Ch6 QM-MolecularStructure [호환 모드] - Yonsei …chem.yonsei.ac.kr/~mhmoon/pdf/General_Oxtoby/Ch6... · 6.5 Born-Oppenheimer Approximation Æseries of calculations gives the

6.1

Chapter 6Quantum Mechanics &

M l l SMolecular Structure

Prof. Myeong Hee Moon

6.2

Chapter Outline

• Quantum picture of the chemical bondSimplest molecule : H +Simplest molecule : H2

Born-Oppenheimer ApproximationElectronic wave functions for H2

+

electronic densit in H +electronic density in H2+

• De-localized Bonds: MOT and LCAOlinear combination of atomic orbitals approximation for H2

+2

homonuclear diatomic molecules : second period atomsheteronuclear diatomic molecules

• Photoelectron spectroscopy for moleculesPhotoelectron spectroscopy for molecules• localized bonds: valence bond model

wave function for electron pair bondsorbital hybridization for polyatomic molecules

• Comparison of LCAO and Valence Bond methods


6.3

6.1 Quantum picture of the chemical bond

• H2+ (molecular ion) : simplest model for single electron molecule.

exact quantum solution of H2+ insight into chemical bondingexact quantum solution of H2 insight into chemical bonding

key to more complex molecules

6.1.1 The Simplest molecule: H2+

The position of electron : (rA, rB, φ) rather than (x,y,z)

Internal potential energyInternal potential energy

eeV ⎟⎟

⎞⎜⎜⎛

+⎟⎟⎞

⎜⎜⎛

+−=111 22

nnen

ABBA

VV

Rrr

+=

⎟⎠

⎜⎝

⎟⎠

⎜⎝ 44 00 πεπε

Attraction repulsion between protons


• Schrodinger eq. ),r ,r,(R BAAB φψ mol

6.4

6.1.2 Born-Oppenheimer Approximation

• 1927, Max Born (Ger) and J. Oppenheimer (USA) Born-Oppenheimer ApproximationBorn Oppenheimer Approximation

foundation for all of molecular quantum mechanics

: nuclei are stationary and electrons move rapidly around them “Frozen Nuclei, Fleeting Electrons” (because nuclei are much heavier than electron)(because nuclei are much heavier than electron)

→ Separate nuclei and electronic motions→ Solve Schrödinger equations for electrons at fixed nuclei positions

then solve Schrödinger equations for nuclei

“molecular orbital: one-electron molecular wave function”


6.5

Born-Oppenheimer Approximation

series of calculations gives the electronic wave function of electron

)R ,r ,(r ABBAl ;φψ e

the electronic wave function of electron

• Born Oppenheimer apprx Ends up with

)(RABnucψSolving nuclear wave function

• Born-Oppenheimer apprx. Ends up with

)R )R ,r ,(r ,r ,r,(R ABucABBAlBAABol (;) nem ψφψφψ ×≈

Boundary conditions: ∞→∞→→ BAl r & r as 0 eψ

Focus first on el wave function,then on energy change of the system as RAB


6.6

6.1.3 Electronic wave functions for H2+

• solutions for are exact. --- first 8 wave functions leψ


6.7

Electronic wave functions for H2+

• Wave functions are identified with four levels: integer either or a subscript g or u and asterisk: integer, either σ or π, a subscript g or u, and asterisk

1. Greek letter : shows the electron probability density distribution1. Greek letter : shows the electron probability density distribution around nucleus

σ angular momentum component =0π angular momentum component = ±h/2p

two wave functions : π wave function has nodal planes

yx pupu 22 ,πππ wave function has nodal planes

δ angular momentum component = ±2h/2pϕ a ngular momentum component = ±3h/2pϕ a ngular momentum component ±3h/2p


6.8

Electronic wave functions for H2+

2. g or u: properties of wave f. changes as we invert the point g p p g pof observation through center of molecule

g sysmetric ; gerade (even)

3. asterisk : antibonding orbital, higher energy

u antisymetric ; ungerade (odd)

g , g gyσg : bonding orbital, σg

* : antibonding orbital

4. Integer : energy levels.1σg : the first of σg wave functions.


6.9

6.1.5 Summary: Key Features of QuantumPicture of Chemical BondingPicture of Chemical Bonding

1. Nuclei are frozen in specific positions and electrons movefleetingly around them “Born-Oppenheimer approx ”fleetingly around them Born-Oppenheimer approx.

2. Molecular orbital is wave function giving amplitude ofelectronic motions and its square give electron densityelectronic motions and its square give electron density

3. Bonding orbital: decreased potential energy and increasedl t d it b t l ielectron density between nuclei

4. Antibonding orbital: increased potential energy and zero amplitude region (i.e. node) between nuclei

5. σ-orbital: no angular momentum and cylindricalsymmetry about internuclear axis

6. π-orbital: angular momentum (±h/2π) and no cylindrical


g ( ) ysymmetry; concentrated amplitude “off the axis”

6.10

Probability Density Distributions


6.11

6.2 De-localized Bonds: Molecular Orbital Theory& Linear Combination of Atomic Orbital App& Linear Combination of Atomic Orbital App.

• MO: LCAO (Linear Combination of Atomic Orbitals)e g H + ion: 1σ = c(ψ + ψ )e.g. H2

+ ion: 1σ = c(ψ1S,A + ψ1S,B)• electron density is delocalized over the intire molecule

to construct approx. MOsto construct approx. MOs

6.2.1 LCAO: Appr. For H2+

BsB

AsAMO CC 11 ϕϕψ +=

CA, CB: relative weights, In case of H2+, CA=CB or CA= - CB

For clarification, atomic orbital is expressed with ϕmolecular orbitals are with σ or s.


6.12

Correlation table showing how six of the exact H2

+ Mos correlated atof the exact H2 Mos correlated at large separations to sums of differences of hydrogen atom orbitals and at short separations toorbitals and at short separations to the atomic orbitals of He+


6.13

LCAO molecular orbital for H2+

• approximate molecular orbitals (MO)

[ ][ ][ ]

BA

Bs

AsABgsgg

RC

RC**

111

)(1

)(1

ϕϕσσ

ϕϕσσ

−=≈

+=≈

[ ] ssABusuu RC 111 )(1 ϕϕσσ =≈


6.14

LCAO molecular orbital for H2+

• distribution of electron probability density

[ ][ ]

BABA

Bs

As

Bs

Asgsg

C

C2222*

112

12

122

1

2)()(][

2)()(][

ϕϕϕϕσ

ϕϕϕϕσ

+

++=

[ ] ssssusu C 11111 2)()(][ ϕϕϕϕσ −+=

di t ib ti f l t b bilit d it f i t ti t• distribution of electron probability density of non-interacting system

[ ]obtained by averaging the probabilities for HA + HB

+ and HA+ + HB

[ ] 21

21

23

2. )()( B

sAsin C ϕϕψ += C3

2 =0.5


6.15

reduced probability to find electron with nodefind electron with node betw nuclei--antibonding

Increased electronIncreased electron density with node betw nuclei -- bonding

Independent AO -- noninteractingnoninteracting


6.16

Energy of H2+ in LCAO approximation

• potential energy of for H2+

Force between nuclei in antibondingeverywhere repulsive

in bonding state, nuclei are attracted toform a bound state at distance (Re) corres

- everywhere repulsive

form a bound state at distance (Re) corres. to lowest potential energy

bond dissociation energy :De

R : attractive & repulsive forces balanceRe: attractive & repulsive forces balanceequilibrium bond length

Predicted: De (predicted) = 1.76 eV t R ( di t d) 1 32Aat RAB (predicted)=1.32A

Actual: De = 2.791 eV at RAB (measured)=1.060AAB ( )

Correlation diagram for H2+ in LCAO app.


Bonding orbital is stabilized by the energy difference

6.17

6.2.2 Homonuclear diatomic molecules:First-Period AtomsFirst Period Atoms

• MO equations for He2+ & He2 by LCAO

[ ] [ ]BABA CC 11)()( HH[ ] [ ][ ] [ ]BA

uBs

Asusu

BAg

Bs

Asgsg

ssCC

ssCC

11)()(

11)()(

11*1

111

+=−=

+=+=

HeHe

HeHe

ϕϕσ

ϕϕσ

Weak bond for He2+ but

No bond for He due to energy diffNo bond for He2 due to energy diff.

Bond Order=1/2 x (# electrons in bonding MO

- # electrons in antibonding MO)

B.O. : He2+ (1/2) & He2 (0)


6.18

Homonuclear diatomic molecules

Higher B.O. – higher bond energy – shorter bond lengths.


6.19

6.2.3 Homonuclear diatomic molecules:Second-Period Atoms

• N2 : at least 7 app. MOs by combination of 1s, 2s, 2p

Second Period Atoms

[ ] [ ] [ ][ ] [ ] [ ][ ] [ ]B

zAz

By

Ay

Bx

Ax

BABAMO

ppCppC

ppCssCssC

2222

222211

54

321

++++

+++++=

ψ

[ ] [ ]zzyy pppp 54

• simplified way to construct MO for multi-electron atoms1 T AO t ib t i ifi tl t b d f ti l if th i1. Two AOs contribute significantly to bond formation only if their

atomic energy levels are very closecan ignor mixing between 1s and valence shell 2s, 2p

2. Two AOs on different atoms contribute significantly to bond formation only if they overlap significantly.y y p g y

Bonding: same phases - constructive interferenceAntibonding: opposite phases - destructive interferenceNonbonding: no overlap or bonding-antibonding cancellation


Nonbonding: no overlap or bonding-antibonding cancellation

6.20

Overlap of orbitals

Sigma (σ) bonds result from the end-on overlap of orbitals.

overlap between s and s orbitals :

• overlap between s and p orbitals l b d h

•overlap between s and s orbitals :

: only by end-on approaches of s orbital to p orbital

• Combination of 2s AOs : same as 1s orbitals

[ ]BAC 22[ ][ ]BA

usu

BAgsg

ssC

ssC

22

22*

2

2

−=

+=

σ

σ


6.21

Overlap of 2p orbitals

• Combinations of the 2p orbitals -- sigma bond

[ ]Bz

Azgpg ppC

z222 −=σ

[ ]Bz

Azupu ppC

z22*

2 +=σ


6.22

π Bonds : side to side overlap

• electron density of bonding orbitals & nonbonding orbitalsx and y directions both.y

[ ][ ]BA

Bx

Axupu

ppC

ppCx

22

22*

2

=

+=

π

π

[ ]xxgpg ppCx

222 −=π

[ ][ ]BA

By

Ayupu

ppC

ppCy

22

22

*

2 +=

π

π

[ ]yygpg ppCy

222 −=π

degenerate are yx pupu 22 & ππ


6.23

Orders of Energy of MO

• 1s orbitals barely overlap – little net effect on bonding Since bonding & antibonding orbitals are all occupiedSince bonding & antibonding orbitals are all occupied.

• energy of π orbital – constant, E or σ orbitals falls rapidly.: from N2 O2 σ orbitals of oxygen falls below π orbital 2 2 yg

paramagnetic


6.24

N F2N2F2


6.25

Paramagnetic vs. diamagnetic

• Paramagnetism of oxygen results from the unpaired electrons in the molecular orbitals.

• Lewis structure shows limitation

but MO shows evidence.


6.26

Prop. of second-period diatomic molecules


6.27

6.2.4 Heteronuclear Diatomic Molecules

• Diatomic molecules : BO, CO, NOMO f th tMO for these, no symmetry

drop g, u

BA CC 22B

BA

As

BB

AAs

sCsC

sCsC

22

22''*

2

2

−=

+=

σ

σ

: if B is more electronegative than ACB>CA for bonding σ.

BOCB CA for bonding σ.

(σ )2(σ* )2(π )4(σ )1(σ2s)2(σ*2s)2(π2p)4(σ2pz)1


6.28

MO of HF

• In the case of NO, 11 V electrons

422 ),()()( nby

nbx

nb ππσσ

bond order : (8-3)/2=2.5 -- paramagnetic

(σ2s)2(σ*2s)2(π2p)4(σ2pz)2(π *2p)1

bond order : (8 3)/2 2.5 paramagnetic

• In the case of HF, 8 V electronsE f 1 & 2 f F E f 1 H: E of 1s & 2s of F << E of 1s H

: overlap of H:1s – F:2s begligible: overlap of H:1s – F:2px of F:2py no y

only H:1s – F:2pz

remainder denoted withσnb πnb (nonbonding)σ , π (nonbonding)no contribution in bonding


bond order : (2-0)/2=1

6.29


6.30

6.3 Photoelectron Spectroscopy for Molecules

• PES confirms the MO descriptions of bonding and measures energy for individual MOs.measures energy for individual MOs.

radiate hν (21.22eV, 58.43 nm) to diatomic gasmeasure KE of emitted photoelectrons

by subtracting KE from light E = measuring IE (IE=-ε by Koopman)by subtracting KE from light E = measuring IE (IE=-ε, by Koopman) however, some of E can be consumed to excite vibrational states

of molecular ion (Evib)

ivib

iiephoton IEEvmh =+−=− )(2

2

1 εν

0,1,2..n =+−=− vibiephoton nhvmh νεν 2

2

1

2


6.31

PES of H2+ ion

H + approaches

n=0

H2+ approaches

dissociation limit

n=0 15.5eV, IE with no vib excitationAs E increases toward 18 eV, the amountof vib excitation of H2

+ ion increases,& spacing bet vib levels becomes smaller.


p gH2

+ approaches dissociation limit

6.32

Vibrational peaks & MO

• Case A : removal of photoelectron from bonding orbital, B.O.of positive ion will be smaller than that of parent molecule.

bond becomes less stiff vib ν becomes lower (PES) : ν = 6.78x1013/s for H2

+ ion vs 12.84x1013/s for H2

• Case B : removal of photoelectron from antibonding orbitalCase B : removal of photoelectron from antibonding orbital

B.O.of positive ion will be larger.bond becomes stiffer vib ν becomes higher (PES)

C C• Case C : removal of photoelectron from nonbonding orbital, no changes in B.O. & vib ν


6.33

PES of HCl

~ 13 eV: from nonbonding orbitalfew peaks

~ 16 eV: numerous peaksfrom bonding orbital


6.34

PES of N2 and O2

N2

O2O2


6.35

6.4 Localized Bonds: Valence Bond Model

• Valence Bond Theory

• Bonds are formed by the overlap of atomic orbitals.

• Stronger covalent bonds are the result of more overlap.

Smaller orbitals overlap more than larger orbitals• Smaller orbitals overlap more than larger orbitals.

• Orbitals with similar sizes overlap more than orbital with mismatched sizes.

• Valence Bond Theory explains bond lengths and bond energies better than VSEPR or Lewis dot structures.

• VB theory does not explain bond angles in molecules – a better model is needed


6.36

6.4.1 Wave function for electron-pair bonds

• single bonds -- σ bondsid H d l l th t tconsider Hydrogen molecule, as the two atoms

approach each other, bond formation

)( BB

AA

ABABBAel rrRCRrr 2121 )()();,( ϕϕψ =

-As the atoms interact, we can not distinguish electrons which belongs to which atom.

at large distances

)( BAABABBA 2121 )()();,( ϕϕψ

gthen, wave function becomes

)( BB

AA

ABABBAel rrRCRrr 21121 )()();,( ϕϕψ =

then simply

)( BB

AA

AB rrRC 122 )()( ϕϕ+

then simply

As two atoms approach(1)]1s (2)g

BABAel sssC 1)2(1)1(1[1 +=ψ


pp

6.37

Simple valence bond model for H2


6.38

valence bond model for F2 & HF

For F2, BABAb d (1)] (2)gBz

Az

Bz

Az

bond ppppC 2)2(22)1(2[1 +=ψ

For HF,

(1)] (2) HFz

HFz

bond spspC 1)2(21)1(2[1 +=ψ


6.39

valence bond model for multiple bonds

• multiple bondsIn case of NIn case of N2,- head-on overlap in z axis (pz) – σ bond

(1)](2) BABAbond C 2)2(22)1(2[ (1)] (2) Bz

Az

Bz

Az

bond ppppC 2)2(22)1(2[1 +=σψ

- side by side overlap in x y axis (p p )- side by side overlap in x, y axis (px, py)

2 π bonds

(1)]R(2)]RR2 BABAbond CC 2)2(2)[(2)1(2)[()1( (1)]R(2)]RR2; ABABABBx

Ax

Bx

Ax

bond ppCppC 2)2(2)[(2)1(2)[(),1( 11 +=πψ


6.40

valence bond model for polyatomic molecules

• Polyatomic molecules : check with VSEPR S N 2 li f VSEPR

• Polyatomic molecules• S.N.=2: linear from VSEPR• Be : 1s22s2

• no unpaired electrons to overlap with 1s of HVBT failed

• S.N.=3: triangular from VSEPR• B : 1s22s2 2p1• B : 1s22s2 2p1

• only one unpaired electron to overlap with 1s of H VBT failed

New concept was neededp: Hybridization of orbitals


6.41

6.4.2 Orbital Hybridization for Polyatomic Moleculesfor Polyatomic Molecules

Hybridization : mix AO’s of central atom to form same number of new hybrid AO’s: mix AO s of central atom to form same number of new hybrid AO s

• SP hybrid atomic orbitals: BeH2 molecule – mix 2s and 2pz orbitals of Be to form new hybrid orbital

wave function of hybrid orbital : χ2 pz y

]22[2

1)(1 zpsr +=χ

]22[2

1)(

2

2 zpsr −=χ


6.42

Hybridization: sp


6.43

Hybriidization of BeH2

• hybrid AO form on the central atom, Be: (1s)2(χ1)1(χ2)1

(3)](4)

(1)] (2)HHbond

HHbond

ssC)(

ssC),(

1)4(1)3([43

1)2(1)1([21 111

χχψ

χχψ σ

+=

+= +

(3)] (4) ssC)( 1)4(1)3([4,3 222 χχψ σ += −


6.44

Hybridization of sp2

• SP2 hybrid atomic orbitals: BH molecule mix 2s and 2p 2p orbitals of B to form new hybrid orbital: BH3 molecule – mix 2s and 2px , 2py orbitals of B to form new hybrid orbital

ypsr 2)2

1(2)( 2/1

1 +=χ

yx ppsr

13

2)2

1(2)

2

3(2)(

2

2/12/12 −+=χ

yx ppsr 2)2

1(2)

2

3(2)( 2/12/1

3 −−=χ


6.45



6.46


• SP3 hybrid atomic orbitals: CH molecule 1

]2222[2

1)(1 zyx pppsr +++=χ

: CH4 molecule –mix 2s and 2px , 2py , 2pz orbitals of C

]2222[2

1)(

]2222[2

1)(

3

2

zyx

zyx

pppsr

pppsr

−−+=

+−−=

χ

χ

]2222[2

1)(4 zyx pppsr −+−=χ


6.47



6.48


• Lone pair electrons – occupy hybrid orbitals.: NH3, H2O


6.49

Hybridization of dsp3


6.50

Hybridization of dsp3


6.51

Sample Problem

What is the hybridization of the central atom in: CH3Cl, H2S, CS2, PCl5?3 2 2 5

• CH Cl – carbon is sp3 hybridized• CH3Cl – carbon is sp hybridized

• H2S – sulfur is sp3 hybridized

CS C• CS2 – C is sp hybridized

• PCl5 – P is sp3d hybridized


6.52

6.5 Comparison of LCAO & VB methods

• LCAO : constructs MO delocalized over molecule by taking linearcombination of AOs.

• VB : quantum mechanical description of localized chemical bond

Comparison for H2• LCAO for H : [ ]BA ss 11 +=σ• LCAO for H2: [ ]sg ss 111 +=σ

(2)]1s (1)][MOBABA

sgsgel sss 1)2(1)1(1[)2()1( 11 ++== σσψ

• VB for H2: (1)1s (2)VBBABAel sss 1)2(1)1(1 +=ψ

by comparing both method for H2

(2)](2) [(1)](2)MOBBAABABAel ssssssss 1)1(11)1(11)2(11)1(1[ +++=ψ

elVBψ --

ionic HH & HH from ++ψ ionic

LCAO includes an ionic contribution to the bond. no evidence.but LCAO good for polar HF.


true chemical bond and molecular structure: between LCAO-VB

6.53

Homework

12, 14, 18, 22, 28, 32, 42,48, 51, 52, 54, 56


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Ch6 QM-MolecularStructure [호환 모드] - Yonsei …chem.yonsei.ac.kr/~mhmoon/pdf/General_Oxtoby/Ch6... · 6.5 Born-Oppenheimer Approximation Æseries of calculations gives the