Factor molecular-state

In spite of the fact that we have introduced the factor of exp(—) in equation (47), our analytical expression for the dipole moment does not have a qualitatively correct asymptotic behaviour for the bond lengths r,—> 00. The function does not converge to the dipole moment of the NH2 fragment if we remove a hydrogen atom. However, neither does it diverge The calculated dipole moment values at large r, are around 2-3 D depending on which dissociation path we use. Obviously, the asymptotic behaviour of the dipole moment is of no importance for the simulations carried out in the present work we are only concerned with molecular states well below dissociation. 

The ( )j(q)Qm (Q) is a set of linearly independent functions the Qm(Q) functions are not orthogonal in Q-space for arbitrary electronic states the overlap integrals Jd Q Qm(Q) Q m (Q) are the well known Franck-Condon factors. The hypothesis is that an arbitrary quantum molecular state is given by the linear superposition jus as in the general case ... 

Using resonant effects in core-level spectroscopic investigations of model chromophore adsorbates, such as bi-isonicotinic acid, on metal-oxide surfaces under UHV condition, even faster injection times have been tentatively proposed [85]. The injection time is observed to be comparable to the core-hole decay time of ca. 5 fs. It is also possible to resolve different injection times for different adsorbate electronic excited states with this technique. While the core-excitations themselves provide a perturbation to the system, and it cannot be ruled out that this influences the detailed interactions, the studies provide some of the first local molecular, state-specific injection time analysis with good temporal resolution in the low femtosecond regime. The results provide information about which factors determine the injection time on a molecular level. 

The natural laws of physics and chemistry which describe the behaviour of matter in the massive and molecular states also, of course, apply to the colloidal state. The characteristic feature of colloid science lies in the relative importance which is attached to the various physicochemical properties of the systems being studied. As we shall see, the factors which contribute most to the overall nature of a colloidal system are ... 

It is noted that this relation only holds in the absence of an external field, when the sets of states are degenerate. Other degeneracies may also occur in the internal molecular states. In this case, the g factors should include those degeneracies. 

Here d is the z projection of the dipole matrix elements for the spinless S PZ transition of an outer electron. The factor (— )sw in equation (38) characterizes the symmetry in the arrangement of atomic dipoles in mixed S-P atomic states. Spin S and parity w of a molecular state are relevant either to the exchange of electrons (with atomic orbital fixed at nuclei) or to the exchange both of electrons and atomic orbitals. The exchange of orbitals (with atomic electrons attached to corresponding nuclei) is accomplished by the product of these two transformations. This explains the appearance of the spin quantum number in (38). 

Typical results are shown in Fig. 6 for U-methane in graphite pores of H =7.5 at T=114 K. At p/ps=l the system is solid-like at this temperature, but a discrete change in density occurs around p ps ca.0.5. The self diffiisivity along axial direction also shows drastic change at this point. Further examination of various characteristics of molecular state such as snapshots, in-plane pair correlations and static structure factors confirmed that this change in density is the result of a phase transition from solid-like state to liquid-like one, or melting. Since the critical condensation condition for this pore is far lower than this transition point to stay around p ps= ca.0.2, the liquid-like state is not on metastable branch but thermodynamically stable. Thus a solid-liquid coexistence point is found for this temperature. 

An interesting di fference is that while in Eq. (3.27) we find the photon frequency as a multiplying factor, in the calculation based on the interaction — IZyi // /ClPy A(ry) we get instead a factor of —the transition frequency between the molecular states involved. For physical processes that conserve energy the two are equal. 

In other words, we seek the expression of the molecular-state g-factor in terms of the local g-factors... 

Hence, the g-factor for a given molecular state becomes... 

Then the molecular-state g-factor becomes expressed as... 

Interestingly, the formula for the molecular-state g-factor gs is similar to that obtained for the atomic-term g-factor gj when spin-orbit coupling was considered (Section 8.5.2) hence... 

Let us consider, as an example, the simplest heterospin dimer with SA = 1/2 (say Cu11) and SB = 1 (say Ni11). According to the above formulae the molecular-state g-factors are... 

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