Assignment 3

3.1 Show that the Hartree products $$ \Psi_a(\mathbf r_1, \mathbf r_2) = \psi_1(\mathbf r_1)\alpha(s_1) \psi_2(\mathbf r_2)\beta(s_2) $$ $$ \Psi_b(\mathbf r_1, \mathbf r_2) = \psi_1(\mathbf r_1)\alpha(s_1) \psi_2(\mathbf r_2)\alpha(s_2) $$ a) are uncorrelated and b) have the same energy as that of the antisymmetrized form of $\Psi_a$, i.e $$ \psi_1(\mathbf r_1)\alpha(s_1) \psi_2(\mathbf r_2)\beta(s_2) - \psi_2(\mathbf r_1)\beta(s_1) \psi_1(\mathbf r_2)\alpha(s_2). \text{(5pts)}$$

3.2 Show that the Fock operator $$ \hat F = \hat h + \hat J + \hat K $$ is invariant to unitary transformation of occupied orbitals. The Coulomb and Exchange operators depend on “$N$” occupied orbitals $$ \hat J = \sum_j^{N} \int dx_2 \chi_j (x_2)\frac{1}{r_{12}}\chi_j(x_2) $$ $$ \hat K = \sum_j^{N} \int dx_2 \chi_j (x_2)\frac{\hat P_{12}}{r_{12}}\chi_j(x_2)\text{. (10 pts)} $$

3.3 For the ground-state of $H_2^-$, provide Slater determinant based on bonding and antibonding spatial MOs denoted by $\psi_g \;\text{and}\; \psi_u$. What is the Hartree-Fock energy in terms of $h,K \;\text{and} \; J$ integrals. In this minimal basis-set, what are the possible excited states and their Hartree-Fock energies? (5 pts).

3.4 For $H_2$ in minimal basis-set, calculate the following (i)$\langle\Psi_0|H|\Psi_0\rangle$ (ii)$\langle\Psi_0|H|\Psi_{1\bar{1}}^{2\bar{2}}\rangle$ (iii)$\langle\Psi_{1\bar{1}}^{2\bar{2}}|H|\Psi_0\rangle$ (iv)$\langle\Psi_{1\bar{1}}^{2\bar{2}}|H|\Psi_{1\bar{1}}^{2\bar{2}}\rangle$ where $|\Psi_0\rangle$ is Hartree-Fock ground state determinant and $\Psi_{1\bar{1}}^{2\bar{2}}$ is doubly excited determinant. (Note: You cannot use the energy expression to directly evaluate all terms, since the “bra” and “ket” states are different. Expand the slater determinants and then evaluate for each case. (10 pts)

3.5 Using HMO theory set up and solve the Huckel secular determinant equation and calculate the $\pi$ electron energy for (i) allyl and (ii) cyclopropenyl molecules. For each case, evaluate HMO energy for the radical, cationic and the anionic states. (5 pts)

3.6 The $U(R)$ curve for a repulsive electronic-state of a diatomic molecule can be approximated by the function $ae^{-bR}-c$, where a,b, and c are positive constants with $a > c$. (Note:This function omits the Van der Waals minimum and fails to go to infinity at R = 0.) Sketch $U$, $\langle T_{el} \rangle$, and $\langle V \rangle$ as functions of R for this function.($V = V_{el} + V_{nn}$). (5 pts)