3周期性边界条件.pptx

1.7 Model System and Interaction PotentialIn most of this course,the microscopic of a system may be specified in terms of the position and momenta of a constituent set of particles.In this case the rapid motion of the electrons have been averaged out.HamiltonianKinetic energyPotential energyAnalysis of the Potential EnergyExternal potentialPair potentialabout 90%Three body contributionIn FCC crystal,up to 10%Effective pair potential1.7.1 Effective pair potential for spherical molecules1.7.1.1 Hard-sphere potentialFor the purposes of investigating general properties of liquids and for comparison with theory,highly idealized pair potentials may be of value.In this section,I will introduce hard-sphere,square-well,Yukawa and Lennard-Jones potentials,etc.is the diameter of hard spheres1.7.1.2 Square-Well potentialSW potential is the simplest one including the attractive forces and can be applied toinert gases and some non-polar substances,etc.1.7.1.3 Yukawa potentialWhen ,it canbe used to model Ar reasonably well.Yukwa potential can also be used to model plasma,colloidal particles,and some electrical interactions.1.7.1.4 Lennard-Jones potentialFor argon:Codes for calculating the total potential V=0.0 DO 100 I=1,N-1 RXI=RX(I)RYI=RY(I)RZI=RZ(I)DO 100 J=I+1,N RXIJ=RXI-RX(J)RYIJ=RYI-RY(J)RZIJ=RZI-RZ(J)RIJSQ=RXIJ*2+RYIJ*2+RZIJ*2 SR2=SIGSQ/RIJSQ SR6=SR2*SR2*SR2 SR12=SR6*2 V=V+SR12-SR6100 CONTINUE V=V*4.0*EPSLONThe coordinate vectors of LJ atoms are stored in three arrays RX(I),RY(I),RZ(I)Calculate1.7.1.5 Potentials for ionsA simple approach to construct potentials for ions is to supplement one of the above pair potentials with the Coulomb charge-charge interaction:Where X may be HS,SW,LJ.The popular one is:1.7.2 potential for macromoleculesThe energy,V,is a function of the atomic positions,R,of all the atoms in the system,these are usually expressed in term of Cartesian coordinates.The value of the energy is calculated as a sum of internal,or bonded terms,which describe the bonds,angles and bond rotations in a molecule,and a sum of external or nonbonded terms.These terms account for interactions between nonbonded atoms or atoms separated by 3 or more covalent bonds.Bonded potentialBond angle-bendBond-strentchTorsion(rotate-along-bond)Angle-bend potentialThe deviation of angles from their reference values is frequently described using a Hookes law or harmonic potential:Bond-stretch potentialTorsion potentialTorsional potentials are almost always expressed as a cosine series expansion.barrier heightDetermines where the torsion angle passes through its minimum value.The number of minimum points in the function as the bond is rotated through 2.1.8 Reduced UnitsWhy use reduced unit?Avoids the possible embarrassment of conducting essentially duplicate simulations.And there are also technical advantages in the use of reduced units due to the simulation box is in the magnitude of molecular scale.Density Temperature Energy Pressure Force Torque Surface tensionTimeReduced unit-continueDiffusion coefficient ViscosityThermal conductivitySI Units:W/(m K)Pa s m2/sTest of unit(For example,viscosity):Reduced unit-continueIt should be pointed that all the reduced units in the previous two slides are based on Lennard-Jones interaction potential.For hard sphere fluid,the reduced units are obtained using kBT to replace.The reduced units for other properties such as chemical potential and heat capacity can be deduced from the units given in this section.Especially,for electrolyte solution,we can useSurface charge density:1.9 Simulation Box and Its Boundary Conditions Computer simulations are usually performed on a small number of molecules,10N10,000.The time taken for a double loop used to evaluate the forces and potential energy is proportional to N 2.Whether or not the cube is surrounded by a containing wall,molecules on the surface will experience quite different forces from molecules in the bulk.It is essential to propose proper methods to overcome the problem of surface effects.1.9.1 Simulation boxxyzCubeHexagonal prismxyzExample:DNA simulation1.9.1 Simulation box-continueTruncated octahedronRhombic dodecahedron1.9.2 Periodic boundary conditionBAHDGFECBAHDGFECIn a cubic box,the cutoff distance is set equal to L/2.Minimum image conventionAEA side view of the box(b)A top view of the boxBDCAEHFGSimulation of molecules in slit-like pore1.9.3 Computer code for periodic boundaries How do we handle periodic boundaries and the minimum image convention in a simulation program?Let us assume,initially,the N molecules in the simulation lie with a cubic box of side BOXL,with the origin at its center,i.e.,all coordinate lie in the range(-BOXL/2,BOXL/2).After the molecules have been moved,we must test the position immediately using a FORTRAN IF statement.IF(RX(I).GT.BOXL2)RX(I)=RX(I)-BOXLIF(RX(I).LT.-BOXL2)RX(I)=RX(I)+BOXLAn alternative code for periodic boundaries An alternative to the IF statement is to use FORTRAN arithmetic functions:RX(I)=RX(I)-BOXL*ANINT(RX(I)/BOXL)The function ANINT(X)returns the nearest integer to X,converting the results back to type REAL.For example,ANINT(-0.49)=0;ANINT(-0.55)=-1 The function ANINT(X)is different from AINT(X).AINT(X)returns the integral part of X.The use of IF statement inside the inner loop,particularly on pipeline machines,is to be avoided.1.9.4 Computer code for minimum image conventionImmediately after calculating a pair separation vector,we apply the code similar to the periodic boundary adjustments.RXIJ=RXIJ-BOXL*ANINT(RXIJ/BOXL)RYIJ=RYIJ-BOXL*ANINT(RYIJ/BOXL)RZIJ=RZIJ-BOXL*ANINT(RZIJ/BOXL)If we use a FORTRAN variable RCUTSQ to represent the square of cutoff distance rc.After the above codes,the following statements would be employed:RIJSQ=RXIJ*2+RYIJ*2+RZIJ*2 IF(RIJSQ.LT.RCUTSQ)THEN compute i-j interaction accumulate energy and force.ENDIFRIJSQ=RXIJ*2+RYIJ*2+RZIJ*2RIJSQI=1.0/RIJSQRIJSQI=CVMGP(RIJSQI,0.0,RCUTSQ-RIJSQ)compute I-j interaction .as a functions of RIJSQI.recommendedThe function CVMGP(A,B,C)is a vector merge statement which returns to the value A if C is non-negative and the value B otherwise.For example:CVMGP(9,0,0)=9 CVMGP(9,8,2)=9 CVMGP(9,8,-1)=8The computer code for other shapes of simulation boxes can be found in program F1.1.9.5 Non-periodic boundary methodsPeriodic boundary conditions are not always used in computer simulation.Why?Some systems,such as liquid droplets or van der Waals clusters,inherently contain a boundary.When simulating inhomongeneous systems or systems that are not at equilibrium,periodic boundary conditions may cause difficulties.In the study of the structural and conformational behavior of macromolecules such as proteins and protein-ligand complexes,the use of periodic boundary conditions would require a prohibitive number of atoms to be included in the simulation.Example for non-periodic boundary conditions-study the active site of an enzyme Reaction zone:r R1.Containing atoms or group with the site of interest.Perform full simulation.Reservoir region:R1rR2,discarded or fixed.Division into reaction zone and reservoir regions in a simulation