“Know how to solve every problem that has been solved.”“What I cannot create, I do not understand.”— Richard FeynmanFinal exam
Quantum Chemistry I — Hartree-Fock from Nothing
30 questions across the whole course, from the variational principle to
correlation. Closed-book and timed (~60 min). No feedback until you
submit — then you get a score, a per-topic breakdown, and a full review with
every answer explained.
One sitting. The clock auto-submits at zero.
Pass mark is 70%. This is for you — grade yourself honestly.
Each run draws the same comprehensive set; retake it until it's automatic.
QC I — Final Exam0/30 answered · 60:00 left
Q1standard
Why does quantum chemistry build orbitals out of Gaussians rather than the more physically-accurate Slater functions?
Q2standard
The Gaussian product theorem states that the product of two s-type Gaussians centred at A and B is…
Q3intro
The variational principle says that for any trial wavefunction ψ, the Rayleigh quotient ⟨ψ|H|ψ⟩ / ⟨ψ|ψ⟩ is…
Q4standard
For hydrogen, the trial orbital ψ(r) = e^{−ζ|r|} with ζ as a parameter makes the variational calculation a…
Q5standard
Why is the Hartree product ψ(r₁,r₂) = φ(r₁)φ(r₂) an unacceptable wavefunction for two electrons?
Q6standard
A Slater determinant automatically encodes the Pauli exclusion principle because…
Q7hard
A single Slater determinant with one α and one β electron in different spatial orbitals (|φ₁α φ₂β|) is…
Q8hard
For H₂'s σ→σ excited states, the triplet lies below* the singlet by exactly 2K_ov = 0.363 hartree (in STO-3G). What pays for the triplet's discount?
Q9standard
The Boys function F₀(x) shows up in which integrals when using a Gaussian basis?
Q10hard
Two Gaussian primitives have exponents α (large) and β (small) on centres A and B. Their product Gaussian sits at P. Where is P?
Q11standard
How does the number of two-electron repulsion integrals (ERIs) scale with the number of basis functions N?
Q12standard
Real-orbital ERIs obey an 8-fold permutation symmetry, e.g. (ab|cd) = (ba|cd) = (cd|ab) = …. Practically, this means a code can…
Q13standard
What does it mean that Hartree-Fock is a mean-field theory?
Q14standard
The correlation energy is defined as…
Q15hard
The Roothaan-Hall equations take the form FC = SCε (a generalized eigenvalue problem) rather than FC = Cε. The overlap matrix S appears because…
Q16standard
Diagonalizing the Roothaan-Hall equations in an N-function basis returns N orbitals. The N/2 with the lowest energies are occupied. What are the rest?
Q17standard
Why must Hartree-Fock be solved by iteration (the SCF cycle) rather than in one shot?
Q18standard
The SCF procedure is best described as…
Q19standard
The density matrix P = 2C_occ C_occᵀ compresses the wavefunction. What does its trace with the overlap matrix, tr(PS), always equal?
Q20hard
Mulliken charges are famously basis-set sensitive — add diffuse functions and the numbers can change wildly. What is the structural reason?
Q21hard
The Coulson-Fischer point of H₂ (≈2.1 bohr in this basis) marks the geometry where…
Q22standard
Pople-Nesbet UHF builds two Fock matrices. In Fα, the exchange term subtracts only Pα — never Pβ. Why?
Q23standard
According to Koopmans' theorem, the ionization potential of a molecule is approximately…
Q24hard
Koopmans' theorem makes two crude approximations, yet its ionization potentials are often surprisingly decent. Why?
Q25hard
Stretching H₂ toward dissociation makes the correlation problem qualitatively harder than it is at equilibrium. Which kind of correlation dominates there?
Q26hard
UHF rescues the H₂ dissociation curve by letting the α and β electrons take different spatial orbitals. What does it pay for this?
Q27hard
Møller-Plesset perturbation theory splits the Hamiltonian as H = H₀ + V. What is H₀, and what is the perturbation?
Q28hard
The MP2 energy is a sum of terms |coupling|² / (εᵢ + εⱼ − εₐ − εᵦ). Which systems should make you distrust it?
Q29standard
What is the Full Configuration Interaction (FCI) wavefunction?