Basic Observations
1.Proteins don’t know biology
The atoms in a protein only react to the forces impinging upon them, they
know nothing about folding, or catalyzing a reaction, only about moving in response
to some force. If the resultant structure confers some advantage to the organism
and that organism has a better survival rate than similar organisms, then the
protein structure will be propagated.
2.Nature has never heard of equilibrium
Equilibrium is not static but results from the balance of dynamic processes.
Consider taking a bag of red candy to this add a bag of blue candy so that the layers
are separate. Now shake the bag and the two colors will slowly mix. No intelligence
necessary. When the mixture in the bag has thoroughly mixed we say the system is
at equilibrium and that the two colors of candy will not separate again. We can
analyze the experiment in terms of equilibrium, but it was the dynamics of shaking
that allowed mixing to occur and that these dynamic processes will continue to
occur for as long as we continue to shake the bag.
3.We can only visualize data in one, two or three dimensions
Real problems are most often larger and we usually try to reduce a problem
to three or less dimensions
Butane – 14 atoms, 42 dimensions in Cartesian space or 3 translational, 3
rotational and 37 vibrational motions .
Shape of butane is most significantly affected by the value of the central
torsion or dihedral angle, which varies from -180 to 180 degrees (or 0 to 360).
If we run a MD simulation and follow the central torsion angle:
We see three states – 60, 180 and 300. Suppose we sample the structure at some
interval, measure the central torsion angle and count the number of times each
value is found:
We “know” the answers for butane – at least we have expectations based on our
theories – butane should have three states and they should be present on the order
of :
Ratio = exp(-dE/RT)
Boltzmann distribution law
If de = 780 cal , R = 1.987 cal/degree and T = ? K
Ratio = 1100/1 then temperature = 60 K
This MD was done with Pcmodel using MMFF94 force field and the temperature was
supposed to be 300K!
Repeat with T = 400 K.
This looks better.
Repeat using Lammps with Reax force field:
Get only one peak at 180 degrees. Problem with lammps and reax force field.
How do the MD results compare with the rotational energy plot you may know:
Given the plot above we could calculate the rate of rotation using the Arrhenius
expression:
Rate = A*exp(-Eact/RT)
If A = 1014 , Eact = 3 kcal and T = 300K
Rate = 3.5 x 1012, or 1/1000 steps
If A = 1014 , Eact = 2 kcal and T = 300K
Rate = 6.5 x 1011, or 1/10000 steps
Suppose we were to take a mole of butane at equilibrium at 300 K and we
were to measure the central dihedral angle of all the molecules – we would expect to
see the equilibrium population given by the Boltzmann distribution. Since the
dynamics run is describing the same population of butane molecules the population
we get from the dynamics run on one molecule must be the same as the equilibrium
population we get snapshot of a large quantity. This equality is what allows us to use
MD simulations to study real systems.
Conclude:
Always check method against known results
Simulations take a long time to reach equilibrium even for “simple”
molecules
Quantum Mechanics
WhyChemistry is the study of matter and matter consists of molecules, molecules
are made of atoms and atoms are composed of nuclei and electrons. This is the
current picture that has developed since 1900. Before then atoms were considered
the smallest particle and it was thought they could be understood using classical,
Newtonian, mechanics. Several experiments required a change in the model used to
describe matter before 1900.
Max Planck discovered a formula that related the intensity of black-body
radiation to temperature. A simple classical harmonic oscillator model could not be
made to reproduce the experimental results. His formula could only be obtained by
assuming that the radiation arises from discrete oscillators that radiate in bursts
and that each burst was an integral multiple of one unit of radiation. Energy was
quantized.
E = h
Where  is the frequency and h is Planck’s constant.
In 1905 Einstein made use of Planck’s relation between energy and
frequency to explain the photoelectric effect. He predicted the kinetic energy of the
electrons ejected from a substance by bombarding light should be given by:
T = hW
This relationship correctly predicts that the energy of the ejected electrons depends
upon the frequency of the light, not the intensity.
In 1912 Einstein and Peter Debye used Planck’s relationship to explain the
heat capacity of many solids.
In 1911 Rutherford used scattering experiments to show that atoms were
composed of a small, positively charged nucleus surrounded by a diffuse cloud of
electrons.
Given the need for quantized energy levels and the vast difference in size of
nuclei and electrons it became clear that classical mechanics would not work and a
search for alternative mathematical descriptions of the atom were sought. It took
about 20 years and several models of the atom were proposed and discarded before
the theory of quantum mechanics (wave mechanics) as devised by Schrodinger, and
equivalently by Heisenberg, Born and Jordan, were finally successful.
There is nothing magical about quantum mechanics – it is a system of equations that
we use to try to describe the results of experiments. If the equations work we use
them, if not we try other models and other equations. Classical models deal with
position and momentum and quantum mechanical models must also deal with
position and momentum and all other observable quantities which leads to the first
postulate of quantum mechanics:
To every observable there corresponds an operator
Classical
x (position)
r (distance)
p (momentum)
E (energy)
QM
x
r
-ih/x
H (Hamiltonian)
These operators operate on functions called wavefunctions which are functions of
position. There are certain conditions imposed on the wavefunctions such as the
function and its first derivatives must be single valued, continuous and finite.
The only possible values which a single measurement of an observable
associated with an operator S* can yield are eigenvalues s of the equation:
S = s
Where s is just a number. You have all seen
H = Ewhich species the energy of a system – Schrodinger’s equation
Since H = T+V (kinetic energy + potential energy) this is the quantum
mechanical equivalent of Newton’s equations.
:
:
C
:
H
:
:
H
:
:
:
C
H
Basics
:
O
:
H
:
:
Model consists of nuclei and electrons
Schrodinger equation describes interactions: HY = EY
Compute everything : structure, properties, reactivity, spectroscopy
Model is simple and complete, no ad hoc assumptions
But:
Can only be solved analytically for hydrogen atom
Calculations are difficult, time and resource consuming
Limited size, scales n4 to n7
In contrast a classical mechanical model would look like:
But:
Model consists of : atoms
Need to define interactions: bonds, angles, dihedrals, non-bonded
Can use any function, math is simpler and faster
Applied to larger systems
Model is messy, requires many assumptions
Need to define types of atoms
Empirical model
So we have two models for chemistry, one based on solid physical principles but
mathematically difficult, and the other less physically accurate but mathematically
more tractable the real question becomes what questions we want to answer.
Structure
Conformational Energies
Reactivity – Thermodynamics and Kinetics
Spectroscopic Properties- uv/vis, ir, nmr
Crystal Packing
Liquid Properties
Binding Energies
Question:
Which tasks require information about electrons?
What model is most appropriate?