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Chapter 9: Problem 3

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 \((a b | c d)\)

Short Answer

Expert verified
The integrals that are equivalent to \((a b | c d)\) are \((a d | c b)\), \((b a | d c)\), \((b a | c d)\) and \((a b | d c)\)

Step by step solution

01

Understanding Permutations in Two-electron Integrals

Given \((a b | c d) = (a d | c b)\). This equation tells that swapping the last two basis functions gives an equivalent integral. Thus, another integral that is equal to \((a b | c d)\) can be found by permuting the basis functions.
02

Identifying Equivalent Integrals

We know that \((a b | c d)=(a d | c b)\). Using this, we can interchange pairs of basis functions \((a, b)\) and \((c, d)\) respectively to find other equivalent integrals. Therefore, the additional equivalent integrals will be \((b a | d c)\), \((b a | c d)\) and \((a b | d c)\).
03

Listing all Possible Permutations

The entire list of equivalent integrals to \((a b | c d)\), garnered from the permutation symmetry of two-electron integrals, will therefore be: \((a b | c d) = (a d | c b) = (b a | d c) = (b a | c d) = (a b | d c)\).

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Most popular questions from this chapter

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 $$\left[\varphi_{4} \varphi_{b} | \varphi_{c} \varphi_{d}\right]=j_{0} \int \varphi_{a}^{*}(1) \varphi_{b}^{*}(2)\left(\frac{1}{r_{12}}\right) \varphi_{c}(1) \varphi_{d}(2) \mathrm{d} r_{1} \mathrm{d} r_{2}$$

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

(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?

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 \(S=0\) \(M_{s}=0\)

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