Molecular Dynamics Integration Time Step Dependence of the Split Integration Symplectic Method on System Density

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1 1922 J. Chem. Inf. Comput. Sc. 2003, 43, Molecular Dynamcs Integraton Tme Step Dependence of the Splt Integraton Symplectc Method on System Densty Dušana Janežč* and Matej Praprotn Natonal Insttute of Chemstry, Hajdrhova 19, 1000 Ljubljana, Slovena Receved July 17, 2003 Ths paper shows that the maxmal sze of the ntegraton tme step of the Splt Integraton Symplectc Method (SISM) for molecular dynamcs (MD) ntegraton, a combnaton of the analytcal soluton of the hgh-frequency harmonc part of the Hamltonan and the numercal soluton of the low-frequency remanng part, depends on the system densty. Ths approach was tested on a system of lnear chan molecules. The numercal results ndcate that the ntegraton tme step used by the SISM s lmted by atoms moton generated by the electrostatc and Lennard-Jones nteractons n the system. As the densty of the system ncreases, the sze of the ntegraton tme step allowed by the SISM thus becomes smaller but remans sgnfcantly larger than possble by standard methods of the same order and complexty. 1. INTRODUCTION Molecular dynamcs (MD) smulaton s a wdely used computatonal method for smulatng the physcal propertes of complex molecular systems. 1 For each atom of the system, the Hamlton equatons of moton are ntegrated to compute the partcles trajectores n phase space, whch s composed of the coordnates and momenta of all the partcles. To perform an MD smulaton of a system wth a fnte number of degrees of freedom, the Hamlton equatons of moton dp dt )- H q, are solved, where H s the Hamltonan, q and p are the coordnate and momentum, respectvely, and d s the number of degrees of freedom. The MD smulaton thus provdes an nsght nto the mcroscopc behavor of complex molecular systems. These systems, whch are studed by MD smulaton, are n the class of classcal many-body problems. The analytcal soluton usually does not exst for these dynamcal systems, and a number of MD algorthms for fndng the numercal soluton have been proposed. 2-7 A major obstacle n the development of effcent algorthms to propagate numercal trajectores of complex molecular systems s that these Hamltonan systems consst of both fast and slow degrees of freedom. Because the fastest degrees of freedom n the system lmt the sze of the allowed ntegraton tme step n standard methods, studyng processes that are several orders of magntude longer than the fastest moton n the system s dffcult despte growng CPU power. Therefore, MD smulaton algorthms that allow the use of larger ntegraton tme steps are requred. Due to ther property of conservng a system s energy, the symmetrc and symplectc ntegrators, whch are also tme reversble, are a sutable choce n achevng ths goal. 8 A new effcent symplectc ntegraton algorthm for MD smulatons of solated lnear molecules usng the splttng * Correspondng author e-mal: dusa@cmm..s. dq dt ) H p ) 1... d (1) of the total Hamltonan nto the hgh-frequency harmonc and low-frequency remanng part was frst ntroduced n ref 9. It was assumed there that bond stretchng satsfactorly descrbes all vbratonal motons of a molecule. The vbratonal moton, whch was descrbed by the hgh-frequency harmonc part of the Hamltonan, was solved analytcally wth the ad of the normal coordnates. In ref 10 the method devsed n ref 9 was extended to also treat systems of lnear molecules. The generalzaton of ths symplectc method was frst ntroduced n ref 11. The crucal dfference from the approach descrbed n refs 9 and 10 was that normal modes wth frequency zero were used for descrbng the rotatonal and translatonal moton of molecules. Ths approach was also extended to a system of lnear molecules for whch t can be assumed that both bond stretchng and angle bendng satsfactorly descrbe all vbratonal moton of molecules. The performance and the propertes of the method, such as the maxmal allowed ntegraton tme step, have not yet been fully explored. In ths artcle, whch bulds upon ref 11, a study of ntegraton tme step dependence on the system densty for SISM 11,12 for MD ntegraton, whch combnes the analytcal soluton of the hgh-frequency harmonc part of the Hamltonan and the numercal soluton of the remanng part, s presented. Because the hgh-frequency degrees of freedom are treated analytcally,.e., ndependently of the sze of the ntegraton tme step, the SISM can employ a sgnfcantly larger ntegraton tme step sze than can be used by the standard leapfrog Verlet (LFV) method METHODOLOGY The SISM, whch s schematcally presented n Fgure 1, s derved n terms of the Le algebrac language. The Hamlton equatons 1 can be wrtten n the form dx dt ) {x, H} ) Lˆ H x (2) where Lˆ H s the Posson bracet operator and x ) (q,p) sa /c034145b CCC: $ Amercan Chemcal Socety Publshed on Web 10/09/2003

2 SPLIT INTEGRATION SYMPLECTIC METHOD J. Chem. Inf. Comput. Sc., Vol. 43, No. 6, matrx of the bond stretchng and angle bendng part of the potental functon. 16 N s the number of atoms n each molecule. Analytcal soluton [propagaton by exp( t/2lˆ H0 )]: the normal coordnates at the begnnng of the ntegraton step, P 0, Q 0, are rotated n phase space by the correspondng vbratonal frequency ω for t/2: [ P Q ] [ ) R P 0 (6) Q 0] R ) [ cos(ω t/2) -ω sn(ω t/2) (1/ω )sn(ω t/2) cos(ω t/2) ] (7) where ω * 0 defnes the vbratons of atoms n each molecule and ω ) 0 defnes translatons and rotatons of molecules. The normal coordnates of the normal modes wth frequency zero (lm xf0 sn x/x ) 1 for ω ) 0) evolve as Fgure 1. Soluton scheme for SISM. vector of the coordnates and momenta of all the partcles. The formula x t+ t ) exp( tlˆ H)x t (3) s the formal soluton of the Hamltonan system (2) n terms of Le operators and represents the exact tme evoluton of a trajectory n phase space from t to t+ t, where t s the ntegraton tme step. 14 Symplectc ntegraton replaces exp- ( tlˆ H) by a product of symplectc maps that approxmate exp( tlˆ H) to a gven order. 8 We splt the Hamltonan H nto two parts 9 H ) H 0 + H r (4) where H 0 s the pure harmonc part and H r s the remanng part of the total Hamltonan. Then the followng second-order approxmaton, nown as the generalzed leapfrog scheme, 15 can be used n (3) x t+ t exp ( t 2 Lˆ H 0) exp( tlˆ H r ) exp ( t 2 Lˆ H 0) x t (5) Ths descrbes how to propagate from one pont n phase space to another. Frst, the system s propagated for a half ntegraton tme step by H 0, then for a whole step by H r, and fnally for another half step by H 0. The whole ntegraton tme step thus combnes the analytcal evoluton of H 0 wth a correcton arsng from the H r performed by numercal ntegraton. Ths ntegraton scheme was used as the bass for the development of the SISM whch reads for each molecule n the system as follows: Preparatory step: at the outset of calculaton, vbratonal frequences and normal modes of H 0, represented by the normal coordnates P, Q ( ) 1,..., 3N), are determned. The ntal normal coordnates are obtaned from the ntal atoms veloctes and the ntal dsplacements of the atoms from ther equlbrum postons by means of the transformatonal matrx A. The columns of A are the egenvectors of the Hessan, the root-mass-weghted second dervatve P ) P 0 Q ) P 0 t 2 + Q 0 (8) (9) Coordnate transformaton: the normal coordnates P, Q are transformed to the Cartesan dsplacement coordnates p, q (m are the atoms' masses): p ) m A P (10) q ) 1 A Q (11) m Numercal soluton [evoluton by exp( tlˆ Hr )]: momenta n the Cartesan coordnates are numercally ntegrated p ) p - t( H r q ) (12) q ) q + t( H r p ) ) q (13) Only one force calculaton per ntegraton step must be performed. Snce H r ) H r (q) and ( H r / p) ) 0, only momenta change n ths part. Bac-transformaton: the Cartesan dsplacement coordnates p, q are bac-transformed to the normal coordnates P, Q 1 P ) A T p (14) m Q ) m A T q (15) Analytcal soluton [propagaton by exp( t/2lˆ H0 )]: the normal coordnates are agan rotated n phase space for t/ 2:

3 1924 J. Chem. Inf. Comput. Sc., Vol. 43, No. 6, 2003 JANEŽIČ AND PRAPROTNIK [ P Q ] ) R [ P Q ] (16) Ths concludes one full SISM ntegraton step, whch s repeated untl the desred number of ntegraton steps s reached. In ths study we used a model MD Hamltonan 8 H ) + b (b - b 0 ) 2 + ϑ (ϑ - ϑ 0 ) 2 + 2m p 2 bonds angles e e j + 4ɛ >j r j >j j[( σ - r j)12 ( σ r j)6] (17) where and j run over all atoms, m s the mass of the th atom, p s the lnear momentum of the th atom, b 0 and ϑ 0 are reference values for bond lengths and angles, respectvely, b and ϑ are correspondng force constants, e denotes the charge on the th atom, r j s the dstance between atoms and j, and ɛ j and σ j are the correspondng constants of the Lennard-Jones potental. The netc energy and the harmonc part of the bond stretchng and angle bendng potental energy of the system n terms of the Cartesan dsplacement coordnates are ncluded n H 0, whle H r s the remanng part of the potental. H 0 descrbes the vbratonal moton of the system as well as the translaton and rotaton of molecules. The moton governed by H 0 s resolved by dagonalzng the Hessan, whch s the root-mass-weghted second dervatve matrx of the bond stretchng and angle bendng part of the potental functon, to obtan the vbratonal frequences and normal mode vectors of H The Hessan depends only on the constant parameters of the smulaton. Ths allows the calculaton of the vbratonal frequences and normal mode vectors of H 0 to be performed only once, at the begnnng of the calculaton. Our approach s dfferent from other splttng methods, e.g., Verlet-I/r-RESPA, 7,15 a multple tme step numercal ntegraton method, n that t analytcally treats hgh-frequency motons usng normal modes. 17 Therefore, the SISM allows a sgnfcantly larger ntegraton tme step to be used n comparson to standard MD ntegraton methods of the same order and complexty. 3. NUMERICAL RESULTS The SISM descrbed here was evaluated on a model system of 256 lnear butadyne molecules of the form H-(-C C- ) 2 -H. All calculatons were performed at three dfferent denstes of the system: F 1 ) g/cm 3, F 2 ) g/cm 3, and F 3 ) g/cm 3. The denstes F 1 and F 2 correspond to the gas phase, and F 3 corresponds to the expermental densty of the lqud at 300 K. 18 The correspondng szes of the smulaton box were 1284 Å, 277 Å, and 31 Å, respectvely. The ntal postons and veloctes of the system atoms were chosen at random. Then the system was equlbrated for 50 ps durng whch the veloctes were scaled, followed by an addtonal 200 ps of the mcrocanoncal equlbraton run to ensure that the veloctes at a temperature of 300 K assume a Maxwell dstrbuton. To obtan physcally and numercally relevant ntal condtons to perform the MD smulaton of a system of Fgure 2. Varaton of Vellard-Baron rotatonal order parameter for lnear molecules durng the mcrocanoncal equlbraton phase of a molecular dynamcs smulaton of the system of 256 butadyne molecules at the system densty of F 3 ) g/cm 3. lnear molecules, the equlbraton was also montored usng the Vellard-Baron rotatonal order parameter for lnear molecules 19 defned as O ) 1 cosγ (18) n )1 where γ s the angle between the current drecton of the molecular axs of the th molecule and the orgnal drecton at the begnnng of the 200 ps long mcrocanoncal equlbraton, and n s the number of molecules n the system. Fgure 2 shows that the order parameter drops from the ntal value of one to zero durng the equlbraton of a system of butadyne molecules at the system densty F 3 ) g/cm 3. Ths ndcates that the molecules are completely rotatonally dsordered and that the lqud state has been reached at equlbrum. The ntal condtons obtaned n ths way were used n all of our computatons. Perodc boundary condtons were mposed to overcome the problem of surface effects; the mnmum mage conventon was used. 1 The Coulomb nteractons were truncated usng force-shfted potental wth a cutoff dstance of r off ) 8.5 Å. 20 The Lennard-Jones nteractons were shfted by addng the term C j r j6 + D j to the potental, where C j and D j were chosen such that the potental and force are zero at r j ) r off. 21 Potental parameters were the same as those taen n ref 11. To demonstrate the effectveness of the SISM, we compared the computatonal performances for the same level of accuracy wth the standard second-order LFV algorthm. 13 Frst, the CPU tmes for the two methods (the SISM and LFV) for 1000 MD steps measured on an AMD Athlon XP processor 22 for dfferent system szes (n) and equal tme steps (1 fs) are gven n Table 1. The results show that the computaton cost per ntegraton step s slghtly larger for the SISM than for the LFV. For larger system szes the cost of computaton per ntegraton step becomes approxmately the same for both methods because the computaton of long-range forces, whch s the same for both methods, prevals over the computaton of extra transformatons requred by the SISM. Therefore the speed up of the n

4 SPLIT INTEGRATION SYMPLECTIC METHOD J. Chem. Inf. Comput. Sc., Vol. 43, No. 6, Table 1: CPU Tme [s] for SISM and LFV for 1000 MD Steps Measured on an AMD Athlon XP Processor for Dfferent System Szes (n) and Equal Integraton Tme Steps (1 fs) n t(sism)[s] t(lfv)[s] t(sism)/t(lfv) Fgure 4. Error n total energy of the system of 256 molecules of butadyne for three dfferent denstes of the system: F 1 ) g/cm 3, F 2 ) g/cm 3, and F 3 ) g/cm 3 usng SISM and LFV for M ) Fgure 3. Dsplacements of atoms from equlbrum postons for an solated molecule of butadyne after 1 ps of smulaton at T ) 300 K computed by SISM and LFV wth dfferent ntegraton tme step szes (0.1 fs, 1.0 fs, and 10.0 fs). SISM over the LFV s manly due to the larger ntegraton tme step allowed by the SISM. Second, the dsplacements of atoms from equlbrum postons were computed for an solated butadyne molecule n the z drecton of the Cartesan coordnate system after a 1 ps long smulaton. The atomc dsplacements were computed by the SISM and LFV wth dfferent szes of the ntegraton tme step (0.1 fs, 1.0 fs, and 10.0 fs). These atomc dsplacements are presented n Fgure 3. The dsplacements computed by the SISM wth a 0.1 fs ntegraton tme step were used as the reference n the comparson wth the dsplacements computed wth dfferent methods and/or szes of the ntegraton tme step. Fgure 3 shows that the dsplacements computed usng the SISM agree for all ntegraton tme step szes. The dsplacements computed by the LFV method, on the contrary, dffer from the reference values even n the case of a 1.0 fs ntegraton tme step, especally for hydrogen atoms. Ths means that the SISM, owng to the analytcal treatment of hgh-frequency vbratons, can correctly descrbe the atomc dsplacements usng a large ntegraton tme step, as opposed to the LFV method. Thrd, the error n total energy, E/E, defned as E E ) 1 M )1 M E 0 - E E 0 (19) where E 0 s the ntal energy, E s the total energy of the system at the ntegraton step, and M s the total number of ntegraton steps, was also montored for both methods. Fgure 4 shows the error n total energy of the model system of butadyne molecules for three dfferent system denstes (F 1 ) g/cm 3, F 2 ) g/cm 3, and F 3 ) g/cm 3 ) usng the LFV and SISM algorthms for M ) It can be observed that for the same level of accuracy, the SISM allows the use of an up to an order of magntude larger ntegraton tme step than the LFV. The maxmal ntegraton tme step allowed s only 0.5 fs for all three denstes F 1, F 2, and F 3 n the case of the LFV, whle t s 4.5 fs for F 1 and F 2 and 3.0 fs for F 3 n the case of the SISM. Wth the growng densty of the system the maxmal allowed ntegraton tme step by the SISM becomes shorter due to the Lennard-Jones and electrostatc nteractons whch are treated numercally, equally, n both methods. The Lennard-Jones and electrostatc nteractons represent the external drvng forces on the nternal moton of the molecules. The effect of the Lennard-Jones and electrostatc nteractons s reflected n the forced oscllatons and ncreased anharmoncty n the case of the denser system, whch leads to a shortenng of the ntegraton tme step. Fourth, to determne the dependence of the qualty of the harmonc approxmaton for the bond stretchng and angle bendng potental on the system densty, we ntroduced a measure for the anharmoncty n the bondng potental energy, defned as M A n ) 1 M )1 b (b - b 0 ) 2 + ϑ (ϑ - ϑ 0 ) bonds angles 2-1 n 3N 2 ω 2 Q 2 j j)1 )1 1 n 3N )1 2 j)1 ω 2 Q j 2 (20) Here, M s the total number of ntegraton steps, n s the number of molecules, and N s the number of atoms n each molecule. The numerator n (20) represents the dfference at the ntegraton tme step between the whole bondng potental energy and the pure harmonc potental energy as expressed by the normal coordnates. The denomnator n (20) s the pure harmonc potental energy. The results presented n Fgure 5 reveal that the anharmonc part of the bondng potental energy depends only on the densty of the system and not on the sze of the ntegraton tme step. From the results n Fgure 5 t can also be

5 1926 J. Chem. Inf. Comput. Sc., Vol. 43, No. 6, 2003 JANEŽIČ AND PRAPROTNIK However, as the densty of the system ncreases, the magntude of the anharmonc forces actng on the system atoms also ncreases, and therefore the sze of the ntegraton tme step allowed by the SISM becomes smaller but remans sgnfcantly larger than possble by the LFV. Fgure 5. Amount of anharmoncty n the bondng potental energy (A n ) for the system of 256 molecules of butadyne for three dfferent denstes of the system: F 1 ) g/cm 3, F 2 ) g/cm 3, and F 3 ) g/cm 3 computed by SISM for M ) Table 2: Average of Quotents of Magntudes of Harmonc and Anharmonc Forces Actng on Atoms of the System of 256 Butadyne Molecules for Three Dfferent Denstes of the System: F 1 ) g/cm 3, F 2 ) g/cm 3, and F 3 ) g/cm 3 Computed by SISM for M ) 1000 Usng t ) 0.5 fs and t ) 3.0 fs Integraton Tme Steps F AVg[ t ) 0.5 fs] AVg[ t ) 3.0 fs] F 1 ) g/cm F 2 ) g/cm F 3 ) g/cm concluded that as the densty of the system ncreases the qualty of the harmonc approxmaton for the bond stretchng and angle bendng potental n terms of the Cartesan dsplacement coordnates deterorates. Ffth, to study the nfluence of the system densty on the sze of the ntegraton tme step allowed by the SISM, we computed the average of the quotents (AVg) of the magntude of the harmonc and the anharmonc forces actng on an atom of the system. Ths quotent, whch s averaged over all the atoms of the system and the total number of ntegraton steps, s defned as M F harm AVg ) 1 M )1 F LJ + F elec + F bonds+angles - F harm (21) where M s the total number of ntegraton steps and. represents the averagng over all the atoms of the system. The numerator n (21) represents the magntude of the harmonc force actng on an atom of the system at the ntegraton step. The harmonc force s derved from the harmonc part of the bond stretchng and angle bendng potental energy n terms of the Cartesan dsplacement coordnates. The denomnator n (21) s the harmonc force subtracted from the total force actng on an atom of the system due to the Lennard-Jones, electrostatc, bond stretchng, and angle bendng potentals, respectvely. Ths dfference represents the total anharmonc force computed n the numercal part of the SISM, whch generates atoms moton that lmts the sze of the ntegraton tme step. The results gven n Table 2 reveal that the bonded harmonc forces wthn molecules are an order of magntude greater than the anharmonc forces n the system at all system denstes. 4. CONCLUSIONS In ths paper the study of the system densty dependence of the sze of the maxmal allowed ntegraton tme step by the Splt Integraton Symplectc Method (SISM) for MD ntegraton was presented. The numercal results of a model system of 256 butadyne molecules showed that as the densty of the system ncreases, the magntude of anharmonc ntermolecular electrostatc and van der Waals forces actng on the system atoms also ncreases, and therefore the sze of the ntegraton tme step, whch s lmted by the atoms moton generated by the ntermolecular forces n the case of the SISM, becomes smaller. The maxmal ntegraton tme step allowed by the standard leapfrog Verlet (LFV) method, however, does not depend on the system densty because t s lmted by the hgh-frequency vbratons of the atoms wthn every molecule of the system. Snce the tme scale for ntramolecular moton s consderably smaller than the tme scale correspondng to the moton generated by the ntermolecular forces regardless of the system densty, the SISM allows an ntegraton tme step sgnfcantly larger than can be used by the standard LFV method, whle retanng the same level of accuracy. ACKNOWLEDGMENT The authors express ther thans to Drs. 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