Lecture 4: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
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Lab 1
Getting started
• Download Lab 1 from the stellar site and follow along.
• Tentative due date: Tuesday 9/23.
• The objective today is to build molecules in various
ways, find optimized minima, carry out conformer
searches, and assess relative accuracy of different
force fields.
• Make sure you have access to an athena machine or
a laptop with Avogadro/openbabel.
• First, we will review some concepts.
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Lab 1
Conformational sampling
• Minimization techniques only find the nearest
minimum – sometimes the local minimum instead
of the global minimum.
• Number of minima grow exponentially with number
of variables.
Multiple minima problem CH3(CH2)n+1CH3 with n
rotatable bonds:
n Conformations
(3n)
Time (1
conf/sec)
1 1 3 sec
5 243 4 min
10 59,049 16 hr
15 14,348,907 166 days
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Lab 1
Conformational sampling
Systematic or grid search method is feasible only
for small systems –iteratively vary rotations by fixed
amount until all have been generated (works for 15-
20 dihedral angles).
Can prune the search for torsions that always lead to
clashing/high energy structures:
First angle
Second angle
Third angle
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Lab 1
Conformational sampling
Genetic algorithms:
-Example of an evolutionary algorithm.
-Start with a population of structures characterized by
“genes” – e.g. 0s and 1s to represent dihedrals.
-Parent structures generate children having mixture of
parent genes.
-Small probability of mutations in the process.
-Fittest, lowest energy species (10%) are carried to
next generation. Other 90% generated by mating the
40% fittest and allowing for mutations.
-typically proceeds for ~100 generations but
population, mutation, breeding, ratio of children/
parents, local optimizations of the structures, etc can
all be varied.
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Lab 1
Conformational sampling
In Avogadro/Openbabel:
• Systematic searching – iteratively vary rotations by fixed
amount until all have been generated. This is hard to do when
we have many rotatable bonds.
• Genetic algorithm – evolutionary algorithm (may not work in
your Avogadro).
• Random rotor— Randomly generate guesses for the torsion
and evaluate the energy
• Weighted rotor—Stochastic: torsion angles are weighted
based on relative energies of generated conformers
(important for large number of rotatable bonds).
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Lab 1
Conformational sampling
Monte Carlo methods:
1) start from a local minimum
2) give a random kick to one or more atoms.
3) New geometry is accepted for next step if it is
lower in energy than the current one.
4) If e-DE/kT is lower than a random number between
0 and 1, then the new geometry is also accepted.
5) If not, next step from old geometry.
6) Step size chosen to be small enough to give
acceptance ratio of ~0.5.
Stochastic methods use larger kicks followed by
minimizations. Can follow low eigenvalues of the
Hessian also.
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Conformational sampling
• Molecular dynamics (later in the course): molecules
have potential and kinetic energy – given high enough
energy, dynamics can sample the PES.
• Simulated annealing: high initial temperature 2000-
3000K followed by dynamics where the temperature is
slowly reduced until trapped in a minimum.
• Distance geometry methods: trial geometries from
lower and upper bounds on distances between all
pairs of atoms. Random numbers between these limits
for trial distances. Then optimize.
• Diffusion methods: slowly modify the PES so that
other minima disappear until only one minimum (the
global minimum) remains. Math expressions are
similar to those for diffusion.
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FFs in avogadro/openbabel
• UFF
– Bond lengths are sum of atomic properties based on atom types plus electronegativity
corrections,
– Bond force constants derived from distance (cubic) and effective charges
– Bending terms are expanded in Fourier series
– Angle force constants derived from law of cosines (so depends on distances) and
electrostatics.
– Torsion is Fourier with V and cos terms.
– vdW is 6-12 L-J and electrostatics are Coulomb (not scaled) – charges from Qeq
(basically a fluctuating charge force field).
• GAFF:
– Generalized Amber Force Field, covered in class.
– Charges from HF/6-31G*
– Force field parameters from MP2/MP4 quantum chemistry.
– Quadratic/Fourier expansions for all constants.
– Gasteiger-Marsili empirical partial charges.
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FFs in avogadro/openbabel
• MMFF94:
– Up to quartic terms in bonds
– Cubic terms in angles
– Stretch-bend cross-terms
– Buffered 14-7 van der Waals
– Scaled (0.75) electrostatics, also a distance buffering term – add 0.05
Angstroms to the actual distance.
– Bond charge increments for charges in electrostatics: neighbors
determine the charges on the atoms.
• Ghemical (Tripos 5.2 based) – deprecated:
– Quadratic bond stretch
– Quadratic angle bending
– Lennard-Jones vdW
– Partial charges from lookup table.
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Optimization algorithms
• Steepest descent: always follow the
gradient.
• Conjugate gradient: use some history
information – the direction is the weighted
average of current and past gradient.
• Molecular dynamics (next lectures):
available for MMFF94 at 300 K/600 K/900 K.
