introduction to elf...as, r. nesper, s. wengert, th. f. fässler, angew. chem., 109, 1893 (1997),...
TRANSCRIPT
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
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100
125
150
1750.027060
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1750.435937
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1750.569998
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1750.708092
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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