Assignment 2

(Due 28/02/2019)

1.1 For the ground-state of H$_2^+$, the simple MO wavefunction is $\sigma_g1s(1) = \phi_{1sa}(1) + \phi_{1sb} (1)$, where $a$ and $b$ denote the two H atoms, and $\phi_{1sa} = \frac{e^{-r_a}}{\sqrt{\pi}}$ is the hydrogen AO. Using confocal elliptical coordinates (see Levine for definition), show that the overlap integral is given by $S_{ab} = e^{-R}(1+R+\frac{R^2}{3})$. Here, $R$ is the internuclear separation. The volume element in confocal elliptic coordinates is $dv = \frac{1}{8}R^3(\xi^2 -\eta^2)d\xi d\eta d\phi$. The following integral formula can be handy: $$ \int_t^{\infty}z^ne^{-az}dz = \frac{n!}{a^{n+1}}e^{-at} \bigg( 1+ at + \frac{a^2t^2}{2!} + … + \frac{a^nt^n}{n!} \bigg), n = 0,1,2,… > 0$$. (8 pts)

1.2 Consider a system (for example, a molecule) consisting of three electrons with spins $\alpha, \alpha, \beta$ occupying three different MOs $\psi_1, \psi_2$ and $\psi_3$ that are spatially different. a) List the possible SDs for this case. b) Check that these Slater determinants (SDs)are not suitable approximate wave functions of such a system [Hint: are they pure spin states?] c) Which wavefunctions would be suitable approximate wavefunctions? [Hint: try to diagonalize the $S^2$ matrix in this SD basis]. (10 pts)

1.3 For the lowest excited-state of H$_2$, which has the configuration ($\sigma_g1s$)$^1$ ($\sigma_u^*1s$)$^1$, i.e. one-electron in each of the two one-electron MO functions, comment on the nature of the dissociation limit. (4 pts)

1.4 For the simple MO and VB wavefunctions of H$_2$, i.e. obtained from using just the 1$s$ AOs of individual hydrogen atoms (separated by internuclear distance $R$), (i) derive the corresponding ground-state one-particle densities $\rho_{MO}(\mathbf x)$ and $\rho_{VB}(\mathbf x)$ for the singlet state. $\mathbf x = (\mathbf r, s)$ denotes the space-spin coordinates. (ii) Show that $\rho_{MO}(\mathbf x)$ is greater than $\rho_{VB}(\mathbf x)$ at the mid-point joining the two nuclei. (8 pts)