Electrons, valence orbitals

Gaussians were originally introduced because they were easier to deal with mathematically than other functions that fitted the orbitals more accurately. They continue to be used because a good approximation to orbitals can be obtained from a reasonably small set. Orbitals of valence electrons (valence orbitals) are often well described by three to five Gaussians. Different sized basis sets containing different numbers of Gaussians are available and can be chosen to suit the problem you want to solve. 

Since the quantum numbers l or L are seldom well-defined in a one-electron valence orbital or a many-electron case (a) basis set, and since, for a linear... 

Fit the parameters of the potential such that the solutions of the Schrddinger (or Dirac) equation produce pseudo-orbitals matching the all-electron valence orbitals. 

In pseudopotential calculations one has a different nodal structure of the pseudovalence orbitals in comparison to the all-electron valence orbitals. (di WdR dr) and its expectation values are neither vanishing nor small. On the contrary, it is assumed and has numerically been shown for Au2 that the main contribution to the relativistic bond-length changes stem from relativistic corrections to the Hellmann-Feynman force resulting from the pseudopotential ... 

The Toulouse quantum-chemistry group created quasirelativistic pseudopotentials based on the atomic structure method developed by Barthelat et al. (1980) and the general procedure of adjustment devised by Durand and Barthelat (1974, 1975). The pseudopotential is constructed in such a way that the pseudo-orbitals coincide best with the all-electron valence orbitals and are smooth in the core region, i.e. an approach which has later been termed norm-conserving (Hamaim et al. 1979) or shape-consistent (Christiansen et al. 1979, Rappe et al. 1981). No parameter sets for the lanthanides and actinides have been published up to now. 

Most electronic valence transitions shift to longer wavelengths at higher pressures drat is, the gap between the highest occupied orbital and lowest unoccupied orbital tends to decrease upon compression. The rates of shift usually are larger (1) for pure materials than for solutes in a solvent and (2) for stronger (more allowed) transitions. However, these correlations are not quantitative, and many transitions shift in the opposite... 

Th is equation is iniportari tin in terprelin g Lh e resii Its tif calculation s. In ah initio an d seni i-em pirical calculation s, atom ic orbitals arc functions of the x, y, and /.coordinates of the electron that closely resemble the valence orbitals of the isolated atoms. 

An extended Huckel calculation is a simple means for modeling the valence orbitals based on the orbital overlaps and experimental electron affinities and ionization potentials. In some of the physics literature, this is referred to as a tight binding calculation. Orbital overlaps can be obtained from a simplified single STO representation based on the atomic radius. The advantage of extended Huckel calculations over Huckel calculations is that they model all the valence orbitals. 

There are several issues to consider when using ECP basis sets. The core potential may represent all but the outermost electrons. In other ECP sets, the outermost electrons and the last filled shell will be in the valence orbital space. Having more electrons in the core will speed the calculation, but results are more accurate if the —1 shell is outside of the core potential. Some ECP sets are designated as shape-consistent sets, which means that the shape of the atomic orbitals in the valence region matches that for all electron basis sets. ECP sets are usually named with an acronym that stands for the authors names or the location where it was developed. Some common core potential basis sets are listed below. The number of primitives given are those describing the valence region. 

Localized Bonds. Because boron hydrides have more valence orbitals than valence electrons, they have often been called electron-deficient molecules. This electron deficiency is partiy responsible for the great interest surrounding borane chemistry and molecular stmcture. The stmcture of even the simplest boron hydride, diborane(6) [19287-45-7] 2 6 sufficientiy challenging that it was debated for years before finally being resolved (57) in favor of the hydrogen bridged stmcture shown. 

Structure. The CO molecule coordinates in the ways shown diagrammaticaHy in Figure 1. Terminal carbonyls are the most common. Bridging carbonyls are common in most polynuclear metal carbonyls. As depicted, metal—metal bonds also play an important role in polynuclear metal carbonyls. The metal atoms in carbonyl complexes show a strong tendency to use ak their valence orbitals in forming bonds. These include the n + 1)5 and the n + l)p orbitals. As a result, use of the 18-electron rule is successflil in predicting the stmcture of most metal carbonyls. 

Boron is a unique and exciting element. Over the years it has proved a constant challenge and stimulus not only to preparative chemists and theoreticians, but also to industrial chemists and technologists. It is the only non-metal in Group 13 of the periodic table and shows many similarities to its neighbour, carbon, and its diagonal relative, silicon. Thus, like C and Si, it shows a marked propensity to form covalent, molecular compounds, but it differs sharply from them in having one less valence electron than the number of valence orbitals, a situation sometimes referred to as electron deficiency . This has a dominant effect on its chemistry. 

Election counting and orbital bookkeeping can easily be checked in these diagrams as each B has 4 valency orbitals (s -F 3p) there should be 4 lines emanating from each open circle likewise, as each B atom contributes 3 electrons in all and each H atom contributes 1 electron, the total... 

The Extended Hiickel model treats all valence electrons within the spirit of the TT-electron model. Each molecular orbital is written as an LCAO expansion of the valence orbitals, which can be thought of as being Slater-type orbitals (to look ahead to Chapter 9). Slater-type orbitals are very similar to hydrogenic ones except that they do not have radial nodes. Once again we can understand the model best by considering the HF-LCAO equations... 

Ab initio ECPs are derived from atomic all-electron calculations, and they are then used in valence-only molecular calculations where the atomic cores are chemically inactive. We start with the atomic HF equation for valence orbital Xi whose angular momentum quantum number is 1 ... 

Naively it may be expected that the correlation between pairs of electrons belonging to the same spatial MO would be the major part of the electron correlation. However, as the size of the molecule increases, the number of electron pairs belonging to different spatial MOs grows faster than those belonging to the same MO. Consider for example the valence orbitals for CH4. There are four intraorbital electron pairs of opposite spin, but there are 12 interorbital pairs of opposite spin, and 12 interorbital pairs of the same spin. A typical value for the intraorbital pair correlation of a single bond is 20kcal/ mol, while that of an interorbital pair (where the two MO are spatially close, as in CH4) is 1 kcal/mol. The interpair correlation is therefore often comparable to the intrapair contribution. 

The chemical bonding occurs between valence orbitals. Doubling the 1 s-functions in for example carbon allows for a better description of the 1 s-electrons. However, the Is-orbital is essentially independent of the chemical environment, being very close to the atomic case. A variation of the DZ type basis only doubles the number of valence orbitals, producing a split valence basis. In actual calculations a doubling of tire core orbitals would rarely be considered, and the term DZ basis is also used for split valence basis sets (or sometimes VDZ, for valence double zeta). 

Again it is instructive to compare this with the traditional MO approach, taking the CH4 molecule as an example. The MO single determinant description (RHF, which is identical to UHF near the equilibrium geometry) of the valence orbitals is in terms of four delocalized orbitals, each occupied by two electrons with opposite spins. The C-H bonding is described by four different, orthogonal molecular orbitals, each expanded in... 

---