Space fixed

At this point the reader may feel that we have done little in the way of explaining molecular synnnetry. All we have done is to state basic results, nonnally treated in introductory courses on quantum mechanics, connected with the fact that it is possible to find a complete set of simultaneous eigenfiinctions for two or more commuting operators. However, as we shall see in section Al.4.3.2. the fact that the molecular Hamiltonian //coimmites with and F is intimately coimected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing tlirough the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the... 

Hj, H2 and H. The pemuitation (12) (where S denotes space-fixed position labels) is defined in this approach as pemuiting the nuclei that are in positions 1 and 2, and the pemuitation (123) as replacing the proton in position 1 by the proton in position 2 etc. With this definition the effect of first doing (12) and then doing (123) can be drawn as... 

Table Al.4.2 The multiplication table of the point group using the space-fixed axis convention (see text).

Nuclear pemuitations in the N-convention (which convention we always use for nuclear pemuitations) and rotation operations relative to a nuclear-fixed or molecule-fixed reference frame, are defined to transfomi wavefunctions according to (equation Al.4.56). These synnnetry operations involve a moving reference frame. Nuclear pemuitations in the S-convention, point group operations in the space-fixed axis convention (which is the convention that is always used for point group operations see section Al.4.2,2 and rotation operations relative to a space-fixed frame are defined to transfomi wavefiinctions according to (equation Al.4.57). These operations involve a fixed reference frame. 

Another distinction we make concerning synnnetry operations involves the active and passive pictures. Below we consider translational and rotational symmetry operations. We describe these operations in a space-fixed axis system (X,Y,Z) with axes parallel to the X, Y, Z) axes, but with the origin fixed in space. In the active picture, which we adopt here, a translational symmetry operation displaces all nuclei and electrons in the molecule along a vector, say. 

We consider rotations of the molecule about space-fixed axes in the active picture. Such a rotation causes the (x, y, z) axis system to rotate so that the Euler angles change... 

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. 

To generalize what we have just done to reactive and inelastic scattering, one needs to calculate numerically integrated trajectories for motions in many degrees of freedom. This is most convenient to develop in space-fixed Cartesian coordinates. In this case, the classical equations of motion (Hamilton s equations) are given... 

The quantum numbers tliat are appropriate to describe tire vibrational levels of a quasilinear complex such as Ar-HCl are tluis tire monomer vibrational quantum number v, an intennolecular stretching quantum number n and two quantum numbers j and K to describe tire hindered rotational motion. For more rigid complexes, it becomes appropriate to replace j and K witli nonnal-mode vibrational quantum numbers, tliough tliere is an awkw ard intennediate regime in which neitlier description is satisfactory see [3] for a discussion of tire transition between tire two cases. In addition, tliere is always a quantum number J for tire total angular momentum (excluding nuclear spin). The total parity (symmetry under space-fixed inversion of all coordinates) is also a conserved quantity tliat is spectroscopically important. 

The electronic Hamiltonian and the comesponding eigenfunctions and eigenvalues are independent of the orientation of the nuclear body-fixed frame with respect to the space-fixed one, and hence depend only on m. The index i in Eq. (9) can span both discrete and continuous values. The q ) form... 

These coupling elements are 3(lVnu — l)-dimensional vectors. If the Cartesian components of Rx in 3(A u — 1) space-fixed nuclear congifuration space are... 

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)... 

Figure 1. The space-fixed (ATZ) and body-fixed (xyz) frames. Any rotation of the coordinate system XYZ) to (xyz) may be performed by three successive rotations, denoted by the Euler angles (a, 3, y), about the coordinate axes as follows a) rotation about the Z axis through an angle a(0 < a < 2n), (b) rotation about the new yi axis through an angle P(0 < P < 7i), (c) rotation about the new zi axis through an angle y(0 Y < 2n). The relative orientations of the initial and final coordinate axes are shown in panel (d).

Figure 2. The space-fixed (XYZ) and body-fixed xyz) frames in a diatomic molecule AB. The nuclei are at A and B, and 1 represents the location of a typical electron. The results of inversions of their SF coordinates are A A, B B, and 1 1, respectively. After one executes only the reinversion of the electronic SF coordinates, one obtains 1 — 1. The net effect is then the exchange of the SF nuclear coordinates alone.

Consider a circular ring with N equally spaced fixed sites, m of which are marked in some fashion say, with a cross x (see figure 8.27). 

These simple relations motivate a more formal approximation in which we first re-expand the interaction potential in a space-fixed ("laboratory-frame") coordinate system as... 

Figure 14. Classical trajectories for the H + H2(v = l,j = 0) reaction representing a 1-TS (a-d) and a 2-TS reaction path (e-h). Both trajectories lead to H2(v = 2,/ = 5,k = 0) products and the same scattering angle, 0 = 50°. (a-c) 1-TS trajectory in Cartesian coordinates. The positions of the atoms (Ha, solid circles Hb, open circles He, dotted circles) are plotted at constant time intervals of 4.1 fs on top of snapshots of the potential energy surface in a space-fixed frame centered at the reactant HbHc molecule. The location of the conical intersection is indicated by crosses (x). (d) 1-TS trajectory in hyperspherical coordinates (cf. Fig. 1) showing the different H - - H2 arrangements (open diamonds) at the same time intervals as panels (a-c) the potential energy contours are for a fixed hyperradius of p = 4.0 a.u. (e-h) As above for the 2-TS trajectory. Note that the 1-TS trajectory is deflected to the nearside (deflection angle 0 = +50°), whereas the 2-TS trajectory proceeds via an insertion mechanism and is deflected to the farside (0 = —50°).

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... 

However, these operators change their form when the reference coordinate system is transformed from space fixed to center of mass. 

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