Ionic optimisation. Georg KRESSE

Transcription

1 Ionic optimisation Geor KRESSE Institut für Materialphysik and Center for Computational Material Science Universität Wien, Sensenasse 8, A-1090 Wien, Austria b-initio ienna ackae imulation G. KRESSE, IONIC OPTIMISATION Pae 1

2 Overview the mathematical problem minimisation of functions rule of the Hessian matrix how to overcome slow converence the three implemented alorithms Quasi-Newton (DIIS) conjuate radient (CG) damped MD strenth, weaknesses a little bit on molecular dynamics G. KRESSE, IONIC OPTIMISATION Pae 2

3 The mathematical problem search for the local minimum of a function f for simplicity we will consider a simple quadratic function f x a bx where B is the Hessian matrix 1 xbx ā 2 B i j 2 f x 1 2 x i x j x B x for a stationary point, one requires i x x f x i f x B x B i j j at the minimum the Hessian matrix must be additionally positive definite x j j G. KRESSE, IONIC OPTIMISATION Pae 3

4 The Newton alorithm educational example start with an arbitrary start point calculate the radient multiply with the inverse of the Hessian matrix and perform a step x 2 B 1 by insertin x B, one immediately reconises that hence one can find the minimum in one step f x 2 in practice, the calculation of B is not possible in a reasonable time-span, and one needs to approximate B by some reasonable approximation G. KRESSE, IONIC OPTIMISATION Pae 4

5 Steepest descent approximate B by the larest eienvalue of the Hessian matrix alorithm (Jacobi alorithm for linear equations) 1. initial uess 2. calculate the radient 3. make a step into the direction of the steepest descent steepest descent x 2 x 1 1 Γmax B 4. repeat step 2 and 3 until converence is reached for functions with lon steep valleys converence can be very slow Γ max Γ min G. KRESSE, IONIC OPTIMISATION Pae 5

6 Γ 2 Speed of converence how many steps are required to convere to a predefined accuracy assume that B is diaonal, and start from 1 Γ B with Γ 1 Γ 3 0 Γ n 1 radient and x 2 after steepest descent step are: B Γ 1 Γ n x 2 1 Γ n 1 Γ 1 1 Γ n Γn Γn G. KRESSE, IONIC OPTIMISATION Pae 6

7 Converence the error reduction is iven by 1 1 Γ 1 Γn x 2 1 Γ/Γ max 1 Γ n Γn Γ Γ Γ Γ Γ Γ 1 1 2Γ/Γ max the error is reduced for each component in the hih frequency component the error vanishes after on step for the low frequency component the reduction is smallest G. KRESSE, IONIC OPTIMISATION Pae 7

8 the derivation is also true for non-diaonal matrices in this case, the eienvalues of the Hessian matrix are relevant for ionic relaxation, the eienvalues of the Hessian matrix correspond to the vibrational frequencies of the system the hihest frequency mode determines the maximum stable step-width ( hard modes limit the step-size ) but the soft modes convere slowest to reduce the error in all components to a predefined fraction ε, k iterations are required 1 Γ min Γ max k ε k ln 1 Γ min Γ max lnε k Γ min Γ max lnε k lnε Γ max Γ min k Γ max Γ min G. KRESSE, IONIC OPTIMISATION Pae 8

9 Pre-conditionin if an approximation of the inverse Hessian matrix is know P the converence speed can be much improved B 1, x N 1 x N λp x N in this case the converence speed depends on the eienvalue spectrum of PB for P B 1, the Newton alorithm is obtained G. KRESSE, IONIC OPTIMISATION Pae 9

10 Variable-metric schemes, Quasi-Newton scheme variable-metric schemes maintain an iteration history they construct an implicit or explicit approximation of the inverse Hessian matrix search directions are iven by 1 Bapproẍ 1 Bapprox the asymptotic converence rate is ive by x number of iterations Γ max Γ min G. KRESSE, IONIC OPTIMISATION Pae 10

11 1 1 Simple Quasi-Newton scheme, DIIS direct inversion in the iterative subspace (DIIS) set of points x i i N and i i N search for a linear combination of x i which minimises the radient, under the constraint α i i 1 i α i x i B i α i B i α i x i x i B i i α i i α i x i radient is linear in it s aruments for a quadratic function i α i G. KRESSE, IONIC OPTIMISATION Pae 11

12 1 Full DIIS alorithm 1. sinle initial point 2. radient 1, move alon radient (steepest descent) x 2 λ 1 3. calculate new radient 2 x 2 4. search in the space spanned by i i N for the minimal radient opt αi i and calculate the correspondin position x opt αi x i 5. Construct a new point x 3 by movin from x opt alon opt x 3 x opt λ opt G. KRESSE, IONIC OPTIMISATION Pae 12

13 1. steepest descent step from to (arrows correspond to radients 0 and 1 ) 2. radient alon indicated red line is now know, determine optimal position opt 3. another steepest descent step form opt alon opt opt 4. calculate radient x 2 now the radient is known in the entire 2 dimensional space (linearity condition) and the function can be minimised exactly a 0 + a 1 x, 1 a 0 +a 1=1 x 1 x 2 opt G. KRESSE, IONIC OPTIMISATION Pae 13

14 Conjuate radient first step is a steepest descent step with line minimisation search directions are conjuated to the previous search directions 1. radient at the current position 2. conjuate this radient to the previous search direction usin: s N x N γ 1 s N x N γ x N x 1 N x 1 N x 1 N x N 3. line minimisation alon this search direction s N 4. continue with step 1), if the radient is not sufficiently small. the search directions satisfy: s N B s M δnm N M the conjuate radient alorithm finds the minimum of a quadratic function with k derees of freedom in k 1 steps exactly G. KRESSE, IONIC OPTIMISATION Pae 14

15 1. steepest descent step from, search for minimum alon steps (crosses, at least one triastep is required) 0 by performin several trial 2. determine new radient 1 and conjuate it to et s 1 (reen arrow) for 2d-functions the radient points now directly to the minimum 3. minimisation alon search direction s 1 x 2 s1 G. KRESSE, IONIC OPTIMISATION Pae 15

16 Asymptotic converence rate asymptotic converence rate is the converence behaviour for the case that the derees of freedom are much lare than the number of steps e derees of freedom but you perform only steps how quickly, do the forces decrease? this depends entirely on the eienvalue spectrum of the Hessian matrix: steepest descent: Γ max fraction ε DIIS, CG, damped MD: Γ max forces to a fraction ε Γmin steps are required to reduce the forces to a Γmin steps are required to reduce the Γ max Γ min are the maximum and minimal eienvalue of the Hessian matrix G. KRESSE, IONIC OPTIMISATION Pae 16

17 ! " x Damped molecular dynamics instead of usin a fancy minimisation alorithms it is possible to treat the minimisation problem usin a simple simulated annealin alorithm reard the positions as dynamic derees of freedom the forces serve as accelerations and an additional friction term is introduced equation of motion ( x are the positions) x 2α x µ usin a velocity Verlet alorithm this becomes v N µ 2 v 1 N 2 2 α F N 1 µ 2 for µ x N 1 x N 1 v N 1 2, this is equivalent to a simple steepest descent step 2 G. KRESSE, IONIC OPTIMISATION Pae 17

18 behaves like a rollin ball with a friction it will accelerate initially, and then deaccelerate when close to the minimum if the optimal friction is chosen the ball will lide riht away into the minimum for a too small friction it will overshoot the minimum and accelerate back for a tool lare friction relaxation will also slow down (behaves like a steepest descent) G. KRESSE, IONIC OPTIMISATION Pae 18

19 '& ' ( Alorithms implemented in VASP additional flas termination DISS IBRION =1 POTIM, NFREE EDIFFG CG IBRION =2 POTIM EDIFFG damped MD IBRION =3 POTIM, SMASS EDIFFG POTIM determines enerally the step size for the CG radient alorithm, where line minisations are performed, this is the size of the very first trial step EDIFFG determines when to terminate relaxation positive values: enery chane between steps must be less than EDIFFG neative values: F i $#% i 1 N ions G. KRESSE, IONIC OPTIMISATION Pae 19

20 ) * + N. ) * + N. DIIS POTIM determines the step size in the steepest descent steps no line minisations are performed!! NFREE determines how many ionic steps are stored in the iteration history set of points x i i 1 +-,,, x i, that minimises the radient NFREE is the maximum N and i i 1 +-,,, for complex problems NFREE can be lare (i.e ) searches for a linear combination of for small problems, it is advisable to count the derees of freedom carefully (symmetry inequivalent derees of freedom) if NFREE is not specified, VASP will try to determine a reasonable value, but usually the converence is then slower G. KRESSE, IONIC OPTIMISATION Pae 20

21 CG the only required parameter is POTIM this parameter is used to parameterise, how lare the trial steps are CG requires a line minisations alon the search direction this is done usin a variant of Brent s alorithm trial step alon search direction (conj. radient scaled by POTIM) x1 xtrial 1 x trial 2 quadratic or cubic interpolation usin eneries and forces at and allows to determine the approximate minimum continue minimisation as lon as approximate minimum is not accurate enouh G. KRESSE, IONIC OPTIMISATION Pae 21

22 ! " two parameters POTIM and SMASS Damped MD v N µ 2 v 1 N 2 2 α F N 1 µ 2 x N 1 x N 1 v N 1 2 α POTIM and µ SMASS POTIM must be as lare as possible, but without leadin to diverence 2 Γmin and SMASS must be set to µ minimal und maximal eienvalues of the Hessian matrix a practicle optimisation procedure: set SMASS=0.5-1 and use a small POTIM of Γmax, where Γmin and Γmax are the increase POTIM by 20 % until the relaxation runs divere fix POTIM to the larest value for which converence was achieved try a set of different SMASS until converence is fastest (or stick to SMASS= ) G. KRESSE, IONIC OPTIMISATION Pae 22

23 1 0 / Damped MD QUICKMIN alternatively do not specify SMASS (or set SMASS 0) this select an alorithm sometimes called QUICKMIN QUICKMIN v new αf αf F v old F F for else v old F 20 if the forces are antiparallel to the velocities, quench the velocities to zero and restart otherwise increase the speed and make the velocities parallel to the present forces I have not often used this alorithm, but it is supposed to be very efficient G. KRESSE, IONIC OPTIMISATION Pae 23

24 Damped MD QUICKMIN my experience is that damped MD (as implemented in VASP) is faster than QUICKMIN but it requires less playin around defective ZnO surface: 96 atoms are allowed to move! relaxation after a finite temperature MD at 1000 K lo(e-e 0 ) damped: SMASS=0.4 quickmin steps G. KRESSE, IONIC OPTIMISATION Pae 24

25 Why so many alorithms :-(... decision chart Really, this is too complicated yes CG no yes close to minimum yes 1 3 derees of freedom no damped MD or QUICKMIN yes no very broad vib. spectrum >20 derees of freedom no DIIS G. KRESSE, IONIC OPTIMISATION Pae 25

26 Two cases where the DIIS has hue troubles force increases alon the search direction riid unit modes i.e. in perovskites (rotation) molecular systems (rotation) X0 X1 DIIS is dead, since it consideres the forces only it will move uphill instead of down in cartesian coordinates the Hessian matrix chanes when the octahedron rotates! G. KRESSE, IONIC OPTIMISATION Pae 26

27 How bad can it et the converence speed depends on the eienvalue spectrum of the Hessian matrix larer systems (thicker slabs) are more problematic (acoustic modes are very soft) molecular system are terrible (week intermolecular and stron intramolecular forces) riid unit modes and rotational modes can be exceedinly soft the spectrum can vary over three orders of manitudes miht be required ionic relaxation can be painful 100 or even more steps to model the behaviour of the soft modes, you need very accurate forces since otherwise the soft modes are hidden by the noise in the forces EDIFF must be set to very small values (10 6 ) if soft modes exist G. KRESSE, IONIC OPTIMISATION Pae 27