S-orbitals

The wave function T i oo ( = 11 / = 0, w = 0) corresponds to a spherical electronic distribution around the nucleus and is an example of an s orbital. Solutions of other wave functions may be described in terms of p and d orbitals, atomic radii Half the closest distance of approach of atoms in the structure of the elements. This is easily defined for regular structures, e.g. close-packed metals, but is less easy to define in elements with irregular structures, e.g. As. The values may differ between allo-tropes (e.g. C-C 1 -54 A in diamond and 1 -42 A in planes of graphite). Atomic radii are very different from ionic and covalent radii. 

An s orbital is spherically symmetrical and can contain a maximum of two electrons with opposed spins. A p orbital has a solid figure-of-eight shape there are three equivalent p orbitals for each principal quantum number they correspond to the three axes of rectangular coordinates. 

The above definitions must be qualified by stating that for principal quantum number I there are only s orbitals for principal quantum number 2 there are only s and p orbitals for principal quantum number 3 there are only s, p and d orbitals for higher principal quantum numbers there are s, p, d and f orbitals. 

The simplest system exliibiting a nuclear hyperfme interaction is the hydrogen atom with a coupling constant of 1420 MHz. If different isotopes of the same element exhibit hyperfme couplings, their ratio is detemiined by the ratio of the nuclear g-values. Small deviations from this ratio may occur for the Femii contact interaction, since the electron spin probes the inner stmcture of the nucleus if it is in an s orbital. However, this so-called hyperfme anomaly is usually smaller than 1 %. 

Flere two electrons occupy the 1 s orbital (with opposite, a and p spins) while the other electron pair resides in 2s-2p polarized orbitals in a maimer that instantaneously correlates their motions. These polarized orbital... 

The teodeocy to aitaia either a half filled or fully filled set of d orbitals at the expense of the outer s orbital is shown by both chromium and copper and should be noted. This apparent irregularity will be discussed in more detail in Chapter 13. 

In elements of Periods 2 and 3 the four orbitals are of two kinds the first two electrons go into a spherically symmetrical orbital—an s orbital with a shape like that shown in Figure 2.7—and the next six electrons into three p orbitals each of which has a roughly doublepear shape, like those shown unshaded in each half of Figure 2.10. 

Slater exponent for the. t type Slater orbitals on atom A and. s orbital Slater exponent for the s type Slater orbitals on atom li. 

The zeroth-order Gaussian function has s-orbital angular symmetry the three first-order iTiiissian functions have p-orbital symmetry. In normalised form these are ... 

We wish to construct linear combinations of the atomic orbitals such that the overall wavefunction meets the Bloch requirement. Suppose the s orbitals in our lattice are labelled X , where the wth orbital is located at position x = na. An acceptable linear combination of these orbitals that satisfies the Bloch requirements is ... 

All of our orbitals have disappeared. How do we escape this terrible dilemma We insist that no two elections may have the same wave function. In the case of elections in spatially different orbitals, say. Is and 2s orbitals, there is no problem, but for the two elechons in the 1 s orbital of the helium atom, the space orbital is the same for both. Here we must recognize an extr a dimension of relativistic space-time... 

This sum describes the polarization of the Is orbital in terms of functions that have PO symmetry by combining an s orbital and po orbitals, one can form a hybrid-like orbital that is nothing but a distorted Is orbital. The appearance of the excited npo orbitals has... 

Each set of p orbitals has three distinct directions or three different angular momentum m-quantum numbers as discussed in Appendix G. Each set of d orbitals has five distinct directions or m-quantum numbers, etc s orbitals are unidirectional in that they are spherically symmetric, and have only m = 0. Note that the degeneracy of an orbital (21+1), which is the number of distinct spatial orientations or the number of m-values. 

For example, in formaldehyde, H2CO, one forms sp hybrids on the C atom on the O atom, either sp hybrids (with one p orbital "reserved" for use in forming the n and 7i orbitals and another p orbital to be used as a non-bonding orbital lying in the plane of the molecule) or sp hybrids (with the remaining p orbital reserved for the n and 7i orbitals) can be used. The H atoms use their 1 s orbitals since hybridization is not feasible for them. The C atom clearly uses its sp2 hybrids to form two CH and one CO a bondingantibonding orbital pairs. 

Draw a plot of the radial probability density (e.g., r2[Rjjj(r)]2 with R referring to the radial portion of the STO) versus r for eaeh of the orthonormal Ei s orbitals found in Exereise 1. 

Plot of the orthogonalized 1 s orbital probability density vs r note there are no nodes. 

Plot of the orthogonalized 3 s orbital probability density vs r note there are two nodes in the 0-5 bohr region but they are not distinguishable as such. A duplicate plot with this nodal region expanded follows. 

An example will help illustrate these ideas. Consider the formaldehyde molecule H2CO in C2v symmetry. The configuration which dominates the ground-state waveflinction has doubly occupied O and C 1 s orbitals, two CH bonds, a CO a bond, a CO n bond, and two 0-centered lone pairs this configuration is described in terms of symmetry adapted orbitals as follows (Iai22ai23ai2lb2 ... 

Two elasses of systems illustrate eases for whieh heterolytie bond dissoeiation lies lower than the homolytie produets. The first involves transition metal dimer eations, M2. Espeeially for metals to the right side of the periodie table, sueh eations ean be eonsidered to have ground-state eleetron eonfigurations with a d d +i eharaeter, where the d eleetrons are not heavily involved in the bonding and the a bond is formed primarily from the metal atom s orbitals. If the a bond is homolytieally broken, one forms X + Y = M (s d +i)... 

In the alkaline earth atom ease, the polarized orbital pairs are formed by mixing the ns and np orbitals (aetually, one must mix in equal amounts of pi, p.i, and po orbitals to preserve overall S symmetry in this ease), and give rise to angular eorrelation of the eleetron pair. Use of an (n+l)s2 CSF for the alkaline earth ealeulation would eontribute in-out or radial eorrelation beeause, in this ease, the polarized orbital pair formed from the ns and (n+l)s orbitals would be radially polarized. 

---