In the Lewis theory of bonding, a covalent bond is a pair of electrons shared between two nuclei. In the more modern view, this has changed relatively little, except that the electron pair is now placed into a molecular orbital, which encompasses both nuclei.
Molecular orbitals differ from atomic orbitals in one very important respect. Atomic orbitals are associated with one and only one nucleus; molecular orbitals are associated with at least two nuclei. Atomic orbitals are always labeled with the Roman alphabet (e.g., s, p, d, f, g), whereas molecular orbitals are always labeled with Greek letters (e.g., σ, π, ∆).
22. Bohr, N. Z. Phys. 1922, 9, 1.
23. Hund, F. Z. Pys. 1925, 33, 345; Linienspektren und periodisches Systeme der Elements (Springer: Berlin, 1927), p. 124.
24. Friedrich Hund (1896-1997) was educated at Gửttingen and Marburg (PhD, Gửttingen, 1922). His aca- demic career began at Gửttingen, and he held positions at Rostock, Leipzig, Jena, and Frankfurt before return- ing to Gửttingen to finish his career. Leading sources for more biographical information on this centenarian scientist can be found at the Leipzig University web site: http://www.uni-leipzig.de/unigeschichte/professoren katalog/leipzig/Hund_67/
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The formation of molecular orbitals from atomic orbitals is usually represented as in- volving overlap of the two atomic orbitals to form the new molecular orbitals. Mathemat- ically, this involves the linear combination (addition or subtraction) of the wave functions for the atomic orbitals according to a strict set of rules: the linear combination of atomic orbitals (LCAO) method. One of the most stringent rules for combining wave functions (orbitals) can be set forth as a “rule of conservation of orbitals.”
During the formation of a set of molecular orbitals from a set of atomic orbitals on two or more atoms, the mathematics mandates that the number of molecular orbitals produced must be exactly equal to the number of atomic orbitals used. Orbitals can be altered but neither created nor destroyed.
Molecular orbitals can be generated by coaxial overlap of orbitals to give cylindrically symmetrical σ and σ* orbitals or by coplanar overlap of orbitals to give π and π* orbitals with mirror plane symmetry. In forming two molecular orbitals by overlap of two atomic orbitals (Figures 4.4 and 4.5), the only permitted combinations of atomic wave functions (designated fa and fb for the two atoms, a and b) are (1) the sum, (1/√2)(fa + fb), termed
§ Answer for Worked Problem:
The 20 valence electrons of the molecule are distributed into this set of orbitals according to the Aufbau principle. This gives the following electron configuration for the molecule, with two unpaired electrons, each in one of the two degenerate n orbitals.
Energy
σ σ∗
π π∗
n
Figure 4.4 The overlap of hydrogen atomic orbitals. Subtracting the wave functions, or out- of- phase overlap, gives an antibonding molecular orbital. Adding the wave functions, or in-phase overlap, gives a bonding orbital.
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in-phase overlap) and (2) the difference, (1/√2)(fa– fb), termed out-of-phase overlap).
The factor 1/√2 is called a normalization factor and simply scales the wave function so that its square is equal to 1.
These two combinations arise from the mathematical constraints on orbital overlap.
The molecular orbitals formed must be orthogonal (i.e., the product of their wave func- tions must be zero), and the sum of the squares of the molecular orbitals must be equal to the sum of the squares of the atomic orbitals from which they are formed. In-phase over- lap gives a molecular orbital lower in energy than either of the atomic orbitals used in its formation; this is the bonding molecular orbital.
Out-of-phase overlap gives a molecular orbital higher in energy than either of the atomic orbitals used in its formation; this is the antibonding molecular orbital. The Aufbau principle, the Pauli Exclusion principle, and Hund’s rule apply to molecules in ex- actly the same way they apply to atoms. Thus, electrons occupy the bonding orbitals first, and antibonding orbitals remain empty.
Up to this point, we have not addressed the effects of orbital energies on the efficiency of orbital overlap. Fortunately, the relationship is both simple and logical—the efficiency of orbital overlap increases as the energies of the two orbitals involved become more closely matched—or more nearly equal. In particular, orbital overlap is most efficient when orbit- als in the same electron shell are involved. A carbon 2p orbital, for example, will form a very strong π bond when it overlaps with the 2p orbital of an oxygen atom, and it will form only a relatively weak π bond when it overlaps with the 3p orbital of a phosphorus atom.
Hybrid Atomic Orbitals: Valence Bond Theory
The methane molecule, CH4, is known to be tetrahedral, and any bonding model that we use to describe the structure of methane should produce this geometric result. However, the atomic orbitals of the valence shell of carbon are not oriented at 109.471 . . .° to each other but rather at 90° to each other. This problem is solved quite elegantly one way by noting that one can inscribe a tetrahedron in a cube, which permits the formation of a tetrahedral molecule from a set of orbitals that are perpendicular to each other. In this way, one can obtain four bonding and four antibonding orbitals, as shown in Figure 4.6.
The result of this approach in methane is one σ-symmetric bonding orbital and its corresponding σ* antibonding orbital as well as a set of three degenerate π-symmetric
1/√2(φa – φb)
antibonding
1/√2(φa + φb)
bonding
Figure 4.5 Out-of-phase coplanar overlap of p orbitals gives an antibonding π*
orbital. In-phase coplanar overlap of p orbitals gives a bonding π orbital.
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σ π
π∗
σ∗
Figure 4.6 Linear combina- tions of atomic orbitals to generate one set of molecular orbitals for methane.
orbitals and their corresponding π* antibonding orbitals.ả Photoelectron spectroscopy of the methane molecule (which gives information about the electron energies of the mole- cule) shows two peaks in the ratio 3:1, which lends support to this model of the methane molecule.25
A similar set of wave functions can be derived from the 12 atomic orbitals of carbon and hydrogen for the 12 LCAO molecular orbitals of the ethylene molecule; these orbitals are shown in Figure 4.7. There is again a pair of unique orbitals with σ and σ* symmetry derived from the carbon 2s orbitals and the hydrogen 1s orbitals, with two sets of two π-symmetry orbitals and their corresponding π* antibonding orbitals. These antibonding orbitals are formed from the carbon 2px orbitals and the hydrogen 1s orbitals, and a corre- sponding set is formed from the carbon 2py orbitals and the hydrogen 1s orbitals. The ethylene molecule also has a pair of unique π and π* orbitals formed from the carbon 2pz orbitals (compare with Figure 4.5).
This LCAO method for describing the bonding in molecules is well adapted to com- puters, but the molecular orbitals rapidly become too complex for most individuals to
25. Hamrin, K.; Johansson, G.; Gelius, U.; Fahlman, A.; Nordling, C.; Siebahn, K. Chem. Phys. Lett. 1968, 1, 613.
ả. Wavefunctions of Simple Molecules from Atomic Basis Orbitals The wave functions of the molecular methane shown above are:
fσ = (1/2√2)(2fs + fa + fb + fc + fd) fπ(x) = (1/2√2)(2fx + fa + fb – fc – fd) fπ(y) = (1/2√2)(2fy + fa – fb + fc – fd) fπ(z) = (1/2√2)(2fz + fa – fb – fc + fd) fπ*(x) = (1/2√2)(2fx – fa – fb + fc + fd) fπ*(y) = (1/2√2)(2fy – fa + fb – fc + fd) fπ*(z) = (1/2√2)(2fz – fa + fb + fc – fd) fσ* = (1/2√2)(2fs – fa – fb – fc – fd)
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visualize without one. The most widely used model for visualizing bonding in molecules describes it in terms of two types of bond: σ bonds, formed by coaxial overlap of atomic orbitals, and π bonds, formed by coplanar overlap of p orbitals. The most stable arrange- ment of electrons in a molecule corresponds to the one in which all atoms are linked by σ bonds, with the minimum number of σ bonds being used. In this model, multiple bonds consist of one—and only one—σ bond, and one or more π bonds. π Orbitals differ from σ orbitals in one very important way; because the p atomic orbitals used in their forma- tion have a nodal plane through the nucleus of the atom, π molecular orbitals also have a nodal plane through the two nuclei—π electrons do not occupy the space directly between the two nuclei. This means that π electrons are always further from the nuclei defining the bond than are σ electrons, so that they are not so tightly bound, and they are more polar- izable and more responsive to external electronic influences. π Bonds are more reactive than σ bonds.
The side-by-side overlap of the p orbitals required to form the π and π* orbitals also imposes another very stringent requirement on molecules containing π bonds. Because π overlap is most effective when the two p orbitals are parallel and coplanar, any rotation of the atoms at the two ends of the π bond relative to each other will weaken, and eventually break, the π bond. Therefore, unless a compound has two π bonds between the same two atoms (i.e., a triple bond), such rotation about the π bond is forbidden.
Problems
4-1 Draw a set of LCAO molecular orbitals for acetylene, HC ≡ CH, similar to those drawn in Figure 4.7 for ethylene.
4-2 Draw a set of molecular orbitals for ethane similar to those in Figure 4.7 for ethylene. (Hint: put the two carbon atoms along the x axis, with two hydrogen atoms in the xz plane.)
The molecular orbitals generated by the LCAO approach above are all delocalized—
every molecular orbital encompasses every nucleus of the molecule. Although this pz
px
py
Figure 4.7 Linear combina- tions of atomic orbitals to generate one set of molecular orbitals for ethylene.
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approach is ideally suited for the generation of molecular orbitals by computer programs, it is rather less intuitive for most organic chemists, who find it much easier to visualize the localized bonds of the Lewis model. The same holds true when it comes to discussing the chemistry of compounds; it is often less confusing to use the simple Lewis model of bond- ing with its localized covalent bonds.
An alternative to this process of forming delocalized molecular orbitals is provided by valence bond theory, which is basically a theoretical form of Lewis bonding theory.
At its core, it involves the overlap of hybrid atomic orbitals to form localized molecu- lar orbitals. The process of hybridization,26 which is one of the simplest models for obtaining directed atomic orbitals capable of forming localized molecular orbitals, in- volves combining the wave functions of orbitals of a single atom according to simple mathematical principles to generate a set of new atomic orbitals. The mathematics of hybridization results in a set of wave functions that are also valid solutions to the Schrửdinger equation and a set of orbitals whose overall symmetry, in terms of the electron probability distribution of the atom, remains unchanged. Hybridization of carbon and other second-row elements always involves a 2s orbital, with one or more 2p orbitals. The unused p orbitals remain unchanged, orthogonal to the hybrid orbit- als. In the discussion that follows, we will refer to the p orbitals by their cartesian designations: x, y, and z.
As we discuss hybridization, it will be easy to get the impression that atoms undergo this real, physical process before they form molecules.27 This is not now generally accepted to be the case, although the case to the contrary has been made quite eloquently by Ala- bugin.28 As with all versions of molecular orbital theory, hybridization theory is a useful model that allows us to rationalize certain properties of the chemical bonds in molecules—in this case, it is the strength, length, and orientation of covalent bonds. It is not a description of a physically real process.
There are three common hybrids formed by second-row elements: sp, sp2, and sp3 hybrid orbitals. Hybrid orbitals are characterized by the hybridization index, which is n for an spn hybrid orbital.29 Although the wave function for an s orbital is the equation for a sphere with no directional component, the wave functions for p orbitals all contain an important directional component (e.g., the directional component of the px orbital causes the wave function to have a value of zero at all points where x = 0). Consequently, all hybrid orbitals have a directional component.
The simplest hybrid orbitals are sp hybrids, formed from the s orbital and a single p orbital. The hybridization of the s orbital with the px orbital gives a pair of wave func- tions that are oriented along the x axis; sp hybrid orbitals are colinear, with the front lobes oriented at 180° to each other (this geometry is described as linear or, occasion- ally, as digonal). Note that the two unused p orbitals (the py and pz orbitals in this case) remain unchanged and are oriented along the y and z axes, orthogonal to the hybrid orbitals.
The set of sp2 hybrid orbitals is formed from the s orbital and two of the degenerate p orbitals. Using the px and py orbitals in the hybridization process gives three degenerate sp2
26. (a) Pauling, L. Proc. Natl. Acad. Sci. 1928, 14, 359; J. Am. Chem. Soc. 1931, 53, 1367; The Nature of the Chemical Bond, 3rd. ed. (Cornell University Press: Ithaca, NY: 1960), p.111ff. (b) Slater, J.C. Phys. Rev. 1931, 37, 481. (c) van Vleck, J.H. J. Chem. Phys. 1933,1, 1778. (d) Hultgren, R. Phys. Rev. 1932, 40, 891.
27. A state known as the “valence state,” an excited form of the atom has been proposed, although it is pointed out that the carbon atom, for example, never attains this state: (a) van Vleck, J.H. J. Chem. Phys. 1933, 1, 177, 219; 1934, 2, 20. (b) Mulliken, R.S. J. Chem. Phys. 1934, 2, 782; J. Phys. Chem. 1952, 56, 295. (c) Moffitt, W.E.
Proc. Roy. Soc. A 1950, 202, 534, 548.
28. (a) Alabugin, I.V.; Manoharan, M. J. Comp. Chem. 2007, 28, 373. (b) Alabugin, I.V.; Manoharan, M.; Pea- body, S.; Weinhold, F. J. Am. Chem. Soc. 2003, 125, 5973.
29. (a) Bingel, W.A.; Lüttke, W. Angew. Chem. Int. Ed. Engl. 1981, 20, 899. (b) Lowry, T.H.; Richardson, K.S.
Mechanism and Theory in Organic Chemistry (Harper & Row: New York, 1976), pp. 43-49.
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hybrids in the xy plane, with one oriented along the y axis, and the other two at 120° to it;
sp2 orbitals lie in a common plane, with the front lobes oriented toward the corners of an equilateral triangle (this geometry is most often referred to as trigonal planar). The unused p orbital (the pz orbital in this case) remains unchanged and lies along the axis orthogonal to the plane of the sp2 orbitals (in this case, the z axis).
The four degenerate sp3 hybrid orbitals derived from the s orbital and the complete set of degenerate p orbitals (the px, py, and pz orbitals) are oriented toward the corners of a tetrahedron that are defined by the Cartesian coordinates (1, 1, 1), (–1, 1, 1), (1, –1, –1), and (–1, 1, –1). This geometry is almost always referred to as tetrahedral.
The projections of the wave functions of the three types of hybrid atomic orbitals for a carbon atom projected on the plane containing the orbital axis are plotted on the same scale in Figure 4.8, which also shows a superimposed plot of projection of the nodal sur- faces on the plane containing the principal axis of the orbital.
Hybrid atomic orbitals still retain some of the character of the pure atomic orbitals from which they are derived. Thus, one can define a property of the orbital, its s character, which for an spn orbital is 1/(1 + n). The amount of s character relates the properties of the hybrid orbital to the extent to which the s orbital wave function contributes to the wave function of the hybrid. All hybrid orbitals have a larger front lobe and a smaller back lobe, with the relative size (i.e., volume) of the front lobe increasing as the s character of the orbital increases. The sp orbital has the largest front lobe and smallest back lobe, whereas the sp3 orbital has the two lobes closer to equal in size.
Figure 4.8 highlights another difference between hybrid orbitals and the atomic orbitals from which they are derived. The nucleus is located on the nodal plane of a p orbital and at the center of the spherical nodes of the s orbitals, but the nodal surface of a hybrid orbital neither passes through the atomic nucleus nor is located symmetrically with respect to it—
the nucleus is located within the back lobe of the hybrid orbital. The diagram in Figure 4.9 shows that the nucleus most closely approaches the nodal surface in the sp3 hybrid and is further from the nodal surface as the s character of the hybrid orbital increases.
sp sp2 sp3
Figure 4.8 Wave functions for hybrid orbitals of carbon projected on a plane containing principal axis of the orbital. Orbital wave function calculations courtesy of Dr. Frederick W. King, Univer- sity of Wisconsin-Eau Claire. Projection diagrams were prepared using Atom in a Box (Dauger Research, Inc.).
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Hybridization has another effect: as the s character of the hybrid orbital increases, the front lobe becomes shorter. That is, the distance from the nucleus to where there is the maximum probability of finding an electron becomes shorter. This, of course, puts the electrons closer to the nucleus, so that they are held more strongly by the nucleus. It is worth remembering at this stage that one defines an increased hold by the nucleus on the electrons as increased electronegativity. In other words, the higher the s character of the hybrid orbitals, the higher the electronegativity of the atom.30 The electronegativity of carbon decreases with hybridization in the order sp > sp2 > sp3. It also means that the length of the bond decreases (and the covalent radius of the atom decreases) as the hybrid- ization changes from sp3 to sp2 to sp.31
The most efficient overlap of the hybrid orbital always occurs through the front lobe, which makes σ bonds formed from hybrid atomic orbitals highly directional. In addition, the hybridization model results in the σ bonds that a second-row atom forms by using hybrid orbitals being stronger than those formed by the same atom from unhybridized p orbitals.32 This means that this model predicts that an atom will form its strongest and most directional σ bonds when it uses hybrid atomic orbitals to form them.ả It is also
30. (a) Sanderson, R.T. J. Am. Chem. Soc. 1983, 105, 2259; J. Chem. Educ. 1988, 65, 112, 223. (b) Walsh, A.D.
Disc. Faraday Soc. 1947, 2, 18.
31. (a) Allen, F.H.; Kennard, O.; Watson, D.G.; Brammer, L.; Orpen, A.G.; Taylor, R. J. Chem. Soc, Perkin Trans. 1 1987, 2, S1-S19. (b) Smith, M.B.; March, J. March’s Advanced Organic Chemistry. Reactions, Mecha- nisms, and Structure, 5th ed. (Wiley-Interscience, New York, 2001), p. 20. (c) Pauling, L. The Nature of the Chemical Bond, 3rd ed. (Cornell University Press: Ithaca, 1960), p. 224.
32. Pauling, L.; Sherman, J. J. Am. Chem. Soc. 1937, 59, 1450.
Figure 4.9 Location of nodal surfaces of s, sp, sp2, and sp3 hybrid orbitals
ả. Hybridization and Molecular Orbital Formation
Hybrid orbitals are formed by the combination of atomic orbitals of the same atom. Mathematically, the process involves the linear combinations of atomic wave functions below. The process leads to degenerate or- bitals whose orientation is determined by the coefficients of the p wave functions used in the hybridization process. X, y, and z represent the px, py, and pz wave functions.
sp sp2 sp3
fa=(1/√2)[s + x] fa=(1/√3)[s + (√2)y] fa=(1/2)[s + x + y + z]
fa=(1/√2)[s + x] fb=(1/√3)[s + (1/√2)x – (√6/2)y] fb=(1/2)[s – x – y + z]
unchanged: y, z fc=(1/√3)[s – (1/√2)x – (√6/2)y] fc=(1/2)[s + x – y – z]
unchanged: z fd (1/2)[s – x + y – z]
unchanged: none
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generally observed that lone pairs on second-row elements tend to occupy hybrid atomic orbitals in preference to unhybridized atomic orbitals unless the atom adjacent to the atom carrying the lone pair is sp2 or sp hybridized.
When one compares the strength of σ bonds formed from hybrid orbitals, the stron- gest bonds are formed from sp hybrid orbitals and the weakest from sp3 hybrid orbitals.
The relative strengths of σ bonds can be related to the s character of the hybrid orbitals used to form them. As the s character of the hybrid atomic orbitals used to form them in- creases, σ bonds become stronger.33 In contrast to the σ bond case, the π-type overlap of hybrid atomic orbitals to form π and π* molecular orbitals is seldom observed. Thus, all π bonds in a molecule are formed from unhybridized p orbitals.
Some important generalizations about hybrid atomic orbitals and molecular orbitals (covalent bonds) hold true for most organic molecules.
1. All σ bonds to carbon in organic compounds are formed from hybrid atomic orbitals, π bonds are formed from unhybridized p orbitals, and lone pairs usually occupy hybrid orbitals.
2. Every atom of a molecule is involved in at least one σ bond.
3. Atoms in a molecule will form the fewest σ bonds consistent with rule 2.
4. The strongest σ bonds are formed from hybrid atomic orbitals with the highest possible s character.
5. All multiple bonds are formed from a single σ bond, with one or more π bonds.
6. The shape of a molecule is determined only by the σ bonding framework only; π bonds have no effect on molecular shape.
The τ Bond Model of Multiple Bonds: Variable Hybridization
Multiple bonds in organic compounds can be described in two ways, the σ/π description, which is the one that will be used in this book, and the t bond system,34 which is actually older and in which the bonds between the carbon atoms are “bent” bonds. This descrip- tion of bonding has been used very effectively to rationalize the bonding in cyclopro- panes,35 where the electron density of the σ bonds is found outside the internuclear axis of the ring, and it has been shown to work just as well as the σ/π model.36
33. (a) Pauling, L. Proc. Natl. Acad. Sci. 1949, 35, 229. (b) Walsh, A.D. Trans. Faraday Soc. 1947, 43, 60. (c) Mulliken, R.S. J. Am. Chem. Soc. 1955, 72, 4493; J. Chem. Phys. 1951, 19, 900. (d) Coulson, C.A. Valence (Oxford Press/Clarendon Press: London, 1952), p. 198.
34. (a) Wintner, C.E. J. Chem. Educ. 1987, 64, 587, and references therein. (b) Palke, W.E. J. Am. Chem. Soc.
1986, 108, 6543. (c) Carroll, F.A. Perspectives in Structure and Mechanism in Organic Chemistry (Brooks/Cole:
Pacific Grove, 1998), p. 47.
35. (a) Coulson, C.A.; Moffitt, W.E. J. Chem. Phys. 1947, 15, 151; Phil. Mag. 1949, 40, 1. (b) Walsh, A.D. Trans.
Faraday Soc., 1949, 45, 179. (c) Hamilton, J.G.; Palke, W.E. J. Am. Chem. Soc. 1993, 115, 4159. (d) de Meijer, A.
Angew. Chem. Int. Ed. Engl. 1979, 18, 809. (e) Wiberg, K.B. In Rappoport, Z., Ed. The Chemistry of the Cyclopro- pyl Group (John Wiley: New York, 1987), ch. 1. (f) Rozsondai, B. In Rappoport, Z., Ed. The Chemistry of the Cyclopropyl Group (John Wiley: New York, 1995), vol. 2, ch. 3.
36. Schultz, P.A.; Messmer, R.P. J. Am. Chem. Soc. 1988, 110, 8258; 1993, 115, 10925, 10943.
The wave functions of the molecular methane generated from hybrid wave functions are of the simplified form:
cσ = (1/√2)(fH + fC) cσ* = (1/√2)(fH – fC)
where fC is the wave function of a carbon hybrid atomic orbital, and fH is the wave function of a hydrogen 1s orbital.
This can be expanded to (e.g., fx is the carbon 2px wave function):
cσa = 1/(2√2)[2fH + fs + fx + fy + fz] cσ*a = 1/(2√2)[2fH – fs – fx – fy – fz] cσb = 1/(2√2)[2fH + fs – fx – fy + fz] cσ*b = 1/(2√2)[2fH – fs + fx + fy – fz] cσc = 1/(2√2)[2fH + fs + fx – fy – fz] cσ*c = 1/(2√2)[2fH – fs – fx + fy + fz] cσd = 1/(2√2)[2fH + fs – fx + fy – fz] cσ*d = 1/(2√2)[2fH – fs + fx – fy + fz]
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