Making Electronic Structure Theory work Ralf Gehrke Fritz-Haber-Institut der Max-Planck-Gesellschaft Berlin, 23 th June 2009 Hands-on Tutorial on Ab Initio Molecular Simulations: Toward a First-Principles Understanding of Materials Properties and Functions Achieving Self-Consistency Density Mixing Linear Mixing, Pulay Mixer, Broyden Mixer Electronic Smearing Fermi-Dirac, Gaussian, Methfessel-Paxton Preconditioning Direct Minimization Spin Polarization Energy Derivatives Local Atomic Structure Optimization Outlook on Global Structure Optimization Outline
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Making Electronic Structure Theory work...2009/06/24 · Broyden-Fletcher-Goldfarb-Shannon (BFGS) Cartesian vs. Internal Coordinates Outlook on Global Structure Optimization Outline
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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
� 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 �
Achieving Self-Consistency
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 �
Achieving Self-Consistency
Density Mixing
Let's try some damping...
Achieving Self-Consistency
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 �
Achieving Self-Consistency
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 �
Achieving Self-Consistency
Density Mixing
?
but we can still do better...
Achieving Self-Consistency
Density Mixing
N2 = 1.1 Å PBE
Achieving Self-Consistency
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
Achieving Self-Consistency
The Pulay Mixer
ninopt��
�n in
� � ���1
R ninopt ��R� �n in
� ���� �R n in���
� take more previous densities into account under the constraint of norm conservation
� main assumption: the charge density residual is linear w.r.t. to the density
P. Pulay, Chem. Phys. Lett. 73 (1980), 393
Achieving Self-Consistency
The Pulay Mixer
ninopt��
�n in
� � ���1
R ninopt ��R� �n in
� ���� �R n in���
�
��l��R nin
opt � R n inopt �����
���0
� take more previous densities into account under the constraint of norm conservation
� 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 �
P. Pulay, Chem. Phys. Lett. 73 (1980), 393
Achieving Self-Consistency
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
P. Pulay, Chem. Phys. Lett. 73 (1980), 393
Achieving Self-Consistency
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
� What to do with Etot and its derivativelocal structural relaxation (steepest-descent, conjugate gradient, BFGS)global structure optimization (Basin-Hopping, Genetic Algorithms,...)
� What to do with Etot and its derivativelocal structural relaxation (steepest-descent, conjugate gradient, BFGS)global structure optimization (Basin-Hopping, Genetic Algorithms,...)