252 Isobaric ensembles
v(d)g =OvgOT, (5.12.17)wherev
(d)
g is a diagonal matrix with the eigenvalues ofvgon the diagonal. The columns
ofOare just the eigenvectors ofvg. Letλα,α= 1, 2 ,3 be the eigenvectors ofvg. Since
vgis symmetric, its eigenvalues are real. In this representation, the three components
ofxiare uncoupled in eqn. (5.12.16) and can be solved independently usingeqn.
(5.12.6). The solution att= ∆tfor each component ofxiis
xi,α(∆t) =xi,α(0)eλα∆t+ ∆tvi,αeλα∆t/^2sinh(λα∆t/2)
λα∆t/ 2. (5.12.18)
Transforming back tori, we find that
ri(∆t) =OTDOri(0) + ∆tOTDO ̃ vi, (5.12.19)where the matricesDandD ̃have the elements
Dαβ= eλα∆tδαβD ̃αβ= eλα∆t/^2 sinh(λα∆t/2)
λα∆t/ 2δαβ. (5.12.20)In a similar manner, eqn. (5.12.14) can be solved forvi(t) and the solution evaluated
att= ∆t/2 with the result
vi(∆t/2) =OT∆Ovi(0) +
∆t
2 miOT∆OF ̃ i, (5.12.21)where the matrices∆and∆ ̃ are given by their elements
∆αβ= e−(λα+bTr[vg])∆t/^2 δαβ∆ ̃αβ= e−(λα+bTr[vg])∆t/^4 sinh[(λα+bTr[vg])∆t/4]
(λα+bTr[vg])∆t/ 4δαβ. (5.12.22)A technical comment is in order at this point. As noted in Section 5.6, ifall nine
elements of the box matrixhare allowed to vary independently, then the simulation
box could execute overall rotational motion, which makes analysis of molecular dy-
namics trajectories difficult. Overall cell rotations can be eliminatedstraightforwardly,
however (Tobiaset al., 1993). One scheme for accomplishing this is to restrict the box
matrix to be upper (or lower) triangular only. Consider, for example, what an upper
triangular box matrix represents. According to eqn. (5.6.2), ifhis upper triangular,
then the vectorahas only one nonzero component, which is itsx-component. Hence,
hlies entirely along thexdirection. Similarly,blies entirely in thex-yplane. Onlyc
has complete freedom. With the base of the box firmly rooted in thex-yplane with
itsavector pinned to thex-axis, overall rotations of the cell are eliminated. The other
option, which is preferable when the system is subject to holonomic constraints, is