Spin-independent

Think of the front carbon atom and its three groups as one fan, and the back carbon atom and its three groups as a different fan. These two fans can spin independently of each other, which gives rise to many different possible conformations. This is why Newman projections are so incredibly powerful at showing conformations. They are drawn in a way that is perfect for showing the various conformations that arise as an individual single bond rotates. 

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

The spin-independent probability of finding one electron at r, and the other simultaneously at r2 is obtained by integrating over the spins. Since the spin functions are... 

These results, as most related results of density functional theory, have direct connections to the fundamental statement of the Hohenberg-Kohntheorem the nondegenerate ground state electron density p(r) of a molecule of n electrons in a local spin-independent external potential V, expressed in a spin-averaged form as... 

When an atomic ion is under the influence of an additional external spin-independent potential vext(f), produced for instance by a polarizable environment, the effective energy of the atomic ion becomes ... 

In this formalism the spin-independent many-body Hamiltonian may be written... 

Here nig denotes the sign of the spin projection (it takes two values, +1 and — 1). By taking into account the cumulant properties, Eqs. (10)-(13) with replaced by F, it can be shown that A must be a real symmetric matrix (A " = A j) with no unique diagonal elements, whereas 11 is a spin-independent (11 = =... 

In Eq. (84), the a- and jS-dependences of have been retained only to emphasize that it is related to these SOs. We recall that If is a spin-independent matrix as a consequence of the antisymmetric properties of cumulant... 

We consider now situations in which a molecule has one or more unpaired electrons hence we may have spin-polarized systems. The spatial orbitals are thus divided into two categories. Namely, those that are double occupied with two electrons of opposite spin (h and n ), called closed shells (cl), and singly occupied ( or nf), called open shells (op). We assume further spin-independent... 

We assume further that the ON of the open shell p is always one (tip = 1). This assumption is trivial for a doublet, but it is more restrictive for higher multi-plet with a corresponding underestimation of the energy. Remember that matrix elements of are nonvanishing only if aU its labels refer to partially occupied NOs therefore A = 0 and II = 0 if we consider a cumulant made up of at least one open-shell level. Since A and II refer only to closed shells, we consider them spin-independent. The sum rule, Eq. (89), becomes... 

Because the iron ions carry a magnetic moment, the Hall data are difficult to interpret. The conventional theory of the Hall effect utilizes a spin-independent resonance (transfer-energy) integral, and an adequate theory incorporating a spin-dependent resonance integral needs to be developed for antiferromagnetic materials. 

The spin-independent part of these equations is identical to the Klein-Gordon equation. If the singularity of V is not stronger than 1/r then,... 

The subtracted radiatively corrected electron factor may be obtained from the subtracted one-loop electron factor in (9.10). To this end, one should restore the radiative photon mass in the one-loop electron factor, and then the polarization operator insertion in the photon line is taken into account with the help of the dispersion integral like one in (3.44) for the spin-independent... 

The book is organized as follows. In the introductory part we briefly discuss the main theoretical approaches to the physics of weakly bound two-particle systems. A detailed discussion then follows of the nuclear spin independent corrections to the energy levels. First, we discuss corrections which can be calculated in the external field approximation. Second, we turn to the essentially two-particle recoil and radiative-recoil corrections. Consideration of the spin-independent corrections is completed with discussion of the nuclear size and structure contributions. A special section is devoted to the spin-independent... 

The Extreme Paschen-Back Effect.—If the magnetic field is very strong, the interactions that cause the spins of the electrons to combine to a resultant spin and the orbital moments to combine to a resultant orbital moment are broken. Then each electron orients its spin independently in the magnetic field, having two possible values, and... 

Probably the best-known approach to the utilization of spin symmetry is that originally developed by Slater and by Fock (see, for example, Hurley [17]). No particular advantage is taken of the spin-independence of the Hamiltonian, at least in the first phase of the construction of the n-particle basis. We take the 1-particle basis to be spin-orbitals — products of orthonormal orbitals (r) and the elementary or-thomormal functions of the spin coordinate a... 

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