Slater determinant

In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system that satisfies anti-symmetry requirements and consequently the Pauli principle by changing sign upon exchange of two electrons (or other fermions).^[1] It is named for John C. Slater, who introduced the determinants in 1929 as a means of ensuring the antisymmetry of a wave function.^[2] But actually the wave function in the determinant form first appeared three years earlier independently in Heisenberg's ^[3] and Dirac's ^[4] papers. The Slater determinant arises from the consideration of a wave function for a collection of electrons, each with a wave function known as the spin-orbital, $\chi (\mathbf {x} )$, where $\mathbf {x}$ denotes the position and spin of a single electron. A Slater determinant containing two electrons with the same spin orbital would correspond to a wave function which is zero everywhere.

Resolution

Two-particle case

The simplest way to approximate the wave function of a many-particle system is to take the product of properly chosen orthogonal wave functions of the individual particles. For the two-particle case with coordinates $\mathbf {x} _{1}$ and $\mathbf {x} _{2}$, we have

$\Psi (\mathbf {x} _{1},\mathbf {x} _{2})=\chi _{1}(\mathbf {x} _{1})\chi _{2}(\mathbf {x} _{2}).$

This expression is used in the Hartree–Fock method as an ansatz for the many-particle wave function and is known as a Hartree product. However, it is not satisfactory for fermions because the wave function above is not antisymmetric under exchange of any two of the fermions, as it must be according to the Pauli exclusion principle. An antisymmetric wave function can be mathematically described as follows:

$\Psi (\mathbf {x} _{1},\mathbf {x} _{2})=-\Psi (\mathbf {x} _{2},\mathbf {x} _{1})$

which does not hold for the Hartree product. Therefore the Hartree product does not satisfy the Pauli principle. This problem can be overcome by taking a linear combination of both Hartree products

$\Psi (\mathbf {x} _{1},\mathbf {x} _{2})={\frac {1}{\sqrt {2}}}\{\chi _{1}(\mathbf {x} _{1})\chi _{2}(\mathbf {x} _{2})-\chi _{1}(\mathbf {x} _{2})\chi _{2}(\mathbf {x} _{1})\}$

$={\frac {1}{\sqrt {2}}}{\begin{vmatrix}\chi _{1}(\mathbf {x} _{1})&\chi _{2}(\mathbf {x} _{1})\\\chi _{1}(\mathbf {x} _{2})&\chi _{2}(\mathbf {x} _{2})\end{vmatrix}}$

where the coefficient is the normalization factor. This wave function is now antisymmetric and no longer distinguishes between fermions, that is: one cannot indicate an ordinal number to a specific particle and the indices given are interchangeable. Moreover, it also goes to zero if any two spin orbitals of two fermions are the same. This is equivalent to satisfying the Pauli exclusion principle.

Generalizations

The expression can be generalised to any number of fermions by writing it as a determinant. For an N-electron system, the Slater determinant is defined as ^[1]

$\Psi (\mathbf {x} _{1},\mathbf {x} _{2},\ldots ,\mathbf {x} _{N})={\frac {1}{\sqrt {N!}}}\left|{\begin{matrix}\chi _{1}(\mathbf {x} _{1})&\chi _{2}(\mathbf {x} _{1})&\cdots &\chi _{N}(\mathbf {x} _{1})\\\chi _{1}(\mathbf {x} _{2})&\chi _{2}(\mathbf {x} _{2})&\cdots &\chi _{N}(\mathbf {x} _{2})\\\vdots &\vdots &\ddots &\vdots \\\chi _{1}(\mathbf {x} _{N})&\chi _{2}(\mathbf {x} _{N})&\cdots &\chi _{N}(\mathbf {x} _{N})\end{matrix}}\right|\equiv \left|{\begin{matrix}\chi _{1}&\chi _{2}&\cdots &\chi _{N}\\\end{matrix}}\right|,$

where in the final expression, a compact notation is introduced: the normalization constant and labels for the fermion coordinates are understood – only the wavefunctions are exhibited. The linear combination of Hartree products for the two-particle case can clearly be seen as identical with the Slater determinant for N = 2. It can be seen that the use of Slater determinants ensures an antisymmetrized function at the outset; other functions are automatically rejected. In the same way, the use of Slater determinants ensures conformity to the Pauli principle. Indeed, the Slater determinant vanishes if the set {χ_i } is linearly dependent. In particular, this is the case when two (or more) spin orbitals are the same. In chemistry one expresses this fact by stating that no two electrons with the same spin can occupy the same spatial orbital. In general the Slater determinant is evaluated by the Laplace expansion. Mathematically, a Slater determinant is an antisymmetric tensor, also known as a wedge product.

A single Slater determinant is used as an approximation to the electronic wavefunction in Hartree–Fock theory. In more accurate theories (such as configuration interaction and MCSCF), a linear combination of Slater determinants is needed.

Arguably, the Slater determinant is the simplest type of fermionic wave function. Not every fermionic wave function can be put in the form of a Slater determinant. The best Slater approximation to a given fermionic wave function is that which maximizes the overlap between the Slater determinant and the target wave function.^[5] The maximal overlap is a geometric measure of entanglement between the fermions.

The word "detor" was proposed by S. F. Boys to refer to a Slater determinant of orthonormal orbitals,^[6] but this term is rarely used.

Unlike fermions that are subject to the Pauli exclusion principle, two or more bosons can occupy the same single-particle quantum state. Wavefunctions describing systems of identical bosons are symmetric under the exchange of particles and can be expanded in terms of permanents.

See also

• Antisymmetrizer
• Electron orbital
• Fock space
• Quantum electrodynamics
• Quantum mechanics
• Physical chemistry
• Hund's rule
• Hartree–Fock method

References

1. Molecular Quantum Mechanics Parts I and II: An Introduction to QUANTUM CHEMISTRY (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0
2. Slater, J.; Verma, HC (1929). "The Theory of Complex Spectra". Physical Review. 34 (2): 1293–1322. Bibcode:1929PhRv...34.1293S. doi:10.1103/PhysRev.34.1293. PMID 9939750.
3. Heisenberg, W. (1926). "Mehrkörperproblem und Resonanz in der Quantenmechanik". Zeitschrift für Physik. 38: 411–426. Bibcode:1926ZPhy...38..411H. doi:10.1007/BF01397160.
4. Dirac, P. A. M. (1926). "On the Theory of Quantum Mechanics". Proceedings of the Royal Society, Series A. 112: 661–677. Bibcode:1926RSPSA.112..661D. doi:10.1098/rspa.1926.0133.
5. Zhang, J M; Kollar, Marcus (2014). "Optimal multiconfiguration approximation of an N-fermion wave function". Physical Review A. 89: 012504. arXiv:1309.1848. Bibcode:2014PhRvA..89a2504Z. doi:10.1103/PhysRevA.89.012504.
6. Boys, S. F. (1950). "Electronic wave functions I. A general method of calculation for the stationary states of any molecular system". Proceedings of the Royal Society. A200: 542.

External links

• Many-Electron States in E. Pavarini, E. Koch, and U. Schollwöck: Emergent Phenomena in Correlated Matter, Jülich 2013, ISBN 978-3-89336-884-6