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Atomic orbital

Atomic orbital

Figure 2.10. Overlap of atomic orbitals to form molecular orbitals in H2+

a a orbitals. This method is called LCAO-MO (Levine, 2000). For example, for the H2+ molecule the lowest-energy atomic orbitals of the constituent hydrogen atoms are 1s. The lowest set of molecular orbitals 1o and 2o* (sometimes labeled as o1s and o*1s) are formed by two possible linear combinations of 1s atomic orbitals of each atom, which describe two different possible modes of overlap of their wave functions. This approach is shown in Figure 2.10.

The plus linear combination describing 1o involves a constructive overlap (similar to constructive interference of light waves) of the electronic wave functions of the individual atoms leading to an increased electron density between the two nuclei. This increased electron density between the two nuclei acts as a spring to bind them and overcome the nuclear-nuclear repulsion. The energy of the resulting molecular orbital 1o is lowered (stabilized) compared to that of the individual atomic orbitals 1sA and 1sB. This is called a bonding molecular orbital. Contrasting this is the minus combination, which leads to a destructive overlap of the wave functions. This results in cancellation of the electron density in the region between the nuclei. The energy of the resulting molecular orbital 2o* is raised compared to that of the constituent atomic orbitals 1s. The 2o* is, therefore, an antibonding molecular orbital. The star symbol as a superscript on the right-hand side represents an antibonding orbital. The symbol o represents the overlap of atomic orbitals along the internuclear axis. In general, the mixing of two atomic orbitals centered on two atoms produces two molecular orbitals. This principle can be used to form molecular orbitals of high erenergies derived from the mixing of higher-energy atomic orbitals (2s, 2p; 3s, 3p, 3d, etc.) centered on individual atoms involved in a bond.

In the case of directional orbitals such as p, there are three p orbitals—px, py, and pz—directed toward the x, y, and z axes, respectively. If z is taken as node

destructive (antibonding) interaction

constructive (bonding) interaction o

Energy

p*antibonding MO

p bonding MO

Figure 2.11. Schematics of p and p* molecular orbital formation by the overlap of two p-type atomic orbitals.

the internuclear axis, then only the pz orbital can overlap along the internuclear direction and thus form the o and o* molecular orbitals. The px and py orbitals can then overlap only laterally (in directions perpendicular to the internuclear axis), forming p and p* molecular orbitals. The formation of the p and p* orbitals by overlap of two atomic orbitals is schematically shown in Figure 2.11. These molecular orbitals are not as stable as the o orbitals and are involved in bonding only when a multiple bond is formed between two atoms. The occupation of a given bonding molecular orbital by two electrons in a spin-paired configuration defines the formation of a bond. The occupation of an o orbital by a pair of electrons defines a single o bond; the electrons involved are called o electrons. The occupation of a p orbital by a pair of electrons defines a p bond; the electrons involved are called p electrons. The various o bonds are shown in Figure 2.12.

Another feature to point out for the bonding and antibonding orbitals is that the energies of these orbitals, obtained as a function of internuclear separation (using the clamped nuclei approximation), behave differently for a o and o* orbital in the H2+ molecule. This is shown in Figure 2.13.

The bonding orbital energy Eo exhibits a minimum corresponding to a bound (stable) state at the internuclear separation Re, called the equilibrium bond length (Levine, 2000). The amount of the energy lowering, De, with respect to the unbound configuration (R = •), is called the equilibrium binding energy or equilibrium dissociation energy, which is needed to dissociate the molecule into its constituent atoms. By contrast, the antibonding orbital

s bonding MO

node

o* antibonding MO Figure 2.12. Examples of the various o bonding.

Figure 2.13. The energies of the bonding and the antibonding orbitals in H2+ obtained as a function of the internuclear separation R.

energy Eo* shows no binding since the energy monotonically increases as the internuclear separation R is decreased. It represents a dissociate state, as opposed to a bound state for the o orbital.

The electronic configurations of a many-electron diatomic molecule are derived by successively filling electrons in molecular orbitals of increasing energy (the Aufbau principle) while observing Pauli's exclusion principle and Hund's rule. One of the greatest triumphs of the MO theory is its prediction of two unpaired electrons for the lowest-energy (ground-state) configuration of the oxygen molecule, with the overall spin S = 1 and thus the spin multiplicity 2S + 1 = 3 (a triplet state). Because a net spin (nonzero spin) gives a a

rise to paramagnetism, the MO theory was thus successful in explaining the observed paramagnetism of the O2 molecule.

The behavior described above applies to a homonuclear diatomic molecule for which the same type of atomic orbitals on two binding atoms combine to form molecular orbitals and the coefficient of mixing between the atomic orbitals on each of the binding atoms is the same. This is the case of a true covalent bond where the electronic probability distribution (electron density) on each atom is the same. In the case of a heteronuclear diatomic molecule, such as HF, the following considerations hold true (Levine, 2000):

• The LCAO-MO method now involves mixing of atomic orbitals, which are energetically similar. For example, the 1s atomic orbital of H combines with the 2pz orbital on F to form the o and o* molecular orbitals for HF (see Figure 2.12).

• In a more accurate description of bonding, more than one atomic orbital on one atom may be involved in forming a linear combination. The mixing coefficient is determined by the variation principle, which imposes the condition that the energy be minimized with respect to the mixing coefficients (which are treated as adjustable parameters).

• Certain atomic orbitals (forming the inner core of an atom) do not significantly mix in bond formation. As a result, the energies of these atomic orbitals in the molecule are basically the same as that in the unbound atomic state. These are called nonbonding molecular orbitals or n orbitals. In the case of HF, the inner 1s orbital of F is a nonbonding orbital.

• The mixing coefficients for atomic orbitals centered on different atoms, when optimized by the variation principle, may not be equal, indicating a higher electron density at a more electronegative atom (such as F in HF). This represents a polar bond where one atom (F) becomes slightly negative (charge S-) due to increased electron density at its site and the other atom (H) becomes slightly positive (charge S+) due to a decrease in the electron density. The charge separation in a polar bond is represented by a dipole moment m = SRe, where Re is the equilibrium bond length and S represents the charge of opposite sign on each atom.

For a polyatomic molecule, the electronic energy is a function of both the bond lengths and bond angles, which define its geometry. Quantum mechanical methods of geometry optimization are often used to predict the geometry. The quantum mechanical method used to solve the electronic Schrodinger equation involves the integration of a differential equation and thus requires solutions of many integrals. Often these integrals are complex and extremely time-consuming. Therefore, a twofold approach is adopted (Levine, 2000):

• The ab initio approach considers all electrons and evaluates all integrals explicitly. This approach requires a great deal of computational time and is used for small or moderately sized molecules. Fortunately, as the speed and efficiencies of computers increase, the size of molecules that can be readily handled by the ab initio method will also increase.

• The semiempirical approaches may simplify the Schrodinger equation by considering only the valence electrons in outer orbitals of the bonding atoms and approximate certain integrals by using adjustable parameters to fit certain experimentally observed physical quantities (such as ioniza-tion energies required to strip electrons from an atom). This approach is more popular, because large molecules and polymers can be treated with relative ease using even a desktop computer.

In the case of polyatomic molecules, one also takes advantage of the symmetry of a molecule to simplify the LCAO-MO approach by making a linear combination of orbitals, which have the same symmetry characteristics of molecules. These symmetry characteristics are defined by the operations (transformations) of various symmetry elements that lead to indistinguishable configurations. These symmetry elements collectively define the point group symmetry of a molecule. Some of these symmetry elements (Levine, 2000; Atkins and dePaula, 2002) are:

• An axis of rotation Cn, such as a sixfold axis of rotation (C6) along an axis perpendicular to the plane of a benzene ring and passing through its center

• A plane of symmetry o, such as the plane of the benzene ring

• A center of inversion i (also called a center of symmetry) such as the center of the benzene ring

As shown in Figure 2.14, inversion of the benzene molecule with respect to its center produces an indistinguishable position. Therefore, it is a centrosym-metric molecule possessing the inversion symmetry. By contrast, the inversion of a chlorobenzene molecule with respect to its center produces a distinguishable configuration (chlorine is now in the down position, as shown in Figure 2.14). Therefore, chlorobenzene is noncentrosymmetric and does not possess inversion symmetry. For molecules possessing inversion symmetry, the

Cl benzene chlorobenzene Cl

Figure 2.14. The structures of a benzene molecule and a chlorobenzene molecule and the inversion operation.

benzene chlorobenzene Cl

Figure 2.14. The structures of a benzene molecule and a chlorobenzene molecule and the inversion operation.

molecular orbitals, which do not change sign under inversion, are labeled as g (gerade). Those that change sign under inversion are labeled as u (ungerade).

In a complete description, the molecular orbitals of a molecule are labeled by the representations of the symmetry point group of a molecule. However, a detailed discussion of this symmetry aspect is beyond the scope of this book. For a simple reading, refer to the physical chemistry book by Atkins and dePaula (2002) or any other physical chemistry text.

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