A Standard Tool for Density-Functional Calculations · A Standard Tool for Density-Functional...

FHI Workshop 2003 1

Jörg Behler

DMol3

A Standard Tool for Density-Functional Calculations

Fritz-Haber-Institut der Max-Planck-Gesellschaft

Berlin, Germany

Paul-Scherrer-Institut,Zürich, 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)

FHI Workshop 2003 3

“DMol” = density functional calculations on molecules

History

• first fragments early 1980’s(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...

FHI Workshop 2003 4

• Gaussian type orbitals (GTO’s)• Slater type orbitals (STO’s)• numerical atomic orbitals (AO’s)

Localized Basis Sets

Most DFT codes use numerical integration (i.e. for XC-part)⇒ in DMol3: numerical techniques used wherever possible⇒ AO‘s 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

FHI Workshop 2003 6

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)

FHI Workshop 2003 7

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

ctio

n

Al 1s

Al 2s

Al 2p

Al 3s

Al 3p

Al+ 3s

Al+ 3p

3d z=5

4f z=7.5

3p z=4

FHI Workshop 2003 8

Atomic orbitals

Basis Set ConvergencePlane waves

quality ⇒ basis set extension ⇒ rcut

quality ⇒ Ecut

extension ⇒ 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

Eto

t / H

a

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

FHI Workshop 2003 10

• 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


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 ϑ and φ


Partition Functions

Partition functions pα are used to rewrite integrals over all space:

( ) ( ) ( ) ( )∫ ∫∑∫∑ == rrrrrrr dpfdfdfα

αα

αα

1. Step: Decomposition into atomic contributions

Enables integration using spherical polar coordinates!

( ) ( )∫ ∑∫=α

αα rrrr dfdf⇒ Sum of atomic integrals


Partition Functions

Example:

Definition: The partition function for a center α:

( ) ( )( )∑

=

ββ

αα rg

rgrp ∑ =α

α 1p

• choose one peaked function gα for each center α

• calculate the partition function for each center α

Normalization:

αg

αR

2

=

α

αα

ρr

gExample:


-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


Numerical IntegrationExample: O atom

points per shell

012345678

0 10 20 30shell number

r / b

ohr

26501101941

( ) 30214 31

=+⋅⋅= zsn

Number of radial shells

Radial Integration Meshes

z = atomic numbers = scaling factor


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


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 each

point • in a molecule or solid a

superposition of atomicmeshes is used

3287 points


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

Eto

t / e

V

fine gridcoarse grid


ElectrostaticsClassical electrostatic contribution in Hamiltonian

[ ] ( ) ( )21

12

21

21 rrrr dd

rJ ∫ ∫=

ρρρ

Hartee-Fock

DFT

( ) ( ) ( ) ( )21

12

2211

1 1

rrrrrr

ddr

PJL L

∫ ∫∑∑= =

= σλνµ

λ σλσµν

χχχχ

Matrix elements

( ) ( ) ( )21

12

211 rrrrr

ddr

J ∫ ∫=ρχχ νµ

µν


Electrostatics

Density-fitting using an auxiliary basis set { }kω

( ) ( ) ( )∑=≈k

kkc rrr ωρρ ~ ( ) Nd =∫ rrρ~

constraint

The basis functions are of the same type as for the wavefunction expansion.

{ }kω

( ) ( ) ( )21

12

211 rrrrr

ddr

cJk

k ∫ ∫∑=ωχχ νµ

µν

auxiliary density

Matrix elements

Common procedure in DFT using localized basis functions:

O(N3)


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 Poisson’s 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!


Electrostatics

2. Step: Multipole expansion of each ρα

( )rlmαρ• truncation of expansion at lmax• reduction to 1-dimensional radial density

1. Step: Partitioning of total density ρ into atomic densities ρα

⇒ like numerical integration

( ) ( ) ( ) φϑφϑρφϑπ

ρ ααα ddrYl

r lmlm ,,,12

141

∫+=

( ) ( ) ( )∑ +≈max

,124,,l

lmlmlm Yrlr φϑρπφϑρ ααα


Electrostatics

Single center Poisson’s equation

( ) ( )φϑπρφϑ αα ,,4,,2 rrV −=∇

2

2

2

22 11

rl

rrr−

∂∂

=∇

( ) ( ) ( )∑=max

,1,,l

lmlmlm YrV

rrV φϑφϑ ααspherical Laplacian

potential expansion

∑= lmαα ρρ

⇒ set of equations for numerical evaluation of all

density decomposition

( )rV lmα

3. Step: Calculation of the potential contributions


Electrostatics

For periodic boundary conditions an Ewald summation is included.

( )φϑα ,,rV ⇒ ( )rVatomic mesh ⇒ full mesh

4. Step: Construction of the total potential fromatomic contributions

O(N2)Calculation of the electrostatic potential:


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


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 ραlm• 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.


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:


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

[ ] [ ] [ ] [ ] nn

N

iXCXCHiiHarris ErdEEfE +−+−= ∑ ∫

=1

3ρρµρρερ

Kohn Sham energy functional

[ ] [ ] [ ] [ ] [ ] nnextXCHKS EEEETE ++++= ρρρρρ


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


• 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 ...


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


• 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. Lüthi, J. Chem. Phys. 100 (1994) 5785

References

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