Atomic orbital shapes

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)... 

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

In the MO-CI language, the correct dissociation of a single bond requires addition of a second doubly excited determinant to the wave function. The VB-CF wave function, on the other hand, dissociates smoothly to the correct limit, the VB orbitals simply reverting to their pure atomic shapes, and the overlap disappearing. 

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. 

Schrodinger s equation required the use of quantum numbers to describe each electron within an atom corresponding to the orbital size, shape, and orientation in space. Later it was found that one needed a quantum number associated with the electron spin. 

Orbitals have a variety of different possible shapes. Therefore, scientists use three quantum numbers to describe an atomic orbital. One quantum number, n, describes an orbital s energy level and size. A second quantum number, I, describes an orbital s shape. A third quantum number, mi, describes an orbital s orientation in space. These three quantum numbers are described further below. The Concept Organizer that follows afterward summarizes this information. (In section 3.3, you will learn about a fourth quantum number, mg, which is used to describe the electron inside an orbital.)... 

Figure 1.7. The symmetry characteristics of (a) s, (b) p, and (c) spn (hybrid) atomic orbitals. The shapes of the electron distributions are similar if one ignores the phases.

Figure 1.9 shows the three-dimensional shape of the electron distributions Pmo2px and symmetry element is called a C axis. An orbital with this kind of symmetry is called a 77 orbital. Atomic orbitals of the s type can form only a molecular orbitals atomic... 

Population of functions other than the monopole does not modify the charge on the pseudo-atom, but redistributes it in a non-spherical manner. Examples of the angular functions showing their symmetry for a selected number of poles are shown in Fig. 2 below. Although one must be careful not to confuse the various multipoles with atomic orbitals, their shapes are the same, i.e. the monopole is a spherical function like an s orbital, the three dipoles resemble p orbitals, etc. 

Orbitals. Atomic orbitals represent the angular distribution of electron density about a nucleus. The shapes and energies of these amplitude probability functions are obtained as solutions to the Schrodinger wave equation. Corresponding to a given principal quantum number, for example n = 3, there are one 3s, three 3p and five 3d orbitals. The s orbitals are spherical, the p orbitals are dumb-bell shaped and the d orbitals crossed dumb-bell shaped. Each orbital can accomodate two electrons spinning in opposite directions, so that the d orbitals may contain up to ten electrons. 

The remarkable accord between the postulates of van t Hoff and Sommer-feld s elliptic orbits must, no doubt have convinced many sceptics of a more fundamental basis of both phenomena to be found in atomic shape. The new quantum theory that developed in the late 1920 s seemed to define such a basis in terms of the magnetic quantum number mi. 

The ECP s are constructed based on the frozen orbital ECP technique (12). In this technique some of the core orbitals are expanded in the valence basis set and frozen in atomic shapes. This reduces the demand on the accuracy of the ECP potentials and the projection operators. One-electron ECP s constructed by this technique for nickel and copper have been shown to give results of quantitative accuracy for surface problems, particularly for hydrogen chemisorption which is treated here (13,14). In the previous studies the one-electron ECP s included a frozen 3s orbital. In the present case, states with a large occupancy of 4p appeared for the s type configurations in a cluster surrounding. 

The quantum number n is primarily responsible for determining the overall energy of an atomic orbital the other quantum numbers have smaller effects on the energy. The quantum number I determines the angular momentum of the orbital or shape of the orbital and has a smaller effect on the energy. The quantum number m/ determines the orientation of the angular momentum vector in a magnetic field, or the position of the orbital in space, as shown in Table 2-3. The quantum number ntg determines the orientation of the electron... 

There are now eight different spatial orbitals, hybrid orbitals, the other four being close to atomic hydrogen s-orbitals. The expansion of each of the VB orbitals in terras of M the basis functions located on the nuclei allows the orbitals to distort from the pure atomic shape. The SCVB wave... 

Although the spatial arrangement of six electron pairs round I-atom is octahedral, due to the presence of two lone pairs of electrons in the axial hybrid orbitals, the shape of IC14- ion gets distorted and becomes square planar as shown in Fig. 

Orbiting, or shape, resonances also play a role in a theory of termolecular recombination developed by Roberts et al. (1969). Their model applies to reactions A + B + M -> AB + M, and assumes that A and B form an excited quasi-bound state, which is de-excited by the third body to form stable AB. Results on H2 + M were compared to those for D2 + M by Roberts and Bernstein (1970). More recently Dickinson et al. (1972) applied the model to ion-atom association in rare gases. 

Just like the guitar string can have different waveforms, the one electron in a hydrogen atom can also have different waveforms, or orbitals. The shapes and sizes for these orbitals are predicted by the mathematics associated with the wave character of the hydrogen electron. Figure 11.6 shows some of them. 

An orbital is a volume of space about the nucleus where the probability of finding an electron is high. Unlike orbits that are easy to visualize, orbitals have shapes that do not resemble the circular paths of orbits. In the quantum mechanical model of the hydrogen atom, the energy of the electron is accurately known but its location about the nucleus is not known with certainty at any instant. The three-dimensional volumes that represent the orbitals indicate where an electron will likely be at any instant. This uncertainty in location is a necessity of physics. 

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