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Methane Symmetry Operations - Table 3
Table 3. Transformation matrices DF1(P) for (eq. 5)
The transposes of these matrices form a representation of the point group Td of species F1. Rows and columns can be labelled by the symbols F1x, F1y, F1z, as indicated for the identity matrix. Matrices other than the identity are labelled below by the permutation-inversion operation to which they correspond.
$$\begin{array}{rc}
&E \\
\begin{array}{c}
F_{1x}\\
F_{1y}\\
F_{1z}
\end{array} & \left[\begin{array}{ccc}
1 &0 &0 \\
0 &1 &0 \\
0 &0 &1
\end{array}\right] \\
& F_{1x} \,~ F_{1y} \,~ F_{1z} \end{array}$$
|
$$\begin{array}{c}
C_3(111)\\
\left[\begin{array}{ccc}
0 &1 &0 \\
0 &0 &1 \\
1 &0 &0
\end{array}\right] \\
(132)
\end{array}$$
|
$$\begin{array}{c}
C_3^2(111)\\
\left[\begin{array}{ccc}
0 &0 &1 \\
1 &0 &0 \\
0 &1 &0
\end{array}\right] \\
(123)
\end{array}$$
|
$$\begin{array}{c}
C_3(-111)\\
\left[\begin{array}{ccc}
0 &0 &-1 \\
-1 &0 &0 \\
0 &1 &0
\end{array}\right] \\
(134)
\end{array}$$
|
$$\begin{array}{c}
C_3^2(-111)\\
\left[\begin{array}{ccc}
0 &-1 &0 \\
0 &0 &1 \\
-1 &0 &0
\end{array}\right] \\
(143)
\end{array}$$
|
$$\begin{array}{c}
C_3(-1-11)\\
\left[\begin{array}{ccc}
0 &1 &0 \\
0 &0 &-1 \\
-1 &0 &0
\end{array}\right] \\
(124)\end{array}$$
|
$$\begin{array}{c}
C_3^2(-1-11)\\
\left[\begin{array}{ccc}
0 &0 &-1 \\
1 &0 &0 \\
0 &-1 &0
\end{array}\right] \\
(142)\end{array}$$
|
$$\begin{array}{c}
C_3(1 -11)\\
\left[\begin{array}{ccc}
0 &0 &1 \\
-1 &0 &0 \\
0 &-1 &0
\end{array}\right] \\
(243)
\end{array}$$
|
$$\begin{array}{c}
C_3^2(1 -11)\\
\left[\begin{array}{ccc}
0 &-1 &0 \\
0 &0 &-1 \\
1 &0 &0
\end{array}\right] \\
(234)
\end{array}$$
|
$$\begin{array}{c}
C_2(x)\\
\left[\begin{array}{ccc}
1 &0 &0 \\
0 &-1 &0 \\
0 &0 &-1
\end{array}\right] \\
(13)(24)
\end{array}$$
|
$$\begin{array}{c}
C_2(y)\\
\left[\begin{array}{ccc}
-1 &0 &0 \\
0 &1 &0 \\
0 &0 &-1
\end{array}\right] \\
(14)(23)
\end{array}$$
|
$$\begin{array}{c}
C_2(z)\\
\left[\begin{array}{ccc}
-1 &0 &0 \\
0 &-1 &0 \\
0 &0 &1
\end{array}\right] \\
(12)(34)\end{array}$$
|
$$\begin{array}{c}
S_4(x) \\
\left[\begin{array}{ccc}
1 &0 &0 \\
0 &0 &-1 \\
0 &1 &0
\end{array}\right] \\
(1432)^*
\end{array}$$
|
$$\begin{array}{c}
S_4^3(x) \\
\left[\begin{array}{ccc}
1 &0 &0 \\
0 &0 &1 \\
0 &-1 &0
\end{array}\right] \\
(1234)^*
\end{array}$$
|
$$\begin{array}{c}
S_4(y)\\
\left[\begin{array}{ccc}
0 &0 &1 \\
0 &1 &0\\
-1 &0 &0
\end{array}\right] \\
(1342)^*
\end{array}$$
|
$$\begin{array}{c}
S_4^3(y)\\
\left[\begin{array}{ccc}
0 &0 &-1 \\
0 &1 &0\\
1 &0 &0
\end{array}\right] \\
(1243)^*
\end{array}$$
|
$$\begin{array}{c}
S_4(z)\\
\left[\begin{array}{ccc}
0 &-1 &0\\
1 &0 &0\\
0 &0 &1
\end{array}\right] \\
(1324)^*
\end{array}$$
|
$$\begin{array}{c}
S_4^3(z)\\
\left[\begin{array}{ccc}
0 &1 &0\\
-1 &0 &0 \\
0 &0 &1
\end{array}\right] \\
(1423)^*
\end{array}$$
|
$$\begin{array}{c}
\sigma_d(011) \\
\left[\begin{array}{ccc}
-1 &0 &0 \\
0 &0 &1 \\
0 &1 &0
\end{array}\right] \\
(24)^*
\end{array}$$
|
$$\begin{array}{c}
\sigma_d(0-11) \\
\left[\begin{array}{ccc}
-1 &0 &0 \\
0 &0 &-1 \\
0 &-1 &0
\end{array}\right] \\
(13)^*
\end{array}$$
|
$$\begin{array}{c}
\sigma_d(101) \\
\left[\begin{array}{ccc}
0 &0 &1 \\
0 &-1 &0\\
1 &0 &0
\end{array}\right] \\
(14)^*
\end{array}$$
|
$$\begin{array}{c}
\sigma_d(10-1) \\
\left[\begin{array}{ccc}
0 &0 &-1 \\
0 &-1 &0\\
-1 &0 &0
\end{array}\right] \\
(23)^*
\end{array}$$
|
$$\begin{array}{c}
\sigma_d(110) \\
\left[\begin{array}{ccc}
0 &1 &0\\
1 &0 &0 \\
0 &0 &-1
\end{array}\right] \\
(34)^*
\end{array}$$
|
$$\begin{array}{c}
\sigma_d(-110) \\
\left[\begin{array}{ccc}
0 &-1 &0\\
-1 &0 &0 \\
0 &0 &-1
\end{array}\right] \\
(12)^*
\end{array}$$
|
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Created September 20, 2016, Updated November 15, 2019