\qquad c_1 + \varepsilon c_2 + c_3 + c_8 = 0\\ 
(3) \qquad c_2 + \varepsilon c_3 + c_4 + c_9 = 0 \\ 
(4) \qquad c_3 + \varepsilon c_4 + c_5 + c_{10}= 0 \\
(5) \qquad c_4 + \varepsilon c_5 + c_6 + c_{11} = 0 \\ 
(6) \qquad c_1 + c_5 + \varepsilon c_6 + c_{12} = 0 \\ 
(7) \qquad c_1 + \varepsilon c_7 + c_8 + c_{12} = 0 \\ 
(8) \qquad c_2 + c_7 + \varepsilon c_8 + c_9 = 0 \\ 
(9) \qquad c_3 + c_8 + \varepsilon c_9 + c_{10} = 0 \\ 
(10) \qquad c_4 + c_9 + \varepsilon c_{10} + c_{11} = 0 \\ 
(11) \qquad c_5 + c_{10} + \varepsilon c_{11} + c_{12} = 0 \\ 
(12) \qquad c_6 + c_7 + c_{11}+\varepsilon c_{12} = 0 \\ 

\end{cases}
\label{eq:10}
$$

As we can see from Fig.\[Fig5\] our cluster is reflection-symmetric in the axes $y$ and $x$, due to the closing bonds 1-6 and 7-12. Hence, wave functions must be symmetric ($S_x, S_y$) or antisymmetric ($A_{x}A_y$) to reflection in the $x$ and y axis. All together there are four possible symmetries of wave functions: $S_{x}S_y, S_{x}A_y, A_{x}S_y$ and $A_{x}A_y$.

First 