Basics of Electronic Structure

We start from the full molecular Hamiltonian, apply the Born–Oppenheimer approximation, discuss antisymmetry and spin, build Hartree products and Slater determinants, and then derive the ideas behind Hartree–Fock and Configuration Interaction (CI). Finally we write the operator algebra and Coulomb/exchange integrals which show how HF works internally.

1. Introduction

We need to solve the time–independent Schrödinger equation

\[ \hat{H}\Psi = E\Psi . \]

with the full molecular Hamiltonian

\[ \hat{H} = -\sum_{i}^{N}\frac{1}{2}\nabla_i^2 -\sum_{A}^{M}\frac{1}{2M_A}\nabla_A^2 -\sum_{i,A}\frac{Z_A}{|\mathbf{r}_i-\mathbf{R}_A|} +\sum_{i<j}\frac{1}{|\mathbf{r}_i-\mathbf{r}_j|} +\sum_{A<B}\frac{Z_A Z_B}{|\mathbf{R}_A-\mathbf{R}_B|}. \]

Here, \(i,j\) label electrons and \(A,B\) label nuclei. The terms represent electronic kinetic energy, nuclear kinetic energy, electron–nuclear attraction, electron–electron repulsion, and nuclear–nuclear repulsion respectively.

1.1 Born–Oppenheimer approximation

We reduce the complexity of the full Hamiltonian with BO approximation, which is basically ignoring the nuclear kinetic energy due to very high \(M_A/m_e\) ratio. So the electrons are now moving under a fixed nuclear geometry and because of this the nuclear repulsion is just a constant with parametric dependence on \(\{\mathbf{R}_A\}\).

Thus our reduced electronic Hamiltonian looks like

\[ \hat{H}_e = -\sum_{i}^{N}\frac{1}{2}\nabla_i^2 -\sum_{i,A}\frac{Z_A}{|\mathbf{r}_i-\mathbf{R}_A|} +\sum_{i<j}\frac{1}{|\mathbf{r}_i-\mathbf{r}_j|}. \]

Thus the entity we now need to find is

\[ \hat{H}_e \Phi_e = E_e \Phi_e , \]

where \(E_e = E_e(\{\mathbf{R}_A\})\) and \(\Phi_e = \Phi_e(\{\mathbf{r}_i\};\{\mathbf{R}_A\})\) now have parametric dependence on \(\{\mathbf{R}_A\}\).

The total energy for fixed nuclei is

\[ E_{\text{tot}} = E_e + E_{nn}, \qquad E_{nn} = \sum_{A<B}\frac{Z_A Z_B}{|\mathbf{R}_A-\mathbf{R}_B|}. \]

2. Antisymmetry principle

The electron needs both spatial and spin description. So we can introduce two spin functions \(\alpha(\omega)\) and \(\beta(\omega)\) such that

\[ \langle \alpha | \beta \rangle = 0. \]

The electron is thus described by two types of coordinates collectively,

\[ \mathbf{x} = (\mathbf{r},\omega). \]

With spin as a fundamental property determining electron statistics, we must incorporate the spin-statistics of electrons in the electronic wavefunction through the antisymmetry principle,

\[ \Psi(\mathbf{x}_1,\ldots,\mathbf{x}_i,\ldots,\mathbf{x}_j,\ldots,\mathbf{x}_N) = - \Psi(\mathbf{x}_1,\ldots,\mathbf{x}_j,\ldots,\mathbf{x}_i,\ldots,\mathbf{x}_N). \]

3. Orbitals, Hartree products, and Slater determinants

3.1 Orbitals and spin orbitals

Orbital → wavefunction for a single particle. A spatial orbital is a function of position vector of a particle, but an electron also needs a spin description and so an orbital wavefunction of a single electron must contain this. We refer to such an orbital as spin orbital. From each spatial orbital \(\phi_i(\mathbf{r})\) we can form two spin orbitals

\[ \chi_{i\alpha}(\mathbf{x}) = \phi_i(\mathbf{r})\alpha(\omega), \qquad \chi_{i\beta}(\mathbf{x}) = \phi_i(\mathbf{r})\beta(\omega). \]

3.2 Hartree products

Let us consider a simple system of non-interacting electrons.

\[ \hat{H} = \sum_{i=1}^{N}\hat{h}(i), \qquad \hat{h}(i)\chi_j(\mathbf{x}_i) = \varepsilon_j \chi_j(\mathbf{x}_i). \]

Now since \(\hat{H}\) is independent one-electron Hamiltonians, we can use separation of variables to suggest the eigenfunctions of \(\hat{H}\) as

\[ \Psi_H(\mathbf{x}_1,\ldots,\mathbf{x}_N) = \chi_i(\mathbf{x}_1)\chi_j(\mathbf{x}_2)\cdots\chi_n(\mathbf{x}_N). \]

This is known as a Hartree product and is an uncorrelated (independent electron) wavefunction since \(|\Psi_H|^2\) factorizes.

In reality, even if we ignore any Coulomb repulsion between electrons, there is an inherent effective repulsion between electrons due to their spins which gives rise to antisymmetry in their wavefunction. This is applied through the constraint of antisymmetry and is the so-called Pauli repulsion / Pauli exclusion principle.

3.3 Slater determinants

To apply antisymmetry, let us look at two 2-electron Hartree products and take their antisymmetric combination:

\[ \Psi(\mathbf{x}_1,\mathbf{x}_2) = \frac{1}{\sqrt{2}} \left[ \chi_i(\mathbf{x}_1)\chi_j(\mathbf{x}_2) - \chi_i(\mathbf{x}_2)\chi_j(\mathbf{x}_1) \right]. \]

This has the form of a determinant. The generalization to \(N\) electrons is the Slater determinant:

\[ \Psi(\mathbf{x}_1,\ldots,\mathbf{x}_N) = \frac{1}{\sqrt{N!}} \begin{vmatrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2) \\ \vdots & \vdots & \ddots & \vdots \\ \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N) \end{vmatrix}. \]

Slater determinants automatically follow the antisymmetry requirement by construction. In shorthand notation, \(\Psi = |\chi_1\chi_2\cdots\chi_N|\), and antisymmetry looks like

\[ |\cdots \chi_m \cdots \chi_n \cdots| = - |\cdots \chi_n \cdots \chi_m \cdots|. \]

This antisymmetry due to exchange between electron coordinates is called exchange-correlation and arises due to Pauli repulsion between two electrons with same spin. Thus correlation between opposite spins is still uncorrelated at this level.

4. Exchange effect and Hartree–Fock approximation

4.1 Exchange effect due to Slater determinants

Consider a 2-electron Slater determinant in which \(\chi_1\) and \(\chi_2\) are occupied. If the two electrons have opposite spin, e.g.

\[ \chi_1(\mathbf{x})=\phi_1(\mathbf{r})\alpha(\omega), \qquad \chi_2(\mathbf{x})=\phi_2(\mathbf{r})\beta(\omega). \]

Then after integrating out spin degrees of freedom, the spatial simultaneous probability becomes just a product of independent probabilities. For same spin, e.g.

\[ \chi_1(\mathbf{x})=\phi_1(\mathbf{r})\alpha(\omega), \qquad \chi_2(\mathbf{x})=\phi_2(\mathbf{r})\alpha(\omega), \]

the probability has the exchange term and if we set \(\mathbf{r}_1=\mathbf{r}_2\), then \(P(\mathbf{r},\mathbf{r})=0\). So two non-interacting electrons with same spin can never be found at the same position. A Fermi hole exists around an electron.

4.2 Hartree–Fock approximation

Our goal is to solve \(\hat{H}_e\Phi_e=E_e\Phi_e\). The simplest guess is to use a single Slater determinant of \(N\) noninteracting electrons:

\[ \Phi_{\mathrm{HF}} = |\chi_1\chi_2\cdots\chi_N|. \]

With variational principle,

\[ E_{\mathrm{HF}} = \frac{\langle \Phi_{\mathrm{HF}}|\hat{H}_e|\Phi_{\mathrm{HF}}\rangle} {\langle \Phi_{\mathrm{HF}}|\Phi_{\mathrm{HF}}\rangle} \ge E_{\mathrm{ground}}. \]

By minimizing \(E_{\mathrm{HF}}\) with respect to the choice of spin orbitals, we obtain the Hartree–Fock equation,

\[ \hat{f}(i)\chi_i(\mathbf{x})=\varepsilon_i\chi_i(\mathbf{x}). \]

The Fock operator is like an effective one-electron operator:

\[ \hat{f}(i)= -\frac{1}{2}\nabla_i^2 -\sum_A \frac{Z_A}{|\mathbf{r}_i-\mathbf{R}_A|} +\hat{V}_{\mathrm{HF}}(i). \]

4.3 SCF procedure

Choose a nuclear geometry \(\{\mathbf{R}_A\}\).
Guess an initial set of \(N\) spin orbitals \(\{\chi_i\}\).
Construct the Fock operator using the current orbitals.
Solve the HF equations to get new orbitals.
Repeat until self-consistency is achieved.

5. Excitations and exact theory (CI)

With the virtual orbitals we can form combinations of singly, doubly, ... excited determinants. This way we can construct all possible determinants which in principle can be used to get the exact ground-state wavefunction to arbitrary accuracy (complete basis of determinants).

\[ |\Psi\rangle = c_0|\Phi_0\rangle + \sum_{ai}c_i^a|\Phi_i^a\rangle + \sum_{abij}c_{ij}^{ab}|\Phi_{ij}^{ab}\rangle +\cdots . \]

This is configuration interaction (CI). The only problem is that it is extremely expensive computationally. The energy difference between exact and Hartree–Fock is known as correlation energy:

\[ E_{\mathrm{corr}} = E_{\mathrm{exact}} - E_{\mathrm{HF}}. \]

6. Operator matrix elements between Slater determinants

The electronic Hamiltonian looks like

\[ \hat{H}_e = \sum_{i=1}^{N}\hat{h}(i) + \sum_{i<j}^{N}\hat{g}(i,j), \qquad \hat{h}(i)= -\frac{1}{2}\nabla_i^2-\sum_A\frac{Z_A}{|\mathbf{r}_i-\mathbf{R}_A|}, \qquad \hat{g}(i,j)=\frac{1}{r_{ij}}. \]

Define one-electron integrals

\[ h_{ij}=\langle \chi_i|\hat{h}|\chi_j\rangle = \int d\mathbf{x}\,\chi_i^*(\mathbf{x})\hat{h}\chi_j(\mathbf{x}), \]

and two-electron integrals

\[ (ij|kl)= \iint d\mathbf{x}_1 d\mathbf{x}_2\, \chi_i^*(1)\chi_j^*(2)\frac{1}{r_{12}}\chi_k(1)\chi_l(2). \]

Coulomb integral comes from \((ij|ij)\) and exchange integral comes from \((ij|ji)\). Using this, for a single determinant the Hartree–Fock energy becomes

\[ E_{\mathrm{HF}} = \sum_m h_{mm} +\frac{1}{2}\sum_{m,n}\left[(mn|mn)-(mn|nm)\right]. \]

We can define Coulomb and exchange operators as

\[ \hat{J}_n \chi_m(\mathbf{x}) = \left(\int d\mathbf{x}'\frac{|\chi_n(\mathbf{x}')|^2}{r_{12}}\right)\chi_m(\mathbf{x}), \qquad \hat{K}_n \chi_m(\mathbf{x}) = \int d\mathbf{x}'\frac{\chi_n^*(\mathbf{x}')\chi_m(\mathbf{x}')}{r_{12}}\chi_n(\mathbf{x}). \]

Then the Fock operator is

\[ \hat{f}=\hat{h}+\sum_n(\hat{J}_n-\hat{K}_n), \qquad \hat{f}\chi_m=\varepsilon_m\chi_m. \]

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