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Chemistry LibreTexts

Direct Product Tables

  • Page ID
    54310
  • Table 1: For \(C_2\), \(C_{2v}\) , \(C_{2 h}\); \(C_3\), \(C_{3 v}\) , \(C_{3 h}\); \(D_3\) , \(D_{3 d}\) ,\( D_{3 h}\) ; \(C_6\) , \(C_{6 v}\) , \(C_{ 6 h}\); \(D_6\), and \(S_{6}\)
      \(A_1\) \(A_2\) \(B_1\) \(B_2\) \(E_1\) \(E_2\)
    \(A_1\) \(A_1\) \(A_2\) \(B_1\) \(B_2\) \(E_1\) \(E_2\)
    \(A_2\) - \(A_1\) \(B_2\) \(B_1\) \(E_1\) \(E_2\)
    \(B_1\) - - \(A_1\) \(A_2\) \(E_2\) \(E_1\)
    \(B_2\) - - - \(A_1\) \(E_2\) \(E_1\)
    \(E_1\) - - - - \(A_1 + [A_2] + E_2\) \(B_1+B_2+E_1\)
    \(E_2\) - - - - - \( A_1 + [A_2] +E_2\)
    Table 2: For \(T\), \(T_{h}\) , \(T_{d}\); \(O\), and \(O_{h}\)
      \(A\) \(B_1\) \(B_2\) \(T\) \(T_2\)
    \(A_1\) \(A_1\) \(A_2\) \(E\) \(T_1\) \(T_2\)
    \(A_2\) - \(A_1\) \(E\) \(T_2\) \(T_1\)
    \(E\) - - \(A_1 + [A_2] + E_2\) \(T_1 + T_2\) \(T_1+T_2\)
    \(T_1\) - - - \(A_1 + E+ [T_1] + T_2\) \(A_2+E + T_1+T_2\)
    \(T_2\) - - - - \(A_1+E +[T_1] +T_2\)
    Table 3: For \(C_{\infty v}\), and \(D_{\infty h}\)
      \(\Sigma^+\) \(\Sigma^-\) \(\Pi\) \(\Delta\) -
    \(\Sigma^+\) \(\Sigma^+\) \(A_2\) \(\Pi\) \(\Delta\) -
    \(\Sigma^-\) - \(\Sigma^+\) \(\Pi\) \(\Delta\) -
    \(\Pi\) - - \( \Sigma^+ + [ \Sigma^- ] + \Delta \) \(\Delta + \Phi\) -
    \(\Delta\) - - - \( \Sigma^+ + [ \Sigma^- ] + \Sigma \) -
    - - - - - -
    Table 4: For \(D_{2}\), and \(D_{2h}\)
      \(A\) \(B_1\) \(B_2\) \(B_3\)
    \(A_1\) \(A\) \(B_1\) \(B_2\) \(B_3\)
    \(B_1\) - \(A\) \(B_3\) \(B_2\)
    \(B_2\) - - \(A\) \(B_1\)
    \(B_3\) - - - \(A\)
    Table 5: For \(C_{6v}\), \(D_6\) and \(D_{6h}\)
      \(A_1\) \(A_2\) \(B_1\) \(B_2\) \(E_1\) \(E_2\)
    \(A_1\) \(A_1\) \(A_2\) \(B_1\) \(B_2\) \(E_1\) \(E_2\)
    \(A_2\) - \(A_1\) \(B_2\) \(B_1\) \(E_1\) \(E_2\)
    \(B_1\) - - \(A_1\) \(A_2\) \(E_2\) \(E_1\)
    \(B_2\) - - - \(A_1\) \(E_2\) \(E_1\)
    \(E_1\) - - - - \(A_1 + [A_2] + E_2\) \(B_1 + [B_2] + E_1\)
    \(E_2\) - - - - - \(A_1 + [A_2] + E_2\)
    Table 6: For \(C_{4v}\), \(D_4\), \(D_{2d}\), and \(D_{4h}\)
      \(A_1\) \(A_2\) \(B_1\) \(B_2\) \(E\)
    \(A_1\) \(A_1\) \(A_2\) \(B_1\) \(B_2\) \(E\)
    \(A_2\) - \(A_1\) \(B_2\) \(B_1\) \(E\)
    \(B_1\) - - \(A_1\) \(A_2\) \(E\)
    \(B_2\) - - - \(A_1\) \(E\)
    \(E\) - - - - \(A_1 + [A_2] + B_1 + B_2\)