Chapter 9

Consider the two-clectron integrals over the basis functions defined in eqn 9.17 . If the basis functions are taken to be real, a number of the integrals are equivalent; for example, \((a b | c d)=(a d | c b)\). Find the other integrals that are equal to \((a b | c d)\)

In a Hartree-Fock calculation on atomic hydrogen using four primitive s-type Gaussian functions (S. Huzinaga, J. Chem. Phys., \(1293,42(1965)\) ), optimized results were obtained with a linear combination of Gaussians with coefficients \(c_{j}\) and exponents \(\alpha\) of 0.50907,0.123317 0.47449,\(0.453757 ; 0.13424,2.01330 ;\) and 0.01906 \(13.3615 .\) Describe how these primitives would be utilized in a \((4 s) /[2 s]\) contraction scheme.

A single Slater determinant is not necessarily an cigenfunction of the total clectron spin operator. Therefore, even within the Hartree-Fock approximation, for the wavefunction \(\Phi_{0}\) to be an eigenfunction of \(S^{2},\) it might have to be expressed as a linear combination of Slater determinants. The linear combination is referred to as a spin-adapted configuration. As a simple example, consider a two-electron system with four possible Slater determinants: $$\begin{array}{l} \Phi_{1}=\left(\frac{1}{2}\right)^{1 / 2} \operatorname{det}\left|\psi_{1}\left(r_{1}\right) \alpha(1) \psi_{2}\left(r_{2}\right) \alpha(2)\right| \\ \Phi_{2}=\left(\frac{1}{2}\right)^{1 / 2} \operatorname{det}\left|\psi_{1}\left(r_{1}\right) \alpha(1) \psi_{2}\left(r_{2}\right) \beta(2)\right| \\ \Phi_{3}=\left(\frac{1}{2}\right)^{1 / 2} \operatorname{det}\left|\psi_{1}\left(r_{1}\right) \beta(1) \psi_{2}\left(r_{2}\right) \alpha(2)\right| \\ \Phi_{4}=\left(\frac{1}{2}\right)^{1 / 2} \operatorname{det}\left|\psi_{1}\left(r_{1}\right) \beta(1) \psi_{2}\left(r_{2}\right) \beta(2)\right| \end{array}$$ (a) Show that the Slater determinants \(\Phi_{1}\) and \(\phi_{4}\) are themselves cigcnfunctions of \(S^{2}\) with cigenvaluc \(2 \hbar^{2}\) (corresponding to \(S=1\) ). (b) From \(\Phi_{2}\) and \(\Phi_{3}\), determine two linear combinations, one of which corresponds to \(S=1, M_{s}=0\) and the other of which corresponds to \(S=0\) \(M_{s}=0\)

Show that the product of an s-type Gaussian centred at \(R_{\text {A with exponent } \alpha_{\text {A and an s-type Gaussian centred at }} R_{\text {nd }}}\) with cxponcnt \(\alpha_{\mathrm{B}}\) can be writtcn in terms of a single s-typc Gaussian centred between \(R_{\lambda}\) and \(R_{B}\)

Prove Brillouin's theorem; that is, show that hamiltonian matrix elements between the HF wavefunction \(\Phi_{0}\) and singly excited determinants are identically zero. Hint. Use the Condon-Slater rules.

Consider two Slater determinants \(\Phi_{1}\) and \(\Phi_{2}\) that differ by only one spinorbital; that is, $$\begin{array}{l} \Phi_{1}=\left\|\cdots \varphi_{m} \varphi_{i} \cdots\right\| \\ \Phi_{2}=\left\|\cdots \varphi_{p} \varphi_{i} \cdots\right\| \end{array}$$ Derive the following Condon-Slater rule: $$\begin{aligned} \left\langle\Phi_{1}|H| \Phi_{2}\right\rangle=&\left\langle\varphi_{m}(1)\left|h_{1}\right| \varphi_{p}(1)\right\rangle+\sum_{i}\left\\{\left[\varphi_{m} \varphi_{i} | \varphi_{p} \varphi_{i}\right]\right.\\\ &\left.-\left[\varphi_{m} \varphi_{i} | \varphi, \varphi_{p}\right]\right\\} \end{aligned}$$ where we have used the notation $$\left[\varphi_{4} \varphi_{b} | \varphi_{c} \varphi_{d}\right]=j_{0} \int \varphi_{a}^{*}(1) \varphi_{b}^{*}(2)\left(\frac{1}{r_{12}}\right) \varphi_{c}(1) \varphi_{d}(2) \mathrm{d} r_{1} \mathrm{d} r_{2}$$

(a) For a CASSCF calculation of the ground-state wavefunction of diatomic \(\mathrm{C}_{2},\) describe a reasonable choice for the distribution of \(\sigma\) and \(\pi\) molecular orbitals into active, inactive and virtual orbitals. (b) How many inactive and active clectrons arc there in the calculation? (c) In an RASSCF calculation, how might the set of active orbitals be further divided?

Use Moller-Plesset perturbation theory to obtain an expression for the ground- state wavefunction corrected to first order in the perturbation.

Use the AM1 and PM3 semiempirical methods to compute the equilibrium bond lengths and enthalpies of formation of (a) ethanol, (b) 1,4 -dichlorobenzene.