Introduction to ELF...AS, R. Nesper, S. Wengert, Th. F. Fässler, Angew. Chem., 109, 1893 (1997),...

Introduction to ELF

A. Savin

Aachen, February 12−13, 2007

http://www.lct.jussieu.fr/pagesperso/savin/

Overview

£ Context

£ ELF

£ What next?

Overview

£ Context

£ ELF

£ What next?

Context

£ Energetic/spatial view in chemistry

£ Quantum mechanics

£ Tools

Context

£ Energetic/spatial view in chemistry

£ Quantum mechanics

£ Tools

Energetic/spatial

Two aspects: energetic and spatial

· Energetic view: shells H¶i L, bonds HDeL, ...· Spatial view: shells (DHr L), bonds (sticks), ...

Spatial (3D) view from the perspective of quantum mechanics

Energetic/spatial

Shells: Orbital energies

Energetic/spatial

Shells: Radial density

r

D

Energetic/spatial

Shells: Radial density

Problems:

· Definition when choice of origin not obvious

· DHr L does not work for several heavy atoms

Energetic/spatial

Bond: Atomization (dissociation) energy

R

De

Energetic/spatial

Need for indicators

· AB : A - B, A+ B-, A+ B- « A- B+?

· ABC : A - B and B - C, or A - C and B - C, or...?

Energetic/spatial

3D: van’t Hoff and Le Bel

Energetic/spatial

Bond: electron pairs

Energetic/spatial

3D, N electrons: Lewis’ cubes

Energetic/spatial

3D, N � 2 el. pairs: Lewis’ tetrahedra

Energetic/spatial

3D, spin: Linnett’s double quartet

Energetic/spatial

What is the 3D image of a bond?

Energetic/spatial

Summary

Interest in spatial approach

· evident

· no unique descriptor

Context

£ Energetic/spatial view in chemistry

£ Quantum mechanics

£ Tools

Quantum mechanicsÈ Y È2

Quantum Mechanics

pΝHWL = ikjj N

Νy{zz ÙW

d x1 ... d xΝ ÙW���d xΝ+1 ... d xN Y* Y

· ikjj N

Νy{zz: electrons cannot be distinguished

· W���

: all space except W

Other definitions

pΝHWL ¹GΝHWL = ÙWGΝHr1, ..., rΝL

pΝHWL = ikjj N

Νy{zz ÙW

d x1 ... d xΝ ÙW���d xΝ+1 ... d xN Y* Y

GΝHWL = ikjj N

Νy{zz ÙW

d x1 ... d xΝ ÙW���ÜW

d xΝ+1 ... d xN Y* Y

For W small, pΝ+1, ... become small, and pΝHWL = GΝHWL - ...

Other definitions

Connection to population

p1HWL = N ÙWd x1 ... d xΝ ÙW

���d xΝ+1 ... d xN Y* Y

G1HWL = ÙWΡHr L = N ÙW

d x1 ÙW���ÜW

d x2 ... d xN Y* Y

Quantum Mechanics

pΝHWL ¹populations

Populations are averagesXN` W\ = ÚΝ=0,N Ν pΝHWLW for which XN` W\ = Ν is, in general, not WΝ

For small volumes, 1 » p0HWL p p1HWL p p2HWL p ...XN` W\ = 0 p0HWL + 1 p1HWL + 2 p2HWL + ... ® p1HWL

Quantum Mechanics

Probability of finding Ν electrons in a sphere

Quantum Mechanics

pΝ(sphere of radius R) HΝ = 0L

0 2 4 6 8 10 12 14

R0

0.2

0.4

0.6

0.8

1

p

Quantum Mechanics

pΝ(sphere of radius R) HΝ = NL

0 2 4 6 8 10 12 14

R

0.2

0.4

0.6

0.8

1

p

pΝ(sphere of radius R) H0 < Ν < NL

0 2 4 6 8 10 12 14

R

0.2

0.4

0.6

0.8

1

p

Quantum Mechanics

pΝHWL, Ν = 0, ... NpΝ HWL in H AIM basin in H2 O

Quantum Mechanics

Population (average) and variance XNW\ = ÚΝ Ν pΝHWL XNW

2\ = ÚΝ Ν2 pΝHWL

Σ2 º XNW2\ - XNW\2

Discrete distribution, not gaussian: not 68% within XNW\ ± Σ2

Quantum Mechanics

pΝHWL, Ν = 0, ... NpΝ HWL in H AIM basin in H2 O

Quantum Mechanics

Infinite number of W having the same XNW\

Spherical shell, between R1 and R2.

Quantum Mechanics

Shells, between R1and R2, having XNW\ = 2

0 2 4 6 8 10 12 14

R10

2

4

6

8

10

12

14

R2

Quantum Mechanics

p2, for W being between R1and R2

0 2 4 6 8 10 12 14

R10

2

4

6

8

10

12

14

R2

Quantum Mechanics

Minimal quality of electronic Y

Slater determinant: F = 1������������!!!!!!N!

ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄj1H1L j2H1L ... jN H1Lj1H2L j2H2L ... jN H2L

... ... ... ...

j1HNL j2HNL ... jN HNLÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ

· Hartree−Fock: minimize energy, XF È H È F\· Kohn−Sham: minimize model energy XF È T + VKS È F\, VKS to yield exact electron density, Ρ = Ú È ji È2

Quantum Mechanics

Localized molecular orbitals

K. Ruedenberg home page

LMOs are not unique

Quantum Mechanics

Strategies to improve Y

CI, MCSCF, CC, ...: Ú cI FI

QMC: Ú cI FI JHr12, ...LGood precision, bad accuracy

Quantum Mechanics

Sensitivity requirements for bond descriptors

Good compromise needed:

· enough sensitivity to yield a reliable interpretation

· not so sensitive to show the errors in Y

Quantum Mechanics

Sensitive descriptor |ÑΡ|/Ρ

M. Kohout, AS, H. Preuss, J. Chem. Phys. 95, 1928 (1991)

Quantum Mechanics

Summary

· ’Universal’ approach

· Connection to bond?

Context

£ Energetic/spatial view in chemistry

£ Quantum mechanics

£ Tools

Tools

Define f HrL Hr Î R3L : interest in 3D Examples in 2D

f HrL Hr Î R3L : isosurfaces (contours)

-2 -1 0 1 2 3-2

-1

0

1

2

3

Tools

Characteristic isosurface (isocontour)

-2 -1 0 1 2 3-2

-1

0

1

2

3

Tools

Characteristic isosurface (isocontour)

-2-1

01

2

3

-2-1

01

23

0.5

1

-2-1

01

2

3

-2-1

01

23

Tools

Bifurcation diagram

0 1

0 1

Tools

Maxima of f HrL Hr Î R3L : characterize f

-2 -1 0 1 2 3-2

-1

0

1

2

3

Tools

Basins of f HrL Hr Î R3L : spatial domains

-2 -1 0 1 2 3-2

-1

0

1

2

3

Tools

Other critical points of f HrL Hr Î R3L : characterize f

-2 -1 0 1 2 3-2

-1

0

1

2

3

Tools

Visualization of 3D functions

M. Schultheiss, PhD thesis

Tools

Visualization of domains in 3D

Tools

Properties of spatial domains

Examples:

· pΝHWL· XNW\

Tools

Optimization of a property of a spatial domain

20 40 60 80

25

50

75

100

125

150

1750.027060

20 40 60 80

25

50

75

100

125

150

1750.435937

20 40 60 80

25

50

75

100

125

150

1750.569998

20 40 60 80

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50

75

100

125

150

1750.708092

20 40 60 80

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1750.737321

Overview

£ Context

£ ELF

£ What next?

ELF

£ History

£ Definitions

£ Properties

ELF

A qualitative definition of ELF

A function which is large where electron pairs localize

0

1

­¯ ­¯ ­¯

ELF

£ History

£ Definitions

£ Properties

History

Reference

A.D. Becke and K.E. Edgecombe

J. Chem. Phys. 92, 5397 (1990)

History

Precursors

Fermi Hole Mobility Function:

W.L. Luken, J.C. CulbersonInt. J. Quantum Chem. S 16, 265, 1982

History

Early precursors

YK. Artmann, 1946H.K. Zimmerman, P. van Rysselberghe, 1949 P2Hr1, r2LLennard−Jones, ~1950

f HrLY. Tal, R.W.F. Bader 1978E. Ludena , 1982B.M. Deb, S.K. Ghosh, 1983,...

History

Further references

· AS, A.D. Becke, J. Flad, R. Nesper, H. Preuss, H.G. von Schnering, (M. Kohout)Angew. Chem. 103, 421 (1991),Angew. Chem. Int. Ed. Engl. 30, 409 (1991)

Testing

History

Further references

· B. Silvi, AS, Nature 371, 683 (1994)

Basins

· AS, B. Silvi, F. Colonna,Can. J. Chem. 74, 1088 (1996)

Basin properties (numbers)

History

Further references

AS, O. Jepsen, J. Flad, O.K. Andersen, H. Preuss, H.G. von Schnering,Angew. Chem. 104, 187 (1992);Angew. Chem. Int. Ed. Engl. 31, 187 (1992).

Several definitions of ELF

All coincide for a single determinant closed shell wavefunction

ELF

£ History

£ Definitions

£ Properties

Definitions

Original definition, probability densityA.D. Becke and K.E. Edgecombe, J. Chem. Phys. 92, 5397 (1990)

P2ÈÈHr1, r2L = 0 + 1����2 È r12 È2 CHr1L + ...; r12 = È r1 - r2 È

in homogenous electron gas: Chom

ELF:

ΗHr L = H1 + @CHr1L � ChomHr LD2L-1

Electron with given spin has more room within a pair

Definitions

Small sphere, probabilityJ.F. Dobson, J. Chem. Phys. 94, 4328 (1991)

ÙWHr1LP2ÈÈHr1, r2L d3 r12 = CHr L Ù0

Rr125 4 Π d r12

Definitions

Small ’breathing’ sphereAS, R. Nesper, S. Wengert, Th. F. Fässler,Angew. Chem., 109, 1893 (1997),Angew. Chem. Int. Ed. Engl. 36, 1809 (1997)

Eliminate electron gas, by fixing R to have the same’number of electrons’ within the sphere WHr1L indepen−dently of r1.

Definitions

pH ­ ¯; WL, pH ­ ¯ ­ ¯; WL?

Same or different?

Definitions

Probability interpretationcf. AS, R. Nesper, S. Wengert, Th. F. Fässler,Angew. Chem., 109, 1893 (1997),Angew. Chem. Int. Ed. Engl. 36, 1809 (1997)

ELF:H1 + p H ­ ¯ ­ ¯; WL � pH ­ ¯; WL2L-1

pH ­ ¯; WL, pH ÈÈ ; WL?

Same or different?

Definitions

Probability interpretationcf.B. Silvi, J. Phys. Chem. 107, 3081 (2003)

AS J.Phys.Chem.Solids 12, 2025 (2004)

ELF:H1 + @pH ÈÈ ; WL � pH ­ ¯; WLD2L-1

Definitions

Density functional approachAS, O. Jepsen, J. Flad, O.K. Andersen,H. Preuss, H.G. von Schnering,Angew. Chem. 104, 187 (1992);Angew. Chem. Int. Ed. Engl. 31, 187 (1992).

ELF:

Related to excess of kinetic energy, due to the Pauli prin−ciple

Definitions

’Experimental approach’M. Kohout, A. Savin, J. Comp. Chem., 18, 1431 (1997)

= DF approach

See also · V. Tsirelson, A. Stash, Chem. Phys. Letters 351, 242 (2002)· D.J. Grimwood, I. Bytheway, D. Jayatilaka, J. Comp. Chem, 24, 470 (2003)

Definitions

Localized MOs È ji È2, É ji � �!!!!Ρ É2

AS J. Mol. Struct. (TheoChem) 727, 127 (2005)

Ρ = Ú È ji È2

Definitions

Invariance (to choice of localization)

âi

É ji � �!!!!Ρ É2 = 1

DefinitionsÉ ji � �!!!!Ρ É2, É ÑIji � �!!!!

ΡM É2

Definitions

Invariance (to choice of localization)

É ÑIji � �!!!!ΡM É2, â

iÉ ÑIji � �!!!!

ΡM É2

Definitions

Transformation to ELF

ELF

£ History

£ Definitions

£ Properties

Properties

How to look at ELF: InI, InII, InIII in In6 S7

Properties

How to look at ELF: InI, InII, InIII in In6 S7

S

Properties

How to look at ELF: InI, InII, InIII in In6 S7

InIII

Properties

How to look at ELF: InI, InII, InIII in In6 S7

InI

Properties

How to look at ELF: InI, InII, InIII in In6 S7

InII

Properties

Covalent/ionic

InII: covalent

InI, InIII, S2-: ionic

Properties

IonicLi+ CN-

Properties

Is H ionic?HCN

Properties

Non−bonded, van der Waals?Ne2 at F2 internuclear distance

Properties

Resonant bond?

Ne2+, Cl2, ...: averaging

Properties

Correlation?

Antibonding, ionic, ... contributions energy stabilizing

Properties

Symmetry operations (LMO, ELF): B6 H62-

A.Burkhardt, U.Wedig, H.G.von Schnering, AS, Z.anorg.allg.Chem.619,437 (1993)

Properties

Symmetry invariance ELF: Si2 H2B. Silvi, I. Fourré, M.E. Alikhani, Monatshefte für Chemie 2005, 136, 855.

Properties

Maxima: bent bonds HC3 H6L

Properties

Maxima HC2 Si2 H8L: bond angle questionAS, H.−J.Flad, J.Flad, H.Preuss, H.G.von Schnering,Angew.Chem.104,185 (1992),Angew.Chem.Int.Ed.Engl.31,185 (1992).

C

Si

C

Si

L ?

Properties

Maxima HC2 Si2 H8L: bond angle problem

C

Si

C

Si

L < 90o

Properties

Maxima HC2 Si2 H8L: ELF

Properties

Maxima HELF in C2 Si2 H8L: angles

Properties

Maxima HC2 Si2 H8L: angles to maxima

C 109o

Properties

Maxima HC2 Si2 H8L: angles to maxima

Si

109o

Properties

Maxima HC2 Si2 H8L: ’broken’ bonds

Properties

Maxima HC2 Si2 H8L: bonds and structure

Properties

Maxima HC2 Si2 H8L: structure

Properties

Maxima HELF in C2 Si2 H8L

Properties

Maxima: in atomic shells

» 1, for s2

» 0.8, for p6

» 0.6, for d10

No absolute measure (superposition)

Properties

Maxima: different for pseudopotentialsM.Kohout,AS,J.Comp.Chem.,18,1431 (1997).

Properties

When is a maximum significant? Na

Properties

When is a maximum significant?

Use bifurcation diagram

0 1

0 1

Properties

When is a basin significant?

Basin is not significant when

maxima are not well separated:

Merge basins

Properties

Numbers from ELF

Beautiful results

Important for molecular dynamicsD. Marx, AS,Angew. Chem. 109, 2168 (1997),Angew. Chem. Int. Ed. Engl. 36, 2077 (1997).

Properties

Numbers from ELF

M. Kohout, AS, Int. J. Quantum Chem. 60, 875 (1996).

Population of "shells", Zn atom:

2.2 8.4 17.2 2.2

Overview

£ Context

£ ELF

£ What next?

What next?

£ Maximization of È Y È2

£ Maximum probability domains

£ Comparison with ELF

A. Scemama,M. Caffarel, AS, J. Comp. Chem. 28, 442 (2007)

What next?

£ Maximization of È Y È2

£ Maximum probability domains

£ Comparison with ELF

Maximization of È Y È2Definition

Positions of electrons for which Y2 is maximal.K. Artmann, 1946H.K. Zimmerman, P. van Rysselberghe, 1949

Maximization of È Y È2Source of Y Quantum Monte Carlo, Hartree−Fock, Kohn−Sham, ...

Maximization of È Y È2Electron arrangement for max of Y2 H2 O (RHF): Lewis’ tetrahedron

Maximization of È Y È2Electron arrangement for a max of Y2

H2 O (correlated): Lewis’ cube

Maximization of È Y È2Electron arrangement for a max of Y2

H2 O (correlated): Linnet’s tetrahedra

Maximization of È Y È2Electron arrangement for a max of Y2

H2 O (correlated): predissociation (Fulde, P2)

Maximization of È Y È2

What is the importance of a maximum?

Maximization of È Y È2

Point ® Region

Maximum probability domains

What next?

£ Maximization of È Y È2

£ Maximum probability domains

£ Comparison with ELF

Maximum probability domains

Definition of the probability of findingΝ electrons in a 3D domain W

pΝ HWL = ikjj N

Νy{zz ÙW

d x1 ... d xΝ ÙW���d xΝ+1 ... d xN Y* Y

· ikjj N

Νy{zz: electrons cannot be distinguished

· W���

: all space except W

Maximum probability domains

Definition of the maximum probability domain

WΝ is a domain W maximizing pΝHWL

Maximum probability domains

Algorithm for finding WΝ

W is constructed from small cubes.

Cubes are add/removed in order to maximize pΝHWL. È Y È2 is estimated in Quantum Monte Carlo, or ...

Maximum probability domains

Technical limitations

· Quality of Y: small changes by correlation?

· Time used for VMC: outer regions not explored

· Convergence in W optimization: do Ws overlap?

· Discretization of Ws: no smooth surfaces

Maximum probability domains

WΝ¹basins

A basin of a local property ¹ property on the basin

Example: · electron density, vs. number of electrons

Maximum probability domains

Wvs and LMOs

Ruedenberg home page

· WΝs have sharp borders, not LMOs

· WΝs " Y, not LMOs

Maximum probability domains

Behavior with symmetry operations

WΝ and LMOs: interchange

ELF basins: invariant

What next?

£ Maximization of È Y È2

£ Maximum probability domains

£ Comparison with ELF

Comparison with ELF

W2: CH4

Comparison with ELF

ELF basin ( and W2): CH4

Comparison with ELF

Space partitioning

Comparison with ELF

W2 in Ne (vs. CH4)

Equivalent Ws in Ne, ¥ number.

Comparison with ELF

ELF in Ne

Comparison with ELF

W2 in C2 H2

Equivalent ’banana bonds’, ¥ number

Comparison with ELF

ELF in C2 H2

Comparison with ELF

W2 in Si2 H2

Only two equivalent solutions

Comparison with ELF

ELF in Si2 H2

Structure of maxima reflects an ’average’

Comparison with ELF

W2 in H2 O

Comparison with ELF

Average number of electrons in W2 and in ELF basin XNW\

W OH lone pair

W2 1.95 1.95ELF 1.58 2.34

1.95 instead of 2: numerical (?)

Comparison with ELF

Ratio pH­ ­, WL � pH­ ¯, WLp­­ �p­¯ OH lone pair

W2 0.16 0.22ELF 0.16 0.27

Difference between directly optimizing the quantity, oradding up small quantities

Comparison with ELF

Do Ws overlap?

Basins do not overlap

Comparison with ELF

Do Ws overlap?Numerical uncertainties HCH4L

Comparison with ELF

W2: FHF- at equilibrium and out of equilibrium

Comparison with ELF

AS, A.D. Becke, J. Flad, R. Nesper, H. Preuss, H.G. von Schnering,Angew. Chem. 103, 421 (1991), Angew. Chem. Int. Ed. Engl. 30, 409 (1991)

ELF in FHF-

Summary

ELF is a wonderful tool

Many applications exist (cf. later talks)

Development possible