Spin-independent orbitals

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. 

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 have so far said little about the nature ofthe space function, S. Earlier we implied that it might be an orbital product, but this was not really necessary in our general work analyzing the effects of the antisymmetrizer and the spin eigenfunction. We shall now be specific and assume that S is a product of orbitals. There are many ways that a product of orbitals could be arranged, and, indeed, there are many of these for which the application of the would produce zero. The partition corresponding to the spin eigenfunction had at most two rows, and we have seen that the appropriate ones for the spatial functions have at most two columns. Let us illustrate these considerations with a system of five electrons in a doublet state, and assume that we have five different (linearly independent) orbitals, which we label a, b,c,d, and e. We can draw two tableaux, one with the particle labels and one with the orbital labels. 

Consider a system of n electrons in a spin state S. We know that there are for n linearly independent orbitals... 

Paramagnetism + 0-0.1 Independent Decreases Spin and orbital motion of electrons on individual atoms... 

Pauli paramagnetism + 10 Independent None Spin and orbital motion of delocalised electrons... 

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... 

In the manganese perovskites, the dominant interactions between d-like orbitals at neighboring Mn atoms are the (180° - ) Mn-O-Mn interactions. The spin-independent resonance integrals describing charge transfer between Mn atoms at positions Ri and Rj are... 

Here, every bonded electron pair, in atomic or hybrid orbitals, contributes every pair of bonds (or bonds involving nonbonded atoms) with spin-independent electrons contributes — every electron pair with parallel spins contributes — Jy-. This expression tells us that bonding lowers the energy, and nonbonded atoms or nonpaired electrons raise the energy. Increased stability will therefore be found when the negative terms in (9) approach zero, as they must when the relevant atoms or electrons move apart. 

Spin-orbit interaction Hamiltonians are most elegantly derived by reducing the relativistic four-component Dirac-Coulomb-Breit operator to two components and separating spin-independent and spin-dependent terms. This reduction can be achieved in many different ways for more details refer to the recent literature (e.g., Refs. 17-21). 

The frozen-core (fc) approach is not restricted to spin-independent electronic interactions the spin-orbit (SO) interaction between core and valence electrons can be expressed by a sum of Coulomb- and exchange-type operators. The matrix element formulas can be derived in a similar way as the Sla-ter-Condon rules.27 Here, it is not important whether the Breit-Pauli spin-orbit operators or their no-pair analogs are employed as these are structurally equivalent. Differences with respect to the Slater-Condon rules occur due to the symmetry properties of the angular momentum operators and because of the presence of the spin-other-orbit interaction. It is easily shown by partial integration that the linear momentum operator p is antisymmetric with respect to orbital exchange, and the same applies to t = r x p. Therefore, spin-orbit... 

The one-center approximation allows for an extremely rapid evaluation of spin-orbit mean-field integrals if the atomic symmetry is fully exploited.64 Even more efficiency may be gained, if also the spin-independent core-valence interactions are replaced by atom-centered effective core potentials (ECPs). In this case, the inner shells do not even emerge in the molecular orbital optimization step, and the size of the atomic orbital basis set can be kept small. A prerequisite for the use of the all-electron atomic mean-field Hamiltonian in ECP calculations is to find a prescription for setting up a correspondence between the valence orbitals of the all-electron and ECP treatments.65-67... 

Older versions of SOCI programs are very I/O intensive because they used to store the Hamiltonian matrix on disk and read it in every iteration step.52,141 Integral-driven direct methods for spin-orbit coupling came up in the mid 1980s123,142 following the original fomulation of direct Cl methods for spin-independent Hamiltonians.143-146 Modern direct SOCI programs can easily handle several million determinants.108,147-151... 

We have seen that with a system of n electrons in a spin state S there are, for n linearly independent orbitals, / (given by Eq. (18)) linearly independent spatial functions that can be constructed from these orbitals. In the present notation the SCVB wave function is written as the general... 

From the conceptual point of view, there are two general approaches to the molecular structure problem the molecular orbital (MO) and the valence bond (VB) theories. Technical difficulties in the computational implementation of the VB approach have favoured the development and the popularization of MO theory in opposition to VB. In a recent review [3], some related issues are raised and clarified. However, there still persist some conceptual pitfalls and misinterpretations in specialized literature of MO and VB theories. In this paper, we attempt to contribute to a more profound understanding of the VB and MO methods and concepts. We briefly present the physico-chemical basis of MO and VB approaches and their intimate relationship. The VB concept of resonance is reformulated in a physically meaningful way and its point group symmetry foundations are laid. Finally it is shown that the Generalized Multistructural (GMS) wave function encompasses all variational wave functions, VB or MO based, in the same framework, providing an unified view for the theoretical quantum molecular structure problem. Throughout this paper, unless otherwise stated, we utilize the non-relativistic (spin independent) hamiltonian under the Bom-Oppenheimer adiabatic approximation. We will see that even when some of these restrictions are removed, the GMS wave function is still applicable. 

The matrix defined in Eq.(47) represents a general, non-relativistic spin-independent TV-electron Hamiltonian given by Eq.(2) in a specifically defined model space. The one-electron orbital space is spanned by N orthonormal localized orbitals 4>j, j = 1,2,..., TV and the TV-electron orbital space is one-dimensional with the basis function... 

Coulomb interactions dominate the electronic structure of molecules. The total spin S2 and Sz are nearly conserved for light atoms. We will consider spin-independent interactions in models with one orbital per site. In the context of tt electrons, the operators a+a and OpCT create and annihilate, respectively, an electron with spin a in orbital p. The Hiickel Hamiltonian is... 

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