Rotational coordinates

A complete treatment of this derivation can be found in Ref. [19]. The first three terms in the kinetic energy operator indicates the presence of a 3D harmonic oscillator, and the final two terms indicate the presence of a 2D rotator (as for the hydrogen atom). A similar conclusion was made by Auberbach et al. [22] where they use a semi-classical quantisation method and the molecule is said to undergo a unimodal distortion and then the semi-classical Hamiltonian is found to be separated into two parts - a harmonic oscillator part with three vibrational coordinates and a rotational part with two rotational coordinates. However, more progress in terms of specifying the wavefunctions of the system can be made by following a different approach. 

Although the parametrization of coordinates in equation (3) is useful to indicate the separation into vibrations and rotations, it does not clearly separate the five nuclear modes into vibrational and rotational parts. However, it must always be possible to move to a set of rotating coordinates gA, formed from a linear combination of the general coordinates gA that do separate into rotational and vibrational coordinates, so that the Hamiltonian in equation (5) can be written as a sum of 7Yrot and Hvib over the rotational coordinates and vibrational coordinates, respectively. The vibrational problem in particular is then easier to solve as the number of variables is reduced. 

For the relatively simple 2D trough of the E Q e JT system, the form of the rotating coordinates was found by inspection [21]. For the 5D space of the /i-modes, this is not so obvious but can still be found with the help of the rotation matrices in Ref. [19]. We define a rotation matrix with elements D A such that 

The positions of the minimum points on the APES can be determined by applying a shift transformation 

Solution of equation (10) is very complicated, but the calculation can be simplified considerably because the energies of the lowest APESs are already known from equation (4) above. The results in the rotating coordinate system are 

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Pack R T and Hirschfelder J O 1968 Separation of rotational coordinates from the W-electron diatomic Schrddinger equation J. Chem. Phys. 49 4009... 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

The main difference between the adiabatic-to-diabatic transformation and the Wigner matrices is that whereas the Wigner matiix is defined for an ordinary spatial coordinate the adiabatic-to-diabatic transformation matrix is defined for a rotation coordinate in a different space. 

The physical parameters that determine under what circumstances the BO approximation is accurate relate to the motional time scales of the electronic and vibrational/rotational coordinates. 

In the rotated coordinate system the H matrix is diagonal, and the function variation in each of the two directions is thus independent of the other variable. 

Equations (5.12a-c) are the Bloch equations in the rotating coordinate frame. 

However, a quite different situation is observed for ene nitroso acetal (423). First, a stable conformation and, correspondingly, new transition states TSR1 and TSR1 with a low barrier (8.8 kJ/mol) appear on the rotation coordinate about the C,N bond at the place of the transition state of enamine (422). Therefore, the barrier to rotation about the C,N bond decreases so that the process is fast on the NMR time scale and cannot be detected by this method. 

The equation for h2 can be derived by re-expressing hi in a rotated coordinate system, then relating the rotated p and d functions back to those in the original coordinate system.) For h2 the spherical coordinates have the values 9 = a, = 0... 

Sutcliffe, B. T., and Tennyson, J. (1991), A General Treatment of Vibration-Rotation Coordinates for Triatomic Molecules, Int. J. Quant. Chem. 39, 183. 

The model of a reacting molecular crystal proposed by Luty and Eckhardt [315] is centered on the description of the collective response of the crystal to a local strain expressed by means of an elastic stress tensor. The local strain of mechanical origin is, for our purposes, produced by the pressure or by the chemical transformation of a molecule at site n. The mechanical perturbation field couples to the internal and external (translational and rotational) coordinates Q n) generating a non local response. The dynamical variable Q can include any set of coordinates of interest for the process under consideration. In the model the system Hamiltonian includes a single molecule term, the coupling between the molecular variables at different sites through a force constants matrix W, and a third term that takes into account the coupling to the dynamical variables of the operator of the local stress. In the linear approximation, the response of the system is expressed by a response function X to a local field that can be approximated by a mean field V ... 

Rotation of the translated factor axes is an eigenvalue-eigenvector problem, the complete discussion of which is beyond the scope of this presentation. It may be shown that there exists a set of rotated factor axes such that the off-diagonal terms of the resulting S matrix are equal to zero (the indicates rotation) that is, in the translated and rotated coordinate system, there are no interaction terms. The relationship between the rotated coordinate system and the translated coordinate system centered at the stationary point is given by... 

A useful equivalent way of looking at the resonance phenomenon is in a coordinate system rotating with the angular frequency of Hi about the z axis which coincides with the direction of Ho, the applied static field 44)-In this rotating coordinate system the effective fields are Hi, which is now time-independent, and a field Ho — u/y, where w is the angular frequency... 

These same rotation matrices arise when the transformation properties of spherical harmonics are examined for transformations that rotate coordinate systems. For example,... 

For example, a projection operator into center-of-mass translation restricts to the molecular center-of-mass coordinates x, y, and z, and a projection operator into rotation restricts it to molecular rotational coordinates. In the case of SD it is usefril to construct projection operators measuring the influence of solvent molecules according to their location relative to the solute by restrictingyto, e.g., the closest molecule, all the molecules within the first solvation shell, etc.. Once we have chosen a particular projection operator,, we can find the projected portion of the influence coefficient... 

For simplicity, we may assume diatomic molecules (characterized by their vibrational and rotational coordinates, r,- = r,r,). For polyatomic molecules, r, stands for the set of normal and rotational coordinates. 

Z, R, 0) and the z axis of the rotated coordinate system are called the direction cosines. These three angles uniquely define the relative orientation between two planes, the principal Z plane and the z plane. The notation used to identify the direction cosines is... 

If the principal stresses had had shear components, which by definition they don t, then, in general, those shear components would have contributed to the stress vector on the rotated z plane. The a vector completely defines the stress state on the rotated z face. However, our objective is to determine the stress-state vector on the z face that aligns with the rotated coordinate system (z,r,G) x--, x-r, and x-e. The a vector itself has no particular value in its own right. Therefore one more transformation from cs to r is required ... 

Fig. A.2 Illustration of the rotation of an orthogonal (z, r, 9) coordinate system to a new set of orthogonal coordinates (z, r, O ). There are three angles between each of the original coordinates (unprimed) to each of the rotated coordinates (primed). The direction cosines are defined as the cosines of these angles. Altogether, the nine directions cosines can be represented in matrix form.

In addition to representing a vector in a rotated coordinate system, the matrix of direction cosines can also be used to transform a tensor (e.g., the stress tensor) into a rotated coordinate system as... 

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