Slater determinant

Last updated November 22, 2023

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.

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 that is zero everywhere.

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.

Definition

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 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}).

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 (\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}},\end{aligned}}

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.

Multi-particle case

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]^[6]

{\begin{aligned}\Psi (\mathbf {x} _{1},\mathbf {x} _{2},\ldots ,\mathbf {x} _{N})&={\frac {1}{\sqrt {N!}}}{\begin{vmatrix}\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{vmatrix}}\\&\equiv |\chi _{1},\chi _{2},\cdots ,\chi _{N}\rangle \\&\equiv |1,2,\dots ,N\rangle ,\end{aligned}}

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 $\{\chi _{i}\}$ 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.

Example: Matrix elements in a many electron problem

Many properties of the Slater determinant come to life with an example in a non-relativistic many electron problem.^[7]

The one particle terms of the Hamiltonian will contribute in the same manner as for the simple Hartree product, namely the energy is summed and the states are independent
The multi-particle terms of the Hamiltonian, i.e. the exchange terms, will introduce a lowering of the energy of the eigenstates

Starting from an Hamiltonian:

{\hat {H}}_{\text{tot}}=\sum _{i}{\frac {\mathbf {p} _{i}^{2}}{2m}}+\sum _{I}{\frac {\mathbf {P} _{I}^{2}}{2M_{I}}}+\sum _{i}V_{\text{nucl}}(\mathbf {r_{i}} )+{\frac {1}{2}}\sum _{i\neq j}{\frac {e^{2}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}+{\frac {1}{2}}\sum _{I\neq J}{\frac {Z_{I}Z_{J}e^{2}}{|\mathbf {R} _{I}-\mathbf {R} _{J}|}}

where $\mathbf {r} _{i}$ are the electrons and $\mathbf {R} _{I}$ are the nuclei and

V_{\text{nucl}}(\mathbf {r} )=-\sum _{I}{\frac {Z_{I}e^{2}}{|\mathbf {r} -\mathbf {R} _{I}|}}

For simplicity we freeze the nuclei at equilibrium in one position and we remain with a simplified Hamiltonian

{\hat {H}}_{e}=\sum _{i}^{N}{\hat {h}}(\mathbf {r} _{i})+{\frac {1}{2}}\sum _{i\neq j}^{N}{\frac {e^{2}}{r_{ij}}}

where

{\hat {h}}(\mathbf {r} )={\frac {{\hat {\mathbf {p} }}^{2}}{2m}}+V_{\text{nucl}}(\mathbf {r} )

and where we will distinguish in the Hamiltonian between the first set of terms as ${\hat {G}}_{1}$ (the "1" particle terms) and the last term ${\hat {G}}_{2}$ which is the "2" particle term or exchange term

{\hat {G}}_{1}=\sum _{i}^{N}{\hat {h}}(\mathbf {r} _{i})

{\hat {G}}_{2}={\frac {1}{2}}\sum _{i\neq j}^{N}{\frac {e^{2}}{r_{ij}}}

The two parts will behave differently when they have to interact with a Slater determinant wave function. We start to compute the expectation values

\langle \Psi _{0}|G_{1}|\Psi _{0}\rangle ={\frac {1}{N!}}\langle \det\{\psi _{1}...\psi _{N}\}|G_{1}|\det\{\psi _{1}...\psi _{N}\}\rangle

In the above expression, we can just select the identical permutation in the determinant in the left part, since all the other N! − 1 permutations would give the same result as the selected one. We can thus cancel N! at the denominator

\langle \Psi _{0}|G_{1}|\Psi _{0}\rangle =\langle \psi _{1}...\psi _{N}|G_{1}|\det\{\psi _{1}...\psi _{N}\}\rangle

Because of the orthonormality of spin-orbitals it is also evident that only the identical permutation survives in the determinant on the right part of the above matrix element

\langle \Psi _{0}|G_{1}|\Psi _{0}\rangle =\langle \psi _{1}...\psi _{N}|G_{1}|\psi _{1}...\psi _{N}\rangle

This result shows that the anti-symmetrization of the product does not have any effect for the one particle terms and it behaves as it would do in the case of the simple Hartree product.

And finally we remain with the trace over the one particle Hamiltonians

\langle \Psi _{0}|G_{1}|\Psi _{0}\rangle =\sum _{i}\langle \psi _{i}|h|\psi _{i}\rangle

Which tells us that to the extent of the one particle terms the wave functions of the electrons are independent of each other and the energy is given by the sum of energies of the single particles.

For the exchange part instead

\langle \Psi _{0}|G_{2}|\Psi _{0}\rangle ={\frac {1}{N!}}\langle \det\{\psi _{1}...\psi _{N}\}|G_{2}|\det\{\psi _{1}...\psi _{N}\}\rangle =\langle \psi _{1}...\psi _{N}|G_{2}|\det\{\psi _{1}...\psi _{N}\}\rangle

If we see the action of one exchange term it will select only the exchanged wavefunctions

\langle \psi _{1}(r_{1},\sigma _{1})...\psi _{N}(r_{N},\sigma _{N})|{\frac {e^{2}}{r_{12}}}|\mathrm {det} \{\psi _{1}(r_{1},\sigma _{1})...\psi _{N}(r_{N},\sigma _{N})\}\rangle =\langle \psi _{1}\psi _{2}|{\frac {e^{2}}{r_{12}}}|\psi _{1}\psi _{2}\rangle -\langle \psi _{1}\psi _{2}|{\frac {e^{2}}{r_{12}}}|\psi _{2}\psi _{1}\rangle

And finally

\langle \Psi _{0}|G_{2}|\Psi _{0}\rangle ={\frac {1}{2}}\sum _{ij}\left[\langle \psi _{i}\psi _{j}|{\frac {e^{2}}{r_{ij}}}|\psi _{i}\psi _{j}\rangle -\langle \psi _{i}\psi _{j}|{\frac {e^{2}}{r_{ij}}}|\psi _{j}\psi _{i}\rangle \right]

which instead is a mixing term, the first contribution is called the "coulomb" term and the second is the "exchange" term which can be written using ${\textstyle \sum _{ij}}$ or ${\textstyle \sum _{i\neq j}}$ , since the Coulomb and exchange contributions exactly cancel each other for $i=j$ .

It is important to notice explicitly that the electron-electron repulsive energy $\langle \Psi _{0}|G_{2}|\Psi _{0}\rangle$ on the antisymmetrized product of spin-orbitals is always lower than the electron-electron repulsive energy on the simple Hartree product of the same spin-orbitals. The difference is just represented by the second term in the right-hand side without the self-interaction terms $i=j$ . Since exchange bielectronic integrals are positive quantities, different from zero only for spin-orbitals with parallel spins, we link the decrease in energy with the physical fact that electrons with parallel spin are kept apart in real space in Slater determinant states.

As an approximation

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.^[8] 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.

Discussion

The word "detor" was proposed by S. F. Boys to refer to a Slater determinant of orthonormal orbitals,^[9] 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.

Related Research Articles

In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's energy spectrum or its set of energy eigenvalues, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known.

The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best-known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and electrons in a molecule can be treated separately, based on the fact that the nuclei are much heavier than the electrons. Due to the larger relative mass of a nucleus compared to an electron, the coordinates of the nuclei in a system are approximated as fixed, while the coordinates of the electrons are dynamic. The approach is named after Max Born and his 23-year-old graduate student J. Robert Oppenheimer, the latter of whom proposed it in 1927 during a period of intense ferment in the development of quantum mechanics.

<span class="mw-page-title-main">Wave function</span> Mathematical description of the quantum state of a system

In quantum physics, a wave function, is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters $ψ$ and $Ψ$ . Wave functions are composed of complex numbers. For example, a wave function might assign a complex number to each point in a region of space. The Born rule provides the means to turn these complex probability amplitudes into actual probabilities. In one common form, it says that the squared modulus of a wave function that depends upon position is the probability density of measuring a particle as being at a given place. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called normalization. Since the wave function is complex-valued, only its relative phase and relative magnitude can be measured; its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables. One has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function $ψ$ and calculate the statistical distributions for measurable quantities.

In physics, an operator is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry. Because of this, they are useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.

In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves modulated by periodic functions. The theorem is named after the physicist Felix Bloch, who discovered the theorem in 1929. Mathematically, they are written

Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields are thought of as field operators, in a manner similar to how the physical quantities are thought of as operators in first quantization. The key ideas of this method were introduced in 1927 by Paul Dirac, and were later developed, most notably, by Pascual Jordan and Vladimir Fock. In this approach, the quantum many-body states are represented in the Fock state basis, which are constructed by filling up each single-particle state with a certain number of identical particles. The second quantization formalism introduces the creation and annihilation operators to construct and handle the Fock states, providing useful tools to the study of the quantum many-body theory.

In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.

In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics.

Time-dependent density-functional theory (TDDFT) is a quantum mechanical theory used in physics and chemistry to investigate the properties and dynamics of many-body systems in the presence of time-dependent potentials, such as electric or magnetic fields. The effect of such fields on molecules and solids can be studied with TDDFT to extract features like excitation energies, frequency-dependent response properties, and photoabsorption spectra.

The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.

In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic field. It is the non-relativistic limit of the Dirac equation and can be used where particles are moving at speeds much less than the speed of light, so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli in 1927.

In 1927, a year after the publication of the Schrödinger equation, Hartree formulated what are now known as the Hartree equations for atoms, using the concept of self-consistency that Lindsay had introduced in his study of many electron systems in the context of Bohr theory. Hartree assumed that the nucleus together with the electrons formed a spherically symmetric field. The charge distribution of each electron was the solution of the Schrödinger equation for an electron in a potential $, derived from the field. Self-consistency required that the final field, computed from the solutions, was self-consistent with the initial field, and he thus called his method the self-consistent field method.$

Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. Spin should not be conceptualized as involving the "rotation" of a particle's "internal mass", as ordinary use of the word may suggest: spin is a quantized property of waves.

Within computational chemistry, the Slater–Condon rules express integrals of one- and two-body operators over wavefunctions constructed as Slater determinants of orthonormal orbitals in terms of the individual orbitals. In doing so, the original integrals involving N-electron wavefunctions are reduced to sums over integrals involving at most two molecular orbitals, or in other words, the original 3N dimensional integral is expressed in terms of many three- and six-dimensional integrals.

<span class="mw-page-title-main">Helium atom</span> Atom of helium

A helium atom is an atom of the chemical element helium. Helium is composed of two electrons bound by the electromagnetic force to a nucleus containing two protons along with two neutrons, depending on the isotope, held together by the strong force. Unlike for hydrogen, a closed-form solution to the Schrödinger equation for the helium atom has not been found. However, various approximations, such as the Hartree–Fock method, can be used to estimate the ground state energy and wavefunction of the atom. Historically, the first such helium spectrum calculation was done by Albrecht Unsöld in 1927. Its success was considered to be one of the earliest signs of validity of Schrödinger's wave mechanics.

In quantum chemistry, Brillouin's theorem, proposed by the French physicist Léon Brillouin in 1934, relates to Hartree–Fock wavefunctions. Hartree–Fock, or the self-consistent field method, is a non-relativistic method of generating approximate wavefunctions for a many-bodied quantum system, based on the assumption that each electron is exposed to an average of the positions of all other electrons, and that the solution is a linear combination of pre-specified basis functions.

This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three possible types of elementary fermions, the other two being the Dirac and the Majorana fermions.

References

1 2 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.
↑ Slater, J. (1929). "The Theory of Complex Spectra". Physical Review. 34 (2): 1293–1322. Bibcode:1929PhRv...34.1293S. doi:10.1103/PhysRev.34.1293.
↑ Heisenberg, W. (1926). "Mehrkörperproblem und Resonanz in der Quantenmechanik". Zeitschrift für Physik. 38 (6–7): 411–426. Bibcode:1926ZPhy...38..411H. doi:10.1007/BF01397160. S2CID 186238286.
↑ Dirac, P. A. M. (1926). "On the Theory of Quantum Mechanics". Proceedings of the Royal Society A. 112 (762): 661–677. Bibcode:1926RSPSA.112..661D. doi: 10.1098/rspa.1926.0133 .
↑ "Slater determinant in nLab". ncatlab.org. Retrieved 2023-11-08.
↑ Szabo, A.; Ostlund, N. S. (1996). Modern Quantum Chemistry. Mineola, New York: Dover Publishing. ISBN 0-486-69186-1.
↑ Solid State Physics - Grosso Parravicini - 2nd edition pp.140-143
↑ Zhang, J. M.; Kollar, Marcus (2014). "Optimal multiconfiguration approximation of an N-fermion wave function". Physical Review A. 89 (1): 012504. arXiv: 1309.1848 . Bibcode:2014PhRvA..89a2504Z. doi:10.1103/PhysRevA.89.012504. S2CID 17241999.
↑ 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 (1063): 542. Bibcode:1950RSPSA.200..542B. doi:10.1098/rspa.1950.0036. S2CID 122709395.

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

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[Atkins-1] 1 2 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. (1929). "The Theory of Complex Spectra". Physical Review. 34 (2): 1293–1322. Bibcode:1929PhRv...34.1293S. doi:10.1103/PhysRev.34.1293.

[3] Heisenberg, W. (1926). "Mehrkörperproblem und Resonanz in der Quantenmechanik". Zeitschrift für Physik. 38 (6–7): 411–426. Bibcode:1926ZPhy...38..411H. doi:10.1007/BF01397160. S2CID 186238286.

[4] Dirac, P. A. M. (1926). "On the Theory of Quantum Mechanics". Proceedings of the Royal Society A. 112 (762): 661–677. Bibcode:1926RSPSA.112..661D. doi: 10.1098/rspa.1926.0133 .

[5] "Slater determinant in nLab". ncatlab.org. Retrieved 2023-11-08.

[Szabo-6] Szabo, A.; Ostlund, N. S. (1996). Modern Quantum Chemistry. Mineola, New York: Dover Publishing. ISBN 0-486-69186-1.

[ReferenceA-7] Solid State Physics - Grosso Parravicini - 2nd edition pp.140-143

[8] Zhang, J. M.; Kollar, Marcus (2014). "Optimal multiconfiguration approximation of an N-fermion wave function". Physical Review A. 89 (1): 012504. arXiv: 1309.1848 . Bibcode:2014PhRvA..89a2504Z. doi:10.1103/PhysRevA.89.012504. S2CID 17241999.

[9] 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 (1063): 542. Bibcode:1950RSPSA.200..542B. doi:10.1098/rspa.1950.0036. S2CID 122709395.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]