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7.5: Electronic States of the H₂ Molecule

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    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.

    This page titled 7.5: Electronic States of the H₂ Molecule is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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