Pauli spin wavefunction

In quantum mechanics, spin is described by an operator which acts on a spin wavefunction of the electron. In the present case this operator describes an angular momentum with two possible eigenvalues along a reference axis. The first requirement fixes commutation rules for the spin components, and the second one leads to a representation of the spin operator by 2 x 2 matrices (Pauli matrices [Pau27]). One has... 

Pauli spin matrices, 91 spinor wavefunction, 90 Perfect gas law, derivation, 42 Periodic table... 

Spin is a further complication. According to the Pauli principle, acceptable wavefunctions must change sign on formal exchange of the coordinates of two electrons [1], This is taken care of by writing the total wavefunction in the form of a determinant (the Slater determinant), in which each MO is multiplied by an appropriate spin wavefunction. 

The array of methods in gamess for treating spin-orbit coupling effects has recently been the subject of two reviews [46,47]. These methods include the full Breit-Pauli spin-orbit operator and approximations to it, primarily developed by Koseki and Fedorov. All of the methods require a multi-reference wavefunction as a starting point. This can be MCSCF, first or second order Cl, or MRPT2. The simplest method is a... 

In order to understand the physical meaning of the equation, the electron mass m and the speed of light c, which are both unity in atomic units, are explicitly written in this and the next sections. The a in Eq. (6.55) is called the Pauli spin matrix (Pauli 1925). This equation is invariant for the Lorentz transformation, because the momentum p = —/ V is the first derivative in terms of space. More importantly, this equation rsqu-irts four-component wavefunctions. 

Notice that the two terms of the spin wavefunction require that electrons 1 and 2 have opposite spin when electron 1 is a, electron 2 is j8, and vice versa. So if nothing else, we ve come up with a fancy way of arriving at a result you already know as the Pauli exclusion principle electrons in the same orbital (the same spatial wave-function) must have different values of m. The spatial wavefunction that puts two electrons in the same orbital if/j is symmetric i/>,(l)i/r,(2). The spin wavefunction in which both electrons have the same m is also symmetric a(l)a(2) or j8(l)/3(2). 

These give rise to the singlet and triplet states, respectively. The antisymmetric af3 — f3a) spin state yields a value for the spin projection quantum number Ms = WTsj + wisj = 0, while the triplet state spin wavefunctions correspond to Ms = — 1 (jSjS), 0 (ajS + 13a), and l(aa). We could have predicted the multiplicity of the ls 2s He atom without ever considering the explicit wavefunctions or their symmetry at all, although we still need the Pauli exclusion principle to exclude cases where both electrons have the same spin and spatial wavefunction. 

Note that the complete wavefunction as written in Eq. (2.47) changes sign if the labels of the electrons (1 and 2) are interchanged. W. PauU pointed out that the wavefunctions of all multielectronic systems have this property. The overall wavefunction invariably is antisymmetric for an interchange of the coordinates (both positional and spin) of any two electrons. This assertion rests on experimental measurements of atomic and molecular absorption spectra absorption bands predicted on the basis of antisymmetric electrOTiic wavefunctirais are seen experimentally, whereas bands predicted on the basis of symmetric electronic wave-functions are not observed. Its most important implication is the Pauli exclusion principle, which says that a given spatial wavefunction can hold no more than two electrons. This follows if an electron can be described completely by specifying its spatial and spin wavefunctions and electrons have only two possible spin wave-functions (a and fi). 

Wavefunctions for more than two electrons can be approximated as linear combinations of all the allowable products of electronic and spin wavefunctions for the individual electrons. Combinations that satisfy the Pauli exclusion principle... 

In actual fact, each level of motional degeneracy [F ] combines with a nuclear spin wavefunction of spin symmetry degeneracy (F ] to give a nondegenerate Pauli allowed state following equation (94) ... 

In Pauli s model, the sea of electrons, known as the conduction electrons are taken to be non-interacting and so the total wavefunction is just a product of individual one-electron wavefuncdons. The Pauli model takes account of the exclusion principle each conduction electron has spin and so each available spatial quantum state can accommodate a pair of electrons, one of either spin. 

Introduction of the half-integral spin of the electrons (values h/2 and —fe/2) alters the above discussion only in that a spin coordinate must now be added to the wavefunctions which would then have both space and spin components. This creates four vectors (three space and one spin component). Application of the Pauli exclusion principle, which states that all wavefunctions must be antisymmetric in space and spin coordinates for all pairs of electrons, again results in the T-state being of lower energy [equations (9) and (10)]. 

For over a decade, the topological analysis of the ELF has been extensively used for the analysis of chemical bonding and chemical reactivity. Indeed, the Lewis pair concept can be interpreted using the Pauli Exclusion Principle which introduces an effective repulsion between same spin electrons in the wavefunction. Consequently, bonds and lone pairs correspond to area of space where the electron density generated by valence electrons is associated to a weak Pauli repulsion. Such a property was noticed by Becke and Edgecombe [28] who proposed an expression of ELF based on the laplacian of conditional probability of finding one electron of spin a at t2, knowing that another reference same spin electron is present at ri. Such a function... 

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 ... 

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 =... 

One needs to learn how to tell which term symbols will be Pauli excluded, and to learn how to write the spin-orbit product wavefunctions corresponding to each term symbol and to evaluate the corresponding term symbols energies. 

Electron motion is more generally formulated in a form of the Schrodinger equation, including the spin in the presence of external fields known as the Pauli equation. This equation is gauge invariant in the sense that a transformation as in (5) also changes the quantum wavefunction as... 

Following the Pauli exclusion principle we must antisymmetrize the wavefunctions when we include the spins of the two electrons. The wavefunction is given by a product of spatial and spin wave functions, i.e. 

UHF Methods. A major drawback of closed-shell SCF orbitals is that whilst electrons of the same spin are kept apart by the Pauli principle, those of opposite spin are not accounted for properly. The repulsion between paired electrons in spin orbitals with the same spatial function is underestimated and this leads to the correlation error which multi-determinant methods seek to rectify. Some improvement could be obtained by using a wavefunction where electrons of different spins are placed in orbitals with different spatial parts. This is the basis of the UHF method,40 where two sets of singly occupied orbitals are constructed instead of the doubly occupied set. The drawback is of course that the UHF wavefunction is not a spin eigenfunction, and so does not represent a true spectroscopic state. There are two ways around the problem one can apply spin projection operators either before minimization or after. Both have their disadvantages, and the most common procedure is to apply a single spin annihilator after minimization,41 arguing that the most serious spin contaminant is the one of next higher multiplicity to the one of interest. 

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