## Methane Symmetry Operations

#### 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*δ_{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

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 (*x*^{2} −
*y*^{2}). These transformation properties are given by (eq. 6)

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

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