Part 2.7: Orbital Diagrams

Part 2.7: Orbital Diagrams

Orbital Diagrams Orbital Interactions Molecular Orbital Theory
Orbital Energies MO Diagrams HF, H2O, CO2, C2H4, NH3, Benzene SALC Hybridization Symmetry and Reactivity

Atomic Orbitals 1s orbital 2p orbital
Atomic Orbital- is a mathematical function (Y) that describes the wave-like behavior of electrons in an atom. Used to calculate the probability (Y2 ) of finding any electron of an atom in any specific region around the atom's nucleus. 1s orbital 2p orbital

Atomic Orbitals Waves can interact-
Atomic Orbital- is a mathematical function (Y) that describes the wave-like behavior of electrons in an atom. Used to calculate the probability (Y2 ) of finding any electron of an atom in any specific region around the atom's nucleus. Waves can interact- constructively = bonding destructively = antibonding not at all = non-bonding

Molecular Orbital Theory
ATOMIC ORBITALS of different atoms combine to create MOLECULAR ORBITALS The number of ATOMIC ORBITALS = the number of MOLECULAR ORBITALS Electrons in these MOLECULAR ORBITALS are shared by the molecule as whole MOLECULAR ORBITALS can be constructed from Linear Combination of Atomic Orbitals (LCAO) LCAO Y = caya + cbyb (for diatomic molecules) BONDING Orbitals have most of the electron density between the two nuclei ANTI-BONDING Orbitals have a node between the two nuclei NONBONDING Orbitals are essentially the same as if it was only one nuclei

Combining Atomic Orbitals
Bonding: Ψ(σ) or Ψ+ = (1/√2 ) [φ(1sa) + φ(1sb) ] Antibonding: Ψ(σ*) or Ψ- = (1/√2 ) [φ(1sa) - φ(1sb) ]

Combining Atomic Orbitals (H2)
Antibonding Bonding

H2 Fe(C5H5)2 2 atoms Only s orbitals Linear interaction Same energy Uniform symmetry 11 relevant atoms s, p, and d orbitals various interactions different energies

Y = caya + cbyb … cnyn Degree of orbital overlap/mixing depends on: Energy of the orbitals The closer the energy, the more mixing. Spatial proximity The atoms must be close enough that there is reasonable orbital overlap. Symmetry Atomic orbitals mix if they have similar symmetries. Strength of the bond depends upon the degree of orbital overlap.

How do we determine orbital energies?
Energy of the Orbitals For heteronuclear molecules: 1. The bonding orbital(s) will reside predominantly on the atom of lower orbital energy (the more electronegative atom). 2. The anti-bonding orbital(s) will reside predominantly on the atom with greater orbital energy (the less electronegative atom). How do we determine orbital energies?

Energy of Orbitals Theoretical calculations Photoelectron spectroscopy
Tabulated data Other peoples UPS/XPS data

Photoelectron Spectroscopy
Ionization occurs when matter interacts with light of sufficient energy (Heinrich Hertz, 1886) (Einstein, A. Ann. Phys. Leipzig 1905, 17, )

Photo-ionization and energy-dispersive analysis of the emitted photoelectrons to study the composition and electronic state of the sample. hνo = I(BE) + Ekinetic X-ray Photoelectron Spectroscopy (XPS) - using soft ( eV) x-ray excitation to examine core-levels. Ultraviolet Photoelectron Spectroscopy (UPS) - using vacuum UV (10-45 eV) radiation from discharge lamps to examine valence levels.

Sample introduction Chamber
Photoelectron Spectrometer X-Ray source Ion source Axial Electron Gun Detector CMA sample SIMS Analyzer Sample introduction Chamber Sample Holder Ion Pump Roughing Pump Slits

Photoelectron Spectrometer

Counts

Diagram for methane (CH4)?
Tabulated Data Miessler and Tarr, Inorganic Chemistry Diagram for methane (CH4)?

Tabulated Data

Y = caya + cbyb … cnyn Degree of orbital overlap/mixing depends on: Energy of the orbitals The closer the energy, the more mixing. Spatial proximity The atoms must be close enough that there is reasonable orbital overlap. Symmetry Atomic orbitals mix if they have similar symmetries. Strength of the bond depends upon the degree of orbital overlap.

Symmetry and Orbital Diagrams
Number of MOs = number of incipient orbitals. This rule could be referred to as “the conservation of orbitals.” Orbitals of the same symmetry mix. Orbital interactions can be bonding, nonbonding or antibonding. There are three basic types of orbital overlap: s (end on interaction), p (side by side approach) and d (off-axis approach). Orbitals with the correct symmetry and most similar energy mix to the greatest extent. J. Chem. Edu. 2004, 81, 997.

Constructing MOs From inspection From Group Theory

Constructing MOs s bond (s, p and d) p bond (p and d) d bond (d) p p p

Constructing MOs (s-s)

Constructing MOs (p-p)

Constructing MOs (d-d)

Simple Diatomics

MO Diagrams from Group Theory
Assign a point group Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to Irreducible Representation Combine central and peripheral orbitals by their symmetry Fill MOs with e- Generate SALCs of peripheral atoms Draw peripheral atoms SALC with central atom orbital to generate bonding/antibonding MOs.

H-F diatomic, H = 1s; F = 2s, 3 x 2p H2O triatomic, H = 2 x 1s; O = 2s, 3 x 2p CO2 p bonding, O = 2s, 3 x 2p; C = 2s, 3 x 2p C2H4 Fragmentation method Benzene Real + Imaginary SALC

HF Orbital Diagram Assign a point group H-F C2v C∞v C2v

HF Orbital Diagram H-F F H C2v A1 Assign a point group
Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 H-F C2v A1 H s orbital F s, px, py and pz orbitals z x F y H GH 1 1 1 1

HF Orbital Diagram H-F F H C2v A1 A1 Assign a point group
Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 H-F C2v A1 H s orbital F s, px, py and pz orbitals A1 z x F y H GFs 1 1 1 1

HF Orbital Diagram H-F F H C2v A1 A1 A1 Assign a point group
Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 H-F C2v A1 H s orbital F s, px, py and pz orbitals A1 A1 z x F y H GFpz 1 1 1 1

HF Orbital Diagram H-F F H C2v A1 A1 B1 A1 Assign a point group
Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 H-F C2v A1 H s orbital F s, px, py and pz orbitals A1 B1 A1 z x F y H GFpx 1 -1 1 -1

HF Orbital Diagram H-F F H C2v A1 A1 B1 B2 A1 Assign a point group
Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 H-F C2v A1 H s orbital F s, px, py and pz orbitals A1 B1 B2 A1 z x F y H GFpy 1 -1 -1 1

HF Orbital Diagram H-F C2v A1 A1 B1 B2 A1 Assign a point group
Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to irreducible representation Combine orbitals by their symmetry H-F C2v A1 H s orbital F s, px, py and pz orbitals A1 B1 B2 A1

HF Orbital Diagram Combine orbitals by their symmetry H s orbital
F s, px, py and pz orbitals

HF Orbital Diagram H H-F F A1 A1 B1 B2 A1
Combine orbitals by their symmetry 1s (A1) pz (A1) px (B1) py (B2) 2s (A1) H H-F F A1 H s orbital F s, px, py and pz orbitals A1 B1 B2 A1

HF Orbital Diagram H H-F F Combine orbitals by their symmetry A1
1s (A1) py (B2) px (B1) pz (A1) px (B1) py (B2) A1 A1 2s (A1) H H-F F

HF Orbital Diagram H H-F F 1 e- 7 e- Fill MOs with e- A1 1s (A1)
py (B2) px (B1) pz (A1) px (B1) py (B2) A1 A1 2s (A1) H H-F 1 e- F 7 e-

e- in MOs Electrons preferentially occupy molecular orbitals that are lower in energy. (Aufbau Principle) If two electrons occupy the same molecular orbital, they must be spin paired. (Pauli Exclusion Principle) When occupying degenerate molecular orbitals, electrons occupy separate orbitals with parallel spins before pairing. (Hund’s Rule)

HF Orbital Diagram H H-F F Fill MOs with e- A1 1s (A1) py (B2) px (B1)
pz (A1) px (B1) py (B2) A1 A1 2s (A1) H H-F F

HF Orbital Diagram Assign a point group
Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to irreducible representation Combine orbitals by their symmetry Fill MOs with e- Generate SALCs of peripheral atoms Draw peripheral atoms SALC with central atom orbital to generate bonding/antibonding MOs. A1 1s (A1) py (B2) px (B1) pz (A1) px (B1) py (B2) A1 A1 2s (A1) H H-F F

HF Orbital Diagram F H-F H Draw orbitals A1 1s (A1) B2 B1 pz py px A1

H-F diatomic, H = 1s; F = 2s, 3 x 2p H2O triatomic, H = 2 x 1s; O = 2s, 3 x 2p CO2 p bonding, O = 2s, 3 x 2p; C = 2s, 3 x 2p C2H4 Fragmentation method NH3/Benzene Real + Imaginary SALC

H2O Orbital Diagram H2O O H H C2v A1 + B1 A1 + B1 Assign a point group
Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 H2O C2v A1 + B1 H s orbitals O s, px, py and pz orbitals z O x y H H GH 2 2 GH A1 + B1

H2O Orbital Diagram H2O O H H C2v A1 + B1 A1 B1 B2 A1
Assign a point group Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 H2O C2v A1 + B1 H s orbitals O s, px, py and pz orbitals A1 B1 B2 A1 z O x y H H

H2O Orbital Diagram H2O C2v A1 + B1 Assign a point group
Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to irreducible representation Combine orbitals by their symmetry H2O C2v A1 + B1 H s orbitals O s, px, py and pz orbitals

HF Orbital Diagram Combine orbitals by their symmetry H s orbital
O s, px, py and pz orbitals

H2O Orbital Diagram 2 x H O A1 + B1 A1 B1 B2 A1
Combine orbitals by their symmetry A1 B1 pz (A1) px (B1) py (B2) H s orbital O s, px, py and pz orbitals A1 + B1 2s (A1) 2 x H O A1 B1 B2 A1

H2O Orbital Diagram 2 x H O H2O Combine orbitals by their symmetry B1
A1 B1 py (B2) pz (A1) px (B1) py (B2) A1 B1 2s (A1) A1 2 x H H2O O

H2O Orbital Diagram 2 x H O H2O 2 e- 6 e- Fill MOs with e- B1 A1 A1 B1
py (B2) pz (A1) px (B1) py (B2) A1 B1 2s (A1) A1 2 e- 2 x H H2O O 6 e-

H2O Orbital Diagram Assign a point group
Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to irreducible representation Combine orbitals by their symmetry Fill MOs with e- Generate SALCs of peripheral atoms Draw peripheral atoms SALC with central atom orbital to generate bonding/antibonding MOs. 2s (A1) pz (A1) py (B2) px (B1) A1 B1 A1 B1 py (B1) A1

Symmetry adapted linear combination of atomic orbitals (SALC)
H2O Orbital Diagram Generate SALCs of peripheral atoms Symmetry adapted linear combination of atomic orbitals (SALC) Use projection operator to generate SALC. Projection operators constitute a method of generating the symmetry allowed combinations. Taking one AO and projecting it out using symmetry. Pi is the projection operator li is the dimension of Gi h is the order of the group i is an irreducible representation of the group R is an operation of the group χi (R) is the character of R in the ith irreducible representation (R) non-symmetry-adapted basis

H2O Orbital Diagram Generate SALCs of peripheral atoms
To generate SALCs, the steps are: group the atomic orbitals in the molecule into sets which are equivalent by symmetry generate the rep. then irr. rep. for each set Use projection operator for one basis

H2O Orbital Diagram O H H A1 + B1 Generate SALCs of peripheral atoms
To generate SALCs, the steps are: group the similar AOs generate the rep. then irr. rep. for each set Use projection operator for one basis z PA1 = 1/4 [((1) E f1 ) + ((1) C2 f1 ) + ((1) sxz f1 ) + ((1) syz f1 )] O x y H H f1 f2 GH = A1 + B1

H2O Orbital Diagram O H H A1 + B1 A1 H1s orbitals
Generate SALCs of peripheral atoms To generate SALCs, the steps are: group the similar AOs generate the rep. then irr. rep. for each set Use projection operator for one basis z PA1 = 1/4 [((1) E f1 ) + ((1) C2 f1 ) + ((1) sxz f1 ) + ((1) syz f1 )] O x y f1 f2 f1 f2 H H f1 f2 PA1 = 1/4 [f1 + f2 + f1 + f2] PA1 = 1/4 [2f1 + 2f2] GH = A1 + B1 PA1 = 1/2 [f1 + f2] A1 H1s orbitals

H2O Orbital Diagram O H H A1 + B1 Generate SALCs of peripheral atoms
To generate SALCs, the steps are: group the similar AOs generate the rep. then irr. rep. for each set Use projection operator for one basis z PB1 = 1/4 [((1) E f1 ) + ((-1) C2 f1 ) + ((1) sxz f1 ) + ((-1) syz f1 )] O x y H H f1 f2 GH = A1 + B1

H2O Orbital Diagram O H H A1 + B1 B1 H1s orbitals
Generate SALCs of peripheral atoms To generate SALCs, the steps are: group the similar AOs generate the rep. then irr. rep. for each set Use projection operator for one basis z PB1 = 1/4 [((1) E f1 ) + ((-1) C2 f1 ) + ((1) sxz f1 ) + ((-1) syz f1 )] O x y f1 f2 f1 f2 H H f1 f2 PB1 = 1/4 [f1 - f2 + f1 - f2] PB1 = 1/4 [2f1 - 2f2] GH = A1 + B1 PB1 = 1/2 [f1 - f2] B1 H1s orbitals

H2O Orbital Diagram O H H H2O Draw SALC with central atom. B1 A1 A1 B1
py (B2) pz (A1) px (B1) py (B2) A1 z B1 O x y H H 2s (A1) A1 H2O

H2O Orbital Diagram Assign a point group
Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to irreducible representation Combine orbitals by their symmetry Fill MOs with e- Generate SALCs of peripheral atoms Draw peripheral atoms SALC with central atom orbital to generate bonding/antibonding MOs. 2s (A1) pz (A1) py (B2) px (B1) A1 B1 A1 B1

Sidenote: Many Electron States
Determining the symmetry of many electron states from the symmetry of the individual one electron wavefunctions. Important for formulating spectroscopic selection rules between orbitals or electronic states. State symmetry found from the direct product of all electron symmetries. B1 A1 B2 A1 B1 A1 H2O

H2O: A1 A1 B2 B2 A1 A1 B2 B2 = A1 B1 A1 A1 B2 B2 A1 A1 B2 B2 = A1 B2 B2 …etc. or closed shell configurations cancel! A1 H2O: A1 A1 B2 B2 A1 A1 B2 B2 = A1 B1 A1 A1 A1 A1 A1 H2O

H2O: A1 A1 B2 B2 A1 A1 B2 B2 = A1 B1 A1 B2 A1 B1 A1 H2O

H2O: A1 A1 B2 B2 A1 A1 B2 B2 = A1 B1 A1 H2O+: A1 A1 B2 B2 A1 A1 B2 = B2 B2 A1 B1 A1 H2O+

H2O: A1 A1 B2 B2 A1 A1 B2 B2 = A1 B1 A1 H2O+: A1 A1 B2 B2 A1 A1 B2 = B2 B2 A1 H2O- = A1 B1 A1 H2O-

H2O: A1 A1 B2 B2 A1 A1 B2 B2 = A1 B1 A1 H2O+: A1 A1 B2 B2 A1 A1 B2 = B2 B2 A1 H2O- = A1 B1 H2O* = B2 A1 H2O*

Sidenote: Spin Multiplicity
H2O H2O+ H2O- H2O* A1 A1 A1 A1 A1 or B2 B2 B2 B2 B2 A1 A1 A1 A1 A1 A1 B2 A1 B2 (2s+1)G1 or 2 Spin Multiplicity: s = 0 s = 1/2 s = 1/2 s = 0 s = 1 1A1 2B2 2A1 1B2 3B2

Ground State Symmetry of H2O is 1A1
H2O Orbital Diagram Assign a point group Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to irreducible representation Combine orbitals by their symmetry Fill MOs with e- Generate SALCs of peripheral atoms Draw peripheral atoms SALC with central atom orbital to generate bonding/antibonding MOs. 2s (A1) pz (A1) py (B2) px (B1) A1 B1 A1 B1 Ground State Symmetry of H2O is 1A1

CO2 Orbital Diagram Assign a point group CO2 D2h D∞h D2h

CO2 Orbital Diagram CO2 D2h Assign a point group
Choose basis function (orbitals) CO2 D2h C s, px, py and pz orbitals O px, py and pz orbitals

Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 CO2 D2h C s, px, py and pz orbitals O px, py and pz orbitals z

Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 CO2 D2h Ag B3u B2u B1u C s, px, py and pz orbitals O px, py and pz orbitals z

Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 CO2 D2h Ag B3u B2u B1u C s, px, py and pz orbitals O px, py and pz orbitals z GOpz 2 2 2 2 GOpz Ag + B1u

Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 CO2 D2h Ag B3u B2u B1u C s, px, py and pz orbitals O px, py and pz orbitals z GOpx 2 -2 2 -2 GOpx B3u + B2g

Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 CO2 D2h Ag B3u B2u B1u C s, px, py and pz orbitals O px, py and pz orbitals z GOpy 2 -2 -2 2 GOpy B2u + B3g

Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to irreducible representation Combine orbitals by their symmetry CO2 D2h Ag B3u B2u B1u C s, px, py and pz orbitals O px py pz B3u + B2g B2u + B3g Ag + B1u

CO2 Orbital Diagram Combine orbitals by their symmetry
C s, px, py and pz orbitals O px, py and pz orbitals

CO2 Orbital Diagram C OCO 2 x O Combine orbitals by their symmetry B1u
px py pz B3u B2u 2p B3u B2u B1u px py pz B2g B3g B2g B3g B1u 2p B3u B2u Ag B3u B2u 2s B1u Ag Ag C OCO 2 x O

CO2 Orbital Diagram Assign a point group
2 x O Ag B3u B2u B1u 2s 2p B2g px py pz B3g OCO Assign a point group Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to irreducible representation Combine orbitals by their symmetry Fill MOs with e-

CO2 Orbital Diagram C OCO 2 x O 4 e- 8 e- Fill MOs with e- B1u Ag px
py pz B3u B2u 2p B3u B2u B1u px py pz B2g B3g B2g B3g B1u 2p B3u B2u Ag B3u B2u 2s B1u Ag Ag 4 e- C OCO 2 x O 8 e-

2 x O Ag B3u B2u B1u 2s 2p B2g px py pz B3g OCO Assign a point group Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to irreducible representation Combine orbitals by their symmetry Fill MOs with e- Generate SALCs of peripheral atoms Draw peripheral atoms SALC with central atom orbital to generate bonding/antibonding MOs.

CO2 Orbital Diagram Generate SALCs of peripheral atoms f2 f1 Ag + B1u
z GOpz Ag + B1u PAg = 1/8 [((1) E f1 ) + ((1) C2 f1 ) + ((1) C2 f1 ) … etc.] PAg = 1/8 [4f f2]

CO2 Orbital Diagram Generate SALCs of peripheral atoms z

CO2 Orbital Diagram Draw SALC with central atom. C OCO 2 x O

Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to irreducible representation Combine orbitals by their symmetry Fill MOs with e- Generate SALCs of peripheral atoms Draw peripheral atoms SALC with central atom orbital to generate bonding/antibonding MOs.

Ethene Orbital Diagram
x z y Two different approaches (D2h) C1 + C2 H1-4 then combine CH2 then combine J. Chem. Edu. 2004, 81, 997

Ethene Orbital Diagram

NH3 Orbital Diagram NH2 C3v A1 + E A1 + E Assign a point group
Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 NH2 C3v A1 + E H s orbitals N s, px, py and pz orbitals z x y GH 3 1 GH A1 + E

NH3 Orbital Diagram NH2 C3v A1 + E A1 E A1 Assign a point group
Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 NH2 C3v A1 + E H s orbitals N s, px, py and pz orbitals A1 E A1 z x y

NH3 Orbital Diagram Combine orbitals by their symmetry H s orbital
N s, px, py and pz orbitals

NH3 Orbital Diagram 3 x H N NH3 Combine orbitals by their symmetry A1
pz (A1) py, px (E) A1 E A1 s (A1) E A1 3 x H NH3 N

NH3 Orbital Diagram 3 x H N NH3 3 e- 5 e- Fill MOs with e- A1 E
pz (A1) py, px (E) A1 E A1 s (A1) E A1 3 e- 3 x H NH3 N 5 e-

NH3 Orbital Diagram Assign a point group
Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to irreducible representation Combine orbitals by their symmetry Fill MOs with e- Generate SALCs of peripheral atoms Draw peripheral atoms SALC with central atom orbital to generate bonding/antibonding MOs. E A1 s (A1) pz (A1) py, px (E)

NH3 Orbital Diagram A1 + E Generate SALCs of peripheral atoms
To generate SALCs, the steps are: group the atomic orbitals in the molecule into sets which are equivalent by symmetry generate the rep. then irr. rep. for each set Use projection operator for one basis z f3 x f1 f2 y GH = A1 + E

NH3 Orbital Diagram Separate classes
Generate SALCs of peripheral atoms Separate classes

NH3 Orbital Diagram Generate SALCs of peripheral atoms z x f3 f1 f2 y

NH3 Orbital Diagram Generate SALCs of peripheral atoms f3 f1 f2
PA1 ≈ ((1) E f1 ) + ((1) C3+f1 ) + ((1) C3-f1 ) + ((1) s1 f1 ) + ((1) s2 f1 ) + ((1) s2 f1 ) f1 f2 f3 f1 f3 f2 PA1 ≈ [f1 + f2 + f3 + f1 + f3 + f2 ] A1 H1s orbitals f3 PA1 ≈ [ 2f1 + 2f2 + 2f3] f1 PA1 ≈ [ f1 + f2 + f3] f2

What about the other E orbital?
NH3 Orbital Diagram Generate SALCs of peripheral atoms PE ≈ ((2) E f1 ) + ((-1) C3+f1 ) + ((-1) C3-f1 ) + ((0) s1 f1 ) + ((0) s2 f1 ) + ((0) s2 f1 ) f1 f2 f3 PA1 ≈ [2f1 - f2 - f3] One of the E orbitals f3 What about the other E orbital? f1 f2

NH3 Orbital Diagram Generate SALCs of peripheral atoms

NH3 Orbital Diagram Generate SALCs of peripheral atoms f3 f3 f1 f1 f2
3 different E SALCS have been generated but they are all similar. Use subtraction or addition to generate new SALC. f3 f3 f1 f2 f1 f2

NH3 Orbital Diagram 3 x H N NH3 Draw SALC with central atom. A1 E
pz (A1) py, px (E) A1 E A1 s (A1) E A1 3 x H NH3 N

NH3 Orbital Diagram Draw SALC with central atom. N 3 x H

NH3 Orbital Diagram Assign a point group
Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to irreducible representation Combine orbitals by their symmetry Fill MOs with e- Generate SALCs of peripheral atoms Draw peripheral atoms SALC with central atom orbital to generate bonding/antibonding MOs.

Benzene MOs and SALC 3 nodes 2 nodes 1 node 0 nodes

C6H6 Orbital Diagram C6H6 D6h Assign a point group
Choose basis function (orbitals) C6H6 D6h only p bonding C pz orbitals

C6H6 Orbital Diagram C6H6 D6h C6 C″2 C′2 Assign a point group
Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 C6H6 D6h only p bonding C pz orbitals C6 C′2 C″2 z axis D6h E 2C6 2C3 C2 3C′2 3C″2 i 2S3 2S6 σh 3 σd 3 σv Гπ 6 -2 -6 2

C6H6 Orbital Diagram C6H6 D6h C6 C″2 C′2 Gp: B2g + E1g + A2u + E2u
Assign a point group Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to irreducible representation C6H6 D6h only p bonding C pz orbitals C6 C′2 C″2 z axis D6h E 2C6 2C3 C2 3C′2 3C″2 i 2S3 2S6 σh 3 σd 3 σv Гπ 6 -2 -6 2 Gp: B2g + E1g + A2u + E2u

C6H6 Orbital Diagram C6H6 D6h C6 C″2 C′2 Gp: B2g + E1g + A2u + E2u
Assign a point group Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to irreducible representation Combine orbitals by their symmetry C6H6 D6h only p bonding C pz orbitals C6 C′2 C″2 D6h E 2C6 2C3 C2 3C′2 3C″2 i 2S3 2S6 σh 3 σd 3 σv Гπ 6 -2 -6 2 Gp: B2g + E1g + A2u + E2u

C6H6 Orbital Diagram Combine orbitals by their symmetry B2g E2u E1g
A2u

C6H6 Orbital Diagram 6 pz orbitals = 6 e- Fill MOs with e- B2g E2u E1g
A2u 6 pz orbitals = 6 e-

C6H6 Orbital Diagram C6H6 D6h Gp: B2g + E1g + A2u + E2u
Assign a point group Choose basis function (orbitals) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to irreducible representation Combine orbitals by their symmetry Fill MOs with e- Generate SALCs of peripheral atoms Draw peripheral atoms SALC with central atom orbital to generate bonding/antibonding MOs. C6H6 D6h only p bonding C pz orbitals Gp: B2g + E1g + A2u + E2u

C6H6 Orbital Diagram Simplify usingC6! C6 D6h
Generate SALCs of peripheral atoms Simplify usingC6! D6h C6

C6H6 Orbital Diagram Simplify usingC6!
Generate SALCs of peripheral atoms Simplify usingC6! D6h C6

C6H6 Orbital Diagram Generate SALCs of peripheral atoms A orbital
To generate SALCs, the steps are: group the similar AOs generate the rep. then irr. rep. for each set Use projection operator for one basis A orbital

C6H6 Orbital Diagram Generate SALCs of peripheral atoms
To generate SALCs, the steps are: group the similar AOs generate the rep. then irr. rep. for each set Use projection operator for one basis

C6H6 Orbital Diagram Generate SALCs of peripheral atoms B orbital
To generate SALCs, the steps are: group the similar AOs generate the rep. then irr. rep. for each set Use projection operator for one basis B orbital

C6H6 Orbital Diagram B ≈ B2g A ≈ A2u
Generate SALCs of peripheral atoms B ≈ B2g B E2 E1 A ≈ A2u A

C6H6 Orbital Diagram Generate SALCs of peripheral atoms
To generate SALCs, the steps are: group the similar AOs generate the rep. then irr. rep. for each set Use projection operator for one basis ok ok What?

C6H6 Orbital Diagram Contain imaginary components; the real component of the linear combination may be realized by taking ± linear combinations:

C6H6 Orbital Diagram Contain imaginary components; the real component of the linear combination may be realized by taking ± linear combinations: For C6 point group: or from Euler’s formula

C6H6 Orbital Diagram Contain imaginary components; the real component of the linear combination may be realized by taking ± linear combinations: divide out and remove prefactor constant (-i√3)

What are the pictorial representation of the SALC’s?
C6H6 Orbital Diagram What are the pictorial representation of the SALC’s?

Projection Operator: Benzene
What are the pictorial representation of the SALC’s? 3 nodes 2 nodes 1 node 0 nodes

Side Note: Orbital Hybridization
In chemistry, hybridization is the concept of mixing atomic orbitals into new hybrid orbitals (with different energies, shapes, etc., than the component atomic orbitals) suitable for the pairing of electrons to form chemical bonds. H C

s + p Hybrid Orbitals Miessler and Tarr, Inorganic Chemistry

s + p + d Hybrid Orbitals

BF3 Hybridization BF3 Steps to determine the hybridization of a bond.
Assign a point group Choose basis function (s bonds) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 D3h s bonds D3h Гs

Reduce to irreducible representation D3h s bonds Гs Gs: A1’ + E’

Compare symmetry of irr. rep. to central atom MOs BF3 D3h B (s) = A1’ B (px)= E’ B (py)= E’ B (pz)= A2” Gs: A1’ + E’

sp2 hybridization (s, px, py)
BF3 Hybridization Steps to determine the hybridization of a bond. Compare symmetry of irr. rep. to central atom MOs z z z z z y y y y y y x x x x x Gs: A1’ + E’ s = A1’ px = E’ py = E’ pz = A2” sp2 hybridization (s, px, py)

Hybridization Steps to determine the hybridization of a bond.
Assign a point group Choose basis function (s bonds) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to irreducible representation Compare symmetry of irr. rep. to central atom MOs

Symmetry and Reactivity
(2 + 2) cycloaddition 2 x ethylene cyclobutane p orbitals s bonds Orbital symmetry is retained during the reaction!

(2 + 2) cycloaddition

Thermal Reaction Photo Reaction 2 bonding + 2 antibonding e- Thermally Forbidden (~115 kcal/mol) 3 bonding + 1 antibonding e- Photochemically Allowed

Part 2.7: Orbital Diagrams

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Part 2.7: Orbital Diagrams

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