Using Symmetry To Generate Molecular Orbital Diagrams

Using Symmetry to GenerateMolecular Orbital Diagrams review a few MO concepts generate MO for XH2, H2O, SF6Formation of a bond occurs when electrondensity collects between the two bonded nuclei(ie., Ψ2 MO large in the region of space betweennuclei)AntibondingMolecular OrbitalSpin-pairingBondingMolecular Orbital1

. the valence electrons draw the atomstogether until the core electrons start to repelone other, and that dominates2

the total number of molecular orbitals (MO’s)generated total number of atomic orbitals(AO’s) combinedAtom AAtom B (4 AO’s)(4 AO’s)Molecule A–B(8 MO’s)in order to overlap .Atom A(4 AO’s) Atom B(4 AO’s)AO’s must have: similar energies identical symmetriesMolecule A–B(8 MO’s)3

Atom AAtom B σ–bonddoesn’t change signupon rotation aboutinternuclear axis Atom AAtom B σ –bond (antibond)σ since still doesn’t change sign uponrotation about internuclear axis Atom AAtom B σ–bonddoesn’t change signupon rotation aboutinternuclear axis Atom AAtom B σ –bond (antibond)antibonding since changes sign uponrotation about axis perpendicular tointernuclear axis4

Atom A Atom Bπ–bondchanges signupon rotation aboutinternuclear axis Atom A Atom B π –bond (antibond)π since still changes sign uponrotation about internuclear axisantibonding since changes sign uponrotation about axis perpendicular tointernuclear axis5

Additional orbital labels, used by your book,describe orbital symmetry with respect to inversion (x,y,z)i (–x,–y,–z)inversion centerAdditional orbital labels, used by your book,describe orbital symmetry with respect to inversioni g gerade, if orbital does notchange phase upon inversionπg*i u ungerade, if orbital doeschange phase upon inversionπu6

Additional orbital labels, used by your book,describe orbital symmetry with respect to inversioni g gerade, if orbital does notchange phase upon inversionσgi u ungerade, if orbital doeschange phase upon inversionσu*the amount of stabilization and destabilization which resultsfrom orbital overlap depends on the type of orbital MOΔΕstab(s-orbital) ΔΕstab(p-orbital)7

the amount of stabilization and destabilization which resultsfrom orbital overlap depends on the type of orbital MOΔΕstab(p-orbital) ΔΕstab(d-orbital)With p-orbitals need to consider two typesof overlap, sigma (σ) and pi ) ΔΕstab(pi)8

Sigma overlap is betterΔstab(sigma) Δstab(pi)One also needs to consider energy matchAOAOAO!Estab(better E match)AO!Estab(poor E match)AOs are closerin energyAOs are furtherapart in energyΔΕstab(better E match) ΔΕstab (poor E match)9

Molecular Orbital Diagramsof more complicated moleculesXH2 (D h )H2O (C2v)SF6 (Oh)XH2 (D h) lineargeneral MOdiagram layouta linear combinationof symmetry adaptedatomic orbitals (LGO’s)correlationlinescentral atom’s (X ‘s)atomic orbitalsH––X––Hmolecular orbitalsatomic orbitalsof terminal atomsH1 H210

a linear combination of symmetry adaptedatomic orbitals (LGO’s) .Taken individually, the hydrogenatoms do not possess thesymmetry properties of theXH2 molecule. Taken as agroup they do, however.LGO (2)H1 –H2LGO (1)H1 what does this mean ?H2atomic orbitalsof terminal atomsH1 H2What are the symmetry properties of this molecule?There are an infinitenumber of theseC2C2H––X––HC .and an infinite numberof mirror planes perpendicularto the C axis, which containthe C2’s.11

Taken individually, this hydrogenatom does not possess C2 symmetryabout axes that are perpendicularto the C .C2H––XC2C Thus, the individualhydrogen atoms do not conformto the molecular D h symmetry.s-orbital symmetry labelXH2’s point groupsymmetry label given toidentify the pz orbitalatom X’s pzof atom X (central atom) atomic orbitals C21–1atom X’s px andpy atomic orbitalssymmetry label given toidentify the px and py orbitalsof atom X (central atom)atom X’ss–orbital12

We can perform symmetry operations on orbitalsas well as molecules We can perform symmetry operations on orbitalsas well as molecules the rows of numbers that follow each orbitals symmetrylabel tell us how that orbital behaves when operated uponby each symmetry element13

C21–1 We can perform symmetry operations on orbitalsas well as molecules the rows of numbers that follow each orbitals symmetrylabel tell us how that orbital behaves when operated uponby each symmetry element a “1” means that the orbital is unchanged by the symmetryoperation a “–1” means that the orbital changes phase as a result ofthe symmetry operation a “0” means that the orbital changes in some other wayas a result of the symmetry operation a “cos θ” occurs with degenerate sets of orbitals (eg. (px, px))that take on partial character of one another upon performanceof a symmetry operation14

s-orbitalC2C2C rotation about the C ,and C2 axes leaves thes–orbital unchangedthus the “1” character in the row highlighted forthe s-orbital, and under the columns headed C ,and C2 C21–115

on the other hand, the pz orbital changesphase upon rotation about any of the C2 axes C21–1rotation about the C2axis changes the phaseof the pz orbital pz–orbitalpz-orbitalC2C2C rotation about the C axis leaves thepz–orbital unchanged16

but a 1 “plus one” under the column headed C C21–1thus the “–1” (negative one) characterin the column headed C2When the px and py are rotated about the C axis,they move, rather than transform into themselves C21–1Thus the “0” in the column labeled 2C 17

inversion, on the other hand, causes both the px andpy to change phase C21–1Thus the “–2” in the column labeled iThe “–2” indicates that both orbitals transform as a setRotation about the C axis moves thepx–orbitalinto a position where it nolonger looks like a px orbitalthe orbitals are definedin terms of a fixed coordinatesystemx-axispx-orbitaly-axisC2HThe z-axis is selected so thatit either coincides with thehighest-fold rotation axis,or the internuclear axis, or bothXC2Hz-axis C For all C2’s except those thatcoincide with the x- and y-axes,rotation about C2causes the px –orbital to takeon some py –orbital character18

Rotation about the C2’s that do not coincide with thex- or y-axes, causes the px–orbital to take onsome py –orbital characterx-axisC2px-orbitalHXHy-axisz-axis C In fact, rotation about the C2 axis that bisectsthe x- and y-axes converts the px–orbital intothe py –orbital. This indicates that these two orbitalsmust be degenerate (ie have identicalenergies) C21–1Thus the “cos φ” term in the column labeled C219

The number in the “E”column tells you howmany orbitals transformtogether as a setinversion, on the other hand, causes boththe px and py to change phase, butmaintain their identities C21–1Thus the “–2” in the column labeled iThe “–2” indicates that both orbitals transform as a setx-axispx-orbitaly-axis inversioncenter, iz-axisinversion thru ichanges the phase of thepx–orbital20

Thus, we have determined the symmetry labels,labels,generally referred to as Mulliken symbols,symbols, forthe s and pz, px,and py-orbitalson the central atom XXH2 (D h )This explains the labels shown forthe central atom X on the leftpza1u(σu)pxpye1u (πu)2sa1g (σg)central atom’s (X ‘s)atomic orbitals21

We determine the symmetry labels for the LGO’s in thesame manner, ie., by examining their symmetry withrespect to the symmetry operations of the D h point groupLGO (2)H1 –H2LGO (1)H1 H2rotation about the C2axes changes the phaseof LGO(2)LGO(2)C2C2C rotation about the C axis leaves LGO(2)unchanged22

.thus it appears that LGO(2) possesses a1usymmetry C21–1 . symmetry identical to that of the central atomX’s pz orbitalXH2 (D h)py px .thus, the symmetry of LGO(2)matches that of the X atom’s pz–orbitalpze1u (πu) a1u(σu)LGO (2) .this interaction issymmetry alloweda1u (σu)LGO (1)2sa1g (σg)central atom’s (X ‘s)atomic orbitals .and LGO(2) will combinewith the X pz–orbital to forma new molecular orbital (MO)atomic orbitalsof terminal atomsH1 H223

rotation about the C2axes leaves LGO(1)unchangedLGO(1)C2C2C rotation about the C axis leaves LGO(1)unchangedOr . .thus it appears that LGO(1) could possess eithera1g , or a2u symmetry C21–1 . a1g would match orbitalson atom X, whereas a2u would not.24

if we examine the symmetry of LGO(1) with respect to bothinversion, and reflection, we should be able to distinguish betweenthese two options C21–1rotation about the C axes leaves LGO(1)unchangedLGO(1)C Reflection thru mirror planePerpendicular to C leavesLGO(1) unchanged25

thus, we can rule out a2u as the symmetry assignment forLGO(1), based on reflection, and improper rotation C21–1inversion leaves LGO(1)unchangedLGO(1)inversioncenter, i 26

thus, we can rule out a1u , and a2u as thesymmetry assignment for LGO(1) C21–1 leaving a1g as the symmetry assignment for LGO(1)XH2 (D h)pz px pye1u (πu)a1u(σu)2sa1g (σg)central atom’s (X ‘s)atomic orbitals .thus, the symmetry of LGO(1)matches that of the X atom’ss–orbitalLGO (2)a1u (σu)a1g (σg)LGO (1)atomic orbitalsof terminal atomsH1 H227

XH2 (D h)pz px pye1u (πu)a1u(σu) .inserting our correlation linesbetween orbitals of the same symmetrya1u * (σu *)a1g* (σg *)e1u (πu)nonbondingLGO (2)a1u (σu)a1g (σg)2sa1g (σg)LGO (1)a1u (σu)central atom’s (X ‘s)atomic orbitalsa1g (σg)atomic orbitalsof terminal atomsH1 H228

’s point group atom X’s p x and p y atomic orbitals symmetry label given to identify the p x and p y orbitals of atom X (central atom) symmetry label given to identify the p z orbital of atom X (central atom) atom X’s p z atomic orbitals atom X’s s–orbital s-orbital symmetry label C 2 1 –1