Molecular Modeling:Molecular Modeling:Molecular VibrationsMolecular Vibrations

C372C372

Introduction to Introduction to Cheminformatics IICheminformatics II

Kelsey ForsytheKelsey Forsythe

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Energy Calculation

Optimization Calculation

Properties Calculation

Vibrations 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

Modeling Potential energy Modeling Potential energy (1-D)(1-D)

€

U(r) U(req ) −dU

dr r= req

(r − req ) +1

2

d2U

dr2

r= req

(r − req )2

€

−1

3

d3U

drr= req

(r − req )3 ....+1

n!

dnU

drn

r= req

(r − req )n€

=

€

≈

Modeling Potential energy Modeling Potential energy (1-D)(1-D)

€

−dU

dr r= req

(r − req ) +

€

U(r) ≈1

2

d2U

dr2

r= req

(r − req )2 ≡1

2kAB (r − req )2

€

U(req )

€

U(r) 1

2

d2U

dr2

r= req

(r − req )2

€

≈

0 at minimum0

Assumptions:Assumptions:Harmonic ApproximationHarmonic Approximation

Determining k?

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

Assumptions:Assumptions:Harmonic ApproximationHarmonic Approximation

E(.65)=3.22E-20JE(.83)=2.13E-20JΔx=.091

Assumptions:Assumptions:Harmonic ApproximationHarmonic Approximation

Assumptions:Assumptions:Harmonic ApproximationHarmonic Approximation

)10395.4(1060.61098.12

1024.11067.1

1029.12

1029.1

121314

1527

23

2

23

2

2

0

−−

−

××≡×==

×=×

×==∴

=>−−−

≡×=

cmcmHz

Hzkgskg

k

mkHO

kskg

dxUd

x

πων

μω

ω

Modeling Potential energy Modeling Potential energy (N-D)(N-D)

€

U(v r ) U(

v r eq ) −

dU

dv r v

r =v r eq

(v r −

v r eq ) +

1

2(v r −

v r eq )T d2U

dv r d

v r

r= req

(v r −

v r eq )

€

−1

3

d3U

dv r d

v r d

v r

r= req

(v r −

v r eq )T (

v r −

v r eq )(

v r −

v r eq )T ....+

1

n!

dnU

dv r n

r= req

(v r −

v r eq )n€

=

€

≈

Modeling Potential energy Modeling Potential energy (N-D)(N-D)

€

−dU

dr r= req

(r − req ) +

€

U(r) ≈1

2(v r −

v r eq )T d2U

dv r d

v r

r= req

(v r −

v r eq ) ≠

1

2kAB (r − req )2

€

U(req )

€

U(r) 1

2

d2U

dr2

r= req

(r − req )2

€

≈

0 at minimum0

Coordinate Coupling Spoils!!!

CoordinatesCoordinatesDegrees of Freedom?Degrees of Freedom?

For N points in spaceFor N points in space 3*N degrees of freedom exist3*N degrees of freedom exist

Cartesian to Center of Mass Cartesian to Center of Mass systemsystem All points related by center/centroid All points related by center/centroid

of massof mass COM ia originCOM ia origin

CoordinatesCoordinatesCenter of Mass SystemCenter of Mass System

3*N degrees of freedom exist3*N degrees of freedom existDOF = iDOF = itranslationtranslation + j + jrotationrotation + k + kvibrationvibration

Linear:Linear: 3N=3 + 2 + k, k=3N-53N=3 + 2 + k, k=3N-5

Non-linearNon-linear 3N=3+3+k, k=3N-63N=3+3+k, k=3N-6

CoordinatesCoordinatesDegrees of Freedom?Degrees of Freedom?

Hydrogen MoleculeHydrogen Molecule CartesianCartesian

rr11=x=x11,y,y11,z,z11

rr22=x=x22,y,y22,z,z22

COM-translational degrees of freedomCOM-translational degrees of freedomx=(mx=(m11xx11+m+m22xx22)/M)/MTT

y=(my=(m11yy11+m+m22yy22)/M)/MTT

z=(mz=(m11zz11+m+m22zz22)/M)/MTT

COM-rotational degrees of freedomCOM-rotational degrees of freedomr,r, - required- required

3(2)-5 = 1 (stretch of hydrogen molecule) 3(2)-5 = 1 (stretch of hydrogen molecule)

Normal ModesNormal Modes

Decouples motion into orthogonal Decouples motion into orthogonal coordinatescoordinates

All motions can be represented in All motions can be represented in terms of combinations of these terms of combinations of these coordinates or modes of motioncoordinates or modes of motion

These normal modes are These normal modes are typically/naturally those of bond typically/naturally those of bond stretching and angle bendingstretching and angle bending

Normal ModesNormal Modes

ProblemProblem

€

A≈

=

∂ 2U

∂r1∂r1

∂ 2U

∂r1∂r2

∂ 2U

∂r1∂r3

.....∂ 2U

∂r1∂rN

∂ 2U

∂r2∂r1

∂ 2U

∂r2∂r2

..............∂ 2U

∂r2∂rN

∂ 2U

∂r3∂r1

∂ 2U

∂r3∂r2

O M

∂ 2U

∂rN∂r1

∂ 2U

∂rN∂r2

∂ 2U

∂rN∂r3

.....∂ 2U

∂rN∂rN

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟€

∂2U

∂ri∂rj

≠ 0 for i ≠ j

Normal ModesNormal Modes

SolutionSolutionr r q q

€

A≈

=

∂ 2U

∂q1∂q1

0 0 0 ........0

0 ∂ 2U

∂q2∂q2

0 0....0

0 0 O 0 0 ..M

0 0 0 ....∂ 2U

∂qN∂qN

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟€

∂2U

∂qi∂q j

= 0 for i ≠ j

Normal ModesNormal Modes

SolutionSolutionr r q q

€

U≈

(v r ) ≈

1

2(v q −

v q eq )T L

≈

T d2U

dv r d

v r

r= req

L≈(v q −

v q eq ) =

1

2K≈

(v q −

v q eq )2

€

L≈

T (v r −

v r eq )T d2U

dv r d

v r

r= req

(v r −

v r eq ) L

≈ ≡ A

≈

off diagonal{

v q = K

≈

diagonal{

v q

q j = c i

i

M

∑ ri

Eigenvalue Problem

Normal ModesNormal Modes

SolutionSolutionr r q q

€

A≈

off diagonal{

v q = K

≈

diagonal{

v q

Eigenvalue Problem

€

q j = c i

i

M

∑ ri

€

Λ≈

= M≈

−1

2 K≈

M≈

−1

2 ≡ frequencies

Normal modes

Normal ModesNormal ModesHydrogenHydrogen

N=#atoms=2N=#atoms=2 # normal modes = ?# normal modes = ?

Linear Linear 3N-5=13N-5=1

Normal ModesNormal ModesAcetyleneAcetylene

N=#atoms=4N=#atoms=4 # normal modes = ?# normal modes = ?

Linear Linear 3N-5=73N-5=7

QM Harmonic oscillator QM Harmonic oscillator ModelingModeling

Need to solve Schrodinger Need to solve Schrodinger Equation for harmonic oscillator Equation for harmonic oscillator

QM Harmonic oscillator QM Harmonic oscillator ModelingModeling

Solutions are Hermite Solutions are Hermite PolynomicalsPolynomicals

QM Harmonic oscillator QM Harmonic oscillator ModelingModeling

EnergiesEnergies

NON-CLASSICAL EFFECTSNON-CLASSICAL EFFECTS QuantizationQuantization EEminmin NOT zero NOT zero

€

En = (n +1

2)hω

QM Harmonic oscillator QM Harmonic oscillator ERRORSERRORS

Molecular MechanicsMolecular Mechanics Error Error parameterizationparameterization

Semi-EmpiricalSemi-Empirical SAM1>PM3>AM1SAM1>PM3>AM1

HFHF Frequencies too highFrequencies too high

– Harmonic approximationHarmonic approximation– No electron correlationNo electron correlation

Correction Correction – Multiply .9Multiply .9 outout

DFT - typically better than semi-empirical DFT - typically better than semi-empirical and HFand HF

IR-SpectraIR-SpectraDiatomic MoleculeDiatomic Molecule

ApplicationApplicationBioMoleculesBioMolecules

Application-Application-Thermodynamics/Thermodynamics/Statistical MechanicsStatistical Mechanics

Equipartition TheoremEquipartition Theorem Heat capacitiesHeat capacities Enthalpy, Entropy and Free EnergyEnthalpy, Entropy and Free Energy

Anharmonic Effects?Anharmonic Effects?

Must calculate higher order Must calculate higher order derivativesderivatives More computational time requiredMore computational time required

SummarySummary