Homework 1
CENG 214
Date posted: Jan 6, 2011
Submission date: Jan 13, 2011
Problem 1: Consider a two-dimensional (planar) configuration of charge Q interacting
with a dipole consisting of charges q and +q separated by a distance l (see Figure).
(a) Show t
Energy Minimization Algorithms
1
Potential energy surface
Energy surface: Multidimensional plot of total energy as a function of all
the degrees of freedom
internal d.o.f = 3N-6 (N>2)
= 3N-5 (N=2)
In general, much more complicated and cannot be visualize
Homework 6
CENG 214
Date posted: March 1, 2011
Submission: March 10, 2011
Problem 1: Consider the following Langevin equation used to model the dynamics of a
single solute particle within a solvent:
dv( t )
= v( t ) + ( t )
dt
6R
is the solvent friction
Homework 6 solutions
Problem 1. The codes for computing the instantaneous temperature vs. simulation time, the
average radial distribution function, and the average velocity autocorrelation function are
provided below.
C
C
Subroutine to calculate the radi
Homework 5
CENG 214
Date posted: Feb 17, 2011
Submission: Mar 1, 2011
Problem 1: Provided with this homework is the cfw_rN, pN data obtained from an NVE molecular
dynamics simulation of liquid methane at a density of 0.39854 g/cm3 and a temperature roughl
Homework 4 solutions
Problem 1
(a) Verlet algorithm provides positions in future time t + dt using current position at time t and
past position at time t - dt :
r(t + t) = 2r(t) r(t t) + (F (t) / m)t 2
.(1)
If the Verlet algorithm is truly reversible, the
Homework 4
CENG 214
Date posted: Feb 8, 2011
Submission: Feb 15, 2011
Problem 1: Newtons equations of motion (NEM) are time reversible, i.e., if suddenly the
velocities of all interacting particles were reversed (v(t) instead of v(t), the particles will
r
Homework 3 solution
Problem 1.
Canonical partition function is given by: Q( N ,V , T ) =
1
(2"mk B T ) 3 N / 2 $ exp[#U / kB T ]dqN .
h N!
For ideal gases with no intermolecular interactions, we can set U = 0, and we get
1
Q( N ,V , T ) = 3 N (2"mkT ) 3 N
Homework 3
CENG 214
Date posted: Jan 27, 2010
Submission: Feb 8, 2010
Problem 1: The canonical partition function is given by
which allows us to compute many thermodynamic properties. For example, the internal energy is
given by
The pressure is given by
T
Homework 2 solutions
Problem 1.
(a) To find stationary points of U(!) = C1(1-cos!) + C2(1-cos2!) + C3(1+cos3!), we differentiate
the function and set it to zero, i.e., U(!) = -sin!(C1 - 3C3 - 4C2cos! + 12C3cos2!) = 0
! sin! = 0, or C1 - 3C3 - 4C2cos! + 12
Homework 2
CENG 214
Date posted: Jan 13, 2011
Submission due: Jan 27, 2011
Problem1: The torsional angle in molecular mechanics (also called the dihedral or twist
angle) is defined as the angle subtended by two planes formed from the consecutive united at
Continuum modeling of solvent
1
Poisson-Boltzmann (PB) equation
PB theory describes the electrostatic interactions between charged
bodies immersed in a solvent with mobile ions
Start from Gausss law of electrostatics
[ (r ) (r )] = 4 (r )
Position depend
Monte Carlo
1
Monte Carlo Simulations
Calculating ensemble averages in stat mech
1
A=
Q
A(p
N
N
N
N
, q ) exp[H (p , q )]dp dq
A=
exp[H (p
1
Z
A (q
N
) exp[ U (q N )]dq N
where
N
, qN )]dpN dqN
N
where
Q=
N
Z=
exp[U (q
N
)]dq N
Monte Carlo (MC) simula
Molecular Mechanics
1
Empirical Force Field Molecular Modeling
Molecular modeling/mechanics methods: describe intramolecular
and intermolecular interactions between atoms using empirically
derived functions that ignore electronic motions and depend only o
Molecular Dynamics
1
Molecular Dynamics (MD)
MD simulates the evolution of classical system in time t, i.e., it
solves the Newtons equations of motion of N interacting
particles (to generate trajectories)
i =
r
Fi
mi
ri =
pi
mi
or
cfw_r(0),p(0),
pi = Fi
w