Basic Concepts

Abstract

In this chapter we examine some basic concepts of quantum chemistry to give a solid foundation for the other chapters. We do not pretend to review all the basics of quantum mechanics but rather focus on some specific topics that are central in the theoretical description of magnetic phenomena in molecules and extended systems. First, we will shortly review the Slater–Condon rules for the matrix elements between Slater determinants, then we will extensively discuss the generation of spin functions. Perturbation theory and effective Hamiltonians are fundamental tools for understanding and to capture the complex physics of open shell systems in simpler concepts. Therefore, the last three sections of this introductory chapter are dedicated to standard Rayleigh–Schrödinger perturbation theory, quasi-degenerate perturbation theory and the construction of effective Hamiltonians.

Problems

1.1

Ordering by spatial or spin part. In the notation of multideterminantal wave functions, one can either respect as much as possible the order of the spatial part in the different determinants, or strictly maintain the order of the spin part. Construct singlet and triplet functions for a two-electrons in two-orbitals case respecting (i) the order of the spatial part and (ii) the order of the spin part of the total wave function.

1.2

Coulomb, exchange or other. Classify the following two-electron integrals as Coulomb, exchange or other integral and assign a relative size (large, medium, or small to the integrals:

$$\begin{aligned} \text {(a) }&\langle {\phi _a(1)\phi _b(2)}|{\frac{1}{r_{12}}}|{\phi _a(1)\phi _b(2)}\rangle \quad&\text {(b) }&\int \frac{\phi _a(1)\phi _b(1)\phi _a(2)\phi _b(2)}{r_{12}}d\tau _1d\tau _2\nonumber \\ \text {(c) }&\langle {\phi _a(1)\overline{\phi }_c(2)}|{\frac{1-\hat{P}_{12}}{r_{12}}}|{\phi _c(1)\overline{\phi }_a(2)}\rangle \quad&\text {(d) }&\langle {\phi _b(1)\phi _c(2)}|{\frac{1}{r_{12}}}|{\phi _c(1)\phi _d(2)}\rangle \nonumber \\ \text {(e) }&\int \phi _a(1)\phi _c(2)\frac{1}{r_{12}}\phi _c(1)\phi _a(2)d\tau _1d\tau _2 \quad&\text {(f)}&\langle {\phi _b(1)\phi _d(2)}|{\frac{\hat{P}_{12}}{r_{12}}}|{\phi _d(1)\phi _b(2)}\rangle \nonumber \end{aligned}$$

$\phi _a$ and $\phi _b$ are centered on site A, $\phi _c$ and $\phi _d$ on site B.

1.3

Perturbation theory. The prototype particle in a box problem is perturbed by a finite potential $V_0$ of width $\gamma$ centered at $x=\frac{1}{2}L$. Calculate the first-order energy correction for the ground state, and the first and second excited states.

Reminder: $\psi ^{(0)}_n(x) = \sqrt{\frac{2}{L}}\sin \frac{n\pi x}{L}$ and $E^{(0)}_n = \frac{hn^2}{8mL^2}$, Assume that $\gamma$ is small enough to consider $\psi ^{(0)}$constant in the $\frac{1}{2}L-\frac{1}{2}\gamma \ldots \frac{1}{2}L+\frac{1}{2}\gamma$ interval.

1.4

Effective Hamiltonians: The following (hypothetical) model Hamiltonian is used to analyze a certain experimental observation

	$|{\varPhi _1}\rangle$	$|{\varPhi _2}\rangle$	$|{\varPhi _3}\rangle$
$\langle {\varPhi _1}|$	0
$\langle {\varPhi _2}|$	$\mu$	$\varDelta _1$
$\langle {\varPhi _3}|$	$\gamma$	$(\gamma -4\mu )/2$	$\varDelta _2$

To get insight in the parameters of the model Hamiltonian an ab initio calculation was performed giving the following multideterminantal wave functions $\varPsi _k$ and energies $E_k$.

	$\varPsi _1$	$\varPsi _2$	$\varPsi _3$	$\varPsi _4$	$\varPsi _5$
$\varPhi _1$	0.4804	0.8486	$-0.0381$	$-0.2147$	0.0387
$\varPhi _2$	0.3203	0.3990	$-0.1391$	$-0.7732$	0.3480
$\varPhi _3$	0.1601	0.0495	0.9468	0.0437	0.2714
$\varPhi _4$	0.8006	0.3397	$-0.1109$	0.4293	$-0.2167$
$\varPhi _5$	0.0000	0.0526	$-0.2656$	$0.4122$	0.8699
E	$-0.50$	$-0.38$	$-0.40$	$-0.36$	$-0.20$

• a. Determine the norm of the projections of $\varPsi _k$ on the model space.

• b. Select the three roots with the largest norm and orthogonalize the projections $\widetilde{\varPsi }_k$

• c. Construct the $3\times 3$ effective Hamiltonian and diagonalize the resulting matrix. Are the eigenvalues of $\hat{H}^{ eff }$ equal to the eigenvalues of $\varPsi _k$?

• d. Determine the value of the model parameters. Is the model Hamiltonian consistent with the ab initio result?