lec 2 - simple bonding theory [compatibility mode]

7/27/2019 Lec 2 - Simple Bonding Theory [Compatibility Mode]

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SIMPLE BONDING THEORYINOCHE3 Lecture 2

Valence Bond Theory

treats the formation of a molecule as

arising from the bringing together ofcomplete atoms which, when they interact,to a large extent, retain their originalcharacter

All bonds are localized

Valence Bond Theory

electrons occupy atomic orbitals ofindividual atoms within a molecule, andthat the electrons of one atom areattracted to the nucleus of another atom.

At a minimum distance (where the electrondensity begins to cause repulsion betweenthe two atoms) the lowest potential energyis acquired, and is considered to be what

holds the two atoms together in a chemicalbond.

Features

Lewis electron-dot diagrams/Lewisstructures (Octet Rule)

Resonance hybrids when none of drawnstructures alone is adequate, lower energydue to delocalization of electrons (biggerbox for particle in a box)

Features

Expanded shells or hypervalent atoms Formal charges

VSEPR Theory Steric number (number of electron pairdomains) is the number of positions occupiedby atoms or lone pairs around a central atom

lp-lp > lp-bp > bp-bp Multiple bonds > single bonds

ClF 3

lp-lp 180 90 120 Cannot be determinedlp-bp 6 at 90 3 at 90

2 at 120 4 at 90 2 at 120

Cannot be determined

bp-bp 3 at 120 2 at 90 1 at 120

2 at 90 2 at 87.5

Axial Cl-F 169.8 pm

Equatorial Cl-F 159.8 pm

Cl

F

F

F Cl

F

F

FClFF

F


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C CH

HH3C

H3C

122.2

115.6

Electronegativity

Pauling bond energies

Mulliken EA and IE Allred & Rochow attraction from Z* Sanderson electron densities Pearson EA and IE Allen valence electron energies Jaffe orbital electronegativities

Electronegativity and Angles

A more electronegative atom pulls bondingelectrons away from the central atom,letting lone pairs spread out, resulting insmaller anglesPF 3 < PCl 3 < PBr 3OSF 2 < OSCl 2 < OSBr 2

Parallels size effects

Electronegativity and Angles

A more electronegative central atom pullsbonding electrons toward itself increasingconcentration of electrons at the center,bp-bp repulsions increase anglesH2O < H 2S < H 2SeNCl3 < PCl 3 < AsCl 3

When opposedN(CH 3)3 110.9 < N(CF 3)3 117.9

Ligand Close Packing

Distance between outer atoms(nonbonded) in molecules determinemolecular shapes

Molecule with the same central atom havethe nonbonded distances between outeratoms constant with angles and lengthschanging.

Ligand Close Packing

VSEPR predicts NF 4- to have the biggerangle (109.5 vs 102.3 )

LCP predicts that FF distance is the same,N-F bond is longer in NF 3 (136 vs 130)

N

F

FF

FN

FF

F


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Polarity

Bonds between atoms of different

electronegativities are polar Depending on the overall structure,

polarity of the bonds can result ininteractions between molecules

Dipole moments not always calculated byadding vectors of bond moments

Qualitatively sufficient

Molecular Orbital theory

allocates electrons to molecular orbitals

formed by the overlap (interaction) ofatomic orbitals

Valence Bond Model in H 2

1 when atoms A and B are far apart,electrons 1 and 2 have no interaction

2 when H atoms are close together,impossible to tell which electron isassociated with which nucleus )()(

)()(

...

1

...

21

21cov

23

22

21

332211cov

spins parallel N

paired spin N

ccc N

ccc

alent

alent

==+=

+++=

+++=

+

Theoreticald = 87 pm, U = 303 kJ mol -1

Experimentald = 74 pm; U = 458 kJ mol-1

Improvements:

electron screening (shielding) Both electrons may be associated with

either nuclei:HA+HB or H A HB+


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3, 4 for each of the ionic form

)([

)]()[(

cov

4321

ionicalent molecule c N

c N

+=+++=+

Theoretical

d = 75 pm, U = 398 kJ mol -1

Experimentald = 74 pm; U = 458 kJ mol -1

1 HA(1)H B(2) 2 HA(2)H B(1) 3 [HA(1)(2)] HB+

4 HA+[HB(1)(2)]

H2 is a resonance hybrid of the fourcontributing resonance or canonicalstructures

HH H+ H H H+

where each does not exist as a separatespecies

each structure is also localized althoughthe combination (hybrid) as a whole maybe viewed as delocalized

MOT

Uses methods of group theory to describebonding in molecules

Symmetry properties and relative energiesof atomic orbitals determine how theyinteract to form molecular orbitals

MOs are then filled with electronsaccording to the same rules used for AOs

Total energy is then compared to gaugestability

Pictorial Approach

Schrodinger equation for electrons inmolecules

Approximate solutions from linearcombination of atomic orbitals (LCAO),sums and differences of atomic wavefunctions


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H2 =

!

Conditions

Symmetry of orbitals must be such that

regions with the same sign of overlap Energies of the atomic orbitals must besimilar (large differences in energies resultin small changes in MO energiescompared with the AOs)

Distance between the atoms must be shortenough to provide good overlap of orbitalsbut not too short for repulsions to interfere

MOs from s orbitals

"() = [ # # $() = [ # % # $

& '

( )*[ ] )()1()1(

2

1)( baba H H ss ++=

[ ] )()1()1(2

1*)( baba H H ss =

= 1* d

MO Types

Bonding molecular orbitals result inincreased concentration of electronsbetween two nuclei and has lower energythan starting atomic orbitals

Antibonding orbitals results in nodes withzero electron density between nucleicaused by cancellation of the wavefunctions and has higher energy

MO Diagram


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More precise calculations show

coefficients of * are slightly larger than for orbital

orbitals

Orbitals symmetric to rotation about the

bond axis are designated orbitals Antibonding orbitals are indicated with an

asterisk * in simpler cases where bondingand antibonding characters are clear

The number of resulting MOs is the sameas the initial number of AOs in the atoms

MOs from p orbitals MOs from p orbitals


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orbitals

The notation for MOs indicates a change

in sign with C 2 (180 ) rotation about thebond axis

Nodes of atomic orbitals become thenodes of the resulting MOsantibonding case MO is similar in

appearance to an expanded d orbital

MOs from d orbitals

orbitals

When atomic orbitals from two parallelplanes and combine side to side they form orbitals

The notation indicates sign changes onC4 rotation (90 )orbitals have no node, orbitals have

one node, orbitals have two nodes (thatinclude the bond axis)

Nonbonding Orbitals

MOs whose energies are essentially thatof the original atomic orbitals When three atomic orbitals satisfy the

requirement for MO formation When atomic orbital symmetries do not match When atomic orbitals have quite different

energies (1 s and 2 s )

Homonuclear DiatomicMolecules

Assuming interactionsonly between AOs ofidentical energies,

second perioddiatomic moleculesshare the generalpattern of MOs


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General Rules The number of molecular orbitals = the

number of atomic orbitals combined Of the MO's, half are bonding (lower energy)

and the other half are anti-bonding (higherenergy)

Electrons enter the lowest orbital available The maximum number of electrons in an

orbital is 2 (Pauli Exclusion Principle) Electrons spread out before pairing up

(Hund's Rule)

Bond Order

Overall number of bonding and

antibonding electrons determine the bondorder (number of bonds)

MOs from the 1 s orbitals have no neteffect on bonding (inner orbitals)

=

orbitalsgantibondinin

electronsof number

orbitalsbondingin

electronsof number order Bond

2

1

Molecule BondOrder

BondEnergy,

eV

BondLength,

O2 2 5.12 1.21F2 1 1.60 1.41

Ne 2 0 Molecule notobserved

Orbital Mixingg(2s) orbital interacts

with the g(2p z) orbitalu*(2s) orbital interacts

with the u*(2p z) orbital Hybridization/mixing Change in the relative

energies of themolecular orbitals

B2, C 2, and N 2 are bestdescribed by a modelthat includeshybridization

Four MOs result from combining four atomicorbitals (two 2 s , and two 2 p z ) that havesimilar energies and appropriate symmetries

c 1 = c 2; c 3 = c 4 for homonuclear molecules Lowest E MO have larger c 1 and c 2 Highest E MO have larger c 3 and c 4

Same symmetries but higher E for upper twoand lower E for two lower orbitals

)2()2()2()2( 4321 baba pc pcscsc =


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Homonuclear DiatomicMolecules

Period 2 Molecular Configurations

Correlation Diagrams

Shows calculated effect of moving two atomstogether from infinite separation to zerointeratomic distance (merged/united atom) As atoms move closer, MOs form At still smaller separation, bonding MOs decrease

in energy while antibonding MOs increase At 0 separation, MOs become AOs of united atom

Actual energies of MOs are intermediatebetween extremes


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Noncrossing Rule

Orbitals of the same symmetry interact so

their energies never cross

Heteronuclear Diatomic MOs

Follow the same general bonding pattern as

homonuclear Greater nuclear charge on one atom lowersits atomic energy levels and shifts theresulting molecular orbital levels(electronegative atom)

Different atomic orbital energies result inMOs with unequal contributions from the AOs

AO closer in energy to an MO contributesmore to the MO (larger c i)


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Carbon monoxide

Frontier Orbitals

Highest occupied molecular orbital, HOMO Lowest unoccupied molecular orbital,

LUMO

CO chemistry with transition metals M-O-C vs M-C-O

HOMO has larger electron density oncarbon because O 2p

zcontributes to more

MOs than C 2p z

Ionic Compounds

As an ion pair, limiting form of polarity inheteronuclear diatomic molecules

Concentration of electrons shifted to themore electronegative atom until it istransferred completely

Ionic Compounds Li +F -

Combination of electrostatic attraction andnon-directional covalent bonding

Formation as a sequence of elementarysteps:Li (s) Li (g)Li (g) Li+ (g) + e -

F 2 (g) F (g)F (g) + e - F- (g)Li+ (g) + F - (g) LiF (g)


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MO theory applied to H 2 Nuclei are placed in their equilibrium

positions then MOs are calculated for theelectrons to occupy MO arises from atomic orbitals interactions

If the symmetries of the atomic orbitals arecompatible

If the region of overlap between atomicorbitals is significant

If the interacting atomic orbitals are relativelyclose in energy

The number of MOs that can be formed

must equal the number of atomic orbitalsof the constituent atoms

MOs have associated energies andelectron distribution follow aufbau principle

MO ( in-phase )MO = N [ 1 + 2]

MO (out-of-phase )*MO = N* [ 1 2]

2

1

)1(2

1*

2

1

)1(2

1

=

+

=

S N

S N

is used to label orbitals that generates nophase change when rotated about theinternuclear axis

* is used when there is a nodal planebetween the nuclei and this plane is tothe internuclear axis. The lack of electrondensity on the nodal plane raisesinternuclear repulsion and destabilizes theMO, making it antibonding

Bond order = [(number of bondingelectrons) (number of antibondingelectrons)]

General result of MO is delocalizedbonding character over the molecularframework

POLYATOMIC MOLECULES


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Orbital Hybiridzation

Mixing

Model Derived spatially directed orbitals for the VB

Theory (localized -bonds) One hybridization scheme is appropriate for an

atom X in a molecule XYn of a particular shape Labeled to reflect contributing atomic orbitals Generated by mixing the characters of atomic

orbitals

sp Hybridization

Linear species, BeCl2 with equivalent Be-

Cl bonds One s atomic orbital and one p atomic

orbital n atomic orbitals produce n hybrid orbitals

( )

( ) pssp

pssp

22

22

2

12

1

=

+=

sp 2 Hybridization

Trigonal planar species, BH 3 withequivalent B-H bonds

y x

y x

x

p pssp

p pssp

pssp

222

222

22

2

1

6

1

3

12

1

6

1

3

1

3

2

3

1

2

2

2

=

+=

+=

sp 3 hybridization

Tetrahedral and related species

)(

2

1

)(2

1

)(2

1

)(2

1

2222

2222

2222

2222

3

3

3

3

z y x

z y x

z y x

z y x

p p pssp

p p pssp

p p pssp

p p pssp

+=

+=

+=

+++=

Other schemes

sp 3 d (dz 2 ) trigonal bipyramidal sp 3 d (dx 2 -y 2 ) square-based pyramidal

sp 3

d 2

octahedral sp 2 d square planar

MO Theory: Ligand Group Orbitals

different treatment for polyatomicmolecules


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Linear XH 2 two 1 s orbitals are taken as a group, the ligand

group orbital (LGO) transforming description from atomic orbitals of

X and H to atomic orbitals of X and LGOs ofH - - - H

the number of LGOs formed = the number ofatomic orbitals used

to generate the other LGOs, consider phases ofseparate orbitals (bonding and antibondingsense)

The MO is then constructed from the

interaction (symmetry consideration) of thevalence atomic orbitals of X and the LGOof H - - - H

Group theory simplifies the selection ofLGOs for larger molecules Starting from identification of point group Only ligand group orbitals that can be

classified within the point group of the wholemolecule are allowed

FHF -

Linear ion with D h symmetry, which issimplified with D 2h

D2h E C2(z) C2(y) C2(x) i (xy) (xz) (yz) _ _

Ag 1 1 1 1 1 1 1 1 x2; y2; z2

B1g 1 1 -1 -1 1 1 -1 -1 Rz xyB2g 1 -1 1 -1 1 -1 1 -1 Ry xzB3g 1 -1 -1 1 1 -1 -1 1 Rx yzAu 1 1 1 1 -1 -1 -1 -1

B1u 1 1 -1 -1 -1 -1 1 1 z

B2u 1 -1 1 -1 -1 1 -1 1 y

B3u 1 -1 -1 1 -1 1 1 -1 x

E C2(z) C2(y) C2(x) i (xy) (xz) (yz)


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E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1

E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1 1

E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1 1 1

E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1 1 1 1

E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1 1 1 1 1

E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1 1 1 1 1 1


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E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1 1 1 1 1 1 1

E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1 1 1 1 1 1 1 1

E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1 1 1 1 1 1 1 1

Ag

E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1

E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1 1


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E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1 1 -1

E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1 1 -1 -1

E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1 1 -1 -1 -1

E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1 1 -1 -1 -1 -1

E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1 1 -1 -1 -1 -1 1

E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1 1 -1 -1 -1 -1 1 1


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Ag 1 1 1 1 1 1 1 1 x2; y2; z2


B1u 1 1 -1 -1 -1 -1 1 1 z

B2u 1 -1 1 -1 -1 1 -1 1 y

B3u 1 -1 -1 1 -1 1 1 -1 x


Ag 1 1 1 1 1 1 1 1 x2; y2; z2


B1u 1 1 -1 -1 -1 -1 1 1 z

B2u 1 -1 1 -1 -1 1 -1 1 y

B3u 1 -1 -1 1 -1 1 1 -1 x

E C2(z) C2(y) C2(x) i (xy) (xz) (yz)

1 1 -1 -1 -1 -1 1 1

B1u Atomic orbitals and group orbitals of thesame symmetry can combine to formmolecular orbitals

In this case there are two A g LGOs

Energy match of the 1 s orbital of H (-13.6eV) is better with the 2 p z (-18.7 eV) thanthe 2 s (-40.2 eV) of fluorine

Polyatomic MO diagrams: central atomorbitals on the left, and group orbitals onthe right, with MOs in the middle


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Lewis approach requires two electrons to

represent a single bond between twoatoms: F-H-F (?)

MO: two electrons in a bonding MOformed by the interaction of all three atoms

For the similarly linear CO 2 the same set ofgroup orbitals are formed by the twooxygens, but the interactions with C nowinclude p orbitals

B3u

B2u

Ag

Ag

B2g

B3g

B1u

B1u


Ag 1 1 1 1 1 1 1 1 x2; y2; z2


B1u 1 1 -1 -1 -1 -1 1 1 z

B2u 1 -1 1 -1 -1 1 -1 1 y

B3u 1 -1 -1 1 -1 1 1 -1 x


Ag 1 1 1 1 1 1 1 1 x2; y2; z2


B1u 1 1 -1 -1 -1 -1 1 1 z

B2u 1 -1 1 -1 -1 1 -1 1 y

B3u 1 -1 -1 1 -1 1 1 -1 x


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2

B3uB2u

Ag

Ag

B2gB3g

B1u

B1u


Ag 1 1 1 1 1 1 1 1 x2; y2; z2


B1u 1 1 -1 -1 -1 -1 1 1 zB2u 1 -1 1 -1 -1 1 -1 1 y

B3u 1 -1 -1 1 -1 1 1 -1 x

B3u

B2u

Ag

Ag

B2g

B3g

B1u

B1u


Ag 1 1 1 1 1 1 1 1 x2; y2; z2


B1u 1 1 -1 -1 -1 -1 1 1 z

B2u 1 -1 1 -1 -1 1 -1 1 yB3u 1 -1 -1 1 -1 1 1 -1 x

B3u

B2uAg

Ag

B2g

B3gB1u

B1u


Ag 1 1 1 1 1 1 1 1 x2; y2; z2


B1u 1 1 -1 -1 -1 -1 1 1 z

B2u 1 -1 1 -1 -1 1 -1 1 y

B3u 1 -1 -1 1 -1 1 1 -1 x


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2

b3u and b 2u ( ) 2 p x - 2p x (- 15.9 )

with 2 p x (-10.7)

b2g and b 3g (non) 2 p x - 2p x and 2p y - 2p y

b3u and b 2u ( 2p x - 2p x with 2p x a g (*) 2p z - 2p z with 2s

b1u (*) 2p z - 2p z with 2 p z

H2O, C 2v Take the H 2O molecule as lying in the yz

plane C

2vcharacter table

C2v E C 2 (z) v(xz) v(yz)A1 +1 +1 +1 +1A2 +1 +1 -1 -1B1 +1 -1 +1 -1B2 +1 -1 -1 +1


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2

2s orbital of O in H 2O: A1

E C 2 v (xz) v (yz)

1 1 1 1

2p x orbital: B 1

E C 2 v (xz) v (yz)

1 -1 1 -1

2p y orbital: B 2

E C 2 v (xz) v (yz)

1 -1 -1 1

2p z orbital: a 1

E C 2 v (xz) v (yz)

1 1 1 1

labels in the first column are the symmetrytypes of orbitals that are permitted withinthe point group

numbers in the column headed E indicatethe degeneracy of each type of orbital inthe point group

each row of numbers following a givensymmetry label indicates how a particularorbital behaves when operated on by eachsymmetry operation. A number 1 indicatesthe orbital is unchanged by the operation,a 1 means that the orbitals changes sign,and a 0 means that the orbital changes insome other way.


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2

systematic reduction of reduciblerepresentations

=c

r icghn

1

Tabular Method

4

=h

C 2v E C 2 v (xz) v (yz

) 2 0 0 2A1 2 0 0 2 4 1A2 2 0 0 2 0 0B 1 2 0 0 2 0 0B 2 2 0 0 2 4 1

E C 2 v (xz) v (yz)

2 0 0 2

A1 1 1 1 1

B 2 1 1 1 1

only two LGOs can be constructed (fromthe two 1 s orbitals)

the symmetry of the LGO must correspondto one of the symmetry types in thecharacter table

= A1 + B 2

the LGOs must possess a 1 and b 2 symmetries

H- - -H LGO: A1

E C 2 v (xz) v (yz)

2 0 0 2


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2

If the two 1 s orbitals

of the Hs aredesignated 1 and 2and the operations ofthe group are appliedon one of the orbitals

E C 2 v ( xz )

v ( yz )

1 2 2 1

The composition of the a 1 LGO is obtained

by multiplying the corresponding A1character with each of the functionsobtained:

(a1 ) = (1 1) + (1 2) + (1 2) + (1 1)= 2 1 + 2 2in-phase combination

the b2 LGO is obtained by multiplying thecorresponding B 2 characters:

(b 2 ) = (1 1)+(1 2) + (1 2) +(1 1)

= 2 1 2 2out-of-phase combination

2s and 2 p both have a1 symmetry andcould interact with a1 LGO forming threeMOs: two bonding and one anti-bonding.The lowest energy a1 is dominated by 2 s contribution because of the separation of2pz .

BH3 z axis coincides with the C 3 , all atoms lie

in the xy plane D

3h


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Td

Symmetry elements f or thegroup Spectroscopy active component

E 8C 3 3C 2 6S 4 6sd Microwave IR RamanA1 1 1 1 1 1 x 2+y2+z2

A2 1 1 +1 -1 -1

E 2 -1 2 0 0 (2z2-x2-y2,

x2-y2)

T1 3 0 -1 1 -1 (R x, R y, R z)

T2 3 0 -1 -1 1 (x, y, z) (xy, xz, yz)

D3hSymmetry elements for the group Spectroscopy act ive component

E 2C 3

(z)3C' 2

h

(xy)2S 3 3v

Microwa

veIR Raman

A'1 1 1 1 1 1 1x2+y2,

z2

A'2 1 1 -1 1 1 -1 R z

E' 2 -1 0 2 -1 0 (x, y) (x2-y2,xy)

A''1 1 1 1 -1 -1 -1A''2 1 1 -1 -1 -1 1 z

E'' 2 -1 0 -2 1 0 (R x, R y) (xz, yz)

D2hS ym me tr y el em en ts for th e g rou p S pe ct ros cop y a ct iv e c om pon en t

E C2(z)C2(y)

C 2(x) i (xy) (xz) (yz)

Microwave IR Raman

Ag 1 1 1 1 1 1 1 1 x 2, y 2, z 2

B1g 1 1 -1 -1 1 1 -1 -1 R z xy

B2g 1 -1 1 -1 1 -1 1 -1 R y xz

B3g 1 -1 - 1 1 1 -1 - 1 1 R x yz

Au 1 1 1 1 -1 -1 -1 -1

B1u 1 1 -1 -1 -1 -1 1 1 z

B2u 1 -1 1 -1 -1 1 -1 1 y

B3u 1 -1 -1 1 -1 1 1 -1 x

Oh

Symmetry elements for the group

E 8C3 6C2 6C4 3C2 =(C4)2 i 6S4 8S6 3h 6d IR Raman

A1g 1 1 1 1 1 1 1 1 1 1 x2+y2+z2

A2g 1 1 -1 -1 1 1 -1 1 1 -1

Eg 2 -1 0 0 2 2 0 -1 2 0 (2z2-x2-y2, x2-y2)

T1g 3 0 -1 1 -1 3 1 0 -1 -1

T2g 3 0 1 -1 -1 3 -1 0 -1 1 (xz, yz, xy)

A1u 1 1 1 1 1 -1 -1 -1 -1 -1

A2u 1 1 -1 -1 1 -1 1 -1 -1 1

Eu 2 -1 0 0 2 -2 0 1 -2 0

T1u 3 0 -1 1 -1 -3 -1 0 1 1 (x, y, z)

T2u 3 0 1 -1 -1 -3 1 0 1 -1

lec 2 - simple bonding theory [compatibility mode]

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