The Linear Combination of Atomic Orbitals

James Richard Fromm

When atoms are brought together to form compounds, their atomic orbitals approach each other sufficiently closely to interact. In the formation of covalent bonds, the electrons in the atomic orbitals on one atom interact with the electrons in the atomic orbitals on another atom to accomplish the sharing of electrons between the atoms. The interaction involves primarily, but not entirely, the interaction of one orbital on one atom with one orbital on another atom. In order to understand molecular geometry, however, it is often necessary to consider the effect of many (ideally all) of the other orbitals on the interacting atoms. One approach which does this is called the "hybridization" of atomic orbitals.

In another section, an extension of the ideas of Lewis structures of molecules called the valence shell electron pair repulsion, or V.S.E.P.R., approach was used to rationalize the observed geometries of many of the simpler inorganic molecules. In this section we shall show that the approach of valence shell electron pair repulsion is an approximation to a more fundamental understanding of the covalent bonding in molecules which is usually called the valence bond theory.

The basic idea of valence bond theory is that a covalent bond is formed by the overlap of atomic orbitals. The two electrons of paired spin which are shared by two bonded atoms lie in an atomic orbital of each of the two atoms. The greater the degree of overlap of the atomic orbitals, the greater will be the degree of sharing and the stronger will be the covalent bond between them. The atomic orbitals can be the original atomic orbitals of the atoms, but often the geometry of these orbitals is such that effective overlap cannot occur in the known geometry of the molecule. Under these circumstances, the atomic orbitals on an atom can reconfigure themselves into a different configuration, and the reconfigured orbitals are said to be hybridized.

The hybridization of atomic orbitals approach, when used in this qualitative and descriptive way, gives little information on energy levels within molecules. It is most easily used with molecules consisting of some arrangement about a single atom, i.e. where a single atom is at the center of symmetry of the molecule. The primary reason for qualitative use of this approach is because it gives information on the molecular geometry. This method of linearly combining atomic orbitals is particularly useful when the bonding in a molecule is essentially determined by one of the atoms in it, unlike the case of molecular oxygen where both of the atoms are significantly involved and no atom is located at the center of symmetry of the molecule. (Such molecules are more easily treated using the molecular orbital approach.)

In 1931, the American physical chemist Linus Pauling demonstrated that the wave functions of the electrons in the orbitals of an atom could be combined mathematically (essentially, by adding together the amplitudes of their waveforms in three-dimensional space--which by mutual constructive and destructive interference will give a different three-dimensional configuration), using the equations of waves, to give sets of equivalent Schroedinger wave functions that we now call hybridized atomic orbitals. For many molecules, hybridization gives a set of atomic orbitals that can overlap more effectively with the atomic orbitals on other atoms in the molecule, thus providing an overall molecular structure which has stronger bonds and is lower in energy. While the mathematical treatment of Pauling is beyond our scope, the results of it are most useful in furthering our understanding of the bonding in molecules.

The principles of the valence bond theory derived from Pauling's analysis can be stated as follows:

In using the hybridization of atomic orbitals approach in this way, we need look only at the empty or half-full orbitals on the central atom. We can look at either the empty or half-full orbitals, not both at the same time. There are only a limited number of types of hybridization commonly found in molecules, and a list is given below. More details of each of them are given in another section.

sp Hybridization

The hybridization of one s orbital and one p orbital on a central atom gives rise to two sp orbitals. Hybridization as sp gives two orbitals, which are in a linear arrangement, that is, 180o apart. An example of sp hybridization is found in HgCl2 (mercury(II) chloride).

sp2 Hybridization

The hybridization of one s orbital and two p orbitals on a central atom gives rise to three sp2 orbitals. Hybridization as sp2 gives three orbitals, which are planar and 120o apart. An example of sp2 hybridization is BCl3.

sp3 Hybridization

The hybridization of one s and all three p orbitals on a central atom gives rise to four sp3 orbitals. Hybridization as sp3 gives four identical orbitals, which are aligned in a tetrahedral configuration and so are 109.5o apart. This very common structure is found in CH4, NH3, NH4+, and CCl4; it is the usual hybridization form of singly-bonded carbon atoms in organic compounds.

In its ground state, the carbon atom has the electronic structure 1s22s22p2. The hybridization requires four half-full orbitals, so the electronic structure of the carbon atom must reach the excited state structure of 1s22s12p3, with one electron in each of the px, py, and pz orbitals, before hybridization can occur. This requires energy--an energy which is more than recovered in the lower-energy more-overlap structures of molecules which are possible with sp3 hybridization.

dsp2 Hybridization

Atoms of the transition metals and of other elements which have d atomic orbitals available to them can use these d orbitals in hybridization. The hybridization of one s, two p, and one d orbital on a central metal atom gives rise to four hybrid dsp2 orbitals. They are square planar in alignment and 90o apart, because the hybrid uses the dxy, s, px and py orbitals, all of which are in the xy plane. An example of a square planar complex ion is Ni(CN)42-.

dsp3 Hybridization

Atoms which have d orbitals available to them can also use them to form other types of hybrid orbitals. The hybridization of one s, one d, and three p orbitals on a central atom gives rise to five dsp3 orbitals. Hybridization as dsp3 gives five orbitals, three equatorial and two axial, because this hybridization uses the dz2, the s, and all three of the p orbitals; one of the p orbitals , pz, is perpendicular to the xy plane. In effect, the trigonal bipyramidal geometry of this hybridization is the trigonal plane of sp2 hybridization with the addition of the z axis component due to the pz and dz2 orbitals. An example of dsp3 hybridization is PCl5, whose geometry is that of a trigonal bipyramid.

d2sp3 Hybridization

Atoms which have two or more d orbitals available to them can use more than one d orbital in a hybridization. The hybridization of two d orbitals, one s orbital, and three p orbitals on a central atom gives rise to six hybrid d2sp3 orbitals. Hybridization as d2sp3 gives six orbitals which are equivalent in energy and geometry. All adjacent orbitals are 90o apart, and all nonadjacent orbitals are 180o apart. This form of hybridization has the geometric structure of an octahedron, with six vertices and eight sides. The orbitals which contribute to the square planar geometry of the equatorial plane of the octahedron are the s, px, py, and dxy atomic orbitals; the z axis component is again due to the pz and the dz2 atomic orbitals. Molecules or ions which have this structure are said to have octahedral structures or symmetry; sulfur hexafluoride and uranium hexafluoride are examples of such molecules.

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Copyright 1997 James R. Fromm