# What is a Slater determinant

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

A Slater determinant is a representation of a many-particle wave function for a system of fermions, which satisfies the anti-symmetry requirement. In other words, that the wave function changes sign on interchange of two particle coordinates (e.g. $$\Psi(\mathbf{x}_1, \mathbf{x}_2) = -\Psi(\mathbf{x}_2, \mathbf{x}_1)$$, letting spin be regarded as another coordinate for the sake of brevity)

Equations

A Slater determinant for a system of N particles takes the form:
$$\Psi(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_N) = \frac{1}{\sqrt{N!}} \left| \begin{matrix} \psi_1(\mathbf{x}_1) & \psi_2(\mathbf{x}_1) & \cdots & \psi_N(\mathbf{x}_1) \\ \psi_1(\mathbf{x}_2) & \psi_2(\mathbf{x}_2) & \cdots & \psi_N(\mathbf{x}_2) \\ \vdots & \vdots && \vdots \\ \psi_1(\mathbf{x}_N) & \psi_2(\mathbf{x}_N) & \cdots & \psi_N(\mathbf{x}_N) \end{matrix} \right|$$

Where $$\psi_i$$ are normalized single-particle wave functions for each respective particle. The exchange of any pair of coordinates or any pair of single-particle wave functions has the effect of exchanging two rows or columns respectively, which leads to a change of sign of the determinant, satisfying anti-symmetry. Also, if any coordinates or single-particle wave functions are identical, two rows or columns will be equal, and the determinant will be identically zero. This consequence of the antisymmetry requirement being the well-known Pauli Principle.

Extended explanation

The Hamiltonian for a many-particle wave function is:
$$\hat{H} = \sum_i^N \frac{\hbar^2}{2 m_i} \nabla_i^2 + V(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_N)$$

If there is no potential $V$ which depends on the coordinates of several particles, the Schrödinger equation is separable into single-particle equations and the resulting wave function written as a product of the single-particle solutions:
$$\Psi(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_N) = \psi_1(\mathbf{x}_1)\psi_2(\mathbf{x}_2)\ldots\psi_N(\mathbf{x}_N),$$ termed a Hartree Product.
(For an electronic system the single-particle solutions are orbitals and will henceforth be referred to as such for brevity, even though the theory applies equally to other fermions)

However, a Hartree Product does not satisfy the anti-symmetry requirement for fermions. For instance, both particles may be in the same orbital, such that $$\Psi(\mathbf{x}_1, \mathbf{x}_2) = \psi(\mathbf{x}_1) \psi(\mathbf{x}_2)$$, which obviously does not change sign on an interchange of coordinates. This also violates the Pauli principle, since two electrons may not exist in the same orbital (when spin is included)

Due to the superposition principle, any linear combination of products of orbitals (with exchanged coordinates) will constitute a solution to the many-particle equation with the same energy. Therefore, we can satisfy the antisymmetry requirement by assuming the solution takes the form of an antisymmetrized product.
For instance: $$\Psi(\mathbf{x}_1, \mathbf{x}_2) = \frac{1}{\sqrt{2}} (\psi_1(\mathbf{x}_1) \psi_2(\mathbf{x}_2) - \psi_1(\mathbf{x}_2)\psi_2(\mathbf{x}_1))$$
which is easily shown to satisfy the antisymmetry requirement, as well as the Pauli principle. These antisymmetrized products were introduced by Slater[1], who showed that they could be conveniently expressed in the determinant form shown above.

Note that since there are many orbitals, a single Slater determinant only represents a single eigenstate of the overall system. A system is thus described by the set of all possible Slater determinants. As eigenstates of the Hamiltonian, these form a compete set.

The interacting case
Most systems of interest, such as the electronic states of atoms and molecules, are interacting and therefore non-separable due to the electron-electron repulsion term in the Hamiltonian.

However, this does not mean that Slater determinants cannot be used to represent the system. As a first approximation, one may describe a system in the ground state in terms of a single Slater determinant of the N lowest orbitals. If this ansatz is plugged into the time-independent electronic Schrödinger equation, one arrives at the Hartree-Fock equations, where each electron "moves" in an averaged field of every other electron. These may be solved variationally using the Self-Consistent-Field method. For this reason the Hartree-Fock approximation is sometimes referred to as the "single-determinant approximation" or the "mean-field approximation". The Hartree-Fock method is the starting point of most quantum chemical methods.

The Hartree-Fock method is exact (in the basis-set limit) for a single determinant of non-interacting (or 'mean-field interacting') electrons. Since the individual Slater determinants form a complete set, they may be used as a convenient basis for calculating the fully-interacting system. The interacting system may be expressed as a linear combination of Slater determinants for the various possible electronic configurations, and solved variationally using SCF. This is referred to as the Configuration Interaction (CI) method. Besides its conceptual simplicity, it has the feature that it may easily and systematically be improved by adding more determinants.

It was shown by Löwdin[2] and independently by Foldy[3] that any antisymmetric function of $n$ variables, which is an eigenfunction of a Hermitian operator, can be expressed by forming the $$\frac{m!}{(m - n)!n!}$$ Slater determinants that may be formed from $m$ single-variable functions. This means that the the CI method is exact in principle (one may simply add determinants until some convergence limit is reached, termed Full-CI). However, it severely limits the usefulness of this method for large systems, as it scales factorially.

In 1965, Kohn and Sham[4] suggested the introduction of Slater determinants into Density Functional Theory when they introduced the concept of a non-interacting 'reference system' of electrons, leaving correlation and exchange energy to be described by the density functional. It was not until later (e.g. [5]) that Slater determinants were introduced into DFT, eventually leading to the utilization of the resulting "exact exchange" for so-called hybrid functionals[6], which has been an extremely successful approach.

Slater determinants are thus utilized in most quantum chemical methods today, both for wave function based methods and density functional methods.

[1] Slater, "The Theory of Complex Spectra", Phys. Rev. 34, 1293 (1929)
[2] Löwdin, "Quantum Theory of Many-Particle Systems I-III", Phys. Rev. 97, 1474, 1490, 1509 (1955)
[3] Foldy, "Antisymmetric Functions and Slater Determinants", J. Math. Phys. 3, 531 (1962)
[4] Kohn, Sham, "Self-Consistent Equations Including Exchange and Correlation Effects", Phys. Rev. 140, A1133 (1965)
[5] Levy, "Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem", PNAS 76(12), 6062 (1979)
[6] Becke, "Density-functional thermochemistry. III. The role of exact exchange", J. Chem. Phys. 98 (7), 5648, (1993)

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