Making Electronic Structure Theory work...2009/06/24 · Broyden-Fletcher-Goldfarb-Shannon (BFGS) Cartesian vs. Internal Coordinates Outlook on Global Structure Optimization Outline

Making Electronic Structure Theory work

Ralf Gehrke

Fritz-Haber-Institut der Max-Planck-GesellschaftBerlin, 23th June 2009

Hands-on Tutorial on Ab Initio Molecular Simulations:Toward a First-Principles Understanding of Materials Properties and Functions

� Achieving Self-ConsistencyDensity Mixing� Linear Mixing, Pulay Mixer, Broyden Mixer

Electronic Smearing� Fermi-Dirac, Gaussian, Methfessel-Paxton

PreconditioningDirect Minimization

� Spin Polarization� Energy Derivatives� Local Atomic Structure Optimization� Outlook on Global Structure Optimization

Outline

� Achieving Self-Consistency� Spin Polarization

Fixed Spin vs. Unconstrained Spin Calculation� Energy Derivatives� Local Atomic Structure Optimization� Outlook on Global Structure Optimization

Outline

Outline� Achieving Self-Consistency� Spin Polarization� Energy Derivatives

Atomic ForcesNumeric vs. Analytic Second DerivativeVibrational Frequencies

� Local Atomic Structure Optimization� Outlook on Global Structure Optimization

Outline� Achieving Self-Consistency� Spin Polarization� Energy Derivatives� Local Atomic Structure Optimization

Steepest DescentConjugate GradientBroyden-Fletcher-Goldfarb-Shannon (BFGS)Cartesian vs. Internal Coordinates

� Outlook on Global Structure Optimization

Outline� Achieving Self-Consistency� Spin Polarization� Energy Derivatives� Local Atomic Structure Optimization� Outlook on Global Structure Optimization

Achieving Self-Consistency

Density Mixing

�hKS��2

2�V eff � n in

�� ,r �

�hKS �l ��l �l �

R nin��nout

� ��n in�� ,r ��n in

��r�� n �r �

� n �� n�2 d3r � � ?

finished !

n in�0�

yes

no

nout��r��

l�l

2 �r �


Density Mixing

charge sloshing !

N2 = 1.1 Å PBE

nin��1 ��nout

��

�hKS��2

2�V eff � n in

�� ,r �

�hKS �l ��l �l �

R nin��nout

� ��n in�� ,r ��n in

��r�� n �r �

� n �� n�2 d3r � � ?

nin��1 �� nout

��1��nin��

finished !

n in�0�

yes

no

nout��r��

l�l

2 �r �


Density Mixing

Let's try some damping...


Density Mixing

�=0.5

N2 = 1.1 Å PBE

nin��1 �� nout

��1��nin��

seems to work !

�hKS��2

2�V eff � n in

�� ,r �

�hKS �l ��l �l �

R nin��nout

� ��n in�� ,r ��n in

��r�� n �r �

� n �� n�2 d3r � � ?

nin��1 �� f �nin

�� ,n in��1� ,n in

��2 �,�,nout�� ,nout

��1 �,nout� j�2 � ,��

finished !

n in�0�

yes

no

nout��r��

l�l

2 �r �


Density Mixingbut we can still do better...

�hKS��2

2�V eff � n in

�� ,r �

�hKS �l ��l �l �

R nin��nout

� ��n in�� ,r ��n in

��r�� n �r �

� n �� n�2 d3r � � ?

nin��1 �� f �nin

�� ,n in��1� ,n in

��2 �,�,nout�� ,nout

��1 �,nout� j�2 � ,��

finished !

n in�0�

yes

no

nout��r��

l�l

2 �r �


Density Mixing

?

but we can still do better...


Density Mixing

N2 = 1.1 Å PBE


The Pulay Mixer

ninopt��

�n in

� � ��1

� take more previous densities into account under the constraint of norm conservation

P. Pulay, Chem. Phys. Lett. 73 (1980), 393


The Pulay Mixer

ninopt��

�n in

� � ��1

R ninopt ��R� �n in

� �� R n in��


� main assumption: the charge density residual is linear w.r.t. to the density



The Pulay Mixer

ninopt��

�n in

� � ��1

R ninopt ��R� �n in

� �� R n in��

�

��l��R nin

opt � R n inopt ��

��0


� main assumption: the charge density residual is linear w.r.t. to the density

� coefficients � are determined by minimizing the residual

system of equations for �



The Pulay Mixer

nin�1 � nin

�2� nnin

opt

�R R�R1�R n in

�1 ��R2�R nin

�2 � �

�R n inopt � R n in

opt � �� R1��1��R2�2

� illustration in one dimension with two densities



The Pulay Mixer

nin�1 � nin

�2� nnin

opt

�R R�R1�R n in

�1 ��R2�R nin

�2 � �

�R n inopt � R n in

opt � �� R1��1��R2�2

nin��1 ��n in

opt�� R n inopt �nin

�1 �

nin�2�

ninopt

n true

xnin�3�

� illustration in one dimension with two densities

� to prevent trapping in subspace, add fraction of residual



The Pulay Mixer


opt� �GR ninopt � �G�A k2

k2�k 02

preconditioning matrix:

� additional preconditioner might accelerate convergence


long-range components of density changes are stronger damped than short-range

G. Kerker, Phys. Rev. B 23 (1981), 3082


The Pulay Mixer

n� � r ��n�� r� n� � r ��n�� r�


opt� �GR ninopt � �G�A k2

k2�k 02

preconditioning matrix:

�R n� ,n� � R n� ,n� � �� d r �� n� �

2�r � � n� �r��nout

� �n in� � , �r ��n in

� �r�

� additional preconditioner might accelerate convergence

long-range components of density changes are stronger damped than short-range

� two spin channels are to be mixed in the case of spin polarization

charge density: spin density:

combine both channels to one matrix !

stable convergenceP. Pulay, Chem. Phys. Lett. 73 (1980), 393

G. Kerker, Phys. Rev. B 23 (1981), 3082


The Broyden Mixer

f �xn�� f �x n�1� f ' �x n��xn�xn�1�

f �xn�1��! 0

xn�1�x n�1f ' �xn�

f �x n�xn�1xn

xn�1

f �xn�

f �xn�1�

secant method


The Broyden Mixer

f �xn�� f �x n�1� f ' �x n��xn�xn�1�

f �xn�1��! 0

xn�1�x n�1f ' �xn�

f �x n�

n in��1 ��n in

��G�� Rn in� ��

� nin��G��Rn in

� ��0

xn�1xnxn�1

f �xn�

f �xn�1�

secant method

approximate Jacobian:

D.D. Johnson, Phys. Rev. B 38 (1988), 12807


The Broyden Mixer

f �xn�� f �x n�1� f ' �x n��xn�xn�1�

f �xn�1��! 0

xn�1�x n�1f ' �xn�

f �x n�


��G�� Rn in� ��


� ��0�

�Gij!G��1 ��G� �!�0

G��1��G� �� n ��G�� R�� R��

xn�1xnxn�1

f �xn�

f �xn�1�

secant method


under-determined

“as little change as possible”



The Broyden Mixer

f �xn�� f �x n�1� f ' �x n��xn�xn�1�

f �xn�1��! 0

xn�1�x n�1f ' �xn�

f �x n�


��G�� Rn in� ��


� ��0�

�Gij!G��1 ��G� �!�0

G��1��G� �� n ��G�� R�� R��

E�w0!G��1��G��!

2��

iwi!� n

�i ��G��1��R�i �!2

xn�1xnxn�1

f �xn�

f �xn�1�

secant method


under-determined

“as little change as possible”

modified Broyden: minimize



Electronic Smearing� level crossing may lead to charge sloshing

EF

�i

��

�i �i

��1� ��2 � .......

EF n�r��l

N occ

�l2 �r �


Electronic Smearing� level crossing may lead to charge sloshing

EF

�i

��

�i �i

��1� ��2 � .......

EF

EF

�i �i �i

EF

f l f l f l1 1 1

� electronic smearing softens the change of electronic density

n�r��l

N occ

�l2 �r �

n�r��lf l�l

2�r�


Electronic Smearing

f � ��E F� �� 1

exp ��E F�"��1Fermi-Dirac1:

f � ��E F� ��1

2 �1�erf ��EF� ��Gaussian2:

f 0�x ��12�1�erf �x ��

f N �x �� f 0� x��m�1

N

Am H2m�1�x �ex2

Methfessel-Paxton3:

N#$%�x �

x��EF�

1 N. Mermin, Phys. Rev. 137 (1965), A14412 C.-L. Fu, K.-H. Ho, Phys. Rev. B 28 (1983), 54803 M. Methfessel, A. Paxton, Phys. Rev. B 40 (1989), 3616


Electronic Smearing

&E { c}, { f }�&E { c}� � &E� f i n0

'0

� Kohn-Sham functional not a variational quantity anymore


Electronic Smearing� Kohn-Sham functional not a variational quantity anymore

� electronic free energy F is variational

&E { c},{ f }�&E { c}� � &E� f i n0

'0

F�E�� S �� ,{ f n}�

entropy term

� F� f i n0

�0


Electronic Smearing

� F� f i n0

�0F�E�� S �� ,{ f n}�

F ��E��0�( O��

2�

O�� 2�N �E��0 (

12�F ��E ��

1N�2

��N�1�F ��E ��

� Kohn-Sham functional not a variational quantity anymore

� electronic free energy F is variational

� systems with continuous density of states at the Fermi edge:

Fermi-Dirac / Gaussian

Methfessel-Paxton

desired total energy E for zero smearing can be backextrapolated

entropy term

&E { c},{ f }�&E { c}� � &E� f i n0

'0

M. Methfessel, A. Paxton, Phys. Rev. B 40 (1989), 3616 M. Gillan, J. Phys.: Condens. Matter 1 (1989), 689

F. Wagner, T. Laloyaux, M. Scheffler, Phys. Rev. B 57 (1998), 2102


Alternative: Direct Minimization

� &E� �)l

� gl �� f l � �hKS��l � )l �

)l��1� �� )l�� gl �

&E�EKS��l�l ��)l )l��1 ��EF ��l f l�N �

EKS

()l *

Etot

gl �

� constrained minimum of Kohn-Sham functional:

� gradient:

� steepest-descent:

Spin Polarization

Fixed Spin vs. Unconstrained Spin

EF�

EF� EF

N��i �

f i� � �i��EF

w �N ��i

f i� � �i

��EF�

w � N ��i

f i�� i

��EF�

w �M��

i� f i

�� fi� �M�N ��N �

total spin is relaxedtotal multiplicity kept fixed

�12�2�V eff

� �r ��)i��i� )i� V XC� �r ��

� EXC

� n� �r�

Spin Polarization

Co3 PBE

obtained with unconstrained spin

fixed spin

obtained multiplicity the lowest one

Fixed Spin vs. Unconstrained Spin

Energy Derivatives

Atomic Forces

E tot�min(c* �E

KS��i��i� ��)i � )i� ��1�� &E (c0 (R�**, (� (R�**, (R�*�

� total energy of the system is the total minimum of the Kohn-Sham-functional under the normalization constraint:

� &E� (c* c�c

0

�0� &E� (�* c�c

0

�0

Energy Derivatives

Atomic Forces

E tot�min(c* �E

KS��i��i� ��)i � )i� ��1�� &E (c0 (R�**, (� (R�**, (R�*�

� total energy of the system is the total minimum of the Kohn-Sham-functional under the normalization constraint:

� since the Kohn-Sham-functional is variational, only the partial derivative remains:

F� ��d E tot

dR��

=� �&E

�R��(c

0*

� &E� (c0*+

=0

�(c0*

� R��

(�*

� &E�(�*+

=0

� (�*

�R�

=� �&E

�R��

�EKS

�R��

�R��i��i� � �)i� )i� ��1�

� &E� (c* c�c

0

�0� &E� (�* c�c

0

�0

Energy Derivatives

The Individual Force Contributions

FHF ,��Z ��,'�

Z,R��R,

R��R, 3�Z��d r n �r � R��r

R��r 3

FPulay ,��2�l� � ��l��R� �hKS��l� �l��

FMP ,�� d r �n � r ��nMP� r��V H n

MP ��R��

FGGA ,��d r n �r�� XC

� � n 2

� � n 2

�R�

� classical Coulomb interaction:

� Pulay-correction due to incomplete or nuclear-dependent basis set:

“moving basis functions”

� Multipole-correction due to “incomplete” Hartree-potential:

“moving multipole components”

� gradient-dependency of exchange-correlation energy in case of a GGA functional:

Energy Derivatives

The Convergence of the Forces� total energy is variational error in the density enters total energy to second order

� atomic forces are not variational error in the density enters total energy to first order

convergence of atomic forces more expensive than total energy

� &E� (c* c�c

0

�0

�F�� (c* c�c

0

��2 &E

�R� � (c* c�c0'0N-N (d=1.0 Å)

total energyforces

Energy Derivatives

Forces and Electronic Smearing

&E { c},{� }, {R }, { f } ,EF�� &E� f i n0

'0

F��d E tot

dR��

=� �&E

�R��

( f *

� &E�( f *+'0

� ( f *�R�

�� &E� EF+

= 0

�EF�R�

'�� &E�R�

� F� f i n0

�0 F�� F�R�

� Kohn-Sham functional not variational w.r.t. the occupation numbers

atomic forces are not consistent with the total energy of the system !

atomic forces are consistent with the electronic free energy

M. Weinert and J.W. Davenport, Phys. Rev. B 45 (1992), 13709 R.M. Wentzcovitch, J.L. Martin and P.B. Allen, Phys. Rev. B 45 (1992), 11372

Energy Derivatives

The Second Derivative and Vibrations

F� �� &E�R��

� first derivative was simple

analytical

Energy Derivatives


F� �� &E�R��


� second derivative more nasty

analytical

e.g. A. Komornicki and G. Fitzgerald, J. Chem. Phys. 98 (2), 1398 (1993)

H� ,�d2 E tot

dR� dR,��d F�dR,

��2 &E

�R� �R,��

(c*

� 2 &E�R� � (c *

�(c0*

�R,

Energy Derivatives


F� �� &E�R��

H� ,�d2 E tot

dR� dR,��d F�dR,

��2 &E

�R� �R,��

(c*

� 2 &E�R� � (c *

�(c0*

�R,

�(c*

�2 E� (c*2

� (c0*

�R��

�2 E�R� �(c *


� second derivative more nasty

� set of coupled equations needs to be solved

analytical

e.g. A. Komornicki and G. Fitzgerald, J. Chem. Phys. 98 (2), 1398 (1993)

Energy Derivatives

The Second Derivative and Vibrationsnumerical

d F�d R,

F� �R , , 0��F��R, , 0��

2�R, ,0

M � -R��ddR�

E �(R*� E �(R*��E �(R0*��12��,

d2 Ed R�d R, (R 0*

u�u, u��R��R� , 0

u� �t ��u� ei. t det �H�,�.2M� ��! 0

vibrations

talk by Gert von Helden, Friday, 26th June, 11:30, “Fingerprinting molecules: Vibrational Spectroscopy, Experiment and Theory”

Local Atomic Structure OptimizationPotential Energy Surface (PES)

E tot � (R�*�

Local Atomic Structure OptimizationPotential Energy Surface (PES)

E tot � (R�*�

Local Atomic Structure Optimization

R� , i�1�R� ,i��i F�� (R� , i*�

Steepest Descent


R� , i�1�R� ,i��i F�� (R� , i*�

Steepest Descent

(R�*

too small �too large �


R� , i�1�R� ,i��i F�� (R� , i*�

Steepest Descent

(R�*


ill-conditioned PES might lead to zig-zag behaviour


R� , i�1�R� ,i��i F�� (R� , i*�

R� , i�1�R� ,i��iG� ,i �(R� , i*� G

� ,i�F� ,i�,iG� ,i�1 ,iFletcher-Reeves�

F� , i/ F� ,i

F� , i�1/ F� ,i�1

Steepest Descent

Conjugate Gradient(R�*


� take information of previous search directions into account

ill-conditioned PES might lead to zig-zag behaviour

,iPolak-Ribiere�

F� , i/ �F� , i�F� ,i�1�F� , i�1/ F� ,i�1

0

� constructions of search directions based upon the assumption that PES is perfectly harmonic

line minimization


The Broyden-Fletcher-Goldfarb-Shanno Method

(R�*

f �x �� f �x i��x�xi�/� f �xi��12�x�xi�/A/�x�x i��

� f �x�� f �xi��A/�x�x i��

� f �x��! 0 x�xi��A��1/� f �x i�

Newton-Method:� additional to the atomic forces, the Hessian is taken into account:

W. H. Press et al., Numerical Recipes, 3rd Edition, Cambridge University Press


The Broyden-Fletcher-Goldfarb-Shanno Method

R� , i�1�R� ,i��iG� ,i �(R� , i*�

(R�*

f �x �� f �x i��x�xi�/� f �xi��12�x�xi�/A/�x�x i��

� f �x�� f �xi��A/�x�x i��

� f �x��! 0 x�xi��A�1/� f �x i�

Newton-Method:

� “Quasi”-Newton-Scheme: inverse Hessian is approximated iteratively

G� ,i�HiF� , i

H0�1 R� , i�1�R� ,i�Hi�1 /�F� , i�F� ,i�1� Hi�1� Hi��

� additional to the atomic forces, the Hessian is taken into account:

� 1 1 perfectly harmonic PES is assumed� line minimization

W. H. Press et al., Numerical Recipes, 3rd Edition, Cambridge University Press


Cartesian vs. Internal Coordinates

d1

d2

2

(R1 ,R2 ,R3 *# (q1 , q2 , q3* q1�d1 q2�d 2 q3�2

q�B �R�R0 �

P. Pulay, G. Fogarasi, F. Pang, J.E. Boggs, J. Am. Chem. Soc. 101, 2550 (1979)

(3N-6) x (3N) 3 not invertable !R�R �q � difficult!


Cartesian vs. Internal Coordinates

d1

d2

2

(R1 ,R2 ,R3 *# (q1 , q2 , q3* q1�d1 q2�d 2 q3�2

E�E0��i)i qi�

12�ijFij qi q j�

16�ijkFijk qi q j qk��

q�B �R�R0 �

P. Pulay, G. Fogarasi, F. Pang, J.E. Boggs, J. Am. Chem. Soc. 101, 2550 (1979)

)i��E�qi

Fijk�� 3E� qi� q j� qk

(3N-6) x (3N) 3 not invertable !R�R �q � difficult!

as “simple” as possible

faster convergence of the local geometry optimizer


Inverse-Distance Coordinates

J. Baker, P. Pulay, J. Chem. Phys. 105 (24), 11100 (1996)

P.E. Maslen, J. Chem. Phys. 122, 014104 (2005)

� appropriate for weekly bound systems

R






R

q� 1R

�E� q��

� E�R/R2

�2E� q2

��2 E� R2

/R4�2 �E� R/R3






R

q� 1R

�E� q��

� E�R/R2

�2E� q2

��2 E� R2

/R4�2 �E� R/R3

R� , i�1�R� ,i��iG� ,i �(R� , i*�

displacements are accelerated for large interatomic distances

Global Structure OptimizationPotential Energy Surface (PES)

E tot � (R�*�

Global Structure OptimizationPotential Energy Surface (PES)

E tot � (R�*�

Global Structure Optimization

D. J. Wales, J. P. K. Doye, M. A. Miller, P. N. Mortenson, and T. R. Walsh, Adv. Chem. Phys. 115, (2000)

Z. Li, and H. A. Scheraga, Proc. Natl. Acad. Sci. U.S.A. 84, 6611 (1987)

Basin-Hopping� random variations of atomic coordinates followed by local structural relaxations

� many different flavours of “trial moves” possible

Global Structure Optimization

further methods...� Genetic Algorithms1:

two “parents” mate and create a “child”

the fittest (energetically lowest) children “survive”

talk by Matt Probert, Friday, 26th June, 9:00, “Structure predictions in materials science”

1D. M. Deaven, K. M. Ho, Phys. Rev. Lett. 75, 288 (1995)

� Minima-Hopping, Basin-Paving, Conformational Space Annealing,...

Summary� How to get Etot in DFT

density-mixing schemes (Pulay, Broyden)direct minimization

� Taking the derivative of Etot

atomic forcesmolecular vibrations

� What to do with Etot and its derivativelocal structural relaxation (steepest-descent, conjugate gradient, BFGS)global structure optimization (Basin-Hopping, Genetic Algorithms,...)

Summary� How to get Etot in DFT

density-mixing schemes (Pulay, Broyden)direct minimization

� Taking the derivative of Etot

atomic forcesmolecular vibrations

� What to do with Etot and its derivativelocal structural relaxation (steepest-descent, conjugate gradient, BFGS)global structure optimization (Basin-Hopping, Genetic Algorithms,...)

Let's do it !!

today, 14:00 – 18:00first practical session

lecture hall

Documents

Making Electronic Structure Theory work...2009/06/24 · Broyden-Fletcher-Goldfarb-Shannon (BFGS) Cartesian vs. Internal Coordinates Outlook on Global Structure Optimization Outline