Question

2. LCAO for the Hydrogen Anion - Atkins 8.2 In class we derived the energy Eigenstates for the \( H_{2}^{+} \)molecules based on a trial wavefunction using the 1s orbitals around the two protons. We termed these orbitals \( \sigma_{g} \) for the ground state and \( \sigma_{u} \) for the next higher energy state. By solving the secular determinant, we found new energy Eigenstates of the form: \[ E_{ \pm}=\frac{\alpha \pm \beta}{1 \pm S_{A B}} \] Show that \( \left.\frac{1}{2}\left(E_{-}+E_{+}\right)-E_{1 s}\right) \) is a positive quantity.To zeroth-order i.e., assuming no electron-electron repulsion, what does the positivity of the quantity above tell you about the stability of the \( H_{2}^{2-} \) anion?

          2. LCAO for the Hydrogen Anion - Atkins 8.2 In class we derived the energy Eigenstates for the \( H_{2}^{+} \)molecules based on a trial wavefunction using the 1s orbitals around the two protons. We termed these orbitals \( \sigma_{g} \) for the ground state and \( \sigma_{u} \) for the next higher energy state. By solving the secular determinant, we found new energy Eigenstates of the form:
\[
E_{ \pm}=\frac{\alpha \pm \beta}{1 \pm S_{A B}}
\]

Show that \( \left.\frac{1}{2}\left(E_{-}+E_{+}\right)-E_{1 s}\right) \) is a positive quantity.To zeroth-order i.e., assuming no electron-electron repulsion, what does the positivity of the quantity above tell you about the stability of the \( H_{2}^{2-} \) anion?

2. LCAO for the Hydrogen Anion - Atkins 8.2 In class we derived the energy Eigenstates for the H2^+molecules based on a trial wavefunction using the 1s orbitals around the two protons. We termed these orbitals σg for the ground state and σu for the next higher energy state. By solving the secular determinant, we found new energy Eigenstates of the form:

E±=(α±β)/(1 ± SA B)

Show that .(1)/(2)(E-+E+)-E1 s) is a positive quantity.To zeroth-order i.e., assuming no electron-electron repulsion, what does the positivity of the quantity above tell you about the stability of the H2^2- anion?

Added by Jennis K.

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