In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electrons (or other fermions).[1] Only a small subset of all possible fermionic wave functions can be written as a single Slater determinant, but those form an important and useful subset because of their simplicity.
\chi(x)
x
The Slater determinant is named for John C. Slater, who introduced the determinant in 1929 as a means of ensuring the antisymmetry of a many-electron wave function,[2] although the wave function in the determinant form first appeared independently in Heisenberg's[3] and Dirac's[4] [5] articles three years earlier.
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
x1
x2
\Psi(x1,x2)=\chi1(x1)\chi2(x2).
This expression is used in the Hartree 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(x1,x2)=-\Psi(x2,x1).
This does not hold for the Hartree product, which therefore does not satisfy the Pauli principle. This problem can be overcome by taking a linear combination of both Hartree products:
\begin{aligned} \Psi(x1,x2)&=
| 1 | |
| \sqrt{2 |
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.
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[6]
\begin{aligned} \Psi(x1,x2,\ldots,xN)&=
| 1 | |
| \sqrt{N! |
where the last two expressions use a shorthand for Slater determinants: The normalization constant is implied by noting the number N, and only the one-particle wavefunctions (first shorthand) or the indices for the fermion coordinates (second shorthand) are written down. All skipped labels are implied to behave in ascending sequence. The linear combination of Hartree products for the two-particle case is identical with the Slater determinant for N = 2. The use of Slater determinants ensures an antisymmetrized function at the outset. In the same way, the use of Slater determinants ensures conformity to the Pauli principle. Indeed, the Slater determinant vanishes if the set
\{\chii\}
See also: Slater-Condon rules. Many properties of the Slater determinant come to life with an example in a non-relativistic many electron problem.[7]
Starting from a molecular Hamiltonian:
ri
RI
Vnucl(r)=-\sumI
| ZIe2 | |
| |r-RI| |
For simplicity we freeze the nuclei at equilibrium in one position and we remain with a simplified Hamiltonian
\hat{H}e=
| N | |
| \sum | |
| i |
\hat{h}(ri)+
| 1 | |
| 2 |
| N | |
| \sum | |
| i\nej |
| e2 | |
| rij |
\hat{h}(r)=
| \hat{p | |
| 2}{2m} |
+Vnucl(r)
and where we will distinguish in the Hamiltonian between the first set of terms as
\hat{G}1
\hat{G}2
\hat{G}1
| N | |
| =\sum | |
| i |
\hat{h}(ri)
\hat{G}2=
| 1 | |
| 2 |
| N | |
| \sum | |
| i\nej |
| e2 | |
| rij |
The two parts will behave differently when they have to interact with a Slater determinant wave function. We start to compute the expectation values of one-particle terms
\langle\Psi0|G1|\Psi0\rangle=
| 1 | |
| N! |
\langle\det\{\psi1...\psiN\}|G1|\det\{\psi1...\psiN\}\rangle
\langle\Psi0|G1|\Psi0\rangle=\langle\psi1...\psiN|G1|\det\{\psi1...\psiN\}\rangle
\langle\Psi0|G1|\Psi0\rangle=\langle\psi1...\psiN|G1|\psi1...\psiN\rangle
And finally we remain with the trace over the one-particle Hamiltonians
\langle\Psi0|G1|\Psi0\rangle=\sumi\langle\psii|h|\psii\rangle
For the two-particle terms instead
\langle\Psi0|G2|\Psi0\rangle=
| 1 | |
| N! |
\langle\det\{\psi1...\psiN\}|G2|\det\{\psi1...\psiN\}\rangle=\langle\psi1...\psiN|G2|\det\{\psi1...\psiN\}\rangle
If we focus on the action of one term of
G2
\langle\psi1(r1,\sigma1)...\psiN(rN,\sigmaN)|
| e2 | |
| r12 |
|det\{\psi1(r1,\sigma1)...\psiN(rN,\sigmaN)\}\rangle=\langle\psi1\psi
|
|\psi1\psi2\rangle-\langle\psi1\psi
|
|\psi2\psi1\rangle
And finally
which instead is a mixing term. The first contribution is called the "coulomb" term or "coulomb" integral and the second is the "exchange" term or exchange integral. Sometimes different range of index in the summation is used since the Coulomb and exchange contributions exactly cancel each other for
i=j
It is important to notice explicitly that the exchange term, which is always positive for local spin-orbitals,[8] is absent in simple Hartree product. Hence the electron-electron repulsive energy
\langle\Psi0|G2|\Psi0\rangle
Most fermionic wavefunctions cannot be represented as a Slater determinant. The best Slater approximation to a given fermionic wave function can be defined to be the one that maximizes the overlap between the Slater determinant and the target wave function.[9] The maximal overlap is a geometric measure of entanglement between the fermions.
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.
The word "detor" was proposed by S. F. Boys to refer to a Slater determinant of orthonormal orbitals,[10] 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.