# Generalized Population Analysis II

#### qcgo

September 04, 2013

## Transcript

2. ### GPA (II) Generalized bonding indices References Outline 1 GPA (II)

QTAIM examples 2 Generalized bonding indices A bit of history Taking advantage of recurrences NAdOs 3 References AMP (QCG@UniOvi) M2 2013 September 2013 1 / 25
3. ### GPA (II) Generalized bonding indices References Generalized Population Analysis (GPA)

Electron Population Distribution Functions Starting point: N electron molecule - R3 = Ω1 + Ω2 + · · · + Ωm Probabilities of events: n1 ∈ Ω1 , n2 ∈ Ω2 ,. . . , nm ∈ Ωm , where n1 + n2 + · · · + nm = N, is: p(S) = N! n1 !n2 ! · · · nm ! D |Ψ|2 dr1 · · · drN D = Multidimensional domain such that: r1 · · · rn1 ∈ Ω1 , rn1 +1 · · · rn1 +n2 ∈ Ω2 ,· · · S = {n1 , n2 , · · · } = Real space resonance structure (RSRS). n 1 3 4 m .... 1 3 5 m 2 Ω Ω Ω Ω Ω Ω n n n n n 2 4 5 c We have devised algorithms to obtain the EDFs Both for 1- and multi-Determinant Ψ’s AMP (QCG@UniOvi) M2 2013 September 2013 2 / 25
4. ### GPA (II) Generalized bonding indices References The Statistical Link PCCP

9, 1087 (2007) Chemical bonding ⇔ Statistics of electron populations. na = na na p(na ) na nb = na,nb na nb p(na , nb ) δa,b = −2cov(na , nb ) If ˜ na = na − na , δa,b = −2 na,nb ˜ na ˜ nb p(na , nb ) large δ ⇐⇒ broad EDF . Example: 2c − 2e case. (2,0) (1,1) (0,2) (2,0) (1,1) (0,2) δ=0 A B <1e> <1e> (2,0) (1,1) (0,2) p(1,A)=1,p(2,B)=1 p(1,A)=1/2,p(2,B)=1/2 δ δ=1 δ=2 Non−correlated Positively Correlated Negatively Correlated AMP (QCG@UniOvi) M2 2013 September 2013 3 / 25
5. ### GPA (II) Generalized bonding indices References QTAIM examples: LiH LiH

Structure, S nα Li nα H nβ Li nβ H p(S) (HF) (CAS) 0 2 1 1 0.003 0.003 1 1 0 2 0.003 0.003 1 1 1 1 0.896 0.890 1 1 2 0 0.048 0.051 2 0 1 1 0.048 0.051 2 0 2 0 0.003 0.002 α and β e’s statistically independent in HF. p(Sα, Sβ) = pα(nα A , nα B ) ⊗ pβ(nβ A , nβ B ) In closed shells, pα ≡ pβ In LiH, pα is a 2e distribution. nα Li nα H p(S) 0 2 0.002 1 1 0.944 2 0 0.053 Extremely correlated same-spin electrons. J. Phys. Chem. A 111, 1084 (2007) J. Chem. Phys. 127, 144103 (2007) J. Chem. Phys. 131, 124125 (2009) Theor. Chem. Acc. DOI:10.1007/s00214-010-0809-4 AMP (QCG@UniOvi) M2 2013 September 2013 4 / 25
6. ### GPA (II) Generalized bonding indices References QTAIM examples: 1Σ+ g

N2 At the HF level, p(S) = pα ⊗ pβ (7,0) (6,1) (5,2) (4,3) (3,4) (2,5) (1,6) (0,7) 0.374 5.9e−4 ~0 ~0 5.9e−4 0.126 0.126 0.374 (3,0) (2,1) (1,2) (0,3) 3 independent 1e p’s (p1 (0, 1) = p1 (1, 0) = 0.5) provide: p3 = p1 ⊗ p1 ⊗ p1 p(3, 0) = p(0, 3) = 0.125, p(2, 1) = p(1, 2) = 0.375 A B N N 4 extremely localized electrons in each basin. 3 α, 3 β fully delocalized electrons. Thus... 3 bonding pairs: δαα = 1.519, δ = 3.038 Correlation (CAS[10,10]) hinders delocalization : δαα = 1.418, δαβ = −0.424, δ = 1.987 AMP (QCG@UniOvi) M2 2013 September 2013 5 / 25
7. ### GPA (II) Generalized bonding indices References QTAIM examples: 3Σ− g

O2 A B O O A B O O At the HF level, p(S) = pα ⊗ pβ ~0 ~0 (8,1) (7,2) (6,3) (5,4) (4,5) (3,6) (2,7) (1,8) 4.7e−4 4.7e−4 0.027 0.027 0.472 0.472 (1,0) (0,1) 0.374 6.2e−4 ~0 ~0 6.2e−4 0.126 0.126 0.374 (3,0) (2,1) (1,2) (0,3) β α (7,0) (6,1) (5,2) (4,3) (3,4) (2,5) (1,6) (0,7) AMP (QCG@UniOvi) M2 2013 September 2013 6 / 25
8. ### GPA (II) Generalized bonding indices References QTAIM examples: 3Σ− g

O2 A B O O A B O O A B O O (2c,3e) (2c,2e) Triplet O Each O atom bears a localized core of 2 αβ pairs, plus 2 localized α electrons. Thus, each O bears a localized triplet state . 1 delocalized αβ pair contributes δ ≈ 1.0 2 independent fully delocalized β electrons, each contributing δ ≈ 0.5: Two (2c,3e) bonds: Pauling’s or modern VB calculations δαα = 0.731, δββ = 1.520 δ = 2.251. δαα = 0.732, δββ = 1.190 δαβ = −0.192, δ = 1.539 Correlated AMP (QCG@UniOvi) M2 2013 September 2013 7 / 25
9. ### GPA (II) Generalized bonding indices References A bit of bonding

indices history Partition electron populations: Mulliken ρ(r) = µν Pµν χ∗ ν (r)χµ (r) N = ρ(r)dr = µν PµνSνµ = Tr(PS) = µ (PS)µµ . Na = µ∈a (PS)µµ ≡ a Tra (PS) Generalizing to general real space analyses: Tra ≡ a . For SDs, Na = a ρ(r)dr = i a dr φ∗ i φi = i Sa ii Also for SDs, Q = PS idempotent: Q2 = Q: TrQ = TrQQ = · · · = TrQn = N And we have a hierarchical partition into pairs, trios, ... of centers. In real space, ρ is idempotent. ρ(r1 , r2 )ρ(r2 , r3 )dr2 = ρ(r1 , r3 ). AMP (QCG@UniOvi) M2 2013 September 2013 8 / 25
10. ### GPA (II) Generalized bonding indices References Systematizing bonding parameters N−center

populations for 1dets N = a Na = a Tra Q Na = Traa Q2 + b=a Trab Q2 = Naa + b=a Nab In real space, Na = i Sa ii , Naa = ij (Sa ij )2, Nab = ij Sa ij Sb ij 2Nab = δab : Wiberg-Mayer bond order. orbital description Real space description N = TrQ = a Na N = ρ(r)dr = ρ(r, r)dr Na = Tra Q Na = a ρ(r, r)dr N = TrQ2 = a,b Nab N = ρ(r1 , r2 )ρ(r2 , r1 )dr1 dr2 Naa = Traa Q2 Naa = a a ρ(r1 , r2 )ρ(r2 , r1 )dr1 dr2 Nab = Trab Q2 Nab = a b ρ(r1 , r2 )ρ(r2 , r1 )dr1 dr2 N = TrQ3 = a,b,c Nabc N = ρ(r1 , r2 )ρ(r2 , r3 )ρ(r3 , r1 )dr1 dr2 dr3 Naaa Naab Path multiplicity: abc,acb Nabc AMP (QCG@UniOvi) M2 2013 September 2013 9 / 25
11. ### GPA (II) Generalized bonding indices References Generalization Q ≡ ρ,

Q2 ≡ ρ(1, 2)ρ(2, 1), etc. ρ(1, 2)ρ(2, 1) = ρ(1)ρ(2) − ρ2 (1, 2) ρ(1, 2)ρ(2, 3)ρ(3, 1) = ρ(1)ρ(2)ρ(3) − 1 2 ˆ Sρ(1)ρ2 (2, 3) + 1 3 ρ3 (1, 2, 3) We have come to cumulants: ρn c (1, . . . , N) ρ2 c (1, 2) = ρxc 2 = ρ(1)ρ(2) − ρ2 (1, 2) General! ρ3 c (1, 2, 3) = ρ(1)ρ(2)ρ(3) − 1 2 ˆ Sρ(1)ρ2 (2, 3) + 1 2 ρ3 (1, 2, 3) cumulants maintain the properties of the SD Qn’s: Extensivity: d1d2 . . . di ρi c (1, 2, . . . i) = N. Recurrence: di ρi c (1, 2, . . . , i) = ρi−1 c (1, 2, . . . , i − 1) cumulants are generators of n-th order ﬂuctuations of the Na center populations: Statistical view of bonding. Na , Nab , Nabc , etc ⇒ multi-center indices. AMP (QCG@UniOvi) M2 2013 September 2013 10 / 25
12. ### GPA (II) Generalized bonding indices References Interpreting multi-center bonding indices

Basin partitioning of cumulant densities from extensivity. ρxc provides a partition of ρ into atomic contributions: a d2 ρ2 c (1, 2) = ρ1 a (1) ≡ Ga (1) (DAFH). a ρ1 a (1) = ρ(1). 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 A B G G A B a d1 ρ1 a (1) = Na . a d1 ρ1 a (1) = Naa , b d1 ρ1 a (1) = Nab But this is just the beginning... In general, we may deﬁne n-center, m-th order partitions of the cumulant densities. 1 2 ˆ Sab a d2 b d3 ρ3 c (1, 2, 3) = ρ1 ab (1) with ab ρ1 ab (1) = ρ(1) 1 2 ˆ Sab a d3 b d4 ρ4 c (1, 2, 3, 4) = ρ2 ab (1, 2) with a,b ρ2 ab (1, 2) = ρ2 c (1, 2). AMP (QCG@UniOvi) M2 2013 September 2013 11 / 25
13. ### GPA (II) Generalized bonding indices References Deﬁning Natural Adaptive Orbitals

Expand densities in orbital basis: Ga (1) = ρ1 a (1) = χtAaχ = i na i φ2 i Ponec’s DNOs. At the SD level, several features may be uncovered Aa ij = χi |χj a = Sa ij and φi |φj a = δij ni 0 ≤ ni ≤ 1 i na2 i = Naa and i na i (1 − na i ) = Na,¯ a In our view: φa ≡ atomic orbitals. If they survive, localized in basin. If not, delocalized. na i ≡ contribution to Na Bonding from DNOs? If delocalized. At SD and with a ∪ ¯ a = R3: na2 i contribution to Naa na i (1 − na i ) contribution to Na¯ a The φ’s are equal, and statistically independent. AMP (QCG@UniOvi) M2 2013 September 2013 12 / 25
14. ### GPA (II) Generalized bonding indices References DNOs Reinterpretation: Fully occupied

φ’s with n’s ≡ probabilities. (exact for SDs) The full EDF pN = i pi 1 !! Example: Two electrons, (φ1 , n1 ), (φ2 , n2 ) p1 1(1, 0) = n1 p1 1(0, 1) = 1 − n1 ⊗ p2 1(1, 0) = n2 p2 1(0, 1) = 1 − n2 =   p2(2, 0) = n1n2 p2(1, 1) = n1(1 − n2) + n2(1 − n1) p2(0, 2) = (1 − n1)(1 − n2)   φ’s are CG statistically independent electrons. Each one contributes additively to δ ⇒ Vxc ⇒ covalency Bonding results from delocalization of single electrons. 00000 00000 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 11111 11111 11111 11111 00000 00000 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 11111 11111 11111 11111 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 δ ni Loc. in Ω 2n(1−n) Loc. in Ω’ Delocalized δ emerges from delocalized single electrons. Electron pairs ≡ Fermi exclusion of spins AMP (QCG@UniOvi) M2 2013 September 2013 13 / 25
15. ### GPA (II) Generalized bonding indices References DNOs H2 α-set: 1

electron β-set: 1 electron (1 × 1)SΩ ab (1 × 1)SΩ ab σg |σg Ω = 0.5 σg |σg Ω = 0.5 n1 = 0.5 n1 = 0.5 φ1 = σg φ1 = σg δi = 0.5 δi = 0.5 set β− α−set α−set set β− He2 α-set: 2 electron β-set: 2 electron (2 × 2)SΩ ab (2 × 2)SΩ ab {σg , σu } 0.5 ≈ 0.5 ≈ 0.5 0.5 0.5 ≈ 0.5 ≈ 0.5 0.5 n1 ≈ 0.0, n2 ≈ 1.0 n1 ≈ 0.0, n2 ≈ 1.0 φ1 ≈ 1sA , φ2 ≈ 1sB φ1 ≈ 1sA , φ2 ≈ 1sB δ1 ≈ 0.0, δ2 ≈ 0.0 δ1 ≈ 0.0, δ2 ≈ 0.0 AMP (QCG@UniOvi) M2 2013 September 2013 14 / 25
16. ### GPA (II) Generalized bonding indices References DNOs Li2 3-electron block

with 1σg , 2σg , 1σu At R = Re , SΩ ab in the 1g, 2g, 1u order:   0.5 0.0 0.5 0.0 0.5 1.57 × 10−3 0.5 1.57 × 10−3 0.5   1 2 3 ni 1.000 0.000 0.500 δi 0.000 0.000 0.500 φi ≈ 1sA ≈ 1sB ≈ 2σg Two localized 1s2 cores, one delocalized pair. α−set AMP (QCG@UniOvi) M2 2013 September 2013 15 / 25
17. ### GPA (II) Generalized bonding indices References DNOs N2 singlet set

α− x2 0.010 0.500 x2 δ 0.500 AMP (QCG@UniOvi) M2 2013 September 2013 16 / 25
18. ### GPA (II) Generalized bonding indices References DNOs O2 triplet x2

0.010 x2 0.010 0.058 x4 β− set set α− x2 0.500 0.500 0.500 δ AMP (QCG@UniOvi) M2 2013 September 2013 17 / 25
19. ### GPA (II) Generalized bonding indices References Deﬁning Natural Adaptive Orbitals

ρ3 c provides a partition of ρ in basin pairs. 1 2 ˆ Sab a b d2d3 ρ3 c (1, 2, 3) = ρ1 ab (1): ab ρ1 ab (1) = ρ(1). d1ρab (1) = Nab c d1 ρab ≡ Nabc ρab = χtAabχ = i nab i φ2 i i nab i = Nab . For SDs Aab = 1 2 (SaSb + SbSa). In our view: φab ≡ two-center bond orbitals. If they delocalize on other basins⇒ multi-center. nab is the additive contribution to Nab General framework. AMP (QCG@UniOvi) M2 2013 September 2013 18 / 25
20. ### GPA (II) Generalized bonding indices References SD relationships Some important

insights if a ∪ ¯ b = R3 and SDs. Sb = 1 − Sa nab = nanb = na(1 − na). max{nab} = 1/4 φa = φb = φab. All insights from usual DAFHs are maintained. If a ∪ ¯ b = R3, even for SDs, nab < 0 are possible. max{nab} still equal to 1/4. φa = φb = φab φab belong to the group of the ab bond. No general need to perform isopycnic localizations. AMP (QCG@UniOvi) M2 2013 September 2013 19 / 25
21. ### GPA (II) Generalized bonding indices References NAdOs in CH4 :

φC, φCH, φHH at |φ| = 0.17 a.u. 0.9999 0.5909 0.4439 0.4439 0.4439 0.0000 0.0004 0.2336 0.0058 0.0058 0.0000 -0.0029 0.0126 0.0002 0.0009 AMP (QCG@UniOvi) M2 2013 September 2013 20 / 25
22. ### GPA (II) Generalized bonding indices References 2-center NAdOs and multiple

bonds C2 H6 , C2 H4 , C2 H2 . |φ| = 0.17 a.u. + 4nab (contrib. to δ) 0.8836 0.9356 0.9504 0.8764 0.9580 0.9580 AMP (QCG@UniOvi) M2 2013 September 2013 21 / 25
23. ### GPA (II) Generalized bonding indices References 2-center NAdOs Correlation effects

LiH RHF, CAS(4,12). nab i RHF CAS nab i CAS Nab... RHF CAS nab 1 0.0074 0.0075 nab 7 -0.0000 Nab 0.0970 0.1042 nab 2 0.0896 0.0986 nab 8 0.0002 Naa 1.9896 1.9932 nab 3 -0.0024 nab 9 0.0002 Nbb 1.8164 1.7984 nab 4 0.0005 nab 10 0.0000 Na 2.0866 2.0974 nab 5 -0.0001 nab 11 0.0000 Nb 1.9134 1.9026 nab 6 -0.0001 nab 12 -0.0001 φ1 φ2 φ3 AMP (QCG@UniOvi) M2 2013 September 2013 22 / 25
24. ### GPA (II) Generalized bonding indices References NAdOs Correlation effects H2

O RHF, CAS(8,14). a, b, c ≡ O, H1, H2. nab i RHF CAS nab i CAS nabc i RHF CAS Nab... RHF CAS nab 1 0.0000 0.0000 nab 9 -0.0119 nabc 1 0.0000 0.0000 Naaa 8.0267 8.0065 nab 2 0.0021 0.0130 nab 10 0.0004 nabc 2 -0.0026 -0.0012 Nbbb 0.0133 0.0333 nab 3 0.3027 0.2990 nab 11 0.0002 nabc 3 0.0052 0.0078 Nabc 0.0028 0.0092 nab 4 0.0074 0.0208 nabc 4 0.0001 0.0019 Naa 8.5550 8.4714 nab 5 0.0117 0.0290 nabc 5 0.0000 0.0007 Nab 0.3240 0.3181 nab 6 -0.0156 nabc 6 -0.0002 Nbb 0.0708 0.1127 nab 7 -0.0009 nabc 7 -0.0002 Na 9.2029 9.1075 nab 8 -0.0158 nabc 8 0.0003 Nb 0.3985 0.4462 2-center 3-center 0.2990 -0.0012 0.0078 0.0019 AMP (QCG@UniOvi) M2 2013 September 2013 23 / 25
25. ### GPA (II) Generalized bonding indices References 3-center functions For SDs,

φabc are obtained by diagonalization of 1 6 ˆ Sabc Sa Sb Sc . Only 3-center delocalized functions provide non-negligible nabc eigenvalues. Example: B2 H6 . Only one |nabc| > 10−4 n = 0.0061 In order to compare, 3 √ n = 0.183. max = 0.333. fairly strong 3-center link. AMP (QCG@UniOvi) M2 2013 September 2013 24 / 25
26. ### GPA (II) Generalized bonding indices References References Large set of

papers by R. Ponec, P. Ziesche, M. y M.S. Giambiagi, R. Bochicccio, L. Lain, etc. J. Chem. Phys. 131, 124125 (2009) Comput. Theor. Chem. doi:j.comptc.2012.09.009 AMP (QCG@UniOvi) M2 2013 September 2013 25 / 25