Chemical bonds from quantum mechanical probabilities

Università della Calabria, June 2007

Overview

£ Approach£ History£ Probabilities: atoms, correlation£ Domains: shapes, multiplicities of solutions£ Numbers: populations, ...

ApproachMaximum probability domains from Quantum Monte Carlo calculations £ Maximum probability domains HWΝL £ Quantum Monte Carlo

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

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 - ...

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 ...

Maximum probability domains

Evolution of W towards WΝ=2

Maximum probability domains

WΝ=2

Quantum Monte Carlo

Common variants · Variational Monte Carlo (VMC) · Diffusion Monte Carlo

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

Quantum Monte Carlo

Evaluation of XY È H È Y\ in VMC XY È H È Y\ = Ù É Y È2 �������H Y

Y

Quantum Monte Carlo

Sampling Ψ2 in H

Quantum Monte Carlo and W

History £ Chemist’s view £ Physicist’s view £ Compromise

Physics vs. Chemistry

http://webpages.marshall.edu/~pricew/chem307/tutorials/comp_chem/intro.html

Chemist’s view

Lewis

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.

Chemist’s view

Lewis’ Cubical Atom

Chemist’s view

Lewis’ Cubical Atom

Lewis’ I· & ·I

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.."

Chemist’s view

Limitations of the Cubical Atom: Electron Pair

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.

Chemist’s view

Linnett’s double−quartet, 1960

­ ¯

Compromise

Attitudes · "Chemical models work " · "Chemists have to learn MO theory !"

Compromise

False contradiction "Chemistry = Make, describe, classify!" (R. Hoppe)Quantum mechanics: universal and can help.

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, ...

Compromise

Electron arrangement for max of Y2 H2 O HRHFL

Compromise

Electron arrangement for a max of Y2

H2 O (correlated): Lewis’ cube

Compromise

Electron arrangement for a max of Y2

H2 O (correlated): Linnett’s tetrahedra

Compromise

Electron arrangement for a max of Y2

H2 O (correlated)

Compromise

What is the importance of a maximum?

Compromise

Point ® Region Maximum probability domains

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

pΝ : Atoms

Atoms

General behavior: Change of pΝ with W

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

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

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

Atoms

pΝHRL: Be

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

Atoms

pΝHRL: Ne

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

Atoms

pΝHRL: Mg

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

Atoms

maxR pΝHRL: Mg

Atoms

Maxima in Mg

Atoms

Multiple maxima in Mg

Atoms

Selecting maxima in Mg

Atoms

Conclusion Atomic shells show up, as maxR pΝHRL

Checked for all Hartree−Fock atoms, up to Kr (Bunge tables)

pΝ : Correlation

· Quality of Y : changes by correlation small?

Correlation

Changes by correlation in Mg

Mg: pΝHsphereL HF vs. VMC (R. Assaraf) Pauli principle satisfied

Correlation

H2HR ® ¥L

1 2

0.25

0.5

0.75

1

pΝ=2HW = half - spaceL HF vs. MCSCF

Correlation

F2HReL

9 10

0.25

0.5

pΝ=2HW = half - spaceL HF , VMC, DMC

Correlation

Conclusion

Role of correlation not clear:

better take it into account, as much as possible

Shapes of W

Quality of the shapes?

Outer regions not explored: fast VMC.

Shapes

W2: CH4

Shapes

ELF basins

1

Shapes

ELF basin ( and W2): CH4

Shapes

Space partitioning by symmetry

Shapes

Deformation: LiH, BH, HF

Shapes

Deformation − dihydrogen bond

LiH LiH..HF HF

Shapes

Different W2 in BH

Several W2

Shapes

Complementarity?

Shapes

Do Ws overlap?Numerical uncertainties HCH4L

Shapes

Do Ws overlap?Overlap in CH5

+: useful for transition state?

Shapes

Do Ws overlap?FHF- at equilibrium and out of equilibrium

Shapes

Conclusions

· 3D optimization problem

· technical limitations / complementarity?

Multiplicity of solutions

Multiplicity

W2 in H2 O: OH and lone pair

Multiplicity

W2 in CH4: CH bonds

Multiplicity

W2 in Ne (vs. CH4)

Multiplicity

W2 in Ne (vs. CH4)

Equivalent Ws in Ne, ¥ number (like LMOs)

Multiplicity

ELF in Ne: lower value for valence shell

Becke, Edgecombe, 1990

Multiplicity

W2 in HCCH: CH, Σ , Π

Multiplicity

W2 in HCCH: CH, banana

¥ many triplets

Multiplicity

W6 in HCCH: CC

Also ELF basin

Multiplicity

HSiSiH

Multiplicity

ELF in Si2 H2

Structure of maxima reflects an ’average’

Multiplicity

W2 in Si2 H2

Multiplicity

W2, W6 in Si2 H2

Multiplicity

HSiSiH: 2 equivalent solutions

Multiplicity

Lewis’ definition of tautomerism Two or more forms of molecules pass readily intoone another and exist together in condition ofmobile equilibrium

Multiplicity

Conclusions

Multiple solutions exist, but are physically significant

Populations, ...

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

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

Populations, ...

Definition of populations ÙW Ρ = ÚΝ Ν pΝHWL = XNW\ average

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

Populations, ...

Population ¹ maximal probabilityH2, for R ® ¥ (W : half−space)

YCovalentHH ... HL IonicHH+ ... H- « H- ... H+L

XNW\ 1 1

p1 HWL 1 0

Populations, ...

Population conservationSpherical shells yielding ÙR1

R2 Ρ = XNW\ = 2

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

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

Populations, ...

Populations in WΝ close to Ν

Zn atom, HF, 4 s2

Method pΝ=2 AIM ELFXNW\ 2.0 - 2.2

Populations, ...

Populations in WΝ close to Ν

YNlone pair] » 3 HELF basinsL, » 2 Hp2L

Populations, ...

Significance of Σ2 = YNW2] - XNW\2

68% of Ν between XNW\ - Σ and XNW\ + Σ? NO!

Populations, ...

Discrete distribution HCH4L

1 2 3 4 5

0.1

0.2

0.3

0.4

Populations, ...

Gaussian distribution?

1 2 3 4 5

0.1

0.2

0.3

0.4

Populations, ...

Discrete distribution: 48% between XNW\ - Σ and XNW\ + Σ

1 2 3 4 5

0.1

0.2

0.3

0.4

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

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 (?)

Populations, ...

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

W2 0.16 0.22ELF 0.16 0.27

Populations, ...

Conclusions

pΝHWΝL preferable to populations, ... :

· countable number of solutions· physical significance

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...

Conclusion

W2: · clear physical significance· useful?

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).