1. The Nuts and Bolts of First-Principles Simulation Durham, 6th-13th December 2001 Lecture 15: Structural Calculations and Pressure CASTEP Developers’ Group with support from the ESF  k Network

2. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 2 Overview of Lecture  Why bother?  What can it tell you?  How does it work?  Damped MD in CASTEP  BFGS in CASTEP  Future directions  Conclusions

3. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 3 Why bother?  Want to find ground state of system Need to minimise energy of electronic structure at fixed ionic positions and then optimise the ionic positions and/or the unit cell shape and size (particularly if external pressure applied) Theoretical minimum depends on choice of pseudopotential, plane-wave cut-off energy, choice of XC functional, etc. Often not exactly the same as experiment! Fully converged calculation should get agreement to better than 1%

4. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 4 What Can It Tell You?  Equilibrium bond lengths and angles  Equilibrium cell parameters  Discriminating between competing structures  Elastic constants  Surface reconstructions  Pressure-driven phase transitions  Starting point for many advanced investigations …

5. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 5 How Does It Work?  Electrons adjust instantly to position of ions so have a multi-dimensional potential energy surface and want to find global minimum Treat as an optimisation problem Simplest approach is steepest descents More physical approach is damped MD More sophisticated approaches are conjugate gradients or BFGS Could use simulated annealing to avoid getting stuck in local minima

6. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 6 Steepest Descents (I)  Simplest minimiser Step downhill in direction of local steepest gradient using trial step length Line minimisation to find the optimal step length Repeat to convergence

7. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 7 Steepest Descents (II) Traversing a long, narrow valley Enlargement of a single step showing the line minimisation in action – the step continues until the local energy starts to rise again whereupon a new direction is selected which is orthogonal to the previous one

8. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 8 Steepest Descents (III)  Advantages Easy to implement Very robust – hard to get it confused! Reliable – will always get to the minima eventually  Disadvantages Often very slow to converge Can get stuck in local minima

9. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 9 Damped MD (I)  Improved Steepest Descents Move ions using velocities as well as forces  second-order equation of motion  more efficient than steepest descents Need to add damping term ‘-  v’ to forces Initially v=0 Use physical insight to get ‘optimal damping’ factor and to adjust the time step s.t. get rapid convergence

10. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 10 Damped MD (II) Over-damped Under-damped Critically damped x t

11. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 11 Damped MD (III)  Advantages Easy to code, robust, much more efficient than either steepest descents or simulated annealing Can do with Car-Parrinello or conjugate gradient minimisation of electrons. If not CP then can accelerate the ab initio part of the calculation using wavefunction extrapolation  Disadvantages Convergence rate depends on damping factor Can get stuck in local minima

12. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 12 Conjugate Gradients (I)  Improved Steepest Descents Moves ions according to the gradient with a line minimisation to find the best step length But not just the locally steepest gradient – constructs a direction which is conjugate to all previous directions so it does not undo earlier minimizations at later times  big improvement in rate of convergence See previous talks on electronic minimisation strategies for details

13. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 13 Conjugate Gradients (II) Traversing a long, narrow valley The initial search direction is given by steepest descents. Subsequent search directions are constructed to be orthogonal to the previous and all prior search directions.

14. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 14 Conjugate Gradients (III)  Advantages Rapid rate of convergence – in a quadratic energy landscape, each iteration should converge one degree of freedom Low storage requirements  Disadvantages More complex to code and cannot use with CP No knowledge of the Hessian explicitly generated Can get stuck in local minima

15. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 15 BFGS (I)  Basic idea: Energy surface around a minima is quadratic in small displacements and so is totally determined by the Hessian matrix A (the matrix of second derivatives of the energy): so if we knew A then could move from any nearby point to the minimum in 1 step!

16. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 16 BFGS (II)  The Problem We do not know A a priori Therefore we build up a progressively improving approximation to A (actually H=A -1 ) as we move the ions according to the BFGS ( Broyden- Fletcher-Goldfarb-Shanno) algorithm. Also known as a quasi-Newton or variable metric method.  Positions updated according to:

17. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 17 BFGS (III)  In a perfectly quadratic system then =1 is the optimal step length. In a real system need a line minimization to find the optimal : F.  X 01 start trial best

18. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 18 BFGS (IV)  Davidon-Fletcher-Powell (DFP) variant Older method Mathematically equivalent to BFGS Less tolerant of round-off error or inexact line minimisation  Direct Hessian Method Calculates A rather than H Uses Cholesky decomposition to keep similar speed but should guarantee that A remains non- singular. Very rare problem in practice.

19. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 19 BFGS (V)  DFP: Hessian updated as  BFGS: Hessian updated as

20. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 20 BFGS (VI)  Advantages Convergence rate similar (or better) than CG Extra physical information generated from Hessian  Disadvantages Complex to code and cannot use with CP Storage of Hessian ~ (number d.o.f.)^2 which is prohibitive for electronic problem but OK for ionic Can get stuck in local minima

21. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 21 Simulated Annealing (I)  Stochastic method Monte-Carlo style exploration of energy landscape Always accept steps that go downhill in energy Sometimes accept uphill steps depending on Boltzman factor combining energy change with current temperature of system Progressively reduce temperature and iterate to ground state

22. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 22 Simulated Annealing (II) U(x) x start stop

23. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 23 Simulated Annealing (III)  Advantages Very robust and reliable Reasonably immune to local minima  Disadvantages Exceedingly slow to converge Need to be careful in cooling rate so as not to quench and trap in local minimum if too fast, and not to waste too much time if too slow. Cannot guarantee will find exact global minimum

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25. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 25 Damped MD in CASTEP (I)  Need to calculate damping  that corresponds to critical damping For simple harmonic oscillator, this can be found from the characteristic frequency  0 For a system with a dominant natural mode then this can be found and used to set  As convergence is approached, the mode with largest  will be converged first whereupon  can be re-evaluated to accelerate the convergence of the slower modes In the same way, the time step of the MD can be linked to the dominant mode  and therefore progressively increased as convergence is approached.

26. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 26 Damped MD in CASTEP (II)  CASTEP has several different algorithms for calculating the optimal  and time step  Whilst it is to be expected that BFGS will be superior, experience has shown that the old CASTEP BFGS can sometimes be beaten Although DMD will require more iterations than BFGS, each one will be cheaper because of wavefunction extrapolation Sometimes it is more stable and succeeds in finding a minimum where old CASTEP BFGS fails  A useful fall-back but not so necessary with new CASTEP and the improved BFGS algorithm? Need more experience to tell!

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28. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 28 BFGS in old CASTEP (I)  Simultaneous optimisation of ions and positions  seeks to minimise the enthalpy H=E+p   Works in space of cell parameters (a,b,c,  ) and ionic positions with optional external pressure  Convergence criteria Simultaneous convergence in enthalpy, RMS force, RMS stress component, RMS displacement of ions but has ‘escape route’ if everything except stress is converged

29. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 29 BFGS in old CASTEP (II)  Problems Uses DFP not BFGS update Uses heuristic for step length rather than rigorous line minimisation Starts with initial H = I (unit matrix) Periodically resets H to I to reduce error accumulation whether strictly necessary or not Does not guarantee that a step will go down in enthalpy

30. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 30 BFGS in new CASTEP (I)  Also seeks to minimise the enthalpy H=E+p   Works in space of fractional ionic positions and strains with optional external pressure Unified approach allows better strain/coordinate coupling  Does rigorous line minimisation to find optimal step length Only recalculate ground-state at the new structure if sufficiently different to the trial structure Also bisection search if the line minimisation fails Also quadratic step if necessary  Builds up H using BFGS update  Only reset H if run out of search directions

31. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 31 BFGS in new CASTEP (II)  Starts with more complex initial guess at H Block diagonal using input/default guess at bulk modulus (B) for cell part and input / default guess at the characteristic frequency (  0 ) for the ionic part Guaranteed to preserve symmetry if so desired  Analyse H to get updated estimates for B and  0  Only allows ‘uphill’ steps if all else has failed (including resetting H)  Stringent convergence criteria Simultaneous convergence in enthalpy, max. modulus of force on any ion, max. component of stress tensor, max. displacement of any ion Over a ‘window’ of successive iterations

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36. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 36 Future Directions  Transition State Search Find saddle point structures not just stable ones  Redundant Internal Coordinates Transform to normal coordinates and then only optimise the non-redundant degrees of freedom Recent theoretical breakthrough in application to crystals/extended systems Need to see how it compares to the new BFGS  Variable cell damped MD

37. Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 37 Conclusions  Two independent algorithms have been implemented within new CASTEP for structural optimisation  Damped MD Can be very efficient depending on  Only works with fixed cell parameters (so far!)  BFGS Can do either/or/both ionic positions and cell parameter optimisation Can re-use calculated B and/or  0 from earlier runs to accelerate convergence in later runs