Space-fixed coordinates

For the interaction between a nonlinear molecule and an atom, one can place the coordinate system at the centre of mass of the molecule so that the PES is a fiinction of tlie three spherical polar coordinates needed to specify the location of the atom. If the molecule is linear, V does not depend on <() and the PES is a fiinction of only two variables. In the general case of two nonlinear molecules, the interaction energy depends on the distance between the centres of mass, and five of the six Euler angles needed to specify the relative orientation of the molecular axes with respect to the global or space-fixed coordinate axes. 

From these relations it follows that is related to the angular momentum modulus, and that the pairs of angle a, P and y, 8 are the azimuthal, and the polar angle of the (J ) and the (L ) vector, respectively. The angle is associated with the relative orientation of the body-fixed and space-fixed coordinate frames. The probability to find the particular rotational state IMK) in the coherent state is... 

Explicit forms of the coefficients Tt and A depend on the coordinate system employed, the level of approximation applied, and so on. They can be chosen, for example, such that a part of the coupling with other degrees of freedom (typically stretching vibrations) is accounted for. In the space-fixed coordinate system at the infinitesimal bending vibrations, Tt + 7 reduces to the kinetic energy operator of a two-dimensional (2D) isotropic haiinonic oscillator. 

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. 

We write them as i / (9) to shess that now we use the space-fixed coordinate frame. We shall call this basis diabatic, because the functions (26) are not the eigenfunction of the electronic Hamiltonian. The matrix elements of are... 

T is a rotational angle, which determines the spatial orientation of the adiabatic electronic functions v / and )/ . In triatomic molecules, this orientation follows directly from symmetry considerations. So, for example, in a II state one of the elecbonic wave functions has its maximum in the molecular plane and the other one is perpendicular to it. If a treatment of the R-T effect is carried out employing the space-fixed coordinate system, the angle t appearing in Eqs. (53)... 

The total number of spatial coordinates for a molecule with Q nuclei and N electrons is 3(Q + N), because each particle requires three cartesian coordinates to specify its location. However, if the motion of each particle is referred to the center of mass of the molecule rather than to the external spaced-fixed coordinate axes, then the three translational coordinates that specify the location of the center of mass relative to the external axes may be separated out and eliminated from consideration. For a diatomic molecule (Q = 2) we are left with only three relative nuclear coordinates and with 3N relative electronic coordinates. For mathematical convenience, we select the center of mass of the nuclei as the reference point rather than the center of mass of the nuclei and electrons together. The difference is negligibly small. We designate the two nuclei as A and B, and introduce a new set of nuclear coordinates defined by... 

The laplacian operators in equation (10.23) refer to the spaced-fixed coordinates Qa with components Qxa, Qya, Qza, so that... 

The matrix elements (8.35) in the uncoupled space-fixed basis can be most easily evaluated if all interaction operators are represented as uncoupled products of spherical tensors, with each tensor defined in the space-fixed coordinate system. Since the Hamiltonian is always a scalar operator, we can write any interaction in the Hamiltonian as a sum... 

Multipolar induction. For the description of the long-range dipole components, we start with the electric potential at the distance R outside the molecule 1 [323, 391]. In a space-fixed coordinate system, the potential is given as... 

To determine the effect on pti of inversion of all space-fixed coordinates, we carry out the inversion in two steps (a) We rotate all the particles (nuclei and electrons) by 180° about the y axis, (b) We then reflect all the electrons in the xz plane. The net effect of these steps is the inversion of all space-fixed nuclear and electronic coordinates. Step (a) rotates the x and z axes (which are rigidly attached to the nuclei) by 180° and has no effect on the xyz coordinates of the electrons or the nuclei see Fig. 4.12a. Step (b) does nothing to the nuclei and therefore nothing to the xyz axes it does convert the coordinate of each electron to — y (Fig. 4.126). 

Fig. 4.1 Space-fixed coordinate system for internal nuclear motion.

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... 

In Section 5.1, we noted that to a good approximation the nuclear motion of a polyatomic molecule can be separated into translational, vibrational, and rotational motions. If the molecule has N nuclei, then the nuclear wave function is a function of 3/V coordinates. The translational wave function depends on the three coordinates of the molecular center of mass in a space-fixed coordinate system. For a nonlinear molecule, the rotational wave function depends on the three Eulerian angles 9, principal axes a, b, and c with respect to a nonrotating set of axes with origin at the center of mass. For a linear molecule, the rotational quantum number K must be zero, and the wave function (5.68) is a function of 6 and only only two angles are needed to specify the orientation of a linear molecule. Thus the vibrational wave function will depend on 3N — 5 or 3N — 6 coordinates, according to whether the molecule is linear or nonlinear we say there are 3N — 5 or 3N — 6 vibrational degrees of freedom. 

B. A. Hess Prof. Jungen, in your talk you emphasized that you don t have to calculate matrix elements of d/dQ or Coriolis coupling. My impression is that this is due to your most appropriate choice of a diabatic basis, which is generally what ab initio quantum chemists do when they want to avoid singularities in the adiabatic basis. On the other hand, the absence of explicit Coriolis coupling matrix elements is due to the transformation to a space-fixed coordinate system. 

Fig. 1. Definition of body-fixed and space-fixed coordinate systems. Refer to Table 1 and to the text for a detailed explanation...

Passing over to a space-fixed coordinate system with z J (Fig. 1.5(d)) and expanding the unit vector dJ over cyclic unit vectors, we obtain three components. Two of these, namely d 1 and d+1, similarly to (1.7) and (1.8), correspond to P- and i -types of transitions, with frequencies wo— and (jJo + Q. respectively. In addition, a new component d° appears which corresponds to Unear osciUations along the 2-axis with frequency cjo. This component of the vector d is connected with a Q-type transition, as a result of which we obtain a change in the electronic state of the molecule, but no change in its rotational quantum number A = J — J" = 0. 

Alternatively, we can view the angles 0 and <]) as defining the orientation of a general rotation axis z", about which a rotation through v / is to be performed, relative to a space-fixed coordinate system x, y, z. 

Figure 12.9 Body-fixed and space-fixed coordinates of a triatomic molecule.

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