Slater determinant

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The Slater determinant (after John C. Slater ) is a method in quantum mechanics for the construction of a wave function for systems consisting of fermions . The resulting wave function then corresponds to the requirements of the Pauli principle and changes its sign when two identical fermions are interchanged. The construction uses the properties of the determinant of a matrix , which also changes the sign when rows or columns are swapped. The resulting wave function is also often referred to as the Slater determinant and z. B. used in quantum chemistry to describe electrons in a molecule .

The construction uses single-particle wave functions (so-called orbitals ), each of which is entered in a column of the underlying matrix. The coordinates of the fermions are then entered line by line as the argument of the functions. The multi-particle wave function that arises when the determinant is solved is then a sum of products of single-particle wave functions and has the property of antisymmetry , which is necessary for the Pauli principle, against the interchange of two indistinguishable fermions.

motivation

In order to solve the Schrödinger equation , it is necessary to find suitable wave functions for the system under consideration. A possible approach of a wave function for N-particle systems, such as B. the electrons in an atom or molecule , is to assume them as the product of wave functions of the individual particles with the coordinates : ${\ displaystyle \ Psi}$ ${\ displaystyle \ chi _ {i} ({\ vec {x}} _ {i})}$ ${\ displaystyle {\ vec {x}} _ {i}}$

${\ displaystyle \ Psi ^ {\ mathrm {approach}} = \ prod _ {i} ^ {N} \ chi _ {i} ({\ vec {x}} _ {i})}$

This approach is also called the "Hartree product". Such a function is not antisymmetric with respect to the exchange of the particles. However, it is possible to generate a wave function as a combination of several Hartree products with interchanged particles.

Example with two particles

Fictitious Slater determinant for two electrons moving on a line between two atoms.

Effects on a Hartree product and a Slater determinant when two particles are exchanged.

The Hartree product with two electrons in two orbitals is and does not meet the requirement . The combination ${\ displaystyle \ Psi ^ {\ mathrm {HP}} ({\ vec {x}} _ {1}, {\ vec {x}} _ {2}) = \ chi _ {1} ({\ vec { x}} _ {1}) \ chi _ {2} ({\ vec {x}} _ {2})}$ ${\ displaystyle \ Psi ^ {\ mathrm {HP}} ({\ vec {x}} _ {1}, {\ vec {x}} _ {2}) = - \ Psi ^ {\ mathrm {HP}} ({\ vec {x}} _ {2}, {\ vec {x}} _ {1})}$

${\ displaystyle {\ begin {aligned} \ Psi ({\ vec {x}} _ {1}, {\ vec {x}} _ {2}) & = {\ frac {1} {\ sqrt {2} }} \ {\ chi _ {1} ({\ vec {x}} _ {1}) \ chi _ {2} ({\ vec {x}} _ {2}) - \ chi _ {1} ( {\ vec {x}} _ {2}) \ chi _ {2} ({\ vec {x}} _ {1}) \} \\ & = {\ frac {1} {\ sqrt {2}} } {\ begin {vmatrix} \ chi _ {1} ({\ vec {x}} _ {1}) & \ chi _ {2} ({\ vec {x}} _ {1}) \\\ chi _ {1} ({\ vec {x}} _ {2}) & \ chi _ {2} ({\ vec {x}} _ {2}) \ end {vmatrix}} \\\ end {aligned} }}$

however, yes. In fact, the function becomes zero when the coordinates of the particles are the same ( ), which fulfills a further requirement of the Pauli principle - that two fermions cannot be described by the same wave function. The prefactor is used for normalization , which is required from the formalism of quantum mechanics for wave functions. ${\ displaystyle {\ vec {x}} _ {1} = {\ vec {x}} _ {2}}$ ${\ displaystyle {\ tfrac {1} {\ sqrt {2}}}}$

It is assumed that two electrons move in only one dimension in a system of two atoms and the position vectors in denote the positions of the electrons on the straight line that connects both atoms. The orbitals are each a normal distribution with one of the atoms in the center. The Hartree product of the two orbitals only has a value significantly different from zero if the electrons are each close to their atoms. The associated Slater determinant also has an amplitude when the two electrons are exchanged - in fact, it then has exactly the negative value. ${\ displaystyle {\ vec {x}} _ {i}}$ ${\ displaystyle \ chi _ {i}}$

The construction as a determinant in this form always creates a permissible wave function, even for more than two electrons. In order to reduce the writing effort, often only the diagonal elements of the determinant are given, the normalization factor is omitted and only either the orbitals or - instead of the coordinates - the indices of the particles are written. The Slater determinant given above could thus e.g. B. written using Dirac notation as

${\ displaystyle | \ Psi \ rangle \ equiv | \ chi _ {1} \ chi _ {2} \ rangle \ equiv | 12 \ rangle}$

Derivation with the antisymmetrization operator

The wave function (eigenfunction of the many-body Hamiltonian) is a product of normalized eigenfunctions of the (interaction-free) one-particle Hamiltonian operator: ${\ displaystyle \ phi}$

{\ displaystyle \ psi \ left (1,2, \ dotsc, N \ right) = A_ {N} \ phi _ {1} (1) \ phi _ {2} (2) \ dots \ phi _ {N} (N)}

The function argument corresponds to the atomic number of the respective electron , e.g. B. . To fulfill the Pauli principle, the antisymmetrization operator is added, i.e. H.: ${\ displaystyle \ phi (i) \ equiv \ phi ({\ vec {r}} _ {i})}$ ${\ displaystyle A_ {N}}$

{\ displaystyle \ psi (1,2, \ dotsc, j, \ dotsc, k, \ dotsc, N) \ equiv - \ psi (1,2, \ dotsc, k, \ dotsc, j, \ dotsc, N) }

Result

The Slater determinant can be written as follows:

{\ displaystyle A_ {N} [\ phi _ {1} (1) \ phi _ {2} (2) \ dots \ phi _ {N} (N)] = {\ frac {1} {\ sqrt {N !}}} {\ begin {vmatrix} \ phi _ {1} (1) & \ phi _ {2} (1) & \ dots & \ phi _ {N} (1) \\\ phi _ {1} (2) & \ phi _ {2} (2) & \ dots & \ phi _ {N} (2) \\\ vdots & \ vdots & \ ddots & \ vdots \\\ phi _ {1} (N) & \ phi _ {2} (N) & \ dots & \ phi _ {N} (N) \ end {vmatrix}} = | \ phi _ {1} \ phi _ {2} \ dots \ phi _ {N } |}

This now includes all combinations. The normalization of the wave function is ensured by the factorial in the denominator. As already mentioned above, the antisymmetry under particle interchange is automatically fulfilled by the realization as a determinant.

For interaction-free many-particle systems this is an eigenstate of the Hamilton operator. This can no longer be assumed for interacting systems.

literature

Attila Szabo, Neil S. Ostlund: Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory . Courier Corporation, 1996, ISBN 978-0-486-69186-2 , pp. 50 ff .
H. Friedrich: Theoretical atomic physics . 2nd Edition. Springer Verlag, Berlin-Heidelberg 1994, ISBN 978-3-540-58267-0 .
T. Fließbach: Quantum Mechanics: Textbook on Theoretical Physics III . 5th edition. Spectrum Akademischer Verlag, Heidelberg 2008, ISBN 978-3-8274-2020-6 .

Individual evidence

^ ^A ^b Attila Szabo, Neil S. Ostlund: Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory . Courier Corporation, 1996, ISBN 978-0-486-69186-2 , pp. 50 .
↑ Peter W. Atkins: Quantum: Terms and Concepts for Chemists . VCH, ISBN 978-0-486-69186-2 .

[Szabo-1] A ^b Attila Szabo, Neil S. Ostlund: Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory . Courier Corporation, 1996, ISBN 978-0-486-69186-2 , pp. 50 .

[2] Peter W. Atkins: Quantum: Terms and Concepts for Chemists . VCH, ISBN 978-0-486-69186-2 .