Molecule-fixed coordinates

We now consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of ineitia, namely, by taking the coordinates (x,y,z) in Figure 1 coincided with the principal axes a,b,c). In order to detemiine the parity of the molecule through inversions in SF, we first rotate all the electrons and nuclei by 180° about the c axis (which is peipendicular to the molecular plane) and then reflect all the electrons in the molecular ab plane. The net effect is the inversion of all particles in SF. The first step has no effect on both the electronic and nuclear molecule-fixed coordinates, and has no effect on the electronic wave functions. The second step is a reflection of electronic spatial coordinates in the molecular plane. Note that such a plane is a symmetry plane and the eigenvalues of the corresponding operator then detemiine the parity of the electronic wave function. 

For molecules that are non-linear and whose rotational wavefunctions are given in terms of the spherical or symmetric top functions D l,m,K, the dipole moment Pave can have components along any or all three of the molecule s internal coordinates (e.g., the three molecule-fixed coordinates that describe the orientation of the principal axes of the moment of inertia tensor). For a spherical top molecule, Pavel vanishes, so El transitions do not occur. 

In a molecule-fixed coordinate system, the multipolar Hamiltonian of ref. 

This is referred as BO ansatz. This ansatz is taken as a variational trial function. Terms beyond the leading order in m/M are neglected m is the electronic and M is nuclear mass, respectively). The problem with expansion (4) is that functions /(r, R) contain except bound states also continuum function since it includes the centre of mass (COM) motion. Variation principle does not apply to continuum states. To avoid this problem we can separate COM motion. The remaining Hamiltonian for the relative motion of nuclei and electrons has then bound state solution. But there is a problem, because this separation mixes electronic with nuclear coordinates and also there is a question how to define molecule-fixed coordinate system. This is in detail discussed by Sutcliffe [5]. In the recent paper by Kutzelnigg [8] this problem is also discussed and it is shown how to derive adiabatic corrections using, as he called it, the Bom-Handy ansatz. There are few important steps to arrive at formula for a diabatic corrections. Firstly, one separates off COM motion. Secondly, (very important step) one does not specify the relative coordinates (which are to some extent arbitrary). In this way one arrives at relative Hamiltonian Hrd [8] with trial wavefunction If we make BO ansatz... 

The derivation of these selection rules proceeds as before, with the following additional considerations. The transition dipole moment s (itrans components along the lab-fixed axes must be related to its molecule-fixed coordinates (that are determined by the nature of the vibrational transition as discussed above). This transformation, as given in Zare s text, reads as follows ... 

We will use the 7D CRS Hamiltonian which has been determined and analyzed in Ref. [10] (DFT/B3LYP, 6-31+G(d,p)). In short, the large-amplitude motion of the H/D atom is restricted to the (x,y) plane of the molecule (cf. Fig. 1). The origin of the molecule-fixed coordinate system is at the center of mass, with the axes pointing along the principal axes of inertia for the enol configuration. The H/D motion couples strongly to 5 in-plane skeleton modes, Q = (Q4, Q, Qu, Q26, Q3o)> which are described in harmonic approximation... 

Now consider fa. We set up the space-fixed and molecule-fixed coordinate systems with a common origin on the internuclear axis, midway between the nuclei, as in Fig. 4.11. (Previously in this chapter, we put the origin at the center of mass, but the difference is of no consequence.) The electronic wave function depends on the electronic spatial and spin coordinates and parametrically on R. The parity operator does not affect spin coordinates, and we shall only be considering transformations of spatial coordinates in this section. 

It might be thought that inverting the molecule-fixed coordinates of all particles is equivalent to inverting the space-fixed coordinates of all particles, but this is not so. The direction of the z axis is defined as going from nucleus a to nucleus b, since the xyz axes are rigidly connected to the nuclear framework when we invert the coordinates of electrons and nuclei, we interchange nuclei a and b, and thereby reverse the direction of... 

Interchanging the nuclear coordinates does not affect R, but it does affect the electronic spatial coordinates since they are defined with respect to the molecule-fixed xyz axes, which are rigidly attached to the nuclei. To find the effect on el of interchanging the nuclear coordinates, we will first invert the space-fixed coordinates of the nuclei and the electrons, and then carry out a second inversion of the space-fixed electronic coordinates only the net effect will be the interchange of the space-fixed coordinates of the two nuclei. We found in the last section that inversion of the space-fixed coordinates of all particles left //e, unchanged for 2+,n+,... electronic states, but multiplied it by —1 for 2, II ,... states. Consider now the effect of the second step, reinversion of the electronic space-fixed coordinates. Since the nuclei are unaffected by this step, the molecule-fixed axes remain fixed for this inversion, so that inversion of the space-fixed coordinates of the electrons also inverts their molecule-fixed coordinates. But we noted in Section 1.19 that the electronic wave functions of homonuclear diatomics could be classified as g or m, according to whether inversion of molecule-fixed electronic coordinates multiplies ptl by + 1 or -1. We conclude that for 2+,2,7,11, IV,... electronic states, i//el is symmetric with respect to interchange of nuclear coordinates, whereas for... 

Fig. 8. C2vF(C2vT)2SRM (a) definition of the internal coordinates -n

The above results may be extended to electronic transitions in the case of symmetric top molecules [156, 374, 402], where, in molecule-fixed coordinates, the rotational state of the molecule is characterized by two values, namely the modulus (J) of the total angular momentum, and the projection (K) of the momentum J on the symmetry axis of the molecule (the x-axis see Fig. 1.4). In the case of an electronic transition where... 

The transformation between the space- and molecule-fixed coordinate systems is thus expressed by... 

The operators in (2.36) are easily re-expressed in the molecule-fixed coordinate system since the V" operator merely becomes the V, operator in the new coordinate system and, as mentioned earlier, Fei,nuci becomes independent of the Euler angles. We must also consider the transformation of the partial differential operators 3/3[Pg.52]

Relationship between operators in space-fixed and molecule-fixed coordinate systems... 

It is tempting to construct spherical tensors from J acting within the molecule-fixed coordinate system. From Table 5.2 the components would be expected to have the form ... 

It is important to distinguish between the space-fixed inversion operator E defined here and the molecule-fixed inversion operator, denoted i. The latter defines the g,u character of functions of molecule-fixed coordinates in appropriate systems (i.e. those with a centre of symmetry) but says nothing about the overall parity of the state. It is therefore a less powerful operator than E. ... 

The effect of space-fixed inversion on the Euler angles and on molecule-fixed coordinates... 

Substituting these results in equation (6.207), we obtain the transformation properties of the molecule-fixed coordinates of a point i under space-fixed inversion ... 

Figure 6.24. The effectofthe space-fixed inversion operator E on the molecule-fixed coordinate system (x, y, z). The molecule-fixed coordinate system is always taken to be right-handed. After the inversion of the electronic and nuclear coordinates in laboratory-fixed space, the (x, y, z) coordinate system is fixed back onto the molecule so that the z axis points from nucleus 1 to nucleus 2 and the y axis is arbitrarily chosen to point in the same direction as before the inversion. As a result, the new values of the Euler angles (ip 6, x ) are related to the original values , 9, x)by

P 2 will, however, affect the molecule-fixed coordinates of the electrons through its effect on the orientation of this axis system. Substitution of equations (6.238) and (6.239) into the transformation equation (6.207) leads directly to the result... 

The molecular physics underlying the nuclear spin-rotation interaction has been discussed by Flygare [107], In the general case of a polyatomic molecule the spin-rotation interaction is represented by a second-rank tensor in a molecule fixed coordinate system x, y, z, the diagonal component in the x direction may be written as the sum of a nuclear part (k labelling the nucleus under consideration) and an electronic part ... 

The first two terms in the purely rotational part of (8.361) are wholly diagonal in our basis set and may be replaced by their respective eigenvalues. The remaining scalar products are expanded in the molecule-fixed coordinate system, q, and in the sum over q we separate the e/ = 0 terms from those with q = 1 (denoted by a superscript prime). We also take note of the anomalous commutation rules for the components of J. Equation (8.361) becomes... 

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