Bader Analysis: Calculating the Charge on Individual...

Bader Analysis: Calculating the Charge onIndividual Atoms in Molecules & Crystals

by Egill Skúlason

Protonated water-layer : q = + 0.64 e

Pt(111) + 7/6 ML Hads : q = - 0.64 e

1

Overview• Introduction

• Electron Density

• Bader Analysis– Critical Points in ρ and their Classification

– Gradient Vector Field of ρ and “Zero-flux” Surfaces

– Laplacian of ρ and the Lewis and VSEPR Models

– Algorithms and Applications

• Mulliken Analysis

• Comparison Between Different Schemes

2

Introduction• Atomic charges in molecules or solids are not

observables and, therefore, not defined by quantummechanical theory.

• The output of quantum mechanical calculations iscontinuous electronic charge density and it is notclear how one should partition electrons amongstfragments of the system such as atoms or molecules.

• Many different schemes have been proposed, somebased on electronic orbitals:– Mulliken population analysis– Density matrix based normal population analysis

and others based on only the charge density:– Bader analysis– Hirshfeld analysis

3

Electron Density of EtheneC = C

H

HH

HPlane containing the2 C and 4 H nuclei

http://www.chemistry.mcmaster.ca/faculty/bader/aim/aim_1.html

Contour Map Portrayed as a projection in the third dimension

cusp

Absolutemaximanot shown

Local maxima in theelectronic charge isat the position of thenuclei

Similar features of the electron density are observed for crystals as for molecules

4

Bader Analysis• The electron density, ρ(x, y, z), of materials is analyzed.• Critical points of ρ(x, y, z) are determined and classified.• The 3D space is divided into subsystems, each usually

containing 1 nucleus (but sometimes none).• The subsystems are separated by “zero-flux” surfaces:

∇ρ(rs) • n(rs) = 0 for every point rs on the surface S(rs)

where n(rs) is the unit vector normal to the surface at rs

• The electron density can either be from experimental data (e.g.X-ray crystallography) or from theoretical data (e.g. ab initiocalculations).

R.F.W. Bader, Atoms in Molecules - A quantum theory, Oxford University Press, New York, 1990.

5

Critical Points and Hessian of ρCritical points of ρ(r): maximum, minimum or saddle where thegradient of ρ(r) vanish (∇ρ(rc) = 0), where

Hessian of ρ at a critical point:

The Hessian matrix is realand symmetric=> we can put it in adiagonal form:

crrzyzxz

zyyxy

zxyxx

)r(A

2

222

2

2

22

22

2

2

c

=

∂

∂

∂∂

∂

∂∂

∂

∂∂

∂

∂

∂

∂∂

∂

∂∂

∂

∂∂

∂

∂

∂

=

ρρρ

ρρρ

ρρρ

=

=

∂

∂

∂

∂

∂

∂

=

3

2

1

2

2

2

2

2

2

00

00

00

crrz00

0y

0

00x

λ

λ

λ

ρ

ρ

ρ

Λ

R.F.W. Bader, Atoms in Molecules - A quantum theory, Oxford University Press, New York, 1990.

eigenvalues = curvatures of ρ

6

Classification of the Critical PointsCP’s are labeled: (rank, signature)

Only 4 possible signature values for critical points of rank = 3 :(3, -3) : Nuclear Attractor (NA)(3, -1) : Bond Critical Point (BCP)(3, +1) : Ring Critical Point (RCP)(3, +3) : Cage Critical Point (CCP)

CCP

BCP

NARCP

B2H6C4H4

7

Gradient Vector Field of the Electron Density• Vector pointing in the direction of maximum increase in ρ• One makes an infinitesimal step in this direction and then

recalculates the gradient to obtain the new direction• By continued repetition of the process, one traces out a

trajectory of ∇ρ(r)• Sets of trajectories terminate where the density is at

maximum (each nucleus)• The space of the molecule is partitioned into basins (atoms)

C = CH

HH

H

Ethene


8

Zero-flux Surfaces• An atom can be defined as a region of real space bounded by

surfaces through which there is no flux in the gradient vector fieldof ρ, meaning that the surface is not crossed by any trajectories of∇ρ(rs)

• An interatomic surface (IAS) satisfies the “zero-flux” boundarycondition:

∇ρ(rs) • n(rs) = 0 for every point rs on the surface S(rs)

where n(rs) is the unit vector normal to the surface at rs• At a point on a dividing surface the

gradient of the electron density hasno component normal to the surface.

Contour map ofNaCl overlaid withtrajectories of ∇ρ

R.F.W. Bader, Atoms in Molecules -A quantum theory, Oxford UniversityPress, New York, 1990.

9

Interatomic Surfaces


• The Bader atoms consist of regions where there is no flux in thegradient vector field of the electronic density.

• Fig: The 2nd-row hydrides; AHn where A = Li, Be, B, C, N, O and F.

Note the change in the size and form of the H atom, from the hydride ionin Li+H- to the positively charged one in H+F-

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Laplacian of ρ and the Lewis & VSEPR Models

Equatorial plane of ClF3

Laplacian:∇2ρ = δ2ρ/δ x2+ δ2ρ/δy2 + δ2ρ/ δz2

∇2ρ < 0 : ρ locally concentrated∇2ρ > 0 : ρ locally depleted

- ∇2ρ

Cl: 3 quantumshells

F: 2 quantumshells

The Laplacian of the electron densityprovides evidence for the localized lonepairs of the VSEPR model

chemistry.mcmaster.ca/faculty/bader/aim/aim_5.html

The Laplacian of ρ recovers the shellstructure of an atom (in agreementwith Lewis Octet Theory)

VSEPR: All electron pairs repel each other.Bonding and lone pairs push apart as faras possible.

Topology of ρ : atoms, bonds and structure.No indication of maxima in ρ correspondingto the electron pairs of the Lewis model

2 Lone pairsBonded chargeconcentration

11

Laplacian of ρ and Lewis Acid & Base

A local charge concentration isa Lewis base or a nucleophile

A local charge depletion is aLewis acid or an electrophile

A chemical reactioncorresponds to thecombination of a ‘lump’ inthe VSCC of the basecombining with the ‘hole’ inthe VSCC of the acid

O=CNon-bonded charge on C

(Lewis base)

BH3Hole on B

(Lewis acid)

Zero envelope of ∇2ρ

chemistry.mcmaster.ca/faculty/bader/aim/aim_5.html

12

Algorithms for Bader Analysis• Commonly used implementations involve:

– Finding the CP’s (accurately) of the charge density where ∇ρ = 0(can involve interpolations between grid points and solving non-linear equations, e.g. Newton’s method or Bisection)

– Construction of the zero-flux surfaces explicitly (expensive)– Integration of the electron density within each region

(e.g. 3D integration from the nucleus)– Packages: AIM2000, TopMoD, InteGriTy, Extreme 94 &

MORPHY– Usually these programs have a nice interface where one can get

nice figures of the electronic density, the gradient vector field, theLaplacian, IAS, CP’s etc.

• However, these algorithms are known to have convergenceproblems in some cases

13

Fast and Robust Algorithm for BaderDecomposition of Charge Density

Charge density grid -> Steepest ascent path -> Bader region -> Sum up

2D : 8 neighboring grid points3D : 26 neighboring grid points

Steepest ascent move to max ∇ρ :

Change in density :

Change in distance :

Code: http://theory.cm.utexas.edu/bader/ Forum: http://theory.cm.utexas.edu/forum/

di, dj, dk are each assigned the values{-1, 0, 1} but excluding di = dj = dk = 0

G. Henkelman, A. Arnaldsson, H. Jónsson, Comp. Mat. Sc. (2006)

14

Fast and Robust Algorithm for BaderDecomposition of Charge Density

• No effort in trying to find CP’s of ρ or the accurate shapeof the IAS.

• No 3D integration (only summation of the charge densityon the grid points for each region)

• Only searching for the local maximum in ρ on the 3D grid.• One can get e.g. partial charges and dipole moments of

individual atoms in molecules or crystals.• Although the dividing surfaces are not found explicitly, it

is easy to render them for visualization after the analysis iscomplete.

• The algorithm scales linearly with number of grid points,(not with e.g. number of atoms or CP’s)

G. Henkelman, A. Arnaldsson, H. Jónsson, Comp. Mat. Sc. (2006)

15

Results: Water MoleculeCharge density: aug-cc-pVDZ basis, Gaussian 98, MP2 level

3 Bader regions found, each containing one atom

G. Henkelman et al., Comp. Mat. Sc. (2006)

16

Results: Boron Clusters in Silicon Crystals

Charge density: DFT, plane wave basis,VASP, ultra-soft PsP, PW9162 Si, 3 B, extra electron added

Total charge of the B3-cluster: 10.5 e-

Overall charge of the cluster is -1.5 eSingle B atom in Si crystal has -0.9 e

Formation of B3 clusters reducesboron electronic activity by ca. 50%


17

Scaling of Effort


Data: Boron cluster insilicon with differentnumber of grid points

Time: 3 - 45 sec on1.8 GHz Athlonbased computer.

3 atom H2O systemrequired the samecomputational effort asthe 65 atom B-Si systemwith similar grid size.

-> The computer timeneither depends upon thenumber of atoms in thesystem nor the bondingtopology.

Scales linearly with number of grid points

18

Bader Analysis at CAMP(from e.g. Dacapo, Siesta or Grid-PAW)

from Dacapo import Dacapofrom ASE.IO.Cube import *

atoms = Dacapo.ReadAtoms('filename.nc')calc = atoms.GetCalculator()dens = calc.GetDensityArray()density = dens * (0.529177)**3

WriteCube(atoms, density, 'filename.cube')

Bader filename.cube

One gets e.g. Charges on each atomDipole moment on each atomBader volume around each atomBader maxima for each atom etc.

19

Bader Analysis from Dacapo Calculations

9.16 e-

OH 0.42 e-0.42 e- H

8.00 e-

OH 0.00 e-0.00 e- H

All electron code (Gaussian): Pseudopotential code (Dacapo):

Partial Charge:O: - 1.16H: + 0.58

Partial Charge:O: - 2.00H: + 1.00

Problems with Bader analysis on O-H groups in pseudopotential codes

We could use Grid-PAW in near future to solve this problem

Protonated water-layer : q = + 0.64

Pt(111) + 7/6 ML Hads : q = - 0.64Hontop : 1.17 e-

However, …


20

Mulliken Method• Can be applied when basis functions centered on atoms are used in the

calculation of the electronic wavefunction of the system.• The charge associated with the basis functions centered on a

particular atom is then assigned to that atom.• This can be a fast and useful way of determining partial charges on

atoms but it has the major drawback that the analysis is sensitive tothe choice of basis set.

• Mulliken analysis is e.g. used in the Gaussian and the Siesta codes.

mulliken: Atomic and Orbital Populations:

Species: CAtom Qatom Qorb 2s 2s 2py 2pz 2px 2py 2pz 2px 2Pdxy 2Pdyz 2Pdz2 2Pdxz 1 4.300 1.822 -0.027 0.504 0.504 1.006 0.072 0.072 0.193 0.054 0.000 0.011 0.054 0.034

Species: OAtom Qatom Qorb 2s 2s 2py 2pz 2px 2py 2pz 2px 2Pdxy 2Pdyz 2Pdz2 2Pdxz 2 5.700 1.039 0.506 1.164 1.164 1.067 0.191 0.191 0.340 0.014 0.000 0.002 0.014 0.007

mulliken: Qtot = 10.000

Output from Siesta calculations on CO molecule:

Data from Mikkel Strange

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Charge Analysis with Hirshfeld, Mulliken & Bader

De Proft et al, Vol. 23, No. 12, J.Comp. Chem. (2002)

Data points: Series of organic molecules, all values in |e|

Usually the points are located in the (-, -) and (+, +) quadrants of the graphs

Atomic charges: Hirshfeld < Mulliken < BaderBa

der

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Charge Analysis with Hirshfeld, Mulliken & Bader

De Proft et al, Vol. 23, No. 12, J.Comp. Chem. (2002)

Hirshfeld

Selected number of hypervalent compounds, all values in |e|

Bader

Charges distribution: Hirshfeld < Mulliken < Bader

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Charge Analysis with Mulliken, NBO &Bader as a Function of Applied Field

Masunov et al., J. Phys. Chem. A, Vol. 105, No. 19, 2001

Fig: Charge on the H atom of HCN(calculated using three differentconventions) as a function of theapplied electric field.GAMESS (HF/D95** level): optimize thegeometries in a dipolar electric field

The charge on the H-atom inHCN is almost the same byall three methods at zeroapplied field.

The sensitivity of the chargeon the H-atom to the changein applied field follows theorder:Bader > Mulliken > NBO

Bader

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Summary• The main features of the Bader analysis

Atoms in Molecules have been presented.• An example of a simple, fast and robust

Bader analysis algorithm has been given.• People at CAMP can start using that Bader

analysis algorithm on their systems.• A comparison of different charge analysis

schemes was presented.

Bader Analysis: Calculating the Charge on Individual...

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