2. 1.1Statistical mechanics and classical mechanics
19
When we do measurements on a physical system, the measured quantities are results of time-averaged action (influence) of all or a large portion of the molecules. The contribution from each molecule is a result of its individual position and momentum at a time. As these can vary widely from ane moment to the next, all properties will in principle fluctuate around their mean values. For macroscopic systems the fluctua-tions are not measurable.
A physical system withj internal degrees of freedom can at every instant be regarded as a point in a 2/-dimensional phase space where the particle positions q=(Ql,Q2, ...,qj) and momentaP=(Pl,P2,...,Pj)constitute the coordinate axes [13, 14]. The physical sys-tem can move through a series of macroscopic states, where each macroscopic state can be realized by a variety of microscopic different arrangements of particle posi-tions and momenta. Each point in phase space is a unique solution of Hamiltons equa-tions. When the system evolves in time, the point representing the state moves in phase space, thus generating a phase trajectory.If the system is at equilibrium or is restricted in any other way, the allowed configurations and momenta are confined to parts of thephase space, and sa is also the trajectory. For an infinite time, the trajec-tory formed will pass through all accessible points in phase space. Knowing the phase trajectory-nearly all information of the system is accessible through relations from classical thermodynamics or statistical mechanics.
[p]
Figure 2.1
Part of a phase space trajectory of a system. q are positions of all' partieies, and p are their conjugated momenta.
[q]
In statistical mechanics the trajectory is represented as ensembles, a number of repli-cas of the system corresponding to the same macroscopic observables, but with dif-ferent microscopic arrangements. If we aim at calculating trajectory averages, we must either have a mechanical model that can be integrated to give the individual mo-tion of the constituent molecules, or we must have some theory that enables us to
cal-20 Chapter2Molecular dynamics
culate trajectory averages directly. The latter is the main subject of statistical me-chanical theories. We relay on the first approach. The mathematical model is pro-vided through equations of motion together with a model for the forces that cause the motion to change. If we can integrate this model, the system can be followed for as lang as we wish.
The equality of results from either of the two approaches is postulated through the ergodic hypothesis, which states t11at given enough time the time average of a property will equal the ensemble average. Then simulations at, say constant en-ergy, number of particles, and volume will provide results corresponding to the microcanonical ensemble.
In a macroscopically sized system, the number of particles is of the order of Avo-gadro's number. To keep track of the individual motions of such an amount of matter is not manageable with the computers of today. The system size must there-fore be reduced drastically. But then one also reduces the dimensionality of the phase space and the number of available configurations that can generate the cho-sen macroscopic state. It is not within reach to follow the system for a macro-scopic portion of time either, and we must be satisfied with a time interval of the order of nanoseconds. We have then arrived at the manageable task of simulating a tiny portion of a (model) system for a limited period of time.
The central questions to ask at this point are, how well does the simulated phase trajectory of the small model system compare to the phase trajectory of the full model system, and how well does the small portion of the calculated trajectory compare to the full trajectory of the small system? And, fin ally, how well does the model system resemble a real system?
The cost to be paid for looking at a small system is that the fluctuations of proper-ties will be large: they decrease as
Nn
with increasing moleeule number N [28].Also the trajectory averages can be systematically displaced. The trajectory being of finite length, cause a danger of not collecting a representative sample of the available microscopic states. Normally, a system size of more than --100 mole-cules is regarded [33] as sufficient for determination of thennodynamic averages and structure. Casulleras and Guardia, 1992 [34] show however that an increase of system size varying from 125 to 512 molecules yield a systematic increase in the self-diffusion coefficient for liquid methanol. They also find the dielectric constant to be dependent of system size. Their study confirms that system size does not in-fluenee significantlyUPOllstructure,
Chapter 2 Molecular dynamics 21 2. 1.2 Equations of motion
From the advent of molecular dynamics simulations in the fifties (Alder and Wain-wright, 1957 [35]), calculations for isolated systems have been the standard ap-proach. Then, to have no mass or energy transfer across system boundaries, mole-cule number N, volume Vand total energy E must be constant. They are the macroscopic fixed quantities defining the state of the system in close relation to the statistical mechanical microcanonical ensemble, or the Nvfi-ensemble. A simulation for which these conditions apply is called an NVE-simulation. If we look at molecules composed of different atoms or group of atoms as constituting a rigid body, the equations of centre-of-mass motion for a classical system of N molecules follow from Newtons 2. law as
and
dr.l _ P,
dI - mi
dPi = F~ntemal
dt l
(2.1)
(2.2) where i=1,2, ..,N.riis centre-of mass position of ith molecule,Pi is linear
momen-f .h . le. and m.i Flntemal. h f f . h . h
turn orzt partic e, an mi1tS masse i 1St e sum o rorces actmg upon t elt
molecule from all molecules j =/::. i. The equations are thus coupled through the forces, and the equations has to be integrated with time t for each molecule simul-taneously. Since the system is isolated, no external forces act upon the system.
For molecules having anisotrope force-fields, one must also integrate the motion with respeet to orientation in space, given by
(2.3) where Liis angular momentum,
;;i
and7
i are the angular velocity of mole-cule i and the total torque on moleeule i from the rest of the moleeules. Since the moment of inertia of a molecule,li relative to a space fixed frame of reference will change as the body rotates, the rotation must be performed relative to axes fixed in the body. Through a standard coordinate transformation [36], Equation (2.3)be-22
comes
Chapter 2 Molecular dynamics
d-;.b
I. ·l ~dt +-;~l x (I. ·l -;~)l = -;.l (2.4) where robis angular velocity relative to the body-fixed axes and li is the inertia tensor of moleeule i. If the body-fixed axes are principal axes, the off-diagonal elements of the inertia tensor vanish, and the differential equations for the compo-nents of the angular velocity are Euler' s equations. The centre of mass of the molecules is taken as the origin of the body axes.
IliåJ~i+(I3i- 12i)w~iOJ~i= 7:li 12iåJ~i+(I1i-13i)wjiw~i= 7:2i
13iåJ~i+(12i- Ili)0J~iw1i== T:3i
(2.5)
where subscripts 1, 2, and 3 refer to the principal axes. The orientation is further described by integrating the Euler angles (<\>, 8,\jf) [36, 37]
e.
l = ros].lcosAl.'Yl+OJs2'lsin,Jo.•'Yl (2.6)where
rofi
are are space fixed angular velocity of molecule i relative the ~-axis.Angular velocities in the space-fixed frame of reference are converted from the body-fixed variables through
--+S _ AT. --+b
Wi - i OJi
whereATis the transpose of the rotation matrix [15]
(2.7)
A= cos
ø
cos1/J - sinrpcosOsin1/J- cos
ø
sin1jJ - sinrpcos ()cos1/J sinøsinOsin rpcos1jJ+cosrpcos ()sin1/J - sinrpsin ()+cos1/Jcos ()cos1/J
- cos ()sin ()
sinO sin1/J sinOcos1jJ
cose
(2.8)
Chapter 2 Molecular dynamics 23
It is also possible to treat the subunits of polyatomic matter as individual objects, each obeying a separate equation of motion. The molecular geometry then enter as constraint forces that keep the moleeule together. This constraint dynamics is par-ticularly useful when the molecular model is flexible [15].
The average energyE of an isolated system of N rigid molecules subject to transla-tions and rotatransla-tions is given by the system Hamiltonian ..H as
(2.9) The first term is the kinetie energy due to the translational motion, the second term is the kinetie energy due to the rotational motion, while U (r) is the potential (configurational) energy in the system. Since the system is isolated, the total en-ergy must be conserved in time according to 1st law of thermodynamics. If the model includes intemal rotation or vibration, these energies must of course be in-cludedin Equation (2.9) to have a constant of the motion.
Raving the mathematical model defined and the internal forces specified, the equations can be solved numerically by some finite-difference scherne, and the system trajectory be calculated for discrete times. A route to solution is described in Chapter 3, while model potentials are described in Chapter 4.