Molecular ModelingMolecular Modeling::Beyond Empirical EquationsBeyond Empirical EquationsQuantum Mechanics RealmQuantum Mechanics Realm

C372C372

Introduction to Introduction to Cheminformatics IICheminformatics II

Kelsey ForsytheKelsey Forsythe

Atomistic Model HistoryAtomistic Model History

Atomic SpectraAtomic Spectra Balmer (1885)Balmer (1885)

Plum-Pudding ModelPlum-Pudding Model J. J. Thomson (circa 1900)J. J. Thomson (circa 1900)

UV Catastrophe-QuantizationUV Catastrophe-Quantization Planck (circa 1905)Planck (circa 1905)

Planetary ModelPlanetary Model Neils Bohr (circa 1913)Neils Bohr (circa 1913)

Wave-Particle DualityWave-Particle Duality DeBroglie (circa 1924)DeBroglie (circa 1924) Uncertainty Principle (Heisenberg)Uncertainty Principle (Heisenberg)

Schrodinger Wave EquationSchrodinger Wave Equation Erwin Schrodinger and Werner Heisenberg(1926) Erwin Schrodinger and Werner Heisenberg(1926)

Classical vs. QuantumClassical vs. Quantum TrajectoryTrajectory Real numbersReal numbers

Deterministic (“The Deterministic (“The value is ___”)value is ___”)

VariablesVariables Continuous energy Continuous energy

spectrumspectrum

WavefunctionWavefunction Complex (Real and Complex (Real and

Imaginary Imaginary components)components)

Probabilistic (“The Probabilistic (“The average value is __ ”average value is __ ”

OperatorsOperators Discrete/Quantized Discrete/Quantized

energyenergy TunnelingTunneling Zero-point energyZero-point energy

Schrodinger’s EquationSchrodinger’s Equation

- Hamiltonian operator- Hamiltonian operator

Gravity? Gravity? €

ˆ H Ψ = EΨ

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ˆ H

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ˆ H = ˆ T + ˆ V

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−h2

2mi

∇ 2

i

N

∑

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Ceie j

ri − rji< j

N

∑

Hydrogen Molecule Hydrogen Molecule HamiltonianHamiltonian

Born-Oppenheimer Approximation (Fix Born-Oppenheimer Approximation (Fix nuclei)nuclei)

Now Solve Electronic ProblemNow Solve Electronic Problem

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−−−+

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ ∇

+∇

+∇

+∇

−=

+=

221221112121

22

21

22

21

2

111111

2ˆ

ˆˆˆ

epepepepppee

e

e

e

e

p

p

p

p

rrrrrrC

mmmmH

VTH

h

212212211121

22

21

2 111111

2ˆ

ˆˆˆ

ppepepepepeee

e

e

eel

nucleinucleielelel

rC

rrrrrC

mmH

VVTH

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−−−+⎭⎬⎫

⎩⎨⎧ ∇

+∇

−=

++= −

h

Electronic Schrodinger Electronic Schrodinger EquationEquation

Solutions:Solutions:

, the basis set, are of a known form , the basis set, are of a known form Need to determine coefficients (cNeed to determine coefficients (cm)

Wavefunctions gives probability of finding Wavefunctions gives probability of finding electrons in space (e. g. s,p,d and f orbitals)electrons in space (e. g. s,p,d and f orbitals)

Molecular orbitals are formed by linear Molecular orbitals are formed by linear combinations of atomic orbitals (LCAO)combinations of atomic orbitals (LCAO)

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Ψ(v r ) = cm ∗Φm (

v r )

m

F

∑

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Φm (v r )

Hydrogen MoleculeHydrogen Molecule

HOMOHOMO

LUMO LUMO

VBT

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ΨHOMO =1

2(φA + φB )

ΨLUMO =1

2(φA − φB )

Hydrogen MoleculeHydrogen Molecule

Bond DensityBond Density

Ab Initio/Ab Initio/DFTDFT

Complete Description!Complete Description! Generic!Generic! Major Drawbacks:Major Drawbacks:

Mathematics can be cumbersomeMathematics can be cumbersome Exact solution only for hydrogenExact solution only for hydrogen

InformaticsInformatics Approximate solution time and storage intensiveApproximate solution time and storage intensive

– Acquisition, manipulation and dissemination problemsAcquisition, manipulation and dissemination problems

Approximate MethodsApproximate Methods SCF (Self Consistent Field) Method (a.ka. SCF (Self Consistent Field) Method (a.ka.

Mean Field or Hartree Fock)Mean Field or Hartree Fock) Pick single electron and average influence of Pick single electron and average influence of

remaining electrons as a single force field (Vremaining electrons as a single force field (V0 external)external)

Then solve Schrodinger equation for single Then solve Schrodinger equation for single electron in presence of field (e.g. H-atom electron in presence of field (e.g. H-atom problem with extra force field)problem with extra force field)

Perform for all electrons in system Perform for all electrons in system Combine to give system wavefunction and Combine to give system wavefunction and

energy (Eenergy (E) Repeat to error tolerance (ERepeat to error tolerance (Ei+1-Ei)

RecallRecall

SchrodingerSchrodinger Equation Equation Quantum vs. ClassicalQuantum vs. Classical Born OppenheimerBorn Oppenheimer Hartree-FockHartree-Fock ( (akaaka

SCF/central field) method SCF/central field) method

Basis SetsBasis Sets

Each atomic orbital/basis function is Each atomic orbital/basis function is itself comprised of a set of standard itself comprised of a set of standard functionsfunctions

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Ψ(v r ) = cm ∗Φm (

v r )

m

F

∑

Φm = Cmje−ζ mj r 2

j

N

∑

STO(Slater Type Orbital):~Hydrogen Atom Solutions

GTO(Gaussian Type Orbital):More Amenable to computation

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Φm ∝−ζ mj r2

Contraction coefficient(Static for calculation)

Expansion Coefficient

Atomic OrbitalLCAO

STO vs. GTOSTO vs. GTO

GTO GTO Improper behavior Improper behavior

for small r (slope for small r (slope equals zero at equals zero at nucleus)nucleus)

Decays too quicklyDecays too quickly

Basis SetsBasis Sets

∑ Φ∗=ΨF

mmm rcr )()(

vv

∑=ΦN

jjmjm C χ

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χ j ∝ e−ζ j r2

Optimized using atomic ab initio calculations

What “we” do!!

Basis Sets Molecular Orbital

Atomic Orbital

GTO/CGTO

STO

PGTO

Gaussian Type OrbitalsGaussian Type Orbitals

PrimitivesPrimitives

Shapes typical of H-atom orbitals Shapes typical of H-atom orbitals (s,p,d etc)(s,p,d etc)

Contracted Contracted Vary only coefficients of valence Vary only coefficients of valence

(chemically interesting parts) in (chemically interesting parts) in calculationcalculation

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χζ ,n,l,m (r,θ,ϕ ) = NYl,m (θ,ϕ )r(2n−2−l )e−ζr 2

Minimum Basis Set (STO-3G)Minimum Basis Set (STO-3G)

The number of basis functions is The number of basis functions is equal to the minimum required to equal to the minimum required to accommodate the # of electrons in accommodate the # of electrons in the systemthe system H(# of basis functions=1)-1sH(# of basis functions=1)-1s Li-Ne(# of basis functions=5) Li-Ne(# of basis functions=5)

1s,2s,2p1s,2s,2pxx, 2, 2yy, 2p, 2pz z

Basis SetsBasis SetsTypes:Types: STO-nG(n=integer)-Minimal Basis SetSTO-nG(n=integer)-Minimal Basis Set

Approximates shape of STO using single Approximates shape of STO using single contraction of n- PGTOs (typically, n=3)contraction of n- PGTOs (typically, n=3)

IntuitiveIntuitive The universe is NOT spherical!!The universe is NOT spherical!!

3-21G (Split Valence Basis Sets)3-21G (Split Valence Basis Sets) Core AOs 3-PGTOsCore AOs 3-PGTOs Valence AOs with 2 contractions, one with Valence AOs with 2 contractions, one with

2 primitives and other with 1 primitive 2 primitives and other with 1 primitive

Basis SetsBasis SetsTypes:Types:

3-21G(*)-Use of d orbital functions (23-21G(*)-Use of d orbital functions (2ndnd row atoms only)-row atoms only)-ad hocad hoc

6-31G*-Use of d orbital functions for 6-31G*-Use of d orbital functions for non-H atomsnon-H atoms

6-31G**-Use of d orbital functions for H 6-31G**-Use of d orbital functions for H as wellas well

ExamplesExamples

CC STO-3G-Minimal Basis SetSTO-3G-Minimal Basis Set

3 primitive gaussians used to model each 3 primitive gaussians used to model each STO STO

# basis functions = 5 (1s,2s,3-2p’s)# basis functions = 5 (1s,2s,3-2p’s) 3-21G basis-Valence Double Zeta3-21G basis-Valence Double Zeta

1s (core) electrons modeled with 3 primitive 1s (core) electrons modeled with 3 primitive gaussiansgaussians

2s/2p electrons modeled with 2 contraction 2s/2p electrons modeled with 2 contraction sets (2-primitives and 1 primitive)sets (2-primitives and 1 primitive)

# basis functions = 8 (1s,2s,6-2p’s)# basis functions = 8 (1s,2s,6-2p’s)

PolarizationPolarization

Addition of higher angular Addition of higher angular momentum functions momentum functions HCNHCN

Addition of p-function to H (1s) basis Addition of p-function to H (1s) basis better represents electron density (ie sp better represents electron density (ie sp character) of HC bondcharacter) of HC bond

Diffuse functionsDiffuse functions

Addition of basis functions with small Addition of basis functions with small exponents (I.e. spatial spread is greater)exponents (I.e. spatial spread is greater) AnionsAnions RadicalsRadicals Excited StatesExcited States Van der Waals complexes (Gilbert)Van der Waals complexes (Gilbert) Ex. Benzene-Dimers (Gilbert)Ex. Benzene-Dimers (Gilbert)

w/o Diffuse functions T-shaped optimumw/o Diffuse functions T-shaped optimum w/Diffuse functions parallel-displaced optimumw/Diffuse functions parallel-displaced optimum

Computational LimitsComputational Limits

Hartree-Fock limitHartree-Fock limit NOT exact solutionNOT exact solution Does not include correlationDoes not include correlation Does not include exchangeDoes not include exchange

Exact Energy*

Basis set size

Correlation/Exchange

BO not withstanding

Correcting ApproximationsCorrecting Approximations

Accounting for Electron Accounting for Electron CorrelationsCorrelations DFT(Density Functional Theory)DFT(Density Functional Theory) Moller-Plesset (Perturbation Theory)Moller-Plesset (Perturbation Theory) Configuration Interaction (Coupling Configuration Interaction (Coupling

single electron problems)single electron problems)

Computational RemindersComputational Reminders

HF typically scales NHF typically scales N44

As increase basis set size accuracy/calculation As increase basis set size accuracy/calculation time increasestime increases

ALL of these ideas apply to any program utilizing ALL of these ideas apply to any program utilizing ab initio techniques NOT just Spartan (Gilbert)ab initio techniques NOT just Spartan (Gilbert)

Quick GuideQuick Guide

BasisBasis STO-3G(minimal STO-3G(minimal

basis)basis)

3-21G-6-311G(split-3-21G-6-311G(split-valence basis)valence basis)

*/***/**

+/+++/++

MeaningMeaning 3 PGTO used for each 3 PGTO used for each

STO/atomic orbitalSTO/atomic orbital Additional basis Additional basis

functions for valence functions for valence electronselectrons

Addition of d-type Addition of d-type orbitals to calculation orbitals to calculation (polarization)(polarization)

** (for H as well)** (for H as well) Diffuse functions (s Diffuse functions (s

and p type) addedand p type) added ++ (for H as well)++ (for H as well)

Modeling Nuclear MotionModeling Nuclear Motion

IR - VibrationsIR - Vibrations NMR – Magnetic SpinNMR – Magnetic Spin Microwave – RotationsMicrowave – Rotations

8.35E-28 8.77567E+14 20568787140 2.03098E-18 1.05374E-188.35E-28 8.77567E+14 20568787140 1.77569E-18 9.66155E-198.35E-28 8.77567E+14 20568787140 1.54682E-18 8.82365E-198.35E-28 8.77567E+14 20568787140 1.34201E-18 8.02375E-198.35E-28 8.77567E+14 20568787140 1.15913E-18 7.26185E-198.35E-28 8.77567E+14 20568787140 9.96207E-19 6.53795E-198.35E-28 8.77567E+14 20568787140 8.51451E-19 5.85205E-198.35E-28 8.77567E+14 20568787140 7.23209E-19 5.20415E-198.35E-28 8.77567E+14 20568787140 6.09973E-19 4.59425E-198.35E-28 8.77567E+14 20568787140 5.10362E-19 4.02235E-198.35E-28 8.77567E+14 20568787140 4.2311E-19 3.48845E-198.35E-28 8.77567E+14 20568787140 3.47061E-19 2.99255E-198.35E-28 8.77567E+14 20568787140 2.81155E-19 2.53465E-198.35E-28 8.77567E+14 20568787140 2.24426E-19 2.11475E-198.35E-28 8.77567E+14 20568787140 1.75987E-19 1.73285E-198.35E-28 8.77567E+14 20568787140 1.35031E-19 1.38895E-198.35E-28 8.77567E+14 20568787140 1.0082E-19 1.08305E-198.35E-28 8.77567E+14 20568787140 7.26787E-20 8.15147E-208.35E-28 8.77567E+14 20568787140 4.99924E-20 5.85247E-208.35E-28 8.77567E+14 20568787140 3.22001E-20 3.93347E-208.35E-28 8.77567E+14 20568787140 1.87901E-20 2.39447E-208.35E-28 8.77567E+14 20568787140 9.29638E-21 1.23547E-208.35E-28 8.77567E+14 20568787140 3.29443E-21 4.56475E-21

Empirical Potential for Hydrogen Molecule

0

2E-19

4E-19

6E-19

8E-19

1E-18

1.2E-18

1.4E-18

0 0.5 1 1.5 2 2.5 3 3.5 4

Modeling Nuclear Motion (Vibrations)Modeling Nuclear Motion (Vibrations)Harmonic Oscillator HamiltonianHarmonic Oscillator Hamiltonian

22

)(2

1

2)(ˆ r

rrH Δ+

∂∂

−=Δ μμ

h