Chm 363 Spring 2017, Exercise Set 2 Main Group Molecular Orbital Diagrams Distortion of Structures Vibrational Analysis Mr. Linck Version 2.0 January 31, 2017 2.1 Building an MO Diagram The first step in building an mo diagram is to determine the orbitals of interest and place them with appropriate energies. Table 1 gives the VOIE of some atoms in the main group to allow you to carry out this procedure. If you were building an mo diagram for Si-Cl, what would be the ordering of orbital energies from most stable to least stable? 2.2 Building an MO Diagram For an mo diagram for H 2 O, what would be the ordering of orbital energies from most stable to least stable? 2.3 Using Symmetry in MO Diagrams Because only orbitals belonging to the same irreducible represenation can interact, it is important to find the symmetry of various orbitals of interest. For those on an atom at the center of symmetry, this is easy; they are listed on the right hand side of the character table. (However, if you are to use the right hand side of the character table, you have to agree to define the z axis as the principle axis of rotation. If you do not, if you insist on being a free spirit, that is fine, but don t look for help from the right hand side.) Find the symmetry of the orbitals on the O atom in water; on the N atom in NH 3. 2.4 Using Symmetry in MO Diagrams Orbitals on atoms not at the center of symmetry generally need to be formed into linear combinations in order to become symmetry adapted. We call these SALCAOs. Either intuition or the projection operator equation is used to find these SALCAOs. Find the SALCAOs for the hydrogen 1s orbitats in a water molecule. 2.5 Using Symmetry in MO Diagrams Find the SALCAOs for the fluorine 2s orbitats in OF 2. Notice any similarity to the orbitals from the last problem? Learn that symmetry is separate from the nature of the orbital.
2.13 2 Figure 1: Application of the Three Orbital Rule for Problem 13-15 2.6 Using Symmetry in MO Diagrams Find the SALCAOs for the fluorine 2p in orbitals in OF 2. HINT: A convenient system of notation is define one axis on the fluorine atom as pointing toward the oxygen atom, a p in orbital. There are then two perpendicular p orbitals left on each fluroine atom; you need to distinguish these. In the case of OF 2 reasonable types might be in plane and perpendicular to plane. 2.7 Using Symmetry in MO Diagrams Find the SALCAOs formed from the 2p orbitals in OF 2. 2.8 Using Symmetry in MO Diagrams Find the SALCAOs for the fluorine 2p in orbitats in square planar PdF 2 4. 2.9 Using Symmetry in MO Diagrams Find the SALCAOs for the fluorine 2p orbitats perpendicular to the bond direction and perpendicular to the plane of the square in square planar PdF 2 4. 2.10 Bonding of Two Orbitals Requires the Same Symmetry Consider the H 2 O molecule. I trust you have the symmetry assigned to the oxygen orbitals and the SALCAOs of the hydrogens. How many orbitals are there of b 2 symmetry? What happens to that orbital in the mo diagram? 2.11 Bonding of Two Orbitals Requires the Same Symmetry More with the H 2 O molecule. How many orbitals are there of b 1 symmetry? What happens to those orbitals in the mo diagram? 2.12 A Vision of the MOs The molecular orbital is a combination of the two atomic orbitals, but their weighting depends on which atomic orbital energy is closest to that of the mo: the ao closest in energy to that of the mo has the greater weighting. Using this rule, draw the two b1 mos in water.
2.13 3 Element E(s) E(p) E(d) H 13.59 Li 5.39 3.5 Be 9.32 5.6 B 14.05 8.3 C 19.43 10.66 N 25.56 13.18 O 32.38 15.85 F 40.17 18.65 Na 5.14 3.09 Mg 7.65 4.28 Al 11.32 5.99 Si 15.89 7.78 P 18.84 9.65 S 22.71 11.62 Cl 25.23 13.67 K 4.34 2.79 Ca 6.11 3.74 Sc 6.65 3.44 8.01. Cu 7.74 3.94 10.4 Zn 9.39 4.66 Ga 12.61 5.93 Ge 16.05 7.54 As 18.94 9.17 Se 21.37 10.82 Br 24.37 12.49 Table 1: Table of VOIE for various elements; energies in ev. D 4h E 2C 4 C 2 2C 2 2C 2 i 2S 4 σ h 2σ v 2σ d A 1g 1 1 1 1 1 1 1 1 1 1 z 2 A 2g 1 1 1-1 -1 1 1 1-1 -1 R z B 1g 1-1 1 1-1 1-1 1 1-1 x 2 -y 2 B 2g 1-1 1-1 1 1-1 1-1 1 xy E g 2 0-2 0 0 2 0-2 0 0 (R x, R y ) (xz, yz) A 1u 1 1 1 1 1-1 -1-1 -1-1 A 2u 1 1 1-1 -1-1 -1-1 1 1 z B 1u 1-1 1 1-1 -1 1-1 -1 1 B 2u 1-1 1-1 1-1 1-1 1-1 E u 2 0-2 0 0-2 0 2 0 0 (x, y)
2.17 4 Figure 2: Application of the Three Orbital Rule for Problem 16 2.13 Bonding of Three Orbitals of the Same Symmetry Figure 1 gives a three orbital situation where A and B are on one fragment and C is on the other. The rule requires that the phases of the orbitals be such that the orbitals A and B have constructive overlap with orbital C. For instance, if B and C are s orbitals, and A is a p orbital, then the positive phase of A must be to the right so that it overlaps constructively with C. Draw the phases of the orbitals if the internuclear axis is the z axis, orbital A is a d yz, orbital B is a p y, and orbital C is a d yz. 2.14 Bonding of Three Orbitals of the Same Symmetry Using the first set of orbitals in the last problem (p, s, and s for the three orbitals A, B, and C), we use perturbation theory (pictorially) to figure out the answers. Here s the argument. Molecular orbital Z is mostly C in character. Since A and B are both above it, they both come into the interaction in a constructive fashion, with more B than A because of the inverse energy relationship between interactions. Sketch the components of this orbital. 2.15 Bonding of Three Orbitals of the Same Symmetry Molecular orbital Y is mostly B in character. Because C is below it in energy, it comes in destructively. How is A (which in the pure fragment (or atom) is orthogonal to B and hence can t interact at all without the presence of C) mixed in? The rule (from perturbation theory) says draw an arrow from A to C and then back to Y. The words that go with this are: A is mixed by C into Y. The arrow follows these words. This is the red dashed arrow in Figure 1. If that arrow crosses a line extended from Y, then the sign with which A enters into the mixture called Y is negative. (If it doesn t cross, the sign is posiive.) The crossing is illustrated by the short red line. Use that logic, and be guided by the blue arrow in the figure, and tell me what molecular orbital X is like. 2.16 Bonding of Three Orbitals of the Same Symmetry Figure 2 gives another possible diagram for three orbitals. The arrows are present. State what the composition of X, Y, and Z are.
2.27 5 Figure 3: Scheme of Levels for the Three Orbital Rule of Problem 17 2.17 Bonding of Three Orbitals of the Same Symmetry Figure 3 gives another possible diagram for three orbitals. State what the composition of X, Y, and Z are. 2.18 Bonding of Three Orbitals of the Same Symmetry We have now looked at all three possible combinations that three orbitals can take. State the rule(s) for finding the appearance of a mo that derives from three ao s of the same symmetry. 2.19 Bonding of Three Orbitals of the Same Symmetry What does the most stable a 1 orbital in the water molecule look like? 2.20 Bonding of Three Orbitals of the Same Symmetry What does the intermediate a 1 orbital in the water molecule look like? 2.21 Bonding of Three Orbitals of the Same Symmetry What does the least stable a 1 orbital in the water molecule look like? 2.22 The Final Step in Builiding an MO Diagram Put electrons into your mo diagram for the water molecule. 2.23 Interpretation of MO Diagrams What is the nature of the highest occupied molecular orbital (the HOMO) in H 2 O? 2.24 Interpretation of MO Diagrams What is the nature of the lowest unoccupied molecular orbital (the LUMO) in H 2 O? 2.25 Buiding a Qualitative MO Diagram Build the mo diagram for NH 3
2.35 6 D 3h E 2C 3 3C 2 σ h 2S 3 3σ v A 1 1 1 1 1 1 1 z 2 A 2 1 1-1 1 1-1 R z E 2-1 0 2-1 0 (x, y) (xy, x 2 -y 2 ) A 1 1 1 1-1 -1-1 A 2 1 1-1 -1-1 1 z E 2-1 0-2 1 0 (xz, yz), (R x, R y ) 2.26 The Appearance of MOs Use projection operators to determine the appearance of the SALCAO e wave functions of the hyrogen atoms in NH 3. HINT: You will need to make your answers orthogonal. 2.27 Buiding a Qualitative MO Diagram Build the mo diagram for planar NH 3. 2.28 The Appearance of MOs Use projection operators to determine the appearance of the SALCAO e wave functions of the hyrogen atoms in planar NH 3. Show that there is clearly bonding between the p x orbital of the N and one of those e members; and between the p y orbital and the other SALCAO. HINTS: You will need to make your answers orthogonal. Also, this kind of linear combination shows up all the time in triangular systems; memorize it. 2.29 Buiding a Qualitative MO Diagram Build the mo diagram for OF 2. HINTS: Consider only p orbitals on all the atoms. Do the F 2p in with appropriate oxygen orbitals first, leave that result alone, then do any interactions with the F 2p. This is justified by the fact that p in orbitals form pseudo-σ bonds which are energetically more stable than π interactions. 2.30 Symmetry Operations in a Linear Molecule Find the symmetry operations present in CO 2. 2.31 Symmetry with Linear Molecules Linear molecules in concept present a problem for our symmetry arguments because the group of symmetry operations is infinite. Fortunately, one can use the character table for such molecules in most cases by a simple inspection procedure. What are the representations for the orbitals on the carbon in CO 2? HINT: Duh? 2.32 Symmetry with Linear Molecules Find the SALCAO of oxygen p in orbitals for CO 2. Reduce that representation by inspection. HINT: You may have to assume some angles of φ to do so. Use ones that are obvious, π/2, π, etc. 2.33 Symmetry with Linear Molecules Find the SALCAO of oxygen p in orbitals for CO 2. HINT: Same as last problem. 2.34 Building an MO Diagram Construct the mo diagram for CO 2.
2.41 7 D h E 2C (φ)... σ v i 2S (φ)... C 2 Σ + g 1 1... 1 1 1... 1 z 2 Σ g 1 1... -1 1 1... -1 R z Π g 2 2cos(φ)... 0 2-2cos(φ)... 0 (R x, R y ) (xz, yz) g 2 2cos(2φ)... 0 2 2cos(2φ)... 0 (x 2 -y 2, xy)... Σ + u 1 1... 1-1 -1... -1 z Σ u 1 1... -1-1 -1... 1 Π u 2 2cos(φ)... 0-2 2cos(φ)... 0 (x, y) u 2 2cos(2φ)... 0-2 -2cos(2φ)... 0 Figure 4: Structures for Descent in Symmetry Problems 2.35 Building an MO Diagram Build an mo diagram for the linear molecule IF 2. Use p orbitals only on both the I and the F. 2.36 Building an MO Diagram Repeat the exercise of the last problem but this time include the s and p orbitals of the I and only the p of the F. Justification? 2.37 Building an MO Diagram Construct the mo diagram for linear H 2 O. 2.38 Steps in Building a Qualitative MO Diagram How many steps do you need to take to build an mo diagram? What are they? 2.39 Symmetry: The Concept and Use of a Descent In Figure 4 are two structures. To which point groups do they belong? HINT: In the left hand structure, all orange balls are in the same plane. In the right hand structure, the ones opposite each other are in the same plane that is normal to the z axis. 2.40 Symmetry: The Concept and Use of a Descent There are 8 symmetry operations in the C 4v point group and only 4 remain when you lower the symmetry to C 2v. Which are they?
2.48 8 Point Group O C 4v E 2C 4 C 2 2σ v 2σ d C 2v E C 2 σ v σ v Table 2: How symmetry operations evolve in the descent from C 4v to C 2v. 2.41 Symmetry: The Concept and Use of a Descent Table 2 suggests a way to represent the results you got in the last problem, where the loopy arrow means that the symmetry operation is not preserved in the system of lower symmetry. Build a similar table for the descent in symmetry from D 3h to C 3v. 2.42 Symmetry: The Concept and Use of a Descent Now here is the critical aspect of this descent. Imagine a function, say a p z orbital on the blue atom of the left hand object in Figure 4. What happens to that function under the rotation C 2? Since what happens to the function is independent of the object (except that the symmetry operation must be present), what happens to the function p z under a rotation of C 2 if it is on the blue atom of the right hand object? 2.43 Symmetry: The Concept and Use of a Descent Our conclusion from the last problem: What happens to the function, and hence the character generated by that function, is independent of the object, as long as both contain the same symmetry operation. What are the characters for the p z function in the left hand object of Figure 4? What are the characters for the p z function in the right hand object of Figure 4? For those operations that are common, do they agree? 2.44 Symmetry: The Concept and Use of a Descent Consider the function xy on the blue atom of Figure 4. What are the characters of this function in the left hand object of the figure? What are they in the right hand object? Do they agree for those symmetry operations that are preserved? 2.45 Symmetry: The Concept and Use of a Descent Here is a more interesting case. What are the characters of the representation generated by the pair of functions x and y for the left hand object in Figure 4? 2.46 Symmetry: The Concept and Use of a Descent Take the characters from the last problem for those symmetry operations that are preserved and reduce those in C 2v symmetry. Does your answer agree with the representations that x and y belong to according to the right hand side of the C 2v character table? 2.47 Review Make an mo diagram for H 2 S using symmetry arguments. The bond angle is about 90 o. How does your answer differ from that of water?
2.52 9 Figure 5: Data for Problem 48. The blue circles are for elements in the first two rows; the yellow ones for elements in the second and third row, for instance, the yellow point for x = 1 is for IE(Na) to IE(Ca). 2.48 Review It is often said that there is a (upper left to lower rigth) diagonal relationship between elements: For instance, O is like Cl. In Figure 5 is a graph giving the ratio of the first ionization energies of an element from the first row with the diagonally related one in the row below. To clarify, the first point (x = 1) is the ratio IE(Li)/IE(Mg), the second IE(Be)/IE(Al), etc. Values near 1.0 (that line is drawn for reference) are consistent with the argument made above. Comment on the values observed. HINT: Your comments might start with a short statement about why you might expect there to be a diagonal relationship in the first place. 2.49 Review Give the Russel-Saunders term symbol (i.e., a designation like 3 F) for the ground state of an ion with a [Ar]3d 4 configuration. 2.50 Review In getting your answer to last problem, you ignored (I hope) the [Ar]. Briefly, why is this allowed? 2.51 Review Find the symmetry operations present in the object in Figure 6 with consideration of the hydrogen atoms (that is, do not average them). HINT: All B-O and H-O bonds are of equal length and all O-B-O and H-O-B angles are the same. All atoms are in the plane of the paper.
2.58 10 Figure 6: Data for Problem 51 2.52 Review Find the irreducible representations generated by four p orbitals located on that atoms at the corners of the square and perpendicular to the plane of a square object. 2.53 Review Find two of the functions that serve as a basis for two of your irreducible representations from the last problem. HINT: In other words, I want you to sketch two of the SALCAO s. 2.54 Review Put an atom in the center of the square of problem 52. Let that atom have only a p orbital perpendicular to the plane of the square. Build an m.o. diagram for this π only problem. HINT: The central atom has a higher VOIE for its p orbital than do the external atoms. 2.55 Review For the molecule SF2, set up the m.o. diagram using symmetry. HINT: Use only the p in orbital on the F s. 2.56 Review Make an mo diagram for CrF 2 as a bent molecule with an F-Cr-F angle of 110 o. Consider only d orbitals on the Cr and only the p in on the F. 2.57 Distortion of Molecules There are two ways to approach this subject. We do the more straightforward first. Consider a linear water molecule. Use the descent in symmetry to determine which symmetry operations of the linear molecule are preserved when it is bent slightly to C 2v symmetry. 2.58 Distortion of Molecules Review your answers for the mo s of a linear water molecule, problem 37. Now when the linear molecule bends, each orbitals set of characters remains the same for those operations that are conserved. In C 2v symmetry, write those preserved characters (from D h ) under the appropriate operation of C 2v. For instance, one of the infinite number of C 2 s in D h becomes the C 2 of C 2v. The character under the π u irreducible presentation (generated by the p x and p y orbitals is 0. So the character we enter under the C 2 of C 2v is a 0. The
2.66 11 linear water molecule has five energy levels (if you include the O 2s, four if not). Do this procedure for all the characters of each of those orbitals (or, in the case of the π u, the orbital pair. BIG HINT: The σ v orbital of C 2v symmetry that is perpendicular to the plane of the molecule is hidden in D h symmetry. It is one of the S (φ), namely the one with φ = 0. 2.59 Distortion of Molecules Reduce the represenations in C 2v symmetry that you generated in the last problem. 2.60 Distortion of Molecules Tha answers that you get from the last problem are the new names of the orbitals of a water molecule in a slightly bent geometry.problem. The issue becomes this: In the linear water it was not possible for the π u orbitals to interact with the σ g + orbital because their symmetries are different. When you bend the molecule, the π u becomes a 1 and b 2, which can now interact with the σ g +, which becomes a 1. Since these orbitals are close together in energy, the interaction is reasonably substantial (especially if the bend is significant) and as a result, the upper goes up and the lower goes down or the orbitals now repel. Does this lead to an net stability of the bent molecule? Explain. 2.61 Computational Proof of Distortion Process Make a rough plot of the energies of each orbital of a water molecule as it bends from linear to some angle as a function of angle of distortion. The result that you get is called a Walsh diagram. 2.62 Structure of Molecules From the last several problems, comment on why water is a distorted molecule. 2.63 Distortion of Molecules Do a Walsh diagram analysis of the distortion of planar ammonia (D 3h ) to pyramidal ammonia (C 3v ). HINT: This is not a quick problem. 2.64 Vibrations in Molecules To deal with the second, the more sophisticated, approach to distortion of molecules, we need to understand vibrations in molecules, or at least the symmetry of those vibrations. We begin that endeavor now. If we put an x, y, and z axis on each atom of a molecule and give them names x i, y i, and z i, (generically called q i ) where i runs over all the atoms in the molecule, then we can describe any possible motion of the molecule by specifying the values of those q i. How would you specify the entire molecule moving in the positive x direction by an amount 0.1Å? That is, what values would you give the q i? 2.65 Symmetry of Vibrations in Molecules We can find the symmetry of the various motions by determining the characters generated by the entire set of q i for a molecule. We are building a representation generated by the functions q i, using character theory. Let s try it for bent water. How many q i are there? What happens to the set when we do the identity operation on them? HINT: Duh?
2.73 12 2.66 Symmetry of Vibrations in Molecules What happens to the set when we do a C 2 operation? Here the two hydrogens are switched in their positions, so they do not contribute to the diagonal of the matrix (and therefore to the character). What happens to the x O, y O, and z O depends on how you define them. So do so in a convenient way for instance, with z O along the C 2 axis. Figure it out. HINT: Do the same with the two planes of symmetry (once you define your axes, you must leave them alone for the rest of the problem. 2.67 Symmetry of Vibrations in Molecules The last problem gives us the characters for a reducible prepresentation generated by all the q i. You need to reduce it. Do so. HINTS: (1) If you made a mistake in your character assignments, you will get fractional values for the number of times Γ i is in Γ red, an impossible situation. Go back and check your characters. (2) I got 3a 1 + a 2 + 3b 1 + 2b 2. 2.68 Symmetry of Vibrations in Molecules The total dimension of the representations from the last problem (or any problem) is 3N, where N is the number of atoms in the molecule. Of those 3N answers, three of them are due to translation of the molecule as a whole see problem 64. The symmetry of these are the same as the symmetry of x, y, and z, and can be found on the right hand side of character tables. To get at the symmetry of the vibrations, we need to subtract from the representation generated by all the coordinates those due to translations. Do so. 2.69 Symmetry of Vibrations in Molecules We now have 3N-3 dimensions left. Of those 3N-3 answers, two (for linear molecules) or three (for nonlinear ones) of them are due to rotation of the molecule as a whole. The symmetry of these are the same as the symmetry of R x,, R y, and R z and can be found on the right hand side of character tables. To get at the symmetry of the vibrations, we need to subtract from the 3N-3 representations those that correspond to rotations. Do so. You now have the symmetry of the 3N-6 vibrations of a water molecule (or any other bent triatomic of C 2v symmetry). 2.70 Symmetry of Vibrations in Molecules Find the symmetry of the vibrations in planar NH 3. 2.71 Symmetry of Vibrations in Molecules Find the symmetry of the vibrations in SF 2. HINT: Duh? 2.72 Observed Vibrational Energy Levels Transitions in Molecules One can excite from a low vibrational level to a higher one using light. The result is the vibrational spectrum that plays such an important role in functional group identification in organic chemistry. Such a transition is allowed in the IR only if the symmetry of the vibration (found in the last several problems) is the same as the symmetry of x, y, or z. How many allowed IR peaks are there in the spectrum of water?
2.81 13 2.73 Observed Vibrational Energy Levels Transitions in Molecules One can excite from a low vibrational level to a higher one using light. There is a second way of exciting a vibrational level. It is called the Raman effect. A transition is allowed in the Raman only if the symmetry of the vibration (found in the last several problems) is the same as the symmetry of q 2 i or q i q j. How many allowed IR peaks are there in the Raman spectrum of water? 2.74 IR and Raman Allowed Peaks For SF 2, how many peaks are IR allowed? How many are Raman allowed? 2.75 IR and Raman Allowed Peaks Consider the molecule Mn(CO) 5 Cl which has C 4v symmetry. In this case we are going to focus on a certain region of the spectrum, the C-O stretching region. This is a very rich region because there is little interference with other vibrations. To do this limited approach, instead of x, y, and z vectors on each atom, we will represent the C-O stretch as a vector between the C and the O. Find the characters of the representation generated by these five vectors. 2.76 IR and Raman Allowed Peaks Reduce the representation from the last problem and ascertain how many IR allowed and how many Raman allowed peaks will be visible for this compound. 2.77 Identification of Geometrical Isomers Consider the pair of isomers, cis and trans-mo(co) 4 Cl 2. Find the number of peaks (i.e., allowed transitions) in the IR spectrum of the cis isomer in the C-O stretching region. 2.78 Identification of Geometrical Isomers Find the number of peaks (i.e., allowed transitions) in the IR spectrum of the trans isomer in the C-O stretching region. Can you distinguish this isomer and the one from the last problem? 2.79 Distortion of Molecules Vibrational Analysis We now return to the sophisticated approach to distortion of molecules. Not only will this method tell us about distortion, it will often indicate the nature of the distortion. To avoid dealing with infinite groups, we deal with planar NH 3. Let s look at just bending of the H-N-H bonds. To bend without stretching, we need a vector on each hydrogen atom that is perpendicular to the plane of the molecule. Put three such vectors on your planar structure and ascertain the irreducible representations generated by those bends. 2.80 Distortion of Molecules When Does Coupling Occur? Figure 7 gives a rough mo diagram for planar NH 3. Be sure you agree with this formulation. Fill in the electrons. What we want to know is can the HOMO interact with the LUMO under a distortion. Why are we most interested in the HOMO and LUMO?
2.86 14 Figure 7: MO Diagram for Planar NH 3 2.81 Distortion of Molecules When Does Coupling Occur? Quantum mechanics tells us that two orbitals can interact if the direct product of the first times that of the second times that symmetry of the vibration that leads to the distortion contains the totally symmetric representation. That is, it involves an integral of the two wave functions and the distorting potential in the form of the vibration. There are two vibrations that bend the molecule and you found their symmetries in problem 79. Do either of those satisfy our requirement? 2.82 Distortion of Molecules When Does Coupling Occur? To learn what our vibration looks like, take the three vectors of problem 79 and use the projection operator formula on them. Does your answer agree with what you know happens to planar NH 3? 2.83 Distortion of Molecules Sketch the Walsh diagram (that which results from a descent in symmetry) for the conversion of a tetrahedral CXYZQ molecule to a D 2d form (where the D 2d form is close to a square planar form). HINTS: Treat the ligands X, Y, Z, and Q as if they are the same to use symmetry to a maximum. We are labeling them differently only so we can see the application to the inversion of an enantiomer into its other form. Also, let the ligand have only a σ donor orbitals. 2.84 Distortion of Molecules Sketch the Walsh diagram (that which results from a descent in symmetry) for the conversion of a square planar form of CXYZQ molecule to a D 2d form (where the D 2d form is close to a tetrahedron). HINT: Same ones as last problem. 2.85 Distortion of Molecules Put your answers from the last two problems together to see the entire path from tetrahedron to square planar to enantomeric tetrahedron. Do you see why this pathway for inversion is difficult?
2.95 15 2.86 Hypervalent Compounds Hypervalent compounds are those that violate the octet rule. There is a peculiar feature of bonding that accounts for their structure, and contrary to the popular misconception, it does not involve d orbitals. We begin with a simple hypothetical example in which the major features are clear. Build a mo diagram for the linear molecule HClH. 2.87 Hypervalent Compounds In the answer to the last problem, the 3s orbital on Cl is unlikely to participate very much in bonding; let s assume that it doesn t. Then how many net bonding orbitals do you have? 2.88 Hypervalent Compounds We can also learn about properties from your answer in problem 86. How many electrons do we have in our system? How many of those electrons are on Cl only orbitals (assuming still that Cl 3s is not bonding because of its stability)? How many electrons are on the hydrogens? How many are in bonding orbitals? To conclude this analysis, which atom is negative? Rule: for stable hypervalent compounds you need to have atoms on the exterior that want electrons more than the central atom does. 2.89 Hypervalent Compounds The molecule PF 5 is a classic example of a hypervalent compound. The molecule has D 3h symmetry. Build a mo diagram for this species. For this treatment use only p orbitals on the P and only p in orbitals on the F. 2.90 Hypervalent Compounds For the analysis of the compound in the last problem it is convenient to separate the in plane bonding from the axial bonding. How many bonding orbitals are there for the axial F? Do the simple calculation to find out the number of bonds per axial F. How many bonding orbitals are there for the equatorial, the in-plane, F? Same calculation for the number of bonds per equatorial F. Which bond do you conclude is longer? 2.91 Hypervalent Compounds A reasonably rigorous calculation of PF 5 finds that the axial P-F bond length is 1.594 and the equatorial bond length is 1.558. Does that support your analysis? 2.92 Hypervalent Compound Predictions Given our knowledge of the atoms that have negative charge, where would you expect that the -CH 3 group in PF 4 CH 3 would reside? In an axial or equatorial position? 2.93 PF 4 CH 3 and NMR Given your answer to the last problem, how many 19 F nmr signals would you expect in the nmr of PF 4 CH 3? HINT: Like H, 19 F has a spin of 1/2. 2.94 Pseudo-Rotation Imagine the PF 4 CH 3 compound with the axial direction up and down and the methyl group in the plane of the paper to the left. Sketch the structure under these conditions.
2.103 16 2.95 Pseudo-Rotation Now carry out the following vibration. Let each of the axial F atoms move to the right and down (keeping the P-F distance about the same, so they move on an approximate circle). At the same time, move the F closest to you to the left and toward you; move the F away from you to the left and further away from you. Convince yourself that this converts the molecule into one in which the two fluorine atoms that were axial are now equatorial and the two fluorine atoms that were equatorial are now axial. This process occurs readily in PF 4 CH 3, so readily that the relatively slow nmr method sees only an average value of the fluorine environment, and hence only one environment. 2.96 Pseudo-Rotation What would you argue about the appearance of the nmr of PF 3 (CH 3 ) 2? Explain your reasoning. HINT: Tricky question. Requires some thought to get it right, though nothing is hard. 2.97 Hypervalence from a Consistent Valence Bond Approach There is no need for expansion of the octet with hypervalent compounds as along as you are willing to use resonance of valence bond structures. Consider HClH. How many total electrons? Make an octet around the Cl with one pair bonding to one of the hydrogen atoms. The last pair of electrons is on the other hydrogen atom without a bond to the Cl. Now resonate the structure. Do you agree that the bond strength should be less than a normal bond? Do you agree that the outer atoms carry negative charge? Both of these conclusion came out of our mo analysis. 2.98 Hypervalence from a Consistent Valence Bond Approach Build a valence bond structure for IF 2. 2.99 Hypervalence from a Consistent Valence Bond Approach Build a valence bond structure for PF 5. Note there is nothing in this valence bond approach that allows two different kinds of fluorine atoms or their different bonding patterns. HINT: You need several resonance structures. 2.100 Hypervalence from a Consistent Valence Bond Approach Build a valence bond structure for SF 6. 2.101 Review The square planar molecule S4 2+ has a S-S bond length of 1.98Å, which is shorter than the normal S-S single bond length of 2.06. Use a π mo theory to account for this result. HINTS: (1) Count electrons carefully. Remember that each sulfur atom in S4 2+ has a lone pair, which you might think comes from the 3s orbital and that there are four σ bonds and four σ* bonds from the in-plane p orbitals. (2) In cases such as this, where there is no atom at the center of symmetry, the SALCAOs are in fact mo s of the system. 2.102 Review Build a bonding model for Te 2+ 3, a triangular compound. HINT: Neglect s orbitals.
2.108 17 2.103 Review When two of the Te3 2+ fragments of the last problem are brought together, face of one triangle to face of the other, the dimer that forms has D 3h symmetry with a bond distance between the triangular faces of 3.13Å(as opposed to 2.68Åwithin the triangular face). Considering just the π orbitals of the Te3 2+ fragment, account for these facts. 2.104 Review Make an mo diagram for B 2 H 6, which has bridging hydrogen atoms. (You may have to look up the structure.) Try it by using sp 3 hybrids on the borons. HINT: This is a hard problem in the sense that you need to consider four orbitals of the same symmetry. You have to invent some logical way of doing that. 2.105 Review Convince yourself that you get approximately the same answer by using s and three p orbitals on each of the two borons. 2.106 Review of MO and A First Look at Octahedral System Build an mo diagram for the hypothetical SH 6. HINT: This molecule has octahedral symmetry. 2.107 Using Projection Operators Sketch the SALCAOs of the hydrogen orbitals in SH 6. HINTS: (1) Use projection operators. (2) Use only the proper symmetry operators and apply logic to determine if the answers are g or u. (3) You will need to orthogonalize the e g SALCAOs to get the proper answers. 2.108 Ligand σ Orbitals in Octahedral Symmetry The answers to the last problem are very important. They occur over and over again in octahedral metal complexes. You should learn them by heart.