molecular modeling: semi-empirical methods c372 introduction to cheminformatics ii kelsey forsythe
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Molecular Modeling: Semi-Molecular Modeling: Semi-Empirical MethodsEmpirical Methods
C372C372
Introduction to Introduction to Cheminformatics IICheminformatics II
Kelsey ForsytheKelsey Forsythe
RecallRecall
Molecular Models
Empirical/Molecular Modeling Semi Empirical Ab Initio/DFT
Neglect ElectronsNeglect Core Electrons
Approximate/parameterize HF IntegralsFull 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 δ=
ij
p
HE
EEEionstabilizatE
=Δ
=
−=−=≡Δ
2
22
β
α
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 0
0 0 E- 0
0 0 0 E-
0 0 0
<=
−
β
αβββαβ
αββαβ
βαβββα 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)
βααβα
βαββαβ
βα
2,,2
0,0
E- 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:
€
α −E1 β 0
β α - E1 β
0 β α - E1
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
a1
a2
a3
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟= 0
E1 = α + 2β
α − (α + 2β ) β 0
β α - (α + 2β ) β
0 β α - (α + 2β )
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
a1
a2
a3
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟= 0
Huckel TheoryHuckel Theory
Matrix multiply :Matrix multiply :
€
α −(α + 2β ) β 0
β α - (α + 2β ) β
0 β α - (α + 2β )
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
a1
a2
a3
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟= 0
Doing the math!
(α − (α + 2β )) * a1 + β * a2 +0 * a3 = 0
β * a1 + (α - (α + 2β )) * a2 + β * a3 = 0
0 * a1 + β * a2 +(α - (α + 2β ))* a3 = 0
3 − 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!€
2β * a1 + β * a2 = 0 → a2 = − 2a1 (1)
β * a1 + 2β * a2 + β * a3 = 0 (2)
β * a2 + 2β * a3 = 0 → 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
*
=++
=
=++++
=ΨΨ
∫∫∫
aaa
draaaaaa
dr
ijji δφφ
φφφφφφ
Huckel TheoryHuckel Theory
Have 3 equations, 3 unknowns.Have 3 equations, 3 unknowns.
2
22,
2
1a3
2
1
2
11
1)1()1(2)1(
1)3()2()1(
122
31
1)3()2()1(
222
222
322
==∴=±=
=++⎪⎭
⎪⎬
⎫
=++
=
=
=++
aa
aaa
aaa
aa
aa
aaa
Huckel TheoryHuckel Theory
3211 2
1*
2
2*
2
1 φφφ ++=Ψ
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 βα 2−
Allylαβα 2+
Huckel TheoryHuckel Theory For lowest energy For lowest energy
Huckel MOHuckel MO
Bonding orbitalBonding orbital
€
a1 =1
2,a2 =
2
2,a3 =
1
2
Huckel TheoryHuckel Theory Next two MO Next two MO
Non-Bonding Non-Bonding orbitalorbital
Anti-bonding Anti-bonding orbitalorbital€
a1=2
2,a2 = 0,a3 = −
2
2
€
a1 =1
2,a2 = −
2
2,a3 =
1
2
Huckel TheoryHuckel Theory Electron DensityElectron Density
The rth atom’s The rth atom’s electron density equal electron density equal to product of # to product of # electrons in filled electrons in filled orbitals and square of orbitals and square of coefficients ofcoefficients 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 jr2
€
q1 = 2 *1
2
⎛
⎝ ⎜
⎞
⎠ ⎟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 2
1*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
707.2
1*
2
2*2
2
1c2on HOMOfor AO oft coefficien
2
2c1on 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 ∑∫∑∫
≠≠
==
2ψρ
{ }jVr
Z
h
i
M
k ik
ki
i
+−∇=
=
∑=1
2
2
1-
nhamiltonia hartree
Fock Fock introduced introduced exchange – exchange – (relativistic (relativistic quantum quantum mechanics)mechanics)
∏
∑
=Ψ
=
N
ii
iihH
ψ
tyseparabili
HF-ExchangeHF-Exchange
For a two electron systemFor a two electron system
Fock modified wavefunctionFock modified wavefunction
Ψ=−=Ψ
−=Ψ
=
≡
=Ψ
-
)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(
αψαψαψαψ
αψαψαψαψ
αψαψαψαψ
αψαψ
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
⎭⎬⎫
=Ψ
Ψ=−=Ψ
−=Ψ=Ψ
βψαψβψαψ
βψαψβψαψ
βψαψβψαψβψαψ
Neglect of Differential Neglect of Differential Overlap (NDO)Overlap (NDO)
CNDO (1965, Pople et al) CNDO (1965, Pople et 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-STO-basis (/S-spectra,/2 d-orbitals)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 ∑∫∑∫
≠≠
==
2ψρ
{ }jVr
Z
h
i
M
k ik
ki
i
+−∇=
=
∑=1
2
2
1-
nhamiltonia hartree
Estimating EnergyEstimating Energy
Want to find c’s so thatWant to find c’s so that
€
Ψ(v r ) = cm ∗Φm (
v r )
m
F
∑
d < E >
dv c
= 0
cm (i
∑ Φm* (
v r ) ˆ H Φ j (
v r )d
v r
Hmj
1 2 4 4 3 4 4 ∫ − E Φmi* (
v r )Φ j (
v r )d
v r
Smj
1 2 4 4 3 4 4 ∫ ) = 0 ∀m
M •v c = 0 }M ij = Φ i
*(v r ) ˆ H Φ j (
v r )∫ − E Φ i
*(v r )Φ j (
v r )∫
M ij = H ij − E Sij
Non - trivial solutions for det(M) = 0
Estimating EnergyEstimating Energy
€
cm (i
∑ Φm* (
v r ) ˆ H Φ j (
v r )d
v r
Hmj
1 2 4 4 3 4 4 ∫ − E Φmi* (
v r )Φ j (
v r )d
v r
Smj
1 2 4 4 3 4 4 ∫ ) = 0 ∀m
€
2 − electron ∝ φμ*Vi j{ }∫ φν dr ∝ φμ
* (1)∫∫ φν* (1)
1
rij
φλ (2)φσ (2)j≠ i
∑Differential Overlap
1 2 4 4 4 4 4 3 4 4 4 4 4
dr
€
1− electron ∝ φμ* Zk
rik
∫k=1
M
∑ φυ 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€
μν λσ( ) =δμνδλσ μμλλ( )
μνμν δ=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)
( ) ( )
( )
( )
γ
λλμμ
γλλμμ
λλμμδδλσμν λσμν
=
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
γγ
λλμμ
γ
λλμμ
≠
≡
≡
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 γ γ depending depending
on orbitals involved (INDO-like)on orbitals involved (INDO-like)
σλνμ
σλνμσλνμ
==
≠≠
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 ElectronsNeglect Core Electrons
Approximate/parameterize HF IntegralsFull Accounting of Electrons
HT EHT CNDO MINDO NDDO MNDO HF
Completeness