Atomic orbitals shapes

In this chapter, we continue our effort to treat topics in physical chemistry in a way that does not skip over the main details but still tries to simplify the presentation. We need to present the derivation for at least a partial answer if a few students ask, Where did that come from However, let us be clear at the outset of this chapter that the desired goal is for smdents to understand pure rotational and vib-rotational spectra of molecules and know where the H atom orbital shapes come from. If you follow the derivation with pencil and paper that may help you understand the... 

Shapes of atomic orbitals play central roles in governing the types of directional bonds an atom can form. 

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. 

Section 1 1 A review of some fundamental knowledge about atoms and electrons leads to a discussion of wave functions, orbitals, and the electron con figurations of atoms Neutral atoms have as many electrons as the num ber of protons m the nucleus These electrons occupy orbitals m order of increasing energy with no more than two electrons m any one orbital The most frequently encountered atomic orbitals m this text are s orbitals (spherically symmetrical) and p orbitals ( dumbbell shaped)... 

The usefulness of spin density surfaces can be seen in the following models of methyl radical, CH3, and allyl radical, CH2=CHCH2. In each case, the surface is shaped somewhat like a 2p atomic orbital on carbon. There are some interesting differences between the two radicals, however. While the unpaired electron is confined to the carbon atom in methyl radical, it is delocalized over the two terminal carbons in allyl radical. 

Consider now a covalent VB function build from atomic orbitals which are allowed to distort from the pure atomic shape. 

The concept of natural orbitals may be used for distributing electrons into atomic and molecular orbitals, and thereby for deriving atomic charges and molecular bonds. The idea in the Natural Atomic Orbital (NAO) and Natural Bond Orbital (NBO) analysis developed by F. Weinholt and co-workers " is to use the one-electron density matrix for defining the shape of the atomic orbitals in the molecular environment, and derive molecular bonds from electron density between atoms. 

The Schrodinger equation can be solved approximately for atoms with two or more electrons. There are many solutions for the wave function, ij/, each associated with a set of numbers called quantum numbers. Three such numbers are given the symbols n, , and mi. A wave function corresponding to a particular set of three quantum numbers (e.g., n = 2, = 1, mi = 0) is associated with an electron occupying an atomic orbital. From the expression for ij/y we can deduce the relative energy of that orbital, its shape, and its orientation in space. 

Click Coached Problems for a self-study module on shapes of atomic orbitals. 

The expressions for a number of other atomic orbitals are shown in Table 1.2a (for R) and Table 1.2b (for Y). To understand these tables, we need to know that each wavefunction is labeled by three quantum numbers n is related to the size and energy of the orbital, / is related to its shape, and mt is related to its orientation in space. 

A note on good practice The concepts of promotion, hybridization, and resonance belong to valence bond theory, not molecular orbital theory. Instead, molecular orbitals are built from all the available atomic orbitals by noting whether or not they have the right shape to overlap with one another. 

Complexes of d- and /-block metals can be described in terms of hybridization schemes, each associated with a particular shape. Bearing in mind that the number of atomic orbitals hybridized must be the same as the number of hybrid orbitals produced, match the hybrid orbitals sp1d, sp fd , and sp d3f to the following shapes (a) pentagonal bipyramidal ... 

The raw output of a molecular structure calculation is a list of the coefficients of the atomic orbitals in each LCAO (linear combination of atomic orbitals) molecular orbital and the energies of the orbitals. The software commonly calculates dipole moments too. Various graphical representations are used to simplify the interpretation of the coefficients. Thus, a typical graphical representation of a molecular orbital uses stylized shapes (spheres for s-orbitals, for instance) to represent the basis set and then scales their size to indicate the value of the coefficient in the LCAO. Different signs of the wavefunctions are typically represented by different colors. The total electron density at any point (the sum of the squares of the occupied wavefunctions evaluated at that point) is commonly represented by an isodensity surface, a surface of constant total electron density. 

FIGURE 16.36 I1ie tear-shaped objects are representations of the six ligand atomic orbitals that are used to build the molecular orbitals of an octahedral complex in ligand field theory. They might represent s- or p-orbitals on the ligands or hybrids of the two. 

In addition to size, an atomic orbital also has a specific shape. The solutions for the Schrodinger equation and experimental evidence show that orbitals have a variety of shapes. A second quantum number indexes the shapes of atomic orbitals. This quantum number is the azimuthal quantum number (1). 

Among atomic orbitals, s orbitals are spherical and have no directionality. Other orbitals are nonspherical, so, in addition to having shape, every orbital points in some direction. Like energy and orbital shape, orbital direction is quantized. Unlike footballs, p, d, and f orbitals have restricted numbers of possible orientations. The magnetic quantum number (fflj) indexes these restrictions. 

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