Applications of Mathematics in Chemistry

Yingbin GeDepartment of Chemistry

Central Washington University

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Some terms that you may see everyday

• Single-Variable Calculus• Multi-Variable Calculus• Differential Equations• Complex Functions• Group Theory• Probability and Statistics• Linear Algebra

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Some terms that chemists see everyday

• Inorganic Chemistry• Organic Chemistry• Biological chemistry• Analytical Chemistry • Physical Chemistry• Quantum Chemistry

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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

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The difference

• The life of a quantum chemist is much easier than that of a mathematician.

• We only solve one equation, the Schrödinger equation:

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For a system with constant energy,

If the system is one-dimensional,

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The equation becomes time-independent:

Or

V(x) is the potential energy.

is the kinetic energy operator;

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If the potential energy is 0,

Or

where

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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.

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What if we put a particle in a box?

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The particle cannot escape from the box.

To satisfy the boundary conditions,

, where n = 1, 2, 3, …

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Application 1: Quantum Teleportation

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Application 1: Quantum Teleportation

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We insert a barrier and split the box into halves.

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Application 1: Quantum Teleportation

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On Earth On the Moon

~400, 000 km

50% 50%

What will happen if we open the box on Earth?

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Application 2: Conjugated Dyes

1D Box Length λ (nm)

Cyanine 556 pm 523

Pinacyanol 834 pm 605

Dicarbocyanine 1112 pm 706

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Application 3: Quantum Dots

Cellular imagingQuantum dots with different sizes

~2nm

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What if the energy barrier is finite?

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Tunneling Effect

More prominent

Hardly noticeable

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Application 4. Scanning Tunneling Microscope

http://www.ieap.uni-kiel.de/surface/ag-kipp/stm/images/stm.jpg

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Application 4. Scanning Tunneling Microscope

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

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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.

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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.

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The first two vibrational states

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The first two vibrational states

The actual vibrational frequencies are ~1014 cycles/second.

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Application 5. Infrared Spectroscopy

• Each molecule has a unique IR spectrum.• My favorite molecule: Vanillin.

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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:

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Group theory in IR spectroscopy

Ethene, C2H4, adopts a D2h point group.

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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.

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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?

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Point Group AnalysisIf 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.

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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:

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HCl IR Spectrum

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Electronic structure of a H atom

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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:

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Selected atomic orbitals of H

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Application 7: Neon Lights from Electron Transitions of Atoms

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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

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Hartree-Fock theory

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

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Exact Solution

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Application 8. Protein folding

and drug design.

Proteins are long chains of amino

acids.

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Molecular dynamics of protein folding

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

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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:

r(t0 t) r0 drdt t0

t 1

2

d2r

dt 2

t0

(t)2

.... +(-1)n 1

n!

dnr

dt nt0

(t)n

• Velocity and acceleration can be obtained similarly.

…

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Molecular Dynamics:Predictor-Corrector Algorithm

r(t0 t) r0 v(t0)t 1

2a(t0) (t)2

v(t t) v0 a(t0)ta(t t) a(t0)

Position, velocity, and acceleration are first predicted using the truncated Taylor Expansion

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Molecular Dynamics:Predictor-Corrector Algorithm

Acceleration is then corrected :

m

drdV

tta

dr

dVmaF

ttrc

ttr

)(

)(

)(

Position, velocity, and acceleration are then updated accordingly. δt is often set to 10-15 sec.

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Molecular dynamics of protein folding

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

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A drug molecule binds to a protein enzyme

http://martin-protean.com/protein-structure.html

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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

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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.

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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‡.