Bader Analysis: Calculating the Charge on Individual Atoms in

Bader Analysis: Calculating the Charge on
Individual Atoms in Molecules & Crystals
by Egill Skúlason
Protonated water-layer : q = + 0.64 e
Pt(111) + 7/6 ML Hads : q = - 0.64 e
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
1
Introduction
• Atomic charges in molecules or solids are not
observables and, therefore, not defined by quantum
mechanical theory.
• The output of quantum mechanical calculations is
continuous electronic charge density and it is not
clear how one should partition electrons amongst
fragments of the system such as atoms or molecules.
• Many different schemes have been proposed, some
based on electronic orbitals:
– Mulliken population analysis
– Density matrix based normal population analysis
and others based on only the charge density:
– Bader analysis
– Hirshfeld analysis
2
Electron Density of Ethene
Plane containing the
2 C and 4 H nuclei
H
H
C=C
H
Local maxima in the
electronic charge is
at the position of the
nuclei
H
Absolute
maxima
not shown
cusp
Contour Map
Portrayed as a projection in the third dimension
Similar features of the electron density are observed for crystals as for molecules
http://www.chemistry.mcmaster.ca/faculty/bader/aim/aim_1.html
3
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 initio
calculations).
4
R.F.W. Bader, Atoms in Molecules - A quantum theory, Oxford University Press, New York, 1990.
Critical Points and Hessian of ρ
Critical points of ρ(r): maximum, minimum or saddle where the
gradient of ρ(r) vanish (∇ρ(rc) = 0), where
Hessian of ρ at a critical point:
The Hessian matrix is real
and symmetric
=> we can put it in a
diagonal form:
eigenvalues = curvatures of ρ
 ∂2ρ
 2
 ∂ x´

Λ= 0



0

0
∂2ρ
∂ y´ 2
0
 ∂2 ρ ∂2ρ
 2
∂ x∂y
 ∂x
 2
∂ ρ ∂2ρ

A( rc ) =
 ∂ y∂x ∂ y 2
 2
2
∂ ρ ∂ ρ
 ∂ z∂x ∂z∂y

∂2ρ 

∂x∂z 

∂2ρ 
∂ y∂z 

∂2ρ 
∂ z 2  r = r
c

0 

 λ1 0 0 



0 
=  0 λ2 0 

0 0 λ 


3
2
∂ ρ
∂ z´ 2  r´ = r
c
R.F.W. Bader, Atoms in Molecules - A quantum theory, Oxford University Press, New York, 1990.
5
Classification of the Critical Points
CP’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
RCP
NA
B2H6
BCP
C4H4
6
Gradient Vector Field of the Electron Density
Ethene
H
H
C=C
H
H
• 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)
7
http://www.chemistry.mcmaster.ca/faculty/bader/aim/aim_1.html
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 field
of ρ, meaning that the surface is not crossed by any trajectories of
∇ρ(rs)
• An interatomic surface (IAS) satisfies the “zero-flux” boundary
condition:
∇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 has
no component normal to the surface.
R.F.W. Bader, Atoms in Molecules A quantum theory, Oxford University
Press, New York, 1990.
Contour map of
NaCl overlaid with
trajectories of ∇ρ
8
Interatomic Surfaces
• The Bader atoms consist of regions where there is no flux in the
gradient 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 ion
in Li+H- to the positively charged one in H+F9
http://www.chemistry.mcmaster.ca/faculty/bader/aim/aim_3.html
Laplacian of ρ and the Lewis & VSEPR Models
F: 2 quantum
shells
Topology of ρ : atoms, bonds and structure.
No indication of maxima in ρ corresponding
to the electron pairs of the Lewis model
The Laplacian of ρ recovers the shell
structure of an atom (in agreement
with Lewis Octet Theory)
Bonded charge
concentration
Laplacian:
∇2ρ = δ2ρ/δ x2+ δ2ρ/δy2 + δ2ρ/ δz2
Cl: 3 quantum
shells
2 Lone pairs
Equatorial plane of ClF3
- ∇2ρ
VSEPR: All electron pairs repel each other.
Bonding and lone pairs push apart as far
as possible.
The Laplacian of the electron density
provides evidence for the localized lone
pairs of the VSEPR model
∇2ρ < 0 : ρ locally concentrated
∇2ρ > 0 : ρ locally depleted
10
chemistry.mcmaster.ca/faculty/bader/aim/aim_5.html
Laplacian of ρ and Lewis Acid & Base
A local charge concentration is
a Lewis base or a nucleophile
Zero envelope of ∇2ρ
A local charge depletion is a
Lewis acid or an electrophile
A chemical reaction
corresponds to the
combination of a ‘lump’ in
the VSCC of the base
combining with the ‘hole’ in
the VSCC of the acid
O=C
Non-bonded charge on C
(Lewis base)
BH3
Hole on B
(Lewis acid)
11
chemistry.mcmaster.ca/faculty/bader/aim/aim_5.html
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 nonlinear 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, the
Laplacian, IAS, CP’s etc.
• However, these algorithms are known to have convergence
problems in some cases
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Fast and Robust Algorithm for Bader
Decomposition of Charge Density
Code: http://theory.cm.utexas.edu/bader/ Forum: http://theory.cm.utexas.edu/forum/
Charge density grid -> Steepest ascent path -> Bader region -> Sum up
2D : 8 neighboring grid points
3D : 26 neighboring grid points
Steepest ascent move to max ∇ρ :
di, dj, dk are each assigned the values
{-1, 0, 1} but excluding di = dj = dk = 0
Change in density :
Change in distance :
13
G. Henkelman, A. Arnaldsson, H. Jónsson, Comp. Mat. Sc. (2006)
Fast and Robust Algorithm for Bader
Decomposition of Charge Density
• No effort in trying to find CP’s of ρ or the accurate shape
of the IAS.
• No 3D integration (only summation of the charge density
on 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 is
complete.
• The algorithm scales linearly with number of grid points,
(not with e.g. number of atoms or CP’s)
14
G. Henkelman, A. Arnaldsson, H. Jónsson, Comp. Mat. Sc. (2006)
Results: Water Molecule
Charge density: aug-cc-pVDZ basis, Gaussian 98, MP2 level
3 Bader regions found, each containing one atom
15
G. Henkelman et al., Comp. Mat. Sc. (2006)
Results: Boron Clusters in Silicon Crystals
Charge density: DFT, plane wave basis,
VASP, ultra-soft PsP, PW91
62 Si, 3 B, extra electron added
Total charge of the B3-cluster: 10.5 eOverall charge of the cluster is -1.5 e
Single B atom in Si crystal has -0.9 e
Formation of B3 clusters reduces
boron electronic activity by ca. 50%
16
G. Henkelman et al., Comp. Mat. Sc. (2006)
Scaling of Effort
Data: Boron cluster in
silicon with different
number of grid points
Time: 3 - 45 sec on
1.8 GHz Athlon
based computer.
3 atom H2O system
required the same
computational effort as
the 65 atom B-Si system
with similar grid size.
Scales linearly with number of grid points
G. Henkelman et al., Comp. Mat. Sc. (2006)
-> The computer time
neither depends upon the
number of atoms in the
system nor the bonding
topology.
17
Bader Analysis at CAMP
(from e.g. Dacapo, Siesta or Grid-PAW)
from Dacapo import Dacapo
from 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 atom
Dipole moment on each atom
Bader volume around each atom
Bader maxima for each atom etc.
18
Bader Analysis from Dacapo Calculations
Problems with Bader analysis on O-H groups in pseudopotential codes
All electron code (Gaussian):
Pseudopotential code (Dacapo):
Partial Charge:
O: - 1.16
H: + 0.58
9.16 e-
O
0.42
e-
H
H 0.42
e-
Partial Charge:
O: - 2.00
H: + 1.00
8.00 e-
O
0.00 e- H
H 0.00 e-
G. Henkelman et al., Comp. Mat. Sc. (2006)
We could use Grid-PAW in near future to solve this problem
However, …
Protonated water-layer : q = + 0.64
Hontop : 1.17 e-
Pt(111) + 7/6 ML Hads : q = - 0.64
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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 to
the choice of basis set.
• Mulliken analysis is e.g. used in the Gaussian and the Siesta codes.
Output from Siesta calculations on CO molecule:
mulliken: Atomic and Orbital Populations:
Species: C
Atom
1
Qatom
Qorb
2s
2s
2py
2pz
2px
2py
2pz
2px
2Pdxy
2Pdyz
2Pdz2
2Pdxz
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
Qatom
Qorb
2s
2s
2py
2pz
2px
2py
2pz
2px
2Pdxy
2Pdyz
2Pdz2
2Pdxz
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
Species: O
Atom
2
mulliken: Qtot =
10.000
20
Data from Mikkel Strange
Charge Analysis with Hirshfeld, Mulliken & Bader
Bader
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 < Bader
21
De Proft et al, Vol. 23, No. 12, J.Comp. Chem. (2002)
Charge Analysis with Hirshfeld, Mulliken & Bader
Selected number of hypervalent compounds, all values in |e|
Hirshfeld
Bader
Charges distribution: Hirshfeld < Mulliken < Bader
22
De Proft et al, Vol. 23, No. 12, J.Comp. Chem. (2002)
Charge Analysis with Mulliken, NBO &
Bader as a Function of Applied Field
The charge on the H-atom in
HCN is almost the same by
all three methods at zero
applied field.
Bader
The sensitivity of the charge
on the H-atom to the change
in applied field follows the
order:
Bader > Mulliken > NBO
Fig: Charge on the H atom of HCN
(calculated using three different
conventions) as a function of the
applied electric field.
GAMESS (HF/D95** level): optimize the
geometries in a dipolar electric field
23
Masunov et al., J. Phys. Chem. A, Vol. 105, No. 19, 2001
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.
24