FHI Workshop 2003 1
Jrg Behler
DMol3 A Standard Tool for Density-Functional
Calculations
Fritz-Haber-Institut der Max-Planck-Gesellschaft
Berlin, Germany
Paul-Scherrer-Institut,Zrich, Switzerland
Bernard Delley
FHI Workshop 2003 2
What is DMol3?
Introduction
all-electron DFT-code basis functions are localized atomic orbitals (LCAO) can be applied to:
free atoms, molecules and clusters solids and surfaces (slabs)
Topic of this talk
Characteristics of the DMol3 approach to DFT (selected aspects)
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DMol = density functional calculations on molecules
History
first fragments early 1980s(full potential electrostatics,numerical atomic orbitals)
total energy functional 1983 electrostatics by partitioning 1986 first release of DMol by Biosym 1988 forces 1991 parallel version 1992 DSolid 1994 geometry optimization 1996 unification of DMol and DSolid
DMol3 1998 molecular dynamics 2002
Main author:Bernard Delley
Further contributions by:J. Andzelm R.D. King-SmithJ. Baker D. EllisG. Fitzgerald many more...
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Gaussian type orbitals (GTOs) Slater type orbitals (STOs) numerical atomic orbitals (AOs)
Localized Basis Sets
Most DFT codes use numerical integration (i.e. for XC-part) in DMol3: numerical techniques used wherever possible AOs can be used
high accuracy (cusp, asymptotic behaviour) high efficiency (few basis functions small matrix size)
Why?
What?
Idea:
FHI Workshop 2003 5
exact DFT-spherical-atomic orbitals of neutral atoms ions hydrogenic atoms
radial functions are calculated in the setup (free atom) implemented as numerically tabulated functions
Properties: maximum of accuracy for a given basis set size infinitely separated atoms limit treated exactly small number of additional functions needed for polarization square integrability, cusp singularities at the nuclei
DMol3 Basis Functions
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minimal: Al 1s, 2s, 2p, 3s, 3p E = -242.234806 Ha
Basis Set Example: Al
dn: Al2+ 3s, 3p E = -242.234845 Ha
dnd: z = 5 3d E = -242.235603 Ha
dnp: z = 4 3p, z = 7.5 4f E = -242.235649 Ha
9 AOs
13 AOs
18 AOs
28 AOs
Atomic energy
Quality test:lowering of total energy by adding basis functions
(variational principle)
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Basis Set Example: AlRadial Basis Functions
-3.00
-2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
0.0 2.0 4.0 6.0 8.0 10.0
r / bohr
Rad
ial B
asis Fun
ction
Al 1s
Al 2s
Al 2p
Al 3s
Al 3p
Al+ 3s
Al+ 3p
3d z=5
4f z=7.5
3p z=4
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Atomic orbitals
Basis Set ConvergencePlane waves
quality basis set extension rcut
quality Ecutextension infinte
Basis test for the oxygen molecule
O basis functionsO 1sO 2sO 2p O2+ 2sO2+ 2pz = 3 3dz = 5 3d
-150.5600
-150.5595
-150.5590
-150.5585
-150.5580
-150.5575
-150.55706 7 8 9 10 11 12 13 14 15
rcut / bohr
Etot / Ha
dndall
alldnd
FHI Workshop 2003 9
BSSE
DMol3: very small BSSE because of nearly perfect basis set for the separated atoms limit=> excellent description of weak bonds (but DFT limitation)
Basis Set Superposition Error
Definition:BSSE is a lowering of the energy when the electrons of each atom spread into the basis functions provided by the other atoms due to an incomplete basis set.
GTO/STO Codes: BSSE can be a serious problem
Plane waves: BSSE does not appear
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no systematic way to improve basis set quality careful tests required to construct a basis set
Summary Basis Functions
very efficient (small) basis fast calculations allows for calculations without periodic boundary conditions or less dense systems (slabs with large vacuum) easy physical interpretation of basis functions (almost) no basis set superposition error different basis sets for different elements possible
Advantages
Disadvantages
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Numerical Integration3 Step approach
1. Decomposition of the total integral in 3D into a sum of independent integrals for atomic contributions partitioning
2. Decomposition of each atomic integral into a radial and an angular part spherical polar coordinates
3. Integration of the angular part on a sphere or decomposition into separate integrations for J and f
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Partition Functions
Partition functions pa are used to rewrite integrals over all space:
( ) ( ) ( ) ( ) == rrrrrrr dpfdfdfa
aa
aa
1. Step: Decomposition into atomic contributions
Enables integration using spherical polar coordinates!
( ) ( ) =a
aa rrrr dfdf Sum of atomic integrals
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Partition Functions
Example:
Definition: The partition function for a center a:
( ) ( )( )=
bb
aa rg
rgrp =a
a 1p
choose one peaked function ga for each center a
calculate the partition function for each center a
Normalization:
ag
aR
2
=
a
aa
rr
gExample:
FHI Workshop 2003 14
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3-3 -2 -1 0 1 2 3
Partition FunctionsTotal function:
Partition functions: Decomposed functions:
Ap Af Bf
Peaked functions gAand gB:
A B
-3 -2 -1 0 1 2 3
A B
A B
Bp
fAg Bg
1
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Numerical IntegrationExample: O atom
points per shell
012345678
0 10 20 30shell number
r / boh
r
26501101941
( ) 30214 31 =+= zsnNumber of radial shells
Radial Integration Meshes
z = atomic numbers = scaling factor
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Numerical IntegrationExample: O atom
Order 17110 points
Order 23194 points
Order 726 points
Order 1150 points
Angular Integration Meshes
(projections of points on sphere into plane)
integration is done on Lebedev Spheres integration scheme with octahedral symmetry
FHI Workshop 2003 17
Numerical IntegrationExample: O atom (cutoff 10 bohr)
Total Grid: Remarks: user defined grids higher order angular schemes are possible
symmetry is used to reduce number of points
a weight is assigned to eachpoint
in a molecule or solid a superposition of atomicmeshes is used
3287 points
FHI Workshop 2003 18
Numerical IntegrationEffect of mesh quality: total energy of the Al atom
-6600.8
-6600.6
-6600.4
-6600.2
-6600.0
-6599.86 8 10 12 14 16 18 20
cutoff radius / bohr
Etot / eV
fine gridcoarse grid
FHI Workshop 2003 19
ElectrostaticsClassical electrostatic contribution in Hamiltonian
[ ] ( ) ( ) 2112
21
21 rrrr dd
rJ =
rrr
Hartee-Fock
DFT
( ) ( ) ( ) ( )21
12
2211
1 1
rrrrrr
ddr
PJL L
= =
= slnml s
lsmn
cccc
Matrix elements
( ) ( ) ( )21
12
211 rrrrr
ddr
J =rcc nm
mn
FHI Workshop 2003 20
Electrostatics
Density-fitting using an auxiliary basis set { }kw
( ) ( ) ( )=k
kkc rrr wrr ~ ( ) Nd = rrr~constraint
The basis functions are of the same type as for the wavefunction expansion.
{ }kw
( ) ( ) ( )21
12
211 rrrrr
ddr
cJk
k =wcc nm
mn
auxiliary density
Matrix elements
Common procedure in DFT using localized basis functions:
O(N3)
FHI Workshop 2003 21
Approach to electrostatics in DMol3: Overview
Electrostatics
Partitioning: decompose the electron density into atomic componentsStep 1
Step 2Projection onto Ylm functions yields multipoles attached to the atoms
Step 3 Solve Poissons equation for each multipole(only 1-dimensional problem)
Step 4 Assemble electrostatic potential from all multipoles and atoms
DMol3: No basis set required for density expansion!
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Electrostatics
2. Step: Multipole expansion of each ra
( )rlmar truncation of expansion at lmax reduction to 1-dimensional radial density
1. Step: Partitioning of total density r into atomic densities ra like numerical integration
( ) ( ) ( ) fJfJrfJp
r aaa ddrYlr lmlm ,,,12
141
+=
( ) ( ) ( ) +max
,124,,l
lmlmlm Yrlr fJrpfJr aaa
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Electrostatics
Single center Poissons equation
( ) ( )fJprfJ aa ,,4,,2 rrV -=
2
2
2
22 11
rl
rrr-
=
( ) ( ) ( )=max
,1,,l
lmlmlm YrVr
rV fJfJ aaspherical Laplacian
potential expansion
= lmaa rr
set of equations for numerical evaluation of all
density decomposition
( )rV lma
3. Step: Calculation of the potential contributions
FHI Workshop 2003 24
Electrostatics
For periodic boundary conditions an Ewald summation is included.
( )fJa ,,rV ( )rVatomic mesh full mesh
4. Step: Construction of the total potential fromatomic contributions
O(N2)Calculation of the electrostatic potential:
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Real space:
Ewald Summation for Monopoles
Reciprocal space:
Point charges
Background charge
Negative Gaussians
Positive Gaussians
0
q
background charge + positive Gaussians
point charges + negative Gaussians
r
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1. Generalization for lattices with multipoles
Extensions to Ewald Summation
2. Extension to monopoles and multipoles of finite extent
DMol3 contains an extension to lattices of point multipoles located at the atomic sites: applied to the ralm computationally as demanding as for point charge lattices
Assumption: multipoles are located inside a radius rcut
Ewald terms with r < rcut have to be modified in the real space part,because explicit calculation of the radial details of the extended charge distribution is required.
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Scaling (localized basis sets)
Electrostatics
O(N4)Hartree Fock
DFT in general O(N3)
DMol3 molecular case O(N2)DMol3 solid case O(N3/2)
Electrostatics in DMol3 scale almost linearly with system size for large systems
O(N2)Real systems:
FHI Workshop 2003 28
The Harris Functional
approximate DFT calculations for very large systems non-selfconsistent energy calculation (1 iteration only) approximated density = superposition of fragment densities (i.e. atomic densities)
J. Harris, Phys. Rev B 31 (1985) 1770.B. Delley et al., Phys. Rev B 27 (1983) 2132.
Original idea of the Harris functional:
Harris energy functional
[ ] [ ] [ ] [ ] nnN
iXCXCHiiHarris ErdEEfE +-+-=
=1
3rrmrrer
Kohn Sham energy functional
[ ] [ ] [ ] [ ] [ ] nnextXCHKS EEEETE ++++= rrrrr
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Total Energy in DMol3
the Harris functional is stationary at the same density as the Kohn-Sham functional and the two are equal in value at this point
the curvature of EHarris about the stationary point is smaller than the curvature of EKS
the density in the Harris functional does not have to be V-representable
Reduction of numerical noise
-= refatomtotbind EEE
DMol3 uses the Harris functional (scf densities)
Realization: subtraction of atomic densities from total densities in integrands
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forces geometry optimization ab initio molecular dynamics and simulated annealing COSMO (COnductor-like Screening MOdel ) transition state search vibrational frequencies Pulay (DIIS) charge density mixing pseudopotentials (optional)
More features ...
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Conclusion Fast
very small basis sets, matrix diagonalization O(N3) no costs for large vacuum or atom / molecules efficient calculation of electrostatics O(N) calculation of Hamilton and overlap matrix
Universal atom, molecule and cluster calculations solids and slabs with periodic boundary conditions
Accurate results comparable to LAPW next talk
Easy to use atoms are given in Cartesian coordinates
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B. Delley, J. Chem. Phys. 92 (1990) 508 B. Delley in Modern Density Functional Theory: A Tool for Chemistry, Theoretical and Computational Chemistry Vol. 2, Ed. by J. M. Seminario and P. Politzer, Elsevier 1995 B. Delley, J. Chem. Phys. 113 (2000) 7756
B. Delley, Comp. Mat. Sci. 17 (2000) 122 J. Baker, J. Andzelm, A. Scheiner, B. Delley, J. Chem. Phys. 101 (1994) 8894 B. Delley, J. Chem. Phys. 94 (1991) 7245 B. Delley, J. Comp. Chem. 17 (1996) 1152 B. Delley, Int. J. Quant. Chem. 69 (1998) 423 B. Delley, J. Phys. Chem. 100 (1996) 6107 B. Delley, M. Wrinn, H. P. Lthi, J. Chem. Phys. 100 (1994) 5785
References
Further Details:
Introduction: