Shell orbitals

Figure Cl. 1.2. (a) Mass spectmm of sodium clusters (Na ), N= 4-75. The inset corresponds to A = 75-100. Note tire more abundant clusters at A = 8, 20, 40, 58, and 92. (b) Calculated relative electronic stability, A(A + 1) - A(A0 versus N using tire spherical electron shell model. The closed shell orbitals are labelled, which correspond to tire more abundant clusters observed in tire mass spectmm. Knight W D, Clemenger K, de Heer W A, Saunders W A, Chou M Y and Cohen ML 1984 Phys. Rev. Lett. 52 2141, figure 1.

In the 6-3IG basis, the inner shell of carbon is represented by 6 primitives and the 4 valence shell orbitals are represented by 2 contracted orbitals each consisting of 4 primitives, 3 contracted and 1 uncontracted (hence the designation 6-31). That gives... 

As illustrated above, the spatial symmetries of these four S = 1 and S = 0 states are obtained by forming the direet produet of the "open-shell" orbitals that appear in this eonfiguration b2 x bi = A2. 

The teehniques used earlier for linear moleeules extend easily to non-linear moleeules. One begins with those states that ean be straightforwardly identified as unique entries within the box diagram. For polyatomie moleeules with no degenerate representations, the spatial symmetry of eaeh box entry is identieal and is given as the direet produet of the open-shell orbitals. For the formaldehyde example eonsidered earlier, the spatial symmetries of the nji and nn states were A2 and Ai, respeetively. 

The spin- and spatial- symmetry adapted N-eleetron funetions referred to as CSFs ean be formed from one or more Slater determinants. For example, to deseribe the singlet CSF eorresponding to the elosed-shell orbital oeeupaney, a single Slater determinant... 

Another family of basis sets, commonly referred to as the Pople basis sets, are indicated by the notation 6—31G. This notation means that each core orbital is described by a single contraction of six GTO primitives and each valence shell orbital is described by two contractions, one with three primitives and the other with one primitive. These basis sets are very popular, particularly for organic molecules. Other Pople basis sets in this set are 3—21G, 4—31G, 4—22G, 6-21G, 6-31IG, and 7-41G. 

Most simple empirical or semi-empirical molecular orbital methods, including all of those used in HyperChem, neglect inner shell orbitals and electrons and use a minimal basis set of valence Slater orbitals. 

Symbols are used to represent aspects of electronic stracture in atoms, and the symbolism is augmented as a student progresses through different levels of study e.g. those used to represent electron shells (K, L, M...) for orbital types (to s, p, d, f, sp, etc. in atoms a, tt, 8, etc. in molecules) for description of electronic states ( Pi, So, A2g, Tig, Cgt g, T2g). Introducing the added complexity of the time dimension, electronic transitions can be represented as shifts between shells, orbitals or states. 

Figure 3. Molecular-orbital diagrams as obtained by the ROHF method. Dashed lines indicate MOs dominated by the metal d-orbitals, the solid lines stand for doubly occupied or virtual ligand orbitals. Orbitals which are close in energy are presented as degenerate the average deviation from degeneracy is approximately 0.01 a.u. In the case of a septet state (S=3), the singly occupied open-shell orbitals come from a separate Fock operator and their orbital energies do not relate to ionization potentials as do the doubly occupied MOs (i.e. Koopmann s approximation). For these reasons, the open-shell orbitals appear well below the doubly occupied metal orbitals. Doubly occupying these gives rise to excited states, see text.

Nakatsuji H, Hirao K (1978) Cluster expansion of the wavefunction. symmetry-adapted-cluster expansion, its variational determination, and extension of open-shell orbital theory. J Chem Phys 68 2053... 

Although the sulfur atom makes use of sp3 hybrid orbitals (from 3s and 3p valence shell orbitals) in... 

C2 and change signs under C2 E, axz, or ayz operations). Although it may not be readily apparent, the py orbital transforms as B2. Using the four symmetry operations for the C2 point group, the valence shell orbitals of oxygen behave as follows ... 

For an octahedral ABS molecule such as SF6, the valence-shell orbitals are considered to be the s, p, and d orbitals of the central atom. It is easy to see that a regular octahedron has a center of symmetry so... 

We begin by writing the Lewis structure. The H atoms are terminal atoms. There are three central atoms and (3 x 1) + 4 + 6 + 4 + (3 x 1) = 20 valence electrons, or 10 pairs. A plausible Lewis structure is drawn at right. Each central atom is surrounded by four electron pairs, requiring sp3 hybridization. The valence-shell orbital diagrams for the atoms follow. 

The valence-shell orbital diagrams for the hybridized central atoms then are ... 

In both structures, the central N is attached to two other atoms, and possesses no lone pairs. The geometry of the molecule thus is linear and the hybridization on this central N is sp. Nb [He]ipfflU 2JUS We will assume that the terminal atoms are not hybridized in either structure. Their valence-shell orbital diagrams are terminal N Na [He] 2J. JiWl O [He] 2i ... 

It is to be expected that the optimum scale factors of orbitals which are singly occupied will show different behaviour from the closed shell orbitals. We have therefore performed an optimisation of the GHOs in the CHO radical in the UHF model. The results are summarised below. 

Kettle and his co-workers (39—42) used a model rather similar to the AOM to discuss stereochemistry. A perturbation approach led to the proportionality of MO energies (relative to the unperturbed orbitals) to squared overlap integrals, as in the AOM. For systems where the valence shell orbitals are evenly occupied, the total stabilization energy shows no angular dependence, suggesting that steric forces determine the equilibrium geometry. 

There exists no uniformity as regards the relation between localized orbitals and canonical orbitals. For example, if one considers an atom with two electrons in a (Is) atomic orbital and two electrons in a (2s) atomic orbital, then one finds that the localized atomic orbitals are rather close to the canonical atomic orbitals, which indicates that the canonical orbitals themselves are already highly, though not maximally, localized.18) (In this case, localization essentially diminishes the (Is) character of the (2s) orbital.) The opposite situation is found, on the other hand, if one considers the two inner shells in a homonuclear diatomic molecule. Here, the canonical orbitals are the molecular orbitals (lo ) and (1 ou), i.e. the bonding and the antibonding combinations of the (Is) orbitals from the two atoms, which are completely delocalized. In contrast, the localization procedure yields two localized orbitals which are essentially the inner shell orbital on the first atom and that on the second atom.19 It is thus apparent that the canonical orbitals may be identical with the localized orbitals, that they may be close to the localized orbitals, that they may be identical with the completely delocalized orbitals, or that they may be intermediate in character. 

For the inner shells the outermost contour is again 0.025 Bohr 3/2. They are much steeper and, therefore, the increment is here 0.2 Bohr-3/2. Three of these inner shell contours are drawn. If the remaining inner shell contours were drawn, the inner part would be solid black. For this reason, the inner shell contours are not drawn beyond the third one and, instead, the value of the inner shell orbital at the position of the nucleus has been written into the diagram. From the figure, it is obvious that the inner shell of lithium is very similar in Li2 and LiH, and in a very practical sense transferable. However, note that the localized inner shell orbital of the lithium atom has a slight negative tail towards the other atom which yields a very small amount of antibinding. 

Fig.4 (see p. 74/75) shows all localized orbitals for the ground state of the BH molecule and the 12 excited state of B2.37) These are again rotationally symmetric orbitals, i.e., sigma type orbitals, and the complete contour surfaces can be obtained by spinning around the indicated axis. In all orbitals shown the outermost contour corresponds to a wavefunction value of 0.025 Bohr-3/2. For all valence shell orbitals the increment from one contour to another is 0.025 Bohr-3/2. For the inner shells the increment is again 0.2 Bohr 3/2, but only three contours and the wavefunction values at the nuclear positions are shown. 

For the inner shell orbitals, too, one finds near-perfect transferability as was the case for lithium. 

The contribution of inner-shell correlation is taken as the difference between the CCSD(T)/MTsmall TAE with and without constraining the inner-shell orbitals to be doubly occupied. 

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