DMol3

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)

FHI Workshop 2003 3

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...

FHI Workshop 2003 4

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

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

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

FHI Workshop 2003 8

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


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


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


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:


-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 / boh

r

26501101941

( ) 30214 31 =+= zsnNumber 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 eachpoint

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

Etot / eV

fine gridcoarse grid


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


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)


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!


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


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


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:


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 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.


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

[ ] [ ] [ ] [ ] nnN

iXCXCHiiHarris ErdEEfE +-+-=

=1

3rrmrrer

Kohn Sham energy functional

[ ] [ ] [ ] [ ] [ ] nnextXCHKS EEEETE ++++= rrrrr


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. Lthi, J. Chem. Phys. 100 (1994) 5785

References

Further Details:

Introduction:

DMol3

Documents

Transcript of DMol3