Nuclear orbital model

A. D. Bochevarov, E. F. Valeev, C. D. Sherrill. The electron and nuclear orbitals model current challenges and future prospects. Mol. Phys., 102 (2004) 111-123. 

In order to retain the orbital model for a many-electron atom, Hartree assumed that each electron came under the influence of the nuclear charge and an average potential due to the remaining electrons. He therefore retained the form of the radial equation for a one-electron atom, equation 12.2, but assumed that the mutual potential energy U was the sum of... 

This experiment established the nuclear model of the atom. A key point derived from this is that the electrons circling the nucleus are in fixed stable orbits, just like the planets around the sun. Furthermore, each orbital or shell contains a fixed number of electrons additional electrons are added to the next stable orbital above that which is full. This stable orbital model is a departure from classical electromagnetic theory (which predicts unstable orbitals, in which the electrons spiral into the nucleus and are destroyed), and can only be explained by quantum theory. The fixed numbers for each orbital were determined to be two in the first level, eight in the second level, eight in the third level (but extendible to 18) and so on. Using this simple model, chemists derived the systematic structure of the Periodic Table (see Appendix 5), and began to... 

The localized molecular orbital model (LMO) (39-41) treats the electrons and nuclei separately. The nuclear contribution is identical to eq. [26] with = Z e, the nuclear charge screened by the irmer shell electrons that are assumed to follow the nuclei. The local units for the contribution from the valence elec-... 

Fig. l.ll. Energy levels of an AX spin system (a) and interaction of the nuclear spins A and X with I = 1/2, involving the bonding electrons (ethane molecular orbital model). 

Neils Bohr (1885-1962) proposed an orbital model of the nuclear atom in which electrons in an atom moved around the nucleus, just as planets move around the sun. 

In spite of the fact that the method of calculation is somewhat different, the HTH model is very close to the Anderson one. This model may be seen as the reduction of the homo-(poly)nuclear Anderson model to the homo-(bi)nuclear case (A = B). The HTH model also focuses on the two-singly occupied magnetic orbitals in the triplet state [Pg.242]

The reduced widths and parity for formation of ground states confirm the sequence of single particle orbits proposed by the nuclear shell model (Sect, 3, and Bethe and Butler ). Excited states of a given nucleus which have nucleon capture probabilities comparable with that found for the ground state may be interpreted as single particle states. Comparison of capture probabilities in dp) and dn) reactions may indicate the first state of T — in a nucleus of r = 0 (see AF , Sect. 52). The relative capture probabilities or reduced widths for two different nuclear states (preferably with the same Q can be used (Auerbach and French ) to determine the intermediate coupling parameter ajK (Sect. 3 and [10 ]). 

In the nuclear shell model, the mutual interaction between the nucleons adds up to a singleparticle average potential consisting of a (spherical) central potential and a spin-orbit interaction. The nucleons are assumed to move independently in this potential. The solutions of the Schrbdinger equation yield single-particle energy levels and wave functions for the individual nucleons (Haxel et al. 1949 Goeppert-Mayer 1949). 

Fig. 5 Schematic representation of the states in the nuclear shell model. The oscillator shells on the left are first split into the individual subshells by deviations of the nuclear potential from the harmonic oscillator, before the spin-orbit interaction creates the groupings of states that produce the correct magic numbers above N = Z = 20. The diagram is schematic and not to scale...

Tong et al. also have found that the appearance of high emission or low emission bands for trinuclear Au(i) complexes [Au3(dcmp)2]X3 (dcmp -bis(dicyclohexylphosphinomethyl)-cyclohe Q lphosphine) (HOMO and LUMO orbitals are plotted in Fig. 6b) is depending on the counter ion X and whether this has close contact or no contact to the complex. DFT calculations assign the emission bands to the three-coordinate Au(i) chain and show that the spectroscopic properties are affected by aurophilic interactions (Au-Au distances of about 3.0 to 3.1 A). DFT calculations of two- to four-nuclear coordinated model complexes present evidence that both absorption and emission enei es are inversely proportional to the number of Au(i) atoms if gold-anion contacts are not close. ... 

Although a separation of electronic and nuclear motion provides an important simplification and appealing qualitative model for chemistry, the electronic Sclirodinger equation is still fomiidable. Efforts to solve it approximately and apply these solutions to the study of spectroscopy, stmcture and chemical reactions fonn the subject of what is usually called electronic structure theory or quantum chemistry. The starting point for most calculations and the foundation of molecular orbital theory is the independent-particle approximation. 

The first step in reducing the computational problem is to consider only the valence electrons explicitly, the core electrons are accounted for by reducing the nuclear charge or introducing functions to model the combined repulsion due to the nuclei and core electrons. Furthermore, only a minimum basis set (the minimum number of functions necessary for accommodating the electrons in the neutral atom) is used for the valence electrons. Hydrogen thus has one basis function, and all atoms in the second and third rows of the periodic table have four basis functions (one s- and one set of p-orbitals, pj, , Pj, and Pj). The large majority of semi-empirical methods to date use only s- and p-functions, and the basis functions are taken to be Slater type orbitals (see Chapter 5), i.e. exponential functions. 

---