# Slater determinant
A **Slater determinant** is a determinant of a matrix of one-electron spin-orbitals used to write an antisymmetric wave function for a system of identical fermions, most often the electrons of an atom or molecule. Each column of the matrix is one spin-orbital, each row holds the coordinates of one electron, and the prefactor \( 1/\sqrt{N!} \) normalizes the result for \( N \) orthonormal spin-orbitals[1](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/The_Live_Textbook_of_Physical_Chemistry_%28Peverati%29/25%3A_Many-Electron_Atoms/25.01%3A_Many-Electron_Wave_Functions) • [2](https://web.stanford.edu/~oas/SI/QM/Atkins05.pdf). It is named after [John C. Slater](https://www.edgepedia.org/john-c-slater), who introduced the form in his 1929 paper "The Theory of Complex Spectra" in *Physical Review*[3](https://journals.aps.org/pr/abstract/10.1103/PhysRev.34.1293).
| Key fact | Detail |
|---|---|
| Definition | \( \Psi(\mathbf{x}_1,\ldots,\mathbf{x}_N) = \frac{1}{\sqrt{N!}} \det[\chi_i(\mathbf{x}_j)] \), columns are spin-orbitals, rows are electron coordinates[1](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/The_Live_Textbook_of_Physical_Chemistry_%28Peverati%29/25%3A_Many-Electron_Atoms/25.01%3A_Many-Electron_Wave_Functions) |
| Normalization | \( 1/\sqrt{N!} \) is the correct factor for a single Slater determinant of \( n \) orthonormal spin-orbitals[2](https://web.stanford.edu/~oas/SI/QM/Atkins05.pdf) |
| Pauli exclusion | If any two spin-orbitals are identical, the determinant has two identical columns and is zero[4](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Physics_%28Ackland%29/09%3A_Indistinguishable_Particles_and_Exchange/9.06%3A_Wavefunction_for_many_spin_one-half_particles) |
| Origin | J. C. Slater, "The Theory of Complex Spectra", *Phys. Rev.* **34**, 1293, published 15 November 1929[3](https://journals.aps.org/pr/abstract/10.1103/PhysRev.34.1293) |
| Role in Hartree-Fock | Fock combined noninteracting-electron wavefunctions in a Slater determinant to satisfy antisymmetry, giving the Hartree-Fock method[5](https://www.ideals.illinois.edu/items/124992/bitstreams/410569/data.pdf) |
| Role in CI | Configuration interaction writes the wavefunction as a linear combination of Slater determinants; full CI grows factorially with molecular size[6](https://www.sciencedirect.com/science/chapter/bookseries/pii/S0065327608605328) |
| Scope | Strictly correct only for noninteracting fermions, but a useful starting point for electrons in atoms or metals[7](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/10%3A_Scattering_Theory/10.04%3A_Identical_Particles-_Symmetry_and_Scattering) |
## Definition and antisymmetry
For \( N \) fermions in one-particle states \( \chi_1, \ldots, \chi_N \), the wave function is
\[ \Psi(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_N) = \frac{1}{\sqrt{N!}} \begin{vmatrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2) \\ \vdots & \vdots & \ddots & \vdots \\ \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N) \end{vmatrix} \]
where each \( \mathbf{x} \) denotes both space and spin coordinates of one particle. A common shorthand writes only the diagonal, \( |\chi_1, \chi_2, \ldots, \chi_N\rangle \)[1](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/The_Live_Textbook_of_Physical_Chemistry_%28Peverati%29/25%3A_Many-Electron_Atoms/25.01%3A_Many-Electron_Wave_Functions). The determinant is exactly the signed sum over all \( N! \) permutations of the electrons among the states, with a factor of \( -1 \) for each exchange needed to reach a given ordering[7](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/10%3A_Scattering_Theory/10.04%3A_Identical_Particles-_Symmetry_and_Scattering).
The antisymmetry follows from a basic property of determinants: exchanging two rows multiplies the determinant by \( -1 \), so swapping the coordinates of any two electrons changes the sign of \( \Psi \)[7](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/10%3A_Scattering_Theory/10.04%3A_Identical_Particles-_Symmetry_and_Scattering). The same structure enforces the [Pauli exclusion principle](https://www.edgepedia.org/pauli-exclusion-principle): if two spin-orbitals are identical, the matrix has two identical columns, and a determinant with two identical columns is zero[4](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Physics_%28Ackland%29/09%3A_Indistinguishable_Particles_and_Exchange/9.06%3A_Wavefunction_for_many_spin_one-half_particles). A two-electron example makes the construction concrete; for helium, placing both electrons in the 1s orbital with opposite spins gives a \( 2 \times 2 \) determinant whose antisymmetric part is carried by the spin combination \( \frac{1}{\sqrt{2}}[\uparrow\downarrow - \downarrow\uparrow] \)[1](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/The_Live_Textbook_of_Physical_Chemistry_%28Peverati%29/25%3A_Many-Electron_Atoms/25.01%3A_Many-Electron_Wave_Functions).
## Origin
Slater introduced the determinantal form in "The Theory of Complex Spectra" (*Physical Review* **34**, 1293), published 15 November 1929 and received 7 June 1929, while he was at Jefferson Physical Laboratory, Harvard University[3](https://journals.aps.org/pr/abstract/10.1103/PhysRev.34.1293). The paper treated atomic multiplets by wave mechanics without group theory, deriving Hund's multiplet classification from theory and computing energy distances between multiplets[3](https://journals.aps.org/pr/abstract/10.1103/PhysRev.34.1293). In it Slater also introduced the F and G parameters, integrals describing the energies of states arising from a given configuration when interconfiguration interaction is neglected[8](https://www.abebooks.com/Theory-Complex-Spectra-Physical-Review-November/18716992452/bd). Contemporaries noted the race aspect of the work: Slater felt he had been outpaced by Dirac, who had published a related result earlier[9](https://wucj.lab.westlake.edu.cn/Others/classic-literature-reading/Between_Accuracy_and_Manageability.pdf).
## Role in Hartree-Fock theory
The Hartree method approximates each electron as moving in the average field of the others, but its product wave function does not satisfy the antisymmetry required of electrons[6](https://www.sciencedirect.com/science/chapter/bookseries/pii/S0065327608605328). Fock resolved this by using a single Slater determinant built from one-particle wave functions, the molecular orbitals, as the variational ansatz, which produces the Hartree-Fock method[5](https://www.ideals.illinois.edu/items/124992/bitstreams/410569/data.pdf) • [10](https://pmc.ncbi.nlm.nih.gov/articles/PMC12645584/). Antisymmetry supplies exchange but not all correlation[11](https://quantumchemistry.net/en/learn/molecular-methods/hartree-fock/).
A useful property of the single-determinant ansatz is invariance under unitary rotation of the occupied orbitals: such rotations leave the determinant's density and total energy unchanged apart from an irrelevant phase, so localized and canonical occupied orbitals describe the same Hartree-[Fock state](https://www.edgepedia.org/fock-state)[11](https://quantumchemistry.net/en/learn/molecular-methods/hartree-fock/).
## Beyond a single determinant
Because a single determinant is exact only for noninteracting fermions, accurate work on real electrons requires electron correlation beyond the Hartree-Fock picture[7](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/10%3A_Scattering_Theory/10.04%3A_Identical_Particles-_Symmetry_and_Scattering). The most broadly applicable method for this is configuration interaction (CI), which expresses the wave function as a linear combination of \( \widehat{S}_z \) eigenfunctions, that is, Slater determinants, or of configuration state functions (CSFs), which are simultaneously eigenfunctions of \( \widehat{S}_z \) and \( \widehat{S}^2 \)[6](https://www.sciencedirect.com/science/chapter/bookseries/pii/S0065327608605328). If all possible \( N \)-electron functions are included, subject to symmetry restrictions, the [Schrödinger equation](https://www.edgepedia.org/schrodinger-equation) is solved exactly within the one-particle basis; but the dimension of this full CI grows factorially with molecular size, so in practice only the most important determinants are selected[6](https://www.sciencedirect.com/science/chapter/bookseries/pii/S0065327608605328).
Matrix elements between determinants are evaluated by the Slater-Condon rules, a standard tool of computational chemistry[12](https://academic.oup.com/book/27876/chapter-abstract/203846747). Work by several groups since 1980 has favored determinants over spin-adapted CSFs because coupling coefficients are easier to compute for determinants[6](https://www.sciencedirect.com/science/chapter/bookseries/pii/S0065327608605328).
## References
1. [Many-Electron Wave Functions, Chemistry LibreTexts (R. Peverati)](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/The_Live_Textbook_of_Physical_Chemistry_%28Peverati%29/25%3A_Many-Electron_Atoms/25.01%3A_Many-Electron_Wave_Functions)
2. [Molecular Quantum Mechanics (Atkins, problem 9.2), Stanford course PDF](https://web.stanford.edu/~oas/SI/QM/Atkins05.pdf)
3. [J. C. Slater (1929). The Theory of Complex Spectra. Phys. Rev. 34, 1293](https://journals.aps.org/pr/abstract/10.1103/PhysRev.34.1293)
4. [Wavefunction for many spin one-half particles, Physics LibreTexts (G. Ackland)](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Physics_%28Ackland%29/09%3A_Indistinguishable_Particles_and_Exchange/9.06%3A_Wavefunction_for_many_spin_one-half_particles)
5. [Thesis discussing Fock's use of the Slater determinant, University of Illinois](https://www.ideals.illinois.edu/items/124992/bitstreams/410569/data.pdf)
6. [The Configuration Interaction Method: Advances in Highly Correlated Approaches, Advances in Quantum Chemistry (ScienceDirect)](https://www.sciencedirect.com/science/chapter/bookseries/pii/S0065327608605328)
7. [Identical Particles: Symmetry and Scattering, Physics LibreTexts (M. Fowler)](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/10%3A_Scattering_Theory/10.04%3A_Identical_Particles-_Symmetry_and_Scattering)
8. [First edition listing of The Theory of Complex Spectra with historical note, Atticus Rare Books](https://www.abebooks.com/Theory-Complex-Spectra-Physical-Review-November/18716992452/bd)
9. [Between Accuracy and Manageability (historical essay on Slater), Westlake University](https://wucj.lab.westlake.edu.cn/Others/classic-literature-reading/Between_Accuracy_and_Manageability.pdf)
10. [Quantum chemistry: from the first steps to linear-scaling electronic structure methods, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC12645584/)
11. [Hartree-Fock from constrained variation, Quantum Chemistry Network](https://quantumchemistry.net/en/learn/molecular-methods/hartree-fock/)
12. [Short Review of Electronic Structure Methods, Oxford Academic](https://academic.oup.com/book/27876/chapter-abstract/203846747)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum formalism and states › Quantum states and wave functions › Wave functions and position-space states › Many-particle wave functions and exchange symmetry*
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