Applications of Mathematics in Chemistry

1. Applications of Mathematics in Chemistry Yingbin Ge Department of Chemistry Central Washington University

2. Some terms that you may see everyday • Single-Variable Calculus • Multi-Variable Calculus • Differential Equations • Complex Functions • Group Theory • Probability and Statistics • Linear Algebra

3. Some terms that chemists see everyday • Inorganic Chemistry • Organic Chemistry • Biological chemistry • Analytical Chemistry • Physical Chemistry • Quantum Chemistry

4. What’s in common • Inorganic Chemistry • Organic Chemistry • Biochemistry • Analytical Chemistry • Physical Chemistry • Quantum Chemistry • Single-Variable Calculus • Multi-Variable Calculus • Differential Equations • Complex Functions • Group Theory • Probability and Statistics • Linear Algebra

5. The difference • The life of a quantum chemist is much easier than that of a mathematician. • We only solve one equation, the Schrödinger equation:

6. For a system with constant energy, If the system is one-dimensional,

7. The equation becomes time-independent: Or is the kinetic energy operator; V(x)is the potential energy.

8. If the potential energy is 0, Or • where

9. The general solution is for • The energy of the particle is E; the magnitude of the momentum is . • The direction of the momentum is probabilistic; the probabilities are proportional to |A+|2 and |A-|2.

10. What if we put a particle in a box?

11. The particle cannot escape from the box. To satisfy the boundary conditions, , where n = 1, 2, 3, …

13. Application 1: Quantum Teleportation

14. Application 1: Quantum Teleportation We insert a barrier and split the box into halves. 14

15. Application 1: Quantum Teleportation 50% 50% ~400, 000 km On the Moon On Earth What will happen if we open the box on Earth? 15 15

16. Application 2: Conjugated Dyes

17. Application 3: Quantum Dots ~2nm Quantum dots with different sizes Cellular imaging

18. What if the energy barrier is finite?

19. Tunneling Effect More prominent Hardly noticeable

20. Application 4. Scanning Tunneling Microscope http://www.ieap.uni-kiel.de/surface/ag-kipp/stm/images/stm.jpg

21. Application 4. Scanning Tunneling Microscope http://prl.aps.org/50years/timeline/Scanning%20tunneling%20microscope http://infiniflux.blogspot.com/

22. How do chemists identify unknown chemicals? • UV-Vis Spectrometry (Conjugated Dyes) • Infrared Spectrometry • Raman Spectroscopy • Nuclear Magnetic Resonance Spectrometry • Mass Spectrometry • All above techniques requires knowledge in mathematics.

23. IR spectrum of hydrogen chloride • HCl is a diatomic molecule; H and Cl are connected by a single bond. • The bond can be approximated as a harmonic oscillator.

24. The first two vibrational states

25. The first two vibrational states The actual vibrational frequencies are ~1014 cycles/second.

26. Application 5. Infrared Spectroscopy • Each molecule has a unique IR spectrum. • My favorite molecule: Vanillin.

27. Not all molecules absorb IR light. • For example, oxygen (O=O) do not absorb IR photons. • The IR absorption intensity is proportional to the squared modulus of the transition dipole moment:

28. Group theory in IR spectroscopy Ethene, C2H4, adopts a D2h point group.

29. Vibrations of Ethene • Ethene, C2H4, has 6 atoms and thus 18 motions. • 3 are translational motions. • 3 are rotational motions. • 12 are vibrations, some are IR active, others not. • If you know ethene’s point group and the symmetry labels for the vibrational modes, then it’s easy to predict which modes will be IR active.

30. Vibrations of Water • Water, has 3 atoms and thus 9 motions. • 3 translational motions. • 3 rotational motions. • 3 vibrational modes. • What is the point group?

31. Point Group Analysis If the symmetry label corresponds to x, y, or z, then its 0  1 transition will be IR active. The 2 A1 symmetry and 1 B2 symmetry vibrational modes of water are IR active.

32. Application 6: Measuring bond length • How do chemists measure the bond length (~10-10 m) of a molecule? • Solve the Schrödinger equation for the 3-D rotation of the molecule:

33. HCl IR Spectrum

34. Electronic structure of a H atom

35. Schrodinger Equation in Polar Coordinates The second derivatives of Ψ with respect to x, y, and z consist of 17, 17, and 7 terms. Fortunately, most terms can be cancelled or combined:

36. Selected atomic orbitals of H

37. Application 7: Neon Lights from Electron Transitions of Atoms

38. Electronic structure of multi-electron systems • Wavefunctions that describe electrons must be anti-symmetric. • Wave functions can be expressed in a Slater determinant. http://kf-lin.elf.stuba.sk/~ballo/piatok/prezentacia/hartree-fock/hf_method.html

39. Hartree-Fock theory http://kf-lin.elf.stuba.sk/~ballo/piatok/prezentacia/hartree-fock/hf_method.html

40. Exact Solution

41. Application 8. Protein folding and drug design.Proteins are long chains of amino acids.

42. Molecular dynamics of protein folding http://www.ks.uiuc.edu/images/ofmonth/2008-05/villin-folding-process.png

43. Molecular Dynamics • Given the initial values of force, velocity, and position for each atom, we can predict the force, velocity, and position for each atom at the first fs(10-15 sec), the second fs, and any other time over the course of MD. • Position can be expanded in a Taylor expansion: … • Velocity and acceleration can be obtained similarly.

44. Molecular Dynamics:Predictor-Corrector Algorithm Position, velocity, and acceleration are first predicted using the truncated Taylor Expansion

45. Molecular Dynamics:Predictor-Corrector Algorithm Acceleration is then corrected : Position, velocity, and acceleration are then updated accordingly. δt is often set to 10-15 sec.

46. Molecular dynamics of protein folding http://www.ks.uiuc.edu/images/ofmonth/2008-05/villin-folding-process.png

47. A drug molecule binds to a protein enzyme http://martin-protean.com/protein-structure.html

48. Questions? • Inorganic Chemistry • Organic Chemistry • Biological chemistry • Analytical Chemistry • Physical Chemistry • Quantum Chemistry • Single-Variable Calculus • Multi-Variable Calculus • Differential Equations • Complex Functions • Group Theory • Probability and Statistics • Linear Algebra