Orbitals localized, electric moments

It is well known that transferability is an important property for the investigation of related systems. In cases when one can define quantities for molecular parts, which are additive and transferable, certain similarities of different molecules can be simply recognized. One-electron properties, as electric moments or kinetic energy contributions derivable from transferable/localized molecular orbitals, are especially useful for the above purpose. 

It is known, that in the framework of the independent particle model the one-electron properties of the system can be written as the sum of contributions from the individual orbitals. The transferability of the one-electron properties is implied by the transferability of the orbitals. The first and higher order electric moments determined for localized molecular orbitals (LMOs) in different systems can thus be used in comparative studies. 

The multipole expansion has already been used in certain quantum chemical calculations [59-65]. As localized orbitals are concentrated in certain spatial region, they can also be represented by their multipole moments. In the following we investigate whether the Coulomb integrals in terms of localized orbitals can be substituted by the multipole expansion of electric moments. 

We decompose the charge distribution of the whole electron system into sum of contributions from localized orbitals. If the localized orbitals do not overlap and their electric moments can be considered as transferable, than it is expected that the Coulomb integrals can be approached by the sum of interaction energies... 

It is interesting to see how these electrical moments, and particularly the dipole moment, can be thought of in terms of contributions from different parts of the molecule. The nuclei are, of course, easily treated but it is helpful to think of the electronic contribution, not in terms of symmetry orbitals like (1.2) but rather as localized orbitals which can be pictured as 0-H bonds and lone pairs. This can be done by performing a unitary transformation upon the wave function specified in terms of components... 

Additional rows and columns should be added to the matrix for more complex compounds with more than two atoms and the formula for the g s becomes much more complex. The matrix also expands when shallow-lying d-orbitals must be taken into accoimt. Simpler structures such as the diamond lattice have a smaller interaction matrix because there is no distinction between cation and anion sites. The matrix may also be modified by effects such as spin-orbit splitting. (Spin-orbit splitting is one of the corrections necessary to an accurate band calculation. It results from the interaction of the electron spin magnetic moment with the dot product of its velocity and the local electric field due to the positive atomic cores of the lattice.) Likewise, greater accuracy can be obtained if additional terms are included in the g values to accoimt for second and higher neighbors. 

I is in general no direct relation between such functions and ionization energies or electron excitation this is because they are not eigenfunctions of a hamiltonian, hence they cannot be associated with an energy. For that reason, we kept the usual designation localized molecular orbitals but with [ the last word in inverted commas orbitals . However, for the interpretation of some other molecular properties, the minimized residual interactions i between quasi-localized molecular orbitals are not very importaint and, so, the direct use of a localized bond description is quite justified. That is the [ Case for properties such as bond energies and electric dipole moments, as well as the features of the total electron density distribution with which those properties are directly associated. 

Ability to transit to the localized orbitals basis set [17,18] and use the additive scheme to obtain the physicochemical properties of molecules (electrical dipole moment, tensor of polarizability), electrooptical characteristics of IR- and Raman spectra (derivative from the first on normal vibration coordinates) [19], and also bond length [20,21]. 

Considerations similar to those made about electric dipole moments apply to other one-electron properties, for instance the nuclear spin-spin coupling constants between non-bonded hydrogen atoms in molecules like methane. These quantities are approximately equal to zero in the simple molecular orbital theory, as it is easily proved by using equivalent orbitals corresponding to the CH bonds instead of the usual delocalized MO s (34). Actually, the nuclear spins of protons cannot interact wta the electrons, since a localized MO cannot be large on two hydrogens at the same time, and correlation should be primarily responsible for all coupling constants, except perhaps for those observed for directly bonded atoms (see Sec. 4). 

The NpT2X2 compounds are usually treated as materials with localized 5f states, and crystalline electric-field effects are held responsible for the depression of the magnetic moments (2.4/tB is expected for the free-ion moment of Np). However, the heavy-fermion behaviour (found in NpCu2Si2) is able to introduce an instability of the localized 5f states. In this sense the reduction in hyperfine field at the Np nucleus could be understood as being due to a partial loss of the orbital moments resulting from 5f-electron delocalization. 

To describe the magnetic properties of amorphous alloys containing rare earth elements with non-zero orbital moment (L 0) the Hamiltonian of eq. (25) is no longer suited. Harris et al. (1973) have proposed a model in which they assume that there is a local uniaxial field of random orientation at each of the rare earth atoms in an amorphous solid. This local uniaxial field of random orientation is closely associated with the presence of an equally random crystalline electric field. The Hamiltonian for this random anisotropy model (RAM) can be written as... 

Fig. 4-46, p. 220, depicts the centroids of charges and the overlap populations. For a tabulation of canonical (CMC) and localized (LMO) orbitals, quadrupole moments, second moments, and electric field gradients, see the original paper [1]. 

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