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
1D Box Length
λ (nm)
Cyanine
556 pm
523
Pinacyanol
834 pm
605
Dicarbocyanine
1112 pm
706

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

21. Application 4. Scanning Tunneling Microscope

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.

39. Hartree-Fock theory

40. Exact Solution

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

42. Molecular dynamics of protein folding

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 sec.

46. Molecular dynamics of protein folding

47. A drug molecule binds to a protein enzyme

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

51. When will a bond break rather than vibrate?
Each vibrational mode of water may absorb IR photons and be excited.
The vibrational energy can be redistributed due to the anharmonicity of the vibrations.
When will a bond eventually accumulate enough energy to break?
Rice, Ramsperger, Kassel (RRK) Theory assumes random distribution of energy quanta among all vibrational modes.

52. Probability of a selected vibrational mode accumulating enough energy (n‡ energy quanta) to break the bond.
Wtotal = (n + s − 1)!/n!(s − 1)!
n is the total number of energy quanta; s is the number of vibrational modes.
W‡ = (n − n‡ + s − 1)! (n − n‡)!(s − 1)!
Prob‡ = W‡/Wtotal
Prob‡ = [(n − n‡ + s − 1)! (n − n‡] / [(n + s − 1)!/n!]
The reaction rate is proportional to Prob‡.