nLab Slater determinant



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In quantum physics and especially in condensed matter theory, Slater determinants are certain wavefunctions expressing the joint quantum state of multiple electrons (or possiby other fermions) as skew-symmetrized products of given single-particle wavefunctions.

Concretely, given a linear basis

(ψ I) I=1 \big( \psi_{I} \big)_{I = 1}^\infty

for the Hilbert space of single-electron wavefunctions, being square-integrable functions on the ambient Euclidean d d -space

ψ iL 2( d;). \psi_i \;\in\; L^2\big( \mathbb{R}^d; \mathbb{C}\big) \,.

a Slater determinant for NN \in \mathbb{N} particles is a function on the NN-fold product space ( d) N\big( \mathbb{R}^d\big)^N of the following determinant-form

Ψ (x 1,,x N)det(ψ I i(x j))=σSym(N)sgn(σ)ψ I 1(x σ(1))ψ I 2(x σ(2))ψ I N(x σ(N)), \Psi_{\mathcal{I}} (\vec x_1, \cdots, \vec x_N) \;\coloneqq\; det \big( \psi_{I_i}(\vec x_j) \big) \;=\; \underset{ \sigma \in Sym(N) }{\sum} sgn(\sigma) \cdot \psi_{I_1}(\vec x_{\sigma(1)}) \cdot \psi_{I_2}(\vec x_{\sigma(2)}) \cdots \psi_{I_N}(\vec x_{\sigma(N)}) \,,

where =(I 1,,I N)\mathcal{I} = \big( I_1, \cdots, I_N\big) is an N N -tuple of indices, with (ψ I 1,ψ I N)\big( \psi_{I_1}, \cdots \psi_{I_N} \big) the corresponding N N -tuple of 1-electron wavefunctions.

(Here Sym(N)Sym(N) denotes the symmetric group of permutations of NN ordered elements, and sgn(σ){±1}sgn(\sigma) \in \{\pm 1\} denotes the signature of a given permutation σ\sigma.)

In fact, for actual electrons the wavefunctions are also functions of their spin, which means, in the non-relativistic case, that the ψ I\psi_I depend also on an argument in {,}\{\uparrow, \downarrow\}, in addition to their dependence on x\vec x, and the corresponding Slater determinant states are obtained by skew-symmetrizing over all of these degrees of freedom.

The point of this construction is that it enforces the skew-symmetry under permutation of position of electrons, which is their characteristic property as fermions. As the multi-index set \mathcal{I} ranges, the corresponding Slater determinants span the Hilbert space of NN-electron quantum states.

In the practice of computing ground states etc. in solid state physics, one tries to use as few multi-indices \mathcal{I} as possible:

In the extreme case, the Hartree-Fock method tries to approximate a multi-electron system by the clever choice of a single Slater determinant. More accurate approximation methods use linear combinations of more and more Slater determinants, as the multi-index set \mathcal{I} ranges. If, in principle, the full space of Slater determinants is used, one speaks of the configuration interaction method.


The construction was maybe first made explicit as eq. (15) in

It is named after:


  • Attila Szabo, Neil S. Ostlund, Sec. 2.2.3 of: Modern Quantum Chemistry – Introduction to Advanced Electronic Structure Theory, Macmillan (1982), McGraw-Hill (1989), Dover (1996) [[pdf]]

  • C. Lanczos, R. C. Clark, G. H. Derrick (eds.), p. 196 in: Mathematical Methods in Solid State and Superfluid Theory, Springer (1986) [[doi:10.1007/978-1-4899-6435-9]]

  • Pablo Echenique, J. L. Alonso, around (33) in: A mathematical and computational review of Hartree-Fock SCF methods in Quantum Chemistry, Molecular Physics 105 (2007) 3057-3098 [[doi:10.1080/00268970701757875]]

See also:

Last revised on May 13, 2022 at 08:12:55. See the history of this page for a list of all contributions to it.