Atomic orbitals spin motion

Each electron in an atom has two possible kinds of angular momenta, one due to its orbital motion and the other to its spin motion. The magnitude of the orbital angular momentum vector for a single electron is given, as in Equation (1.44), by... 

In Eq. (2.30), F is the Fock operator and Hcore is the Hamiltonian describing the motion of an electron in the field of the spatially fixed atomic nuclei. The operators and K. are operators that introduce the effects of electrons in the other occupied MOs. Hence, when i = j, J( (= K.) is the potential from the other electron that occupies the same MO, i ff IC is termed the exchange potential and does not have a simple functional form as it describes the effect of wavefunction asymmetry on the correlation of electrons with identical spin. Although simple in form, Eq. (2.29) (which is obtained after relatively complex mathematical analysis) represents a system of differential equations that are impractical to solve for systems of any interest to biochemists. Furthermore, the orbital solutions do not allow a simple association of molecular properties with individual atoms, which is the model most useful to experimental chemists and biochemists. A series of soluble linear equations, however, can be derived by assuming that the MOs can be expressed as a linear combination of atomic orbitals (LCAO)44 ... 

Magnetic properties are due to the orbital and spin motions of electrons in atoms. The relation between the magnetic dipole moment p and the angular momentum J of an electron of charge e and mass m can be expressed as... 

Although the one-dimensional model bears little resemblance to any real molecular system, many of its features carry over to cases of practical interest. Suppose we consider three-dimensional motion in a central field as in atoms. The orbitals or single-electron functions now become atomic orbitals and can be classified in the usual manner as Is, 2a,. . . 2p, 3p,. . . 3d,. Suppose we are dealing with an atom in which there are two electrons of the same spin (a, say) occupying the 2a and 2p orbitals (inner shells being ignored for the present). Then the antisymmetric product function is... 

The fluorescence of aromatic molecules is quenched (diminished in intensity) partially or completely by heavy-atom substituents such as —As(OH)2, Br, and I, and by certain other groups such as —NH2, —CHO, —COR, and nitrogen in six-membered heterocyclic rings (e.g., quinoline). Each of these substituents has the ability to cause mixing of the spin and orbital electronic motions of the aromatic system. Spin-orbital coupling destroys the concept of molecular spin as a well-defined property of the molecule and thereby enhances the probability or rate of singlet —> triplet intersystem crossing. 

The upward arrow denotes one of the two possible spinning motions of the electron. (Alternatively, we could have represented the electron with a downward arrow.) The box represents an atomic orbital. 

Since a magnetic field is produced by an electric current, which consists in a flow of electrons along a conductor, the motion of free electrons in atomic orbits may fairly be supposed to give rise to a similar effect. It is also a reasonable hypothesis that the spin of an electron should be associated with a magnetic moment, and so it proves. 

First prindple quantum chemical methods, whether wave function based ( ab initio ) or density based, are aimed at solving the electronic Schrddinger equation without any reference to adjustable parameters or empirical data. In their standard form, they invoke the Bom-Oppenhdmer separation of electronic and nuclear motion and employ a nonrelativistic Hamiltonian which does not include any explicit reference to spin-dependent terms. Many quantum chemical methods are based on the variational prindple which, for computational convenience, is implemented in algebraic form via either one-electron functions built from linear combinations of atomic orbitals or n-electron functions constructed from Slater determinants. [11, 12]... 

In the Dirac relativistic equation, spin is naturally included. It is possible to identify the energy terms in the Dirac equation and include them in the ordinary Hamiltonian as magnetic terms. In particular, the spin-orbit coupling term is important. This term is physically due to interactions between the electron spin and its motional spin around the atomic nucleus. Spin-orbit coupling increases in importance for heavy atoms. Transitions or curve crossings are no longer spin forbidden. 

Horizontal lines on the diagram indicate the energies of several atomic orbitals. Note that the p orbitals are split into two levels that differ only slightly in energy. The classical view rationalizes this difference by invoking the idea that an electron spins on an axis and that the direction of the spin may be in either the same direction as its orbital motion or the opposite direction. Both the spin and the orbital motions create magnetic fields as a result of the rotation of the charge on the electron. The two fields interact in an attractive... 

First consider an atom with only one electron outside the closed shells. The atom is situated in a homogeneous magnetic field B, which is weak. The meaning of the word "weak will be clarified later. Associated with the orbital and spin motions there are magnetic moments, see (2.9), i.e.. 

There are three components of a coupling constant arising from nucleus-electron interactions. First, the magnetic moment of one nucleus interacts with the field produced by orbital motion of the electrons, which in turn interacts with a second nuclear moment. Secondly, there is a dipole interaction involving the electron spin magnetic moments. The final contribution arises from the spins of the electrons in the orbitals that have a non-zero probability of being at the nucleus (and are therefore derived from the s atomic orbitals). This last term, known as the Fermi contact term, is by far the most important for proton-proton couplings, but for other nuclei the situation is not so simple. 

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