The geometrical operations can be transformed into algebraic operations by asking: what are the new coordinates of a particle originally placed at the position x,y,z and then subjected to one of the geometrical symmetry operations P? If the convention for rotations is fixed by defining the effect of C_{n}(u) on a particle to be a rotation of the particle (active operation) through an angle -2π/n about the direction u, using the right-hand thumb rule to determine the sign of the rotation, then we find that the new coordinates of the particle x^{P}_{new} , y^{P}_{new} , z^{P}_{new} are given as functions of the old by an equation of the form
$$\left[\begin{array}{c} x_{\rm new}^P(x,y,z)\\ y_{\rm new}^P(x,y,z)\\ z_{\rm new}^P(x,y,z)\end{array}\right] = [\quad D(P)\quad ] \left[\begin{array}{c} x\\ y\\ z \end{array}\right] ~ ,$$
(eq. 1)
where the matrices D(P) for each point group operation P in section 2.1 are given in full in Table 2.
Note that the functions x_{new} , y_{new} , z_{new} defined by (eq. 1) and Table 2 happen to be extremely simple. Each of them has one of the six forms (+x, -x, +y, -y, +z, -z), regardless of which point group operation is considered. These very simple transformation properties make it relatively easy for the reader to verify many of the algebraic statements in this article.
It may seem strange to have chosen a convention in which a particle is rotated through an angle -2π/n rather than +2π/n. It will turn out later that vibrational displacement vectors are subjected to this negative rotation, while the molecule-fixed axis system is subjected to the corresponding positive rotation [12, 13].
Conventions for the application of a point group operation P to a function f of the variables x,y,z are fixed by the following definition
$$P \, f (x,y,z) \equiv f \left(x_{\rm new}^P(x,y,z), ~ y_{\rm new}^P(x,y,z), ~ z_{\rm new}^P(x,y,z) \right) = g(x,y,z) ~ .$$
(eq. 2)
Note that the transformed function g is still a function of the original variables x,y,z; it was obtained from the untransformed function f (x,y,z) by substituting everywhere x^{P}_{new}(x,y,z) for x, etc. The effect of two point group operations in succession is defined by (eq. 3)
$$Q\,P\,f (x,y,z) \equiv Q\,g (x,y,z)= h(x,y,z) ~ ,$$
(eq. 3)
where g is obtained from f as in (eq. 2) and h is obtained from g(x,y,z) by substituting everywhere x^{Q}_{new}(x,y,z) for x, etc.
It can be seen that the results of the operation Q P correspond to replacing x in the function f(x,y,z), by x^{QP}_{new}(x,y,z), etc., where
$$\begin{eqnarray*} \left[ \begin{array}{c} x_{\rm new}^{QP}(x,y,z)\\ ~\\ y_{\rm new}^{QP}(x,y,z)\\ z_{\rm new}^{QP}(x,y,z) \end{array}\right] &=& Q\,P \left[\begin{array}{c} x\\ y \\ z\end{array} \right] = Q[\quad D(P)\quad ] \left[\begin{array}{c} x\\ y\\ z \end{array}\right] = [\quad D(P)\quad ] Q \left[\begin{array}{c} x\\ y\\ z \end{array}\right] \\ &=& [\quad D(P)\quad ] ~ [\quad D(Q)\quad ] ~ \left[\begin{array}{c} x\\ y\\ z \end{array}\right] ~ . \end{eqnarray*}$$
(eq. 4)
Because the order of the matrices D(P)D(Q) is opposite to the order of the operators Q P, the transposes of the matrices D(P) form a representation [22] of the group T_{d}, not the matrices themselves. Avoiding this slight complication would have required adopting a form of (eq. 1) written in terms of (x,y,z) row vectors post multiplied by the D(P)'s.