"What are the electrons really doing in molecules?" *

Transcription

1 Basics of Electron Frank R. Wagner Max-Planck-Institut für Chemische Physik fester Stoffe, Dresden, Germany. "What are the electrons really doing in molecules?" * * Robert S. Mulliken, lecture title, in accepting the Gilbert. Lewis award, "Chemical Bonding" Winterschool in Theoretical Chemistry, , Helsinki.

2 Basics of Electron Frank R. Wagner "What are the electrons really doing in molecules?" * * Robert S. Mulliken, lecture title, in accepting the Gilbert. Lewis award, "... old-fashioned chemical concepts, which at first seemed to have their counterparts in MQM, the more accurate the calculations became the more the concepts tended to vanish into thin air.... should we try to keep these concepts or should they be relegated to chemical history? mong such concepts are electronegativity, hybridization, population analysis, charges on atoms, even the idea of orbitals...."* * Robert S. Mulliken, honorary lecture, "Symposium on tomic and Molecular Quantum Theory (dedicated to R. S. Mulliken)", Sanibel Island, Florida "Chemical Bonding" Winterschool in Theoretical Chemistry, , Helsinki.

3 Basics of Electron Frank R. Wagner "What are the electrons really doing in molecules?" * * Robert S. Mulliken, lecture title, in accepting the Gilbert. Lewis award, "... old-fashioned chemical concepts, which at first seemed to have their counterparts in MQM, the more accurate the calculations became the more the concepts tended to vanish into thin air...."* * Robert S. Mulliken, honorary lecture, "Symposium on tomic and Molecular Quantum Theory (dedicated to R. S. Mulliken)", Sanibel Island, Florida "ccounting for the chemical bond in terms of straightforward recognition in the wavefunction is still an open problem. In this view, if you want it out, you have to put it in." but: "It could, perhaps, prevent us seeing more deeply into the nature of things."* * Brian T. Sutcliffe, The Development of the Idea of a Chemical Bond, IJQC 58, 645 (1996). "Chemical Bonding" Winterschool in Theoretical Chemistry, , Helsinki.

4 General Firm quantum mechanical foundation, refinement and development of chemical bonding concepts in physical space.

5 General Firm quantum mechanical foundation, refinement and development of chemical bonding concepts in physical space. Pure QM methods for position space analysis of chemical bonding: Topological nalysis (of local properties of the electron distribution) of the electron density: toms in Molecules (QTIM) information from the (un-)correlated.... ELI (electron indicator): tomic Shells, Bonds and Lone Pairs information from the (un-)correlated.

6 General Firm quantum mechanical foundation, refinement and development of chemical bonding concepts in physical space. Pure QM methods for position space analysis of chemical bonding: Topological nalysis (of local properties of the electron distribution) of the electron density: toms in Molecules (QTIM) information from the (un-)correlated.... ELI (electron indicator): tomic Shells, Bonds and Lone Pairs information from the (un-)correlated. Statistical analysis of the correlation of electronic motion in macroscopic regions: localization and delocalization indices, fluctuation of electronic populations: static and dynamical charges, effective bond orders information from the (un-)correlated 1- and.

7 General Firm quantum mechanical foundation, refinement and development of chemical bonding concepts in physical space. Pure QM methods for position space analysis of chemical bonding: Topological nalysis (of local properties of the electron distribution) of the electron density: toms in Molecules (QTIM) information from the (un-)correlated.... ELI (electron indicator): tomic Shells, Bonds and Lone Pairs information from the (un-)correlated. Statistical analysis of the correlation of electronic motion in macroscopic regions: localization and delocalization indices, fluctuation of electronic populations: static and dynamical charges, effective bond orders information from the (un-)correlated 1- and. Energetical analysis of "Interacting Quantum toms" (IQ) atomic self and deformation energies, interatomic interaction energies information from the (un-)correlated 1- and.

8 Pair density, Fermi correlation and Electron The electron density: a scaled probability density (unit: bohr -3 ) * ( r) ds 1 dx dx ( x1,, x) ( x1,, x) x n =(r n,s n ) times the probability per unit volume to find an electron of any spin in small volume element dv around r. ρ = Ψ Ψ The basic ingredient of DFT. Basics

9 Pair density, Fermi correlation and Electron The electron density: a scaled probability density (unit: bohr -3 ) * ρ ( r) = ds dx dx Ψ( x,, x ) Ψ ( x,, x ) x n =(r n,σ n ) times the probability per unit volume to find an electron of any spin in small volume element dv around r. The basic ingredient of DFT. Basics The pair density: a scaled probability density (unit: bohr -6 ) * r1 r ds1 ds dx3 dx x1 x x1 x ρ (, ) = ( 1) Ψ(,, ) Ψ (,, ) (-1) times the probability per unit volume to find an electron of any spin in small volume element dv 1 around r 1 and simultaneously another electron of any spin in dv around r. The basic ingredient of ab initio methods.

10 ver. electronic population of region Ω ( Ω ) = ρ( r) dr * ( Ω ) = dr ds dx dx Ψ( x, x ) Ψ ( x, x ) Ω Basics Extract electronic behavior from density matrices inside and outside of Ω Ω

11 ver. electronic population of region Ω ( Ω ) = ρ( r) dr * ( Ω ) = dr ds dx dx Ψ( x, x ) Ψ ( x, x ) Ω Basics Extract electronic behavior from density matrices inside and outside of Ω is in fact a sum of n-electron events: n e - inside Ω, the remaining e - outside! * Pn( Ω ) = dx1 dxn dx ' n+ 1 dx ( ' Ψ x1, x) Ψ ( x1, x) n!( n)! Ω Ω Ω Ω inside Ω outside Ω Event probabilities (Daudel) P n

12 ver. electronic population of region Ω ( Ω ) = ρ( r) dr * ( Ω ) = dr ds dx dx Ψ( x, x ) Ψ ( x, x ) Ω Basics Extract electronic behavior from density matrices inside and outside of Ω is in fact a sum of n-electron events: n e - inside Ω, the remaining e - outside! * Pn( Ω ) = dx1 dxn dx ' n+ 1 dx ( ' Ψ x1, x) Ψ ( x1, x) n!( n)! Ω Ω Ω Ω inside Ω outside Ω Event probabilities (Daudel) P n ( ) Ω= n P ( Ω) n= 0 P 0 only important for the normalization. n n= 0 P ( Ω= ) 1 n average value of a discrete random variable X: X = x p i= 1 i i

13 Example 1: e -, two spacefilling regions, ( Ω= ) 1 one event: ( ) 100% Ω 1 Ω ( ) Ω = 0 P + 1 P + P = = 1 Ω= n Pn ( Ω ) n= 0

14 Example 1: e -, two spacefilling regions, one event: ( ) 100% Ω = 0 P + 1 P + P = = 1 two events: ( ) 1 Ω= Ω 1 Ω Ω 1 Ω 50% 50% + ( Ω 1) = P P P = = 1

15 Example 1: e -, two spacefilling regions, one event: ( ) 100% Ω = 0 P + 1 P + P = = 1 two events: ( ) 1 Ω= Ω 1 Ω Ω 1 Ω 50% 50% + ( Ω 1) = P P P = = 1 How to distinguish between the two cases? Fluctuation of the aver. population: Variance of a discrete random variable i with probability function p i ( ) σ ( ; Ω ) = ( Ω) ( Ω) = ( ) p i i i

16 Example 1: e -, two spacefilling regions, one event: ( ) 100% Ω = 0 P + 1 P + P = = 1 two events: ( ) 1 Ω= Ω 1 Ω Ω 1 Ω 50% 50% + ( Ω 1) = P P P = = 1 How to distinguish between the two cases? Fluctuation of the aver. population: ( ) σ Ω = Ω Ω ( ; ) ( ) ( ) n n n n= 1 n= 1 n= 1 ( Ω= ) n P ( Ω= ) n( n 1) P ( Ω ) + np ( Ω) = dr dr ρ ( r, r ) + ( Ω) Ω 1 1 Ω n= 0 D (Ω), distinct ( ) σ ( ; Ω ) = n( n 1) Pn + ( Ω) ( Ω) Remark: distinct pairs: (1,), (,1) 1 non-distinct pair: {1, }

17 Example 1: e -, two spacefilling regions, one event: 100% two events: ( ) 1 Ω= 50% 50% ( ) Ω 1 Ω Ω = 0 P + 1 P + P = = 1 + ( Ω 1) = P P P = = 1 n= 0 ( ) σ ( ; Ω ) = n( n 1) Pn + ( Ω) ( Ω) D( Ω ) = 0 P0 + 0 P1 + 0= 0 σ ( ; Ω ) = = 0!!! distinct 0, in the present case

18 Example 1: e -, two spacefilling regions, one event: 100% two events: ( ) 1 Ω= 50% 50% ( ) Ω 1 Ω Ω = 0 P + 1 P + P = = 1 + ( Ω 1) = P P P = = 1 n= 0 ( ) σ ( ; Ω ) = n( n 1) Pn + ( Ω) ( Ω) D( Ω ) = 0 P0 + 0 P1 + 0= 0 σ ( ; Ω ) = = 0 distinct!!! 0, in the present case ( Ω ) = 1 distinct D ( Ω ) = 0 P + 0 P = σ ( ; Ω ) = σ ( ; Ω ) = ( Ω)!!! =1

19 Basics Decomposition of the pair density into pairwise spin contributions: ββ x1 x r1 r 1 r1 r 1 ρ (, ) = ρ (, ) ( ω ) ( ω ) + ρ (, ) β( ω ) β( ω ) + β β r1 r 1 r1 r 1 + ρ (, ) ( ω ) β( ω ) + ρ (, ) β( ω ) ( ω ) The same-spin pair density: normalized to ( -1)distinct pairs. r1 r ρ r1 r D = d d (, ) = ( 1) The opposite-spin pair density: normalized to β distinct pairs. β β r1 r ρ r1 r D = d d (, ) = β The spinless pair density: normalized to ( - 1)distinct pairs. D = dr dr ρ ( r, r ) = ( 1) 1 1

20 Basics Correlation in statistical sense probability of the combined event vs. probability of the product of single events, B = B + f B correlation factor f = 0, B = B statistically independent events else: statistically correlated events

21 Basics Pair density and correlation: the correlation factor f σσ (r 1, r ) Correlation in statistical sense Coulomb and exchange correlation: ρ ( r1, r) = ρ( r1) ρ( r) + ρ( r1) ρ( r) f ( r1, r) D = ρ ( r1, r) = ρ( r1) ρ( r) + ρ ( r1, r) xc

22 Basics Pair density and correlation: the correlation factor f σσ (r 1, r ) Correlation in statistical sense Coulomb correlation: β ρ ( r1, r) = ρ( 1) ρβ( ) ρ f β ρβ r r + ( r1) ( r) ( r1, r) D β = 0 β

23 Basics Pair density and correlation: the correlation factor f σσ (r 1, r ) Coulomb correlation: β ρ ( 1, ) = ρ Correlation in statistical sense r r ( r1) ρβ( r) + ρ( r1) ρβ( r) ( r1, r) D β = 0 Fermi-correlation: β f β ρ ( r1, r) = ρ( r1) ρ( r) 1 + f ( 1, ) r r = ρ ( r) ρ ( r ) + ρ ( r) ρ ( r ) f ( r, r ) D = "correlation of electronic motion", "Fermi hole part of the pair density", "exchange-correlation hole density" (same-spin part) Exclusion of self-pairing! ρ x r1 r (, )

24 Basics Pair density and correlation: the correlation factor f σσ (r 1, r ) Coulomb correlation: β ρ ( 1, ) = ρ Correlation in statistical sense r r ( r1) ρβ( r) + ρ( r1) ρβ( r) ( r1, r) D β = 0 Fermi-correlation: β f β ρ ( r1, r) = ρ( r1) ρ( r) 1 + f ( 1, ) r r = ρ ( r) ρ ( r ) + ρ ( r) ρ ( r ) f ( r, r ) D = ( r1, r) 1 f ( r, r) = 1 and ρ ( r1, r1) = 0 f Exclusion of self-pairing! due to Pauli principle!

25 Electron and the Correlation of Electronic Motion 1 = f 1 ρ ( r, r ) ρ ( r) ρ ( r ) ρ ( r) ρ ( r ) ( r, r ) f r1 r ( ) 1 (, ) Ω = r1 r ρ r1 r = Ω + r1 r ρ r1 ρ r r1 r Ω Ω Ω Ω = ( ( )) F Ω + ( Ω) D ( ) d d (, ) ( ) d d ( ) ( ) f (, )

26 1 = f 1 ρ ( r, r ) ρ ( r) ρ ( r ) ρ ( r) ρ ( r ) ( r, r ) f r1 r ( ) Ω = r1 r ρ r1 r = Ω + r1 r ρ r1 ρ r r1 r Ω Ω Ω Ω = ( ( )) F Ω + ( Ω) D ( ) d d (, ) ( ) d d ( ) ( ) f (, ) total space: Electron and the Correlation of Electronic Motion 1 1 dr dr ρ ( r, r ) = ( ) σ (, Ω ) = D ( Ω) ( Ω ) + ( Ω) 0 1 (, ) fluctuation of is zero!

27 1 = f 1 ρ ( r, r ) ρ ( r) ρ ( r ) ρ ( r) ρ ( r ) ( r, r ) f r1 r ( ) Ω = r1 r ρ r1 r = Ω + r1 r ρ r1 ρ r r1 r Ω Ω Ω Ω = ( ( )) F Ω + ( Ω) D ( ) d d (, ) ( ) d d ( ) ( ) f (, ) total space: Electron and the Correlation of Electronic Motion 1 1 dr dr ρ ( r, r ) = ( ) σ (, Ω ) = D ( Ω) ( Ω ) + ( Ω) 0 F 1 (, ) fluctuation of is zero! ( Ω+ ) ( Ω) 0 F ( Ω) ( Ω) F ( Ω) ( Ω) number of Fermi holes

28 1 = f 1 ρ ( r, r ) ρ ( r) ρ ( r ) ρ ( r) ρ ( r ) ( r, r ) f r1 r ( ) Ω = r1 r ρ r1 r = Ω + r1 r ρ r1 ρ r r1 r Ω Ω Ω Ω = ( ( )) F Ω + ( Ω) D ( ) d d (, ) ( ) d d ( ) ( ) f (, ) total space: 1 1 dr dr ρ ( r, r ) = ( ) σ (, Ω ) = D ( Ω) ( Ω ) + ( Ω) 0 therefore: Electron and the Correlation of Electronic Motion F ( ) ( ) D ( Ω ) = ( Ω ) + F ( Ω) ( Ω) ( Ω) ( Ω) ( Ω) 1 (, ) fluctuation of is zero! limiting number of pairs in Ω (lower limit)

29 1 = f 1 ρ ( r, r ) ρ ( r) ρ ( r ) ρ ( r) ρ ( r ) ( r, r ) f r1 r ( ) Ω = r1 r ρ r1 r = Ω + r1 r ρ r1 ρ r r1 r Ω Ω Ω Ω = ( ( )) F Ω + ( Ω) D ( ) d d (, ) ( ) d d ( ) ( ) f (, ) total space: 1 1 dr dr ρ ( r, r ) = ( ) σ (, Ω ) = D ( Ω) ( Ω ) + ( Ω) 0 sub-space Ω: Electron and the Correlation of Electronic Motion ( ) ( ) D ( Ω ) = ( Ω ) + F ( Ω) ( Ω) ( Ω) If the whole -correlation of electronic motion F (Ω) is contained in region Ω, the lowest possible value for D (Ω) is obtained : F ( Ω ) = ( Ω) F ( ) D ( Ω= ) ( Ω) ( Ω) ( Ω) ( Ω) 1 (, ) fluctuation of is zero! σ ( ; Ω= ) 0 i.e., the av. number of -pairs is obtained from the av. number of -electrons. complete localization of the electron(s) within Ω.

30 ρ ( r, r ) ρ ( r) ρ ( r ) ρ ( r) ρ ( r ) ( r, r ) 1 ( r1, r) 1 = f 1 ( ) Ω = r1 r ρ r1 r = Ω + r1 r ρ r1 ρ r r1 r Ω Ω Ω Ω = ( ( )) F Ω + ( Ω) D ( ) d d (, ) ( ) d d ( ) ( ) f (, ) total space: 1 1 "correlation of electronic motion", "Fermi hole part of the pair density" dr dr ρ ( r, r ) = ( ) σ (, Ω ) = D ( Ω) ( Ω ) + ( Ω) 0 sub-space Ω: ( ) ( ) D ( Ω ) = ( Ω ) + F ( Ω) ( Ω) ( Ω) If more pairs D (i.e., less holes F ) are found in region Ω: due to pairing with electrons outside of Ω: F ( ) ( Ω) ( Ω) f fluctuation of is zero! F ( Ω ) > ( Ω) D ( Ω ) > ( Ω) ( Ω) σ ( ; Ω ) > 0 incomplete localization of the electron(s) within Ω The number of same-spin pairs in Ω indicates the of the electrons inside!

31 Basics For a region Ω where: (Bader, Stephens, 1974) σ (, Ω ) = 0 the Fermi hole is fully contained in region Ω. the electrons are maximally localized in that region: localized Fermi hole implies a localized electron. ( Ω ) = ( Ω) Relation between the correlation of electronic motion of same spin electrons, i.e., the Fermi hole part of the pair density and electron.

32 Basics For a region Ω where: (Bader, Stephens, 1974) σ (, Ω ) = 0 the Fermi hole is fully contained in region Ω. the electrons are maximally localized in that region. localized Fermi hole implies a localized electron. Relation between the correlation of electronic motion of same spin electrons, i.e., the Fermi hole part of the pair density and electron. Electron in a certain spatial region is related to the behavior of the Fermi hole. The local behavior of electron is related to the curvature of the Fermi hole.

33 * r1 r ds1 ds dx3 dx x1 x x1 x ρ (, ) = ( 1) Ψ(,, ) Ψ (,, ) Localization Index : σσ ' σσ ' r1 r ρ r1 r Ω Ω D ( ) d d (, ) Ω = σσ ' σσ ' B r1 r ρ r1 r Ω ΩB σ = dr ρ σ () r Ω D (, ) d d (, ) Ω Ω = Delocalization Index: (ESI) Fluctuation of : Basics Extract electronic behavior from density matrices Use information about the non-local pairing of same-spin electrons: λ Ω = Ω Bader, Stephens, JCS 1975, 97, Fradera, usten, Bader, J. Phys. Chem. 1999, 103,304. ( ) D ( ) number of e - localized in Ω δ( Ω, Ω ) = ( D ( Ω, Ω )) B B B number of e - delocalized/shared between Ω and Ω B. 1 ( Ω ) = ( Ω ) = ( Ω, ΩB) B σ λ δ macroscopic regions Ω e - Ω B

34 Basics Extract electronic behavior from density matrices Use information about the non-local pairing of same-spin electrons: Localization Index : Delocalization Index: (Electron Sharing Index) Fluctuation of : Sum rules: λ Ω = Ω ( ) D ( ) number of e - localized in Ω δ( Ω, Ω ) = ( D ( Ω, Ω )) B B B number of e - delocalized/shared between Ω and Ω B. 1 ( Ω ) = ( Ω ) = ( Ω, ΩB) B σ λ δ 1 = λ( Ω ) + δ( Ω, ΩB) B = λ( Ω ) + δ( Ω, Ω ) B, B>

35 Chemical Bonding, Electron Pairing and Electron Sharing "The facts just brought forward strongly suggest that instead of treating the electron pair as a unit bond, we should regard a single bonding electron as the natural unit bond. The one-electron bond is a single bonding electron, the electron-pair bond is two bonding electrons symmetrically related." R.S. Mulliken, Bonding Power of Electrons and Theory of Valence, Chem. Rev. 9, (1931). "It is often said that the chemical bond is related to spin pairing. However, spin only plays a role in relation to the Pauli principle. The direct interaction of the spins (an interaction of two magnetic dipoles) is entirely negligible...." W. Kutzelnigg, The Physical Origin of the Chemical Bond, Ed. Z. B. Maksic, Theoretical Models of Chemical Bonding, Part, Springer, "... the sharing of a single electron establishes a chemical bond. The two-electron bond is essentially the cumulative result of the effects of each electron being shared individually between atoms (tempered, of course, by the effect of the interelectronic repulsion)." K. Ruedenberg, M. W. Schmidt, Why Does Electron Sharing Lead to Covalent Bonding? Variational nalysis. J. Comput. Chem. 8, (007).

36 Example : e -, two spacefilling regions, H molecule ( Ω ) = 1 5% β + 5% 5% 5% Monodeterminantal wavefunction ϕg 1 ω1 ϕg β ω Ψ ( x,x) = ( r) ( ) ( r) ( ) 1 = ϕg(1) ϕg() ϕg(1) ϕ () g = C ( s(1) sb (1))( s() sb() ) ( s(1) sb( 1) )( s() sb ()) = C s (1) s () + s (1) s () + s (1) s () + s (1) s () [ B B B B s ( 1) s () s (1) s () s (1) s ( ) s ( 1) () B B B B s ] 5% 5% 5% 5%

37 Example : e -, two spacefilling regions, H molecule ( Ω ) = 1 5% β + 5% 5% 5% Monodeterminantal wavefunction Spinaveraged: = n n = 0 n P ( ) D ( ) = 0 λ = = D ( ) = n ( n 1) P n = = 1 n = = 0.5 = 0.5 D (, B) = n n P = = 0.5 B n, n n, n = 0 δ (, B) = ( D (, B)) = (11 0.5) = 1.0 B B B

38 Example : e -, two spacefilling regions, H molecule ( Ω ) = 1 5% β + 5% 5% 5% Monodeterminantal wavefunction Spinresolved: same-spin = npn n = 0 D ( ) = n( n 1) Pn n = 0 λ ( ) = D ( ) + = = 0.5 = = 0. 0 = 0 = 0.5 D (, B) = n n P = 0. 0 = 0 B n, n n, nb= 0 (, B) ( B D (, B)) δ = = (0.5 0) = 0.5 B

39 Example : e -, two spacefilling regions, H molecule ( Ω ) = 1 5% β + 5% 5% 5% Monodeterminantal wavefunction Spinresolved: = β = 0.5 β, β β β λ ( ) = ( D ( )) = β β ( ) = n, n β n, n = 0 β opp.-spin D n n P = D ( ) β = 0 β β β D (, B) = n n P = D = = 0.5 β (, B) B n, n β n, nb = 0 β, β β β (, B) 4( B D (, B)) δ = = 4( ) = 0 B = = 0.5

40 Example 3: e -, two spacefilling regions, H molecule 30% 30% β + ( Ω ) = 1 Correlated wavefunction, schematical 0% 0% + decrease the weight of "ionic events" (exagg.) Ψ ( x,x) = C ϕ ( r) ( ω ) ϕ ( r) β( ω ) + C ϕ ( r) ( ω ) ϕ ( r) β( ω ) 1 1 g 1 1 g u 1 1 u 1 ( (1) (1))( () ()) ( (1) (1))( () ()) = C 1 s sb s sb s sb s s B + C ( s(1) sb(1) )( s() sb() ) ( s(1) sb(1) )( s() sb() ) = C s (1) s () + s (1) s () + s (1) s () + s (1) s () [ [ B B s (1) s () s (1) s () s (1) s () s (1) s () B B + C s (1) s () + s (1) s () s (1) s () s (1) s () B B s (1) s () s (1) s ( ) + s (1) s () + s (1) s () B B B B B B B B B B ] ] C 1 > 0 C < 0

41 Example 3: e -, two spacefilling regions, H molecule 30% 30% β + ( Ω ) = 1 Correlated wavefunction, schematical Spinaveraged: = n n = 0 n P ( ) D ( ) 0% 0% = 0 + = = 1 decrease the weight of "ionic events" (exagg.) D ( ) = n( n 1) P = = 0.4 n n λ = = = 0.6 D (, B) = n n P = = 0.6 B n, n n, n = 0 δ (, B) = ( D (, B)) = ( ) = 0.8 B B B

42 Example 3: e -, two spacefilling regions, H molecule 30% 30% β + ( Ω ) = 1 Correlated wavefunction, schematical Spinresolved: same-spin 0% 0% = npn n = 0 D ( ) = n( n 1) Pn n = 0 λ ( ) = D ( ) + = = 0.5 = = = 0 = 0.5 D (, B) = n n P = = 0 B n, n n, nb= 0 (, B) ( B D (, B)) δ = = (0.5 0) = 0.5 B decrease the weight of "ionic events" (exagg.)

43 Example 3: e -, two spacefilling regions, H molecule 30% 30% β + ( Ω ) = 1 Correlated wavefunction, schematical Spinresolved: opp.-spin = β = 0.5 β, β β β 0% 0% λ ( ) = ( D ( )) + β β ( ) = n, n β n, n = 0 D n n P = ( ) decrease the weight of "ionic events" (exagg.) β = 0.1 = = 0.0 β β β D (, B) = n n P = D = = 0.30 β (, B) B n, n β n, nb = 0 β, β β β (, B) 4( B D (, B)) δ = = 4 ( ) = 0. B β = D ( )

44 Uncorrelated (Hartree-Fock) vs. correlated Wavefunction summary: H molecule, schematical δ λ, ββ, ββ λ δ β, β β, β ( ) (, B) λ ( ) + λ ( B) + δ (, B) = ( ) (, B) β, β β, β β, β λ ( ) + λ ( B) + δ (, B) = 0 HF corr. WF λ λ λ,, ( ) ββ β β = ( ) + ( ),, (, B) ββ β β = (, B) + (, B) δ δ δ λ( ) + λ( B) + δ(, B) =

45 Basics Correlated WF Hartree-Fock LI : ββ β β λ( Ω ) = λ ( Ω ) + λ ( Ω ) + λ ( Ω ) + λ ( Ω ) ( ) D λ σσ ( Ω ) = σσ σσ ( Ω ) λ ( Ω ) = D ( Ω ) σσ ' σ σ ' σσ ' lim r 0 σ dto. σσ λ ' ( Ω ) = 0 DI : ββ β β B B B B B δ( Ω, Ω ) = δ ( Ω, Ω ) + δ ( Ω, Ω ) + δ ( Ω, Ω ) + δ ( Ω, Ω ) ( D ) ( D ) σσ σ σ σσ B B B δ ( Ω, Ω ) = ( Ω, Ω ) σσ ' σ σ ' σσ ' B B B δ ( Ω, Ω ) = ( Ω, Ω ) 0 dto. 0 σσ δ ' ( Ω, Ω ) = 0 B Ll+DI : = λ( Ω ) + δ( Ω, Ω ) 1 B B, ββ, ββ λ δ B λ δ B, B>, B> β, β β, β λ ( Ω ) + δ ( Ω, Ω B) = 0, B> = ( Ω ) + ( Ω, Ω ) = ( Ω ) + ( Ω, Ω )

46 Concrete calculations: Hartree-Fock vs. Configuration Interaction Pair density in MO representation LI : DI : Ll+DI : λ( Ω ) = D ( Ω ) = D S ( Ω ) S ( Ω ) 1-det. approx. ij, kl ij kl i, j, k, l δ ( Ω, Ω B) = ( B D ( Ω, Ω B) ) = B Dij, kl Sij( Ω) Skl( ΩB) i, j, k, l 1-det. approx. * * r1 r Dij, kl i r1 j r1 k r i r i, j, k, l ρ (, ) = φ ( ) φ ( ) φ ( ) φ ( ) S ( Ω ) = dr φ ( r) φ ( r) ij Ω i j 1 = λ( Ω ) + δ( Ω, ΩB) B occ. = Sij ( Ω) ij occ. = S ( Ω ) S ( Ω ) i, j ij ij B

47 Concrete calculations: Hartree-Fock level H : S ii ( Ω ) = dr φ g ( r) φ g ( r) = 0.5 Ω Ν =1 λ (Ω ) = 0.5 λ(ω ) = 0.5 λ ββ (Ω ) = 0.5 ( λ ) δ( Ω, Ω ) = ( Ω ) B δ(ω,ω B ) = ( ) = 1.0 He : Ν = S(Ω ) = 1σ g 1σ u 1σ g 1σ u S ii = 0.5, allways (symmetric dimer) S ij = ± 0.5, perfect overlap in half space λ(ω ) =.0 λ (Ω ) = 1.0 λ ββ (Ω ) = 1.0 δ(ω,ω B ) = (.0-.0) = 0.0 Li : Ν =3 S(Ω ) = 1σ g 1σ u σ g 1σ g 1σ u σ g λ(ω ) =.5 s radial node! <1s/s> Ω =0 λ (Ω ) = 1.5 λ ββ (Ω ) = 1.5 δ(ω,ω B ) = (3.0-.5) = 1.0

48 Concrete calculations: Hartree-Fock level Li : Ν =3 S(Ω ) = 1σ g 1σ u σ g 1σ g 1σ u σ g ( λ ) δ( Ω, Ω ) = ( Ω ) B λ (Ω ) = 1.5 λ(ω ) =.5 λ ββ (Ω ) = 1.5 s radial node! δ(ω,ω B ) = (3.0-.5) = 1.0 Be : Ν =4 S(Ω ) = 1σ g 1σ u σ g σ u 1σ g 1σ u σ g σ u λ(ω ) = 3.6 λ (Ω ) = 1.81 λ ββ (Ω ) = 1.81 δ(ω,ω B ) = ( ) = 0.76 significant 3σ u (p z ) mixing, <s/p z > Ω = 0 cf. He

49 Concrete calculations: Hartree-Fock level : Ν =7 1σ g 1σ u σ g σ u 3σ g 1π u 1π u 1σ g 1σ u σ g λ(ω ) = 5.48 λ (Ω ) =.74 λ ββ (Ω ) =.74 S(Ω ) = σ u δ(ω,ω B ) = ( ) = σ g 1π u 1π u orbital contributions: δ(ω,ω B ) = 1.04 (σ) +.0 (π) <p x /p y > Ω = 0, <p z /p x > Ω =0 <s/p x > Ω Remind: we are still talking about electron pairing and delocalization, the orbital overlaps are just the way how this is calculated at HF level!

50 LI : DI : Ll+DI : Concrete calculations: Hartree-Fock vs. Configuration Interaction Pair density in MO representation λ( Ω ) = D ( Ω ) = D S ( Ω ) S ( Ω ) 1-det. approx. ij, kl ij kl i, j, k, l δ ( Ω, Ω B) = ( B D ( Ω, Ω B) ) = B Dij, kl Sij( Ω) Skl( ΩB) i, j, k, l 1-det. approx. * * r1 r Dij, kl i r1 j r1 k r i r i, j, k, l ρ (, ) = φ ( ) φ ( ) φ ( ) φ ( ) S ( Ω ) = dr φ ( r) φ ( r) 1 = λ( Ω ) + δ( Ω, ΩB) B occ. = Sij ( Ω) ij Basics ij Ω i j occ. = S ( Ω ) S ( Ω ) i, j ij ij B

51 Calculation results: 3 Correlated WF vs. Hartree-Fock WF mol. (Ω ) λ( ) λ σσ λ σσ δ(ω,ω B ) δ σσ λ( ) cum. δ(ω,ω B ) cum. δ cum. cum. λ( ) δ(ω,ω B ) H Li Be F e from SRCI-SD or MRCI-SD, cc-pvqz or better, 3 FRW, Bezugly, Kohout, in preparation. Compare to: correlated wavefunction 1, Hartree-Fock δ σσ HF r1 r = 1 r1 r1 1 r r 1 r1 r 1 r r1 ρ (, ) ρ ( ; ) ρ ( ; ) ρ ( ; ) ρ ( ; ) ρ ( r, r ) = ρ ( r; r) ρ ( r ; r ) ρ ( r; r ) ρ ( r ; r) + λ ( r, r ) X. Fradera, M. usten, R.F.W. Bader, J. Phys. Chem. 1999, 103, 304. R.C. Bochicchio, L. Lain,. Torre, Chem. Phys. Lett. 003, 374, 567. E.Matito, M. Solà, P. Salvador, M. Duran, Faraday Discuss. 007, 135, 35. -particle cumulant, the irreducible -particle part of the -matrix, the "true" correlation part

52 bout interaction energies within the QTIM-IQ framework:!!! ( ) ( B B B B) en ee en en ee nn = E T V V V V V V = E self large xc 1 = Basics > B + > B small ρ ( r, r ) ρ( r) ρ( r ) ρ ( r, r ) Eint = Vcl + Vxc = ( V + V + V ) + V cl Binding energies: wrt. free atom energies E B B bind = def + int > B E E E E B int B B B, with: Edef = Eself E0 def def def E = E ( CR) + E ( CT) 0 B B B B Cee ne ne xc def = self def ( ) = ( ) 0 E ( CR) E E ( ) E CT E E [ ] =Ω QTIM 1 dr r ( r, r ) xc 1 d ρ 1 B r1 ( ) E ( ) = E ( n) + ( n) E ( n+ 1) E ( n)... under investigation since a few years only:. Martín Pendás, M. Blanco et al.

53 * r1 r ds1 ds dx3 dx x1 x x1 x ρ (, ) = ( 1) Ψ(,, ) Ψ (,, ) = dr ρσ () r Ω Localization Index : σ σσ r1 r ρ r1 r Ω Ω D ( ) d d (, ) Ω = σ σσ B r1 r ρ r1 r Ω Ω D (, ) d d (, ) Ω Ω = Delocalization Index: Fluctuation of : B Basics Extract electronic behavior from density matrices Use information about the non-local pairing of same-spin electrons: λ Ω = Ω ( ) D ( ) number of e - localized in Ω Bader, Stephens, JCS 1975, 97, Fradera, usten, Bader, J. Phys. Chem. 1999, 103,304. δ( Ω, Ω ) = ( D ( Ω, Ω )) B B B number of e - delocalized/shared between Ω and Ω B. 1 ( Ω ) = ( Ω ) = ( Ω, ΩB) B σ λ δ macroscopic regions Ω e - Ω B

54 μ Basics Extract electronic behavior from density matrices Use information about the local pairing of same-spin electrons: * r1 r ds1 ds dx3 dx x1 x x1 x ρ (, ) = ( 1) Ψ(,, ) Ψ (,, ) = dr ρ ( r) μ σ σ σσ μ r1 r ρ r1 r μ μ D ( ) d d (, ) = microscopic regions μ e - μ B ELI... tomorrow!