Week 2

Basics of MD Simulations

•Lecture 3: Force fields (empirical potential functions among atoms)

Boundaries and computation of long-range interactions

•Lecture 4: Molecular mechanics (T=0, energy minimization)

MD simulations (integration algorithms, constrained dynamics,

ensembles)

Empirical force fields (potential functions)

Problem: Reduce the ab initio potential energy surface among the atoms

ji kji

kjijii UUU ),,(),( 32 RRRRRR

To a classical many-body interaction in the form

ieji ji

jii E

ezzU R

RRR

2

Such a program has been tried for water but so far it has failed to produce a

classical potential that works. In strongly interacting systems, it is difficult to

describe the collective effects by summing up the many-body terms.

Practical solution: Truncate the series at the two-body level and assume

(hope!) that the effects of higher order terms can be absorbed in U2 by

reparametrizing it.

iiii

iii

jiji

U

R

RR

qqU

)0(

3

21

ˆ)ˆ(31

,

EE

pRRpE

Ep

RR

pol

pol

Coul

Interaction of two atoms at a distance R can be decomposed into 4 pieces

1. Coulomb potential (1/R)

2. Induced polarization (1/R2)

3. Dispersion (van der Waals) (1/R6)

4. Short range repulsion (eR/a)

The first two can be described using classical electromagnetism

1. Coulomb potential:

2. Induced polarization:

Dipole field

Total polarization int. (Initial and final E fieldsin iteration)

Non-bonded interactions

aRAeU /sr

2/)( jiijjiij

62 43 RU disp

Here corresponds to the depth of the potential at the minimum; 21/6σ

Combination rules for different atoms:

The last two interactions are quantum mechanical in origin.

Dispersion is a dipole-dipole interaction that arises from quantum

fluctuations (electronic excitations to non-spherical states)

Short range repulsion arises from Pauli exclusion principle (electron clouds

of two atoms avoid each other), so it is proportional to the electron density

The two interactions are combined using a 12-6 Lennard-Jones (LJ)

potential 6124 RRU LJ

12-6 Lennard-Jones potential (U is in kT, r in Å)

3 3.5 4 4.5 5-0.5

0

0.5

1

1.5

2

2.5

r

U(r

)

AkT 3,41

Because the polarization interaction is many-body and requires iterations,

it has been neglected in most force fields.

The non-bonded interactions are thus represented by the Coulomb and

12-6 LJ potentials.

Model RO-H (Å) HOH qH (e) (kT) (Å) (D) (T=298 C)

SPC 1.0 109.5 0.410 0.262 3.166 2.27 65±5

TIP3P 0.957 104.5 0.417 0.257 3.151 2.35 97±7

Exp. Gas 1.86 80Ab initio Water 3.00

SPC: simple point charge

TIP3P: transferable intermolecular potential with 3 points

Popular water models (rigid)

There are hundreds of water models in the market, some explicitly

including polarization interaction. But as yet there is no model that

can describe all the properties of water successfully.

SPC and TIP3P have been quite successful in simulation of biomolecules,

and have become industry standards.

However, the mean field description of the polarization interaction is bound

to break in certain situations:

1. Ion channels, cavities

2. Interfaces

3. Divalent ions

To deal with these, we need polarizable force fields.

Grafting point polarizabilities on atoms and fitting the model to data has not

worked well. Ab initio methods are needed to make further progress.

Covalent bonds

In molecules, atoms are bonded together with covalent bonds, which

arise from sharing of electrons in partially occupied orbitals. In order to

account for the dipole moment of a molecule in a classical

representation, partial charges are assigned to the atoms. If the bonds

are very strong, the molecule can be treated as rigid as in water

molecule. In most large molecules, however, the structure is quite flexible

and this must be taken into account for a proper description of the

molecules. This is literally done by replacing the bonds by springs. The

nearest neighbour interactions involving 2, 3 and 4 atoms are described

by harmonic and periodic potentials

02

02

0 cos122

ijklijkl

ijklijk

ijkijk

ijij

rij

bond nVk

rrk

U

bond stretching bending torsion (not very good)

Interaction of two H atoms in the ground (1s) state can be described

using Linear Combinations of Atomic Orbitals (LCAO)

)1()1( 21 scsc BA

)1()1(2

1ss BA

From symmetry, two solutions with lowest and highest energies are:

+ Symmetric

- Anti-symmetric

The R dependence of the potential energy is approximately given by the

Morse potential

2)( 01 RRe eDU

Morse

20 )(

21

RRkUbond

Where

De: dissociation energy

Re: equilibrium bond distance

controls the width of the potential

Classical representation:

eDk 2/

k

Electronic wave functions for the 1s, 2s and 2p states in H atom

H2 molecule:

In carbon, 4 electrons (in 2s and 2p) occupy 4 hybridized sp3 orbitals

pxpypxs

pxpypxs

pxpypxs

pxpypxs

2

1

21

21

2

1

4

3

2

1

Thus carbon needs 4 bonds to have a stable configuration.

Nitrogen has an extra electron, so needs 3 bonds.

Oxygen has 2 extra electrons, so needs 2 bonds.

Tetrahedral structure of sp3

20bend )(

21 ijkkU

Bond stretching and bending interactions are quite strong and well

determined from spectroscopy. Torsion is much weaker compared to

the other two and is not as well determined. Following QM calculations,

they have been revamped recently (e.g., CMAP corrections in the

CHARMM force field), which resulted in a better description of proteins.

For a complete description of flexibility, we need, besides bond stretching,

bending and torsion. The former can be described with a harmonic form:

0torsion cos1 ijklnVU

While the latter is represented with a periodic function

An in-depth look at the force fields

Three force fields, constructed in the eighties, have come to dominate

the MD simulations

1. CHARMM (Karplus @ Harvard)

Optimized for proteins, works well also for lipids and nucleic acids

2. AMBER (Kollman & Case @ UCSF)

Optimized for nucleic acids, otherwise quite similar to CHARMM

3. GROMOS (Berendsen & van Gunsteren @ Groningen)

Initially optimized for lipids and did not work very well for proteins

(smaller partial charges in the carbonyl and amide groups) but it has

been corrected in the more recent versions.

The first two use the TIP3P water model and the last one, SPC model.

They all ignore the polarization interaction. Polarizable versions have

been under construction for over a decade but no working code yet.

Charm parameters for alanine

Partial charge (e)

ATOM N -0.47 |

ATOM HN 0.31 HN—N

ATOM CA 0.07 | HB1

ATOM HA 0.09 | /

GROUP HA—CA—CB—HB2

ATOM CB -0.27 | \

ATOM HB1 0.09 | HB3

ATOM HB2 0.09 O=C

ATOM HB3 0.09 |

ATOM C 0.51

ATOM O -0.51

Total charge: 0.00

Bond lengths :

kr (kcal/mol/Å2) r0 (Å)

N CA 320. 1.430 (1)

CA C 250. 1.490 (2)

C N 370. 1.345 (2) (peptide bond)

O C 620. 1.230 (2) (double bond)

N H 440. 0.997 (2)

HA CA 330. 1.080 (2)

CB CA 222. 1.538 (3)

HB CB 322. 1.111 (3)

1. NMA (N methyl acetamide) vibrational spectra

2. Alanine dipeptide ab initio calculations

3. From alkanes

20)( rrkU r bond

Bond angles :

k (kcal/mol/rad2) 0 (deg)

C N CA 50. 120. (1)

C N H 34. 123. (1)

H N CA 35. 117. (1)

N CA C 50. 107. (2)

N CA CB 70. 113.5 (2)

N CA HA 48. 108. (2)

HA CA CB 35. 111. (2)

HA CA C 50. 109.5 (2)

CB CA C 52. 108. (2)

N C CA 80. 116.5 (1)

O C CA 80. 121. (2)

O C N 80. 122.5 (1)

20)( kUangle

Total 360 deg.

Total 360 deg.

Basic dihedral configurations

trans cisDefinition of the dihedral

angle for 4 atoms A-B-C-D

)cos(12 0 n

VU n

dihedDihedrals:

Dihedral parameters:

Vn (kcal/mol) n 0 (deg) Name

C N CA C 0.2 1 180.

N CA C N 0.6 1 0.

CA C N CA 1.6 1 0.

H N CA CB 0.0 1 0.

H N CA HA 0.0 1 0.

C N CA CB 1.8 1 0.

CA C N H 2.5 2 180.

O C N H 2.5 2 180.

O C N CA 2.5 2 180.

)cos(12 0 n

VU n

dihed

–helix structure

-helix

~ 57o

~ 47o

(Sasisekharan)

When one of the 4 atoms is not in a chain (e.g. O bond with C in CA-C-N),

then an improper dihedral interaction is used to constrain that atom.

Most of the time, off-the-chain atom is constrained to a plane using:

)cos(1 00 VU imp

CA

C O

N

Where is the angle C=O bond makes

with the CA-C-N plane, and = 180

which enforces a planar configuration.

Boundaries

In macroscopic systems, the effect of boundaries on the dynamics of

biomolecules is minimal. In MD simulations, however, the system size

is much smaller and one has to worry about the boundary effects.

• Using nothing (vacuum) is not realistic for bulk simulations.

• Minimum requirement: water beyond the simulation box must be

treated using a continuum representation (reaction field). An

intermediate zone is treated using stochastic boundary conditions.

• Most common solution: periodic boundary conditions.

The simulation box is replicated in all directions just like in a crystal.

The cube and rectangular prism are the obvious choices for a box

shape, though there are other shapes that can fill the space (e.g.

hexagonal prism and rhombic dodecahedron).

Periodic boundary conditions in two dimensions: 8 nearest neighbours

Particles in the box freely move to the next box, which means they

appear from the opposite side of the same box.

In 3-D, there are 26 nearest neighbours.

Treatment of long-range interactions

Problem: the number of non-bonded interactions grows as N2 where N is

the number of particles. This is the main computational bottle neck that

limits the system size. A simple way to deal with this problem is to

introduce a cutoff radius for pairwise interactions (together with a

switching function), and calculate the potential only for those within the

cutoff sphere. This is further facilitated by creating non-bonded pair lists,

which are updated every 10-20 steps.

• For Lennard-Jones (6-12) interaction, which is weak and falls rapidly,

this works fine and is adapted in all MD codes.

• Coulomb interaction, however, is very strong and falls very slowly.

[Recall the Coulomb potential between two unit charges, U=560 kT/r (Å)]

Hence use of any cutoff is problematic and should be avoided.

This problem has been solved by introducing the Ewald sum method.

All the modern MD codes use the Ewald sum method to treat the long-

range interactions without cutoffs.

Here one replaces the point charges with Gaussian distributions, which

leads to a much faster convergence of the sum in the reciprocal (Fourier)

space.

The remaining part (point particle – Gaussian) falls exponentially in real

space, hence can be computed easily using a cutoff radius (~10 Å).

The mathematical details of the Ewald sum method is given in the

following appendix. Please read it to understand how it works in practice.

N

j ij

ji

N

iii

qqU

11

')(,)(2

1

n nrrrr Coul

Appendix: Ewald summation method

The coulomb energy of N charged particles in a box of volume V=LxLyLz

Here with integer values for n.

The prime on the sum signifies that i=j is not included for n=0.

Ewald’s observation (1921): in calculating the Coulomb energy of ionic

crystals, replacing the point charges with Gaussian distributions leads to

a much faster convergence of the sum in the reciprocal (Fourier) space.

The remaining part (point particle – Gaussian) falls exponentially in real

space, hence can be computed easily.

),,( zzyyxx LnLnLnnr

rq

q )(rpoint

''

)'(4 32 rd

rr

r

For a point charge q at the origin:

function error the iswhere

GaussGauss

xu

r

r

duex

rr

qrd

eq

qrdeq

0

3'

2/3

3

32/3

3

2

22

22

2)erf(

)erf(''

)(

)(,)(

rrr

rr

The Poisson equation and it’s solution:

When the charge q has a Gaussian distribution:

)erf()(

)erf(2

'2

)(

2

14

''4'''

41

00

'

22/3

3

'2/3

3

2

2

2/3

3

2

2

222

22

22

22

rrq

rqdueqdreqr

eqrdrd

drreqdrrdr

d

eqrdr

dr

ru

rr

r

rr

r

r

r

r

This result is obtained most easily by integrating the Poisson equation

Integrate 0 to r :

)erfc()erf(1)( ijij

jij

ij

ji r

r

qr

r

q r

)( GausspointGausspoint

22re

For the second part, the potential due to a charge qj at rj is given by:

Writing the charge density as

Where erfc(x) is the complementary error function which falls as

Thus choosing 1/ about an Angstrom, this potential converges quickly.

Typically, it is evaluated using a cutoff radius of ~10 Å, so the original box

with N particles is sufficient for this purpose (with the nearest image

convention). The direct (short-range) part of the energy of the system is:

),erfc()(,)(2

1

11ij

N

j ij

ji

N

iii r

r

qqU

rrdirect

23 4

)(~)()(~k

qqrdeq i krk rk

Vi

zzyyxxi

rdeff

LnLnLnefV

f

3)()(~

,,2,)(~1

)(

rk

n

rk

rk

kkr

The Gaussian part converges faster in the reciprocal (Fourier) space

hence best evaluated as a Fourier series

The Poisson equation in the Fourier space

)(~4)(~

)(~14)(~1

2

2

kk

kkn

rk

n

rk

k

eV

eV

ii

For a point charge q at the origin:

)()(2/3

3

32/3

3

2222

22

)(~

zkykxkizyx

ir

zyxeedzdydxq

rdeeq

rkk

22 4ke

When the charge q has a Gaussian distribution:

Multiply each integral by to compete the square

Each Gaussian integral then gives

22 42

4)(~ ke

k

q k

The corresponding potential in the Fourier space

N

j

kij eeq

kj

1

42

224)(~ rkkGauss

N

j

kij

N

jj

i

N

jjV

i

N

jj

eeq

eqerd

eqerd

eq

j

j

j

j

1

4

12/3

33

)(

12/3

33

)(

12/3

3

22

22

22

22

)(~

)(

rk

rrrk

rrr

n

rk

rrr

n

n

n

k

r

space all

Gauss

Gauss

The Gaussian charge density for the periodic box

Which yields for the potential

0

422

0

)(

1,

42

1

22

22

)(~4

2

1

4

2

1

)(2

1

n

n

rrk

r

k

iN

ji

kji

N

iii

ekkV

eek

qq

V

qU

ji

Gaussrecip

0

)(

1

42

0

2241

)(~1)(

n

rrk

n

rkkr

jiN

j

kj

i

eek

q

V

eV

GaussGauss

Transforming back to the real space

The reciprocal space (long-range) part of the system’s energy:

N

ii

N

iii

q

rqU

q

rrq

1

2

1

)(21

2)0(

)erf()(

selfself

self

Gauss

r

r

This energy includes the self-energy of the Gaussian density which needs

to be removed

So the total energy is: selfrecipdirecttotal UUUU

Molecular mechanics

Molecular mechanics deals with the static features of biomolecular

systems at T=0 K, that is, particles do not have any kinetic energy.

[Note that in molecular dynamics simulations, particles have an average

kinetic energy of (3/2)kT, which is substantial at room temp., T=300 K]

Thus the energy is given solely by the potential energy of the system.

Two important applications of molecular mechanics are:

1. Energy minimization (geometry optimization):

Find the coordinates for the configuration that minimizes the potential

energy (local or absolute minimum).

2. Normal mode analysis:

Find the vibrational excitations of the system built on the absolute

minimum using the harmonic approximation.

Energy minimization

The initial configuration of a biomolecule whether experimentally

determined or otherwise does not usually correspond to the minimum

of the potential energy function. It may have large strains in the

backbone that would take a long time to equilibrate in MD simulations,

or worse, it may have bad contacts (i.e. overlapping van der Waals

radii), which would crash the MD simulation in the first step!

To avoid such mishaps, it is a standard practice to minimize the energy

before starting MD simulations.

Three popular methods for analytical forms:

• Steepest descent (first derivative)

• Conjugate gradient (first derivative)

• Newton-Raphson (second derivative)

Steepest descent:

Follows the gradient of the potential energy function U(r1, …,rN) at each

step of the iteration

where i is the step size. The step size can be adjusted until the minimum

of the energy along the line is found. If this is expensive, a single step is

used in each iteration, whose size is adjusted for faster convergence.

• Works best when the gradient is large (far from a minimum), but tends

to have poor convergence as a minimum is approached because the

gradient becomes smaller.

• Successive steps are always mutually perpendicular, which can lead

to oscillations and backtracking.

nNini

nii

ni

ni U ),,(, 1

1 rrffrr

A simple illustration of steepest descent with a fixed step size in

a 2D energy surface (contour plots)

Conjugate gradient:

Similar to steepest descent but the gradients calculated from previous

steps are used to help reduce oscillations and backtracking

(For the first step, is set to zero)

• Generally one finds a minimum in fewer steps than steepest descent,

e.g. it takes 2 steps for the 2D quadratic function, and ~n steps for nD.

• But conjugate gradient may have problems when the initial

conformation is far from a minimum in a complex energy surface.

• Thus a better strategy is to start with steepest descent and switch to

conjugate gradient near the minimum.

nNini

ni

nin

ini

ni

nii

ni

ni U ),,(,, 121

2

11 rrff

fδfδδrr

Newton-Raphson:

Requires the second derivatives (Hessian) besides the first.

Predicts the location of a minimum, and heads in that direction.

To see how it works in a simple situation, consider the quadratic 1D

case

In general

For a quadratic energy surface, this method will find the minimum in one

step starting from any configuration.

a

b

a

baxx

f

fxx

a

bxfcbxaxxf

x 22

2

"

'

20',)(

000min

min2

0

nlk

Nkl

ni

ni

ni

ni

ni xx

U

),,(][,][ 1

211 rr

HfHrr

• Construction and inversion of the 3Nx3N Hessian matrix is

computationally demanding for large systems (N>100).

• It will find a minimum in fewer steps than the gradient-only methods in

the vicinity of the minimum.

• But it may encounter serious problems when the initial conformation

is far from a minimum.

• A good strategy is to start with steepest descent and then switch to

alternate methods as the calculations progress, so that each algorithm

operates in the regime for which it was designed.

Using the above methods, one can only find a local minimum.

To search for an absolute minimum, Monte Carlo methods are more

suitable. Alternatively, one can heat the system in MD simulations,

which will facilitate transitions to other minima.

Normal mode analysis

Assume that the minimum energy of the system is given by the 3N

coordinates, {r0i}. Expanding the potential energy around the

equilibrium configuration gives

Ignoring the constant energy, the leading term is that of a system of

coupled harmonic oscillators. In a normal mode, all the particles in the

system oscillate with the same frequency . To find the normal modes,

first express the 3N coordinates as {xi, i=1,…,3N}.

0}{

1 1

0

}{1

001

0

0

21

0})({),,(

jjji

N

i

N

jii

i

N

iiiiN

i

i

U

UUU

rrrr

rrrrr

r

r

The potential energy becomes

where the spring constants are given by the Hessian as

Introducing the 3Nx3N diagonal mass matrix M

The secular equation for the normal modes is given by

}{

2

0ixji

ij xxU

K

03

1

3

1

031 2

1),,( jjij

N

i

N

jiiN xxKxxxxU

),,,,,,,,,diag( 222111 NNN mmmmmmmmm M

022/12/1 IMKM

For a 3Nx3N matrix, solution of the secular equation will yield 3N

eigenvalues, i and the corresponding eigenvectors,

Of these, 3 correspond to translations and 3 to rotations of the system.

Thus there are 3N-6 intrinsic vibrational modes of the system.

At a given temperature T, the motion of the i’th coordinate is given by

The mean square displacement of the coordinates and atoms:

)cos()(

2/13

72 jjij

N

j jii t

m

kTtx

223

213

23

2

23

72

2

)()()()(

2

1)(

nnnn

ij

N

j jii

xxx

m

kTx

r

A simple example: normal modes of water molecule

Water molecule has 9-6=3 intrinsic vibrations, which correspond to

symmetric and anti-symmetric stretching of H atoms and bending.

Because of H-bonding, water molecules in water cannot freely rotate

but rather librate (wag or twist).

Wave numbers: 3652 cm-1 3756 cm-1 1595 cm-1 (200 cm-1~1 kT)

Excitation energies >> kT, which justifies the use of a rigid model for water.

Normal modes of a small protein BPTI (Bovine pancreatic tyripsin inhibitor)

Some characteristic frequencies (in cm-1)Stretching Bending Torsion

H-N: 3400-3500 H-C-H: 1500 C=C: 1000

H-C: 2900-3000 H-N-H: 1500 C-O: 300-600

C=C, C=O: 1700-1800 C-C=O: 500 C-C: 300

C-C, C-N: 1000-1250 S-S-C: 300 C-S: 200

Applications to domain motions of proteins

• Many functional proteins have two main configurations called, for

example, resting & excited states, or open & closed states.

• Proteins can be crystallized in one of these states – the other

configuration need to be found by other means.

• The configurational changes that connect these two states usually

involve large domain motions that could take milli to micro seconds,

which is outside the reach of current MD simulations.

• Normal mode analysis can be used to identify such collective motions

in proteins and predict the missing state that is crucial for description of

the protein function.

• Examples: 1. Gating of ion channels (open & closed states)

2. Opening and closing of gates in transporters

Molecular dynamics

In MD simulations, one follows the trajectories of N particles according

to Newton’s equation of motion:

where U(r1,…, rN) is the potential energy function consisting of bonded

and non-bonded interactions (Coulomb and LJ 6-12).

We have already discussed force fields and boundary conditions in some

detail. Here, we will consider:

• Integration algorithms, e.g., Verlet, Leap-frog

• Initial conditions and choice of the time step

• Constrained dynamics for rigid molecules (SHAKE and RATTLE)

• MD simulations at constant temperature and pressure

),,(,2

2

Niiiiii Udt

dm rrFFr

2

2

)()()(2)(

2

)()()()(

tm

tttttt

tm

tttttt

i

iiii

i

iiii

Frrr

Fvrr

2

2

)()()()(

)()()(

tm

tttttt

tm

tttt

i

iiii

i

iii

Fvrr

Fvv

Integration algorithms

Given the position and velocities of N particles at time t, a straightforward

integration of Newton’s equation of motion yields at t+t

In practice, variations of these equations are implemented in MD codes

In the popular Verlet algorithm, one eliminates velocities using the

positions at t-t,

Adding with eq. (2), yields:

(1)

(2)

This is especially useful in situations where one is interested only in the

positions of the atoms. Velocities can be calculated from

Some drawbacks of the Verlet algorithm:

• Positions are obtained by adding a small quantity (order t2) to large

ones, which may lead to a loss of precision.

• Velocity at time t is available only at the next time step t+t

• It is not self starting. At t=0, there is no position at t-t. It is usually

estimated using

)]()([1

)(

)]()([2

1)(

tttt

t

ttttt

t

iii

iii

rrv

rrv

or

tt iii )0()0()( vrr

tttttt

tm

ttttttt

iii

i

iiii

)2/()()(

)()2/()()( 2

vrr

Fvrr

tttttt

tm

ttttt

iii

i

iii

)2/()()(

)()2/()2/(

vrr

Fvv

In the Leap-frog algorithm, the positions and velocities are calculated

at different times separated by t/2

To show its equivalence to the Verlet algorithm, consider

Subtracting the two equations yields the Verlet result.

If required, velocity

at t is obtained from:)]2/()2/([

21

)( ttttt iii vvv

To iterate these equations, we need to specify the initial conditions.

• The initial configuration of biomolecules can be taken from the Protein

Data Bank (PDB) (if available).

• In addition, membrane proteins need to be embedded in a lipid bilayer.

VMD has a facility that will perform this step with minimal effort.

• All the MD codes have facilities to hydrate a biomolecule, i.e., fill the

void in the simulation box with water molecules at the correct density.

• Ions can be added at random positions. Alternatively, VMD solves the

Poisson-Boltzmann equation and places the ions where the potential

energy are at minimum.

After energy minimization, these coordinates provide the positions at t=0.

Initial velocities are sampled from a Maxwell-Boltzmann distribution:

kTvmkT

mvP ixi

iix 2exp

2)( 2

2/1

Choosing the time step:

In choosing a time step, one has to compromise between two conflicting

demands:

• In order to obtain statistically significant results and access biologically

relevant time scales, one needs long simulation times, which requires

long time steps

• To avoid instabilities, conserve energy, etc., one needs to integrate

the equations as accurately as possible, which requires using short

time steps.

System Types of motion Recom. time step (fs)

Atoms translation 10 fs

Rigid molecules … + rotation 5 fs

Flex. mol’s, rigid bonds … + torsion 2 fs

Completely flex. mol’s ….+ vibrations 1 - 0.5 fs

)(2 jiijii rrF

Constrained dynamics

MD simulations are carried out most efficiently using the 3N Cartesian

coordinates of the N atoms in the system. This is fine if there are no rigid

bonds among the atoms in the system. For rigid molecules (e.g. water),

one needs to impose constraints on their coordinates so that their rigid

geometry is preserved during the integration. This is achieved using

Lagrange multipliers. Consider the case of a rigid diatomic molecule,

where the bond length between atoms i and j is fixed at d, which imposes

the following constraint on their coordinates

The force on atom i due to this constraint can be written as

0)( 22 djiij rr

ijiijjj FrrF )(2

)]()([2)(

)()(2)(

)]()([2)(

)()(2)(

22

22

tttm

tm

tttttt

tttm

tm

tttttt

jijj

jjjj

jiii

iiii

rrF

rrr

rrF

rrr

Note that

Thus the total constraint force on the molecule is zero, and the constraint

forces do not do any work. Incorporating the constraint forces in the

Verlet algorithm, the positions of the atoms are given by

To simplify, combine the known first three term on the r.h.s. as

)]()([2

)(')(

)]()([2

)(')(

2

2

tttm

ttt

tttm

ttt

jij

jj

jii

ii

rrrr

rrrr

(1)

(2)

22

2 )]()([11

2)(')(' dtttmm

tt jiji

ji

rrrr

A third equation is given by the constraint condition

222)()()()( dtttttt jiji rrrr

Substituting eqs. (1) and (2) above yields

This quadratic equation can be easily solved to obtain the Lagrange

multiplier . Substituting back in eqs. (1) and (2), one finds the positions

of the atoms at t+t.

For a rigid many-atomic molecule, one needs to employ a constraint for

every fixed bond length and angle. (Bond and angle constraints among

three atoms (i, j, k) can be written as three bond constraints.)

Assuming k constraints in total, this will lead to k coupled quadratic

equations, which are not easy to solve.

A common approximation is to exploit the fact that is small, hence the

quadratic terms in can be neglected. This leads to a system of k linear

equations, which can be solved by matrix inversion.

In the SHAKE algorithm, a further simplification is introduced by solving

the constraint equations sequentially one by one. This process is then

iterated to correct any constraint violations that arise from the neglect of

the coupling in the constraint equations.

SHAKE is commonly used in fixing the bond lengths of H atoms.

Another well-known constraint algorithm is RATTLE, which is designed

for the velocity-Verlet integrator, where velocities as well as positions are

corrected for constraints.

MD simulations at constant temperature and pressure

MD simulations are typically performed in the NVE ensemble, where all 3

quantities are constant. Due to truncation errors, keeping the energy

constant in long simulations can be problematic. To avoid this problem,

the alternative NVT and NPT ensembles can be employed. The

temperature of the system can be obtained from the average K. E.

Thus an obvious way to keep the temperature constant at T0 is to scale

the velocities as:

)(),()( 0 tTTtvtv ii

NkTK23

Because K. E. has considerable fluctuations, this is a rather crude method.

A better method which achieves the same result more smoothly is the

Berendsen method, where the atoms are weakly coupled an external

heat bath with the desired temperature T0

If T(t) > T0 , the coefficient of the coupling term is negative, which

invokes a viscous force slowing the velocity, and vice-versa for T(t) < T0

Similarly in the NPT ensemble, the pressure can be kept constant by

simply scaling the volume. Again a better method (Langevin piston), is

to weakly couple the pressure difference to atoms using a force as in

above, which will maintain the pressure at the desired value (~1 atm).

dt

d

tT

Tm

dt

dm i

iiiiir

Fr

1)(

02

2

Monte Carlo (MC)

Metropolis algorithm:

Given N particles interacting with a potential energy function U(r1,…, rN)

Probability:

• Assume some initial configuration with energy U0

• Move the particles by small increments to new positions with energy U1

• If U1 < U0, accept the new configuration

• If U1 > U0 , select a random number r between 0 and 1, and accept the

new configuration if rkTUU /)(exp 01

kTUeP /

• Keep iterating until the minimum energy is found