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How and why does the sign of the electronic wavefunction changes when it is taken around a contour?

For example, suppose the initial wavefunction is $f(s;S_0)$ at nuclear configuration $S_0$ and now we follow some closed curve/loop and come again back at $S_0$) So we should expect that our wavefunction is still $f(s;S_0)$ but if the loop contains a point (say, $P$) where the potential energy surfaces become degenerate then the sign of the electronic wavefuntion changes and we get $-f(s;S_0)$. Here the point $P$ corresponds to the nuclear configuration at which the two or more potential energy surfaces intersect, which means that the two or more electronic wavefunctions are giving the same energy values at that nuclear configuration.

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