localized? The pair function ρ2 (x 1 , x 2 ; x1 , x2 ) = N(N − 1) dx3 . . . xN Ψ∗(x 1 , x 2 , . . . , xN )Ψ(x1 , x2 , . . . , xN ) If the electrons were not correlated objects, the pair density would only depend on the 1-particle density: ρind 2 (x1 , x2 ) = N−1 N ρ(x1 )ρ(x2 ) The Pauli principles force ρ2 (x 1 , x 2 ; x1 x2 ) = −ρ2 (x 2 , x 1 ; x1 , x2 ), so ρ2 (x1 , x1 ) = 0 If a reference electron (ER) is at x1 , the conditional density that any other be found at x2 is ρ(x2 |x1 ) = ρ2 (x1 ,x2 ) ρ(x1 ) The exchange-correlation hole is deﬁned as hxc (x2 |x1 ) = ρ(x2 |x1 ) − ρ(x2 ) dx2 hxc (x2 |x1 ) = −1, hxc (x2 → x1 |x1 ) = −ρ(x1 ). AMP (QCG@UniOvi) M2 2013 Sep 2013 3 / 24
to same spin (Fermi) and opposite spin (Coulomb) electrons. We thus divide the exchange-correlation hole, hxc (x2 |x1 ) = hxc (r2 (s2 = s1 )|x1 ) + hxc (r2 (s2 = s1 )|x1 ) = hF (r2 |r1 ) + hC (r2 |r1 ). Only The Fermi hole is responsible from exclusion dr2 hF (r2 |r1 ) = −1, hF (r2 → r1 |r1 ) = −ρ(r1 ), dr2 hC (r2 |r1 ) = 0. The FH is negative deﬁnite, deep around the RE, and goes to zero quickly as we get far from it. If the RE is far from the molecular frame, hF will be shallow close to it, and since it integrates to −1, it will leave the RE behind. AMP (QCG@UniOvi) M2 2013 Sep 2013 4 / 24
Ω. Then Pn(Ω), determines the probability of ﬁnding a given number of electrons, n, in Ω is: Pn(Ω) = N n Ω dx1 . . . Ω dxn Ω dxn+1 . . . Ω dxNΨ∗(x1, . . . , xN)Ψ(x1, . . . , xN). Since the total number of electrons in N, n=N n=1 Pn(Ω) = 1, NΩ = n=N n=1 nPn(Ω), With this σ2(Ω) = n=N n=1 (n − NΩ)2Pn(Ω) = n=N n=1 n2Pn(Ω) − n=N n=1 nPn(Ω) 2 . σ2(Ω) may only vanish if there exists just one term in the sum, n = l ⇒ Pl(Ω) = 1, NΩ = l. We dﬁne hte pair population of a region, DΩ,Ω = 1 2 Ω dx1 Ω dx2 ρ2(x1, x2) = n=N n=1 n(n − 1) 2 Pn(Ω). σ2(Ω) = 2DΩ,Ω − NΩ(NΩ − 1), AMP (QCG@UniOvi) M2 2013 Sep 2013 6 / 24
DΩ,Ω = l(l − 1)/2. We say that we have a pure pair population. There is spatial localization since knowing the number of electrons and its pairing just requires ρ1 y ρ2 in that region. If the variance is not null, 2DΩ,Ω > NΩ(NΩ − 1). Using the hxc deﬁnition, σ2(Ω) = NΩ + Ω dx1ρ(x1) Ω dx2hxc(x2|x1) = NΩ + FΩ,Ω, wher FΩ,Ω is a measure of the correlation in region Ω. If hxc is completely localized in Ω, hxc(x2 / ∈ Ω|x1 ∈ Ω) = 0, Ω dx2hxc(x2|x1) = −1, σ2(Ω) = 0, y so complete localization of the hole implies a pure pair population. We will get a perfect localization if the hole is completely localized in a a region, independently of the position of the RE. If the space is made up of regions with completely localized holes, interbasin repulsion is only classical. Thus hole localization is equivalent to electron localization AMP (QCG@UniOvi) M2 2013 Sep 2013 7 / 24
regions ΩA and ΩB are completely localized, then in the joint region ΩA ∪ ΩB we ﬁnd that σ2(ΩA ∪ ΩB) = FΩA,ΩB = 0, FΩA,ΩB = 2 ΩA dx1ρ(x1) ΩB dx2hxc(x2|x1), where we have used A ↔ B. FΩA,ΩB is thus a measure of inter-basin correlation. We thus introduce the interbasin pair population DΩA,ΩB . When FΩA,ΩB is negligible, DΩA,ΩB = NΩA NΩB , the number of order pairs. These leads to the following indices Sinde |FAA | < NA , we introduce the localization index, λA = |FAA | and the delocalization index, δAB = |FAB | FAA + 1 2 B=A FAB = −NA. AMP (QCG@UniOvi) M2 2013 Sep 2013 8 / 24
ﬁeld with localization information? Seminal work of Becke & Edgecombe (JCP 92, 5398 (1990)) Leads to the Electron Localization Function (ELF) Let Pαα be the same spin conditional 2-particle probability for a 1det, Pαα(r, r ) = ραα 2 (r, r ) ρα(r) = ρα(r ) − |ρα 1 (r, r )|2 ρα(r) = ρα(r ) − | α i ψ∗ i (r )ψi (r)|2 ρα(r) Pαα for a RE at r and another at distance s admits a Taylor expansion, Pαα(r, s) = 1 3 α i |∇ψi (r)|2 − |∇ρα(r)|2 4ρα(r) s2 + . . . = 1 3 Dα(r)s2 + . . . This is calibrated with Dα(r) for the homogeneous electron gas of the same density, Dα h (r) = 3 5 (6π2)2/3ρα(r)5/3 AMP (QCG@UniOvi) M2 2013 Sep 2013 10 / 24
is deﬁned as χBE (r) = Dα(r) Dα h (r) , and in order to get a reasonable scaled quantity, a Lorenztian projection is made, ELF(r) = η(r) = 1 1 + χ2 BE (r) This map preserves the topology of χ (number and type of CPs) η = 1/2 ≡ fully delocalized system (electron gas). η = 1 ≡ fully localized system. AMP (QCG@UniOvi) M2 2013 Sep 2013 11 / 24
undeﬁned in KS DFT. Savin et. al. found a way our. t(r) ≥ 1 8 |∇ρ(r)|2 ρ(r) , the equality being valid for a bosonic system, Thus the excess kinetic energy is called the Pauli density, tP (r) = t(r) − 1 8 |∇ρ(r)|2 ρ(r) Again, if we take the value for the uniform electron gas, th (r) = cF ρ(r)5/3, cF = 3 10(3π2)2/3 , we may deﬁne the Savin kernel, χS (r) = tP (r) th (r) , ηS = 1 1 + χ2 S formally equivalent to BE for a closed shell. ηS ≈ 1 for bosonic behavior (lone pairs, bonds...) The interpretation of ELF is full of subtleties. AMP (QCG@UniOvi) M2 2013 Sep 2013 12 / 24
the standard topological tools have been used. Localization domains An f−localization domain is a region of η > f. If it contains only 1 attractor, irreducible. Localization domains in CF4 with f = 0.75 → 0.885 (a-c), Li, LiF, LiH Silvi & Savin, Nature 371, 683 (1994). AMP (QCG@UniOvi) M2 2013 Sep 2013 13 / 24
domains split as CPs are traversed. These are the bips, bond interaction points. This process may be followed by constructing a tree, the bifurcation diagram, where we see the different entities of a system arise: molecules, cores, lone pairs, valences, etc. Solid N2 . N2 N2 N2 V(N) C(N) V(N) V(N,N) 0.0050 0.7187 0.1334 AMP (QCG@UniOvi) M2 2013 Sep 2013 14 / 24
of a valence basin (V) is the number of core (C) basins with which it shares a separatrix (a proton is counted as formal core) Monosynaptic: shells, lone pairs Polisynaptic: multicenter bonds. Densities may also be integrated in ELF basins (care with T). Charges: Those of cores, LPs, and disynaptic basins related to the number of pairs. variances, covariance: Fluctuaction (λ = σ2/N) Base Ω N(Ω) σ2 λ 6-311++G(3df,2p) C(O) 2.10 0.36 0.17 H2 O V(H1,O) 1.65 0.78 0.47 V(O) 2.30 1.08 0.47 Rules: populations of monsynaptic and disynaptic greater and smaller than 2. etc AMP (QCG@UniOvi) M2 2013 Sep 2013 15 / 24
real space. For continuous distributions, a sample volume Vi given by region µi centered around position ai such that the integral of the control function has a ﬁxed value ω. w−restricted space partitioning (RSP) is the decomposition of volume such that the integral of the control function in each region is ω. If ω is small enough, the RSP regions are small: micro-cells. Integrals may be replaced by polynomials in terms of Taylor expansions. AMP (QCG@UniOvi) M2 2013 Sep 2013 18 / 24
µ-cells, allows us to sample with another function fs , such that adding the integral of fs over all the cells gives the total F fs will provide a discrete distribution of values ζi . depending on the control function fc . If fs = fc , the sampling has only one value, ω. If fs =const, the sampling will mimic fs for small ω. On decreasing ω, the density of samples will increase, its value will decrease: Rescaling so that there is a ω → 0 limit. ω = µi dr1 . . . µi fc (r1 , . . . , rm ) drm = tc (ai ) Vϑc i + εc (ai ) with tc depending on the ﬁrst non-vanishing Taylor term. AMP (QCG@UniOvi) M2 2013 Sep 2013 19 / 24
ρ2 as control and sample functions (both possibilities). ELI is the rescaled discrete distribution of electron populations (ELI-D, ΥD ) or electron pairs (ELI-q, Υq ) Same-spin Electron Pairs Due to Pauli, the ﬁrsti r2 Taylor term for the same spin pair population is Dσσ i ≈ 1 2 dr1 µ ((r2 − a) · ∇r2 )2ρσσ 2 (r1 , r2 ) r→a dr2 A bit of algebra leads to Dσσ i ≈ 1 12 V8/3 i gσ(ai ), where gσ(ai ) is the Fermi-hole curvature at the µ-cell. gσ(ai ) = σ i<j σ k<l Pij,kl [φi (ai )∇φj (ai )−φj (ai )∇φi (ai )] [φk (ai )∇φl (ai )−φl (ai )∇φk (ai )] AMP (QCG@UniOvi) M2 2013 Sep 2013 21 / 24
12ω gσ(ai) 3/8 : ˜ VD(r) = lim ω→0 1 ω3/8 Vi = 12 gσ(r) 3/8 . Υσ D (µi) = 1 ω3/8 µ ρσ 1 (r) dr = Υσ D (ai) ≈ 1 ω3/8 ρσ 1 (ai) Vi and, after rescaling, ˜ Υσ D (r) = lim ω→0 {Υσ D (µi)} = ρσ 1 (r) ˜ VD(r) . Υα D in Ar atom using the basis of Clementi and Roetti; black bars: discrete distribution of charges from the integration of ρα 1 over the µ-cells of the ωRSP restricted to enclose 10−8 αα-pairs 1 2 3 4 5 6 r / a 0 1.0 2.0 q / 10-3 e- AMP (QCG@UniOvi) M2 2013 Sep 2013 22 / 24
by ΥD (r)Υ3/8 q (r) = 1. At the 1-det level gσ(r) = ρσ(r)Dα(r) where Dα is the numerator of the ELF kernel. The ELF kernel is proportional to ELI-q. ELI-D and ELF (1/(1 + χ2)) have the same topology. ELI-q for a CAS(8,14)/VDZ calculation of CH4 . 0.91-localization domains of Υq . The hydrogen positions are marked by grey spheres AMP (QCG@UniOvi) M2 2013 Sep 2013 23 / 24
Physics 1990, 92, 5397-5403. A. Savin, A.D. Becke, J. Flad, R. Nesper, H. Preuss, G. von Schnering. Angewandte Chemie-International Edition in English 1991, 30, 409-412. B Silvi, A. Savin. Nature 1994, 371, 683. R. F. W. Bader, S. Johnson, T. H. Tang, P. L. A. Popelier Journal of Physical Chemistry 1996, 100, 15398-15415. M. Kohout, A. Savin, International Journal of Quantum Chemistry 1996, 60, 875-882. M. Kohout. International Journal of Quantum Chemistry 2004, 97, 651-658. M. Kohout, K. Pernal, F. R. Wagner, Y. Grin, Theoretical Chemistry Accounts 2004, 112, 453-459. A. Ormeci, H. Rosner F. R. Wagner M. Kohout, Y Grin, Journal of Physical Chemistry A 2006, 110, 1100-1105. AMP (QCG@UniOvi) M2 2013 Sep 2013 24 / 24