chemical bonds from quantum mechanical probabilities · 2007. 6. 22. · lewis on early quantum...
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Chemical bonds from quantum mechanical probabilities
Università della Calabria, June 2007
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Overview
£ Approach£ History£ Probabilities: atoms, correlation£ Domains: shapes, multiplicities of solutions£ Numbers: populations, ...
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ApproachMaximum probability domains from Quantum Monte Carlo calculations £ Maximum probability domains HWΝL £ Quantum Monte Carlo
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Maximum probability domains
Definition of WΝ WΝ is the W maximizing the probability to find Ν elec−trons in W ,
pΝHWL = K NΝ
O ÙW d x1 ... d xΝ ÙW���d xΝ+1 ... d xN Y* Y
· K NΝ
O: electrons cannot be distinguished
· W���
: all space except W
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Maximum probability domains
pΝHWL ¹2−RDM, GΝHWLpΝHWL = K N
ΝO ÙW d x1 ... d xΝ ÙW
���d xΝ+1 ... d xN Y* Y
GΝHWL = K NΝ
O ÙW d x1 ... d xΝ ÙW���
ÜWd xΝ+1 ... d xN Y* Y
For W small, pΝ+1, ... become small, and pΝHWL = GΝHWL - ...
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Maximum probability domains
Algorithm for finding WΝ W is constructed from small cubes. Cubes are added/removed in order to maximizepΝHWL. È Y È2 is obtained in Quantum Monte Carlo, or ...
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Maximum probability domains
Evolution of W towards WΝ=2
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Maximum probability domains
WΝ=2
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Quantum Monte Carlo
Common variants · Variational Monte Carlo (VMC) · Diffusion Monte Carlo
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Quantum Monte Carlo
Variational Monte Carlo (VMC)
Y = YHp1, p2, ...L, e.g., Fdet Pi<j f Irij , p1, p2, ..M p1, p2, ... are obtained by minp1,p2... XY È H È Y\ XY È H È Y\ evaluated numerically
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Quantum Monte Carlo
Evaluation of XY È H È Y\ in VMC XY È H È Y\ = Ù É Y È2 �������H Y
Y
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Quantum Monte Carlo
Sampling Ψ2 in H
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Quantum Monte Carlo and W
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History £ Chemist’s view £ Physicist’s view £ Compromise
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Physics vs. Chemistry
http://webpages.marshall.edu/~pricew/chem307/tutorials/comp_chem/intro.html
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Chemist’s view
Lewis
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Chemist’s view
Lewis on early Quantum Mechanics, 1916Indeed it seems hardly likely that much progress can be made in the solu −tion of difficult problems relating to chemical combination by assigning inadvance definite laws of force between the positive and negative constitu −ents of an atom ...... a study of the mathematical theory of electrons leads, I believe, irresist −ibly to the conclusion that Coulomb’s law of inverse squares must fail atsmall distances.
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Chemist’s view
Lewis’ Cubical Atom
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Chemist’s view
Lewis’ Cubical Atom
Lewis’ I· & ·I
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Chemist’s view
Limitations of the Cubical Atom: Electron Pair
Lewis: "With the cubical structure it is not only impossible to represent thetriple bond, ..."
Lewis: "Assuming now,at least in such very small atoms as that ofcarbon,that each pair of electrons has a tendency to be drawn together.."
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Chemist’s view
Limitations of the Cubical Atom: Electron Pair
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Chemist’s view
Linnet’s double−quartet, 1960 (spin) ... the octet should be treated as two groups of four, rather as four pairs, aswas done by Lewis......it will be supposed, as theory would suggest , that each group of four elec−trons will tend to have a disposition round the nucleus which is approxi −mately that of a regular tetrahedron.
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Chemist’s view
Linnett’s double−quartet, 1960
¯
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Compromise
Attitudes · "Chemical models work " · "Chemists have to learn MO theory !"
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Compromise
False contradiction "Chemistry = Make, describe, classify!" (R. Hoppe)Quantum mechanics: universal and can help.
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Compromise
Quantum chemistry K. Artmann, 1946H.K. Zimmerman, P. van Rysselberghe, 1949
Positions of electrons for which Y2 is maximal. · Pauling ~ 1930, · Lennard−Jones, ~1950: P2Hr1, r2L
· LMO· Fermi hole ® FHMF, ELF, ...
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Compromise
Electron arrangement for max of Y2 H2 O HRHFL
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Compromise
Electron arrangement for a max of Y2
H2 O (correlated): Lewis’ cube
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Compromise
Electron arrangement for a max of Y2
H2 O (correlated): Linnett’s tetrahedra
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Compromise
Electron arrangement for a max of Y2
H2 O (correlated)
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Compromise
What is the importance of a maximum?
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Compromise
Point ® Region Maximum probability domains
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Maximum probability domains
Definition of WΝ WΝ is the W maximizing the probability to find Ν elec−trons in W ,
pΝHWL = K NΝ
O ÙW d x1 ... d xΝ ÙW���d x3 ... d xN Y* Y
· K NΝ
O: electrons cannot be distinguished
· W���
: all space except W
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pΝ : Atoms
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Atoms
General behavior: Change of pΝ with W
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Atoms
Change of pΝ=0 with W (sphere radius, R)
0 2 4 6 8 10 12 14
0.0
0.2
0.4
0.6
0.8
1.0
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Atoms
Change of pΝ=N with W (sphere radius, R)
0 2 4 6 8 10 12 14
0.0
0.2
0.4
0.6
0.8
1.0
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Atoms
Change of pΝH¹0,NL with W (sphere radius, R)
0 2 4 6 8 10 12 140.0
0.2
0.4
0.6
0.8
1.0
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Atoms
pΝHRL: Be
0 1 2 3 4 5 60.0
0.2
0.4
0.6
0.8
1.0
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Atoms
pΝHRL: Ne
0 1 2 3 4 5 60.0
0.2
0.4
0.6
0.8
1.0
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Atoms
pΝHRL: Mg
0 1 2 3 4 5 60.0
0.2
0.4
0.6
0.8
1.0
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Atoms
maxR pΝHRL: Mg
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Atoms
Maxima in Mg
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Atoms
Multiple maxima in Mg
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Atoms
Selecting maxima in Mg
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Atoms
Conclusion Atomic shells show up, as maxR pΝHRL
Checked for all Hartree−Fock atoms, up to Kr (Bunge tables)
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pΝ : Correlation
· Quality of Y : changes by correlation small?
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Correlation
Changes by correlation in Mg
Mg: pΝHsphereL HF vs. VMC (R. Assaraf) Pauli principle satisfied
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Correlation
H2HR ® ¥L
1 2
0.25
0.5
0.75
1
pΝ=2HW = half - spaceL HF vs. MCSCF
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Correlation
F2HReL
9 10
0.25
0.5
pΝ=2HW = half - spaceL HF , VMC, DMC
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Correlation
Conclusion
Role of correlation not clear:
better take it into account, as much as possible
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Shapes of W
Quality of the shapes?
Outer regions not explored: fast VMC.
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Shapes
W2: CH4
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Shapes
ELF basins
1
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Shapes
ELF basin ( and W2): CH4
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Shapes
Space partitioning by symmetry
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Shapes
Deformation: LiH, BH, HF
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Shapes
Deformation − dihydrogen bond
LiH LiH..HF HF
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Shapes
Different W2 in BH
Several W2
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Shapes
Complementarity?
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Shapes
Do Ws overlap?Numerical uncertainties HCH4L
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Shapes
Do Ws overlap?Overlap in CH5
+: useful for transition state?
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Shapes
Do Ws overlap?FHF- at equilibrium and out of equilibrium
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Shapes
Conclusions
· 3D optimization problem
· technical limitations / complementarity?
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Multiplicity of solutions
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Multiplicity
W2 in H2 O: OH and lone pair
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Multiplicity
W2 in CH4: CH bonds
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Multiplicity
W2 in Ne (vs. CH4)
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Multiplicity
W2 in Ne (vs. CH4)
Equivalent Ws in Ne, ¥ number (like LMOs)
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Multiplicity
ELF in Ne: lower value for valence shell
Becke, Edgecombe, 1990
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Multiplicity
W2 in HCCH: CH, Σ , Π
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Multiplicity
W2 in HCCH: CH, banana
¥ many triplets
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Multiplicity
W6 in HCCH: CC
Also ELF basin
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Multiplicity
HSiSiH
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Multiplicity
ELF in Si2 H2
Structure of maxima reflects an ’average’
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Multiplicity
W2 in Si2 H2
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Multiplicity
W2, W6 in Si2 H2
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Multiplicity
HSiSiH: 2 equivalent solutions
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Multiplicity
Lewis’ definition of tautomerism Two or more forms of molecules pass readily intoone another and exist together in condition ofmobile equilibrium
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Multiplicity
Conclusions
Multiple solutions exist, but are physically significant
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Populations, ...
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Populations, ...
Definition of populations ÙW Ρ
ΡHr L= XÚi=1,N ∆Hri - r L\= PHe1; r L + PHe2; r L + ... + PHeN; r LÙW ΡHr L ¹ pΝ=1HWL, for finite W
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Populations, ...
p1HWL ¹ PHe1; WL· p1HWL: probability to find one and only one electron in W
· PH1; WL : probability to find electron 1 in W
1
2
1 2
Counts for PH1,WLand for p1HWL Counts for PH1,WL
not for p1HWL
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Populations, ...
Definition of populations ÙW Ρ = ÚΝ Ν pΝHWL = XNW\ average
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Populations, ...
Average ¹ maximum: H atom
1 2 3 4r
0.1
0.2
0.3
0.4
0.5
D
rmax = 1 bohrXr \ = 1.5 bohr
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Populations, ...
Population ¹ maximal probabilityH2, for R ® ¥ (W : half−space)
YCovalentHH ... HL IonicHH+ ... H- « H- ... H+L
XNW\ 1 1
p1 HWL 1 0
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Populations, ...
Population conservationSpherical shells yielding ÙR1
R2 Ρ = XNW\ = 2
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Populations, ...
R1, R2 such that ÙR1
R2 Ρ = XNW\ = 2
0 2 4 6 8 10 12 14
0
2
4
6
8
10
12
14
R1
R2
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Populations, ...
pΝ=2 for W a spherical shell
0 2 4 6 8 10 12 14
0
2
4
6
8
10
12
14
R1
R2
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Populations, ...
Populations in WΝ close to Ν
Zn atom, HF, 4 s2
Method pΝ=2 AIM ELFXNW\ 2.0 - 2.2
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Populations, ...
Populations in WΝ close to Ν
YNlone pair] » 3 HELF basinsL, » 2 Hp2L
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Populations, ...
Significance of Σ2 = YNW2] - XNW\2
68% of Ν between XNW\ - Σ and XNW\ + Σ? NO!
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Populations, ...
Discrete distribution HCH4L
1 2 3 4 5
0.1
0.2
0.3
0.4
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Populations, ...
Gaussian distribution?
1 2 3 4 5
0.1
0.2
0.3
0.4
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Populations, ...
Discrete distribution: 48% between XNW\ - Σ and XNW\ + Σ
1 2 3 4 5
0.1
0.2
0.3
0.4
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Populations, ...
p2HWL for W2 and for ELF basins
1 2 3 4 5
0.1
0.2
0.3
0.4
OH
1 2 3 4 5
0.1
0.2
0.3
0.4
lone pair
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Populations, ...
Average number of electrons in W2 and in ELF basin XNW\
W OH lone pairW2 1.95 1.95ELF 1.58 2.34
1.95 instead of 2: numerical (?)
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Populations, ...
Ratio pH , WL � pH ¯, WLp �p¯ OH lone pair
W2 0.16 0.22ELF 0.16 0.27
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Populations, ...
Conclusions
pΝHWΝL preferable to populations, ... :
· countable number of solutions· physical significance
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Summary
maxW p2HWL Þ several W2 « bonds, lone pairs, ...
· W2: sharp borders, equivalent by symmetry· can be used for correlated wave functions, same simple interpretation· WΝ ¹basins, or loges, except...
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Conclusion
W2: · clear physical significance· useful?
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References
A. Scemama, M. Caffarel, ASJ. Comp. Chem. 28, 442 (2007) "90 Years of Chemical Bonding"
· AS, K.D. Sen, ed., Reviews of modern quantum chemistry: A celebration of the contributions of Robert G. Parr, World Scientific , Singapore, 2002, p.43.· E. Chamorro, P. Fuentealba, AS, J. Comp. Chem. 24, 496 (2003).· E. Cancès, R. Keriven, F. Lodier , AS, Theor. Chem. Acc. 111, 373 (2004).· A. Gallegos , R. Carbó−Dorca , F. Lodier , E. Cancès, AS, J. Comp. Chem. 26, 455 (2005).· AS, J. Chem. Sci. 117, 473 (2005).