Spin-same-orbit

Foldy-Wouthuysen transformation the spin-same orbit h and spin-other orbit... 

From the four-component Dirac-Coulomb-Breit equation, the terms [99]—[102] can be deduced without assuming external fields. A Foldy-Wouthuysen transformation23 of the electron-nuclear Coulomb attraction and collecting terms to order v1 /c1 yields the one-electron part [99], Similarly, the two-electron part [100] of the spin-same-orbit operator stems from the transformation of the two-electron Coulomb interaction. The spin-other-orbit terms [101] and [102] have a different origin. They result, among other terms, from the reduction of the Gaunt interaction. 

A slightly different situation may be related to the electron spin-same orbit interaction which can be considered as a blend of Fj type operators and gradient derivative elements. 

Here //sso and //soo denote the spin-same orbit (SSO) and spin-other-orbit (SOO) contributions. 1 and Si stand for the orbital momentum and the spin momentum operators of the th electron, and rlA = r, — R/t is the distance between the electron i and the nucleus A. The angular momentum operator of the electron i calculated with respect to the nucleus A at the position is defined as iA = (r, P.i) / p,- Similarly, ry = r, — r represent the distance between the electrons i and j ly = (r,- rj) x p, symbolizes the orbital momentum operator of the electron i with respect to the position of the electron j. Boldface printed... 

One recognizes the two-electron Darwin term and the two-electron spin-(same)-orbit term (see also sections 2.12 and 4.6). 

The spin-orbit term involves the interaction of the spin of the electron with its own orbital angular momentum around the other electrons, and is often called the spin-own-orbit interaction or spin-same-orbit interaction. 

The two-electron operator is given in the nuclear frame and not in the reference of either electron. The spin-orbit coupling due to the relative motion of elecrons therefore splits into two parts The total interaction is the coupling of the spin of a selected reference electron with the magnetic field induced by a second electron. The spin-same orbit (SSO) and spin-other orbit (SOO) contributions arise from the motion of the reference electron and the other electron, respectively, relative to the nuclear frame and are carried by the Coulomb and Gaunt terms, respectively. For most molecular application it suffices to include the Coulomb term only, thus defining the Dirac-Coulomb Hamiltonian, but for the accurate calculation of molecular spectra the Gaunt term should be included as well. 

A variation on the HF procedure is the way that orbitals are constructed to reflect paired or unpaired electrons. If the molecule has a singlet spin, then the same orbital spatial function can be used for both the a and P spin electrons in each pair. This is called the restricted Hartree-Fock method (RHF). 

In addition to being negatively charged electrons possess the property of spin The spin quantum number of an electron can have a value of either +5 or According to the Pauli exclusion principle, two electrons may occupy the same orbital only when... 

Pauli exclusion principle (Section 1 1) No two electrons can have the same set of four quantum numbers An equivalent expression is that only two electrons can occupy the same orbital and then only when they have opposite spins PCC (Section 15 10) Abbreviation for pyndimum chlorochro mate C5H5NH" ClCr03 When used in an anhydrous medium PCC oxidizes pnmary alcohols to aldehydes and secondary alcohols to ketones... 

Pauli exclusion principle (Section 1.3) No more than two electrons can occupy the same orbital, and those two must have spins of opposite sign. 

The Pauli exclusion principle has an implication that is not obvious at first glance. It requires that only two electrons can fit into an orbital, since there are only two possible values of m,. Moreover, if two electrons occupy the same orbital, they must have opposed spins. Otherwise they would have the same set of four quantum numbers. 

With the next element, carbon, a complication arises. In which orbital should the sixth electron go It could go in the same orbital as the other 2p electron, in which case it would have to have the opposite spin, [. It could go into one of the other two orbitals, either with a parallel spin, f, or an opposed spin, Experiment shows that there is an energy difference among these arrangements. The most stable is the one in which the two electrons are in different orbitals with parallel spins. The orbital diagram of the carbon atom is... 

This is the most general form of a spin orbital, but if the Hamiltonian does not contain the spin explicitly, it may be more convenient to try to introduce simplified spin orbitals which contain only one nonvanishing component and hence are of either pure a or character. Corresponding to the idea of the doubly occupied orbitals, the spin orbitals are often constructed in pairs simply by multiplying the same orbital tp(r) with a and ft, respectively. 

The correlation error can, of course, be defined with reference to the Hartree scheme but, in modem literature on electronic systems, one usually starts out from the Hartree-Fock approximation. This means that the main error is due to the neglect of the Coulomb correlation between electrons with opposite spins and, unfor-tunetely, we can expect this correlation error to be fairly large, since we force pairs of electrons with antiparallel spins together in the same orbital in space. The background for this pairing of the electrons is partly the classical formulation of the Pauli principle, partly the mathematical fact that a single determinant in such a case can... 

The kinetic energy in the Hartree-Fock scheme is evidently too low, owing to the fact that we have assumed the existence of a simplified uncorrelated motion, whereas the particles in reality have much more complicated movements because of their tendency to avoid each other. The potential energy, on the other hand, comes out much too high in the HF scheme essentially due to the fact that we have compelled a pair of electrons with opposite spins together in the same orbital in space. 

This theorem follows from the antisymmetry requirement (Eq. II.2) and is thus an expression for Pauli s exclusion principle. In the naive formulation of this principle, each spin orbital could be either empty or fully occupied by one electron which then would exclude any other electron from entering the same orbital. This simple model has been mathematically formulated in the Hartree-Fock scheme based on Eq. 11.38, where the form of the first-order density matrix p(x v xx) indicates that each one of the Hartree-Fock functions rplt y)2,. . ., pN is fully occupied by one electron. 

The wave function, constructed from the atomic orbitals must be antisymmetric with respect to interchange of electrons in order to satisfy the Pauli exclusion principle, having different spin quantum numbers (a and J3) for two electrons which are in the same orbital. 

In other cases, discussed below, the lowest electron-pair-bond structure and the lowest ionic-bond structure do not have the same multiplicity, so that (when the interaction of electron spin and orbital motion is neglected) these two states cannot be combined, and a knowledge of the multiplicity of the normal state of the molecule or complex ion permits a definite statement as to the bond type to be made. 

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