Molecular Modeling: Semi-Molecular Modeling: Semi-Empirical MethodsEmpirical Methods

C372C372Introduction to Introduction to

Cheminformatics IICheminformatics IIKelsey ForsytheKelsey Forsythe

RecallRecall

Molecular Models

Empirical/Molecular Modeling Semi Empirical Ab Initio/DFT

Neglect Electrons Neglect Core ElectronsApproximate/parameterize HF Integrals Full Accounting of Electrons

Huckel TheoryHuckel Theory AssumptionsAssumptions

Atomic basis set - parallel 2p orbitalsAtomic basis set - parallel 2p orbitals No overlap between orbitals, No overlap between orbitals, 2p Orbital energy equal to ionization potential of 2p Orbital energy equal to ionization potential of

methyl radical (singly occupied 2p orbital)methyl radical (singly occupied 2p orbital) The The stabilization energy is the difference stabilization energy is the difference

between the 2p-parallel configuration and the 2p between the 2p-parallel configuration and the 2p perpendicular configurationperpendicular configuration

Non-nearest interactions are zeroNon-nearest interactions are zero

)( ijijS

ij

p

HE

EEEionstabilizatE

2

22

Ex. Benzene (C3H6)Ex. Benzene (C3H6) One p-orbital per carbon One p-orbital per carbon

atom -atom - asis size = 6 asis size = 6 Huckel determinant isHuckel determinant is

0,0

E- 0 0 0 E- 0 0 0

0 0 E- 0 00 0 E- 0 0 0 0 E-

0 0 0

E

Ex. Allyl (C3H5)Ex. Allyl (C3H5) One p-orbital per carbon One p-orbital per carbon

atom -atom - basis size = 3 basis size = 3 Huckel matrix isHuckel matrix is

Resonance stabilization Resonance stabilization same for allyl cation, same for allyl cation, radical and anion (NOT radical and anion (NOT found experimentally)found experimentally)

2,,2

0,0E- 0

E- 0

E

E

Huckel TheoryHuckel Theory Molecular Orbitals?Molecular Orbitals?

# MO = #AO# MO = #AO 3 methylene radical orbitals as AO basis set3 methylene radical orbitals as AO basis set

Procedure to find coefficients in LCAO expansionProcedure to find coefficients in LCAO expansion Substitute energy into matrix equationSubstitute energy into matrix equation Matrix multiplyMatrix multiply Solve resulting N-equations in N-unknownsSolve resulting N-equations in N-unknowns

Huckel TheoryHuckel Theory Substitute energy into matrix equation:Substitute energy into matrix equation:

E1 0 - E1 0 - E1

a1a2a3

0

E1 2

( 2) 0

- ( 2)

0 - ( 2)

a1a2a3

0

Huckel TheoryHuckel Theory Matrix multiply :Matrix multiply :

( 2) 0

- ( 2)

0 - ( 2)

a1a2a3

0

Doing the math!

( ( 2))* a1 + * a2 +0 *a3 = 0

* a1 + ( - ( 2))* a2 + * a30

0 *a1 + * a2 +( - ( 2))* a303 equations,3 unknowns NOT

Huckel TheoryHuckel Theory Matrix multiply :Matrix multiply :

(1) and (3) reduce problem to 2 equations, 3 (1) and (3) reduce problem to 2 equations, 3 unknowns unknowns

Need additional constraint!Need additional constraint!

2 * a1 + * a2 = 0 a2 2a1 (1)

* a1 + 2 * a2 + * a30 (2)

* a2 + 2 * a30 a2 2a3 (3)

122

31)3()1(

aa

aa

Huckel TheoryHuckel Theory Normalization of MO:Normalization of MO:

Gives third equation, NOW have 3 equations, 3 unknowns.Gives third equation, NOW have 3 equations, 3 unknowns.

1)3()2()1(

* If

1*3*2*1**3*2*1

1

222

*

321*3

*2

*1

*

aaa

draaaaaa

dr

ijji

222,

21a3

21

211

1)1()1(2)1(

1)3()2()1(

122

31222

222

aa

aaa

aaa

aa

aa

Huckel TheoryHuckel Theory Have 3 equations, 3 unknowns.Have 3 equations, 3 unknowns.

222,

21a3

21

211

1)1()1(2)1(

1)3()2()1(

122

311)3()2()1(

222

222

322

aa

aaa

aaa

aa

aaaaa

Huckel TheoryHuckel Theory

3211 21*

22*

21

2p orbitals

Aufbau PrincipalAufbau Principal Fill lowest energy orbitals Fill lowest energy orbitals

firstfirst Follow Pauli-exclusion Follow Pauli-exclusion

principleprinciple

Carbon

Aufbau PrincipalAufbau Principal Fill lowest energy Fill lowest energy

orbitals firstorbitals first Follow Pauli-Follow Pauli-

exclusion exclusion principleprinciple 2

Allyl 2

Huckel TheoryHuckel Theory For lowest energy For lowest energy

Huckel MOHuckel MO

Bonding orbitalBonding orbital

a112

,a2 22

,a312

Huckel TheoryHuckel Theory Next two MO Next two MO

Non-Bonding Non-Bonding orbitalorbital

Anti-bonding Anti-bonding orbitalorbital

a1 22

,a2 0,a3 22

a112

,a2 22

,a312

Huckel TheoryHuckel Theory Electron DensityElectron Density

The rth atom’s electron The rth atom’s electron density equal to density equal to product of # electrons product of # electrons in filled orbitals and in filled orbitals and square of coefficients ofsquare of coefficients of

ao on that atomao on that atom Ex. AllylEx. Allyl

Electron density on c1 Electron density on c1 corresponding to the corresponding to the occupied MO is occupied MO is

qr n j *j a jr

2

q1 2 * 12

2

.5, qc 2 1 qc 3 .5

Total 2

Central carbon is neutral with partial positive charges on end carbons

j=1

2

1 21*2

q

Huckel TheoryHuckel Theory Bond Order Bond Order

prs n j *j a jr * a js

Ex. AllylEx. AllylBond order between c1-c2 Bond order between c1-c2 corresponding to the corresponding to the occupied MO isoccupied MO is

707.21*

22*2

21c2on HOMOfor AO oft coefficien

22c1on HOMOfor AO oft coefficien

2

12

12

11

1

p

a

a

n

Corresponds to partial pi bond due to delocalization over threecarbons (Gilbert)

Beyond One-Electron Beyond One-Electron FormalismFormalism

HF methodHF method Ignores electron Ignores electron

correlationcorrelation Effective Effective

interaction interaction potentialpotential

Hartree Product-Hartree Product-

drr

drr

jVij ij

j

ij ij

ji

2

jVrZ

h

i

M

k ik

ki

i

1

2

21-

nhamiltonia hartree

Fock Fock introduced introduced exchange – exchange – (relativistic (relativistic quantum quantum mechanics)mechanics)

N

ii

iihH

tyseparabili

HF-ExchangeHF-Exchange For a two electron systemFor a two electron system

Fock modified wavefunctionFock modified wavefunction

- )2()2(*)1()1()1()1(*)2()2(ˆ

)1()1(*)2()2()2()2(*)1()1(

SIGN IN CHANGE NO )1()1(*)2()2()2()2(*)1()1(ˆ ˆ

)2()2(*)1()1(

baba

baba

baba

ba

P

P

operatoryPermutivitP

Slater DeterminantsSlater Determinants Ex. Hydrogen moleculeEx. Hydrogen molecule

tDeterminan )2()2( )2()2( )1()1( )1()1(

- )2()2(*)1()1()1()1(*)2()2(ˆ

)1()1(*)2()2()2()2(*)1()1()2()2(*)1()1(

Slater

P

ba

ba

baba

baba

ba

Neglect of Differential Neglect of Differential Overlap (NDO)Overlap (NDO)

CNDO (1965, Pople et CNDO (1965, Pople et al) al)

MINDO (1975, Dewar )MINDO (1975, Dewar )

MNDO (1977, Thiel)MNDO (1977, Thiel)

INDO (1967, Pople et al)INDO (1967, Pople et al)

ZINDOZINDO

SINDO1SINDO1

STO-basis (/S-spectra,/2 d-orbitals)STO-basis (/S-spectra,/2 d-orbitals)

/1/2/3, organics/1/2/3, organics

/d, organics, transition metals/d, organics, transition metals

OrganicsOrganics

Electronic spectra, transition Electronic spectra, transition metalsmetals

1-3 row binding energies, 1-3 row binding energies, photochemistry and transition photochemistry and transition metalsmetals

Semi-Empirical MethodsSemi-Empirical Methods SAM1SAM1

Closer to # of Closer to # of ab initioab initio basis functions (e.g. basis functions (e.g. d orbitals)d orbitals)

Increased CPU timeIncreased CPU time G1,G2 and G3G1,G2 and G3

Extrapolated Extrapolated ab initioab initio results for organics results for organics ““slightly empirical theory”(Gilbert-more ab slightly empirical theory”(Gilbert-more ab

initio than semi-empirical in nature)initio than semi-empirical in nature)

Semi-Empirical MethodsSemi-Empirical Methods AM1AM1

Modified nuclear repulsion terms model to Modified nuclear repulsion terms model to account for H-bonding (1985, Dewar et al)account for H-bonding (1985, Dewar et al)

Widely used today (transition metals, Widely used today (transition metals, inorganics)inorganics)

PM3 (1989, Stewart)PM3 (1989, Stewart) Larger data set for parameterization Larger data set for parameterization

compared to AM1compared to AM1 Widely used today (transition metals, Widely used today (transition metals,

inorganics)inorganics)

What is Differential What is Differential Overlap?Overlap?

When solving HF equations we When solving HF equations we integrate/average the potential energy integrate/average the potential energy over all other electronsover all other electrons

Computing HF-matrix introduces 1 and 2 Computing HF-matrix introduces 1 and 2 electron integrals electron integrals

drr

drr

jVij ij

j

ij ij

ji

2

jVrZ

h

i

M

k ik

ki

i

1

2

21-

nhamiltonia hartree

Estimating EnergyEstimating Energy

Want to find c’s so thatWant to find c’s so that

(r ) cm m (r )m

F

d E dc

0

cm (i m

* (r ) ˆ H j (r )dr

Hmj

E mi* (r ) j (

r )dr Smj

) 0 m

M c 0 M ij i*(r ) ˆ H j (

r ) E i*(r ) j (

r ) M ij H ij E Sij

Non - trivial solutions for det(M) = 0

Estimating EnergyEstimating Energy

cm (i m

* (r ) ˆ H j (r )dr

Hmj

E mi* (r ) j (

r )dr Smj

) 0 m

2 electron *Vi j dr

* (1) * (1) 1

rij

(2) (2)ji

Differential Overlap

dr

1 electron * Zk

rik

k1

M

dr

Complete Neglect of Complete Neglect of Differential Overlap (CNDO)Differential Overlap (CNDO)

Basis set from valence STOs Basis set from valence STOs Neighbor AO overlap integrals, S, are assumed zero or Neighbor AO overlap integrals, S, are assumed zero or Two-electron terms constrain atomic/basis orbitals (But Two-electron terms constrain atomic/basis orbitals (But

atoms may be different!)atoms may be different!)

Reduces from NReduces from N44 to N to N2 2 number of 2-electron integralsnumber of 2-electron integrals Integration replaced by algebraIntegration replaced by algebra

IP from experiment to represent 1, 2-electron orbitalsIP from experiment to represent 1, 2-electron orbitals

S

Complete Neglect of Complete Neglect of Differential Overlap (CNDO)Differential Overlap (CNDO)

OverlapOverlap

Methylene radical (CHMethylene radical (CH22*) *) Interaction same in singlet (spin paired) and triplet state Interaction same in singlet (spin paired) and triplet state

(spin parallel)(spin parallel)

= orbital-pin 1

spin one parallel, electrons of Overlap

:=

spin paired electrons of Overlap

:

2

2

Parallel

Paired

Intermediate Neglect of Intermediate Neglect of Differential Overlap (INDO)Differential Overlap (INDO)

Pople et al (1967)Pople et al (1967) Use different Use different

values for values for interaction of interaction of different orbitals different orbitals on same atom s.t. on same atom s.t. MethyleneMethylene

222

22

,,

2

,

2

= orbital-pin 1

spin one parallel, electrons of Overlap

:

=

spin paired electrons of Overlap

:

spsppsp

spsp

Parallel

Paired

Neglect of Diatomic Differential Neglect of Diatomic Differential Overlap (NDDO)Overlap (NDDO)

Most modern methods (MNDO, AM1, PM3) Most modern methods (MNDO, AM1, PM3) All two-center, two-electron integrals All two-center, two-electron integrals

allowed iff:allowed iff:

Recall, CINDO-Recall, CINDO- Allow for different values of Allow for different values of depending depending

on orbitals involved (INDO-like)on orbitals involved (INDO-like)

and

and center atomic sameon are and center atomic sameon are and

Neglect of Diatomic Differential Neglect of Diatomic Differential Overlap (NDDO)Overlap (NDDO)

MNDO – Modified NDDOMNDO – Modified NDDO AM1 – modified MNDOAM1 – modified MNDO PM3 – larger parameter space used PM3 – larger parameter space used

than in AM1 than in AM1

Performance of Popular Performance of Popular Methods (Methods (C. CramerC. Cramer))

Sterics - MNDO overestimates steric crowding, Sterics - MNDO overestimates steric crowding, AM1 and PM3 better suited but predict planar AM1 and PM3 better suited but predict planar structures for puckered 4-, 5- atom ringsstructures for puckered 4-, 5- atom rings

Transition States- MNDO overestimates (see Transition States- MNDO overestimates (see above)above)

Hydrogen Bonding - Both PM3 and AM1 better Hydrogen Bonding - Both PM3 and AM1 better suitedsuited

Aromatics - too high in energy (~4kcal/mol)Aromatics - too high in energy (~4kcal/mol) Radicals - overly stableRadicals - overly stable Charged species - AO’s not diffuse (only Charged species - AO’s not diffuse (only

valence type)valence type)

SummarizeSummarize

Molecular Models

Empirical/Molecular Modeling Semi Empirical Ab Initio/DFT

Neglect Electrons Neglect Core ElectronsApproximate/parameterize HF Integrals Full Accounting of Electrons

HT EHT CNDO MINDO NDDO MNDO HF

Completeness