Anisotropy of the potential

If V were isotropic, i.e. if atom C would have C...AB interaction energy independent of r, then of course we might say that there is no coupling between the rotation of C and the rotation of AB. We would have then a conservation law separately for the first and the second angular momentum and the corresponding commutation rules (cf. Chapter 2 and Appendix F) 

Therefore, the wave function of the total system would be the eigenfunction of P and 4 as well as ofand The corresponding quantum numbers / = 0,1,2. 

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Rotational state distributions of the OH(A) product for v = 0 to 3 have also been determined. Highly rotationally excited OH(A,v = 0,1) products are dominant as in the ground state, indicating that the angular anisotropy of the potential is also very important to the production of these product states on the H2O B lA state surface. The vibrational distribution... 

The anisotropy of the potential shifts the position of r toward smaller values than for the isotropic potential [see Eq. (48)]. Now Q(r ) follows as... 

More complicated anisotropies of the potential are, for example, encountered in the association of two linear dipoles. Adiabatic channel potential curves for this case have been calculated and expressed analytically in Ref. 16. More systematic studies, also comparing SACM and trajectory results, were reported in Ref. [36], One may as well consider open-shell effects for example, the association of two open-shell HO radicals in their lowest rotational state was treated in Ref. 37. Figure 12 shows the lowest rovi-bronic adiabatic channel potential curves for this system. The ultimate goal... 

The analysis of the IFC behaviour, for the different solids, will proceed along two lines the comparison of the strength and anisotropy of the potential well that each atom feels when the other atoms are kept fixed and the decomposition of the IFCs in their dipole-dipole part and the remaining part. Each material has its own characteristics, in term of the respective strength of ionic and covalent bonding, and their balance. 

Electron scattering from molecules is receiving increasing attention, and theoretically it can be treated by calculation of the static potential (the interaction potential of an electron with the unperturbed charge distribution). Ab initio calculations for N2 using wavefunctions varying between minimal-basis and near-HF quality have been reported by Truhlar et a/.,237 and compared with semi-empirical INDO calculations. The anisotropy of the potential is only correctly described if tf-functions are included in the basis set. 

A. He/H2 and Ha.—It is convenient to consider first these species. Tsapline and Kutzelnigg375 have applied the IEPA-PNO method, previously described, to the ground state of the He/Ha system. The van der Waals minimum was computed, using a gaussian lobe basis set with carefully optimized exponents. The collinear arrangement with a depth of 21 K was found for the van der Waals minimum, with a saddle point of 14 K for the Czv geometry. The computed surface was compared with experiment and with the R 6 term. The anisotropy of the potential is larger than that predicted asymptotically. 

It should be stressed that for a fixed set of quantum numbers jA, kA, jB, kB, jAB, J, M, and K running from — min(J,jAB) to + min(7, jAB) the basis functions of Eq. (1-266) span the same space as the basis functions of Eq. (1-263) with / running from J — jAB to 7 + jAB. This means that the Hilbert spaces spaned by the basis functions (1-263) and (1-266) are isomorphic. Consequently, the final quantum states (eigenvalues and eigenvectors) will be the same in both bases. The specific choice of the mathematical form of the Hamitonian, Eq. (1-261) or (1-265), and consequently, of the basis depends on the anisotropy of the potential energy surface. 

As can be seen, the first-order correction for the anisotropy of the potential well in the liquid introduces a temperature-dependent correction to the relaxation rate but does not essentially change the density variation of this rate. 

Finally, we observe that, in principle, one may truncate the expansion (15) after a few terms and thus model the anisotropy of the potential (the term 1 = 0, 0, 0 is the isotropic part). Simple parameterized rPP functions for the expansion coefficients vt(rPP ) can then be introduced and the parameters can be fitted empirically. The latter procedure is similar to the empirical way of obtaining atom-atom model potentials and the same questions can be raised regarding the validity of the resulting potentials. 

The outer molecular space is partitioned by nodal surfaces into different portions. In the molecular plane on the side of the hydrogens, there is a positive region where the approach of a positively charged reactant is disfavored. On the opposite side W (r) is negative and the approach of a positive charged reactant is favored. The anisotropy of the potential is similar to that produced by a dipole, but the picture given... 

In this chapter, we present an ab initio study of the potential energy surface and stability of the Lia+CX Sg" ") alkali dimer interacting with the xenon atom in different radial geometries and for six angles from 0° to 90°. In Sect. 16.2, the ab initio calculation method is presented. Section 16.3 reports the results of calculation and analysis of the interesting and unusual feature of the strong interaction and anisotropy of the potential. Finally, we present our conclusions in Sect. 16.4. 

To understand the J dependence of the reaction cross-section fully one also needs to consider the so-called reorientation effect. The anisotropy of the potential originates the action of torques on the molecule, which reorients the molecular axis towards the angle of minimal potential energy, i.e. towards the reactive area - the reactive spot. 

An overview of the current lattice dynamics methods is given in Sect. 2.2 (see also Ref [27]). In practically all cases, up to now, these methods have used semi-empirical intermolecular potentials, mostly of the atom-atom type. The parameters occurring in these potentials are usually fitted to the properties of interest, such as the lattice structure, the cohesion energy and the phonon frequencies. This procedure hides the flaws which are present in the intermolecular potentials as well as in the lattice dynamics method. In studies of solid [49-53], solid [41, 54-56] and solid [57] ab initio potentials have been used, however, which contain detailed information on the anisotropy of the potential and, in the case of O, also on its spin-dependence. Illustrative results of these studies are described in Sect. 2.3. The final Sect. 2.4, shows some typical phenomena occurring in more complex molecular crystals, such as phonon-vibron mixing, the dispersion and shifts of vibron bands and the effects of isotopic substitution, i.e. changes of nuclear masses, on the lattice- and internal vibrations. These phenomena are illustrated by results obtained on solid tetra-cyano-ethene [58] and on several chlorinated-benzene crystals [59, 60]. 

Lattice dynamics studies of the disordered j -phase are more scarce because, obviously, the standard harmonic method and the SCP method cannot be applied to this phase (although in some studies the harmonic method has still been used for the translational phonons, while neglecting the anisotropy of the potential.) Most calculations on have been made by classical Monte Carlo or Molecular Dynamics methods, using semiempirical atom-atom or quadrupole-quadrupole potentials. In our group [50, 52] we have investigated the motions in and the a — jS order/ disorder phase transition by means of the MF, RPA and TDH methods, using the same spherically expanded anisotropic ab initio potential which yields accurate properties for a-N2. 

It is reasonable to relate this kind of behavior with the orientational anisotropy of the potentials however, a number of studies demonstrate water-like anomalies in fluids that interact through spherically symmetric potentials [19-47],... 

The only atom-nonlinear molecule system whose excited states have been studied in any detail is Ar-H20. This system is actually quite weakly anisotropic the anisotropy of the potential splits and shifts the H2O free-rotor levels, but the free-rotor quantum numbers are... 

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