Part 27 Orbital Diagrams 1 Orbital Diagrams Orbital Interactions Molecular Orbital Theory Orbital Energies MO Diagrams HF H 2 O CO 2 C 2 H 4 NH 3 Benzene SALC Hybridization Symmetry and Reactivity ID: 763683
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Part 2.7: Orbital Diagrams 1
Orbital Diagrams Orbital Interactions Molecular Orbital Theory Orbital EnergiesMO DiagramsHF, H2O, CO2, C2H4, NH3, BenzeneSALCHybridizationSymmetry and Reactivity 2
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. Atomic Orbitals 1s orbital 2p orbital 3
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. Atomic Orbitals 1s orbital 2p orbital 4
Atomic Orbitals constructively = bonding destructively = antibonding not at all = non-bonding 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 ( Y 2 ) of finding any electron of an atom in any specific region around the atom's nucleus. 5
ATOMIC ORBITALS of different atoms combine to create MOLECULAR ORBITALS The number of ATOMIC ORBITALS = the number of MOLECULAR ORBITALSElectrons in these MOLECULAR ORBITALS are shared by the molecule as whole MOLECULAR ORBITALS can be constructed from Linear Combination of Atomic Orbitals (LCAO) Molecular Orbital Theory Y = c aya + c b y b ( for diatomic molecules) LCAO 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 nuclei6
Combining Atomic Orbitals Bonding: Ψ ( σ ) or Ψ + = ( 1/√2 ) [ φ (1s a ) + φ (1s b ) ] Antibonding : Ψ ( σ *) or Ψ - = ( 1/√2 ) [ φ (1s a ) - φ (1s b ) ] 7
Combining Atomic Orbitals (H 2 ) Antibonding Bonding 8
Combining Atomic Orbitals H 2 Fe(C 5 H 5 ) 2 2 atoms Only s orbitals Linear interaction Same energy Uniform symmetry 11 relevant atoms s, p, and d orbitals various interactions different energies 9
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. Combining Atomic Orbitals Strength of the bond depends upon the degree of orbital overlap. Y = c a y a + c b y b … c n y n 10
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). Energy of the Orbitals How do we determine orbital energies? 11
Energy of Orbitals Theoretical calculations Photoelectron spectroscopy Tabulated data Other peoples UPS/XPS data 12
Photoelectron Spectroscopy Ionization occurs when matter interacts with light of sufficient energy (Heinrich Hertz, 1886 ) ( Einstein, A. Ann. Phys. Leipzig 1905, 17, 132-148.) 13
Photo-ionization and energy-dispersive analysis of the emitted photoelectrons to study the composition and electronic state of the sample . X-ray Photoelectron Spectroscopy (XPS) - using soft (200-2000 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. Photoelectron Spectroscopy h ν o = I(BE) + E kinetic 14
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 15
Photoelectron Spectrometer 16
Photoelectron Spectroscopy Counts 17
Photoelectron Spectroscopy Counts 18
19
Miessler and Tarr , Inorganic Chemistry Tabulated Data Diagram for methane (CH 4 )? 20
http://en.wikipedia.org/wiki/Ionization_energy Tabulated Data 21
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. Combining Atomic Orbitals Strength of the bond depends upon the degree of orbital overlap. Y = c a y a + c b y b … c n y n 22
Symmetry and Orbital Diagrams J. Chem. Edu . 2004, 81, 997. 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. 23
Constructing MOs From inspection From Group Theory 24
Constructing MOs s bond (s, p and d) d d p bond (p and d) p p p d d bond (d) 25
Constructing MOs (s-s) 26
Constructing MOs (p-p) 27
Constructing MOs (d-d) 28
Simple Diatomics 29
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 symmetryFill MOs with e - Generate SALCs of peripheral atoms Draw peripheral atoms SALC with central atom orbital to generate bonding/ antibonding MOs. 30
MO Diagrams from Group Theory H-F diatomic, H = 1s; F = 2s, 3 x 2p H 2 O triatomic, H = 2 x 1s; O = 2s, 3 x 2p CO 2 p bonding, O = 2s, 3 x 2p; C = 2s, 3 x 2p C 2 H 4 Fragmentation method Benzene Real + Imaginary SALC 31
32 HF Orbital Diagram Assign a point group C 2v C ∞v H-F C 2v
33 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 F H z x y G H 1 1 1 1 H s orbital F s, p x , p y and p z orbitals H-F C 2v A 1
34 HF Orbital Diagram H-F 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 C 2v F H z x y G Fs 1 1 1 1 H s orbital F s, p x , p y and p z orbitals A 1 A 1
35 H s orbital F s, px, py and pz orbitals HF Orbital Diagram H-F 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 C 2v F H z x y G Fpz 1 1 1 1 A 1 A 1 A 1
36 H s orbital F s, px, py and pz orbitals HF Orbital Diagram H-F 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 C 2v F H z x y G Fpx 1 -1 1 -1 A 1 A 1 A 1 B 1
37 H s orbital F s, px, py and pz orbitals HF Orbital Diagram H-F 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 C 2v F H z x y G Fpy 1 -1 -1 1 A 1 A 1 A 1 B 1 B 2
38 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 H s orbital F s, p x , p y and p z orbitals H-F C 2v A 1 A 1 A 1 B 1 B 2
HF Orbital Diagram Combine orbitals by their symmetry H s orbital F s, p x , p y and pz orbitals39
HF Orbital Diagram Combine orbitals by their symmetry H F 2s (A 1 ) H s orbital F s, p x , p y and p z orbitals A 1 A 1 A 1 B 1 B 2 p z (A 1 ) p y (B 2 ) p x (B 1 ) 1s (A 1 ) H-F 40
HF Orbital Diagram Combine orbitals by their symmetry H F 2s (A 1 ) p z (A 1 ) p y (B 2 ) p x (B 1 ) 1s (A 1 ) A 1 A 1 A 1 p y (B 2 ) p x (B 1 ) H-F 41
HF Orbital Diagram Fill MOs with e - H F 2s (A 1 ) p z (A 1 ) p y (B 2 ) p x (B 1 ) 1s (A 1 ) A 1 A 1 p y (B 2 ) p x (B 1 ) 1 e - 7 e - A 1 H-F 42
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 ) 43
HF Orbital Diagram Fill MOs with e - H F 2s (A 1 ) p z (A 1 ) p y (B 2 ) p x (B 1 ) 1s (A 1 ) A 1 A 1 p y (B 2 ) p x (B 1 ) H-F A 1 44
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 (A 1 ) 45 HF Orbital Diagram H F p z (A 1 ) p y (B 2 ) p x (B 1 ) 1s (A 1 ) A 1 A 1 A 1 p y (B 2 ) p x (B 1 ) H-F
HF Orbital Diagram Draw orbitals 2s (A 1 ) p z p y p x 1s (A 1 ) A 1 A 1 B 2 B 1 F H z x y F F H F F F F H F F F H A 1 H-F 46
H-F diatomic, H = 1s; F = 2s, 3 x 2p H 2 O triatomic , H = 2 x 1s; O = 2s, 3 x 2p CO 2 p bonding, O = 2s, 3 x 2p; C = 2s, 3 x 2p C 2 H 4 Fragmentation method NH 3 /Benzene Real + Imaginary SALC MO Diagrams from Group Theory 47
48 H 2 O 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 O H z x y G H 2 0 2 0 H s orbitals O s, p x , p y and p z orbitals H 2 O C 2v H A 1 + B 1 G HA1 + B1
49 H 2 O 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 O H z x y H s orbitals O s, p x , p y and p z orbitals H 2 O C 2v H A 1 + B 1 A 1 A 1 B 1 B 2
50 H 2 O 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 H s orbitals O s, p x , p y and p z orbitals H 2 O C 2v A 1 + B 1
HF Orbital Diagram Combine orbitals by their symmetry H s orbital O s, p x , p y and pz orbitals51
H 2 O Orbital DiagramCombine orbitals by their symmetry2 x H O 2s (A 1 ) H s orbital O s, px, py and pz orbitals A1 + B 1 A1 A 1 B 1 B 2 p z (A 1 ) p y (B 2 ) p x (B 1 ) A 1 B1 52
H 2 O Orbital DiagramCombine orbitals by their symmetry2 x H O 2s (A 1 ) p z (A1) p y (B2) px (B 1) A1 B1 H2 O A 1 A 1 A 1 B 1 B 1 p y (B 2 ) 53
H 2 O Orbital DiagramFill MOs with e-2 x H O 2s (A 1 ) p z (A1) p y (B2) px (B 1) A1 B1 H2 O A 1 A 1 A 1 B 1 B 1 p y (B 2 ) 2 e - 6 e - 54
55 H 2 O 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 (A 1 ) p z (A 1 ) p y (B 2 ) p x (B 1 ) A 1 B 1 A 1 A 1 B 1 B 1 p y (B 1 ) A 1
H 2 O Orbital DiagramGenerate SALCs of peripheral atoms 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. Symmetry adapted linear combination of atomic orbitals (SALC) P i is the projection operator li is the dimension of Gih is the order of the groupi is an irreducible representation of the group R is an operation of the groupχi (R) is the character of R in the ith irreducible representation(R) non-symmetry-adapted basis 56
H 2 O Orbital DiagramGenerate 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 57
H 2 O Orbital DiagramGenerate 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 O H z x y H f 1 f 2 A 1 + B 1 G H = P A1 = 1/4 [((1) E f 1 ) + ((1) C 2 f 1 ) + ((1) s xz f 1 ) + ((1) s yz f1 )] 58
H 2 O Orbital DiagramGenerate 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 O H z x y H f 1 f 2 A 1 + B 1 G H = P A1 = 1/4 [((1) E f 1 ) + ((1) C 2 f 1 ) + ((1) s xz f 1 ) + ((1) s yz f 1 )] 59
P A1 = 1/4 [((1) E f1 ) + ((1) C2 f1 ) + ((1) sxz f1 ) + ((1) syz f1 )] H 2 O 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 O H z x y H f 1 f 2 A 1 + B 1 G H = 60
H 2 O Orbital DiagramGenerate 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 O H z x y H f 1 f 2 A 1 + B 1 G H = P A1 = 1/4 [((1) E f 1 ) + ((1) C 2 f 1 ) + ((1) s xz f 1 ) + ((1) s yz f1 )]f1 f2 f 1 f 2 P A1 = 1/4 [ f 1 + f 2 + f 1 + f 2 ] P A1 = 1/4 [2 f 1 + 2 f 2 ] P A1 = 1/2 [ f 1 + f 2 ] A 1 H1s orbitals 61
H 2 O Orbital DiagramGenerate 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 O H z x y H f 1 f 2 A 1 + B 1 G H = P B1 = 1/4 [((1) E f 1 ) + ((-1) C 2 f 1 ) + ((1) s xz f 1 ) + ((-1) s yz f1 )] 62
H 2 O Orbital DiagramGenerate 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 O H z x y H f 1 f 2 A 1 + B 1 G H = P B1 = 1/4 [((1) E f 1 ) + ((-1) C 2 f 1 ) + ((1) s xz f 1 ) + ((-1) s yz f1 )] 63
H 2 O Orbital DiagramGenerate 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 O H z x y H f 1 f 2 A 1 + B 1 G H = P B1 = 1/4 [((1) E f 1 ) + ((-1) C 2 f 1 ) + ((1) s xz f 1 ) + ((-1) s yz f1 )] 64
H 2 O Orbital DiagramGenerate 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 O H z x y H f 1 f 2 A 1 + B 1 G H = f 1 f 2 f 1 f 2 P B1 = 1/4 [ f 1 - f 2 + f 1 - f 2 ] P B1 = 1/4 [2 f 1 - 2 f 2 ] P B1 = 1/2 [ f 1 - f 2 ] B 1 H1s orbitals P B1 = 1/4 [((1) E f 1 ) + ((-1) C 2 f 1 ) + ((1) s xz f 1 ) + ((-1) s yz f 1 )] 65
H 2 O Orbital Diagram Draw SALC with central atom. 2s (A 1 ) p z (A 1) p y (B2) px (B 1) A1 B1 H 2 O A 1 A 1 A 1 B 1 B 1 p y (B 2 ) O H z y H x 66
67 H 2 O 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 (A 1 ) p z (A 1 ) p y (B 2 ) p x (B 1 ) A 1 B 1 A 1 A 1 A 1 B 1 B 1 p y (B 2 )
Sidenote : Many Electron States H 2OA1 A 1 A 1 B 1 B 1 B 2 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. 68
H 2 O A1 A 1 A 1 B 1 B 1 B 2 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. Sidenote: Many Electron States 69
H 2 O A 1 A 1 A 1 B 1 B 1 B 2 H 2 O: A 1 A 1 B 2 B 2 A 1 A 1 = A 1 B 2 B 2 A 1 A 1 B 2 …etc. or closed shell configurations cancel! A 1 A 1 A 1 A 1 B 2 B 2 = A 1 B 2 B 2 H 2 O: A 1 A 1 B 2 B 2 A 1 A 1 = A 1 B 2 B 2 A 1 Sidenote: Many Electron States 70
H 2 O A 1 A 1 A 1 B 1 B 1 B 2 H 2 O: A 1 A 1 B 2 B 2 A 1 A 1 = A 1 B 2 B 2 Sidenote: Many Electron States 71
H 2 O + A 1 A 1 A 1 B 1 B 1 B 2 H 2 O: A 1 A 1 B 2 B 2 A 1 A 1 = A 1 B 2 B 2 H2O+:A1 A 1 B 2 B 2 A 1 A 1 = B 2 B 2 Sidenote: Many Electron States 72
H 2 O - A 1 A 1 A 1 B 1 B 1 B 2 H 2 O: A 1 A 1 B 2 B 2 A 1 A 1 = A 1 B 2 B 2 H2O+:A 1 A 1 B 2 B 2 A 1 A 1 = B 2 B 2 H 2 O - = A 1 Sidenote: Many Electron States 73
H 2 O * A 1 A 1 A 1 B 1 B 1 B 2 H 2 O: A 1 A 1 B 2 B 2 A 1 A 1 = A 1 B 2 B 2 H2O+:A1 A 1 B 2 B 2 A 1 A 1 = B 2 B 2 H 2 O - = A 1 Sidenote: Many Electron States H 2 O * = B 2 74
(2s+1) G 1 or 2 A1 A 1 B 2 H 2 O A 1 H 2 O + B 2 H 2 O - A 1 Sidenote : Spin Multiplicity H 2 O * B 2 A 1 A 1 B 2 A 1 A 1 B 2 A 1 A 1 B 2 Spin Multiplicity: s = 0 1 A 1 s = 1/2 2 B 2 s = 1/2 2 A 1 s = 0 1 B 2 A 1 A 1 B 2 or s = 1 3 B 2 75
76 H 2 O 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 (A 1 ) p z (A 1 ) p y (B 2 ) p x (B 1 ) A 1 B 1 A 1 A 1 A 1 B 1 B 1 p y (B 2 ) Ground State Symmetry of H 2 O is 1 A 1
MO Diagrams from Group Theory H-F diatomic, H = 1s; F = 2s, 3 x 2p H 2 O triatomic , H = 2 x 1s; O = 2s, 3 x 2p CO 2 p bonding, O = 2s, 3 x 2p; C = 2s, 3 x 2p C 2 H 4 Fragmentation method NH 3 /Benzene Real + Imaginary SALC 77
78 CO 2 Orbital Diagram Assign a point group D2h D ∞h CO 2 D 2h
79 CO 2 Orbital Diagram Assign a point group Choose basis function ( orbitals ) C s, p x , py and pz orbitalsO px , py and p z orbitals CO2 D 2h
80 CO 2 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 z C s, p x , p y and p z orbitals O p x , p y and p z orbitals CO 2 D 2h
81 CO 2 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 A g z C s, p x , p y and p z orbitals O p x , p y and p z orbitals CO 2 D 2h B 3u B 2u B 1u
82 CO 2 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 A g z C s, p x , p y and p z orbitals O p x , p y and p z orbitals CO 2 D 2h B 3u B 2u B 1u GOpz22000022GOpzAg + B1u
83 CO 2 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 A g z C s, p x , p y and p z orbitals O p x , p y and p z orbitals CO 2 D 2h B 3u B 2u B 1u GOpx2-200002-2GOpxB3u + B2g
84 CO 2 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 A g z C s, p x , p y and p z orbitals O p x , p y and p z orbitals CO 2 D 2h B 3u B 2u B 1u GOpy2-20000-22GOpyB2u + B3g
85 CO 2 Orbital DiagramAssign 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 A g C s, p x , p y and p z orbitals CO 2 D 2h B 3u B 2u B 1u O p x p y pzB2u + B3gB3u + B2gAg + B1u
CO 2 Orbital Diagram Combine orbitals by their symmetry C s, px, py and pz orbitals O p x, p y and pz orbitals 86
A g CO 2 Orbital DiagramCombine orbitals by their symmetry C 2 x O A g B 3u B 2u B 1u 2s 2p 2p B 2g B 3u p x p y p z B 2u B 3g A g B 1u p x p y p z B 2g B 3g OCO A g B 1u B 1u B 3u B 2u B 3u B 2u 87
88 CO 2 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 - C 2 x O A g B 3u B 2u B 1u 2s 2p 2p B 2g B 3u p x p y p z B 2u B 3g A g B 1u p x p y p z B 2g B 3g OCO A g A g B 1u B 1u B 3u B 2u B 3u B 2u
CO 2 Orbital Diagram Fill MOs with e- C 2 x O A g B 3u B 2u B 1u 2s 2p 2p B 2g B 3u p x p y p z B 2u B 3g A g B 1u p x p y p z B 2g B 3g OCO A g A g B 1u B 1u B 3u B 2u B 3u B 2u 4 e - 8 e - 89
90 CO 2 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. C 2 x O A g B 3u B 2u B 1u 2s 2p 2p B 2g B 3u p x p y p z B 2u B 3g A g B 1u p x p y p z B 2g B 3g OCO A g A g B 1u B 1u B 3u B 2u B 3u B 2u
91 CO 2 Orbital DiagramGenerate SALCs of peripheral atoms z G Opz A g + B 1u f 1 f 2 P Ag = 1/8 [((1) E f 1 ) + ((1) C 2 f 1 ) + ((1) C 2 f 1 ) … etc.] P Ag = 1/8 [4 f 1 + 4f2]
92 CO 2 Orbital DiagramGenerate SALCs of peripheral atoms z
CO 2 Orbital DiagramDraw SALC with central atom. C 2 x O OCO 93
94 CO 2 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.
H-F diatomic, H = 1s; F = 2s, 3 x 2p H 2Otriatomic, H = 2 x 1s; O = 2s, 3 x 2pCO2p bonding, O = 2s, 3 x 2p; C = 2s, 3 x 2pC2H4Fragmentation methodNH3/BenzeneReal + Imaginary SALC MO Diagrams from Group Theory 95
Two different approaches (D 2h ) Ethene Orbital Diagram C 1 + C 2 H 1-4 then combine CH2then combine z y x J. Chem. Edu . 2004 , 81, 997 96
Ethene Orbital Diagram 97
Ethene Orbital Diagram 98
Ethene Orbital Diagram 99
H-F diatomic, H = 1s; F = 2s, 3 x 2p H 2Otriatomic, H = 2 x 1s; O = 2s, 3 x 2pCO2p bonding, O = 2s, 3 x 2p; C = 2s, 3 x 2pC2H4Fragmentation methodNH3/BenzeneReal + Imaginary SALC MO Diagrams from Group Theory 100
101 NH 3 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 z x y G H 3 0 1 H s orbitals N s, p x , p y and p z orbitals NH 2 C 3v A 1 + E G H A 1 + E
102 NH 3 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 z x y H s orbitals N s, p x , p y and p z orbitals NH 2 C 3v A 1 + E A 1 E A 1
NH 3 Orbital Diagram Combine orbitals by their symmetry H s orbital N s, p x , py and pz orbitals103
E A 1 NH3 Orbital Diagram Combine orbitals by their symmetry 3 x H N s (A 1 ) p z (A1) p y, px (E) E A 1 NH 3 A 1 A 1 E 104
E A 1 NH3 Orbital Diagram 3 x H N s (A 1 ) p z (A1) py, p x (E) E A 1 NH 3 A 1 A 1 E Fill MOs with e - 3 e - 5 e - 105
106 NH 3 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 A 1 s (A 1 ) p z (A 1 ) p y , p x (E) E A 1 A 1 A 1 E
NH 3 Orbital DiagramGenerate 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 f 1 f 2 A 1 + E G H = z x y f 3 107
Generate SALCs of peripheral atoms NH 3 Orbital Diagram Separate classes 108
Generate SALCs of peripheral atoms NH 3 Orbital Diagram f 1 f 2 z x y f 3 109
Generate SALCs of peripheral atoms NH 3 Orbital Diagram P A1 ≈ ((1) E f 1 ) + ((1) C 3+f 1 ) + ((1) C3-f 1 ) + ((1) s1 f1 ) + ((1) s 2 f1 ) + ((1) s2 f 1 ) f 1 f 2 f 3 P A1 ≈ [ f 1 + f 2 + f 3 + f 1 + f 3 + f 2 ] P A1 ≈ [ 2f1 + 2f2 + 2f3]A1 H1s orbitalsf1 f 3 f 2 P A1 ≈ [ f 1 + f 2 + f 3 ] f 1 f 2 f 3 110
Generate SALCs of peripheral atoms NH 3 Orbital Diagram P E ≈ ((2) E f 1 ) + ((-1) C 3+f 1 ) + ((-1) C3-f 1 ) + ((0) s1 f1 ) + ((0) s 2 f1 ) + ((0) s2 f 1 ) f 1 f 2 f 3 P A1 ≈ [2 f 1 - f 2 - f 3 ] One of the E orbitals 0 0 0 f 1 f 2 f 3 What about the other E orbital? 111
Generate SALCs of peripheral atoms NH 3 Orbital Diagram 112
Generate SALCs of peripheral atoms NH 3 Orbital Diagram f 1 f 2 f 3 3 different E SALCS have been generated but they are all similar. Use subtraction or addition to generate new SALC. f 1 f 2 f 3 113
E A 1 NH3 Orbital Diagram 3 x H N s (A 1 ) p z (A1) py, p x (E) E A 1 NH 3 A 1 A 1 E Draw SALC with central atom. 114
NH 3 Orbital DiagramDraw SALC with central atom. 3 x H N 115
116 NH 3 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.
H-F diatomic, H = 1s; F = 2s, 3 x 2p H 2Otriatomic, H = 2 x 1s; O = 2s, 3 x 2pCO2p bonding, O = 2s, 3 x 2p; C = 2s, 3 x 2pC2H4 Fragmentation methodNH3/Benzene Real + Imaginary SALC MO Diagrams from Group Theory 117
Benzene MOs and SALC 0 nodes 1 node 2 nodes 3 nodes 118
119 C 6 H6 Orbital Diagram Assign a point group Choose basis function (orbitals) C 6 H 6 D 6h only p bonding C p z orbitals
120 C 6 H6 Orbital DiagramC6H6 D 6h only p bonding C p z orbitals 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 D 6h E 2C 6 2C 3 C 2 3C′ 2 3C″ 2 i 2S 3 2S 6 σ h 3 σ d 3 σ v Г π z axis C′ 2 C″ 2 C 6 6 0 0 0 -2 0 0 0 0 -6 0 2
121 C 6 H6 Orbital DiagramC6H6 D 6h only p bonding C p z orbitals D 6h E 2C 6 2C 3 C 2 3C′ 2 3C″ 2 i 2S 3 2S 6 σ h 3 σ d 3 σ v Г π z axis G p : B 2g + E 1g + A 2u + E 2u C′ 2 C″ 2 C 6 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 6 0 0 0 -2 0 0 0 0 -6 0 2
122 C 6 H6 Orbital DiagramC6H6 D 6h only p bonding C p z orbitals D 6h E 2C 6 2C 3 C 2 3C′ 2 3C″ 2 i 2S 3 2S 6 σ h 3 σ d 3 σ v Г π 6 0 0 0 -2 0 0 0 0 -6 0 2 C′ 2 C″ 2 C 6 G p : B 2g + E 1g + A 2u + E 2u 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
C 6 H 6 Orbital DiagramCombine orbitals by their symmetry E 1g B 2g A 2u E 2u 123
C 6 H 6 Orbital Diagram E 1g B 2g A 2u E 2u Fill MOs with e - 6 p z orbitals = 6 e - 124
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. 125 C 6 H 6 Orbital Diagram C 6 H 6 D 6h only p bonding C p z orbitals G p : B 2g + E 1g + A2u + E2u
Simplify usingC 6 ! C6H6 Orbital Diagram Generate SALCs of peripheral atoms C 6 D 6h 126
Simplify usingC 6 ! C 6 D 6h C 6 H 6 Orbital Diagram Generate SALCs of peripheral atoms 127
C 6 H 6 Orbital DiagramGenerate 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 A orbital 128
C 6 H 6 Orbital DiagramGenerate 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 129
C 6 H 6 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 B orbital 130
C 6 H 6 Orbital Diagram E 1 B A E 2 Generate SALCs of peripheral atoms B ≈ B 2g A ≈ A 2u 131
C 6 H 6 Orbital DiagramGenerate 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? 132
Contain imaginary components; the real component of the linear combination may be realized by taking ± linear combinations: C 6 H 6 Orbital Diagram 133
Contain imaginary components; the real component of the linear combination may be realized by taking ± linear combinations: For C 6 point group: or from Euler’s formula C 6 H 6 Orbital Diagram 134
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) C 6 H 6 Orbital Diagram 135
What are the pictorial representation of the SALC’s? C 6 H 6 Orbital Diagram 136
What are the pictorial representation of the SALC’s? C 6 H 6 Orbital Diagram 137
Projection Operator: Benzene What are the pictorial representation of the SALC’s? 0 nodes 1 node 2 nodes 3 nodes 138
Orbital Diagrams Orbital Interactions Molecular Orbital Theory Orbital EnergiesMO DiagramsHF, H2O, CO2, C2H4, NH3, BenzeneSALCHybridizationSymmetry and Reactivity 139
H-F diatomic, H = 1s; F = 2s, 3 x 2p H 2Otriatomic, H = 2 x 1s; O = 2s, 3 x 2pCO2p bonding, O = 2s, 3 x 2p; C = 2s, 3 x 2pC2H4Fragmentation methodNH3/BenzeneReal + Imaginary SALC MO Diagrams from Group Theory 140
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. C H 141
Miessler and Tarr , Inorganic Chemistry s + p Hybrid Orbitals 142
s + p + d Hybrid Orbitals 143
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 BF 3 Hybridization Steps to determine the hybridization of a bond. BF 3 D 3h s bonds D 3h Г s 3 0 1 3 0 1 144
Reduce to irreducible representation BF 3 Hybridization Steps to determine the hybridization of a bond. BF 3 D 3h s bonds Г s 3 0 1 3 0 1 G s : A 1 ’ + E ’ 145
Compare symmetry of irr . rep. to central atom MOs BF3 Hybridization Steps to determine the hybridization of a bond. BF 3 D 3h G s : A 1 ’ + E’ B (s) = A 1 ’ B ( p x )= E ’ B ( p y )= E ’ B ( p z )= A 2 ” 146
Compare symmetry of irr . rep. to central atom MOs BF 3 Hybridization Steps to determine the hybridization of a bond. G s : A 1 ’ + E ’ s = A 1 ’ z y x z y x z y x p x = E ’ p y = E ’ z y x y z y x p z = A 2 ” sp 2 hybridization (s, p x , p y ) 147
Orbital Diagrams Orbital Interactions Molecular Orbital Theory Orbital EnergiesMO DiagramsHF, H2O, CO2, C2H4, NH3, BenzeneSALCHybridizationSymmetry and Reactivity 148
Hybridization 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 Steps to determine the hybridization of a bond. 149
(2 + 2) cycloaddition 150 Symmetry and Reactivity p orbitals 2 x ethylene cyclobutane s bonds Orbital symmetry is retained during the reaction!
(2 + 2) cycloaddition 151 Symmetry and Reactivity
Symmetry and Reactivity 2 bonding + 2 antibonding e - Thermally Forbidden(~115 kcal/mol) 3 bonding + 1 antibonding e -Photochemically Allowed Thermal Reaction Photo Reaction
Orbital Diagrams Orbital Interactions Molecular Orbital Theory Orbital EnergiesMO DiagramsHF, H2O, CO2, C2H4, NH3, BenzeneSALCHybridizationSymmetry and Reactivity 153