Intra-atomic correlations

Hurley, A. C., Proc. Phys. Soc. A69, 49, (i) On the method of atoms in molecules. II. An intra-atomic correlation correction." Modification of Moffitt s method. 

In the case of a valence MCSCF calculation the difference between the optimised orbitals and these atomic RHF orbitals simply represents the way in which the atoms are distorted by the molecular environment. Thus, this difference is closely related to the idea of atoms in molecules (1). However, here, the atoms are represented only at the RHF level, and the difference concerns only the orbitals, not the intra- atomic correlation. 

To put it on a more quantitative basis, Johansson uses the expression for the critical point of the Mott transition as reformulated by Hubbard in terms of the bandwidth W[ and the polar state formation energy Uh (or effective intra-atomic correlation) (Eq. (36)). 

A 6-configuration curve is also shown which excludes intra-atomic correlation terms. 

The comparison of spectral line shapes computed on the basis of the ab initio dipole surface of He-Ar with absorption measurements has demonstrated the soundness of the data. The agreement indicates that exchange effects due to intra-atomic correlation and higher-order dispersion terms contribute significantly to the induced dipole. However,... 

Recent work improved earlier results and considered the effects of electron correlation and vibrational averaging [278], Especially the effects of intra-atomic correlation, which were seen to be significant for rare-gas pairs, have been studied for H2-He pairs and compared with interatomic electron correlation the contributions due to intra- and interatomic correlation are of opposite sign. Localized SCF orbitals were used again to reduce the basis set superposition error. Special care was taken to assure that the supermolecular wavefunctions separate correctly for R —> oo into a product of correlated H2 wavefunctions, and a correlated as well as polarized He wavefunction. At the Cl level, all atomic and molecular properties (polarizability, quadrupole moment) were found to be in agreement with the accurate values to within 1%. Various extensions of the basis set have resulted in variations of the induced dipole moment of less than 1% [279], Table 4.5 shows the computed dipole components, px, pz, as functions of separation, R, orientation (0°, 90°, 45° relative to the internuclear axis), and three vibrational spacings r, in 10-6 a.u. of dipole strength [279]. 

Also, large differential intra-atomic correlation errors between different atomic states cause a quite incorrect zero th order picture of the spectrum. 

A simpler molecule which exhibits the same features as NiH, of large intra-atomic correlation effects, is FH. The SCF configuration for this molecule is... 

On the contrary, the valence of the actinides varies and there is some uncertainty as to the configuration in the metals. In particular, in the first half of the series, the number of 5/electrons can vary with temperature and pressure and is not an integer. Moreover, the magnetism is present in the metals only from the middle of the series (5). On the other hand, in the case of ionic and covalent compounds, a strong intra-atomic correlation between electrons from the 5/ states has been found and the experimental data clearly show that the 5/states have an atomic-like character (4) indeed, the 5/ electrons in the actinide compounds present a magnetic behaviour which is similar to that of the 4/ electrons in the rare earth compounds. 

In the MO pairs description the intra-atomic correlations are not sorted out from the inter-atomic ones. Thus, in a H.F. -j- MO... 

C.I. calculation or in <1, calculations most of the effort would go into obtaining the large intra-atomic correlations. In the equivalent orbital description one may deal with just the inter-atomic pairs, Eq. (167), using Eq. (157b) [E being a. small quantity compared to the total E, one would then have to calculate A< >, Eq. (157c), too]. Intra-atomic correlations (as in He) are insensitive to the details of atomic orbitals (AO) and Fj. Hence, even at smaller r s where the AO s are distorted into rj s, they are expected to cancel the purely atomic values at r = oo. 

This was recognized by Mof tt and led to his theory of atoms in molecules . However, he tried to obtain even the valence shell correlations from free atoms. Since valence electrons in a molecule do not preserve their atomic character, the idea of empirical intra-atomic correlation corrections applies properly only to inner shells (see also Section XXV). 

Coulomb repulsion between the electrons in the f shell and are much more sensitive to changes in the local environment than the spin-orbit transitions. They are therefore a useful probe of the way intra-atomic correlations are affected by the metallic state, particularly in the actinides. Nevertheless, there are relatively few investigations of Coulomb transitions in metals since their energies all exceed 500 meV and most exceed 1 eV, as can be seen in fig. 1. The lowest energies are found in Pr, Sm " and Tm, all of which have now been studied by neutrons (Taylor et al. 1988, Needham 1989, Osborn et al. 1990). Although Coulomb transitions are non-dipolar, they can have appreciable cross-sections at intermediate values of momentum transfer, and are, in many cases, stronger than the dipolar (i.e., J—>J 1) cross-sections at the same k. 

The recent observations of Coulomb transitions in actinide intermetallic compounds (McEwen et al. 1990, Osborn et al. 1990) are of particular interest because the 5f electrons are on the boundary between localised and itinerant behaviour. The neutron results give direct evidence of the persistence of strong intra-atomic correlations in, e.g., the heavy-fermion compound UPtj. The theoretical challenge is to reconcile this with the substantial evidence that the f electrons also form a coherent Fermi liquid (Fulde et al. 1988, Zwicknagl 1988). 

In LDA, the electron correlations are taken into account only by a mean field approximation which utilizes the correlation enei of the uniform electron gas. In the Ce compounds where the 4f electrons are believed to be itinerant in the ground state, such as in CeSns, the topology of the Fermi surface can be described by the band structure calculated in LDA. However, the strong intra-atomic correlation effect between the 4f electrons should be considered for consistent explanations of the Fermi surface, the electronic specific heat coefficient and the cyclotron efifective mass. Beyond LDA, there are two approaches by which the correlation effect between the 4f electrons is taken into account in an explicit way. One is p-f mixing theory and the other is renormalized band theory. 

The calculated and experimental equilibrium distances (Re)> dissociation energies (Dg), and harmonic frequencies (o) ) for H2 and OH are listed in Table 1. While the errors in Rq and 0)0 are small (about 3%), the errors in the calculated De s are substantial, 14,3 kcal/mol (about 13%) for H2 and 11.3 kcal/mol (about 11%) for OH. These errors are, however, typical of P0L CI calculations the underestimation of the dissociation energy is a deficiency of the P0L CI wavefunction arising from the neglect of the dynamical (inter-and intra-atomic) correlations of the electrons on the two atoms. Fortunately, the errors in the calculated D s are comparable for H2 and OH, leading to an error in the calculated reaction energy defect... 

---