Covalent Bonding & Molecular Orbital Theory

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1 Covalent Bonding & Molecular Orbital Theory Chemistry 754 Solid State Chemistry Dr. Patrick Woodward Lecture #16 References - MO Theory Molecular orbital theory is covered in many places including most general inorganic chemistry texts. The material for this lecture (along with many of the figures) was taken from the following two texts: Orbital Interactions in Chemistry Thomas Albright, Jeremy K. Burdett & Myung-Hwan Whangbo, Wiley & Sons, New York (1985). Chemical Bonding in Solids Jeremy K. Burdett, Oxford University Press, Oxford (1995).

2 Questions to Consider Why is H 2 O bent rather than linear? Why is NH 3 pyramidal rather than planar? Why are Sn and Pb metals, while Si and Ge are semiconductors? Why are the π electrons delocalized in benzene (C 6 H 6 ) and localized in cyclobutadiene (C 4 H 4 )? In oxides, chalcogenides and halides explain the following coordination preferences: Cu 2+ & Mn 3+ distorted octahedral environment Ni 2+ and Fe 3+ regular octahedral environment Pd 2+ and Pd 2+ square planar environment Pb 2+, Sn 2+, Bi 3+, Sb 3+ asymmetric coordination environment MO Diagram for H 2 The number of MO s is equal to the number of atomic orbitals. ach MO can hold 2 electrons (with opposite spins). The antibonding MO has a nodal plane between atoms and to the bond. As the spatial overlap increases ψ 1 (bonding MO) is stabilized and ψ 2 (antibonding MO) is destabilized. The destabilization of the antibonding MO is always greater than the stabilization of the bonding MO. In the diagrams at the top and bottom the solid line denotes the electron density from MO theory and the dashed line the electron density from superimposing to atomic orbitals.

3 1 st Order MO Diagram for O 2 The 2s orbitals have a lower energy than the 2p orbitals. The σ-bonds have a greater spatial overlap than the π-bonds. This leads to a larger splitting of the bonding and antibonding orbitals. The 2p x and 2p y π-interaction produces to two sets of degenerate orbitals. The MO s have symmetry descriptors, σ g+, σ u+, π g, π u within point group D h. Mixing is allowed between MO s s of the same symmetry. In O 2 there are 12 valence electrons and each of the 2pπ * orbitals (π g ) are singly occupied. Thus the bond order = 2, and O 2 is paramagnetic. 2 nd Order MO Diagram for O 2 (N 2 ) A more accurate depiction of the bonding takes into account mixing of of MO s with the same symmetry (σ + ( g & σ u+ ). The consequences of this 2 nd order effect are: The lower energy orbital is stabilized while the higher energy orbital is destablized. The s and p character of the σ MO s becomes mixed. The mixing becomes more pronounced as the energy separation decreases.

4 Heteronuclear Case & lectronegativity i The atomic orbitals of the more electronegative atom are lowered. The splitting between bonding and antibonding MO s now has an ionic (( i ) and a covalent (( c ) component. The ionic component of the splitting (( i ) increases as the electronegativity difference increases. The covalency and the covalent stabilization/destabilization decrease as the electronegativity difference increases. The orbital character of the more electronegative atom is enhanced in the bonding MO and diminished in the antibonding MO. Linear AX 2 (H 2 O) MO Diagram In linear H 2 O the O 2s and O 2p z orbitals could form σ-bonds to H, while the O 2p x & 2p y orbitals would be non-bonding.

5 Bent AX 2 (H 2 O) MO Diagram In bent H 2 O the O 2s σ orbital and the O 2p x orbital are allowed to mix by symmetry, lowering the energy of the O 2p x orbital. Now there is only one non-bonding orbital (O 2p y ) Walsh Diagrams & 2 nd Order JT Distortions HOMO Walsh Diagram Shows how the MO levels vary as a function of a geometrical change. Walsh s Rule A molecule adopts the structure that best stabilizes the HOMO. If the HOMO is unperturbed the occupied MO lying closest to it governs the geometrical preference. 2 nd Order Jahn-Teller Dist. A molecule with a small energy gap between the occupied and unoccupied MO s is susceptible to a structural distortion that allows intermixing between them.

6 Covalent Bonding & the Structure of Cristobalite Idealized β-cristobalite (SiO 2 ) Actual β-cristobalite (SiO 2 ) Space Group = Fd3m (Cubic) Si-O- -O-Si = 180 sp bonding at O 2-, 2 nonbonding O 2p orbitals Space Group = I-42d (Tetragonal) Si-O- -O-Si = 147 sp 2 bonding at O 2- Walsh Diagram for NH 3 HOMO In the planar (D 3h ) form the HOMO is a non-bonding O 2p orbital (a 2 ) containing 2 electrons. In the pyramidal (C 3v form the N 2s H 1s σ * orbital (a 1 ) can mix with the nonbonding O 2p orbital. Stabilizing the HOMO. 3v )

7 Tetrahedral AX 4 (CH 4 ) MO Diagram Notice that while both the 2s and 2p orbitals on Carbon are involved in bonding, in a perfect tetrahedron mixing of the s (a 1 ) and p (t 2 ) orbitals is forbidden. C 2p 2s Pb t * 2 a * 1 t 2 a 1 t * 2 Diamonds and Lead 2p 2s Structure & Properties of the Group 14 lements lement Structure g (ev) C Diamond 5.5 Si Diamond 1.1 Ge Diamond 0.7 α-sn Diamond 0.1 β-sn Tetragonal Metal Pb FCC Metal 6p 6s t 2 a 1 * a 1 6p 6s As you go proceed down the group the tendency for the s-orbitals to become involved in bonding diminishes. This destabilizes tetrahedral coordination and semiconducting/insulating behavior.

8 2 nd Order JT Distortion in PbO Pb 6s HOMO In both polymorphs of PbO (red PbO,, the tetragonal form is shown above) the Pb 2+ ions adopt a very asymmetric coordination environment. The driving force for this is to lower the energy of the filled, antibonding Pb 6s orbitals,, by mixing with an empty Pb 6p orbital. Such mixing is forbidden by symmetry in tetrahedral and octahedral coordination, so a distortion to a lower symmetry leading to the formation of the so-called stereoactive electron lone pair occurs. Such distortions are common for main group ions with their valence s electrons (Tl( +, Bi 3+, Sn 2+, Sb 3+, etc.). This distortion is similar to the one seen in NH 3. Benzene (C 6 H 6 ) Cyclic Polyenes Cyclobutadiene (C 4 H 4 ) Consider two cyclic C n H n systems. The sketches to the left show the phases of the C 2p z orbitals that are responsible for π-interactions. In each system there are n π-mo s. The lowest energy orbital has no nodes (all orbitals in phase) while the highest energy state has the maximum number (n/2). In C 6 H 6 there is a large HOMO- LUMO gap and the e 1g orbitals are fully occupied. In C 4 H 4 the e g orbital HOMO is ½ occupied (triplet ground state).

9 1 st Order Jahn-Teller Distortion in C 4 H 4 In practice cyclobutadiene does not form a regular square (D 4h ), but undergoes a distortion to a rectangular shape (D 2h ). This stabilizes one of the HOMO s (which becomes doubly occupied) and destabilizes the other (which becomes empty). This leads to formation of two localized double bonds. Hence, C 4 H 4 is said to be antiaromatic. 1 st Order Jahn-Teller Dist. A non-linear molecule with an incompletely filled degenerate HOMO is susceptible to a structural distortion that removes the degeneracy. Octahedral Coordination The diagram to the left shows a MO diagram for a transition metal octahedrally coordinated by σ-bonding ligands.. (π-bonding( has been neglected) Note that in an octahedron there is no mixing between s, p and d-orbitals orbitals. For a main group metal the same diagram applies, but we neglect the d- orbitals. The t 2g orbitals (d xy,d yz,d xz ) are π- antibonding (not shown), while the e g orbitals (d z2,d y2-y2 ) are σ- antibonding.. The latter are higher in energy since the spatial overlap of the σ-interaction is stronger.

10 Square Planar Coordination The diagram to the left shows a MO diagram for a transition metal in square planar coordination. (π-bonding( has been neglected) Among the changes the most important is that now the s and d z2 orbitals can mix, which stabilizes the d z2 and removes the degeneracy of the e g orbitals. Transition metals with electron counts that lead to partially filled e g orbitals (HS d 4, d 8 & d 9 in particular) will be prone to undergo distortions from octahedral toward square planar. The d 8 ions Pd 2+ and Pt 2+ have a strong preference for sq. planar coordination, but with Ni 2+ the crystal field splitting is usually too small to overcome the spin pairing energy and octahedral coordination results. Jahn-Teller Distortions: The long and the short of it. The Jahn-Teller theorem tells us there should be a distortion when the e g orbitals of a TM octahedral complex are partially occupied, but it doesn t tell us what type of distortion should occur. To a first approximation two choices give the same energetic stabilization. 2 long + 4 short bonds stabilizes the d z2 orbital 2 short + 4 long bonds stabilizes the d x2-y2 orbital.

11 Distortions in d 9 & d 10 Halides Short bonds drawn with solid lines. Long bonds drawn with dotted lines. In practice Cu 2+ (d 9 ) and Mn 3+ (HS) almost always take the 2 long + 4 short distortion,, and the distortions are usually considerably larger with Cu 2+. In contrast d 10 ions, such as Hg 2+ adopt very large 2 short + 4 long distortions (in many cases the distortion is so large that the coordination is essentially linear). For example consider the bond distances in CuBr 2 (4 2.40Å, Å) ) and HgBr 2 (4 3.23Å, Å), both of which adopt distorted CdI 2 structures. Why is this so? Why do d 10 ions distort at all? Jahn-Teller Distortions d z2 -s Mixing The empty ns s orbital is of appropriate symmetry to mix with the ( (n-1)d)d z2 orbital, but not with the ( (n-1)d)d x2-y2 orbital. This dictates the details of the dist. d 9 case (Cu 2+ ): The d z2 -s mixing favors preferential occupation of the d z2 orbital (2 long + 4 short favored) d 10 case (Hg 2+ ): The d z2 -s mixing is largest when the energy separation between the two is minimized ( ( 2 > 1 ). (2 short + 4 long favored)

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