Two electrons are assigned to the same spatial orbital in an N-electron Slater determinant. What is the value of the resulting wavefunction?
ADouble the amplitude of the single-occupancy case, since both electrons contribute
BZero — the state does not exist
CSymmetric rather than antisymmetric, so the wavefunction must be renormalized
DUnchanged — the normalization factor 1/√N! absorbs the double occupancy
Two electrons in the same orbital means two identical rows in the determinant matrix. A determinant with two identical rows is always zero. This is the Pauli exclusion principle emerging as a mathematical identity from the determinant structure — not as an additional physical postulate imposed on top. The wavefunction doesn't become small or ill-defined; it is exactly zero, meaning the state simply cannot exist.
Question 2 Multiple Choice
A Slater determinant captures which aspects of electron-electron interactions in a many-electron system?
BNeither exchange nor correlation — it treats electrons as fully independent
CExchange interactions exactly, but not electron correlation
DCorrelation exactly, but exchange only approximately via the antisymmetric prefactor
The Slater determinant captures exchange effects exactly — the antisymmetry requirement and Pauli exclusion follow directly from the determinant structure. However, it assumes electrons move in independent orbitals, missing the dynamic correlation: electrons avoid each other beyond what Pauli requires. The gap between Hartree-Fock (Slater determinant) energy and the exact ground-state energy is the correlation energy, and closing this gap is the central challenge of modern quantum chemistry.
Question 3 True / False
A Slater determinant automatically enforces the Pauli exclusion principle without it being imposed as a separate physical postulate.
TTrue
FFalse
Answer: True
Correct. The antisymmetry requirement is built into the determinant's mathematical structure: swapping two electrons (swapping two columns) changes the sign of the wavefunction, satisfying the fermionic antisymmetry condition. And two electrons in the same orbital (identical rows) gives a determinant of zero — Pauli exclusion is a consequence of linear algebra, not an additional rule bolted on to the formalism.
Question 4 True / False
The Slater determinant provides an exact description of the N-electron ground state in Hartree-Fock theory because it correctly accounts for most electron-electron interactions.
TTrue
FFalse
Answer: False
The Slater determinant is the best single-determinant approximation to the many-electron wavefunction, but it is not exact. It correctly handles exchange (Pauli exclusion) via its antisymmetric structure, but it treats electrons as moving in independent average fields — missing correlation, the tendency of electrons to dynamically avoid each other beyond what Pauli requires. The correlation energy (exact energy minus Hartree-Fock energy) is always negative and nonzero for real systems.
Question 5 Short Answer
Explain why swapping two electrons in a Slater determinant changes the sign of the wavefunction, and what this property has to do with the Pauli exclusion principle.
Think about your answer, then reveal below.
Model answer: Swapping two electrons corresponds to swapping two columns in the determinant matrix. A fundamental property of determinants is that exchanging any two columns changes the sign — this directly implements fermionic antisymmetry. The connection to Pauli exclusion is: if two electrons occupy the same orbital, two rows of the matrix are identical. A determinant with two identical rows is zero. So the same property that enforces sign-flip under exchange also makes double-occupancy states vanish entirely.
The elegance of the Slater determinant is that both antisymmetry and Pauli exclusion follow from the same mathematical object. No new physics is needed — just the properties of determinants applied to a matrix of single-particle orbitals.