4、分子模拟的数学基础ppt课件

1、11.9 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,

2、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.21.9.1 Simulation boxxyzCubeHexagonal prismxyzExample:DNA simulation31.9.1 Simulation box-continueTruncated octahedronRhomb

3、ic dodecahedron41.9.2 Periodic boundary conditionBAHDGFEC5BAHDGFECIn a cubic box, the cutoff distance is set equal to L/2. Minimum image convention6AEA side view of the box(b) A top view of the boxBDCAEHFGSimulation of molecules in slit-like pore71.9.3 Computer code for periodic boundaries How do we

4、 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 m

5、oved, 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)+BOXL8An 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

6、 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 m

7、achines, is to be avoided.91.9.4 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 F

8、ORTRAN variable RCUTSQ to represent the square of cutoff distance rc. After the above codes, the following statements would be employed:10RIJSQ=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=CVMG

9、P(RIJSQI, 0.0, RCUTSQ-RIJSQ) compute I-j interaction .as a functions of RIJSQI. recommended11The 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

10、code for other shapes of simulation boxes can be found in program F1.121.9.5 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 inho

11、mongeneous 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 prohibit

12、ive number of atoms to be included in the simulation.13Example 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

13、 zone and reservoir regions in a simulation14Chapter 2 Statistical MechanicsWhy talk about statistical mechanics? Computer simulation generates information at the microscopic level: atomic and molecular positions, velocities etc. It is the statistical mechanics that can be used as a tool to converse

14、 this very detailed information into macroscopic properties: pressure, internal energy etc.MicroscopicStatistical MechanicsMacroscopic152.1 Basic Statistical MechanicsLet us consider a system (microcanonical ensemble) containing N particles and their energy levels are 1, 2, If there are n1 particles

15、 in the energy level 1, n2 particles in 2 and so on, then there are W ways in which this distribution can be achieved:1212( ,)!/(!)W n nNn nThe most favorable distribution is the one with highest ways and this corresponds to configuration with just one particle in each energy level (W = N!). However

16、, there are two constraints on the system.iiinEandiinN16Probability density and partition function From maximising the entropy of the system, we can derived the Boltzmann distribution which gives the number of particles ni in each energy level i asexp(/)exp(/)iiBiBink TNk TProbability density of the

17、 ensemblePartition function Q17Relation between thermodynamic potential and the molecular partition function enslnensQ Because the energy level depends on the position and momenta, we may writeProbability density: (,)NNprPartition function:exp(,)NNNNQdp drE prEnergy:(,)iENNpr182.2 Summary of various

18、 ensembles Micro-canonicalCanonicalIsothermal-isobaricGrand canonicalFixed thermodynamicvariables E, V, N N, V, T N, P, T , V, TSystemsisolatedclosedclosedopenedthermodynamic potential-S/kBF/kBTG/kBT-PV/kBTEach combination of 3N positions and 3N momenta defines a point in the 6N-dimensional phase sp

19、ace; an ensemble can thus be considered to be a collection of the points in phase space.19Partition functions for various ensembles Microcanonical ensemble:311( , )!NVENQd dHEN hr pr p Canonical ensemble311exp( , )/)!NVTBNQd dHk TN hr pr p Isothermal-isobaric ensemble301exp ( , )/!NPTBNQdVd dHPVk TN

20、 h Vr pp r Grand ensemble31exp(/)exp( , )/!VTBBNNQN k Td dHk TN hr pp r202.3 Sampling from ensembles2.3.1 Time averages and ensemble averagesIn experiment, the instantaneous value of the propertyA can be written as , and true averagevalue is)(),(ttANNrpdtttAAtNN0ave)(),(1limrpTime averageBoltzmann a

21、nd Gibbs developed statistical mechanics, in which a single system evolving in time is replaced by a large number of replications of the system that are considered simultaneously.),(),(NNNNNNAddArprprpEnsemble average21Ergodic hypothesisIn accordance with the ergodic hypothesis, ensemble average is

22、equal to the time average.AAaveMiNNtittitAMA100)(),(1rpMD SimulationMiNAMA1)(1rMC SimulationEvery quantum state of a many-body system with energy E is equally likely to be occupied.-One of the axioms in statistical mechanics.222.3.2 Calculation of simple thermodynamic properties Internal energy()NNE

23、KVpr（）1MiiEUEM Heat capacityVVUCT222/VBCEEk T222()EEEE22()/VBCEEk TMore accurate Derivation: w1.doc23Generalized equipartition principle Making the approximation that a classical description is adequate, we may write the Hamiltonian H of a system of N molecules as( , )( )( )HKVq ppqA set of coordina

24、tesA set of momenta In the canonical ensemble, we can derive(/)/kBkAHqk TAq(/)/kBkAHpk TAp Above formulae are valid (to O(N-1) in any ensemble.As a special case, A=pk, or qk, we have(/)kkBpHpk T(/)kkBqHqk T24 TemperatureIn a canonical ensemble the total temperature is constant. In the microcanonical

25、 ensemble, however, the temperature will fluctuate. The temperatureis directly related to the kinetic energy of the system as follows:21(3)22NiBCiik TKNNmpTotal momentum of imassTheorem of the equipartition of energyDegrees of freedomNumber of constraints25 PressureLet qk in generalized equipartition principle be , thenir1( )3iBVk Tirrq( )totiV irqf13totiiBk Tr f1