Orbitals shapes

Hartree-Fock orbital Relatively accurately calculated orbital shapes. 

Figure 8-2 The li Orbital Shape and a Gaussian Approximation (dotted line).

Moreover, many studies use the nature and shape of moleeular orbitals for interpreting eleetronie effeets. However, it should be kept in mind that the total energy of a moleeule is invariant with respeet to the oeeupied moleeular orbitals. In other words, oeeupied moleeular orbitals may strongly mix with eaeh other without ehanging the energy of the system. Therefore, the interpretation of orbital shapes usually is not a unique souree of information. 

The value of / correlates with the number of preferred axes in a particular orbital and thereby identifies the orbital shape. According to quantum theoiy, orbital shapes are highly restricted. These restrictions are linked to energy, so the value of the principal quantum number ( ) limits the possible values of /. The smaller U is, the more compact the orbital and the more restricted its possible shapes ... 

Historically, orbital shapes have been identified with letters rather than numbers. These letter designations correspond to the values of / as follows ... 

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. 

The chemical properties of atoms are determined by the behavior of their electrons. Because atomic electrons are described by orbitals, the Interactions of electrons can be described in terms of orbital interactions. The two characteristics of orbitals that determine how electrons interact are their shapes and their energies. Orbital shapes, the subject of this section, describe the distribution of electrons in three-dimensional space. Orbital energies, which we describe in Chapter 8, determine how easily electrons can be moved. 

The shapes of orbitals strongly influence chemical interactions. Hence, we need to have detailed pictures of orbital shapes to understand the chemistry of the elements. 

The chemistry of all the common elements can be described completely using s, p, and d orbitals, so we need not extend our catalog of orbital shapes to the f orbitals and beyond. 

Another influence on the magnitude of the crystal field splitting is the position of the metal in the periodic table. Crystal field splitting energy increases substantially as valence orbitals change from 3 d to 4d to 5 d. Again, orbital shapes explain this trend. Orbital size increases as n increases, and this means that the d orbital set becomes... 

Figure 4.2 shows thep and d orhitals. Thep orhitals are dumh-hell shaped, and all hut one d orbital have four lohes. The orbital shapes represent electron probabilities. The chance of finding an electron within the boundary of an orbital is approximately 90%. 

Figure 7.10 Different types of hybridization and the resulting orbital shapes.

A molecular orbital (MO) is an orbital resulting from the overlap and combination of atomic orbitals on different atoms. An MO and the electrons in it belong to the molecule as a whole. Molecular orbitals calculations are used to develop (1) mathematical representations of the orbital shapes, and (2) energy level diagrams for the molecules. 

To get the orbital shape it is necessary to draw contour surfaces of constant probability density. For the py orbital the cross section in the yz plane is first obtained and rotated about the y-axis. In the yz plane, (j) = 7t/2, sin > = 1, and hence... 

Figure 2.2 Valence 2p NAOs for ground 3P and first excited1 If states of the C atom, showing the weak dependence of orbital shape on electron configuration.

Qualitatively the isomorphism character is defined by geometrical similarity of orbital shapes responsible for isomorphism. At the same time, the more similar are the extensions, trajectories and inclination angles of such orbitals, the more perfect is isomorphism. 

The second quantum number describes an orbital s shape, and is a positive integer that ranges in value from 0 to (n - 1). Chemists use a variety of names for the second quantum number. For example, you may see it referred to as the angular momentum quantum number, the azimuthal quantum number, the secondary quantum number, or the orbital-shape quantum number. 

Regardless of its name, the second quantum number refers to the energy sublevels within each principal energy level. The name that this hook uses for the second quantum number is orbital-shape quantum number (i), to help you remember that the value of 1 determines orbital shape. (You will see examples of orbital shapes near the end of this section.)... 

To identify an energy sublevel (type of orbital), you combine the value of n with the letter of the orbital shape. For example, the sublevel with n = 3 and 7 = 0 is called the 3s sublevel. The sublevel with n = 2 and 7=1 is the 2p sublevel. 

Within the atom, electrons behave as waves. Different shapes and sizes of these waves are possible around the nucleus. These are known as orbitals . The simplest orbital is spherical, but more complex orbital shapes are possible. Any orbital, irrespective of its size or shape, can hold a maximum of two electrons. 

The first shell or energy level out from the nucleus is called the K shell or energy level and contains a maximum of two electrons in the s orbital— that is, K = s2, where the K represents the shell number (or principle quantum number), the s describes the orbital shape of the angular momentum quantum number, and the 2 is the maximum number of electrons that the s orbital can contain. This particular sequence is K = s2, which means K shell contains 2 electrons in the s orbital. This is the sequence for the element helium. Look up helium in the text for more information. 

We focus in this Section on particular aspects relating to the direct interpretation of valence bond wavefunctions. Important features of a description in terms of modem valence bond concepts include the orbital shapes (including their overlap integrals) and estimates of the relative importance of the different stmctures (and modes of spin coupling) in the VB wavefunction. We address here the particular question of defining nonorthogonal weights, as well as certain aspects of spin correlation analysis. 

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