Methane Symmetry Operations - Representation Matrices

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 F₂. Their traces occur as characters in the last row of Table 1.

Matrices whose transposes form an F₁ 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 = −iħδ_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 A₁ and A₂ 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 (2z² − x²y²) and

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

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.