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Physical Measurement Laboratory

Methane Symmetry Operations - Table 4

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Table 4. Transformation matrices DE(P) for (eq. 6)

The transposes of these matrices form a representation of the point group Td of species E. Rows and columns can be labelled by the symbols Ea, Eb, as indicated for the identity matrix. Matrices other than the identity are labelled below by the permutation-inversion operation to which they correspond.

$$\begin{array}{rc}
&E \\
\begin{array}{c}
E_a\\ 
E_b
\end{array} & \left[\begin{array}{rr}
1 &0\\
0 &1
\end{array}\right] \\
 & E_a \,~ E_b \end{array}$$
 
$$\begin{array}{c}
C_3(111)\\
\left[\begin{array}{rr}
-{1\over2} &+{\sqrt{3\over4}} \\
-\sqrt{3\over4} &-{1\over2}
\end{array}\right] \\
(132)
\end{array}$$
 
$$\begin{array}{c}
C_3^2(111)\\
\left[\begin{array}{rr}
-{1\over2} &-{\sqrt{3\over4}} \\
+\sqrt{3\over4} &-{1\over2}
\end{array}\right] \\
(123)
\end{array}$$
 
$$\begin{array}{c}
C_3(-111)\\
\left[\begin{array}{rr}
-{1\over2} &-{\sqrt{3\over4}} \\
+\sqrt{3\over4} &-{1\over2}
\end{array}\right] \\
(134)
\end{array}$$
 
$$\begin{array}{c}
C_3^2(-111)\\
\left[\begin{array}{rr}
-{1\over2} &+{\sqrt{3\over4}} \\
-\sqrt{3\over4} &-{1\over2}
\end{array}\right] \\
(143)
\end{array}$$
 
$$\begin{array}{c}
C_3(-1-11)\\
\left[\begin{array}{rr}
-{1\over2} &+{\sqrt{3\over4}} \\
-\sqrt{3\over4} &-{1\over2}
\end{array}\right] \\
(124)\end{array}$$
 
$$\begin{array}{c}
C_3^2(-1-11)\\
\left[\begin{array}{rr}
-{1\over2} &-{\sqrt{3\over4}} \\
+\sqrt{3\over4} &-{1\over2}
\end{array}\right] \\
(142)\end{array}$$
 
$$\begin{array}{c}
C_3(1 -11)\\
\left[\begin{array}{rr}
-{1\over2} &-{\sqrt{3\over4}} \\
+\sqrt{3\over4} &-{1\over2}
\end{array}\right] \\
(243)
\end{array}$$
 
$$\begin{array}{c}
C_3^2(1 -11)\\
\left[\begin{array}{rr}
-{1\over2} &+{\sqrt{3\over4}} \\
-\sqrt{3\over4} &-{1\over2}
\end{array}\right] \\
(234)
\end{array}$$
 
$$\begin{array}{c}
C_2(x)\\
\left[\begin{array}{rr}
1 &0 \\
0 &1
\end{array}\right] \\
(13)(24)
\end{array}$$
 
$$\begin{array}{c}
C_2(y)\\
\left[\begin{array}{rr}
1 &0 \\
0 &1
\end{array}\right] \\
(14)(23)
\end{array}$$
 
$$\begin{array}{c}
C_2(z)\\
\left[\begin{array}{rr}
1 &0 \\
0 &1 
\end{array}\right] \\
(12)(34)\end{array}$$
 
$$\begin{array}{c}
S_4(x) \\
\left[\begin{array}{rr}
-{1\over2} &-{\sqrt{3\over4}} \\
-\sqrt{3\over4} &+{1\over2}
\end{array}\right] \\
(1432)^*
\end{array}$$
 
$$\begin{array}{c}
S_4^3(x) \\
\left[\begin{array}{rr}
-{1\over2} &-{\sqrt{3\over4}} \\
-\sqrt{3\over4} &+{1\over2}
\end{array}\right] \\
(1234)^*
\end{array}$$
 
$$\begin{array}{c}
S_4(y)\\
\left[\begin{array}{rr}
-{1\over2} &+{\sqrt{3\over4}} \\
+\sqrt{3\over4} &+{1\over2}
\end{array}\right] \\
(1342)^*
\end{array}$$
 
$$\begin{array}{c}
S_4^3(y)\\
\left[\begin{array}{rr}
-{1\over2} &+{\sqrt{3\over4}} \\
+\sqrt{3\over4} &+{1\over2} 
\end{array}\right] \\
(1243)^*
\end{array}$$
 
$$\begin{array}{c}
S_4(z)\\
\left[\begin{array}{rr}
1 &0\\
0 &-1  
\end{array}\right] \\
(1324)^*
\end{array}$$
 
$$\begin{array}{c}
S_4^3(z)\\
\left[\begin{array}{rr}
1 &0 \\ 
0 &-1
\end{array}\right] \\
(1423)^*
\end{array}$$
 
$$\begin{array}{c}
\sigma_d(011) \\
\left[\begin{array}{rr}
-{1\over2} &-{\sqrt{3\over4}} \\
-\sqrt{3\over4} &+{1\over2}
\end{array}\right] \\
(24)^*
\end{array}$$
 
$$\begin{array}{c}
\sigma_d(0-11) \\
\left[\begin{array}{rr}
-{1\over2} &-{\sqrt{3\over4}} \\
-\sqrt{3\over4} &+{1\over2}
\end{array}\right] \\
(13)^*
\end{array}$$
 
$$\begin{array}{c}
\sigma_d(101) \\
\left[\begin{array}{rr}
-{1\over2} &+{\sqrt{3\over4}} \\
+\sqrt{3\over4} &+{1\over2}
\end{array}\right] \\
(14)^*
\end{array}$$
 
$$\begin{array}{c}
\sigma_d(10-1) \\
\left[\begin{array}{rr}
-{1\over2} &+{\sqrt{3\over4}} \\
+\sqrt{3\over4} &+{1\over2}
\end{array}\right] \\
(23)^*
\end{array}$$
 
$$\begin{array}{c}
\sigma_d(110) \\
\left[\begin{array}{rr}
1 &0 \\
0 &-1 
\end{array}\right] \\
(34)^*
\end{array}$$
 
$$\begin{array}{c}
\sigma_d(-110) \\
\left[\begin{array}{rr}
1 &0 \\
0 &-1 
\end{array}\right] \\
(12)^*
\end{array}$$

 

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Created September 20, 2016, Updated November 15, 2019