Phase switch MC generalised coordinates

We define a set of coordinates that describe each molecule in the system with respect to its own reference configuration. Monte Carlo moves made on the molecule can be written in terms of the reference structure plus this set of generalised coordinates. The generalised coordinates represent the collective Monte Carlo moves that have been applied to each molecule during the simulation.

They are independent of the phase and if applied to another reference state will result in the same configuration as if the original Monte Carlo moves had been directly applied to that reference state. That is, if we run a Monte Carlo simulation of phaseα for some arbitrary number of cycles, we can calculate a set of generalised coordinates that describe the change from the initial configuration to the current configuration, so that if these are then applied to a completely different starting configuration or phase, we would recover the configuration that we would have gotten if the exact same Monte Carlo moves had been made to this new system instead. Having the coordinates independent of each other is important as one can reconstruct the move history associated with each coordinate without having to include information from the move history of other coordinates.

Our system hasN particles in the system and Nmol =N/4 molecules. Each molecule has its own set of generalised coordinates {G}, where the curly braces

denote the set across molecules.

Translations are represented by the change in fractional displacement of the first bead in each molecule.

{∆s}= (h−1{r₁}α)−(h−1{rref₁ }α) (5.2) where{r₁}α_{denotes the set of coordinates of the first bead in a molecule in phaseα,} {rref₁ }α _{denotes the equivalent for the reference coordinates, andh}−1 _{is the inverse} cell matrix of the simulation box. Fractional co-ordinates are used so that the displacement is independent of the simulation box. This is needed because the cell matrix contains the basis vectors for the coordinate system in fractional coordinates. Rigid molecule coordinates are defined as the rotation from the reference configuration to the current configuration; we define a molecule-centred coordinate system for each molecule. The axes are defined asa= (e₁, e₂, e₃) withe₁ = ˆr₁₂, the vector between first and second beads in a molecule. e₂ is defined as the component ofr₂₃ vector that is perpendicular to r₁₂

e₂ =r₂₃−(r₁₂·r₂₃)ˆr₂₃ (5.3) and the third axis is the cross-product of the other two axes, e₃ = e₁ ×e₂. This gives us an orthogonal coordinate systems for each molecule in the system, {a} = {(e1, e2, e3)}.

A set of quaternionsq = (q0, q1, q2, q3) is defined over all molecules as {a}={q}{aref}{q−1}. (5.4) This defines the quaternions needed to transform each molecule from the reference coordinates to the current coordinates.

The internal representation of torsional movement,{θ}, is defined as the change in torsion angle from the reference to the current system about the ˆr₂₃ axis for each molecule. The angle is calculated as

θ= atan2(|[b_i×b_a]×[b_a×b_f],[b_i×b_a]·[b_a×b_f]), (5.5) whereb_a= ˆr₂₃ is the rotation axis,b_i is the initial torsion vector andb_f is the final torsion vector; see Figures 5.3 and 5.4.

Figure 5.3: Depiction of the torsion angle in the butane model, bond length and angle are fixed so the rotation is on a circle with axis ˆr₂₃.

Figure 5.4: Depiction of the torsion angle in the butane model looking top-down onto the ˆr₂₃ axis.

well. This can simply be defined as the change in cell matrix ∆h, given by

∆h=h−href (5.6)

whereh is the current matrix andhref is the reference matrix.

We must label the two systems we are calculating the free energy difference between, byαandγ. We must also differentiate between the system in which we are currently creating the moves and calculating the probability, and the system that they are just being applied to; these will be referred to as the active and passive respectively. To symbolically denote this, a superscript + will be used for active, and a superscriptfor passive. Therefore a vectorbin the active system specifically will be denoted asb+ and the same vector in the passive system will beb.

This leaves us with a set of generalised coordinates

{G}= ({∆s},{q},{θ},{∆H}), (5.7) which are independent of each other and only describe the change in configuration due to the MC moves applied.

5.3.1 Degrees of freedom

All of the degrees of freedom in the system should be accounted for in the internal coordinates. For the butane model at constant pressure, we start with 3N+6 degrees of freedom. Each of theN particles can move in three dimensions, and the 3×3 cell matrix has 6 degrees of freedom, because some of the values are not independent. For a system at constantN and constant pressure P, this can be written as

3NmolNbead+ 6, (5.8)

where Nmol is the number of molecules in the system and Nbead is the number of beads within a molecule so that N = Nmol×Nbead. For butane Nbead = 4. The numbers of degrees of freedom are given in Table 5.1.

Degrees of Freedom

Cell box 6

Translation 3Nmol

Rotation 3Nmol

Torsion (Nbead−3)Nmol

Constrained bond lengths −(Nbead−1)Nmol Constrained bond angles −(Nbead−2)Nmol

Table 5.1: The numbers of degrees of freedom for each type of motion in the system.

These can all be combined to give

(Nbead−1)Nmol+ (Nbead−2)Nmol + (Nbead−3)Nmol+ 3Nmol+ 3Nmol+ 6 = NbeadNmol−Nmol+NbeadNmol−2Nmol +

NbeadNmol−3Nmol+ 3Nmol+ 3Nmol+ 6 = 3NbeadNmol+ 6,

recovering the total number of degrees of freedom. This shows that our generalised coordinates account for all degrees of freedom in our model.