√

z y x First let ’ s consider the sulfur orbitals we need to consider their symmetry and, we need to consider their energy The fluorines lie along the axes

The d–orbitals point along the axes point between the axes

triply degenerate doubly degenerate triply degenerate notice how the degeneracy of the sulfur ’ s five d–orbitals is “ lifted ” upon interaction with the six F ’ s What symmetry do these orbitals have? Sulfur s-orbital Sulfur p-orbitals Sulfur d-orbitals (  -type) the central atom ’ s atomic orbitals Sulfur d-orbitals (  ) (originally all five d-orbitals were degenerate)

5 Upon interaction with the F ’ s, the d–orbitals will no longer be equivalent, but divide up into one set of three (t 2g ), and another set of two (e g ) d yz d xy 3d xz d z2 d x2–y2 egeg t 2g Sulfur AO ’ s (atomic orbitals) egeg t 2g

If a symmetry element exists which interconverts two or more orbitals, then the symmetry related orbitals are degenerate (i.e., they are energetically equivalent)

Therefore, E(d xz ) = E(d yz ) d xy d xz d yz ie, they are degenerate -d xy ≠d x2–y2

Therefore, E(d xy ) = E(d xz ) d xy d xz d yz ie, they are degenerate -d xy ≠d x2–y2 C4C4

triply degenerate Sulfur d-orbitals (  -type) the central atom ’ s atomic orbitals

What about the relative energies of these orbitals? Most importantly, how does their energy compare with the fluorine orbitals? 3s pxpx pypy 3p z Sulfur AO ’ s (atomic orbitals) 2s pxpx pypy 2p z Fluorine ’ s AO ’ s too low in E to interact with the sulfur orbitals a 1g t 1u d yz d xy 3d xz d z2 d x2–y2 egeg t 2g What about the symmetry of the fluorine AO ’ s? How do the AOs combine to form LGO ’ s in this molecule?

11 Symmetry Adapted Orbitals (p. 808, Shriver & Atkins)  – type

12 Symmetry Adapted Orbitals (p. 805, Shriver & Atkins) 12

a 1g egeg egeg t 1u The Fluorine LGO ’ s (  –bonding only) pzpz pxpx pypy s antibonding MO is pictured here d z2 d x2–y2 LGO(1) LGO(2) LGO(3) LGO(4) LGO(5) LGO(6) t 1u Sulfur p–orbitals have t 1u symmetry & thus match LGO(2)–LGO(4) a 1g Sulfur s–orbital has a 1g symmetry & thus matches LGO(1) two of the Sulfur d–orbitals e g have e g symmetry & thus match LGO(5) & LGO(6) overlaps with the sulfur ’ s

Alternate way to figure out symmetry species Use group theory – identify symmetry species of appropriate valence orbitals on metal for that point group, e.g., O h point group 4s orbital is A 1g ; 3d z 2 and 3d x 2 –y 2 are E g 14

Alternate way to figure out symmetry species Identify symmetry species of sigma bonding orbitals on ligands for the molecule’s point group, e.g., O h point group has six identical ligands at right angles to each other Set up reducible representation: 15 OhOh E8C 3 6C’ 2 6C 4 3C 2 i6S 4 8S 6 3h3h 6d6d  red

Alternate way to figure out symmetry species Decompose the reducible representation:  reducible = T 1u + E g + A 1g As in standard MO theory, mix orbitals on the metal with orbitals in the ligands that are of the same symmetry species There may be more than one set of orbitals on the metal that are the same symmetry species 16

17 Counting electrons: S has 6 valence electrons F has 7 valence electrons, but only 1 is in an AO that will be “shared” Total = 12 valence e –

sulfur orbitals Which sulfur orbitals do the fluorine p x and p y fluorine p x and p y –orbitals interact with?

p y (fluorine) & d xy (sulfur) The p y (fluorine) & d xy (sulfur) orbitals are configured for  – overlap

p y (fluorine) & d xy (sulfur) The p y (fluorine) & d xy (sulfur) orbitals are configured for  – overlap  –type orbitals

d yz another LGO with t 2g t 2g symmetry lies in the yz plane z y  –type orbitals

The sulfur d yz only interacts with fluorines that lie along the y– and z–axes d yz another LGO with t 2g t 2g symmetry lies in the yz plane z y

ML n complexes Metal-ligand bonds are treated such that the ligand acts as a Lewis acid and donates an electron pair to the metal This means that (monodentate) ligands contribute 2 electrons The metal is considered to have the number of electrons the complex’s formula suggests, e.g., in [Ag(NH 3 ) 4 ] +, the silver has a 1+ charge and thus 46 electrons 23

ML n complexes Nevil Sidgwick (1927) came up with an extension of the Lewis’s octet rule, called the effective atomic number (EAN) rule: the sum of the electrons on the metal plus the electrons donated from the ligands must be 36, 54 or 86. Example: [Ag(NH 3 ) 4 ] + : Ag + has 46 electrons, the four ammonia ligands contribute 8 electrons for a total of 54 e – 24

ML n complexes There are exceptions: [Cr(NH 3 ) 6 ] 3+ has 33 electrons and is stable. This helps populate the MO diagrams – each ligand orbital should have a pair of electrons, and the metal orbitals should contain the number of valence electrons the charge on the metal suggests 25

26 Cr 3+ 6 NH 3