Electron spin and antisymmetry

All electrons are characterized by a spin quantum number. The electron spin function is an eigenfunction of the operator and has only two eigenvalues, h/2 the spin eigenfunctions 

Knowing these aspects of quantum mechanics, if we were to construct a ground-state Hartree-product wave function for a system having two electrons of the same spin, say a, we would write 

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These defects of the Hartree SCF method were corrected by Fock (Section 4.3.4) and by Slater2 in 1930 [8], and Slater devised a simple way to construct a total wavefunction from one-electron functions (i.e. orbitals) such that will be antisymmetric to electron switching. Hartree s iterative, average-field approach supplemented with electron spin and antisymmetry leads to the Hartree-Fock equations. 

Brickstock, A., and Pople, J. A., Phil. Mag. 44, 697, The spatial correlation of electrons in atoms and molecules. III. The influence of spin and antisymmetry on the correlation of electrons/ A discussion is made of the refinement needed, if the interelectronic repulsion is taken into account beyond the single determinant, b. 

So far, we have not seen any very spectacular consequences of electron spin and the Pauli principle. In the hydrogen and heUum atoms, the spin factors in the wave functions and the antisymmetry requirement simply affect the degeneracy of the levels but... 

We now reconsider the helium atom from the standpoint of electron spin and the antisymmetry requirement. In the perturbation treatment of helium in Section 9.3, we found the zeroth-order wave function for the ground state to be li(l)li(2). To take spin into account, we must multiply this spatial function by a spin eigenfunction. We therefore consider the possible spin eigenfunctions for two electrons. We shall use the notation o (l )a(2) to indicate a state where electron 1 has spin up and electron 2 has spin up o (l) stands for... 

The Pauli antisymmetry principle is a requirement a many-electron wavefunction must obey. A many-electron wavefunction must be antisymmetric (i.e. changes sign) to the interchange of the spatial and spin coordinates of any pair of electrons i and/, that is ... 

To understand the main idea behind DFT, consider the following. In the absence of magnetic fields, the many-electron Hamiltonian does not act on the electronic spin coordinates, and the antisymmetry and spin restrictions are directly imposed on the wave function (r j, v j,..., rvyv). Within the Bom-Oppenheimer approximation,... 

The antisymmetry of many-electron spin-orbital products places constraints on any acceptable model wavefunction, which give rise to important physical consequences. For example, it is antisymmetry that makes a function of the form I Isa Isa I vanish (thereby enforcing the Pauli exclusion principle) while I lsa2sa I does not vanish, except at points ri and 1 2 where ls(ri) = 2s(r2), and hence is acceptable. The Pauli principle is embodied in the fact that if any two or more columns (or rows) of a determinant are identical, the determinant vanishes. Antisymmetry also enforces indistinguishability of the electrons in that Ilsals(32sa2sp I =... 

A little further discussion on electron spin is in order now. Spin orbitals are necessary because an electron possesses a spin quantum number (-l-j or — ). In the absence of a magnetic held, the up and down spins are energetically degenerate, or indistinguishable. The Pauli exclusion principle says that electronic wave functions must be antisymmetric (they change sign) under the interchange of any two electrons. Because of this antisymmetry, two electrons are not allowed to occupy the same quantum state. 

The above studies all involved only one and two electron systems. And with the exception of the Be high spin excited states (61) none required the use of "Fermi statistics" (wavefunction antisymmetry) in the Monte Carlo simulations. This is of course a prerequisite for multi-electron systems. We have recently carried out REP-QMC simulations on some three-electron systems. Aluminum is probably the simplest. In Table III we show energies for two states of Al and also for Al. ... 

The symmetric and antisymmetric squares have special prominence in molecular spectroscopy as they give information about some of the simplest open-shell electronic states. A closed-shell configuration has a totally symmetric space function, arising from multiplication of all occupied orbital symmetries, one per electron. The required antisymmetry of the space/spin wavefunction as a whole is satisfied by the exchange-antisymmetric spin function, which returns Fq as the term symbol. In open-shell molecules belonging to a group without... 

As we have just implied, solutions to the many-electron scattering problem, like solutions to the many-electron bound-state problems of quantum chemistry, are obtained in terms of products of one-electron functions, subject to constraints of spin, exchange antisymmetry (the Pauli principle), and possibly spatial (point... 

However the valence bond model seems inextricably bound up with the idea of electron spin indeed the chemical bond is often qualitatively described as due to the pairing of electron spins. What is certainly true is that the largest contributions to a description of electron-pair bonds are functions which have a singlet spin function for each bond what is not so obvious is that the bonding is due to this spin pairing. Ideas of electron spin have become entangled with the Pauli principle and antisymmetry. 

There is no question of an antisymmetry requirement here since we arc only dealing with a single electron but, in anticipation of future generalisations, let us extend this result by introducing an (algebraic) spin space of two functions a(s) and /3 s) which represent the two possible directions of electron spin ... 

Here Xi(0 X<( >) > (>0 represents the spin-orbital of electron i. The type of wavefunction represented by Eq. [5] is not complete because it does not (1) account for the indistinguishability of electrons and (2) satisfy the Pauli principle, which requires that if the coordinates of electrons i and / are interchanged in the above wavefunction, the wavefunction must change sign. To account for indistinguishability and ensure antisymmetry, the spin-orbitals for a closed-shell atom are arranged as a Slater determinant. 

The second intrusion of the electron spin came through a non-energetic, symmetry requirement, the so-called Eermi-Dirac statistics for systems of identical, half-integer spin particles, which results in total antisymmetry of the Schrbdinger wave function in a combined space and spin coordinate domain. This entails the Pauli exclusion principle (1925) in the framework of the independent-particle, Slater-determinantal model. The expression of atomic and molecular wave functions as iinear combinations of Slater determinants has been the basis of most of the subsequent methodologies of quantum chemistry, thermodynamics, and spectroscopy. 

We call a function like (10.39) a spin-orbital. A spin-orbital is the product of a one-electron spatial orbital and a one-electron spin function. If we were to take g(l) = ls(l)a(l), this would make the first and second columns of (10.37) identical, uid the wave function would vanish. This is a particular case of the Pauli exclusion principle No two electrons can occupy the same spin-orbital. Another way of stating this is to say that no two electrons in an atom can have the same values for all their quantum numbers. The Pauli exclusion principle is a consequence of the more general Pauli-principle antisymmetry requirement and is less satisfying than the antisymmetry statement, since the exclusion principle is based on approximate (zeroth-order) wave functions. We therefore take g(l) = ls(l)/3(l), which puts two electrons with opposite spin in the Is orbital. For the spin-orbit tl h, we cannot use either ls(l)a(l) or ls(l)/3(l), since these choices make the determinant vanish. We take /i(l) = 2s(l)a(l), which gives the familiar Li ground-state configuration Is and the zeroth-order wave function... 

A spin-orbital is the product of a one-electron spatial wave function and a one-electron spin function. An approximate wave function for a system of electrons can be written as a Slater determinant of spin-orbitals. Interchange of two electrons interchanges two rows in the Slater determinant, which multiplies the wave function by -1, ensuring antisymmetry. In such an approximate wave function, no two electrons can be assigned to the same spin-orbital. This is the Pauli exclusion principle and is a consequence of the Pauli-principle antisymmetry requirement. 

The atomic Hamiltonian (t of (11.1) (which omits spin-orbit interaction) does not involve spin and therefore commutes with the total-spin operators 5 and S. The fact that commutes with ft is not enough to show that the atomic wave functions are eigenfunctions of 5 The Pauli antisymmetry principle requires that each tp must be an eigenfunction of the exchange operator with eigenvalue —1 (Section 10.3). Hence must also commute with if we are to have simultaneous eigenfunctions of H, S, and ic. Problem 11.16 shows that [. , 4fc] = 0, so the atomic wave functions are eigenfunctions of We have = S S + l)feV each atomic state can be characterized by a total-electronic-spin quantum number S. 

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