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Chapter 5
Molecular Orbitals
Molecular orbital theory uses group theory to describe the bonding in molecules; it comple- ments and extends the introductory bonding models in Chapter 3. In molecular orbital theory the symmetry properties and relative energies of atomic orbitals determine how these orbitals interact to form molecular orbitals. The molecular orbitals are then occupied by the available electrons according to the same rules used for atomic orbitals as described in Sections 2.2.3 and 2.2.4. The total energy of the electrons in the molecular orbitals is compared with the initial total energy of electrons in the atomic orbitals. If the total energy of the electrons in the molecular orbitals is less than in the atomic orbitals, the molecule is stable relative to the separate atoms; if not, the molecule is unstable and predicted not to form. We will first describe the bonding, or lack of it, in the first 10 homonuclear diatomic molecules (H 2 through Ne 2 ) and then expand the discussion to heteronuclear diatomic molecules and molecules having more than two atoms. A less rigorous pictorial approach is adequate to describe bonding in many small mole- cules and can provide clues to more complete descriptions of bonding in larger ones. A more elaborate approach, based on symmetry and employing group theory, is essential to under- stand orbital interactions in more complex molecular structures. In this chapter, we describe the pictorial approach and develop the symmetry methodology required for complex cases.
5.1 Formation of Molecular Orbitals from Atomic Orbitals
As with atomic orbitals, Schrödinger equations can be written for electrons in molecules. Approximate solutions to these molecular Schrödinger equations can be constructed from linear combinations of atomic orbitals (LCAO) , the sums and differences of the atomic wave functions. For diatomic molecules such as H 2 , such wave functions have the form
= ca c a + cb c b
where is the molecular wave function, c a and c b are atomic wave functions for atoms a and b, and ca and cb are adjustable coefficients that quantify the contribution of each atomic orbital to the molecular orbital. The coefficients can be equal or unequal, positive or negative, depending on the individual orbitals and their energies. As the distance between two atoms is decreased, their orbitals overlap, with significant probability for electrons from both atoms being found in the region of overlap. As a result, molecular orbitals form. Electrons in bonding molecular orbitals have a high probability of occupying the space between the nuclei; the electrostatic forces between the electrons and the two positive nuclei hold the atoms together. Three conditions are essential for overlap to lead to bonding. First, the symmetry of the orbitals must be such that regions with the same sign of c overlap. Second, the atomic orbital energies must be similar. When the energies differ greatly, the change in the energy of electrons upon formation of molecular orbitals is small, and the net reduction in energy 117117
118 Chapter 5 | Molecular Orbitals
of the electrons is too small for significant bonding. Third, the distance between the atoms must be short enough to provide good overlap of the orbitals, but not so short that repul- sive forces of other electrons or the nuclei interfere. When these three conditions are met, the overall energy of the electrons in the occupied molecular orbitals is lower in energy than the overall energy of the electrons in the original atomic orbitals, and the resulting molecule has a lower total energy than the separated atoms.
5.1.1 Molecular Orbitals from s Orbitals
Consider the interactions between two s orbitals, as in H 2. For convenience, we label the atoms of a diatomic molecule a and b , so the atomic orbital wave functions are c(1 s (^) a ) and c(1 s (^) b ). We can visualize the two atoms approaching each other, until their electron clouds overlap and merge into larger molecular electron clouds. The resulting molecular orbitals are linear combinations of the atomic orbitals, the sum of the two orbitals and the difference between them. In general terms for H (^2)
1 s 2 = N^3 ca c 11 s a 2 + cb c 11 s b 2 4^ =
3 c 11 s a 2 + c 11 s b 2 4 1 H a + H b 2
and 1 s* 2 = N^3 ca c 11 s a 2 - cb c 11 s b 2 4^ =
3 c 11 s a 2 - c 11 s b 2 4 1 H a - H b 2
where N = normalizing factor, so 1 * d t = 1
ca and cb = adjustable coefficients
In this case, the two atomic orbitals are identical, and the coefficients are nearly identical as well.*^ These orbitals are depicted in Figure 5.1. In this diagram, as in all the orbital diagrams in this book (such as Table 2.3 and Figure 2.6), the signs of orbital lobes are indicated by shading or color. Light and dark lobes or lobes of different color indicate opposite signs of . The choice of positive and negative for specific atomic orbitals is arbitrary; what is important is how they combine to form molecular orbitals. In the dia- grams in Figure 5.2 , the different colors show opposite signs of the wave function, both
More precise calculations show that the coefficients of the s (^) orbital are slightly larger than those for the s orbital; but for the sake of simplicity, we will generally not focus on this. For identical atoms, we will use c (^) a = c (^) b = 1 and N = 112. The difference in coefficients for the s and s^ orbitals also results in a larger change in energy (increase) from the atomic to the s^ molecular orbitals than for the s orbitals (decrease). In other words, E s* 7 E s, as shown in Figure 5.1.
s^ = s
(^1) Cc 11 sa 2 - c 11 sb 2D √ 2
1 s (^) a 1 sb
1 sa 1 sb
¢ E s*
¢ E s
s
s*
overlap 1 sa 1 sb
1 √ 2
s = s
Cc 11 s (^) a 2 + c 11 sb 2D
E
Figure 5.1 Molecular Orbitals from Hydrogen 1s Orbitals. The s molecular orbital is a bonding molecular orbital, and has a lower energy than the original atomic orbitals, since this combination of atomic orbitals results in an increased concen- tration of electrons between the two nuclei. The s* orbital is an antibonding orbital at higher energy since this combination of atomic orbitals results in a node with zero electron density between the nuclei.
120 Chapter 5 | Molecular Orbitals
nuclear attraction. Nonbonding orbitals are also possible. The energy of a nonbonding orbital is essentially that of an atomic orbital, either because the orbital on one atom has a symmetry that does not match any orbitals on the other atom or the orbital on one atom has a severe energy mismatch with symmetry-compatible orbitals on the other atom. The s (sigma) notation indicates orbitals that are symmetric to rotation about the line connecting the nuclei:
z
C 2 C 2
z
s* from s orbital s* from p (^) z orbital
An asterisk is frequently used to indicate antibonding orbitals. Because the bonding, nonbonding, or antibonding nature of a molecular orbital is not always straightforward to assign in larger molecules, we will use the asterisk notation only for those molecules where bonding and antibonding orbital descriptions are unambiguous. The pattern described for H 2 is the usual model for combining two orbitals: two atomic orbitals combine to form two molecular orbitals, one bonding orbital with a lower energy and one antibonding orbital with a higher energy. Regardless of the number of orbitals, the number of resulting molecular orbitals is always the same as the initial number of atomic orbitals; the total number of orbitals is always conserved.
5.1.2 Molecular Orbitals from p Orbitals
Molecular orbitals formed from p orbitals are more complex since each p orbital contains separate regions with opposite signs of the wave function. When two orbitals overlap, and the overlapping regions have the same sign, the sum of the two orbitals has an increased electron probability in the overlap region. When two regions of opposite sign overlap, the combination has a decreased electron probability in the overlap region. Figure 5.1 shows this effect for the 1 s orbitals of H 2 ; similar effects result from overlapping lobes of p orbit- als with their alternating signs. The interactions of p orbitals are shown in Figure 5.2. For convenience, we will choose a common z axis connecting the nuclei and assign x and y axes as shown in the figure. When we draw the z axes for the two atoms pointing in the same direction,^ the pz orbitals subtract to form s and add to form s orbitals, both of which are symmetric to rotation about the z axis, with nodes perpendicular to the line that connects the nuclei. Interactions between px and py orbitals lead to p and p* orbitals. The p (pi) notation indicates a change in sign of the wave function with C 2 rotation about the bond axis:
z z
C 2 C 2
As with the s orbitals, the overlap of two regions with the same sign leads to an increased concentration of electrons, and the overlap of two regions of opposite signs leads to a node of zero electron density. In addition, the nodes of the atomic orbitals become the
*The choice of direction of the z axes is arbitrary. When both are positive in the same direction , the difference between the p (^) z orbitals is the bonding combination. When the positive z axes are chosen to point toward each other, , the sum of the pz orbitals is the bonding combination. We have chosen to have the p (^) z orbitals positive in the same direction for consistency with our treatment of triatomic and larger molecules.
5.1 Formation of Molecular Orbitals from Atomic Orbitals | 121
nodes of the resulting molecular orbitals. In the p* antibonding case, four lobes result that are similar in appearance to a d orbital, as in Figure 5.2(c). The px , py , and pz orbital pairs need to be considered separately. Because the z axis was chosen as the internuclear axis, the orbitals derived from the pz orbitals are symmetric to rotation around the bond axis and are labeled s and s* for the bonding and antibonding orbitals, respectively. Similar combinations of the py orbitals form orbitals whose wave functions change sign with C 2 rotation about the bond axis; they are labeled p and p. In the same way, the px orbitals also form p and p orbitals. It is common for s and p atomic orbitals on different atoms to be sufficiently similar in energy for their combinations to be considered. However, if the symmetry properties of the orbitals do not match, no combination is possible. For example, when orbitals overlap equally with both the same and opposite signs, as in the s + px example in Figure 5.2(b), the bonding and antibonding effects cancel, and no molecular orbital results. If the symmetry of an atomic orbital does not match any orbital of the other atom, it is called a nonbonding orbital. Homonuclear diatomic molecules have only bonding and antibonding molecular orbitals; nonbonding orbitals are described further in Sections 5.1.4, 5.2.2, and 5.4.3.
5.1.3 Molecular Orbitals from d Orbitals
In the heavier elements, particularly the transition metals, d orbitals can be involved in bonding. Figure 5.3 shows the possible combinations. When the z axes are collinear, two dz^2 orbitals can combine end-on for s bonding. The dxz and dyz orbitals form p orbitals. When atomic orbit- als meet from two parallel planes and combine side to side, as do the dx^2 - y^2 and dxy orbitals with collinear z axes, they form 1 d 2 delta orbitals (Figure 1.2). (The d notation indicates sign changes on C 4 rotation about the bond axis.) Sigma orbitals have no nodes that include the line connecting the nuclei, pi orbitals have one node that includes the line connecting the nuclei, and delta orbitals have two nodes that include the line connecting the nuclei. Again, some orbital interactions are forbidden on the basis of symmetry; for example, pz and dxz have zero net over- lap if the z axis is chosen as the bond axis since the pz would approach the dxz orbital along a dxz node (Example 5.1). It is noteworthy in this case that px and dxz would be eligible to interact in a p fashion on the basis of the assigned coordinate system. This example emphasizes the importance of maintaining a consistent coordinate system when assessing orbital interactions.
e x a m p l e 5. Sketch the overlap regions of the following combination of orbitals, all with collinear z axes, and classify the interactions.
pz dxz s dyz
s and dz^2 s and dyz
dz^2 s
p (^) z and dxz
no interaction s interaction no interaction
EXERCISE 5.1 Repeat the process in the preceding example for the following orbital combinations, again using collinear z axes. px and dxz pz and dz 2 s and dx (^2) - y 2
5.2 Homonuclear Diatomic Molecules | 123
5.2.1 Molecular Orbitals
Although apparently satisfactory Lewis electron-dot structures of N 2 , O 2 , and F 2 can be drawn, the same is not true with Li 2 , Be 2 , B 2 , and C 2 , which violate the octet rule. In addi- tion, the Lewis structure of O 2 predicts a double-bonded, diamagnetic (all electrons paired) molecule ( O O), but experiment has shown O 2 to have two unpaired electrons, making it paramagnetic. As we will see, the molecular orbital description predicts this paramagnet- ism, and is more in agreement with experiment. Figure 5.5 shows the full set of molecular orbitals for the homonuclear diatomic molecules of the first 10 elements, based on the energies appropriate for O 2. The diagram shows the order of energy levels for the molecular orbitals, assuming significant interactions only between atomic orbitals of identical energy. The energies of the molecular orbitals change in a periodic way with atomic number, since the energies of the interacting atomic orbitals decrease across a period ( Figure 5.7 ), but the general order of the molecular orbitals remains similar (with some subtle changes, as will be described in several examples) even for heavier atoms lower in the periodic table. Electrons fill the molecular orbitals according to the same rules that govern the filling of atomic orbitals, filling from lowest to highest energy (aufbau principle), maximum spin
2 p
2 s 2 s
2 p
p u p u
p g * p g *
s u *
s g
1 s 1 s
s u * s g
s u *
s g
Figure 5.5 Molecular Orbitals for the First 10 Elements, Assuming Significant Interactions Only between the Valence Atomic Orbitals of Identical Energy.
124 Chapter 5 | Molecular Orbitals
multiplicity consistent with the lowest net energy (Hund’s rules), and no two electrons with identical quantum numbers (Pauli exclusion principle). The most stable configuration of electrons in the molecular orbitals is always the configuration with minimum energy, and the greatest net stabilization of the electrons. The overall number of bonding and antibonding electrons determines the number of bonds (bond order):
Bond order =
c a
number of electrons in bonding orbitals
b - a
number of electrons in antibonding orbitals
b d
It is generally sufficient to consider only valence electrons. For example, O 2 , with 10 electrons in bonding orbitals and 6 electrons in antibonding orbitals, has a bond order of 2, a double bond. Counting only valence electrons, 8 bonding and 4 antibond- ing, gives the same result. Because the molecular orbitals derived from the 1 s orbitals have the same number of bonding and antibonding electrons, they have no net effect on the bond order. Generally electrons in atomic orbitals lower in energy than the valence orbitals are considered to reside primarily on the original atoms and to engage only weakly in bonding and antibonding interactions, as shown for the 1 s orbitals in Figure 5.5; the difference in energy between the s g and s u ^ orbitals is slight. Because such interactions are so weak, we will not include them in other molecular orbital energy level diagrams. Additional labels describe the orbitals. The subscripts g for gerade , orbitals symmetric to inversion, and u for ungerade , orbitals antisymmetric to inversion (those whose signs change on inversion), are commonly used.^ The g or u notation describes the symmetry of the orbitals without a judgment as to their relative energies. Figure 5.5 has examples of both bonding and antibonding orbitals with g and u designations.
e x a m p l e 5. 2 Add a g or u label to each of the molecular orbitals in the energy-level diagram in Figure 5.2. From top to bottom, the orbitals are s u *, p g *, p u , and s g.
EXERCISE 5.2 Add a g or u label to each of the molecular orbitals in Figure 5.3(a).
5.2.2 Orbital Mixing
In Figure 5.5, we only considered interactions between atomic orbitals of identical energy. However, atomic orbitals with similar, but unequal, energies can interact if they have appropriate symmetries. We now outline two approaches to analyzing this phenomenon, one in which we first consider the atomic orbitals that contribute most to each molecular orbital before consideration of additional interactions and one in which we consider all atomic orbital interactions permitted by symmetry simultaneously. Figure 5.6(a) shows the familiar energy levels for a homonuclear diatomic molecule where only interactions between degenerate (having the same energy) atomic orbitals are considered. However, when two molecular orbitals of the same symmetry have similar energies, they interact to lower the energy of the lower orbital and raise the energy of the higher orbital. For example, in the homonuclear diatomics, the s g 12 s 2 and s g 12 p 2 orbit- als both have s g symmetry (symmetric to infinite rotation and inversion); these orbitals
*See the end of Section 4.3.3 for more details on symmetry labels.
126 Chapter 5 | Molecular Orbitals
on the basis of s @ p mixing, gain either a slightly bonding or slightly antibonding character and contribute in minor ways to the bonding. Each orbital must be considered separately on the basis of its energy and electron distribution.
5.2.3 Diatomic Molecules of the First and Second Periods
Before proceeding with examples of homonuclear diatomic molecules, we must define two types of magnetic behavior, paramagnetic and diamagnetic. Paramagnetic com- pounds are attracted by an external magnetic field. This attraction is a consequence of one or more unpaired electrons behaving as tiny magnets. Diamagnetic compounds, on the other hand, have no unpaired electrons and are repelled slightly by magnetic fields. (An experimental measure of the magnetism of compounds is the magnetic moment , a concept developed in Chapter 10 in the discussion of the magnetic properties of coordina- tion compounds.) H 2 , He 2 , and the homonuclear diatomic species shown in Figure 5.7 will now be discussed. As previously discussed, atomic orbital energies decrease across a row in the Periodic Table as the increasing effective nuclear charge attracts the electrons more strongly. The result is that the molecular orbital energies for the corresponding homonuclear
s u* 12 p 2
s u* 12 p 2
p g* 12 p 2
s u* 12 s 2
p g* 12 p 2
p u 12 p 2
p u 12 p 2
s g 12 p 2
s g 12 s 2
s u* 12 s 2
s g 12 s 2
s g 12 p 2
Li (^2) 1 0
Be (^2) 0 0
B 2 1 2
C (^2) 2 0
N 2 3 0
O 2 2 2
F 2 1 0
Ne 2 0 0
Bond order Unpaired e-
Figure 5.7 Energy Levels of the Homonuclear Diatomics of the Second Period.
5.2 Homonuclear Diatomic Molecules | 127
diatomics also decrease across the row. As shown in Figure 5.7, this decrease in energy is larger for s orbitals than for p orbitals, due to the greater overlap of the atomic orbitals that participate in s interactions.
H 2 [ S g^2 (1s)] This is the simplest diatomic molecule. The MO description (Figure 5.1) shows a single s orbital containing one electron pair; the bond order is 1, representing a single bond. The ionic species H 2 +^ , with a single electron in the a s orbital and a bond order of 12 , has been detected in low-pressure gas-discharge systems. As expected, H 2 +^ has a weaker bond than H 2 and therefore a considerably longer bond distance than H 2 (105.2 pm vs. 74.1 pm).
He 2 [ S g^2 S u2(1s)]* The molecular orbital description of He 2 predicts two electrons in a bonding orbital and two in an antibonding orbital, with a bond order of zero—in other words, no bond. This is what is observed experimentally. The noble gas He has no significant tendency to form diatomic molecules and, like the other noble gases, exists in the form of free atoms. He 2 has been detected only in very low-pressure and low-temperature molecular beams. It has an extremely low binding energy,^1 approximately 0.01 J/mol; for comparison, H 2 has a bond energy of 436 kJ/mol.
li 2 [ S g^2 (2s)] As shown in Figure 5.7, the MO model predicts a single Li i Li bond in Li 2 , in agreement with gas-phase observations of the molecule.
Be 2 [ S g^2 S u2(2p)]* Be 2 has the same number of antibonding and bonding electrons and consequently a bond order of zero. Hence, like He 2 , Be 2 is an unstable species.*
B 2 [ P u^1 P u^1 (2p)] Here is an example in which the MO model has a distinct advantage over the Lewis dot model. B 2 is a gas-phase species; solid boron exists in several forms with complex bonding, primarily involving B 12 icosahedra. B 2 is paramagnetic. This behavior can be explained if its two highest energy electrons occupy separate p orbitals, as shown. The Lewis dot model cannot account for the para- magnetic behavior of this molecule. The energy-level shift caused by s @ p mixing is vital to understand the bonding in B 2. In the absence of mixing, the s g (2 p ) orbital would be expected to be lower in energy than the p u (2 p ) orbitals, and the molecule would likely be diamagnetic.**^ However, mixing of the s g (2 s ) orbital with the s g (2 p ) orbital (Figure 5.6b) lowers the energy of the s g (2 s ) orbital and increases the energy of the s g 12 p 2 orbital to a higher level than the p orbitals, giving the order of energies shown in Figure 5.7. As a result, the last two electrons are unpaired in the degenerate p orbitals, as required by Hund’s rule of maximum multiplicity, and the molecule is paramagnetic. Overall, the bond order is one, even though the two p electrons are in different orbitals.
*Be 2 is calculated to have a very weak bond when effects of higher energy, unoccupied orbitals are taken into account. See A. Krapp, F. M. Bickelhaupt, and G. Frenking, Chem. Eur. J ., 2006 , 12 , 9196. **This presumes that the energy difference between s g 12 p 2 and p u 12 p 2 would be greater than c (Section 2.2.3), a reliable expectation for molecular orbitals discussed in this chapter, but sometimes not true in transition metal complexes, as discussed in Chapter 10.
5.2 Homonuclear Diatomic Molecules | 129
The extent of mixing is not sufficient in O 2 to push the s g (2 p ) orbital to higher energy than the p u 12 p 2 orbitals. The order of molecular orbitals shown is consistent with the photoelectron spectrum, discussed in Section 5.2.4.
F 2 [ S g^2 P u^2 P u^2 P g2* P g2(2p)]* The MO model of F 2 shows a diamagnetic molecule having a single fluorine–fluorine bond, in agreement with experimental data.
The bond order in N 2 , O 2 , and F 2 is the same whether or not mixing is taken into account, but the order of the s g (2 p ) and p u (2 p ) orbitals is different in N 2 than in O (^2) and F 2. As stated previously and further described in Section 5.3.1, the energy difference between the 2 s and 2 p orbitals of the second row main group elements increases with increasing Z , from 5.7 eV in boron to 21.5 eV in fluorine. As this difference increases, the s @ p interaction (mixing) decreases, and the “normal” order of molecular orbitals returns in O 2 and F 2. The higher s g (2 p ) orbital (relative to p u (2 p )) occurs in many heteronuclear diatomic molecules, such as CO, described in Section 5.3.1.
Ne (^2) All the molecular orbitals are filled, there are equal numbers of bonding and antibonding electrons, and the bond order is therefore zero. The Ne 2 molecule is a transient species, if it exists at all.
One triumph of molecular orbital theory is its prediction of two unpaired electrons for O 2. Oxygen had long been known to be paramagnetic, but early explanations for this phenomenon were unsatisfactory. For example, a special “three-electron bond”^5 was proposed. The molecular orbital description directly explains why two unpaired electrons are required. In other cases, experimental observations (paramagnetic B 2 , diamagnetic C 2 ) require a shift of orbital energies, raising s g above p u , but they do not require major modifications of the model.
Bond lengths in Homonuclear Diatomic molecules Figure 5.8 shows the variation of bond distance with the number of valence electrons in second-period p -block homonuclear diatomic molecules having 6 to 14 valence electrons. Beginning at the left, as the number of electrons increases the number in bonding orbitals also increases; the bond strength becomes greater, and the bond length becomes shorter.
Bond distance (pm)
10 11 12 13 14 Valence electrons
6 7 8 9
100
110
120
B 2
C 2
C 22 - N 2 + O 2
O 2 -
O 22 - F 2
O 2
N (^2)
130
140
150
160
Figure 5.8 Bond Distances of Homonuclear Diatomic Molecules and Ions.
130 Chapter 5 | Molecular Orbitals
This continues up to 10 valence electrons in N 2 , where the trend reverses, because the additional electrons occupy antibonding orbitals. The ions N 2 +^ , O 2 +^ , O 2 -^ , and O 22 -^ are also shown in the figure and follow a similar trend. The minimum in Figure 5.8 occurs even though the radii of the free atoms decrease steadily from B to F. Figure 5.9 shows the change in covalent radius for these atoms (defined for single bonds), decreasing as the number of valence electrons increases, primarily because the increasing nuclear charge pulls the electrons closer to the nucleus. For the elements boron through nitrogen, the trends shown in Figures 5.8 and 5.9 are similar: as the covalent radius of the atom decreases, the bond distance of the matching diatomic molecule also decreases. However, beyond nitrogen these trends diverge. Even though the covalent radii of the free atoms continue to decrease (N 7 O 7 F), the bond distances in their diatomic molecules increase (N 2 6 O 2 6 F 2 ) with the increasing population of antibonding orbitals. In general the bond order is the more important factor, overriding the covalent radii of the component atoms. Bond lengths of homonuclear and heteronuclear diatomic species are given in Table 5..
5.2.4 Photoelectron Spectroscopy
In addition to data on bond distances and energies, specific information about the energies of electrons in orbitals can be determined from photoelectron spectroscopy.^6 In this tech- nique, ultraviolet (UV) light or X-rays eject electrons from molecules: O 2 + h n(photons) S^ O 2 +^ + e- The kinetic energy of the expelled electrons can be measured; the difference between the energy of the incident photons and this kinetic energy equals the ionization energy (bind- ing energy) of the electron: Ionization energy = hv (energy of photons) - kinetic energy of the expelled electron UV light removes outer electrons; X-rays are more energetic and can remove inner electrons. Figures 5.10 and 5.11 show photoelectron spectra for N 2 and O 2 , respectively, and the relative energies of the highest occupied orbitals of the ions. The lower energy peaks (at the top in the figure) are for the higher energy orbitals (less energy required to remove electrons). If the energy levels of the ionized molecule are assumed to be essentially the same as those of
Covalent Radius (pm)
5 6 7 Valence electrons
3 4
70
74
78
B
C
N O F
Figure 5.9 Covalent Radii of 82 Second-Period Atoms.
132 Chapter 5 | Molecular Orbitals
Nitrogen s g 12 p (^2 ) π g +
N 2 +^ terms
(^2) ∑ u
p u 12 p 2
(^2) π u
s u* 12 s (^2) +
Figure 5.10 Photoelectron Spectrum and Molecular Orbital Energy Levels of N 2. Spectrum simulated by Susan Green using FCF program available at R. L. Lord, L. Davis, E. L. Millam, E. Brown, C. Offerman, P. Wray, S. M. E. Green, J. Chem. Educ., 2008 , 85 , 1672 and data from “Constants of Diatomic Molecules” by K.P. Huber and G. Herzberg (data prepared by J.W. Gallagher and R.D. Johnson, III) in NiST Chemistry WebBook, NiST Standard reference Database Number 69, Eds. P.J. Linstrom and W.G. Mallard, National Insti- tute of Standards and Technol- ogy, Gaithersburg MD, 20899, http://webbook.nist.gov, (retrieved July 22, 2012).
Oxygen p g* 12 p (^2 2) ∑ g
O 2 +^ terms
(^4) ∑ u (^2) ∑ u
p u 12 p 2
(^2) π g s u* 12 s 2 -
(^4) π g s g 12 p 2 -
Figure 5.11 Photoelectron Spectrum and Molecular Orbital Energy Levels of O 2. Spectrum simulated by Susan Green using FCF program available at R. L. Lord, L. Davis, E. L. Millam, E. Brown, C. Offerman, P. Wray, S. M. E. Green, J. Chem. Educ., 2008 , 85 , 1672 and data from “Constants of Diatomic Molecules” by K.P. Huber and G. Herzberg (data prepared by J.W. Gallagher and R.D. Johnson, III) in NiST Chemistry WebBook, NiST Standard reference Database Number 69 , Eds. P.J. Linstrom and W.G. Mallard, National Institute of Standards and Technology, Gaithersburg MD, 20899, http:// webbook.nist.gov, (retrieved July 22, 2012).
5.3 Heteronuclear Diatomic Molecules | 133
are much more closely spaced than electronic levels, any collection of molecules will include molecules with different vibrational energies even when the molecules are in the ground electronic state. Therefore, transitions from electronic levels can originate from different vibrational levels, resulting in multiple peaks for a single electronic transition. Orbitals that are strongly involved in bonding have vibrational fine structure (multiple peaks); orbitals that are less involved in bonding have only a few peaks at each energy level.^8 The N 2 spectrum indicates that the p u orbitals are more involved in the bonding than either of the s orbitals. The CO photoelectron spectrum ( Figure 5.13 ) has a similar pattern. The O 2 photoelectron spectrum (Figure 5.11) has much more vibrational fine structure for all the energy levels, with the p u levels again more involved in bonding than the other orbitals. The photoelectron spectra of O 2 and of CO show the expected order of energy levels for these molecules.^8
5.3 Heteronuclear Diatomic Molecules
The homonuclear diatomic molecules discussed Section 5.2 are nonpolar molecules. The electron density within the occupied molecular orbitals is evenly distributed over each atom. A discussion of heteronuclear diatomic molecules provides an introduction into how molecular orbital theory treats molecules that are polar, with an unequal distribution of the electron density in the occupied orbitals.
5.3.1 Polar Bonds
The application of molecular orbital theory to heteronuclear diatomic molecules is similar to its application to homonuclear diatomics, but the different nuclear charges of the atoms require that interactions occur between orbitals of unequal energies and shifts the resulting molecular orbital energies. In dealing with these heteronuclear molecules, it is necessary to estimate the energies of the atomic orbitals that may interact. For this purpose, the orbital potential energies, given in Table 5.2 and Figure 5.12 , are useful.*^ These potential ener- gies are negative, because they represent attraction between valence electrons and atomic nuclei. The values are the average energies for all electrons in the same level (for example, all 3 p electrons), and they are weighted averages of all the energy states that arise due to electron–electron interactions discussed in Chapter 11. For this reason, the values do not
*A more complete listing of orbital potential energies is in Appendix B-9, available online at pearsonhighered. com/advchemistry
Li
H
He
Potential energy (eV)
Ne
F
O
N
Na Mg Al
Al
Si
Si
P
P
S
S
Cl
Cl
Ar
10 15 20 Atomic number
0 5
Ar
C BeB B C N
O
F
Ne
2 s
2 p
3 s
3 p
1 s
0 Figure 5.12^ Orbital Potential Energies.
5.3 Heteronuclear Diatomic Molecules | 135
1 p*
2 p
6 a 1
2 e 1
1 e 1
5 a 1
4 a 1
3 a 1
1 p* LUMO
HOMO
3 s*
3 s
2 s* 2 s
2 s
C CO O
2 s
1 p 1 p
2 p
X^2 ©+
A^2 ß
B^2 ©+
E (eV)
Figure 5.13 Molecular Orbitals and Photoelectron Spectrum of CO. Molecular orbitals 1s and 1s*^ are from the 1s orbitals and are not shown. (Photoelectron spectrum reproduced with permission from J. L. Gardner, J. A. R. Samson, J. Chem. Phys., 1975, 62 , 1447.)
136 Chapter 5 | Molecular Orbitals
each atom (such as the 2 pz ) are identical. In heteronuclear diatomic molecules, such as CO and HF, the atomic orbitals have different energies, and a given MO receives unequal con- tributions from these atomic orbitals; the MO equation has a different coefficient for each of the atomic orbitals that contribute to it. As the energies of the atomic orbitals get farther apart, the magnitude of the interaction decreases. The atomic orbital closer in energy to an MO contributes more to the MO, and its coefficient is larger in the wave equation.
Carbon monoxide The most efficient approach to bonding in heteronuclear diatomic molecules employs the same strategy as for homonuclear diatomics with one exception: the more electronegative element has atomic orbitals at lower potential energies than the less electronegative ele- ment. Carbon monoxide, shown in Figure 5.13, shows this effect, with oxygen having lower energies for its 2 s and 2 p orbitals than the matching orbitals of carbon. The result is that the orbital interaction diagram for CO resembles that for a homonuclear diatomic (Figure 5.5), with the right (more electronegative) side pulled down in comparison with the left. In CO, the lowest set of p orbitals ( 1 p in Figure 5.13) is lower in energy than the lowest s orbital with significant contribution from the 2 p subshells ( 3 s in Figure 5.13); the same order occurs in N 2. This is the consequence of significant interactions between the 2 pz orbital of oxygen and both the 2 s and 2 pz orbitals of carbon. Oxygen’s 2 pz orbital (-15.85 eV) is intermediate in energy between carbon’s 2 s (-19.43 eV) and 2 pz 1 - 10.66 eV 2 , so the energy match for both interactions is favorable. The 2 s orbital has more contribution from (and is closer in energy to) the lower energy oxygen 2 s atomic orbital; the 2 s^ orbital has more contribution from (and is closer in energy to) the higher energy carbon 2 s atomic orbital.^ In the simplest case, the bonding orbital is similar in energy and shape to the lower energy atomic orbital, and the antibond- ing orbital is similar in energy and shape to the higher energy atomic orbital. In more complicated cases, such as the 2 s^ orbital of CO, other orbitals (the oxygen 2 pz orbital) also contribute, and the molecular orbital shapes and energies are not as easily predicted. As a practical matter, atomic orbitals with energy differences greater than about 10 eV to 14 eV usually do not interact significantly. Mixing of the s and s^ levels, like that seen in the homonuclear s g and s u orbitals, causes a larger split in energy between the 2 s^ and 3 s, and the 3 s is higher than the 1 p levels. The shape of the 3 s orbital is interesting, with a very large lobe on the carbon end. This is a consequence of the ability of both the 2 s and 2 pz orbitals of carbon to interact with the 2 pz orbital of oxygen (because of the favorable energy match in both cases, as mentioned previously); the orbital has significant contributions from two orbitals on carbon but only one on oxygen, leading to a larger lobe on the carbon end. The pair of electrons in the 3 s orbital most closely approximates the carbon-based lone pair in the Lewis structure of CO, but the electron density is still delocalized over both atoms. The px and py orbitals also form four molecular orbitals, two bonding ( 1 p) and two antibonding ( 1 p). In the bonding orbitals the larger lobes are concentrated on the side of the more electronegative oxygen, reflecting the better energy match between these orbitals and the 2 px and 2 py orbitals of oxygen. In contrast, the larger lobes of the p* orbitals are on carbon, a consequence of the better energy match of these antibonding orbitals with the 2 px
*Molecular orbitals are labeled in different ways. Most in this book are numbered within each set of the same sym- metry 11 s g , 2s g and 1s u , 2s u 2. In some figures of homonuclear diatomics, 1 s g and 1 s u MOs from 1 s atomic orbitals are understood to be at lower energies than the MOs from the valence orbitals and are omitted. It is noteworthy that interactions involving core orbitals are typically very weak; these interactions feature sufficiently poor overlap that the energies of the resulting orbitals are essentially the same as the energies of the original atomic orbitals.
