#### 2.3 Representation Matrices

The transposes of the matrices *D(P)* given in Table 2 form a representation [22] of the point group *T*_{d} belonging to the symmetry species *F*_{2}. Their traces occur as characters in the last row of Table 1.

Matrices whose transposes form an *F*_{1} representation of T_{d} can be generated by considering transformation properties of the three components of the angular momentum of a particle. The linear momenta *p*_{x} , p_{y} , p_{z} , conjugate to *x,y,z*, which occur in the angular momentum expressions, transform under the point group operations just as the coordinates x,y,z themselves do, i.e., according to (eq. 1) with the D(P) taken from Table 2. Quantum mechanically, this result arises because equations of the form p_{i }q_{j} − *q*_{j }p_{i} = −*i*ħδ_{ij}, with *i,j* = *x,y,z*, must remain true after performing the variable changes corresponding to a point group operation. The angular momentum transformation properties obtained by the use of (eq. 1) and Table 2 are thus

$$\begin{eqnarray*} \left[ \begin{array}{c} (yp_z - zp_y)_{\rm new}\\ ~\\ (zp_x - xp_z)_{\rm new}\\ (xp_y - yp_x)_{\rm new} \end{array}\right] = P \left[\begin{array}{c} (yp_z - zp_y)\\ (zp_x - xp_z)\\ (xp_y - yp_x)\end{array} \right] = [ ~ D^{F_1}(P) ~ ] \left[\begin{array}{c} (yp_z - zp_y)\\ (zp_x - xp_z)\\ (xp_y - yp_x)\end{array}\right] ~ , \end{eqnarray*}$$

with the matrices *D*^{F1}(*P*) taken from Table 3.

Matrices corresponding to the one-dimensional representations *A*_{1} and *A*_{2} each contain a single element, equal to the character indicated in Table 1.

Matrices whose transposes form an *E* representation of *T*_{d} can be generated by considering the transformation properties of the somewhat less intuitively meaningful functions (2*z*^{2} − *x*^{2}*y*^{2}) and

$$\sqrt{3}(x^2-y^2)$$

These transformation properties are given by (eq. 6)

$$\begin{eqnarray*} \left[ \begin{array}{c} (2z^2-x^2-y^2)_{\rm new}\\ ~\\ \sqrt{3}(x^2 - y^2)_{\rm new} \end{array}\right] = P \left[\begin{array}{c} (2z^2-x^2-y^2)\\ \sqrt{3}(x^2 - y^2) \end{array} \right] = [ D^E(P) ] \left[\begin{array}{c} (2z^2-x^2-y^2)\\ \sqrt{3}(x^2 - y^2)\end{array}\right] ~ ,\end{eqnarray*}$$

with the matrices *D*^{E}(P) taken from Table 4.

It is common in the methane literature to discuss not only the transformation properties of *x,y,z* but also the transformation properties of the spherical tensor forms +2^{−½}(x + iy), +z, −2^{−½}(x − iy). It can be shown, by applying *P* to both sides of the following equation, that the transformation matrices for any functions *f,g,h*, defined by

$$\begin{eqnarray*} \left[ \begin{array}{c} f\\ g\\ h\end{array}\right] = [ \quad U \quad ] \left[\begin{array}{c} x\\ y\\ z\end{array}\right] ~ , \end{eqnarray*}$$

where *U* is not a function of *x,y,z*, are equal to *U D*^{F2}(P) U^{ −1}.