Vector derivative, vibration-rotation

The result of all of the vibrational modes contributions to la (3 J-/3Ra) is a vector p-trans that is termed the vibrational "transition dipole" moment. This is a vector with components along, in principle, all three of the internal axes of the molecule. For each particular vibrational transition (i.e., each particular X and Xf) its orientation in space depends only on the orientation of the molecule it is thus said to be locked to the molecule s coordinate frame. As such, its orientation relative to the lab-fixed coordinates (which is needed to effect a derivation of rotational selection rules as was done earlier in this Chapter) can be described much as was done above for the vibrationally averaged dipole moment that arises in purely rotational transitions. There are, however, important differences in detail. In particular. 

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. 

The Bloch equation gives the time derivative of the density matrix p in terms of its commutator with the Hamiltonian for the system, and the decay rate matrix T. Each of the matrices, p, H, and T are n x n matrices if we consider a molecule with n vibration-rotation states. We so ve this equation by rewriting the n x n square matrix p as an n -element column vector. Rgwrit ng p in this way transforms the H and V matrices into an n x n complex general matrix R. We obtain... 

In all these cases the Hamiltonian form of the kinetic energy was derived by procedures that are fundamentally similar to the original method for ordinary rigid molecules used by Wilson and Howard1,2 A presentation of their treatment is now found as an essential part of most textbooks on vibration-rotation spectroscopy49-52 and their notation is therefore assumed to be a widely accepted standard. For convenience we adopt a similar notation here, and in particular we shall use vectors of our ordinary three-dimensional space when discussing atomic positions, velocities and momenta. 

Eckart System (ES). It is well-known how the axis convention proposed by Eckart39 enters the standard vibration-rotation theory2 49 52) in the alternative method of deriving the kinetic energy the Eckart conditions are used in formulating rotational s-vectors. For this purpose we may proceed exactly as in the PAS example above. 

Here i//0 is the ground vibrational wave function and ij/ is the wavefunction corresponding to the first excited vibrational state of the th normal mode /< is the electric dipole moment operator Qj is the normal coordinate for the /th vibrational mode the subscript 0 at derivative indicates that the term is evaluated at the equilibrium geometry. The related rotational strength or VCD intensity is determined by the dot product between the electric dipole and magnetic dipole transition moment vectors, as given in (2) ... 

To derive the selection rules for the allowed transitions between vibration, inversion and rotation states, we must express Hz in terms of the components Ha (a = X, y, z) of the dipole moment vector with respect to the molecule-fixed axes xyz,i.e. 

These equations are similar to the conditions formulated by Malhiot and Ferigle56 for s°-vectors of the equilibrium configuration. The general conditions were derived earlier54 by explicitly considering the invariance of vibrational coordinates under translations and rotations. 

Generally in molecular beam studies, both beams have comparable velocities and intersect one another at 90°, and thus the CM velocity vector points at a wide angle intermediate between the two beams. Measurement of the displacement of the laboratory angular distribution of products from the centre-of-mass vector enables an estimate of the velocity of the products to be derived. Reaction products have been velocity analysed (e.g. see refs. 8 and 231) and the results support the view that the product relative translational energy is usually within ca. 1 kcal mole of the reactant relative translational energy. Most of the alkali metal reactions studied to date are exothermic, thus the products must be internally excited. It is believed [8] that, for most reactions, the internal excitation consists mainly of vibrational excitation however, the partition of the vibrational energy between, for example, KI and CH3 is as yet unknown. There are a few exceptions, e.g. the K + HBr reaction where KBr is rotationally excited rather than vibrationally excited [8], and the... 

The r-vectors are derived from the atomic coordinates and principal axes of inertia determined by diagonalization of the matrix of inertia (eq. (13.27)). By forming the matrix product P FP, the translation and rotational directions are removed from the force constant matrix, and consequently the six (five) trivial vibrations become exactly zero (within the numerical accuracy of the machine). 

D has been derived from the Stark effect of four rotational transitions in the microwave region. The dipole moment vector points along the intermediate axis of inertia b (= C2 axis of symmetry). For vibrationally excited states, presumably belonging to the torsional vibration, a small variation of [i was observed = A3 (v=1), 1.47 (v = 2), 1.48 D(v = 3) [1]. For [i (v = 0), see also the table [2] of selected dipole moments. Theoretical moments from ab initio [3, 4] and CNDO/2 [5, 6] calculations lie between 0.26 and 0.80 D. For the atomic charges involved, see the original papers [3 to 6]. 

It is required to express either pair of molecular velocities before and after the interaction in terms of the other pair, and of two independent geometrical variables b and (j>) in order to complete the specification of the encounter. The original Boltzmann equation derivation considers elastic collisions in free space between two spin-less molecules of equal mass. However, due to the major interest in multicomponent mixtures, the theory outline consider elastic collisions between two spin-less mono-atomic molecules in an ideal gas mixture. The theory may be useful even if the molecules are not mono-atomic, provided that their states of internal motion (i.e., rotation and vibration) are not affected by the collisions. The two molecules under consideration are treated as point particles with respective masses m and m2. In the laboratory frame, the incoming molecule positions are denoted by ri and r2, and the particle velocities are indicated by ci and C2. The corresponding positions and velocities after the encounter are r j, and c, c, respectively. The classical trajectories for two interacting molecules presented in the laboratory system frame are viewed in Fig. 2.1. It is supposed that the particle interaction is determined by conservative potential interaction forces only. Any external forces which might act on the molecules are considered negligible compared to the potential forces involved locally in the collision. The relative position vectors in the laboratory frame are defined by ... 

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