PH 548 Atomistic Simulation Techniques Lectures: Lab: 2-0-2-6 Tuesday 12-1 PM Thursday 12-1 PM Monday 2-5 PM P. K. Padmanabhan
PH548:
Two Excellent Books
How to do it?
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) Advance positions & velocities of each atom: Taylor Expansion: too crude to use it as such!!
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
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:
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!
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.)
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 )
Applications?
Ion Channel s In NASICON s Supriya Roy Na 3 Zr 2 Si 2 PO 12 Supriya Roy and PKP to be published.
Structure of Liquids! Carbonates in Solution
Understanding Experimental Spectrum Prof. Marx, Theothem, Ruhr-Universitaet, Bochum, Germany
Advanced Simulation Techniques: Simulating Chemical Reaction Dissociation of H 2 CO 3 in gas-phase H 2 CO 3 =H 2 O + CO 2
Materials science Cond. Matter Physics Chem. Phys./ Phys.Chem Atomic/Mol. Phys. Classical Mechanics Statistical Mechanics Solid State Phys.
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 = 119.8 k B (k B = 1.3806488 10-23 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.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 + 0.0 write(1,*) i*a + a/2, j*a + a/2, k*a + 0.0 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
!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
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.)
NaCl
NaCl
MD simulations on NaCl Melting Transition of NaCl Molten NaCl Nucleation and growth from melt/water Dissolution in water Influence on water h-bonding
Interatomic potentials Alkali Halides
Salts of alkaline earth metals, CaF2 F - MSD Ca 2+ CaF2 time σ = 2 Nq D / V k B T PRB, 43, 3180,1991.
Salts of alkaline earth metals, CaF2
CaF2/BaF2 - interface
AgI Vashishta-Rahman potential
β-agi (Wurtzite) α-agi (bcc)
Ion Channel s In NASICON s Supriya Roy Na 3 Zr 2 Si 2 PO 12 Supriya Roy and PKP to be published.
MD simulation of molecular systems O2, CO2, H2O, NH3, CH4, C2H6, C6H6, proteins
Diatomic Molecule U= U intra + U inter U intra = )2 Harmonic U inter = + 4ε ( σ σ )
U intra Morse potential
Carbon dioxide U intra = ) 2 + θ θ ) 2 U inter = + 4ε ( σ σ ) C q= 0.6512 e O q= -0.3256 e σ =2.800 (A) ε= 0.2340 (kj/mol) σ =3.028 (A) ε= 0.6683 (kj/mol) = 1.162 (A) k bs = 4421.5 kj/m/a θ = 180 k bb = 451.9 kj/m Cygan et. al., J. Phys. Chem C, 116, 13079 (2012).
Dihedral Angle potentials
WHATT ERR?
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 3.166 0.65 1.0 +0.41-0.82 109.47 http://www1.lsbu.ac.uk/water/water_models.html
Lorentz-Berthelot rule σ ij = σ i + σ j ) ε ij ε i ε j U C-Ow = + 4ε ( σ, σ, ) C q c = 0.6512 e σ c = 2.800 (A) ε c = 0.234 (kj/mol) O q Ow = -0.82 e σ Ow = 3.166 (A) ε Ow = 0.650 (kj/mol)
Flexible SPC -model U intra = ) 2 + U inter = + 4ε ( σ σ ) U intra : k HH = 1374.0 kj/mol/a 2 ; r HH = 1.633 (A); D OH = 420.0 kj/mol; ρ = 2.566 (A -1 ) r e = 1 (A) U inter : O: q= -0.82 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 +0.5200-1.0400 104.52 52.26 TIP5P 3.12000 0.6694 0.9572 0.70 +0.2410-0.2410 104.52 109.47 http://www1.lsbu.ac.uk/water/water_models.html
Model Dipole moment e Dielectric constant self-diffusion, 10-5 cm 2 /s Average configurational energy, kj mol -1 Density maximu m, C SPC 2.27 65 3.85-41.0-45 7.3 TIP3P 2.35 82 5.19-41.1-91 9.2 TIP4P 2.18 53 3.29-41.8-25 4.4 TIP5P 2.29 81.5 2.62-41.3 +4 6.3 Expansion coefficient, 10-4 C -1
Ewald Summation
-Frenkel & Smith