Assignment 4

3.1 Starting from the Hartree-Fock-Roothan equations, what are the approximations necessary to reduce to the CNDO, INDO and MNDO methods. Arrange the methods in increasing order of computational effort. (7 pts)

3.2 For the Fe(CO)$_5$ molecule, enumerate the total number of CGTOs that constitute the 6-31G and STO-3G basis sets. Try to answer this without and with the help of the basis-set website (https://www.basissetexchange.org/). How many spatial MOs will be occupied and how many will be empty? (4 pts)

3.3 For the hydrogen atom, plot the radial wavefunction for (i) the exact 1$s$ AO, and (ii) approximate AOs using CGTOs (a)STO-3G and (b) STO-6G. Check if these AOs are normalized. (4 pts)

3.4 In the HF-SCF procedure, at the stationary point, $\chi_i $ and $\varepsilon_i$ and eigenfunctions and eigenvalues of the the one-electron Fock operator, $$ \hat F (r_1) \chi_i (r_1) = \varepsilon_i \chi_i (r_1) \text{.}$$ Show that the Slater Determinant, $| \Phi_0 \rangle = | \chi_1 \chi_2 … \chi_N| $ is an eigenfunction of the following many-electron operator: $$ \hat H_0 = \sum_{i=1}^{N} \hat F (r_i) \text{.}$$ Determine the corresponding eigenvalue for $\hat H_0$ acting on $| \Phi_0 \rangle $. (5 pts)