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

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

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

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

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

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

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

If the potential energy is 0, Or where

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.

What if we put a particle in a box?

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

Application 1: Quantum Teleportation

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

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

Application 2: Conjugated Dyes 1D Box Length λ (nm) Cyanine 556 pm 523 Pinacyanol 834 pm 605 Dicarbocyanine 1112 pm 706

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

What if the energy barrier is finite?

Tunneling Effect More prominent Hardly noticeable

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

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

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.

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.

The first two vibrational states

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

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

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:

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

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.

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?

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.

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:

HCl IR Spectrum

Electronic structure of a H atom

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:

Selected atomic orbitals of H

Application 7: Neon Lights from Electron Transitions of Atoms

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

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

Exact Solution

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

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

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.

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

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.

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

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

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

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.

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‡.