# 7.5: Electronic States of the H₂ Molecule


Electrons are fermions with spin $$\frac{1}{2}$$, so the gerade state can be double occupied, as can the ungerade state (four states in all, same as two $$1s$$ orbitals for each ion). The second electron changes the structure of the wavefunction. Staying within LCAO, and ignoring spin, we can label basis states as, e.g. $$u^1_{1s} ({\bf r_2})$$ indicating the first electron on the second atom. The electrons are indistinguishable, so the total wavefunctions (spin times spatial) must be eigenstates of parity and the exchange operator $$\hat{P}_{12}$$ which switches the electron labels, e.g. $$\hat{P}_{12}u^1_{1s} ({\bf r_2}) = u^2_{1s} ({\bf r_2})$$. They are fermions, hence antisymmetric: $$P = −1$$.

Assuming both electrons are $$1s$$ and in the bonding $$g$$ state, and ignoring their interaction, the LCAO 1s$$^2$$ spatial wavefunction is

$\psi ({\bf r_1},{\bf r_2}) = [u^1_{100} ({\bf r_1}) + u^1_{100}({\bf r_2})][u^2_{100}({\bf r_1}) + u^2_{100}({\bf r_2})] \nonumber$

This must be combined with a spin eigenfunction $$\uparrow \uparrow$$, $$\downarrow \downarrow$$, $$(\uparrow \downarrow + \downarrow \uparrow )$$, or $$(\uparrow \downarrow − \downarrow \uparrow )$$, where the first arrow represents the spin state $$(m_s = \pm 1)$$ of the first electron. Since the spatial wavefunction is symmetric under label exchange, in fact it must be combined with the antisymmetric spin wavefunction $$\uparrow \downarrow − \downarrow \uparrow$$ to give the overall wavefunction in spin and space.

$\psi ({\bf r_1},{\bf r_2}, s_1, s_2) = [u^1_{100} ({\bf r_1}) + u^1_{100}({\bf r_2})][u^2_{100}({\bf r_1}) + u^2_{100}({\bf r_2})] [\uparrow \downarrow − \downarrow \uparrow] \nonumber$

This wavefunction describes two electrons, and is non-degenerate.

The second electron also adds an electron-electron repulsion to the Hamiltonian, which can be treated by perturbation theory.

$\Delta E = \langle \psi ({\bf r_1}, {\bf r_2})| e^2 /4\pi \epsilon_0 |{\bf r_1} − {\bf r_2}| |\psi ({\bf r_1}, {\bf r_2}) \rangle \nonumber$

There is a lot of subtlety here, since the electrons don’t interact with themselves, only with each other, and we must avoid double-counting the interaction of 1-2 and 2-1. We’ll return to this in more detail later in the context of Helium.

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