Multi-Electron Atoms & Chemical Bonding in Solids

1. Indistinguishability & The Permutation Operator

In quantum mechanics, identical particles (such as electrons) are fundamentally indistinguishable. Consider a system of two identical particles described by coordinate vectors $\xi_1 = (\vec{r}_1, \sigma_1)$ and $\xi_2 = (\vec{r}_2, \sigma_2)$ combining spatial and spin coordinates. The permutation operator $\hat{P}_{12}$ exchanges the particles:

Because $\hat{P}_{12}^2 = \hat{I}$, its eigenvalues are $\lambda = \pm 1$. The Symmetrization Postulate partitions all physical particles in nature into two disjoint classes:

• Bosons (Integer Spin $S = 0, 1, 2, \dots$): Symmetric wavefunctions under exchange:
  $$\Psi(\xi_2, \xi_1) = + \Psi(\xi_1, \xi_2)$$
• Fermions (Half-Integer Spin $S = 1/2, 3/2, \dots$): Antisymmetric wavefunctions under exchange:
  $$\Psi(\xi_2, \xi_1) = - \Psi(\xi_1, \xi_2)$$

2. The Pauli Exclusion Principle & Slater Determinants

Electrons are spin-$1/2$ fermions, so the total wavefunction of an $N$-electron system must be completely antisymmetric under the exchange of any pair of electrons $i$ and $j$. If the electrons occupy single-particle spin-orbitals $\chi_{\alpha}(\xi) = \phi_a(\vec{r}) \chi_s(\sigma)$, the total antisymmetric wavefunction is represented by a Slater Determinant:

If two electrons occupy the identical quantum state ($\alpha_1 = \alpha_2$), two columns of the determinant are identical, causing $\Psi \equiv 0$. This establishes the Pauli Exclusion Principle: no two electrons can occupy the same quantum state simultaneously.

1. Helium Hamiltonian & Electron-Electron Repulsion

The non-relativistic Hamiltonian of the neutral Helium atom ($Z = 2$) with nucleus at the origin is:

The total two-electron wavefunction factors into spatial and spin parts: $\Psi(\xi_1, \xi_2) = \psi(\vec{r}_1, \vec{r}_2) \chi_{\text{spin}}(1, 2)$. Total antisymmetry requires:

• Singlet State ($S = 0$, Para-Helium): Antisymmetric spin $\chi_{0,0} = \frac{1}{\sqrt{2}}(\alpha_1 \beta_2 - \beta_1 \alpha_2)$ coupled to a symmetric spatial wavefunction:
  $$\psi_+(\vec{r}_1, \vec{r}_2) = \frac{1}{\sqrt{2}} [\phi_a(\vec{r}_1)\phi_b(\vec{r}_2) + \phi_b(\vec{r}_1)\phi_a(\vec{r}_2)]$$
• Triplet State ($S = 1$, Ortho-Helium): Symmetric spin ($M_S = +1, 0, -1$) coupled to an antisymmetric spatial wavefunction:
  $$\psi_-(\vec{r}_1, \vec{r}_2) = \frac{1}{\sqrt{2}} [\phi_a(\vec{r}_1)\phi_b(\vec{r}_2) - \phi_b(\vec{r}_1)\phi_a(\vec{r}_2)]$$

2. Direct and Exchange Integrals

Evaluating the expectation value of the electron-electron Coulomb repulsion $\hat{V}_{ee}$ in first-order perturbation theory:

where:

• Direct Coulomb Integral $J$: Classical electrostatic repulsion between the two electron charge clouds:
  $$J = \iint |\phi_a(\vec{r}_1)|^2 \frac{e^2}{4\pi\varepsilon_0 r_{12}} |\phi_b(\vec{r}_2)|^2 d^3r_1 d^3r_2 > 0$$
• Exchange Integral $K$: Purely quantum mechanical energy shift arising from the overlap of the two single-particle orbitals:
  $$K = \iint \phi_a^*(\vec{r}_1) \phi_b^*(\vec{r}_2) \frac{e^2}{4\pi\varepsilon_0 r_{12}} \phi_b(\vec{r}_1) \phi_a(\vec{r}_2) d^3r_1 d^3r_2 > 0$$

The total energy eigenvalues are: $E_{\text{singlet}} = E_0 + J + K$ and $E_{\text{triplet}} = E_0 + J - K$. The triplet state (ortho-helium) lies lower in energy than the singlet state (para-helium) by the exchange splitting $\Delta E = 2K$. In the triplet state, $\psi_-(\vec{r}, \vec{r}) = 0$, meaning electrons avoid each other in space, reducing Coulomb repulsion.

1. The Central Field Approximation

In multi-electron atoms and solids with $N$ electrons, the exact many-body Schrödinger equation cannot be solved analytically. In the central field approximation, each electron moves independently in an effective spherically symmetric potential $V_{\text{eff}}(r)$ created by the nucleus plus the spherically averaged charge distribution of all other $N-1$ electrons:

2. The Hartree Self-Consistent Field (SCF) Method

Douglas Hartree (1928) formulated an iterative variational procedure. Assuming a trial product wavefunction $\Psi = \phi_1(\vec{r}_1)\phi_2(\vec{r}_2)\dots\phi_N(\vec{r}_N)$, electron $i$ experiences an electrostatic potential produced by the charge density $\rho_j(\vec{r}') = -e |\phi_j(\vec{r}')|^2$ of all other electrons:

The self-consistent iteration cycle proceeds as follows:

1. Guess an initial set of radial wavefunctions $\{\phi_j^{(0)}\}$.
2. Compute the electronic charge density $\rho^{(0)}(\vec{r})$ and the Hartree potential $V_i^{(0)}(\vec{r})$.
3. Solve the single-particle Schrödinger equations to obtain a new set of wavefunctions $\{\phi_j^{(1)}\}$.
4. Repeat steps 2–3 iteratively until the input and output potentials converge within a numerical tolerance: $|V^{(k+1)} - V^{(k)}| < \delta$.

3. The Hartree-Fock Method & Non-Local Exchange Potential

The simple Hartree product fails to satisfy the Pauli principle. Fock (1930) replaced the product with a fully antisymmetric Slater determinant, introducing an additional non-local exchange potential $V_{\text{ex}}$:

The exchange operator acts only between electrons with parallel spins, creating a surrounding depletion zone known as the Fermi hole (or exchange hole).

1. Hund's Rules for Atomic Ground States

For multi-electron open subshells ($p^n, d^n, f^n$), electrostatic repulsion and spin-orbit coupling determine the energetic ordering of spectroscopic term symbols $^{2S+1}L_J$ via Hund's three empirical rules:

1. Rule 1 (Maximize Spin Multiplicity $S$): The ground state term has the maximum total spin $S$ permitted by the Pauli exclusion principle (minimizes Coulomb repulsion by maximizing exchange stabilization).
2. Rule 2 (Maximize Total Orbital Angular Momentum $L$): For a given $S$, the ground state has the maximum total orbital angular momentum $L$ (electrons orbit in the same sense, minimizing spatial close encounters).
3. Rule 3 (Spin-Orbit Coupling $J$):
   • If the subshell is less than half full, the lowest energy level has minimum total angular momentum: $J = |L - S|$.
   • If the subshell is more than half full, the lowest energy level has maximum total angular momentum: $J = L + S$.

2. Characteristic X-Ray Spectra & Moseley's Law

When high-energy electrons eject an inner-core atomic electron, vacancies are filled by radiative transitions from outer shells, producing sharp characteristic X-ray lines:

• $K_\alpha$ line: transition from $n = 2$ ($L$-shell) to $n = 1$ ($K$-shell).
• $K_\beta$ line: transition from $n = 3$ ($M$-shell) to $n = 1$ ($K$-shell).
• $L_\alpha$ line: transition from $n = 3$ ($M$-shell) to $n = 2$ ($L$-shell).

Henry Moseley (1913) discovered that the frequency $\nu$ of characteristic X-rays is related linearly to the atomic number $Z$:

For the $K_\alpha$ line, $b = 1$ (the remaining $1s$ electron screens one unit of nuclear charge), yielding:

3. Molecular Orbital Theory & LCAO Covalent Bonding

In solids, atomic orbitals overlap to form extended molecular orbitals. In the simplest diatomic system ($\text{H}_2^+$ and $\text{H}_2$), Linear Combination of Atomic Orbitals (LCAO) yields two molecular orbitals from atomic hydrogen $1s$ states $\phi_A$ and $\phi_B$:

• Bonding Orbital ($\sigma_g$): $\psi_+ = \frac{1}{\sqrt{2(1+S)}} (\phi_A + \phi_B)$. Large electron probability density $|\psi_+|^2$ builds up between the two positively charged nuclei, screening their Coulomb repulsion and lowering the total electronic energy.
• Antibonding Orbital ($\sigma_u^*$): $\psi_- = \frac{1}{\sqrt{2(1-S)}} (\phi_A - \phi_B)$. Nodal plane ($\psi = 0$) midway between the nuclei pushes electron density away, increasing nuclear repulsion and raising the energy.