REPRESENTATIONS OF ORBITALS

So far we have emphasized orbital energies, but the wave function also provides information about an electron s probable location in space. Let s examine the ways in which we can picture orbitals because their shapes help us visualize how the electron density is distributed around the nucleus. 

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G [H-Xe] Split valence 2 sets of functions in the valence region provide a more accurate representation of orbitals. Use for very large molecules for which 6-31G(d) is too expensive. 9 2 6D... 

A complete specification of how an atom s electrons are distributed in its orbitals is called an electron configuration. There are three common ways to represent electron configurations. One is a complete specification of quantum numbers. The second is a shorthand notation from which the quantum numbers can be inferred. The third is a diagrammatic representation of orbital energy levels and their occupancy. 

Similar to spin adaptation each 2-RDM spin block may further be divided upon considering the spatial symmetry of the basis functions. Here we assume that the 2-RDM has already been spin-adapted and consider only the spatial symmetry of the basis function for the 2-RDM. Denoting the irreducible representation of orbital i as T, the 2-RDM matrix elements are given by... 

Representation of orbital spinors of symmetric molecules in terms of relativistic double groups [6]. 

It should be noticed that lower case Mulliken symbols are used to indicate the irreducible representations of orbitals. The upper case Mulliken symbols are reserved for the description of the symmetry properties of electronic states. 

Simple balloon-type representations of orbitals were presented in the previous chapter without comment. A further example is given in Figure 2.5b or 2.5c. The boundary where... 

Fig. 3.3 Schematic representation of orbital (upper) and state (lower) energy diagrams of superexchange (left) and hopping (right) mechanisms of photoinduced electron transfer in DBA...

Fig. 4 Schematic representation of orbital energies for (a) Ti02 monovacancy in a monoclinicaUy distorted octahedron with a d ion state (b) Hf02 monovacancy and in a divacancy monovacancy is d occupancy state, divacancy in a d occupancy state. Electron occupancies are all in high spin states...

FIGURE 1.21 Representation of orbital mixing in sp hybridization. Mixing of one s orbital with three p orbitals generates four sp hybrid orbitals. Each sp hybrid orbital has 25% s character and 75% p character. The four sp hybrid orbitals have their major lobes directed toward the corners of a tetrahedron, which has the carbon atom at its center. 

In order to emphasize that 0 is a continuous function we have extended boundary surfaces in representations of orbitals to the nucleus, but for p orbitals, this is strictly not true if we are considering Ki95% of the electronic charge. 

Fig. 3-2. Schematic representation of orbital overlap in end-on metal complexes of N2.

The representation of orbital overlap in Figure 3-2 corresponds qualitatively to that in arenediazonium ions, as discussed in our book on aromatic diazo compounds (Sects. 8.3 and 8.4) the C —N o bond in Ar-N is also stabilized by n electron donation from the Pt orbitals of the aryl group to the diazonio group. 

Fig. I. I. Schematic representation of orbital configurations in the water molecule, (a) according to the simplified scheme of (i.i) in which 0(2s) and O(zpz) electrons do not participate in bond formation, (6) with approximately tetrahedral hybridization. The actual calculated electron density is shown in fig. 1.3.

Although we must not lose sight of the fact that wave-functions are mathematical in origin, most chemists find such functions hard to visualize and prefer pictorial representations of orbitals. The boundary surfaces of the s and three p atomic orbitals are shown in Figure 1.9. The... 

In addition to basis set expansions, there are various numerical methods for parameterizing orbitals including numerical basis sets of the form (p(r) = Yim(r)f(r), in which the radial function,/fr) does not have an analytical form, but is evaluated by a spline procedure [117]. Numerical orbitals may be more flexible than STO or GTO basis sets, but their use is more computationally demanding. Wavelet representations of orbitals [118] are exceptionally flexible as well and have an intriguing multiresolution property wavelet algorithms adaptively increase the flexibility of the orbital in regions where the molecular energy depends sensitively on the precision of the orbital and use coarser descriptions where precision is less essential. 

Another issue is development of methods suitable for very large systems. The challenges here are associated with necessity for more efficient and accurate representation of orbitals, more robust elimination of finite size errors for solids and surface calculations with periodic boundary conditions. Advancement in this area also imply certain progress in QMC-based dynamical methods. The issue of QMC approaches that treat nuclear and electronic problem on the same footing remains open. 

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