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1、1,1.7 Model System and Interaction Potential,In 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.,Hamiltonian,Kinetic energy,Potential energy
2、,2,Analysis of the Potential Energy,External potential,Pair potential about 90%,Three body contribution In FCC crystal, up to 10%,Effective pair potential,3,1.7.1 Effective pair potential for spherical molecules,1.7.1.1 Hard-sphere potential,For the purposes of investigating general properties of li
3、quids 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 spheres,4,1.7.1.2 Square-Well potential,SW potential is the simplest one including th
4、e attractive forces and can be applied to inert gases and some non-polar substances, etc.,5,1.7.1.3 Yukawa potential,When , it can be used to model Ar reasonably well.,Yukwa potential can also be used to model plasma, colloidal particles, and some electrical interactions.,6,1.7.1.4 Lennard-Jones pot
5、ential,For argon:,7,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-SR6 100 CONTINUE V=V*4.0*EPSLON,The coordinate v
6、ectors of LJ atoms are stored in three arrays RX(I), RY(I), RZ(I),Calculate,8,1.7.1.5 Potentials for ions,A 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:,9,
7、1.7.2 potential for macromolecules,The 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
8、 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.,10,Bonded potential,Bond angle-bend,Bond-strentch,Torsion(rotate-along-bond),11,Angle-bend potential,The deviation of
9、angles from their reference values is frequently described using a Hookes law or harmonic potential:,12,Bond-stretch potential,13,Torsion potential,Torsional potentials are almost always expressed as a cosine series expansion.,barrier height,Determines where the torsion angle passes through its mini
10、mum value.,The number of minimum points in the function as the bond is rotated through 2.,14,1.8 Reduced Units,Why 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 simul
11、ation box is in the magnitude of molecular scale.,Density,Temperature,Energy,Pressure,Force,Torque,Surface tension,Time,15,Reduced unit-continue,Diffusion coefficient,Viscosity,Thermal conductivity,SI Units:,W/(m K),Pa s,m2/s,Test of unit (For example, viscosity):,16,Reduced unit-continue,It should
12、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
13、units given in this section. Especially, for electrolyte solution, we can use,Surface charge density:,17,1.9 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 a
14、nd 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.,18,1.9.1 Simulatio
15、n box,x,y,z,Cube,Hexagonal prism,x,y,z,Example: DNA simulation,19,1.9.1 Simulation box-continue,Truncated octahedron,Rhombic dodecahedron,20,1.9.2 Periodic boundary condition,B,A,H,D,G,F,E,C,21,B,A,H,D,G,F,E,C,In a cubic box, the cutoff distance is set equal to L/2.,Minimum image convention,22,A,E,A
16、 side view of the box,(b) A top view of the box,Simulation of molecules in slit-like pore,23,1.9.3 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
17、 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)-BOXL IF(RX(I).LT.-BOXL2) RX(I)=RX(I)+BOXL,24,An a
18、lternative 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 fun
19、ction 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.,25,1.9.4 Computer code for minimum image convention,Immediately after calculating a pair separation vector, we apply t
20、he 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:,
21、26,RIJSQ=RXIJ*2+RYIJ*2+RZIJ*2 IF(RIJSQ.LT.RCUTSQ)THEN compute i-j interaction accumulate energy and force. ENDIF,RIJSQ=RXIJ*2+RYIJ*2+RZIJ*2 RIJSQI=1.0/RIJSQ RIJSQI=CVMGP(RIJSQI, 0.0, RCUTSQ-RIJSQ) compute I-j interaction .as a functions of RIJSQI. ,recommended,27,The function CVMGP(A,B,C) is a vecto
22、r 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)=8,The computer code for other shapes of simulation boxes can be found in program F1.,28,1.9.5 Non-periodic boundary methods,Periodic boundary
23、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
24、 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.,29,Example for non-periodic boundary conditions-study the active site
25、of an enzyme,Reaction zone: r R1. Containing atoms or group with the site of interest. Perform full simulation.,Reservoir region: R1rR2 atoms are fixed at initial position or restrained to initial positions using a harmonic potential.,rR2, discarded or fixed.,Division into reaction zone and reservoir regions in a simulation,