Assignment 5

5.1 (Taken from Levine) For H$_2$O molecule, the five lowest-energy spatial MOs in terms of AOs are $$ \psi_1 = 1.0 \phi_{O1s} + 0.015\phi_{O2s} + 0.003\phi_{O2p_z} - 0.004(\phi_{H1_{1s}} + \phi_{H2_{1s}}) \\ \psi_2 = -0.027 \phi_{O1s} + 0.820\phi_{O2s} + 0.132\phi_{O2p_z} + 0.152(\phi_{H1_{1s}} + \phi_{H2_{1s}}) \\ \psi_3 = 0.624\phi_{O2p_y} + 0.424(\phi_{H1_{1s}} - \phi_{H2_{1s}}) \\ \psi_4 = -0.026\phi_{O1s} -0.502\phi_{O2s} + 0.787\phi_{O2p_z} + 0.264 (\phi_{H1_{1s}} + \phi_{H2_{1s}}) \\ \psi_5 = \phi_{O2p_x}. $$ For this configuration, calculate the individual AO electronic population, Gross electronic population on each atom, and the net charges. The AOs on O atom are mutually orthogonal, while the other relevant overlap integrals are $$ \langle \phi_{H1_{1s}}| \phi_{O_{1s}}\rangle = \langle \phi_{H2_{1s}} | \phi_{O_{1s}} \rangle = 0.054 \text{,} \; \langle \phi_{H1_{1s}}| \phi_{O_{2s}}\rangle = \langle \phi_{H2_{1s}} | \phi_{O_{2s}} \rangle = 0.471 \\ \langle \phi_{H1_{1s}}| \phi_{O_{2p_y}}\rangle = -\langle \phi_{H2_{1s}} | \phi_{O_{2p_y}} \rangle = 0.319 \text{,} \; \langle \phi_{H1_{1s}}| \phi_{O_{2p_z}}\rangle = \langle \phi_{H2_{1s}} | \phi_{O_{2p_z}} \rangle = 0.247 \\ \langle \phi_{H1_{1s}}| \phi_{H2_{1s}}\rangle=0.238. \text{(5 pts)} $$

5.2 For He in 6-31G basis-set, assume that the two s-type functions are orthogonal to each other. The AOs hence are also like the MOs, i.e. $\psi_1 = \phi_{1s}$ and $\psi_2 = \phi_{2s}$ . For these two MOs, use Slater-Condon rules and the $\Phi_0$ and $\Phi_{1\bar1}^{2\bar2}$ configurations to verify that the CI matrix is $$ \begin{pmatrix} 2h_{11} + J_{11} & K_{12} \\ K_{12} & 2h_{22} + J_{22} \\ \end{pmatrix}.\text{(5 pts)} $$

5.3 For H$_2$ in a minimal basis-set, but with infinite nuclear separation and using only $\Phi_0$ and $\Phi_{1\bar1}^{2\bar2}$ configurations, derive the CI wavefunctions corresponding to the ground and excited-states, i.e. solve for $c_0$ and $c_{1\bar1}^{2\bar2}$ for $\Psi = c_0\Phi_0 + \Phi_{1\bar1}^{2\bar2}$. Comment on the ionic/covalent nature of the two CI wavefunction by expanding the MOs of the two Slater determinants in AOs. (5pts)