis Z electric field vector. This assumption involves a maximum interaction of the IR and UV laser pulses with the system. Recalling that the time-dependent interaction potential is given by the scalar product of the electric field vector and the dipole vector, i.e. (t) /j, cos 9, it is clear that for field polarizations perpendicular to the molecular axis [9 = 90°) the interaction of the IR laser pulse with the anion vanishes, and for any molecular orientation different from 0= 0° or 180° the interaction is less efficient. Consider now an ensemble of randomly oriented FHF molecules, as in Fig. 4.13(c). Since the UV pulse is tuned to match the energy gap between anion and neutral... 

Asymptotically, when the separation between the collision partners is so large that the electrostatic interaction potential is negligibly small, the wavefunction of the combined system is most conveniently expanded in an uncoupled basis, namely a product of the wavefunction of the open-shell atom, iLSjn >, multiplied by the wavefunction which describes the relative orbital motion or the two collision partners, I Cm 2> Here j=L+S is the total ai ar momentum of the open-shell atom (e,g. j=l for Ip atoms, j=0, 1, or 2 for atoms) and mj and mQ denote the projections of j and C, respectively, along the space-fixed Z-axis. For collisions involving atoms... 

For simplicity, we will assume that the single-particle states are plane waves, as in the jellium model. We denote the interaction potential in real space as yp ""(r), where r is the relative position of the two electrons (in an isotropic solid would be a function of the relative distance r = r only). Taking the center of mass of the two electrons to be fixed, consistent with the assumption that they have opposite momenta, we arrive at the following Schrodinger equation for their relative motion ... 

Even within the limitations of fixed conformation, early force fields were deficient. The predicted unit cell dimensions were too small by more than 20 percent This situation was remedied by adjusting nonbonded potentials describing the fluorine-fluorine and carbon-fluorine interactions until predicted unit cell dimensions adequately reproduced experimentally measured values. This fitting was actually carried out using structural information from several polymers and was not an ad hoc parameterization for PTFE. It should also be pointed out that reproducing PTFE cell dimensions was a process of achieving an approximately correct interchain spacing for an approximate helix since the actual crystalline structure was not known at that time. Subsequently, it has become... 

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