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
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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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• 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 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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The equation becomes time-independent:
Or is the kinetic energy operator;
V(x) is the potential energy.
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Or
where
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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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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
We insert a barrier and split the box into halves.

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

Application 2: Conjugated Dyes
Cyanine
Pinacyanol
1D Box
Length
556 pm
834 pm
λ (nm)
523
605
Dicarbocyanine 1112 pm 706
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Application 3: Quantum Dots
~2nm
Quantum dots with different sizes
Cellular imaging
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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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• 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 actual vibrational frequencies are ~10 14 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, C
2
H
4
, adopts a D
2h point group.
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• Ethene, C
2
H
4
, 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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• 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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If the symmetry label corresponds to x, y, or z, then its 0  1 transition will be IR active.
The 2 A
1 symmetry and 1 B
2 modes of water are IR active.
symmetry vibrational
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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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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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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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
• 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 ( t
0
  t )  r
0
 dr dt t
0
  t 
1
2 d 2 r dt 2 t
0
 (  t ) 2  …
.... + (-1) n
1 n !
d n r dt n
 (  t ) n t
0
• Velocity and acceleration can be obtained similarly.
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
Molecular Dynamics:
Predictor-Corrector Algorithm
Position, velocity, and acceleration are first predicted using the truncated Taylor Expansion r ( t
0
  t )  r
0
 v ( t
0
)   t 
1
2 a ( t
0
)  (  t ) 2 v ( t  a ( t 
 t )  v
0
 t )  a ( t
 a ( t
0
0
)
)   t
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Molecular Dynamics:
Predictor-Corrector Algorithm
Acceleration is then corrected :
F
 ma
  dV dr r ( t
  t ) a c
( t
  t )

 dV dr m r ( t
  t )
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.
W total
= (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 ‡ /W total
Prob ‡ = [(n − n ‡ + s − 1)! (n − n ‡ ] / [(n + s − 1)!/n!]
The reaction rate is proportional to Prob ‡ .
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