Contents

# Contents

## 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.

## 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:

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$]$