Force Fields and Simulations Flashcards
For a diatomic molecule the potential energy only depends on
one variable - the bond length
Plotting oscillator potential to approximate potential energy represents
how often energy varies as a function of the distortion of the bond length from its equilibrium value
1 d potential energy (E) =
f(l) = 1/2 * k (l-l0)^2 for bond length l, min bond length l0 and bond force constant k
If potential energy is a function of more than one variable we have a…
potential energy surface
potential energy wells correspond to
lowest energy states for reactants and products
Saddle point on PES correspond to
transition state for reaction
hypersurface depends on
many variables
molecules spend most of the time in
potential wells on PES
energy minima on PES are
STABLE CONFORMATIONS
Anti conformation (180) is more stable than gauche (60) because
reduced steric hindrance
PES calculated by considering
rotation about backbone dihedral angles
To calculate potential energy of molecular system write
as a sum of all the different contributions from bond stretching/bending, dihedral angle distortions and nonbonded interaction (eg steric repulsion and van der Waals) and electrostatic interaction
Can use computational algorithms to minimise
steric energy and find stable conformations corresponding to minima on the PES
High force constant for bond corresponds to
stiff bonds where distortion from equilibrium value corresponds to large energies
Energy in torsional motion of dihedral angle =
sum up to number of terms (n) in series expansion of: (force constants / 2)*(1+cos(nangle))
2 neutral atoms approaching at long distances
attract each other through dispersion interactions and repel at close distances
Lennard jones potential
epsilon is well depth and sigma is distance where attractive and repulsive energies cancel equation on cheat sheet
Energy due to electrostatic interactions can be given by
a charge-charge interaction equation on cheat sheet
in some force fields electrostatic interactions can be represents by
bond dipoles. equation on cheat sheet
Steric energy should not be compared between
molecules. Compared between different conformation of a molecule instead.
Individual force fields are not normally needed because
so similarin different molecules so can use the same parameters
making force fields:
using computational fitting procedures. using experimental data, using other calculations eg monte carlo (details on pg 10-11)
Structure with the lowest energy is the most stable and therefore the
most abundant structure.
IF first derivative of energy is 0 and second derivative is > 0
miminmum
Formal derivation of minimisation algorithm
start at x. x(min) = x + delta x. differential of x(min) = 0. expand as taylor series. equation. Find x(min) as roughly x - E’/E’’.
For higher dimensions replace E’ by
vector (force vector)
For higher dimensions replace E’’ by
hessian matrix, giving a vector of small changes for each iteration.
Boltzmann Law gives
relative population of states with different energies.
If more than 2 states in system we can use boltzman distribution to find
probability of each state
partition function is the
sum over boltzmann factors for each energy level and energy of lowest energy state is subtracted from each energy level
To find probability of different conformations:
Find energy relative to lowest energy state for each conformation. Work our Boltzmann factor for each conformation. Sum up boltzmann factor to give partition function. Work out probability of each level using partition. Us relative energy to get correct partition function.
Uses of Molecular mechanics (3)
finding global energy minima, conformational searching and modelling of interactions between molecules.
Advantages of MM over QM: (2):
MM faster, possible for large molecules, simple organic molecule stuctures are better for MM as poor representation of dispresion in Qm
Disadvantages of MM over QM: (3)
No account of electrons, not good where electronic effects dominate over steric effects, and polarisability not taken into account in most force fields.
Limitations of MM(4):
in large molecules, the number of accessible conformations is great, the number of molecular arrangements grows exponentially as the number of molecules increases. if only considering energy minima then ignore entropy and free energy, molecules are moving all the time so sampling diff energy states.
Average energy =
average over representation number of states. equation on cheat sheet.
Velocity Verlet algorithm
breaks time down into a sequence of small discrete time-steps (delta t). method on page 18. delta t is very small and should be 1/20 of the fastest motion in system.
Leapfrog algorithm
velocities are half a time step behind the positions but then leap ahead by half a time step. Algorithm of pg 18
Monte carlo simulation
uses random numbers to make small changes from current configuration, calculating Blotzmann factor for energy change and acceptingrejecting the change. diagram on pg 19
MC vs MD (5):
Both generate energy states according to their Bolztmann weights. Only coordinates enter MC, md has velocities and forces. Md good for big molecules. More good.free programs available for MD.
LARGE molecules hard to simulate via MC because
hard to generate MC mobes that sample conformational space efficiently
Mc has many advantages over MD such as
use of special MC moves and use of thermodynamic ensembles for simulation. Even possible to have MC moves whree molecules appear/disappear. Move can be used to speed up simulation of mixing or demixing in a binary mixture of 2 liquids.
When simulating bulk systems with many molecules avoid
serious problem caused by fact we can only simulate a few hundred molecules (at most a few thousand).
For a small system surface molecules dominate over
bulk molecules
Periodic boundary conditions
employed alongside minimum image convention
when simulating in the bulk it is usually necessary to
conserve certain quantities that define the thermodynamic ensemble used. important ensembles on page 21
Thermodyanmic properties can be obtained using
MC and MD techniques. examples on page 22
increasing the temperature in MC will
change the moves that are accepted with additional moves to higher energy configurations being accepted
For MD temperature can be obtained from velocities in a simulation by the
equipartition principle equation on page 22
dihedral angle distributions plot the
frequency of dihedral angles occuring during a simulation.
Can take natural log of the dihedral angle distribution to obtain the
torsional free energy
Radial distribution function g(r) shows
the average structure in a liquid. 1st peak corresponds to solvation shell, 2nd solvation shell is less well defined. g(r) decays to 1 at long distances as no long range structure in a liquid
Bulk diffusion coefficient can be estimated in MD simulation from the
mean squared displacement at long times or the velocity autocorrection function equation on page 23
Molecular mechanics is for single molecules and
solids and ‘clusters’
MC and MD for
single molecules and fluids
