nLab Slater determinant

Idea

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

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

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

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

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

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

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

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

In fact, for actual electrons the wavefunctions are also functions of their spin, which means, in the non-relativistic case, that the $\psi_I$ depend also on an argument in $\{\uparrow, \downarrow\}$, in addition to their dependence on $\vec x$, 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 $N$-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.

Related concepts

• Pauli exclusion principle

References

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

• Paul A. M. Dirac, On the theory of quantum mechanics, Proceedings of the Royal Society 112 762 (1926) $[$di:10.1098/rspa.1926.0133$]$

It is named after:

• John C. Slater, The Theory of Complex Spectra, Phys. Rev. 34 (1929) 1293 $[$doi:10.1103/PhysRev.34.1293$]$

Review:

• 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:

• Wikipedia, Slater determinant