2. Molecular Modeling: Molecular Vibrations C372 Introduction to Cheminformatics II Kelsey Forsythe

3. Next Time Energy Calculation Optimization Calculation Properties Calculation VibrationsRotations

4. Modeling Nuclear Motion (Vibrations) Harmonic Oscillator Hamiltonian

5. Modeling Potential energy (1-D)

6. 0 at minimum 0

7. Assumptions: Harmonic Approximation Determining k?

8. Assumptions: Harmonic Approximation E(.65)=3.22E-20J E(.83)=2.13E-20J  x=.091

9. Assumptions: Harmonic Approximation

11. Modeling Potential energy (N-D)

12. 0 at minimum 0 Coordinate Coupling Spoils!!!

13. Coordinates Degrees of Freedom? For N points in space For N points in space 3*N degrees of freedom exist 3*N degrees of freedom exist Cartesian to Center of Mass system Cartesian to Center of Mass system All points related by center/centroid of mass All points related by center/centroid of mass COM ia origin COM ia origin

14. Coordinates Center of Mass System 3*N degrees of freedom exist 3*N degrees of freedom exist DOF = i translation + j rotation + k vibration Linear: Linear: 3N=3 + 2 + k, k=3N-5 3N=3 + 2 + k, k=3N-5 Non-linear Non-linear 3N=3+3+k, k=3N-6 3N=3+3+k, k=3N-6

15. Coordinates Degrees of Freedom? Hydrogen Molecule Hydrogen Molecule Cartesian r 1 =x 1,y 1,z 1 r 2 =x 2,y 2,z 2 Cartesian r 1 =x 1,y 1,z 1 r 2 =x 2,y 2,z 2 COM-translational degrees of freedom x=(m 1 x 1 +m 2 x 2 )/M T y=(m 1 y 1 +m 2 y 2 )/M T z=(m 1 z 1 +m 2 z 2 )/M T COM-translational degrees of freedom x=(m 1 x 1 +m 2 x 2 )/M T y=(m 1 y 1 +m 2 y 2 )/M T z=(m 1 z 1 +m 2 z 2 )/M T COM-rotational degrees of freedom r,  - required COM-rotational degrees of freedom r,  - required 3(2)-5 = 1 (stretch of hydrogen molecule) 3(2)-5 = 1 (stretch of hydrogen molecule)

16. Normal Modes Decouples motion into orthogonal coordinates Decouples motion into orthogonal coordinates All motions can be represented in terms of combinations of these coordinates or modes of motion All motions can be represented in terms of combinations of these coordinates or modes of motion These normal modes are typically/naturally those of bond stretching and angle bending These normal modes are typically/naturally those of bond stretching and angle bending

17. Normal Modes Problem Problem

18. Normal Modes Solution r  q Solution r  q

19. Normal Modes Solution r  q Solution r  q Eigenvalue Problem

20. Normal Modes Solution r  q Solution r  q Eigenvalue Problem Normal modes

21. Normal Modes Hydrogen N=#atoms=2 N=#atoms=2 # normal modes = ? # normal modes = ? Linear Linear 3N-5=1 3N-5=1

22. Normal Modes Acetylene N=#atoms=4 N=#atoms=4 # normal modes = ? # normal modes = ? Linear Linear 3N-5=7 3N-5=7

23. QM Harmonic oscillator Modeling Need to solve Schrodinger Equation for harmonic oscillator Need to solve Schrodinger Equation for harmonic oscillator

24. QM Harmonic oscillator Modeling Solutions are Hermite Polynomicals Solutions are Hermite Polynomicals

25. QM Harmonic oscillator Modeling Energies Energies NON-CLASSICAL EFFECTS NON-CLASSICAL EFFECTS Quantization Quantization E min NOT zero E min NOT zero

26. QM Harmonic oscillator ERRORS Molecular Mechanics Molecular Mechanics Error  parameterization Error  parameterization Semi-Empirical Semi-Empirical SAM1>PM3>AM1 SAM1>PM3>AM1 HF HF Frequencies too high Frequencies too high –Harmonic approximation –No electron correlation Correction Correction –Multiply.9  out DFT - typically better than semi-empirical and HF DFT - typically better than semi-empirical and HF

27. IR-Spectra Diatomic Molecule

28. Application BioMolecules

29. Application- Thermodynamics/Statistical Mechanics Equipartition Theorem Equipartition Theorem Heat capacities Heat capacities Enthalpy, Entropy and Free Energy Enthalpy, Entropy and Free Energy

30. Anharmonic Effects? Must calculate higher order derivatives Must calculate higher order derivatives More computational time required More computational time required

31. Summary