Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 199 Sto chastic Classical Molecular Dynamics Coupled to Functional Density Theory: Applications to Large Molecular Systems K. C. Mundim Institute of Physics, Federal University of Bahia, Salvador, Ba, Brazil and D. E. Ellis Dept. of Chemistry and Materials Research Center Northwestern University, Evanston IL 60208, USA Received 07 Decemb er, 1998 Ahybrid approach is describ ed, which combines sto chastic classical molecular dynamics and rst principles DensityFunctional theory to mo del the atomic and electronic structure of large molecular and solid-state systems. The sto chastic molecular dynamics using Gener- alized Simulated Annealing GSA is based on the nonextensive statistical mechanics and thermo dynamics. Examples are given of applications in linear-chain p olymers, structural ceramics, impurities in metals, and pharmacological molecule-protein interactions. scrib e each of the comp onent pro cedures. I Intro duction In complex materials and biophysics problems the num- b er of degrees of freedom in nuclear and electronic co or- dinates is currently to o large for e ective treatmentby purely rst principles computation. Alternative tech- niques whichinvolvea hybrid mix of classical and quan- tum metho dologies can provide a p owerful to ol for anal- ysis of structure, b onding, mechanical, electrical, and sp ectroscopic prop erties. In this rep ort we describ e an implementation which has b een evolved to deal sp ecif- ically with protein folding, pharmacological molecule do cking, impurities, defects, interfaces, grain b ound- aries in crystals and related problems of 'real' solids. As in anyevolving scheme, there is still muchroom for improvement; however the guiding principles are simple: to obtain a lo cal, chemically intuitive descrip- tion of complex systems, which can b e extended in a systematic way to the nanometer size scale. The computational scheme is illustrated in Fig. 1, show- Figure 1. Flowchart describing the hybrid MD+DV+GSA ing classical Molecular Dynamics MD, Monte Carlo approach. MC-GSA sto chastic sampling, and Discrete Varia- tional DV DensityFunctional DF quantum mechan- ics pro cedures coupled together. We next brie y de- 200 K.C. Mundim and D. E. Ellis II Classical Dynamics, Sto chas- metho ds. - In MM, it is very simple to intro duce time evolu- tic Metho ds, and Quantum tion; Clusters - In MM, it is p ossible to intro duce the temp erature as an external p erturbation, and trace its e ects. Two imp ortant questions arise which are unfavor- I I.1 Classical Metho dology: Molecular Dynam- able to the MM metho ds: First, there do not exist well- ics de ned rules to evaluate the force constants; and sec- ond, in order to cho ose the b est force eld is necessary The idea of molecular mo deling is an attempt to de- to have a priori some considerable knowledge ab out the scrib e the quantum chemical b onds in terms of a clas- molecular system. In molecular mechanics the atom is sical force eld in Newton's equations. In a Molecular represented by a spherical b o dy with a particle mass Mechanics MM mo del the atomic b onds are repre- equal, in general, to the resp ective atomic mass. In sented by springs joining the atoms; i.e., the molecule to day's molecular mechanics, several force eld mo dels is assumed to b e a collection of masses and springs. have b een prop osed. In general the molecular energy In this case is necessary nd a set of parameters, for p otential function related with the classical force eld this force eld, that t the quantum atomic interac- can b e expressed by the following sum [3],[4]; tions with sucient accuracy. Ideally one hop es that such re nements will eventually lead toward a uni ed computational mo del that can successfully mimic ob- V = V + V + V + V + V + V 1 H b tor i tor p C vdW served molecular prop erties. Academic research e orts and the pharmaceutical industry interest in developing N H X 1 new comp ounds in biological molecular systems have 2 K r r V = Hn n o H 2 stimulated the app earance of di erent computational n co des based on classical force elds. The rst molecular dynamics applications was p er- N X 1 2 K V = formed by Alder and Wainwright, [1] who used a p er- n n o b 2 n fectly elastic hard sphere mo del to represent the atomic interactions. One of the most widely used force eld N ones is called MM1 Molecular Mechanics, prop osed X 1 2 V = K tor i n n o by Allinger [2]. At the moment there are an uncount- 2 n able numb er of MM computational co des using a classi- cal force eld. Each one of these uses a particular force N X 1 eld to describ e di erent molecular prop erties and to 2 K [1 + cosm ] V = n n n tor p 2 t some exp erimental results. n Reasons which justify the increasing use of MM can b e listed, as for example: Nc X 1 q q i j Vc = -Possibly the chief reason is the relatively short 4 r ij i<j computational time, which for MM metho ds increases 2 as M , where M is the numb er of atoms in the molecule. " Nv In contrast, the use of the ab-initio quantum metho ds X C ij C ij 12 6 V = vdW 12 6 in such molecular systems is computationally impracti- r r ij ij i<j cable b ecause the computer time to evaluate the inter- 4 electronic repulsion integrals increases as N or more Where, in summary, the rst term of eq. 1 V H rapidly, when correlation is taken into account, where represent the energy necessary to stretch or compress N is the numb er of basis orbitals. Normally, there are the atomic b ond; Vb is the p otential contribution that at least several basis functions p er atomic orbital shell- represents the b ond-angle b ending interaction; V , tor i 1s, 2s, 2p, etc.. V are the torsional contributions that represent tor p the harmonic dihedral b ending interaction and the si- - The MM metho d and its results are conceptually nusoidal dihedral torsion interactions. The last terms easier to understand than quantum mechanical QM Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 201 V and V represent the non-b onded interactions the time t + t. In the same pro cedure the forces and C vdW such as Coulombic repulsion, hydrogen b onding and the new velo cities are used to calculate the new atomic van der Waals interactions. Each of these p otential en- p osition at time t + t, and so on. ergy functions represents a molecular deformation from Once the force on each atom has b een calculated for a reference geometry. The interactions given in equa- time t, the p osition of each atom at some later time t tion 1 are available in our Molecular Mechanics co de, + t is then given by and are typical of other currently available co des. To ! ! ! ! 2 study molecular prop erties in solid systems, additional r r + v t+ a t 3 = t t t t+t p otentials have b een intro duced, including Born-typ e ! ! a where = /m . Here t = n t, and t 0.001 pico F exp onential repulsion and Morse p otentials. ! r may b e taken seconds. The initial co ordinates t=0 Molecular dynamics MD calculations consist of from exp erimental data or may b e randomized; the analyzing the evolution of the molecular system with ! v are derived from the Maxwell- initial velo cities t=0 time. In this case the atoms are continuously moving, Boltzmann distribution, in general, and temp erature T the b onds are vibrating, the angles are b ending and is controlled by a dynamic scaling pro cedure. A co oling the whole molecule is rotating. In MD, successive con- or annealing pro cedure of mo difying T provides one way gurations of the system are generated byintegrating to approach stationary p oints, and p ossibly the equi- Newton's equations of motion. The result is a tra jec- librium state of the system. Once the p otentials are tory that sp eci es how the p ositions and velo cities of calculated for each new geometry, the new forces and the atoms vary with time. The tra jectory is obtained p ositions of each atom can again found. This pro cedure by solving the following di erential equations; is rep eated until convergence in the energy is reached. ! ! 2 There are di erent algorithms or numerical metho ds d r F i i 2 = 2 available to solve di erential equation 2. All meth- dt m i ! ! o ds are based on nite di erences and solve the equa- r where is the force over atom i along the co ordinate F i tion step by step in time. Often the step size is taken and m is the atomic mass. i constant, but adaptive metho ds have sometimes b een Equation 2 is solved by a di erence nite tech- found useful. nique with continuous p otential mo dels 1, which are The simplest and most straightforward but unfor- generally assumed to b e pairwise additive. The essen- tunately not suciently accurate approachistouse tial idea is that the integration is broken down into the Taylor expansion based on equation 2.