16.2: Tablas de caracteres

Última actualización
Guardar como PDF

Page ID: 81253

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

EN CONSTRUCCIÓN

Grupos de baja simetría (\(C_1, \; C_s, \; C_i\))

\(\begin{array}{|l|c|l|l|} \hline \bf{C_1} & \ E & h=1\\ \hline A_1 & 1 \\ \hline \end{array} \nonumber\)

\(\begin{array}{|l|lc|ll|} \hline \bf{C_s} & E & \sigma_h & h=2& \\ \hline A & 1 & 1 & x, y , R_z & x^2, y^2, z^2, xy \\ A' & 1 & -1 & z, R_x, R_y & yz, xz \\ \hline \end{array}\nonumber\)

\(\begin{array}{|l|cc|ll|} \bf{C_1} & E & i & h=3 & \hline \\ \hline A_g & 1 & 1 & R_x, R_y, R_z & x^2, y^2, z^2, xy, xz, yz \\ A_u & 1 & -1 & x, y, z & \\ \hline \end{array} \nonumber\)

Los grupos\(C_n\)

\(\begin{array}{|l|cc|ll|} \hline \bf{C_2} & E & C_2 & h=2 & \\ \hline A & 1 & 1 & z, R_z & x^2, y^2, z^2, xy \\ B & 1 & -1 & x, y , R_x, R_y & yz, xz \\ \hline \end{array} \nonumber\)

\(\begin{array}{|l|c|ll|} \hline \bf{C_3} & E \: \: \: \: \: C_3 \: \: \: \: \: C_3^2 & h = 3 & \\ \hline A & 1 \: \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: \: 1 & x, R_z & x^2 + y^2, z^2 \\ E & \begin{Bmatrix} 1 & \varepsilon & \varepsilon^* \\ 1 & \varepsilon^* & \varepsilon \end{Bmatrix} & (x, y), \; (R_x, R_y) & (x^2-y^2, xy), \; (xz, yz) \\ \hline \end{array} \\ \nonumber \\ \)
\(\varepsilon = e^{(2\pi i)/3}\)

\(\begin{array}{|l|c|ll|} \hline \bf{C_4} & E \: \: \: \: \: C_4 \: \: \: \: \: C_2 \: \: \: \: \: C_4^3 &h=4& \\ \hline A & 1 \: \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: \: 1 & z, R_z & x^2 + y^2, z^2 \\ B & 1 \: \: \: \: -1 \: \: \: \: \: \: \: \: 1 \: \: \: \: -1 & & x^2 - y^2, xy \\ E & \begin{Bmatrix} 1 & i & -1 & -i \\ 1 & -i & -1 & i \end{Bmatrix} & (x, y), \;(R_x, R_y) & (yz, xz) \\ \hline \end{array} \)

Los grupos\(C_{nv}\)

\(\begin{array}{|l|cccc|ll|} \hline \bf{C_{2v}} & E & C_2 & \sigma_v(xz) & \sigma_v'(yz) & h=4 & \\ \hline A_1 & 1 & 1 & 1 & 1 & z & x^2, y^2, z^2 \\ A_2 & 1 & 1 & -1 & -1 & R_z & xy \\ B_1 & 1 & -1 & 1 & -1 & x, R_y & xz \\ B_2 & 1 & -1 & -1 & 1 & y, R_x & yz \\ \hline \end{array}\)

\(\begin{array}{|l|ccc|ll|} \hline \bf{C_{3v}} & E & 2C_3 & 3\sigma_v & & \\ \hline A_1 & 1 & 1 & 1 & z & x^2 + y^2, z^2 \\ A_2 & 1 & 1 & -1 & R_z & \\ E & 2 & -1 & 0 & x, y, R_x, R_y & x^2 - y^2, xy, xz, yz \\ \hline \end{array}\)

Los grupos\(C_{nh}\)

\[\begin{array}{l|cccc|l|l} C_{2h} & E & C_2 & i & \sigma_h & & \\ \hline A_g & 1 & 1 & 1 & 1 & R_z & x^2, y^2, z^2, xy \\ B_g & 1 & -1 & 1 & -1 & R_x, R_y & xz, yz \\ A_u & 1 & 1 & -1 & -1 & z & \\ B_u & 1 & -1 & -1 & 1 & x, y & \end{array} \label{30.9}\]

\[\begin{array}{l|c|l|l}C_{3h} & E \: \: \: \: \: C_3 \: \: \: \: \: C_3^2 \: \: \: \: \: \sigma_h \: \: \: \: \: S_3 \: \: \: \: \: S_3^5 & & c = e^{2\pi/3} \\ \hline A & 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 & R_z & x^2 + y^2, z^2 \\ E & \begin{Bmatrix} 1 & \: \: c & \: \: c^* & \: \: 1 & \: \: c & \: \: c^* \\ 1 & \: \: c^* & \: \: c & \: \: 1 & \: \: c^* & \: \: c \end{Bmatrix} & x, y & x^2 - y^2, xy \\ A' & 1 \: \: \: \: \: \: 1 \: \: \: \: \: \: 1 \: \: \: \: -1 \: \: \: \: -1 \: \: \: \: \: -1 & z & \\ E' & \begin{Bmatrix} 1 & c & c^* & -1 & -c & -c^* \\ 1 & c^* & c & -1 & -c^* & -c \end{Bmatrix} & R_x, R_y & xz, yz \end{array} \label{30.10}\]

Los grupos\(D_n\)

\[\begin{array}{l|cccc|l|l} D_2 & E & C_2(z) & C_2(y) & C_2(x) & & \\ \hline A & 1 & 1 & 1 & 1 & & x^2, y^2, z^2 \\ B_1 & 1 & 1 & -1 & -1 & z, R_z & xy \\ B_2 & 1 & -1 & 1 & -1 & y, R_y & xz \\ B_3 & 1 & -1 & -1 & 1 & x, R_x & yz \end{array} \label{30.11}\]

\[\begin{array}{l|ccc|l|l} D_3 & E & 2C_3 & 3C_2 & & \\ \hline A_1 & 1 & 1 & 1 & & x^2 + y^2, z^2 \\ A_2 & 1 & 1 & -1 & z, R_z & \\ E & 2 & -1 & 0 & x, y, R_x, R_y & x^2 - y^2, xy, xz, yz \end{array} \label{30.12}\]

Los grupos\(D_{nd}\)

\[\begin{array}{|l|ccccc|ll|} \hline D_{2d} & E & 2S_4 & C_2 & 2C_2' & 2\sigma_d & & \\ \hline A_1 & 1 & 1 & 1 & 1 & 1 & & x^2 + y^2, z^2 \\ A_2 & 1 & 1 & 1 & -1 & -1 & R_z & \\ B_1 & 1 & -1 & 1 & 1 & -1 & & x^2 - y^2 \\ B_2 & 1 & -1 & 1 & -1 & 1 & z & xy \\ E & 2 & 0 & -2 & 0 & 0 & (x, y), (R_x, R_y) & (xz, yz) \\ \hline \end{array} \label{30.14}\]

\[\begin{array}{l|cccccc|l|l} D_{3d} & E & 2C_3 & 3C_2 & i & 2S_6 & 3\sigma_d & & \\ \hline A_{1g} & 1 & 1 & 1 & 1 & 1 & 1 & & x^2 + y^2, z^2 \\ A_{2g} & 1 & 1 & -1 & 1 & 1 & -1 & R_z & \\ E_g & 2 & -1 & 0 & 2 & -1 & 0 & R_x, R_y & x^2 - y^2, xy, xz, yz \\ A_{1u} & 1 & 1 & 1 & -1 & -1 & -1 & & \\ A_{2u} & 1 & 1 & -1 & -1 & -1 & 1 & z & \\ E_u & 2 & -1 & 0 & -2 & 1 & 0 & x, y & \end{array} \label{30.15}\]

Los Grupos\(D_{nh}\)

\ [\ begin {array} {|c|rrrrrrrrr|cc|}\ hline\ bf {D_ {2h}} & E & C_2 (z) & C_2 (y) &C_2 (x) & i &\ sigma (xy) &\ sigma (xz) &\ sigma (yz) & h=8 &\
\ hline A_ {g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2,\; y^2,\; z^2\\
B_ {1g} & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & R_z & xy\\
B_ {2g} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & R_y & zx\\
B_ {3g} & 1 & -1 & 1 & -1 & -1 & -1 & 1 & 1 & R_x & amp; yz\\
A_ {u} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & -1 & &\
b_ {1u} & 1 & 1 & -1 & -1 & -1 & 1 & 1 & z &\\
b_ {2u} & 1 & -1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & amp; y &\\
B_ {3u} & 1 & -1 & 1 & -1 & 1 & 1 & 1 & -1 & -1 & x &\\
\ hline\ end {array}\]

\ [\ begin {array} {|c|rrrrrrr|cc|}\ hline\ bf {D_ {3h}} & E & 2C_3 & 3C_2 &\ sigma_h & 2S_3 & 3\ sigma_v & h=8 &
\\\ hline A_ {1} '& 1 & 1 & 1 & 1 & & x2+y^y^2,\; z^2\\
A_ {2}' & 1 & 1 & -1 & 1 & 1 & 1 & ; -1 & R_z &\\
E' & 2 & -1 & 0 & 2 & -1 & 0 & (x,\; y) & (x^2-y^2,\; xy)\\
A_ {1}” & 1 & 1 & -1 & -1 & -1 & & -1 & &\
A_ {2}” & 1 & 1 & 1 & R_z &\\ A_ {2}” & 1 & 1 & R_z &\\
E” & 2 & -1 & 0 & -2 & 1 & 0 & (R_x,\; R_y) & (xz,\; yz)\\
\ hline\ end {array}\]

\ begin {array} {|c|rrrrrrrrrrrr|cc|}
\ hline\ bf {D_ {4h}} & E & 2C_4 & C_2 & 2C_2' & 2C_2" & i & 2S_4 &\ sigma_h & 2\ sigma_v & 2\ sigma_d & h=16 &
\\ hline A_ {1g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,\; z^2\\
A_ {2g} & 1 & 1 & 1 & -1 & 1 & 1 & 1 & -1 & -1 & r_z &\\
B_ {1g} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & 1 & -1 & & x^2-y^2\\ B_ {2g} & 1 & 1 & 1 & 1 & 1 & -1 & x^2-y^2\
B_ {2g} & & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & 1 & & xy\\
E_ {g} & 2 & 0 & 2 & 0 & 2 & 0 & -2 & 0 & 0 & (R_x,\; R_y) & (xz,\; yz)\\
A_ {1u} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & &\\
A_ {2u} & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & z &\\
B_ {1u} & 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & & &\
B_ {2u} & amp; 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & & &\\
E _ {u} & 2 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & & (x,\; y)\
\ hline\ end {array}

\ begin {array} {|c|rrrrrrrrr|cc|}
\ hline\ bf {D_ {5h}} & E & 2C_5 & 2C_5 & 2C_5^2 & 5C_2 &\ sigma _h & 2S_5 & 2S_5^2 & 5\ sigma_h & h=20 &
\\ hline A_ {1} 'y 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & & x^2+y^2,\; z^2\\ A_ {2} '& 1 & 1 & 1 & -1 & 1 & 1 & 1 & -1 & r_z &\\
E_ {1}' & 2 & 2cos (72 ^ {\ circ}) & 2cos (144 ^ {\ circ}) & 0 & 2 & 2cos (72 ^ {\ circ}) & 2cos (144 ^ {\ circ}) & 0 & (x,\; y) &\\
E_ {2} 'y 2 & 2cos (144 ^ { \ circ}) & 2cos (72 ^ {\ circ}) & 0 & 2 & 2cos (144 ^ {\ circ}) & 2cos (72 ^ {\ circ}) & 0 & & (x^2-y^2,\; xy)\\
A_ {1}” & 1 & 1 & 1 & -1 & -1 & -1 & & -1 & &\\
A_ {2}” & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & z &\\
E_ {1}” & 2 & 2cos (72 ^ {\ circ}) & 2cos (144 ^ {\ circ}) & 0 & -2 & -2cos (72 ^ {\ circ}) & -2cos (144 ^ {\ circ}) & 0 & (R_x,\; R_y) & (xz,\; yz)\\
E_ {2}” y 2 y 2cos (144 ^ {\ circ}) y 2cos (72 ^ {\ circ}) y amp; 0 & -2 & -2cos (144 ^ {\ circ}) & -2cos (72 ^ {\ circ}) & 0 & &\\
\ hline\ end {array}

Grupos de alta simetría

\ [\ begin {array} {|c|cccccccc|c|c|}
\ hline\ bf {D_ {\ infty h}} &\ mathrm {E} & 2\ mathrm {C} _ {\ infty} ^ {\ phi} &... &\ infty\ sigma_v & i & 2S_ {\ infty} ^ {\ phi} &... &\ infty C_2 & &\\
\ hline
{A} _ {1g} & 1 & 1 &... & 1 & 1 & 1 &... & 1 & & x^ {2} +y^ {2},\, z^ {2}\\
{A} _ {2g} & 1 &... & -1 & 1 & 1 &... & -1 & R_z &\\
{E} _ {1g} & 2 & 2\ cos\ phi &... & 0 & 2 & -2\ cos\ phi &... & 0 & (R_z,\, R_y) & (xz,\, yz)\\
{E} _ {2g} & 2 & 2\ cos 2\ phi &... & 0 & 2 & 2\ cos2\ phi &... & 0 & & (x^ {2} -y^ {2},\, xy)\\
... &... &... &... &... &... &... &... &... & &\\
{A} _ {1u} & 1 &... & 1 & -1 & -1 &... & -1 & z &\\
{A} _ {2u} & 1 &... & -1 & -1 & -1 &... & 1 & &\\
{E} _ {1u} & 2 & 2\ cos\ phi &... & 0 & -2 & 2\ cos\ phi &... & 0 & (x,\, y) &\\
{E} _ {2u} & 2 & 2\ cos 2\ phi &... & 0 & -2 & -2\ cos2\ phi &... & 0 & &\\
... &... &... &... &... &... &... &... &... & &\\
\ hline
\ end {array}\]

\(C_{\infty v}\) and \(D_{\infty h}\)

\[\begin{array}{l|cccccccc|l|l} D_{\infty h} & E & 2C_\infty^\Phi & \ldots & \infty \sigma_v & i & 2S_\infty^\Phi & \ldots & \infty C_2 & & \\ \hline \Sigma_g^+ & 1 & 1 & \ldots & 1 & 1 & 1 & \ldots & 1 & & x^2 + y^2, z^2 \\ \Sigma_g^- & 1 & 1 & \ldots & -1 & 1 & 1 & \ldots & -1 & R_z & \\ \Pi_g & 2 & 2cos \Phi & \ldots & 0 & 2 & -2cos \Phi & \ldots & 0 & R_x, R_y & xz, yz \\ \Delta_g & 2 & 2cos 2\Phi & \ldots & 0 & 2 & 2cos 2\Phi & \ldots & 0 & & x^2 - y^2, xy \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & & \\ \Sigma_u^+ & 1 & 1 & \ldots & 1 & -1 & -1 & \ldots & -1 & z & \\ \Sigma_u^- & 1 & 1 & \ldots & -1 & -1 & -1 & \ldots & 1 & & \\ \Pi_u & 2 & 2cos \Phi & \ldots & 0 & -2 & 2cos \Phi & \ldots & 0 & x, y & \\ \Delta_u & 2 & 2cos 2\Phi & \ldots & 0 & -2 & -2cos 2\Phi & \ldots & 0 & & \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & & \end{array} \label{30.16}\]

\(S_n\) groups

\[\begin{array}{l|c|l|l} S_4 & E \: \: \: \: \: S_4 \: \: \: \: \: C_2 \: \: \: \: \: S_4^3 & & \\ \hline A & 1 \: \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: \: 1 & R_z & x^2 + y^2, z^2 \\ B & 1 \: \: \: \: -1 \: \: \: \: \: \: \: \: 1 \: \: \: \: -1 & z & x^2 - y^2, xy \\ E & \begin{Bmatrix} 1 & i & -1 & -i \\ 1 & -i & -1 & i \end{Bmatrix} & x, y, R_x, R_y & xz, yz \end{array} \label{30.17}\]

\[\begin{array}{l|c|l|l} S_6 & E \: \: \: \: \: C_3 \: \: \: \: \: C_3^2 \: \: \: \: \: i \: \: \: \: \: S_6^5 \: \: \: \: \: S_6 & & c=e^{2\pi/3} \\ \hline A_g & 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 & R_z & x^2 + y^2, z^2 \\ E_g & \begin{Bmatrix} 1 & \: \: c & \: \: c^* & \: \: 1 & \: \: c & \: \: c^* \\ 1 \: \: & \: \: c^* & \: \: c & \: \: 1 & \: \: c^* & \: \: c \end{Bmatrix} & R_x, R_y & x^2 - y^2, xy, xz, yz \\ A_u & 1 \: \: \: \: \: \: 1 \: \: \: \: \: \: 1 \: \: \: \: -1 \: \: \: \: -1 \: \: \: \: \: -1 & z & \\ E_u & \begin{Bmatrix} 1 & c & c^* & -1 & -c & -c^* \\ 1 & c^* & c & -1 & -c^* & -c \end{Bmatrix} & x, y & \end{array} \label{30.18}\]

Grupos cúbicos

\[\begin{array}{l|c|l|l} T & E \: \: \: 4C_3 \: \: \: 4C_3^2 \: \: \: 3C_2 & & c=e^{2\pi/3} \\ \hline A & 1 \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: 1 & & x^2 + y^2, z^2 \\ E & \begin{Bmatrix} 1 & c & c^* & 1 \\ 1 & c* & c & 1 \end{Bmatrix} & & 2z^2 - x^2 - y^2, x^2 - y^2 \\ T & 3 \: \: \: \: \: 0 \: \: \: \: \: \: \: 0 \: \: \: -1 & R_x, R_y, R_z, x, y, z & xy, xz, yz \end{array} \label{30.19}\]

\[\begin{array}{l|ccccc|l|l} T_d & E & 8C_3 & 3C_2 & 6S_4 & 6\sigma_d & & \\ \hline A_1 & 1 & 1 & 1 & 1 & 1 & & x^2 + y^2, z^2 \\ A_2 & 1 & 1 & 1 & -1 & -1 & & \\ E & 2 & -1 & 2 & 0 & 0 & & 2z^2 - x^-2 - y^2, x^2 - y^2 \\ T_1 & 3 & 0 & -1 & 1 & -1 & R_x, R_y, R_z & \\ T_2 & 3 & 0 & -1 & -1 & 1 & x, y, z & xy, xz, yz \end{array} \label{30.20}\]

Colaboradores y Atribuciones

Claire Vallance (University of Oxford)

Modified by Kathryn Haas (khaaslab.com)