Quantum chemistry Study Guide

📖 Core Concepts Quantum Chemistry – Uses quantum mechanics to calculate electronic contributions to molecular properties (structures, spectra, thermodynamics). Born–Oppenheimer Approximation – Treats nuclei as fixed while solving the electronic Schrödinger equation; the electronic wavefunction is parametrized by nuclear positions. Schrödinger Equation (Electronic) – The foundational equation whose solution (exact only for H atom) gives the electronic energy and wavefunction. Valence‑Bond (VB) Theory – Bonds are formed by overlapping half‑filled atomic orbitals; emphasizes orbital hybridization and resonance. Molecular‑Orbital (MO) Theory – Electrons occupy delocalised orbitals that are linear combinations of all atomic orbitals in the molecule. Sigma (σ) vs Pi (π) Bonds – σ: head‑on overlap (s–s, s–p, p–p); π: side‑by‑side p‑p overlap, lower overlap, higher energy. Density Functional Theory (DFT) – Uses electron density rather than wavefunction; Kohn–Sham formalism splits total energy into kinetic, external, exchange, and correlation terms. Potential Energy Surface (PES) – A scalar field giving the electronic energy as a function of nuclear coordinates; central to adiabatic dynamics. RRKM Theory – Estimates unimolecular reaction rates from PES features (energy, density of states, transition‑state properties). Landau–Zener Transition – Gives the probability of hopping between two coupled adiabatic PESs at an avoided crossing. --- 📌 Must Remember Exact analytical solution of the non‑relativistic Schrödinger equation: hydrogen atom only. Born–Oppenheimer = electronic problem solved at fixed nuclear geometry. σ bond = strongest, formed by direct overlap; π bond = weaker, formed by side‑by‑side p overlap. MO theory = delocalised electrons; VB theory = localized bonds. DFT scaling: computational cost ≲ $n^{3}$ (where $n$ = number of basis functions). RRKM rate depends on (1) density of states of the activated complex, (2) number of ways energy can be distributed. Landau–Zener probability $P = \exp\!\left[-\dfrac{2\pi H{12}^{2}}{\hbar v |\Delta F|}\right]$ (where $H{12}$ = coupling, $v$ = nuclear velocity, $\Delta F$ = difference in slopes of the diabatic curves). --- 🔄 Key Processes Electronic Structure Calculation (generic workflow) Choose approximation method (HF, DFT, semi‑empirical, CC, etc.). Build basis set → form molecular integrals. Solve the electronic Schrödinger equation → obtain orbitals/energy. Post‑processing: compute properties (geometry, spectra, thermodynamics). Hartree–Fock Self‑Consistent Field (SCF) Guess initial orbitals. Form Fock matrix $F$ using current density. Diagonalise $F$ → new orbitals. Iterate until energy change < convergence threshold. Kohn–Sham DFT Cycle (mirrors HF SCF) Guess electron density $\rho(\mathbf{r})$. Build Kohn–Sham potential $v{\text{KS}} = v{\text{ext}} + v{\text{H}} + v{\text{xc}}$. Solve Kohn–Sham equations → new orbitals → new $\rho$. Iterate to self‑consistency. RRKM Rate Estimation Locate transition state (saddle point on PES). Compute vibrational frequencies → density of states $ρ(E)$. Apply RRKM expression $k(E) = \dfrac{N^\ddagger(E - E0)}{h\,ρ(E)}$, where $N^\ddagger$ = sum of states of TS, $E0$ = activation energy. --- 🔍 Key Comparisons VB vs MO Theory VB: localized bonds, emphasizes hybridisation; good for intuitive bonding pictures. MO: delocalised orbitals over whole molecule; better for spectroscopy and excited‑state descriptions. Quantum Dynamics vs Molecular Dynamics Quantum: solves full Schrödinger equation for nuclei → includes tunnelling, zero‑point energy. Classical (MD): integrates Newton’s equations; neglects quantum nuclear effects. Adiabatic vs Non‑Adiabatic Dynamics Adiabatic: single PES, nuclei move on one surface; Born–Oppenheimer holds. Non‑adiabatic: multiple coupled PESs, require vibronic coupling, surface hopping (Landau–Zener). HF vs DFT HF: explicit electron–electron exchange, no correlation; scales $\sim n^{4}$. DFT: includes exchange‑correlation via functionals; typically $n^{3}$ scaling, often more accurate for bulk properties. --- ⚠️ Common Misunderstandings “DFT gives exact energies because it uses density.” – DFT is exact in principle, but practical functionals are approximations; errors can be systematic. “Sigma bonds are always stronger than pi bonds.” – Generally true for simple diatomics, but conjugation or aromaticity can give delocalised π systems comparable stability. “Born–Oppenheimer means nuclei never move.” – It only decouples electronic and nuclear motion; nuclear dynamics are later treated on the PES. “Hartree–Fock includes electron correlation.” – HF includes exchange exactly but neglects dynamic correlation; post‑HF methods (MP2, CCSD) add it. --- 🧠 Mental Models / Intuition Orbitals as “rooms” – In VB, each bond is a private room (localized); in MO, all electrons share a common open‑plan living space (delocalised). Scaling picture – Think of $n$ as “rooms” you must clean; HF needs to check every pair of rooms ($n^2$), DFT only needs to sweep each room once ($n$) plus a modest extra step → $n^{3}$. Potential Energy Surface – Visualise a landscape of hills (high energy) and valleys (stable geometries). Adiabatic dynamics = walking on a single valley; non‑adiabatic = moving between adjacent valleys through a pass (avoided crossing). --- 🚩 Exceptions & Edge Cases Heavy‑element chemistry – Relativistic effects become significant; standard non‑relativistic Hamiltonian fails. Strongly correlated systems (e.g., transition metal complexes) – DFT or HF may give qualitatively wrong results; multi‑reference or coupled‑cluster needed. Highly excited states – Born–Oppenheimer breaks down; non‑adiabatic couplings dominate. --- 📍 When to Use Which Small molecules, high accuracy required → Post‑HF (CCSD(T)) or full CI (if feasible). Medium‑sized organic molecules, balance of cost/accuracy → Hybrid DFT (e.g., B3LYP) or MP2. Very large systems, polymers, biomolecules → Pure DFT (GGA) or semi‑empirical methods. Exploring reaction pathways → HF/DFT geometry optimisations → compute PES → apply RRKM or surface‑hopping. Studying non‑adiabatic processes (photochemistry, conical intersections) → Multi‑state DFT or CASSCF + Landau–Zener surface‑hopping. --- 👀 Patterns to Recognize Linear combination of atomic orbitals (LCAO) → any MO diagram will show constructive (bonding) and destructive (antibonding) combinations. σ > π overlap → look for head‑on overlap in bond length trends (shorter bonds → more σ character). Scaling blow‑up → If method’s cost grows faster than $n^{3}$, expect it to become prohibitive beyond 50–100 atoms. Avoided crossing signatures – Energy curves approach but never cross; Landau–Zener probability becomes significant when the gap is small and nuclear velocity is high. --- 🗂️ Exam Traps “DFT scales as $n^{4}$.” – Only pure Hartree–Fock and many post‑HF methods have $n^{4}$ scaling; DFT is $\le n^{3}$. “Exact solution exists for all one‑electron systems.” – Only the hydrogen atom (single electron, single nucleus) has a closed‑form solution; He⁺ is a one‑electron system but still requires approximation because of nuclear charge screening in many‑electron environments. “Sigma bonds always involve s orbitals.” – σ bonds can also be formed from p–p or hybrid orbitals (sp, sp², sp³). “RRKM applies to bimolecular reactions.” – RRKM is specifically for unimolecular reactions; bimolecular rates require transition‑state theory with two‑reactant PES. “Non‑adiabatic dynamics ignores nuclear motion.” – It explicitly treats coupled nuclear–electronic motion; the term “mixed quantum‑classical” still propagates nuclei classically. ---