Character 1: Dimension: 2 Multiplicity: 2, i.e. the following SALCs are not necessarily eigen vectors of the Hamiltonian $H$ of the molecule but $H$ has at most two distinct eigen values on this space: \begin{eqnarray*} b_1&=&\tfrac1{\sqrt6}(s_1+s_2+s_3+s_4+s_5+s_6),\\ b_2&=&\tfrac1{\sqrt6}(-p_{1x}-p_{4x}) +\tfrac1{\sqrt{24}}(-p_{2x}+p_{3x}+p_{5x}-p_{6x}) +\tfrac1{\sqrt8}(-p_{2y}-p_{3y}+p_{5y}+p_{6y}) \end{eqnarray*} Character 2: Dimension: 0 Multiplicity: 0
Character 3: Dimension: 1 Multiplicity: 1, i.e. the following is an eigen vector for $H$: $$ b_3=\tfrac1{\sqrt6}(p_{1y}+p_{4y}) +\tfrac1{\sqrt{24}}(-p_{2y}-p_{3y}-p_{5y}-p_{6y}) +\tfrac1{\sqrt8}(p_{2x}-p_{3x}+p_{5x}-p_{6x}) $$ Character 4: Dimension: 0 Multiplicity: 0
Character 5: Dimension: 1 Multiplicity: 1, i.e. the following is an eigen vector for $H$ : $$ b_4=\tfrac1{\sqrt6}(p_{1y}-p_{4y}) +\tfrac1{\sqrt{24}}(p_{2y}-p_{3y}-p_{5y}+p_{6y}) +\tfrac1{\sqrt8}(-p_{2x}-p_{3x}+p_{5x}+p_{6x}) $$ Character 6: Dimension: 1 Multiplicity: 1, i.e. the following is an eigen vector for $H$: $$ b_5=\tfrac1{\sqrt6}(p_{1z}-p_{2z}+p_{3z}-p_{4z}+p_{5z}-p_{6z}) $$ Character 7: Dimension: 2 Multiplicity: 2, i.e. the following SALCs are not necessarily eigen vectors for $H$ and $H$ has at most two distinct eigen values on this space: \begin{eqnarray*} b_6&=&\tfrac1{\sqrt6}(-s_1+s_2-s_3+s_4-s_5+s_6),\\ b_7&=&\tfrac1{\sqrt6}(p_{1x}-p_{4x}) +\tfrac1{\sqrt{24}}(p_{2x}+p_{3x}+p_{4x}+p_{6x}) +\tfrac1{\sqrt8}(p_{2y}-p_{3y}+p_{5y}-p_{6y}) \end{eqnarray*} Character 8: Dimension: 1 Multiplicity: 1, i.e. the following is an eigen vector for $H$: $$ b_8=\tfrac1{\sqrt6}(p_{1z}+p_{2z}+p_{3z}+p_{4z}+p_{5z}+p_{6z}) $$ Character 9: Dimension: 2 Multiplicity: 1, i.e. the following are eigen vectors for $H$: \begin{eqnarray*} b_9&=&\tfrac12(p_{1z}+p_{3z}-p_{4z}+p_{6z}),\\ b_{10}&=&\tfrac1{\sqrt3}(-p_{2z}-p_{5z}) +\tfrac1{\sqrt{12}}(p_{1z}+p_{3z}-p_{4z}-p_{6z}) \end{eqnarray*} Character 10: Dimension: 6 Multiplicity: 3, i.e. the following SALCs are not necessarily eigen vectors for $H$ and $H$ has at most three distinct eigen values on this space: \begin{eqnarray*} b_{11}&=&\tfrac12(-s_1+s_3+s_4-s_6)\\ b_{12}&=&\tfrac12(-p_{1x}-p_{4x})+\tfrac14(-p_{3x}-p_{6x})+\tfrac{\sqrt3}4(p_{3y}+p_{6y})\\ b_{13}&=&\tfrac12(-p_{1y}-p_{4y})+\tfrac14(-p_{3y}-p_{6y})+\tfrac{\sqrt3}4(-p_{3x}-p_{6x})\\ b_{14}&=&\tfrac1{\sqrt{12}}(s_1+s_3-s_4-s_6)+\tfrac1{\sqrt3}(s_2-s_5)\\ b_{15}&=&\tfrac1{\sqrt{48}}(p_{1x}+p_{3x}+p_{4x}+p_{6x})+\tfrac14(-p_{1y}+p_{3y}-p_{4y}+p_{6y})+\tfrac1{\sqrt3}(p_{2x}+p_{5x})\\ b_{16}&=&\tfrac14(p_{1x}-p_{3x}+p_{4x}-p_{6x})+\tfrac1{\sqrt{48}}(p_{1y}+p_{3y}+p_{4y}+p_{6y})+\tfrac1{\sqrt3}(p_{2y}+p_{5y}) \end{eqnarray*} Character 11: Dimension: 2 Multiplicity: 1, i.e. the following are eigen vectors for $H$: \begin{eqnarray*} b_{17}&=&\tfrac12(p_{1z}-p_{3z}+p_{4z}-p_{6z}),\\ b_{18}&=&\tfrac1{\sqrt3}(p_{2z}+p_{5z}) +\tfrac1{\sqrt{12}}(-p_{1z}-p_{3z}-p_{4z}-p_{6z}) \end{eqnarray*} Character 12: Dimension: 6 Multiplicity: 3, i.e. the following SALCs are not necessarily eigen vectors for $H$ and $H$ has at most three distinct eigen values on this space: \begin{eqnarray*} b_{19}&=&\tfrac12(-s_1+s_3-s_4+s_6)\\ b_{20}&=&\tfrac12(-p_{1x}+p_{4x})+\tfrac14(-p_{3x}+p_{6x})+\tfrac{\sqrt3}4(p_{3y}-p_{6y})\\ b_{21}&=&\tfrac12(-p_{1y}+p_{4y})+\tfrac14(-p_{3y}+p_{6y})+\tfrac{\sqrt3}4(-p_{3x}+p_{6x})\\ b_{22}&=&\tfrac1{\sqrt{12}}(-s_1-s_3-s_4-s_6)+\tfrac1{\sqrt3}(s_2+s_5)\\ b_{23}&=&\tfrac1{\sqrt{48}}(-p_{1x}-p_{3x}+p_{4x}+p_{6x})+\tfrac14(p_{1y}-p_{3y}-p_{4y}+p_{6y})+\tfrac1{\sqrt3}(p_{2x}-p_{5x})\\ b_{24}&=&\tfrac14(-p_{1x}+p_{3x}+p_{4x}-p_{6x})+\tfrac1{\sqrt{48}}(-p_{1y}-p_{3y}+p_{4y}+p_{6y})+\tfrac1{\sqrt3}(p_{2y}-p_{5y}) \end{eqnarray*} Maximal number of distinct eigen values of $H$: 16. The eigen vectors: $b_3$ and $b_4$ are linear combinations of $p_x$- and $p_y$ orbitals and the six eigen vectors: $b_5,b_8,b_9,b_{10},b_{17},b_{18}$ are only linear combinations of the $p_z$-orbitals. The remaining 16 eigen vectors are linear combinations of $s$-, $p_x$ and $p_y$-orbitals.