Official websites use .gov
A .gov website belongs to an official government organization in the United States.
Secure .gov websites use HTTPS
A lock (
) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.
Methane Symmetry Operations - Table 2
Table 2. Transformation matrices D(P) for (eq. 1)
The transposes of these matrices form a representation of the point group Td of species F2. Rows and columns can be labelled by the symbols F2x, F2y, F2z, as indicated for the identity matrix. Matrices other than the identity are labelled below by the permutation-inversion operation to which they correspond.
$$\begin{array}{rc}
&E \\
\begin{array}{c}
F_{2x}\\
F_{2y}\\
F_{2z}
\end{array} & \left[\begin{array}{ccc}
1 &0 &0 \\
0 &1 &0 \\
0 &0 &1
\end{array}\right] \\
& F_{2x} \,~ F_{2y} \,~ F_{2z} \end{array}$$
|
$$\begin{array}{c}
C_3(111)\\
\left[\begin{array}{ccc}
0 &1 &0 \\
0 &0 &1 \\
1 &0 &0
\end{array}\right] \\
(132)
\end{array}$$
|
$$\begin{array}{c}
C_3^2(111)\\
\left[\begin{array}{ccc}
0 &0 &1 \\
1 &0 &0 \\
0 &1 &0
\end{array}\right] \\
(123)
\end{array}$$
|
$$\begin{array}{c}
C_3(-111)\\
\left[\begin{array}{ccc}
0 &0 &-1 \\
-1 &0 &0 \\
0 &1 &0
\end{array}\right] \\
(134)
\end{array}$$
|
$$\begin{array}{c}
C_3^2(-111)\\
\left[\begin{array}{ccc}
0 &-1 &0 \\
0 &0 &1 \\
-1 &0 &0
\end{array}\right] \\
(143)
\end{array}$$
|
$$\begin{array}{c}
C_3(-1-11)\\
\left[\begin{array}{ccc}
0 &1 &0 \\
0 &0 &-1 \\
-1 &0 &0
\end{array}\right] \\
(124)\end{array}$$
|
$$\begin{array}{c}
C_3^2(-1-11)\\
\left[\begin{array}{ccc}
0 &0 &-1 \\
1 &0 &0 \\
0 &-1 &0
\end{array}\right] \\
(142)\end{array}$$
|
$$\begin{array}{c}
C_3(1 -11)\\
\left[\begin{array}{ccc}
0 &0 &1 \\
-1 &0 &0 \\
0 &-1 &0
\end{array}\right] \\
(243)
\end{array}$$
|
$$\begin{array}{c}
C_3^2(1 -11)\\
\left[\begin{array}{ccc}
0 &-1 &0 \\
0 &0 &-1 \\
1 &0 &0
\end{array}\right] \\
(234)
\end{array}$$
|
$$\begin{array}{c}
C_2(x)\\
\left[\begin{array}{ccc}
1 &0 &0 \\
0 &-1 &0 \\
0 &0 &-1
\end{array}\right] \\
(13)(24)
\end{array}$$
|
$$\begin{array}{c}
C_2(y)\\
\left[\begin{array}{ccc}
-1 &0 &0 \\
0 &1 &0 \\
0 &0 &-1
\end{array}\right] \\
(14)(23)
\end{array}$$
|
$$\begin{array}{c}
C_2(z)\\
\left[\begin{array}{ccc}
-1 &0 &0 \\
0 &-1 &0 \\
0 &0 &1
\end{array}\right] \\
(12)(34)\end{array}$$
|
$$\begin{array}{c}
S_4(x) \\
\left[\begin{array}{ccc}
-1 &0 &0 \\
0 &0 &1 \\
0 &-1 &0
\end{array}\right] \\
(1432)^*
\end{array}$$
|
$$\begin{array}{c}
S_4^3(x) \\
\left[\begin{array}{ccc}
-1 &0 &0 \\
0 &0 &-1 \\
0 &1 &0
\end{array}\right] \\
(1234)^*
\end{array}$$
|
$$\begin{array}{c}
S_4(y)\\
\left[\begin{array}{ccc}
0 &0 &-1 \\
0 &-1 &0\\
1 &0 &0
\end{array}\right] \\
(1342)^*
\end{array}$$
|
$$\begin{array}{c}
S_4^3(y)\\
\left[\begin{array}{ccc}
0 &0 &1 \\
0 &-1 &0\\
-1 &0 &0
\end{array}\right] \\
(1243)^*
\end{array}$$
|
$$\begin{array}{c}
S_4(z)\\
\left[\begin{array}{ccc}
0 &1 &0\\
-1 &0 &0\\
0 &0 &-1
\end{array}\right] \\
(1324)^*
\end{array}$$
|
$$\begin{array}{c}
S_4^3(z)\\
\left[\begin{array}{ccc}
0 &-1 &0\\
1 &0 &0 \\
0 &0 &-1
\end{array}\right] \\
(1423)^*
\end{array}$$
|
$$\begin{array}{c}
\sigma_d(011) \\
\left[\begin{array}{ccc}
1 &0 &0 \\
0 &0 &-1 \\
0 &-1 &0
\end{array}\right] \\
(24)^*
\end{array}$$
|
$$\begin{array}{c}
\sigma_d(0-11) \\
\left[\begin{array}{ccc}
1 &0 &0 \\
0 &0 &1 \\
0 &1 &0
\end{array}\right] \\
(13)^*
\end{array}$$
|
$$\begin{array}{c}
\sigma_d(101) \\
\left[\begin{array}{ccc}
0 &0 &-1 \\
0 &1 &0\\
-1 &0 &0
\end{array}\right] \\
(14)^*
\end{array}$$
|
$$\begin{array}{c}
\sigma_d(10-1) \\
\left[\begin{array}{ccc}
0 &0 &1 \\
0 &1 &0\\
1 &0 &0
\end{array}\right] \\
(23)^*
\end{array}$$
|
$$\begin{array}{c}
\sigma_d(110) \\
\left[\begin{array}{ccc}
0 &-1 &0\\
-1 &0 &0 \\
0 &0 &1
\end{array}\right] \\
(34)^*
\end{array}$$
|
$$\begin{array}{c}
\sigma_d(-110) \\
\left[\begin{array}{ccc}
0 &1 &0\\
1 &0 &0 \\
0 &0 &1
\end{array}\right] \\
(12)^*
\end{array}$$
|
Contacts
-
PML webmaster
Created September 20, 2016, Updated November 15, 2019