Orbitals multipole

Atomic orbitals Multipole distribution Number of charges... 

Molecular energies and structures Energies and structures of transition states Bond and reaction energies Molecular orbitals Multipole moments... 

Molecular dipole moments are often used as descriptors in QPSR models. They are calculated reliably by most quantum mechanical techniques, not least because they are part of the parameterization data for semi-empirical MO techniques. Higher multipole moments are especially easily available from semi-empirical calculations using the natural atomic orbital-point charge (NAO-PC) technique [40], but can also be calculated rehably using ab-initio or DFT methods. They have been used for some QSPR models. 

The following types of multipole distributions are used to represent the four types of atomic orbital products. 

Multipole analysis with high-resolution X-ray data for [Ni(thmbtacn)]2+ was carried out to determine the electron configuration in the C3 symmetry-adapted orbitals of the Ni ion, confirming a higher occupancy of the crystal field-stabilized t2g orbitals relative to the destabilized eg orbitals. This is interpreted in terms of a predominantly ionic metal-ligand interaction.1424... 

Etchebest, C., R. Lavery, and A. Pullman. 1982. The Calculations of Molecular Electrostatic Potential from a Multipole Expansion Based on Localized Orbitals and Developed at Their Centroids Accuracy and Applicability for Macromolecular Computations. Theor. Chim. Acta 62, 17. 

There are two approaches to map crystal charge density from the measured structure factors by inverse Fourier transform or by the multipole method [32]. Direct Fourier transform of experimental structure factors was not useful due to the missing reflections in the collected data set, so a multipole refinement is a better approach to map charge density from the measured structure factors. In the multipole method, the crystal charge density is expanded as a sum of non-spherical pseudo-atomic densities. These consist of a spherical-atom (or ion) charge density obtained from multi-configuration Dirac-Fock (MCDF) calculations [33] with variable orbital occupation factors to allow for charge transfer, and a small non-spherical part in which local symmetry-adapted spherical harmonic functions were used. 

Many authors [8-10] have demonstrated that the CP method undercorrects the BSSE. Moreover, Karlstrom and Sadlej [11] pointed out that addition of the partner orbitals to the basis set of a molecule not only lowers its energy, in accordance with the variation principle, but also affects the monomer properties (multipole moments and polarizabilities). Latajka and Scheiner [12] found that in a model ion-neutral system such as Li" -OH2, this secondary BSSE can be comparable in magnitude to the primary effect at both SCF and MP2 levels. The same authors also underlined the strong anisotropy of secondary error [13]. 

Atomic density functions are expressed in terms of the three polar coordinates r, 6, and multipole formalism, the density functions are products of r-dependent radial functions and 8- and -dependent angular functions. The angular functions are the real spherical harmonic functions ytm (8, ), but with a normalization suitable for density functions, further discussed below. The functions are well known as they describe the angular dependence of the hydrogenic s, p, d,f... orbitals. 

The coefficients n, have to obey the condition n, f, imposed by Poisson s electrostatic equation, as pointed out by Stewart (1977). The radial dependence of the multipole density deformation functions may be related to the products of atomic orbitals in the quantum-mechanical electron density formalism of Eq. (3.7). The ss, sp, and pp type orbital products lead, according to the rules of multiplication of spherical harmonic functions (appendix E), to monopolar, dipolar, and quadrupolar functions, as illustrated in Fig. 3.6. The 2s and 2p hydrogenic orbitals contain, as highest power of r, an exponential multiplied by the first power of r, as in Eq. (3.33). This suggests n, = 2 for all three types of product functions of first-row atoms (Hansen and Coppens 1978). 

The Relation Between the Occupancies of Transition-Metal Valence Orbitals and the Multipole Population Parameters... 

The d-orbital occupancies are derived from the experimental multipole populations by the inverse expression (Holladay et al. 1983)... 

The Matrix M 1 Relating c/-Orbital Occupancies / to Multipole Populations Ptmp (Eq. 10.9)... 

All the above methods are somehow based on an orbital hypothesis. In fact, in the multipolar model, the core is typically frozen to the isolated atom orbital expansion, taken from Roothan Hartree Fock calculations (or similar [80]). Although the higher multipoles are not constrained to an orbital model, the radial functions are typically taken from best single C exponents used to describe the valence orbitals of a given atom [81]. Even tighter is the link to the orbital approach in XRCW, XAO, or VOM as described above. Obviously, an orbital assumption is not at all mandatory and other methods have been developed, for example those based on the Maximum Entropy Method (MEM) [82-86] where the constraints/ restraints come from statistical considerations. 

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