Lecture 15: Structural Calculations and Pressure

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 ψk Network

Nuts and Bolts 2001 Lecture 15: Structural Calculations and Pressure 2

Overview of Lecture

r Why bother?

r What can it tell you?

r How does it work?

r Damped MD in CASTEP

r BFGS in CASTEP

r Future directions

r Conclusions


Why bother?

r Want to find ground state of systeml 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)

l Theoretical minimum depends on choice of pseudopotential, plane-wave cut-off energy, choice of XC functional, etc.

l Often not exactly the same as experiment! Fully converged calculation should get agreement to better than 1%


What Can It Tell You?

r Equilibrium bond lengths and angles

r Equilibrium cell parameters

r Discriminating between competing structures

r Elastic constants

r Surface reconstructions

r Pressure-driven phase transitions

r Starting point for many advanced investigations …


How Does It Work?

r Electrons adjust instantly to position of ions so have a multi-dimensional potential energy surface and want to find global minimuml Treat as an optimisation problem

l Simplest approach is steepest descents

l More physical approach is damped MD

l More sophisticated approaches are conjugate gradients or BFGS

l Could use simulated annealing to avoid getting stuck in local minima


Steepest Descents (I)

r Simplest minimiser

l Step downhill in direction of local steepest gradient using trial step length

l Line minimisation to find the optimal step length

l Repeat to convergence


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


Steepest Descents (III)

r Advantages

l Easy to implement

l Very robust – hard to get it confused!

l Reliable – will always get to the minima eventually

r Disadvantages

l Often very slow to converge

l Can get stuck in local minima


Damped MD (I)

r Improved Steepest Descents

l Move ions using velocities as well as forces e second-order equation of motion e more efficient than steepest descents

l Need to add damping term ‘-γv’ to forces

l Initially v=0

l Use physical insight to get ‘optimal damping’ factor and to

adjust the time step s.t. get rapid convergence


Damped MD (II)

Over-damped

Under-dampedCritically damped

x

t


Damped MD (III)

r Advantagesl Easy to code, robust, much more efficient than either steepest

descents or simulated annealing

l Can do with Car-Parrinello or conjugate gradient minimisation of electrons.

l If not CP then can accelerate the ab initio part of the calculation using wavefunction extrapolation

r Disadvantagesl Convergence rate depends on damping factor



Conjugate Gradients (I)

r Improved Steepest Descents

l Moves ions according to the gradient with a line minimisation to find the best step length

l 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 e big improvement in rate of convergence

l See previous talks on electronic minimisation strategies for details


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 andall prior search directions.


Conjugate Gradients (III)

r Advantages

l Rapid rate of convergence – in a quadratic energy landscape, each iteration should converge one degree of freedom

l Low storage requirements

r Disadvantages

l More complex to code and cannot use with CP

l No knowledge of the Hessian explicitly generated



BFGS (I)r Basic idea:

l 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):

l so if we knew A then could move from any nearby point to the minimum in 1 step!

( ) ( )minmin

2

1

2

1

2

11

2

2

1XXAXX

A

−••−=

∂∂∂

∂∂∂

∂∂∂

∂∂∂

=

T

NNN

N

E

xxE

xxE

xxE

xxE

δ

O


BFGS (II)

r The Problem

l We do not know A a priori

l 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.

l Also known as a quasi-Newton or variable metric method.

r Positions updated according to:

iii

iii

FHX

XXX

=∆∆+=+ λ1


BFGS (III)

r In a perfectly quadratic system then λ=1 is the optimal step length. In a real system need a line minimization to

find the optimal λ:

F.∆∆X

λ0 1

start

trial

best


BFGS (IV)

r Davidon-Fletcher-Powell (DFP) variantl Older method

l Mathematically equivalent to BFGS

l Less tolerant of round-off error or inexact line minimisation

r Direct Hessian Methodl Calculates A rather than H

l Uses Cholesky decomposition to keep similar speed but should guarantee that A remains non-singular. Very rare problem in practice.


BFGS (V)

r DFP: Hessian updated as

r BFGS: Hessian updated as

( ) ( )( ) ( )

( )( ) ( )( )( ) ( )111

111

11

111

−−−

−−−

−−

−−−

−•−−⊗−

−

−•−−⊗−

−=

iiiii

iiiiii

iiii

iiiii

DFPi

FFHFF

FFHFFH

FFXX

XXXXHH

( ) ( )[ ]( )

( ) ( )( )( )

( ) ( )111

11

11

1

111

−−−

−−

−−

−

−−−

−•−−

−−•−

−=

⊗−•−+=

iiiii

iii

iiii

ii

iiiiiDFPi

BFGSi

FFHFF

FFH

FFXX

XXU

UUFFHFFHH


BFGS (VI)

r Advantages

l Convergence rate similar (or better) than CG

l Extra physical information generated from Hessian

r Disadvantages

l Complex to code and cannot use with CP

l Storage of Hessian ~ (number d.o.f.)^2 which is prohibitive for electronic problem but OK for ionic



Simulated Annealing (I)

r Stochastic method

l Monte-Carlo style exploration of energy landscape

l Always accept steps that go downhill in energy

l Sometimes accept uphill steps depending on Boltzman factor combining energy change with current temperature of system

l Progressively reduce temperature and iterate to ground state

≥

−−

<=

−−

−

11

1

exp

1

iiii

ii

i EET

EE

EE

p


Simulated Annealing (II)

U(x)

x

start

stop


Simulated Annealing (III)

r Advantages

l Very robust and reliable

l Reasonably immune to local minima

r Disadvantages

l Exceedingly slow to converge

l 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.

l Cannot guarantee will find exact global minimum


Damped MD in CASTEP (I)

r Need to calculate damping γ that corresponds to critical damping

l For simple harmonic oscillator, this can be found from the characteristic

frequency ω0

l For a system with a dominant natural mode then this can be found and

used to set γl As convergence is approached, the mode with largest ω will be converged

first whereupon γ can be re-evaluated to accelerate the convergence of the slower modes

l In the same way, the time step of the MD can be linked to the dominant

mode ω and therefore progressively increased as convergence is approached.


Damped MD in CASTEP (II)

r CASTEP has several different algorithms for calculating the optimal γ and time step

r Whilst it is to be expected that BFGS will be superior, experience has shown that the old CASTEP BFGS can sometimes be beatenl Although DMD will require more iterations than BFGS, each one will be

cheaper because of wavefunction extrapolation

l Sometimes it is more stable and succeeds in finding a minimum where old CASTEP BFGS fails

r A useful fall-back but not so necessary with new CASTEP and the improved BFGS algorithm? Need more experience to tell!


BFGS in old CASTEP (I)

r Simultaneous optimisation of ions and positionse seeks to minimise the enthalpy H=E+pΩ

r Works in space of cell parameters (a,b,c,α,β,γ) and ionic positions with optional external pressure

r Convergence criterial Simultaneous convergence in enthalpy, RMS force, RMS stress

component, RMS displacement of ions but has ‘escape route’ if everything except stress is converged


BFGS in old CASTEP (II)

r Problems

l Uses DFP not BFGS update

l Uses heuristic for step length rather than rigorous line minimisation

l Starts with initial H = I (unit matrix)

l Periodically resets H to I to reduce error accumulation whether strictly necessary or not

l Does not guarantee that a step will go down in enthalpy


BFGS in new CASTEP (I)r Also seeks to minimise the enthalpy H=E+pΩr Works in space of fractional ionic positions and strains with optional

external pressurel Unified approach allows better strain/coordinate coupling

r Does rigorous line minimisation to find optimal step lengthl Only recalculate ground-state at the new structure if sufficiently different to the trial

structure

l Also bisection search if the line minimisation fails

l Also quadratic step if necessary

r Builds up H using BFGS update

r Only reset H if run out of search directions


BFGS in new CASTEP (II)

r Starts with more complex initial guess at Hl Block diagonal using input/default guess at bulk modulus (B) for cell part

and input / default guess at the characteristic frequency (ω0) for the ionic part

l Guaranteed to preserve symmetry if so desired

r Analyse H to get updated estimates for B and ω0

r Only allows ‘uphill’ steps if all else has failed (including resetting H)

r Stringent convergence criterial Simultaneous convergence in enthalpy, max. modulus of force on any ion,

max. component of stress tensor, max. displacement of any ion

l Over a ‘window’ of successive iterations


Future Directions

r Transition State Search

l Find saddle point structures not just stable ones

r Redundant Internal Coordinates

l Transform to normal coordinates and then only optimise the non-redundant degrees of freedom

l Recent theoretical breakthrough in application to crystals/extended systems

l Need to see how it compares to the new BFGS

r Variable cell damped MD


Conclusions

r Two independent algorithms have been implemented within new CASTEP for structural optimisation

r Damped MDl Can be very efficient depending on γl Only works with fixed cell parameters (so far!)

r BFGSl Can do either/or/both ionic positions and cell parameter

optimisation

l Can re-use calculated B and/or ω0 from earlier runs to accelerate convergence in later runs

Lecture 15: Structural Calculations and Pressure

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Transcript of Lecture 15: Structural Calculations and Pressure