Sapienza, P. le A. Moro 2, , Roma, Italy Published online: 23 Nov 2009.

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1 This article was downloaded by: [University of Tennessee, Knoxville] On: 26 December 2014, At: 22:17 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Molecular Physics: An Internional Journal the Interface Between Chemistry and Physics Publicion details, including instructions for authors and subscription informion: Constrained reaction coordine dynamics for systems with constraints IVAN COLUZZA a, MICHIEL SPRIK b & GIOVANNI CICCOTTI c a AMOLF-FOM, Kruislaan 407, 1098, Amsterdam, SJ, The Netherlands b Department of Chemistry, University of Cambridge, Cambridge, CB2 1EW, UK c Istituto Nazionale di Fisica della Meria and, Dipartimento di Fisica dell'universita' di Roma La Sapienza, P. le A. Moro 2, , Roma, Italy Published online: 23 Nov To cite this article: IVAN COLUZZA, MICHIEL SPRIK & GIOVANNI CICCOTTI (2003) Constrained reaction coordine dynamics for systems with constraints, Molecular Physics: An Internional Journal the Interface Between Chemistry and Physics, 101:18, , DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the informion (the Content ) contained in the publicions on our plform. However, Taylor & Francis, our agents, and our licensors make no representions or warranties whsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publicion are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of informion. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whsoever or howsoever caused arising directly or indirectly in connection with, in relion to or arising out of the use of the Content. This article may be used for research, teaching, and prive study purposes. Any substantial or systemic reproduction, redistribution, reselling, loan, sub-licensing, systemic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found

2 MOLECULAR PHYSICS, 2003, VOL. 101, No. 18, , Taylor & Francis 0 Taylor 6Francir Group Constrained reaction coordine dynamics for systems with constraints IVAN COLUZZA*, MICHIEL SPRIK2 and GIOVANNI CICCOTT13 AMOLF-FOM, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands Department of Chemistry, University of Cambridge, Cambridge CB2 lew, UK 3 Istituto Nazionale di Fisica della Meria and Dipartimento di Fisica dell universita di Roma La Sapienza, P. le A. Moro 2, Roma, Italy (Received 21 April 2003; revised version accepted 30 May 2003) In the context of molecular dynamics simulions of rare events, the applicion of constraints on a suitable reaction coordine has often been found useful for sampling of the free energy barrier. The efficiency of these calculions is hampered by geometrical difficulties, reled to the metric factor and inertial forces. Some years ago Mulders et al. [1996, J. chem. Phys., 104, suggested a way to simplify the approach. Their idea was demonstred shortly afterwards by Sprik and Ciccotti [1998, J. chem. Phys., 109, The present paper extends these results to vector reaction coordine and molecular systems modelled with holonomic constraints. 1. Introduction Sampling of rare events occurring time scales much longer than the durion of a numerical simulion (in real time for molecular dynamics and effective time for Monte Carlo) is one of the major challenges in computer simulion. The prime example of such an infrequent process is an actived chemical reaction in solution. The reactive event must be forced to occur which requires some form of interference with the dynamics of the system. This problem has been the object of intensive research and several approaches are available [I]. Perhaps the least intrusive method is specifying both the initial and final ste in terms of some order parameter and sampling the trajectories connecting them. This is the principle behind the transition ph sampling method developed by the group of David Chandler, who worked out a rigorous scheme to implement this idea for finite temperure systems such as liquids. For a recent review of this see [2]. Because of its basis in fundamental stistical mechanics this approach also brought a number of conceptual advances, such as a demonstrion th a transition ste in fluctuing systems must be regarded as an ensemble of microscopic stes rher than a single point. Reled to this is the issue of the distinction *Author for correspondence. coluzza@amolf.ni between order parameters and the reaction coordine. Order parameters are well-defined structural (or energetic) functions of the configurion, measuring the order of the local minima between which a reaction takes place, whereas the reaction coordine describes the detailed reaction ph and can be a highly compliced function. An alternive approach, taking this distinction into account, uses the dynamics to obtain an estime of the free energy associed with a number of predetermined order parameters. This informion is then used to bias the potential so as to drive the system away from this region in configurion space minimal increase in free energy [3, 41. The requirements for compution of free energies of activion and the reaction free energies are in practice not the same. While for compution of activion energy determinion of the correct transition ste is essential, reaction energies can often be calculed by thermodynamic integrion involving one or more approprie order parameters. If the order parameter also provides a good description of the reaction coordine, the maximum in the free energy profile can be used as an estime for the activion energy. A convenient scheme for carrying out thermodynamic integrion in molecular dynamics simulion is fixing the order parameter by constraint methods and determining the mean force as the average of a mechanical force over an adjusted Molecular Physics ISSN print/issn online Taylor & Francis Ltd DOI: /

3 2886 I. Coluzza et al. ensemble, the so-called blue-moon ensemble. As suggested in [5], this mechanical force is closely reled to the force of constraint. For the case th the reaction coordine is the only constraint imposed on the system, it was demonstred in [6] th the average force of constraint and the mean force differ by a geometrical correction term. In the present paper we return to this problem and consider the general case of a system subject to an arbitrary number of constraints, including, for example, the rigid molecular geometries used for the modelling of complex molecular systems. 2. Mean force and force of constraint Thermodynamic integrion is used for the compution of relive free energies from the derivives of free energy with respect to some control parameter. Applied to the compution of free energies along a reaction ph the integrion variable is the reaction coordine 6 defining the reaction ph. This integrion variable, however, is not an independent parameter in the Hamiltonian but a function (possibly a vector function) < = <(r) of the configurional degrees of freedom r. For a given value < = 6 it can be given a probabilistic interpretion to the free energy W(q) = -ks T In P& ) + k~ T In P,, where P, is a suitable normalizion constant and Pg is the probability density given by (1) W(< ) can be interpreted as the reversible work needed to bring the system from some reference ste to 6. By thermodynamic differentiion, we see th W(6 ) is given by where (...I<) <(r) = ( : is the conditional average evalued If < is a low probability value for the reaction coordine, any direct sampling scheme like Monte Carlo (MC) or molecular dynamics (MD) fails because the spontaneous occurrence of these stes is a rare event in the simulion. In 1989, Carter et a]. [8] introduced the blue-moon ensemble method. In th work the (4) conditional average, equion (3), was estimed by a time average over a constrained trajectory with the reaction coordine fixed a specified value, <(r) = 6. From [8] we obtain the relion between the conditional and the constrained averages of an observable U(r) where (...)er is the average over the ensemble in which the reaction coordine is constrained. The factor D is det 2 D=det Z where det 2 is the determinant of the submrix of the inverse metric tensor (M- ), corresponding to the constrained variables present in the system (i.e. both the molecular model and the reaction coordines), while det Z is the determinant of the corresponding M- submrix of the molecular constraints alone. Applying the blue-moon method to the mean force, equion (3), one obtains the following expression for the constrained average of a H/a< ($IF.) ( D- /2 [ ( au/ a<) - $(( a In det M/ 36) +(a In det Z/ a<))]) - < (0 /2)< This expression, in addition to the potential term, contains two inertial terms. The first is the logarithmic derivive of the determinant of the metric tensor M, defined as where p is the mass tensor, with the particle masses mi as diagonal elements and zero off-diagonal elements, and J is the Jacobian of the transformion from the Cartesian coordines to a full set of generalized coordines, including the expressions of the molecular constraints, u, and of the reaction coordine, 6, as variables. The second is the logarithmic derivive of the det 2. Equion (7) has some unpleasant feures. First, we must introduce a complete set of new independent variables in order to calcule the partial derivives. Second, the calculion of the logarithmic terms requires the full knowledge of the Jacobian mrix. In this work we derive a new expression for this formula which overcomes these problems, without loss (7)

4 Constrained reaction coordine dj mamics for systems with constraints 2887 of generality. In doing this we extend a result of Sprik and Ciccotti [6], restricted to systems without molecular constraints and with the scalar reaction coordine. The main idea is the use of the Hamiltonian equion of motion for the coordines $ [5], (9) to substitute the conditional average of a'h/a$ with the constrained average of the Lagrange multipliers At. This gives the expression for the mean force as x (a In det 2)/ikj - (a In det Z)/ikj ( a=o, g with YT and W defined by which contains only explicit quantities since all the derivives are now made with respect to the Cartesian coordines (e.g. Yactd = xzl (l/mi)(aa,/ar,)(acd/ari)). The problem of systems with multiple constraints has already been studied in a different approach by den Otter and Briels [8-lo], while the simpler case of no other constraints but a vectorial reaction coordine has been derived by Sergi et al. [19]. The structure of the paper is as follows: in section 3 we write down the formalism and derive the main result equion (10). Section 4 contains some concluding remarks. 3. Formalism and results In order to derive our formula we will go through four steps corresponding to four subsections. In the first step (section 3.1) we will introduce the formalism and some useful properties of the mechanical and stistical quantities. In section 3.2 we define the ensemble constrained on the and a coordines, and the ensemble conditioned on the reaction coordines and constrained on the molecular constraints. In section 3.3 we derive equion (7) by integring over the momenta the microscopic components of the force along the reaction coordines in the two ensembles, and by correcting for the difference between them. In section 3.4 we derive equion (10) by substituting the total force along the reaction coordines with the corresponding Lagrange multiplier Notions and definitions The system we consider consists of N interacting particles subjected to Nu mechanical constraints described by the following function of the Cartesian coordines ano = (a,,..., UN,). The system is described by the Cartesian coordines rn = (rl,..., r N) and a Lagrangian of standard form L(rN,P) = ~(r"') - V(rN), (12) where K is the kinetic energy and V is the potential energy function. The reaction coordines tnt = (41,..., e ~ are ~ functions ) of the Cartesian coordines c = c(rn). The 6 and a functions, for given values and a*, define the (3N- Nt - N,)-dimensional hypersurface, ((.") = r, a(rn) = a*, in the configurional space. Supplementing $ and a with a set of independent coordines q = qu, (Y = 1,..., 3N - No - Nt, we obtain a set of generalized coordines in one-to-one correspondence with the Cartesian coordines N -24s {q,a,c} = ( 4J) 9 (13) where t = {a, $}. The Lagrangian L' of the system in the u coordines is 1 L'(u, U) = -UTMli 2 - V(U). (14) In equion (14) M is the mass-weighted metric tensor with elements: The elements of M and M-' can be grouped in different block forms as I I1 111

5 2888 I. Coluzza et al. Second, considering representions 11, 111, IV of the mrices M and M-', the Fixman theorem [12] reads, det A = det Zdet M, det 2 = det Ldet M, det k' = det Zdet M. (26) (27) (28) 111 (174 Alternively reordering the vector u as ii = { (q,(), a}, we can have a new represention of the metric tensor as IV To discuss the stistical properties of our system in the two given representions, we change to phase space and use the Hamiltonian language. The Hamiltonian in the (rn, pn) represention is written as while defining we can build the Hamiltonian H' coordines as V in the (u,p") The mrix M and its submrices have some useful properties. First, imposing M-'M = 1 within the second represention (11) of M in equion (16 b) we obtain: Equion (26) is demonstred (the others are obtained proceeding along similar lines) by writing Taking the determinant we get det A = det 2 det M. (29) Third, we want a factorizion for det2 and explicit expressions for W-' and YT. Considering the mrix 2, equion (1 7), with and applying the Fixman theorem to the determinant of 2 we have det 2 = det Z det W-', while we get W-' by multiplying 2-' with 2, >- WL= 1 XZ+ YZT= 1 XI+ YL=O ytz+ wt=o Y~I+ Taking the bottom elements in equion (34), w-' = L + w-' YTI, w-1 yt = -1Tz-l. Substituting W-' YT from equion (36) equion (35) we have (YT comes from equion (36), where we substitute back W-')

6 Constrained reaction coordine dynamics for systems with constraints 2889 Finally, combining equion (32) with equion (37) we find for det 2 the expression det 2 = det Zdet W-' = det 2detL det (1 - ITZ-'IL-'), (38) which factorizes the determinant in a way useful for future simplificions Construction of the ensembles As already mentioned in the introduction, in this subsection we will give the expressions of the density probabilities for various ensembles appearing in our derivion. First we need the expressions for the Hamiltonians belonging to the various situions. Using the formalism just introduced we have th the first ensemble (say the M-R ensemble) is obtained by imposing the constraint conditions u = a = 6-6' = 4 = 0. The system of equion (21) becomes - pq = Aq; p' = 2Tq, or, substituting q = k 'pq in the second equion, p' = 2'TA"-Ip4. (39) Here and in the following means th the indiced quantity is computed with the constraint conditions, u = 6 == < - <' = 6 = 0, sisfied. Equion (39) gives the a and c components of the momenta when the constraints are applied. The constrained Hamiltonian since p;m-lp, = pqta-'pq as explained in Appendix. For the system where only the u constrained (ensemble M) the condition for momenta p"" is (40) the are the The Hamiltonians from equions (40) and (42) yield the probability density of the two systems, p,(rn,pn) and po(rn,pn), respectively Pr(rN,PN)drN dpn N N = p(r,p )6(u)B(t- c1)6(p' - p') drn dpn, (43) N N,P )drndpn = p(r N,p N )6(a)6(p' -jt')drn dpn, (44) where p(rn, pn) is the probability density of the system without constraints. In order to integre over the momenta of the two ensembles, it is convenient to factorize the probability densities in the reduced configurional term (function of the particle coordines rn) and the (conditional) momentum probability density defined as (45) In the next paragraphs we will derive these terms in the two ensembles. In the following we will need the density probability, pu(rn,pnit), conditioned to 6 = $ which is defined as Ensemble constrained on ( and B (M-R) Following [13] we first look the reduced term, given by Calculing the Gaussian integral over the momentum we obtain (det Z)"'6(c)B(t - 6') exp [-pv(rn)] drn p:(p) dp = f drn(det 2)'/*8(u)6(5 - P) exp (-pv) ' (41) Next we get the conditional momentum space probability density p:(pnlrn) as the rio between the total probability density pr and the reduced term pi(rn) as 1 = -pqct2'-1(q, ff = 0, 4)pqC + V'(q, ff = 0). 2 (42) exp (-ph)6(pr - F) =C (det 2)'/' exp(-pv) 9 (49)

7 2890 I. Coluzza et al. where C is the rio between the normalizion factor of p:(p) and of pr(p,pn) as derived in equion (40) in [6]: where the conditional momentum space integral is given by Inserting C in equion (49) we have the fully explicit expression for the conditional momentum probability density p: (pn I rn) : (3N-Nr-N~)/2 (det p det 2)-'12 x exp(-bk)8(pr -p). (51) Ensemble constrained only on a (M) We now look the system constrained over the coordines a and conditioned by 6. Again we factorize the distribution pu(p,pni() in the reduced term p;(rnl(), analogous to p:(rn), p'(pi') (det z)'/28(a)8(( -(') exp(-bv) = JdrN(det Z)'/28(a)8(( - p) exp(-bv) ' (52) and in the conditional momentum space probability density pc,(pnjp, e), analogous to p&(pnlrn), x exp(-bk)g(p" -p"). (53) 3.3. Mean force In this subsection we start integring some auxiliary quantities over the momentum space in the ensembles M-R and M. Then we apply the stic blue-moon theorem to compute the mean force (defined in the ensemble M) using the ensemble M-R Integrion in the ensemble M-R According to the factorizion of the probability density equion (49, the average (D-'/2(aK/))r in the M-R ensemble is = (;)(3N-*.-Nt)/2 (det M det 2)-'/2 ak x /dp" exp(-j3k)-8(pr -p'), (55) and we have substituted the conditional probability density p:(pnlrn) with its expression from equion (51). Using the Fixman theorem equion (26) we can substitute det A to the product det M det 2 (3N- No - Nt)/2 (det A)-'/2 x /exp(-/3k)-8(pr ak -p')dpu x 2 / dpq exp [- (pia-'pq)] 1,2 a(det A) 'j2 = --(det A)- B a In (det A)'/2 = -kbt Reapplying Fixman's theorem in reverse order, we can write the full constrained average as ). (57) Adding the potential contribution to the mean force in the constrained ensemble we get ( D-l/2!?!)r=b-l/2 (?+Z)) + alndetm alndet Integrion in the ensemble M Using the factorizion of the probability density pu(rn,pn), the average of the generalized force over the (-conditioned ensemble is given by +

8 + Constrained reaction coordine dynamics for systems with constraints Making the explicit calculion of the integral over the momentum Rearranging the terms we obtain (216) = U (D-'/2)r x /dpn exp(-pk)h('f ak -p")- 1 falndetz alndetm\ we have the correct expression for the mean force, ((ax/)l6), as a function only of the conformional coordines: - av 1 alndetm alndetz (a Applicion of the stic blue-moon theorem From the blue-moon theorem [7], (61) we get the unbiasing factor, D = det Z/det Z, which converts a configurional average from the M to the M- R ensemble. Applying it to equion (61) we obtain the right-hand side of equion (61) as - + (D-l/2)r )I)'. (63) Comparing the previous equion with equion (58) we can write which gives a slight generalizion of the blue-moon theorem Final result The results th we have obtained until now are exact but need more work to be of practical use since we still have to evalue the partial derivives of the rs with respect to the vector t, which requires the change of coordines rn + u. This is precisely wh we want to avoid for a high-dimensional system. Following [5] we replace the generalized force a'h/ using the Hamiltonian equion of motion as ($16) where h, is the Lagrange multiplier associed with the constraint t. Considering the term D-1/2pr we have 1 (D-'/2[(d In D)/dtlps), (D-l/2)7 The first term in the central equion is zero because it is the average of a total time derivive of a function with odd time parity. Applying the chain rule, we can transform the time derivive into a quadric form involving the momenta ar =1.;16) U - ( D-'/2)7 (64)

9 2892 I. Coluzza et al. which has to be averaged over the constrained ensemble. Using the well-known properties of the multidimensional Gaussian integrals we can calcule the conditional momentum space integral of the product puprt. Remembering the second represention of the mrix M (16 b ) and equion (25) we have (pqp'. lrn) = (pqpqt Ir") A-l B' = kb TB', r r (prpttirn) T = ~ '~~-'(pqpq~~r") r A-'H =~BTB'~A-'B'=-~BTZ -I F IT B', (70) which, multiplying by M-', becomes From the condition M-'M = 1, we have FlTB' + 2A' = 1, EB' + F'A' = 0. Multiplying equion (72) by -F'Z-l we get (72) (73) -F'z-'F'TB' - F'A' = -&"2-', (74) F'A' = -EB'. (75) Substituting the second in the first equion we have Inserting the previous equion in equion (69) and recalling th the elements of F' are Fid = CEl(l/mJ (aqc/3ri>(d/&,), we obtain -prt d In D dt = -(T) aind F'2-I Let us now bring T/arj and 2-1 inside the square brackets d In D N N N,+Nt a In D -ptt =-kbtrr c - dt i=l j=1 /=I &j By definition of the submrix 2 the term inside the round brackets is the identity mrix in the subspace of the ts, so writing equion (78) as a column vector we get d p r dt 1 aind a In D +kbtf. = -kb T2-l cc,., i=l j=l (79) Substituting equion (79) in equion (67) we get the final expression for the mean force on the t variables - where p and v are (Dqr a In det 2 a In det Z v=kgt - kbt (82) The vector v compenses the terms in the round brackets in the right-hand side of equion (go), leaving us with the Lagrange multiplier Ar and p where

10 Constrained reaction coordine dj inamics for systems with constraints 2893 The objective of our work is to calcule the generalized mean force relive to the vector <. To do th let us go back to the represention of the mrix 2-' in the following form The generalized mean force relive to 6 is then (8 lntt 2 - i3 In det Z )I): (86) ki Using for W and YT the expressions in equion (37) and equion (38), we can finally write the final form of the mean force Equion (87) is the final result of our work. It extends the Sprik-Ciccotti equion of [6] to a system with mechanical constraints. Since equion (87) depends only on explicit variables, it is possible to evalue the mean force on the M-R ensemble without introducing a non-physical change of the variables. 4. Discussion The work of Carter et al. [7] introduced the bluemoon ensemble in the molecular dynamics simulions. The first applicion of the blue-moon ensemble, in the form of equion (7), was the calculion of the conditional average of the mean force associed with a simple reaction coordine. Equion (7) can be very difficult to evalue. First, we must introduce a complete set of generalized variables to give an explicit expression to the partial derivives appearing there. Second, the calculion of the logarithmic term requires the full knowledge of the Jacobian mrix. We have derived a general solution to those problems, following the idea of Mulders et al. [5], as confirmed in the restricted derivion given by Sprik and Ciccotti [6]. The main result, given in equion (87), shows th the main advantage of this approach is obtained the price of substituting the Lagrange multipliers into the derivive of the potential with respect to the reaction coordines: all the correcting inertial terms, appearing in equion (7), are then substituted with terms easy to compute explicitly without defining new generalized variables. In spite of this, the presence of the inverse of the Z mrix in equion (87) can leave us with a numerical problem whenever the number of connected molecular constraints is high, since the mrix inversion requires a number of operions th grows with N3. We would like to thank A. Alavi, R. Kapral and D. Frenkel for the useful discussions. G. Ciccotti is greful to Daan Frenkel for kindly providing Spinoza support for a visit to AMOLF. Appendix In this appendix we will demonstre th, when the constraints D = C+ = 0 are sisfied, This equion is very important to define the Hamiltonian of the constrained system and consequently to evalue the Gaussian integrals in sections 3.2 and 3.3. The demonstrion is based on the relion between the submrices of the metric tensor M. The explicit form of the product ptm-'p, is T IT ptm-'pu =PiEpq +PT Pq +PiF$, +PT2pT. (A2) Remembering equion (39) we can express equion (A2) only in terms of the momenta pq p;f~-~p, =pq T EP, + p;~-l~'~ftpq +pq TF tb f ~ -'p, + p: A-I B'2B'A- Pq. ('43) At this point we need the relions between the submrices of M as obtained by imposing MM-' = 1 to the second represention of the metric tensor (equion (1 6 b ) and equion (1 7 b )) AE+BfFfT= 1 AFt+Bf2=0 MM-1 z l?"e+afft=o BtTFt+A2= 1 Combining the submrices of the first line with the relion in equion (25) we find E - Ff2-'F'T = A-'. ('45)

11 2894 I. Coluzza et al. Substituting equion (25) and (A5) in the expanded kinetic energy equion (A3) we get =p:(e - F'2-'PlT)p, =p:a-'p,. References [l] FRENKEL, D., and SMIT, B., 2002, Understanding Molecular Simulions (London: Academic Press). [2] BOLHUIS, P. G., CHANDLER, D., DELLAGO, C., and GEISSLER, P. L., 2002, Annu. Rev. phys. Chem., 53, 291. [3] WANG, F., and LANDAU, D. P., 2001, Phys. Rev. E, 64, [4] LAIO, A., and PARRINELLO, M., 2002, Proc. Nl. Acad. Sci. (USA), 20, [5] MULDERS, T., KRUGER, P., SWEGAT, W., and SCHLITTER, J., 1996, J. chem. Phys., 104, [6] SPRIK, M., and CICCOTTI, G., 1998, J. chem. Phys., 109, [7] CARTER, E. A,, CICCOTTI, G., HYNES, J. T., and KAPRAL, R., 1989, Chem. Phys. Lett., 156, 472. [8] DEN OTTER, W. K., and BRIELS, W. J., 2000, Molec. Phys., 98, 773. [9] DEN OTTER, W. K., and BRIELS, W. J., 1998, J. chem. Phys., 109, [lo] DEN OTTER, W. K., and BRIELS, W. J., 2000, J. chem. Phys., 112, [I 11 SERGI, A,, CICCOTTI, G., FALCONI, M., DESIDERI, A., and FERRARIO, M., 2002, J. chem. Phys., 116, [12] FIXMAN, M., 1974, Proc. Nl. Acad. Sci. (USA), 71, [I31 CICCOTTI, G., and FERRARIO, M., 1998, Classical and Quantum Dynamics in Condensed Phase Simulions, edited by B. J. Berne, G. Ciccotti and D. F. Coker (Singapore: World Scientific Publishing), Chap. 7.

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