Operator vibration-rotational

For a diatomie speeies, the vibration-rotation (V/R) kinetie energy operator ean be expressed as follows in terms of the bond length R and the angles 0 and (j) that deseribe the orientation of the bond axis relative to a laboratory-fixed eoordinate system ... 

Symmetry tools are used to eombine these M objeets into M new objeets eaeh of whieh belongs to a speeifie symmetry of the point group. Beeause the hamiltonian (eleetronie in the m.o. ease and vibration/rotation in the latter ease) eommutes with the symmetry operations of the point group, the matrix representation of H within the symmetry adapted basis will be "bloek diagonal". That is, objeets of different symmetry will not interaet only interaetions among those of the same symmetry need be eonsidered. 

The half-width (at half-height) and the shift of any vibrational-rotational line in the resolved spectrum is determined by the real and imaginary parts of the related diagonal element TFor linear molecules the blocks of the impact operator at k = 0,2 correspond to Raman scattering and that at k = 1 to IR absorption. The off-diagonal elements in each block T K, perform interference between correspond-... 

Here, ej f are the vibration-rotation energies of the initial (anion) and final (neutral) states, and E denotes the kinetic energy carried away by the ejected electron (e.g., the initial state corresponds to an anion and the final state to a neutral molecule plus an ejected electron). The density of translational energy states of the ejected electron is p(E) = 4 nneL (2meE) /h. We have used the short-hand notation involving P P/p to symbolize the multidimensional derivative operators that arise in the non BO couplings as discussed above ... 

Before returning to the non-BO rate expression, it is important to note that, in this spectroscopy case, the perturbation (i.e., the photon s vector potential) appears explicitly only in the p.i f matrix element because this external field is purely an electronic operator. In contrast, in the non-BO case, the perturbation involves a product of momentum operators, one acting on the electronic wavefimction and the second acting on the vibration/rotation wavefunction because the non-BO perturbation involves an explicit exchange of momentum between the electrons and the nuclei. As a result, one has matrix elements of the form (P/ t)Xf > in the non-BO case where one finds lXf > in the spectroscopy case. A primary difference is that derivatives of the vibration/rotation functions appear in the former case (in (P/(J.)x ) where only X appears in the latter. 

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... 

Recall that homonuclear diatomic molecules have no vibration-rotation or pure-rotation spectra due to the vanishing of the permanent electric dipole moment. For electronic transitions, the transition-moment integral (7.4) does not involve the dipole moment d hence electric-dipole electronic transitions are allowed for homonuclear diatomic molecules, subject to the above selection rules, of course. [The electric dipole moment d is given by (1.289), and should be distinguished from the electric dipole-moment operator d, which is given by (1.286).] Analysis of the vibrational and rotational structure of an electronic transition in a homonuclear diatomic molecule allows the determination of the vibrational and rotational constants of the electronic states involved, which is information that cannot be provided by IR or microwave spectroscopy. (Raman spectroscopy can also furnish information on the constants of the ground electronic state of a homonuclear diatomic molecule.)... 

Here, Q is the projector on the bound subspace and P projects onto the open, continuum channels. The intramolecular coupling is written as V+ U so that, as before, U is any additional coupling brough about by external perturbations. The equation H = Hq + V+U, where Ho is the zero-order Hamiltonian of the Rydberg electron and so includes only the central part of the potential due to the core plus the motion (vibration, rotation) of the core, uncoupled to the electron. The perturbations V + U can act within the bound subspace, as the operator Q(V+l/)Q is not necessarily diagonal and is the cause of any intramolecular dynamics even in the absence of coupling to the continuum. The intramolecular terms can also couple the bound and dissociative states. 

In vibration-rotation theory, the /., / and contributions to the contact-transformed Hamiltonian are commonly evaluated directly from the relationships (7.59), (7.63), (7.65) and (7.66). This is because the particularly simple commutation relationships which exist between the normal coordinate operator Q, its conjugate... 

Using standard methods to transform differential operators, Hougen attempted to achieve as great a separation of electronic, vibrational, rotational and spin coordinates as possible. The resultant operator representing the rotational kinetic energy has the form... 

The primes denote successive derivatives of the operator Oe with respect to R at the equilibrium internuclear separation Re. The values of the coefficients an, as well as Be and < >, are known from analysis of the vibration rotation spectrum [102],... 

R is the intemuclear distance and // is the reduced mass, so that the first term represents the vibrational motion of the nuclei. R is the angular momentum operator for rotation ofthe nuclear framework. TheinteractionpotentialfortheHe... Ar+ system, V(R, ra), is a function of the intemuclear distance R and the electron coordinates ra we will discuss the details in due course. The problem was set up in a Hund s case (e) basis... 

The large Einstein radiative coefficients [225] and the widely spaced vibration-rotation quantum states make HF peculiarly prone to stimulated emission, and a large proportion of the chemical lasers which have been reported operate on lines in the infrared bands of this molecule [224], H-atom abstraction reactions by F and F-atom abstraction by H are both normally exothermic, and HF is quite generally produced in a vibrational distribution giving rise to oscillation. However, the systems are complex frequently both types of reaction occur, and the details of the vibrational distribution resulting from chemical reaction are difficult to evaluate. 

N. C. Handy, The derivation of vibration-rotation kinetic energy operators, in internal... 

J. Pesonen, Vibration-rotation kinetic energy operators A geometric algebra approach. J. Chem. Phys. 114, 10598-10607 (2001). 

The dependence of rag on the internal coordinates is not restricted by requirements other than the center of mass conditions (2.4) and that Eq. (2.6) is invertible. In expressing the rag functions we may therefore also consider how the final Hamiltonian is influenced, so that we obtain an operator of optimum suitability characterized by e.g. rapid convergence of the perturbing terms. In this respect there are two particular concerns, the vibration-rotation interaction and the potential energy expansion. 

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