Materials Genomics
Unit 2: QM Postulates, Solvable Systems, Multi-Electron Atoms

Prof. Dr. Philipp Pelz

FAU Erlangen-Nürnberg

Where We Stand

Recap of Unit 1

• Classical physics fails at the atomic scale: blackbody, photoelectric, Compton, double slit
• Schrödinger wave mechanics solves the hydrogen atom as an eigenvalue problem
• Born: \(|\psi|^2\) is a probability density
• Stern-Gerlach: angular momentum is quantized; spin exists
• Postulates 1–4: Hilbert space, Hermitian operators for observables, eigenvalues as outcomes, \(P(a_n)=|\langle\phi_{a_n}|\psi\rangle|^2\)

What Unit 2 Adds

• The remaining postulates: state collapse and time evolution
• The mathematical machinery: orthogonal decomposition, Hamilton operator, spin formalism
• Approximation toolkit: variational principle and perturbation theory
• A tour of exactly solvable 1D systems: free particle, harmonic oscillator, square wells
• The jump from one electron to many: helium, indistinguishability, antisymmetry
• The molecular Hamiltonian, Born-Oppenheimer, Hartree products, Slater determinants, LCAO

Lecture Roadmap

Part I — Formalism (postulates completed)

• Operators and orthogonal decomposition
• Hamilton operator, time-dep. and time-indep. SE
• Spin
• Averages, variance
• Variational principle
• Perturbation theory

Part II — Solvable models & many electrons

• Free particle, harmonic oscillator, wells
• Helium and the failure of independent electrons
• Indistinguishability and the exchange principle
• Molecular Hamiltonian and Born-Oppenheimer
• Hartree products, Slater determinants
• LCAO

Operators and Orthogonal Decomposition

Hermitian Operators Form a Basis

A Hermitian operator \(\hat{A}\) has eigenstates \(|\phi_{A_n}\rangle\) with eigenvalues \(A_n\):

\[\hat{A}\,|\phi_{A_n}\rangle = A_n\,|\phi_{A_n}\rangle\]

The eigenstates form an orthonormal basis of the Hilbert space:

\[\langle \phi_{A_m} | \phi_{A_n} \rangle = \delta_{mn}\]

• Real eigenvalues \(\Rightarrow\) valid measurement outcomes
• Completeness \(\Rightarrow\) any state can be represented in this basis

Spectral / Orthogonal Decomposition

Any quantum state \(|\psi\rangle\) admits the expansion

\[|\psi\rangle = \sum_n c_n\, |\phi_{A_n}\rangle, \qquad c_n = \langle \phi_{A_n} | \psi \rangle\]

The expansion coefficients carry probabilistic meaning:

\[P(A_n) = |c_n|^2\]

— consistent with Postulate 4.

This is the orthogonal (or spectral) decomposition of \(|\psi\rangle\).

Why This Matters

Note

The orthogonal decomposition is the bridge between abstract Hilbert-space states and concrete measurement statistics.

• Diagonalising an observable \(\hat{A}\) = solving the measurement problem for \(\hat{A}\)
• The resolution of identity \(\sum_n |\phi_n\rangle\langle\phi_n| = \mathbb{1}\) is reused throughout QM
• Algorithmically: every QM simulation reduces to a (very large) eigenvalue problem

Postulate 5 — State After Measurement

Measurement does not just report an eigenvalue — it changes the state.

If a measurement of \(\hat{A}\) yields the eigenvalue \(a_n\), the post-measurement state is the corresponding eigenstate:

\[|\psi\rangle \;\longrightarrow\; |\phi_{a_n}\rangle\]

Often called wavefunction collapse or projection postulate:

\[|\psi\rangle \;\longrightarrow\; \frac{\hat{P}_n |\psi\rangle}{\| \hat{P}_n |\psi\rangle \|}, \qquad \hat{P}_n = |\phi_{a_n}\rangle\langle\phi_{a_n}|\]

Repeating the same measurement immediately reproduces \(a_n\) with probability \(1\).

The Hamilton Operator

From Classical Energy to the Hamiltonian

For \(N\) classical particles, total energy = kinetic + potential:

\[E = E_{\rm kin} + V = \sum_{j=1}^{N} \frac{p_j^2}{2 m_j} + V\]

The correspondence principle promotes this to a quantum operator:

\[\hat{H} = \sum_{j=1}^{N} \frac{\hat{p}_j^{\,2}}{2 m_j} + V(t,\hat{\mathbf{x}}_1,\ldots,\hat{\mathbf{x}}_N)\]

The energy eigenvalue problem reads

\[\hat{H}\,|\phi_n\rangle = E_n\,|\phi_n\rangle\]

Position Representation

Substitute \(\hat{p}_j = -i\hbar\,\nabla_j\) to obtain the position-space Hamiltonian:

\[\hat{H}(t, \mathbf{x}_1,\ldots,\mathbf{x}_N) = -\sum_{j=1}^{N} \frac{\hbar^2}{2 m_j}\,\Delta_j + V(t, \mathbf{x}_1,\ldots,\mathbf{x}_N)\]

The stationary Schrödinger equation becomes a partial differential equation:

\[\left(-\sum_{j=1}^{N} \frac{\hbar^2}{2 m_j}\,\Delta_j + V(t, \mathbf{x}_1,\ldots,\mathbf{x}_N)\right)\psi = E\,\psi\]

For time-independent \(V\), this is the stationary Schrödinger equation; eigenfunctions \(\phi_n\) are stationary states.

Postulate 6 — Time Evolution

The full state evolves according to the time-dependent Schrödinger equation:

\[i\hbar\,\frac{\partial}{\partial t}|\psi(t)\rangle = \hat{H}\,|\psi(t)\rangle\]

For time-independent \(\hat{H}\), separate \(|\psi(t)\rangle = |\psi_t(t)\rangle\,|\psi_x\rangle\):

\[\frac{\partial |\psi_t\rangle}{\partial t} = -i\,\frac{E}{\hbar}\,|\psi_t\rangle \quad\Rightarrow\quad |\psi_t(t)\rangle = C\,e^{-iEt/\hbar}\]

Energy eigenstates evolve only by a phase:

\[|\psi(t)\rangle = \sum_n c_n\,e^{-i E_n t/\hbar}\,|\phi_n\rangle\]

Superposition Principle

If \(|\psi_1\rangle, |\psi_2\rangle, \ldots, |\psi_K\rangle\) solve the Schrödinger equation, so does

\[|\psi\rangle = \sum_{j=1}^{K} c_j\,|\psi_j\rangle, \qquad c_j \in \mathbb{C}\]

• A direct consequence of the linearity of \(\hat{H}\)
• Distinguishes QM from classical mechanics
• Underpins interference, entanglement, and the entire quantum computing programme

Note

Energy eigenstates are convenient because they propagate by simple phase factors \(e^{-iE_n t/\hbar}\) — but any complete orthonormal basis works.

Spin

Spin as an Intrinsic Quantum Number

• Stern-Gerlach: silver atoms split into two beams — angular momentum is quantized
• Treated formally like angular momentum, but no classical analogue
• The electron is not literally spinning — spin is intrinsic
• For the electron, the spin quantum number is

\[s = \tfrac{1}{2}, \qquad m_s = \pm\tfrac{1}{2}\]

Two spin states, denoted \(|\alpha\rangle\) (“spin up”) and \(|\beta\rangle\) (“spin down”).

Spin Operators and Pauli Matrices

Spin- \(\tfrac{1}{2}\) operators in the \(|\alpha\rangle, |\beta\rangle\) basis:

\[\hat{S}_x = \tfrac{\hbar}{2}\sigma_x,\quad \hat{S}_y = \tfrac{\hbar}{2}\sigma_y,\quad \hat{S}_z = \tfrac{\hbar}{2}\sigma_z\]

Pauli matrices:

\[\sigma_x = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix},\;\; \sigma_y = \begin{pmatrix}0 & -i\\ i & 0\end{pmatrix},\;\; \sigma_z = \begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}\]

\(\hat{S}_z\) eigenstates: \(\hat{S}_z|\alpha\rangle = +\tfrac{\hbar}{2}|\alpha\rangle\), \(\hat{S}_z|\beta\rangle = -\tfrac{\hbar}{2}|\beta\rangle\).

The components do not commute: \([\hat{S}_x,\hat{S}_y] = i\hbar\,\hat{S}_z\) — only one component is sharp at a time.

Electron Wavefunction with Spin

The full electronic wavefunction is a tensor product of a spatial and a spin part:

\[\psi = \psi_x(\mathbf{r})\,\chi(s)\]

Common shorthand for one-electron spin-orbitals:

\[\phi(\mathbf{r}, s) = \psi_x(\mathbf{r})\,|\alpha\rangle \quad \text{or} \quad \psi_x(\mathbf{r})\,|\beta\rangle\]

• Doubles the size of the one-electron Hilbert space
• Crucial for the exchange principle later in this lecture
• Ground state of helium: both electrons share spatial orbital, spins paired

Averages, Variance, and Variational Methods

Expectation Values

Quantum mechanics is probabilistic — we summarise outcomes statistically.

For an observable \(\hat{A}\) in state \(|\psi\rangle\), the expectation value is

\[\langle \hat{A}\rangle = \langle \psi | \hat{A} | \psi\rangle = \sum_n A_n\,P(A_n)\]

The variance measures the spread:

\[\text{Var}(\hat{A}) = \langle \hat{A}^2\rangle - \langle \hat{A}\rangle^2\]

For energy: \(\langle \hat{H}\rangle\) is the energy expectation; eigenstates have zero variance.

The Variational Principle — Setup

Suppose \(|\psi\rangle\) is any normalized state, \(\langle\psi|\psi\rangle = 1\).

Decompose into eigenstates of \(\hat{H}\):

\[|\psi\rangle = \sum_n c_n\,|\phi_{H_n}\rangle\]

Compute the energy expectation:

\[\langle\psi|\hat{H}|\psi\rangle = \sum_n E_n\,|c_n|^2 = \sum_n E_n\,P(E_n)\]

The Variational Principle — Inequality

Since \(E_0 \le E_n\) for all \(n\):

\[\langle\psi|\hat{H}|\psi\rangle = \sum_n E_n\,P(E_n) \;\ge\; E_0 \sum_n P(E_n) = E_0\]

Note

Variational principle. For any normalized trial state \(|\psi\rangle\),

\[\langle \psi|\hat{H}|\psi\rangle \;\ge\; E_0.\]

Equality iff \(|\psi\rangle\) is the ground state.

For unnormalised trial states, use the Rayleigh quotient:

\[\frac{\langle\psi|\hat{H}|\psi\rangle}{\langle\psi|\psi\rangle} \;\ge\; E_0\]

Variational Principle in Practice

Pick a trial wavefunction \(|\psi_{\text{trial}}(\alpha)\rangle\) depending on parameters \(\alpha\), then minimise:

\[\frac{\partial}{\partial \alpha}\!\left(\frac{\langle\psi_{\text{trial}}|\hat{H}|\psi_{\text{trial}}\rangle}{\langle\psi_{\text{trial}}|\psi_{\text{trial}}\rangle}\right) = 0\]

• “Lower energy is always better” — bounded below by the true ground state
• Workhorse of quantum chemistry: Hartree-Fock, DFT (Kohn-Sham), CI, CC
• Convergence to \(E_0\) depends on the flexibility of the trial space

Perturbation Theory — Setup

When \(\hat{H}\) cannot be solved exactly, decompose into a solvable part plus a small correction:

\[\hat{H} = \hat{H}_0 + \epsilon\,\hat{H}_1\]

• \(\hat{H}_0\): unperturbed (exactly diagonalisable) Hamiltonian
• \(\hat{H}_1\): perturbation (e.g. external field, electron-electron interaction)
• \(\epsilon\): formal small parameter

Expand both energies and states in powers of \(\epsilon\):

\[E_n = \sum_{j=0}^{\infty}\epsilon^{j}\,E_n^{(j)}, \qquad |\psi_n\rangle = \sum_{j=0}^{\infty}\epsilon^{j}\,|\psi_n^{(j)}\rangle\]

Order-by-Order Equations

Insert into \(\hat{H}|\psi_n\rangle = E_n|\psi_n\rangle\) and collect powers of \(\epsilon\):

\[\epsilon^0:\quad \hat{H}_0|\psi_n^{(0)}\rangle = E_n^{(0)}|\psi_n^{(0)}\rangle\]

\[\epsilon^1:\quad \hat{H}_0|\psi_n^{(1)}\rangle + \hat{H}_1|\psi_n^{(0)}\rangle = E_n^{(0)}|\psi_n^{(1)}\rangle + E_n^{(1)}|\psi_n^{(0)}\rangle\]

Project onto \(\langle\psi_n^{(0)}|\) and use \(\langle\psi_n^{(0)}|\hat{H}_0 = \langle\psi_n^{(0)}|E_n^{(0)}\) to obtain:

\[\boxed{\;E_n^{(1)} = \langle\psi_n^{(0)}|\hat{H}_1|\psi_n^{(0)}\rangle\;}\]

— the first-order energy correction is just the expectation of the perturbation in the unperturbed state.

Limitations and Higher Orders

• The choice of \(\hat{H}_0\) is not always obvious — sometimes one starts from a numerical solution
• The perturbation must indeed be small relative to \(\hat{H}_0\)
• Convergence, boundedness, and even asymptotic usefulness are not guaranteed

The standard form is stationary Rayleigh-Schrödinger perturbation theory.

For time-dependent perturbations (transitions, optical absorption) one uses Dirac (time-dependent) perturbation theory — beyond this lecture.

Exactly Solvable Systems

The Free Particle

Set \(V(x) = 0\). The 1D Schrödinger equation reads

\[-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\,\psi\]

Solutions are plane waves with continuous wavevector \(k\in\mathbb{R}\):

\[\psi_k(x) = \frac{1}{\sqrt{2\pi}}\,e^{ikx}, \qquad E = \frac{\hbar^2 k^2}{2m} \ge 0\]

The spectrum is continuous and non-negative — a sharp contrast to bound states.

Plane Waves are Not Square-Integrable

\[\langle\psi_k|\psi_k\rangle = \int_{-\infty}^{\infty}\frac{1}{2\pi}\,dx \;\to\; \infty\]

• Plane waves are not normalisable — not strictly elements of \(L^2(\mathbb{R})\)
• They live in a rigged Hilbert space / are delta-normalised
• Physical free particles are wave packets: superpositions of plane waves

\[\psi(x,t) = \int dk\; \tilde\psi(k)\,e^{i(kx - \omega(k)\,t)}, \qquad \omega(k) = \frac{\hbar k^2}{2m}\]

Wave packets disperse — the group velocity gives the classical particle velocity.

The 1D Harmonic Oscillator

Potential of a quadratic restoring force:

\[V(x) = \tfrac{1}{2}k\,x^2 = \tfrac{1}{2}m\omega^2 x^2, \qquad \omega = \sqrt{k/m}\]

The spectrum is equidistant with quantum number \(n = 0,1,2,\ldots\):

\[E_n = \hbar\omega\!\left(n + \tfrac{1}{2}\right)\]

• Echoes Planck’s \(E = nh\nu\) — but with a zero-point energy \(\tfrac{1}{2}\hbar\omega\)
• Linchpin model for vibrations, phonons, photons, and optical lattices

Harmonic Oscillator — Wavefunctions

The eigenfunctions involve Hermite polynomials \(H_n\):

\[\psi_n^{\rm harm}(x;m,\omega) = \frac{1}{\sqrt{2^n n!}}\!\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\!H_n\!\left(\sqrt{\tfrac{m\omega}{\hbar}}\,x\right)\exp\!\left(-\tfrac{m\omega}{2\hbar}x^2\right)\]

\[H_n(x) = (-1)^n\,e^{x^2}\,\frac{d^n}{dx^n}\,e^{-x^2}\]

Each level alternates parity: \(H_0=1\) (even), \(H_1\propto x\) (odd), \(H_2\propto 4x^2-2\) (even), …

Ladder Operators

Define creation/annihilation operators

\[\hat{a} = \sqrt{\tfrac{m\omega}{2\hbar}}\!\left(\hat{x} + \tfrac{i}{m\omega}\hat{p}\right), \qquad \hat{a}^{\dagger} = \sqrt{\tfrac{m\omega}{2\hbar}}\!\left(\hat{x} - \tfrac{i}{m\omega}\hat{p}\right)\]

with canonical commutator \([\hat{a},\hat{a}^{\dagger}] = 1\).

Then \(\hat{H} = \hbar\omega\!\left(\hat{a}^{\dagger}\hat{a} + \tfrac{1}{2}\right)\) and

\[\hat{a}^{\dagger}|n\rangle = \sqrt{n+1}\,|n+1\rangle, \qquad \hat{a}|n\rangle = \sqrt{n}\,|n-1\rangle\]

Algebraic derivation of the spectrum without solving any ODE — same idea reused for phonons, photons and the entire formalism of second quantisation.

Why the Harmonic Oscillator Matters

• Equally spaced levels — Planck’s quantisation, IR/Raman vibrational spectra
• Zero-point energy \(\tfrac{1}{2}\hbar\omega\) — atoms still fluctuate at \(T=0\) K
• Any smooth potential near a minimum looks harmonic — lattice phonons, molecular vibrations, lattice dynamics
• Underpins second quantisation: field quanta are oscillator excitations

Note

The harmonic oscillator is arguably the most important model in physics — almost every weakly excited bosonic system reduces to it.

The d-Dimensional Oscillator

Separable potential

\[V(\mathbf{x}) = \frac{m}{2}\sum_{i=1}^{d}\omega_i^2\,x_i^2\]

Eigenfunctions factorise; eigenenergies sum:

\[\psi_{n_1,\ldots,n_d}(\mathbf{x}) = \prod_{i=1}^{d}\psi_{n_i}^{\rm harm}(x_i;m,\omega_i)\]

\[E = \sum_{i=1}^{d}\hbar\omega_i\!\left(n_i + \tfrac{1}{2}\right)\]

Most well-behaved separable potentials reduce similarly — the 1D building block does the heavy lifting.

The Infinite Well — Setup

A particle confined to a box:

\[V(x) = \begin{cases}0, & 0 < x < L \\ \infty, & \text{otherwise}\end{cases}\]

Boundary conditions \(\psi(0) = \psi(L) = 0\) select sine modes:

\[\psi_n(x) = \sqrt{\tfrac{2}{L}}\,\sin\!\left(\tfrac{n\pi x}{L}\right), \qquad n = 1,2,\ldots\]

Energy spectrum scales as \(n^2\):

\[E_n = \frac{n^2 \pi^2 \hbar^2}{2 m L^2}\]

Infinite Well — Lessons

• Discrete spectrum \(\{E_n\}\) from confinement alone
• Spectrum is not equidistant (unlike harmonic oscillator)
• Zero-point energy \(E_1 \ne 0\) — uncertainty principle: a localised particle cannot have \(\langle p^2\rangle = 0\)
• Spacings increase with \(n\); spacings shrink as the box grows (\(L\to\infty\) recovers continuous spectrum)

The infinite well is the prototype for quantum dots, semiconductor nanowires, and “particle-in-a-box” molecular models.

Excursus: The Finite Well

A more physical bound system:

\[V(x) = \begin{cases}-V_0, & |x| < a \\ 0, & |x| \ge a\end{cases}\]

Bound states have \(-V_0 < E < 0\). Wavefunctions are oscillatory inside, exponentially decaying outside:

\[\psi(x) = \begin{cases}A\,e^{\kappa x}, & x < -a \\ B\cos(kx) + C\sin(kx), & |x| < a \\ D\,e^{-\kappa x}, & x > a\end{cases}\]

\[k = \sqrt{\tfrac{2m(E + V_0)}{\hbar^2}}, \qquad \kappa = \sqrt{\tfrac{-2mE}{\hbar^2}}\]

Finite Well — Quantisation and Tunnelling

Matching \(\psi\) and \(\psi'\) at \(\pm a\) yields the transcendental conditions

\[k\,\tan(ka) = \kappa \quad\text{(even)}, \qquad -k\,\cot(ka) = \kappa \quad\text{(odd)}\]

• Only a finite number of bound states exist
• Bound-state spacings are not equidistant
• Above the well: a continuum of scattering states
• The decaying tails \(\propto e^{-\kappa|x|}\) leak outside the well — purely quantum tunnelling

Tunnelling is essential for STM, alpha decay, fusion, Josephson junctions, and chemical reaction rates.

Multi-Electron Atoms

Why Helium is Hard

Helium (\(Z=2\), 2 electrons): the simplest non-trivial atom. The Hamiltonian, in the nucleus’ frame, reads

\[\hat{H}_{\rm He} = -\frac{\hbar^2}{2m_e}(\Delta_{\mathbf{r}_1} + \Delta_{\mathbf{r}_2}) + \frac{e^2}{4\pi\varepsilon_0}\!\left[(-Z)\!\left(|\mathbf{r}_1|^{-1} + |\mathbf{r}_2|^{-1}\right) + |\mathbf{r}_1 - \mathbf{r}_2|^{-1}\right]\]

Note

The electron-electron repulsion \(|\mathbf{r}_1 - \mathbf{r}_2|^{-1}\) couples the two electrons. The Schrödinger equation \(\hat{H}_{\rm He}|\psi\rangle = E|\psi\rangle\) has no closed-form solution — we must approximate.

Independent Electrons (Naive Estimate)

Drop the \(|\mathbf{r}_1 - \mathbf{r}_2|^{-1}\) term — the equation separates:

\[|\psi_{\rm He}(\mathbf{r}_1,\mathbf{r}_2)\rangle = |\psi_1(\mathbf{r}_1)\rangle\,|\psi_2(\mathbf{r}_2)\rangle\]

Each electron is hydrogen-like with nuclear charge \(Z\):

\[E_{n_1, n_2}^{\rm He} = E_{n_1} + E_{n_2}\]

For ground state \(n_1 = n_2 = 1\):

\[E_{1,1}^{\rm He} = -2 Z^2\,(13.6\ \text{eV}) = -108.8\ \text{eV}\]

Experimental total ionisation energy: \(-78.93\ \text{eV}\). Error \(\approx 30\ \text{eV}\) (\(\approx 40\%\)) — too large.

Effective Charge / Screening

Refine by introducing an effective nuclear charge:

\[Z_{\rm eff} = Z - S\]

• Inner electron screens the nuclear charge seen by the outer electron
• Crude choice \(S = 1\) for helium gives

\[E_{1,1}^{\rm He} = -(1 + Z^2)(13.6\ \text{eV}) = -68.0\ \text{eV}\]

— much closer to experiment

Screening intuition shows up everywhere: Slater rules, pseudopotentials, Kohn-Sham orbitals.

Indistinguishable Particles

The product ansatz \(|\psi_1(\mathbf{r}_1)\rangle|\psi_2(\mathbf{r}_2)\rangle\) has a conceptual flaw:

• Electrons are fundamentally indistinguishable — same mass, charge, spin
• No measurement can label which electron is “1” or “2”
• Labels \(1, 2\) are mere mathematical bookkeeping for coordinates \(\mathbf{r}_1, \mathbf{r}_2\)

The probability density must be invariant under particle exchange:

\[|\psi(\mathbf{r}_1, \mathbf{r}_2)|^2 = |\psi(\mathbf{r}_2, \mathbf{r}_1)|^2\]

The Exchange Principle

Invariance of \(|\psi|^2\) allows at most a phase under exchange:

\[\psi(\mathbf{r}_2,\mathbf{r}_1) = e^{i\alpha}\,\psi(\mathbf{r}_1,\mathbf{r}_2)\]

Applying twice must return to the original state: \(e^{2i\alpha} = 1 \;\Rightarrow\; e^{i\alpha} = \pm 1\).

• Symmetric (\(+\)): bosons (photons, \(^4\)He, phonons)
• Antisymmetric (\(-\)): fermions — electrons, protons, neutrons

For electrons:

\[\psi(\mathbf{r}_2, \mathbf{r}_1) = -\psi(\mathbf{r}_1, \mathbf{r}_2)\]

Antisymmetrisation and the Pauli Exclusion

The naive product \(|\psi_a(\mathbf{r}_1)\rangle|\psi_b(\mathbf{r}_2)\rangle\) is not antisymmetric. Form

\[|\psi^-\rangle = \tfrac{1}{\sqrt{2}}\Big(|\psi_a(\mathbf{r}_1)\rangle|\psi_b(\mathbf{r}_2)\rangle - |\psi_b(\mathbf{r}_1)\rangle|\psi_a(\mathbf{r}_2)\rangle\Big)\]

Indeed \(\psi^-(\mathbf{r}_2,\mathbf{r}_1) = -\psi^-(\mathbf{r}_1,\mathbf{r}_2)\) — and \(\langle\psi^-|\psi^-\rangle = 1\).

If both electrons occupy the same spin-orbital \(\psi_a\):

\[|\psi^-\rangle = \tfrac{1}{\sqrt{2}}(|\psi_a\rangle|\psi_a\rangle - |\psi_a\rangle|\psi_a\rangle) = 0\]

— this is the Pauli exclusion principle: no two electrons share the same spin-orbital.

Molecules

The Molecular Hamiltonian

Nuclei at \(\mathbf{R}_i\) (\(N_{\rm at}\) total), electrons at \(\mathbf{r}_i\) (\(N_e\) total):

\[\hat{H} = \underbrace{-\sum_{i}\!\frac{\hbar^2}{2 M_i}\Delta_{\mathbf{R}_i}}_{T_{\rm nuc}}\,\underbrace{-\sum_{i}\!\frac{\hbar^2}{2 m_e}\Delta_{\mathbf{r}_i}}_{T_{\rm el}}\,\underbrace{-\frac{e^2}{4\pi\varepsilon_0}\!\sum_{i,j}\!\frac{Z_i}{|\mathbf{R}_i - \mathbf{r}_j|}}_{V_{\rm en}}\]

\[\;+\; \underbrace{\tfrac{1}{2}\frac{e^2}{4\pi\varepsilon_0}\!\sum_{i\ne j}\!\frac{1}{|\mathbf{r}_i - \mathbf{r}_j|}}_{V_{\rm ee}}\;+\;\underbrace{\tfrac{1}{2}\frac{e^2}{4\pi\varepsilon_0}\!\sum_{i\ne j}\!\frac{Z_i Z_j}{|\mathbf{R}_i - \mathbf{R}_j|}}_{V_{\rm nn}}\]

Nuclear kinetic + electron kinetic + electron-nuclear + electron-electron + nuclear-nuclear.

Born-Oppenheimer Approximation

Mass disparity: \(M_i \gg m_e\) (\(M_{\rm proton}/m_e \approx 1836\)) \(\Rightarrow\) nuclei are much slower than electrons.

Born-Oppenheimer:

• Treat nuclear positions \(\{\mathbf{R}_i\}\) as fixed parameters
• Drop nuclear kinetic energy \(T_{\rm nuc}\) in the electronic problem
• Solve the electronic Schrödinger equation for the electrons in the static field of the nuclei
• Nuclei feel an effective potential \(E_{\rm el}(\{\mathbf{R}_i\}) + V_{\rm nn}\) — the potential energy surface (PES)

The Electronic Hamiltonian

After Born-Oppenheimer, the electronic Hamiltonian is

\[\hat{H}_{\rm el} = -\sum_{i=1}^{N_e}\!\frac{\hbar^2}{2 m_e}\Delta_{\mathbf{r}_i} - \frac{e^2}{4\pi\varepsilon_0}\!\left(\sum_{i,j}\!\frac{Z_i}{|\mathbf{R}_i - \mathbf{r}_j|} + \tfrac{1}{2}\sum_{i\ne j}\!\frac{1}{|\mathbf{r}_i - \mathbf{r}_j|}\right) + V_{\rm nn}\]

• For fixed \(\{\mathbf{R}_i\}\), \(V_{\rm nn}\) is just a constant energy shift
• The remaining electronic Schrödinger equation is what we approximate (HF, DFT, CC, …)
• Solving on a grid of \(\{\mathbf{R}_i\}\) traces out the PES \(\to\) geometry optimisation, vibrations, reaction paths, MD on PES

Born-Oppenheimer — Caveats

• Excellent for ground-state structure of most molecules and solids
• Breaks down at conical intersections, near level crossings, in non-adiabatic dynamics
• Vibronic coupling mixes electronic and nuclear motion (Jahn-Teller, photochemistry)
• Hydrogen tunnelling, ultrafast spectroscopy: nuclear quantum effects matter

Note

Most of computational materials science operates inside Born-Oppenheimer. Knowing where it fails is essential when interpreting simulations as data.

Wavefunction Approximations

The Hartree Product

A first guess for an \(N\)-electron wavefunction: assume independence and multiply one-electron orbitals \(\phi_j\):

\[\psi(\mathbf{r}_1,\ldots,\mathbf{r}_N) = \prod_{j=1}^{N}\phi_j(\mathbf{r}_j)\]

• Easy to construct, easy to interpret
• Each electron sees an average field from the others (Hartree’s self-consistent field)
• But: violates the exchange principle for fermions — not antisymmetric under particle exchange

We need a properly antisymmetric many-electron ansatz.

The Slater Determinant

Build a fully antisymmetric many-electron wavefunction as a determinant of one-electron spin-orbitals:

\[\psi(\mathbf{r}_1,\ldots,\mathbf{r}_N) = \frac{1}{\sqrt{N!}}\det\!\begin{pmatrix}\phi_1(\mathbf{r}_1) & \phi_2(\mathbf{r}_1) & \cdots & \phi_N(\mathbf{r}_1)\\ \phi_1(\mathbf{r}_2) & \phi_2(\mathbf{r}_2) & \cdots & \phi_N(\mathbf{r}_2)\\ \vdots & \vdots & \ddots & \vdots\\ \phi_1(\mathbf{r}_N) & \phi_2(\mathbf{r}_N) & \cdots & \phi_N(\mathbf{r}_N)\end{pmatrix}\]

• Swapping two rows (\(\equiv\) exchanging two electrons) flips the sign — automatic antisymmetry
• Two equal columns (\(\equiv\) two electrons in the same spin-orbital) make the determinant vanish — Pauli exclusion built in

Slater Determinants in Practice

Common compact notation drops the prefactor visually using vertical bars:

\[\psi(\mathbf{r}_1,\ldots,\mathbf{r}_N) = \frac{1}{\sqrt{N!}}\begin{vmatrix}\phi_1(\mathbf{r}_1) & \cdots & \phi_N(\mathbf{r}_1)\\ \vdots & \ddots & \vdots\\ \phi_1(\mathbf{r}_N) & \cdots & \phi_N(\mathbf{r}_N)\end{vmatrix}\]

• The single-Slater ansatz is the foundation of Hartree-Fock
• Configuration Interaction (CI) and Coupled Cluster (CC): linear / exponential combinations of Slater determinants
• DFT (Kohn-Sham): a fictitious non-interacting Slater determinant reproducing the true density

Linear Combination of Atomic Orbitals

LCAO — The Idea

For molecules and solids, the molecular wavefunction should look “atom-like” near each nucleus.

Ansatz: expand each molecular orbital in atomic orbitals \(\psi_j^a\):

\[|\psi\rangle = \sum_{j=1}^{N_b} c_j\,|\psi_j^a\rangle = \mathbf{c}^{T}\boldsymbol{\psi}^a\]

• \(N_b\): number of basis functions
• Coefficients \(c_j\): how strongly each atomic orbital contributes
• Smaller \(N_b\) = cheaper computation; larger \(N_b\) = better approximation

LCAO — The Generalised Eigenvalue Problem

Substitute the LCAO ansatz into the Schrödinger equation. Atomic orbitals on different atoms are generally not orthogonal \(\Rightarrow\) generalised eigenvalue problem:

\[\boxed{\;\mathbf{H}\,\mathbf{c} = E\,\mathbf{S}\,\mathbf{c}\;}\]

• Hamiltonian matrix: \(H_{ij} = \langle\psi_i^a|\hat{H}|\psi_j^a\rangle\)
• Overlap matrix: \(S_{ij} = \langle\psi_i^a|\psi_j^a\rangle\)
• Coefficient vector \(\mathbf{c}\): solution of the generalised eigenproblem

Reduces a PDE (infinite-dim. eigenproblem) to a finite matrix problem of size \(N_b\) — the central simplification underlying nearly all quantum-chemistry codes.

LCAO — Quality, Choices, and Reach

• Quality depends on how well the atomic basis spans the true molecular orbitals
• Basis sets (Gaussian, Slater-type, plane waves, numerical AO) trade cost vs accuracy
• LCAO + Slater determinant + variational principle = Hartree-Fock
• LCAO is the engine behind tight-binding, DFT codes like Quantum ESPRESSO, VASP, ORCA, Gaussian, …
• Generates exactly the kind of energies, forces, and orbitals that feed ML pipelines in materials genomics

Wrap-Up

Unit 2 — Key Takeaways

• Postulates 5–6 complete QM: state collapse on measurement, Schrödinger time evolution
• Hermitian operators give the spectral / orthogonal decomposition — the basis of all measurement statistics
• The Hamiltonian governs both stationary and dynamical quantum mechanics
• Spin is intrinsic, \(s = 1/2\), encoded by Pauli matrices, indispensable for fermionic statistics
• Variational principle + perturbation theory are the two universal approximation engines
• Free particle, harmonic oscillator, and well potentials are the building blocks of all later models
• Multi-electron problems demand antisymmetry \(\Rightarrow\) Slater determinants \(\Rightarrow\) Pauli exclusion
• Born-Oppenheimer decouples electrons from nuclei and defines the PES
• LCAO turns the molecular Schrödinger PDE into a generalised matrix eigenproblem

Outlook to Unit 3

• Hartree-Fock: variational principle on a single Slater determinant of LCAO orbitals
• Recovers the mean-field picture — but misses electron correlation
• Density Functional Theory (Kohn-Sham): shift the unknown from the wavefunction to the electron density \(n(\mathbf{r})\)
• Generates the bulk of computed materials data: total energies, forces, band structures, formation energies
• These outputs become the labels and features for ML in materials genomics

Continue

• ← Previous: Unit 01 — What is Materials Genomics?
• → Next: Unit 03 — Quantum Chemistry Methods (HF, MP, CC, DFT)
• All courses

References