PH 548 Atomistic Simulation Techniques

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1 PH 548 Atomistic Simulation Techniques Lectures: Lab: Tuesday 12-1 PM Thursday 12-1 PM Monday 2-5 PM P. K. Padmanabhan

2 PH548:

3 Two Excellent Books

4 How to do it?

5 1 2 F 12 F 23 3 F i = F ij j i F 1 = F 12 + F 13 F 13 F 2 = F 21 + F 23 F = U ij Three Atoms ij F 3 = F 31 + F 32 Newton s II nd Law: a = F / i i m i

Progress in time 1 1 3 2 2 3 x(t) x(t+ t) y(t) y(t+ t) z(t) z(t+ t) t ~ 1-5 fs (10-15 sec)

6 Progress in time x(t) x(t+ t) y(t) y(t+ t) z(t) z(t+ t) t ~ 1-5 fs (10-15 sec) Advance positions & velocities of each atom: Taylor Expansion: too crude to use it as such!!

7 1 3 2 F 12 F 23 F 13 t+δt 1 st MD Step Atoms move forward in time! 1. Calculate 2. Update 3. Update t ~ 1-5 fs (10-15 sec) Continue this procedure for several lakhs of Steps. Or as much as you can afford! The main O/P of MD is the trajectory. 3 1 t+2δt 2 nd MD Step 2

8 Verlet Scheme: A good Integrator Newton s equations are time reversible, Summing the two equations, Now we have advanced our atoms to time t+ t!! Velocity of the atoms:

9 The missing ingredient Forces? Force is the gradient of potential: Gravitational Potential: Gm m 1 r 2 U = too weak, Neglect it!! The predominant inter-atomic forces are Coulombic in origin! 1 q q 1 = 4πε r U 2 0 However, this pure monopole interaction need not be present!

10 Interatomic forces for simple systems 1. Lennard-Jones Potential: (non-bonded interactions) 2. Born-Mayer (Tosi-Fumi) Potential: Instantaneous dipoles Gives an accurate description of inert gases (Ar, Xe, Kr etc.) Faithful in describing pure ionic solids (NaCl, KCl, NaBr etc.)

11 The Lennard-Jones Potential for Ar: Å (k B ) Pauli s repulsion F = U ij Should be known apriori. ij r(nm) dispersion interaction F F x ij x ij U = x 12σ = 4ε ( 14 r i U = r 12 i r x i 6σ 8 r 6 )( x i x j )

12 Applications?

13 Ion Channel s In NASICON s Supriya Roy Na 3 Zr 2 Si 2 PO 12 Supriya Roy and PKP to be published.

14 Structure of Liquids! Carbonates in Solution

15 Understanding Experimental Spectrum Prof. Marx, Theothem, Ruhr-Universitaet, Bochum, Germany

16 Advanced Simulation Techniques: Simulating Chemical Reaction Dissociation of H 2 CO 3 in gas-phase H 2 CO 3 =H 2 O + CO 2

17 Materials science Cond. Matter Physics Chem. Phys./ Phys.Chem Atomic/Mol. Phys. Classical Mechanics Statistical Mechanics Solid State Phys.

18 Lab-Session: 1 (Monday, 11 th Aug., 2-4 pm) Assignments: 1a: Generation of FCC lattice Make 4x4x4 unit cells of the FCC lattice of Argon. Unit cell parameters 5.26 A (angstrom). 1b: Calculation of total potential energy Argon atoms interact through Lennard-Jones potential, with ε = k B (k B = J K -1 ), and σ = 3.4 Å. Read the positions of the Ar atoms from the output of a) and calculate the total potential energy of the system.

Lab session -I!Generation of coordinates of FCC lattice (4x4x4 unit cells) program fcc implicit none integer::i,j,k, n =4! No of unit cells real::a=5.26 open(unit=1,file="fcc.

20 Lab session -I!Generation of coordinates of FCC lattice (4x4x4 unit cells) program fcc implicit none integer::i,j,k, n =4! No of unit cells real::a=5.26 open(unit=1,file="fcc.dat") do i = 0, n - 1 do j = 0, n - 1 do k = 0, n - 1 write(1,*) i*a + 0.0, j*a + 0.0, k*a write(1,*) i*a + a/2, j*a + a/2, k*a write(1,*) i*a + 0.0, j*a + a/2, k*a + a/2 write(1,*) i*a + a/2, j*a + 0.0, k*a + a/2 end do end do end do end program fcc

21 !Potential Energy of fcc lattice (4x4x4 unit cells ) allocate(x(no_atom),y(no_atom),z(no_atom)) sigma6 = sigma**6; sigma12 = sigma6**2; eps4 = 4* eps; pot_en =0.0 do i=1,no_atom-1 do j=i+1,no_atom dx = x(i) - x(j) ; dy = y(i) - y(j); dz= z(i)-z(j); r2=dx**2+dy**2+dz**2 r6= r2*r2*r2 r12= r6 * r6 fact= eps4*(sigma12/r12 - sigma6/r6) pot_en = pot_en + fact end do end do print*,"potential energy=",pot_en

22 Interatomic forces for simple systems 1. Lennard-Jones Potential: (non-bonded interactions) 2. Born-Mayer (Tosi-Fumi) Potential: Instantaneous dipoles Gives an accurate description of inert gases (Ar, Xe, Kr etc.) Faithful in describing pure ionic solids (NaCl, KCl, NaBr etc.)

23 NaCl

24 NaCl

25 MD simulations on NaCl Melting Transition of NaCl Molten NaCl Nucleation and growth from melt/water Dissolution in water Influence on water h-bonding

26 Interatomic potentials Alkali Halides

28 Salts of alkaline earth metals, CaF2 F - MSD Ca 2+ CaF2 time σ = 2 Nq D / V k B T PRB, 43, 3180,1991.

29 Salts of alkaline earth metals, CaF2

30 CaF2/BaF2 - interface

31 AgI Vashishta-Rahman potential

32 β-agi (Wurtzite) α-agi (bcc)

33 Ion Channel s In NASICON s Supriya Roy Na 3 Zr 2 Si 2 PO 12 Supriya Roy and PKP to be published.

34 MD simulation of molecular systems O2, CO2, H2O, NH3, CH4, C2H6, C6H6, proteins

35 Diatomic Molecule U= U intra + U inter U intra = )2 Harmonic U inter = + 4ε ( σ σ )

36 U intra Morse potential

37 Carbon dioxide U intra = ) 2 + θ θ ) 2 U inter = + 4ε ( σ σ ) C q= e O q= e σ =2.800 (A) ε= (kj/mol) σ =3.028 (A) ε= (kj/mol) = (A) k bs = kj/m/a θ = 180 k bb = kj/m Cygan et. al., J. Phys. Chem C, 116, (2012).

40 Dihedral Angle potentials

41 WHATT ERR?

42 Simple Point Charge (SPC) - model Berendsen, Postma, Gunsteren and Hermans, in Pullman (ed.), Intermolecular Forces (Reidel, Dordrecht, 1981) p331. A rigid model U inter = + 4ε o ( σ o σ o ) σ Å ε kj mol -1 l 1 Å q 1 (e) q 2 (e) θ

Lorentz-Berthelot rule σ ij = σ i + σ j ) ε ij ε i ε j U C-Ow = + 4ε ( σ, σ, ) C q c = 0.

43 Lorentz-Berthelot rule σ ij = σ i + σ j ) ε ij ε i ε j U C-Ow = + 4ε ( σ, σ, ) C q c = e σ c = (A) ε c = (kj/mol) O q Ow = e σ Ow = (A) ε Ow = (kj/mol)

44 Flexible SPC -model U intra = ) 2 + U inter = + 4ε ( σ σ ) U intra : k HH = kj/mol/a 2 ; r HH = (A); D OH = kj/mol; ρ = (A -1 ) r e = 1 (A) U inter : O: q= e σ =3.166 (A) ε= 0.65 (kj/mol) H: q= 0.41 e σ =0.0 (A) ε= 0.0 (kj/mol) K TOUKAN AND A.RAHMAN,PHYS. REV. B Vol. 31(2) 2643 (1985)

4 & 5 site models (TIP4P & TIP5P) σ Å ε kj mol -1 l 1 Å l 2 Å q 1 (e) q 2 (e) θ φ TIP4P 3.15365 0.6480 0.9572 0.15 +0.5200-1.

45 4 & 5 site models (TIP4P & TIP5P) σ Å ε kj mol -1 l 1 Å l 2 Å q 1 (e) q 2 (e) θ φ TIP4P TIP5P

46 Model Dipole moment e Dielectric constant self-diffusion, 10-5 cm 2 /s Average configurational energy, kj mol -1 Density maximu m, C SPC TIP3P TIP4P TIP5P Expansion coefficient, 10-4 C -1

49 Ewald Summation

52 -Frenkel & Smith

Lecture C2 Microscopic to Macroscopic, Part 2: Intermolecular Interactions. Let's get together.

Lecture C2 Microscopic to Macroscopic, Part 2: Intermolecular Interactions Let's get together. Most gases are NOT ideal except at very low pressures: Z=1 for ideal gases Intermolecular interactions come