Translational states

It has long been known from statistical mechanical theory that a Bose-Einstein ideal gas, which at low temperatures would show condensation of molecules into die ground translational state (a condensation in momentum space rather than in position space), should show a third-order phase transition at the temperature at which this condensation starts. Nonnal helium ( He) is a Bose-Einstein substance, but is far from ideal at low temperatures, and the very real forces between molecules make the >L-transition to He II very different from that predicted for a Bose-Einstein gas. 

The frequency (number per second) of /transitions from all g. degenerate initial internal states and from the p. d E. initial external translational states is equal to tire reverse frequency from the g degenerate final internal states and the pyd final external translational states. The detailed balance relation between the forward and reverse frequencies is therefore... 

The number of translational states per unit volume dR with directed energies E = (E, p) in the range [E, E +... 

Solutions to a Schrodinger equation for this last Hamiltonian (7) describe the vibrational, rotational, and translational states of a molecular system. This release of HyperChem does not specifically explore solutions to the nuclear Schrodinger equation, although future releases may. Instead, as is often the case, a classical approximation is made replacing the Hamiltonian by the classical energy ... 

This Hamiltonian is used in the Schrodinger equation for nuclear motion, describing the vibrational, rotational, and translational states of the nuclei. Solving the nuclear Schrodinger equation (at least approximately) is necessary for predicting the vibrational spectra of molecules. 

In this form, which is analogous to Eq. (26) in the photon absorption case, the rate is expressed as a sum over the neutral molecule s vibration-rotation states to which the specific initial state having energy , can decay of (a) a translational state density p multiplied by (b) the average value of an integral operator A whose coordinate representation is... 

E. The degrees of freedom for A include the particle in the box translational motion along the reaction coordinate which replaces the so-called imaginary frequency. The spacing between particles in the box states is small and depends on the length l of the box and reduced mass i of the transition state. The density of the translational states corresponding to an energy x in the reaction coordinate is... 

Mac Manus JP, Graber T, Luebbert G, Preston E, Rasquinha I, et al. 2004. Translation-state analysis of gene expression in mouse brain after focal ischemia. J Gereb Blood Flow Metab 24 657. 

Figure 8.6 Collisions of ultracold molecules in a quasi-2D geometry. Presented are the rates of the loss of molecules from an optical lattice trap occurring due to chemical reactions. The squares represent the reactions of molecules prepared in the same (translational and internal) quantum states the circles are for collisions of molecules in different translational states but the same internal states the triangles are for molecules in different internal states. Adapted with permission from Ref. [1].

For the rare gas mixtures, i) and /) represent the initial and final translational states. The integration is over positive frequencies so that only the terms corresponding to E, < Ef survive,... 

For rare gas mixtures, hcvft — Ef — E, is the energy difference between final and initial translational states. The limitation of the integration to... 

It is clear that the interaction potential is an essential part of the two-particle Hamiltonian and thus of the translational state of the supermolecule the interaction potentials of initial and final state differ if a molecule undergoes vibrational excitation. For example, for a diatom-diatom pair like H2-H2, the translational state is determined by the vibrational average of the potential,... 

The subscripts s = j, m, v, j2,m2,V2 and t = A,wv, t denote molecular and translational states, respectively the j, m refer to rotational states and the v to vibrational states, the subscripts 1 and 2 refer to molecules 1 and 2, and a prime denotes final states. The normalized translational Boltzmann factor may be written as... 

We have not attempted to exhibit in great detail the effects of the rotational excitations on the induced dipole components B and those of vibrational excitation on the interaction potential because this was done elsewhere for similar systems [151, 63,295,294], The significance of the j,f corrections is readily seen in the Tables and need not be displayed beyond that. The vibrational influence is displayed in Fig. 6.20 first and second spectral moments are strongly affected, especially at high temperatures, similar to that which was seen earlier for H2-He [294], Fig. 6.23. The close agreement of the measurements of the rotovibrational collision-induced absorption bands of hydrogen with the fundamental theory shown above certainly depends on proper accounting for the rotational dependences of the induced dipole moment, and of the vibrational dependences of the final translational states of the molecular pair. 

Next, we assume that the translational states ) and [Pg.339]

The quantity 2/u/( + — y ) 1/2 /h is the density of translational states per unit length of the reaction coordinate [eqn. (1)], 2( +— i y)lfi v2 is the translational velocity, the reciprocal lifetime of the complex per unit length of the reaction coordinate, the leading factor of selects only those trajectories which lead to product formation, and i is the reduced mass associated with the reaction coordinate. 

The values of coordinates and velocities (or momenta) at some point in the reactant and product asymptote, respectively, can be used to define the boundary conditions for solving the equations of motion describing an elementary process (18,43). The coordinates Qk = Q, Q2 = Qi and velocities Vl = V, V2 = VJ2j at the end points refer to both the internal and translational degrees of freedom the choice of the end points thus determines the channel as well as the internal (quantized) and translational states of reactants and products. 

It should be remarked that for the described catenanes, but also for most of the catenane-based machines developed so far, the repeated switching between the two translational states does not need to occur through a full rotation. In fact, because of... 

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