513.001 Molecular and Solid State Physics

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Physical
Constants

Periodic System
of Elements

      

Molecular hydrogen

The molecular orbitals of H2 can be calculated using a linear combination of atomic orbitals. The two molecular orbitals with the lowest energy are:

\[ \begin{equation} \large \psi_+(\vec{r})=\frac{1}{\sqrt{2}}\left(\phi_{1s}(\vec{r}-\vec{r}_A)+\phi_{1s}(\vec{r}-\vec{r}_B)\right), \end{equation} \] \[ \begin{equation} \large \psi_-(\vec{r})=\frac{1}{\sqrt{2}}\left(\phi_{1s}(\vec{r}-\vec{r}_A)-\phi_{1s}(\vec{r}-\vec{r}_B)\right). \end{equation} \] Here $\vec{r}_A$ is the position of one of the protons and $\vec{r}_B$ is the position of the other proton. The function $\phi_{1s}(\vec{r})$ is the 1s atomic orbital of hydrogen.

(a) What is the two-electron wavefunction for the ground state of H2?

(b) What is the two-electron wavefunction for the first excited state of H2?

(c) How could you calculate the energies of the ground state and the first excited state?