Ab initio molecular dynamics : BO

Transcription

1 School on First Principles Simulations, JNCASR, 2010 Ab initio molecular dynamics : BO Vardha Srinivasan IISER Bhopal

2 Why ab initio MD Free of parametrization and only fundamental constants required. Bond forming and breaking processes can be treated accurately. Transferable unlike many classical MD methods. On the fly computation of potential energy surface, so avoids dimensionality bottleneck.

3 Quantum dynamics i Ψ(r, R; t) =ĤΨ(r, R; t) t r electrons R (RI) ions Ĥ = 2 2M I I 2 2 2m e e 2 + ˆV e e + ˆV I I + ˆV e I electron-electron repulsion Ψ(r, R; t) Complicated Full quantum dynamics quite messy and not needed in most cases. ion-ion repulsion electron-ion attraction

4 Decoupling electrons and ions Ĥ e = 2 2m e 2 e + ˆV e e + ˆV I I + ˆV e I First solve for clamped nuclei Ĥ e ψ k (r; R) =E k (R)ψ k (r; R) Instantaneous representation for full solution Ψ(r, R; t) = l ψ l (r; R)χ l (R; t) Integrate out r dependence dr ψ k (r; R) ĤΨ(r, R; t)

5 Decoupling electrons and ions i t χ(r; t) = 2 2 I + E k (R) 2M I Exact non-adiabatic coupling C kl = drψk 2 2 I 2M I C kl = C k δ kl C kl =0 Adiabatic Approx. ψ k + 1 M I χ(r; t)+ l C kl χ l (R; t) drψk [ i I ] [ i I ] leads to the Born-Oppenheimer approximation

6 Born-Oppenheimer Dynamics i t χ(r; t) = 2 2 I + E k (R) 2M I χ(r; t) Effective Kinetic energy potential energy Dynamics of ions does not change the state of the electronic sub-system. Thus, Ψ(r, R; t) =χ(r; t)ψ(r; R) Ek(R) defines an energy eigenvalue for every configuration R - Potential Energy Surface (of the k th electronic eigenstate) Classical ions would move on the ground-state surface.

7 Classical Dynamics for ions Let s define a classical Lagrangean for B-O dynamics L BO = 1 2 M IṘ2 I E BO (R) Equations of motion are obtained by d dt ṘL BO R L BO =0 Thus we have M I R = I E BO (R) E BO = ψ 0 Ĥe ψ 0 =minψ Ĥe ψ ψ The potential energy at every instant is obtained by solving the electronic problem variationally.

8 DFT Energy and Forces E BO (R) =E DFT (R)+E II (R) =E tot Where the first term is just the DFT energy computed from the ground-state density n(r) by methods discussed earlier in this school. Hellmann-Feynman forces : F i = I E BO = drn(r) I V ext (r) I E II (R) External potential on electrons

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11 B-O Molecular Dynamics At every time step we need to perform a minimization to reach self consistency. This could be time-consuming (especially around 1985 where iterative diagonalization schemes were not employed for first-principles calculations). The accuracy of the simulation critically depends on the accuracy of the SCF minimization. (Energy drifts) The performance depends on the algorithms used to extrapolate the wavefunctions from the previous steps.

12 B-O Molecular Dynamics BO MD of butadiene molecule by two different routes to guess initial wavefunction at each instant P. Pulay and G. Fogarasi, Chem. Phys. Lett. 386, 272 (2004). Energy drifts are a concern in BO MD but can be reduced by choosing good extrapolation schemes and other clever ways of propagating the wavefunction.

13 Choosing parameters for MD A large time step means lesser overall computational cost. But need to sample fastest ionic motion. Roughly, dt ~ dtmax where dtmax=1/ωmax=period of fastest phonon. SCF convergence at each step should be very good to conserve energy, avoid systematic drifts and ensure accurate forces. Error on DFT energy is quadratic in SCF error of charge density whereas error on forces is linear.

14 Some extensions

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16 &control calculation='md' restart_mode='from_scratch', pseudo_dir = '$PSEUDO_DIR/', outdir='$tmp_dir/', dt=20, nstep=100, 1 a.u. = 2.4 * s disk_io='high' / &system ibrav= 2, celldm(1)=10.18, nat= 2, ntyp= 1, ecutwfc = 8.0, nosym=.true. / &electrons conv_thr = 1.0d-8 mixing_beta = 0.7 / &ions pot_extrapolation='second-order' wfc_extrapolation='second-order' / ATOMIC_SPECIES Si Si.pz-vbc.UPF ATOMIC_POSITIONS Si Si K_POINTS {automatic}

17 iteration # 2 ecut= 8.00 Ry beta=0.70 Davidson diagonalization with overlap ethr = 4.32E-10, avg # of iterations = 2.0 total cpu time spent up to now is 0.10 secs End of self-consistent calculation k = ( 113 PWs) bands (ev): ! total energy = Ry Harris-Foulkes estimate = Ry estimated scf accuracy < 2.8E-09 Ry convergence has been achieved in 2 iterations Forces acting on atoms (Ry/au): atom 1 type 1 force = atom 2 type 1 force = Total force = Total SCF correction = Entering Dynamics: iteration = 3 time = pico-seconds ATOMIC_POSITIONS (alat) Si Si kinetic energy (Ekin) = Ry temperature = K Ekin + Etot (const) = Ry Linear momentum : Writing output data file pwscf.save second order wave-functions extrapolation

18 References Most of the equations have been taken from D. Marx and J. Hutter, Ab initio molecular dynamics, Cambridge University Press, 2009 Part of the material has been adapted from P. Giannozzi s tutorial on QE The original CP paper is R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985).