In this section we wish to relate the point group operations to the permutation-inversion group operations. This can be accomplished with the least amount of misunderstanding by using algebraic rather than geometric arguments, i.e., by considering an equation relating the laboratory-fixed coordinates of the nuclei to the rotational and vibrational coordinates used in molecular wave functions [12, 13]. In this article we shall take this equation to have the form
$$\mbox{$R$}_i = \mbox{$R$}+S^{-1}(\chi\theta\phi) (\mbox{$a$}_i+\mbox{$d$}_i) ~ .$$
(eq. 9)
R_{i} is a column vector containing the three laboratory-fixed coordinates X_{i}, Y_{i}, Z_{i} of nucleus i. R is a column vector containing the three laboratory-fixed coordinates of the center-of-mass of the nuclei. S is a 3 × 3 rotation matrix relating the orientation of the molecule-fixed axes x,y,z, to the orientation of the corresponding laboratory-fixed axes X,Y,Z. We fix conventions for the Eulerian angles χ θ φ by taking S to have the form
$$\begin{eqnarray*} &~& \qquad\qquad X \qquad\qquad\qquad\qquad Y \qquad\qquad\qquad Z \\ \begin{array}{r} x\\ S(\chi\theta\phi) = y\\ z \end{array} &~& \left[\begin{array}{ccc} c\chi \, c\theta \, c\phi -s\chi \, s\phi & c\chi \, c\theta \, s\phi +s\chi \, c\phi & -c\chi \, s\theta\\ -s\chi \, c\theta \, c\phi -c\chi \, s\phi & -s\chi \, c\theta \, s\phi +c\chi \, c\phi & s\chi \, s\theta\\ s\theta \, c\phi & s\theta \, s\phi & c\theta\end{array} \right] ~ , \end{eqnarray*}$$
(eq. 10)
where cχ = cosχ, sχ = sinχ, etc. The quantity a_{i} is a column vector containing the three molecule-fixed coordinates of nucleus i at equilibrium, and d_{i} is a column vector containing the three molecule-fixed components of the instantaneous displacement vector of nucleus i from its equilibrium position.
Equation (9) can be understood most easily by reading it from the right-hand side, as illustrated in Fig. 2. We consider the nuclei to be brought into their instantaneous positions in the laboratory by four steps. (a) Each nucleus is placed at its equilibrium position a_{i}, as in Fig. 2a. (b) Each nucleus is displaced from its equilibrium position by an amount d_{i}, as in Fig. 2b. (c) The entire molecular framework is rotated using the rotation matrix S(χθφ), as in Fig. 2c. (d) The entire molecular framework is translated by the vector R, as in Fig. 2d.
For completeness, though they will not be referred to again in this article, the center-of-mass and Eckart conditions defining (explicitly) the vector R and (implicitly) the rotational angles (χθφ), are given in (eq. 11).
$$\begin{eqnarray*} \Sigma_i m_i (\mbox{$R$}_i - \mbox{$R$}) &=&\mbox{$0$}\\ \Sigma_i m_i \mbox{$a$}_i \times S(\chi\theta\phi) \cdot (\mbox{$R$}_i - \mbox{$R$}) &=&\mbox{$0$} ~ . \end{eqnarray*}$$
(eq. 11)
The qualitative ideas required to relate the point-group and permutation-inversion operations are as follows. We seek a set of transformations for the dynamical variables comprising vibration-rotation-translation phase space, such that after the d_{i}, χ, θ, φ, and R on the right-hand side of (eq. 9), have been replaced by the (d_{i})_{new}, χ_{new}, θ_{new}, φ_{new}, and R_{new}, the new right-hand side will be consistent with a left-hand side in which R_{i} has been replaced by (R_{i})_{new} = ±R_{j}, with a sign choice and subscript choice in agreement with the prescription of the permutation-inversion operation under consideration.
It is sometimes convenient, when a given symmetry operation P is to be applied to the dynamical variables of configuration space, to visualize it as the product of three separate operators ^{1}P, ^{2}P, ^{3}P, which act on the vibrational, rotational, and translational variables, respectively. While we shall not use this notation explicitly in Section 4.2 and Section 4.3, we shall deal separately and in turn with vibrational, rotational, and translational transformations. When considerations involving translation of the molecule as a whole are not important, the operator ^{3}P can be ignored, as has been done in Fig. 3 and Fig. 4 illustrating the vibrational and rotational transformations discussed below.