Generalized Population Analysis I

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1. Generalized Population Analysis (I) Ángel Martín Pendás Quantum Chemistry Group Universidad de Oviedo Spain Madrid 2013 AMP (QCG@UniOvi) M2 2013 Sep 2013 1 / 6
2. Content 1 Generalized Population Analysis Basic Aspects Overlap Bonding Indices AMP (QCG@UniOvi) M2 2013 Sep 2013 2 / 6
3. Basic Aspects 1-Det Orbital model: Ψ = |φ1 . . . φN | Linear Variational Procedure: φi = µ ciµ χµ Minimal Basis: µ ≡ electron. χµ atom centered. χν |χµ = Sνµ is key. Example: H+ 2 . χ1 ≡ 1sa , χ2 ≡ 1sb : φ = 1 √ 2(1+S12) (1sa + 1sb ){α, β}. Ψ ≡ φ{α, β}. How to define bonding parameters? Partitioning charges: Mulliken analysis ρ(r1 , r1 ) = N Ψ∗(r1 , . . . , rN )Ψ(r1 , . . . , rN )dr2 . . . drN ρ(r, r) = ρ(r) = i φ∗ i (r)φi (r) = µν Pµν χ∗ ν (r)χµ (r) Pµν = i ciµc∗ iν . N = ρ(r)dr = µν Pµν Sνµ = Tr(PS) = µ (PS)µµ . Na = µ∈a (PS)µµ PS = 1 2(1 + S) 1 1 1 1 1 S S 1 = 1/2 1/2 1/2 1/2 Building orthogonal χ’s, φi = µ ciµ χµ Na = µ∈a Pµµ = ˆ Pa ρ(r)dr = a ρ(r)dr = i a dr φ∗ i φi = i Sa ii AMP (QCG@UniOvi) M2 2013 Sep 2013 3 / 6
4. Bonding ≡ Overlap −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −4 −2 0 2 4 6 ψ / a0 −3/2 z / a0 Symmetrically orthogonalized atomic basis for the H+ 2 molecule Q = PS ≡ P idempotent: Q2 = Q: TrQ = TrQQ = · · · = TrQn = N Partition in pairs, trios, ... of centers. N = µ Qµµ = µν QµνQνµ . Qµν = i ciµ c ∗ iν Na = µ∈a Qµµ = Tra Q = µ∈a,ν∈a QµνQνµ + µ∈a,ν / ∈a QµνQνµ Na = Traa Q2 + b=a Trab Q2 = Naa + b=a Nab Na = i Sa ii Sa ij = µ∈a ciµ cjµ Naa = ij (Sa ij )2 Nab = ij Sa ij Sb ij AMP (QCG@UniOvi) M2 2013 Sep 2013 4 / 6
5. Bonding ≡ Overlap −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −4 −2 0 2 4 6 ψ / a0 −3/2 z / a0 Symmetrically orthogonalized atomic basis for the H+ 2 molecule Q = PS ≡ P idempotent: Q2 = Q: TrQ = TrQQ = · · · = TrQn = N Partition in pairs, trios, ... of centers. N = µ Qµµ = µν QµνQνµ . Qµν = i ciµ c ∗ iν Na = µ∈a Qµµ = Tra Q = µ∈a,ν∈a QµνQνµ + µ∈a,ν / ∈a QµνQνµ Na = Traa Q2 + b=a Trab Q2 = Naa + b=a Nab Na = i Sa ii Sa ij = µ∈a ciµ cjµ Naa = ij (Sa ij )2 Nab = ij Sa ij Sb ij AMP (QCG@UniOvi) M2 2013 Sep 2013 4 / 6
6. Bonding ≡ Overlap −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −4 −2 0 2 4 6 ψ / a0 −3/2 z / a0 Symmetrically orthogonalized atomic basis for the H+ 2 molecule Q = PS ≡ P idempotent: Q2 = Q: TrQ = TrQQ = · · · = TrQn = N Partition in pairs, trios, ... of centers. N = µ Qµµ = µν QµνQνµ . Qµν = i ciµ c ∗ iν Na = µ∈a Qµµ = Tra Q = µ∈a,ν∈a QµνQνµ + µ∈a,ν / ∈a QµνQνµ Na = Traa Q2 + b=a Trab Q2 = Naa + b=a Nab Na = i Sa ii Sa ij = µ∈a ciµ cjµ Naa = ij (Sa ij )2 Nab = ij Sa ij Sb ij AMP (QCG@UniOvi) M2 2013 Sep 2013 4 / 6
7. Bonding ≡ Overlap −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −4 −2 0 2 4 6 ψ / a0 −3/2 z / a0 Symmetrically orthogonalized atomic basis for the H+ 2 molecule Q = PS ≡ P idempotent: Q2 = Q: TrQ = TrQQ = · · · = TrQn = N Partition in pairs, trios, ... of centers. N = µ Qµµ = µν QµνQνµ . Qµν = i ciµ c ∗ iν Na = µ∈a Qµµ = Tra Q = µ∈a,ν∈a QµνQνµ + µ∈a,ν / ∈a QµνQνµ Na = Traa Q2 + b=a Trab Q2 = Naa + b=a Nab Na = i Sa ii Sa ij = µ∈a ciµ cjµ Naa = ij (Sa ij )2 Nab = ij Sa ij Sb ij AMP (QCG@UniOvi) M2 2013 Sep 2013 4 / 6
8. Bonding ≡ Overlap −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −4 −2 0 2 4 6 ψ / a0 −3/2 z / a0 Symmetrically orthogonalized atomic basis for the H+ 2 molecule Q = PS ≡ P idempotent: Q2 = Q: TrQ = TrQQ = · · · = TrQn = N Partition in pairs, trios, ... of centers. N = µ Qµµ = µν QµνQνµ . Qµν = i ciµ c ∗ iν Na = µ∈a Qµµ = Tra Q = µ∈a,ν∈a QµνQνµ + µ∈a,ν / ∈a QµνQνµ Na = Traa Q2 + b=a Trab Q2 = Naa + b=a Nab Na = i Sa ii Sa ij = µ∈a ciµ cjµ Naa = ij (Sa ij )2 Nab = ij Sa ij Sb ij AMP (QCG@UniOvi) M2 2013 Sep 2013 4 / 6
9. Bonding ≡ Overlap −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −4 −2 0 2 4 6 ψ / a0 −3/2 z / a0 Symmetrically orthogonalized atomic basis for the H+ 2 molecule Q = PS ≡ P idempotent: Q2 = Q: TrQ = TrQQ = · · · = TrQn = N Partition in pairs, trios, ... of centers. N = µ Qµµ = µν QµνQνµ . Qµν = i ciµ c ∗ iν Na = µ∈a Qµµ = Tra Q = µ∈a,ν∈a QµνQνµ + µ∈a,ν / ∈a QµνQνµ Na = Traa Q2 + b=a Trab Q2 = Naa + b=a Nab Na = i Sa ii Sa ij = µ∈a ciµ cjµ Naa = ij (Sa ij )2 Nab = ij Sa ij Sb ij AMP (QCG@UniOvi) M2 2013 Sep 2013 4 / 6
10. Bonding ≡ Overlap ≡ Delocalization −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −4 −2 0 2 4 6 ψ / a0 −3/2 z / a0 Symmetrically orthogonalized atomic basis for the H+ 2 molecule Nab = 0 ⇔ ciµ ciν = 0 en a, b ⇔ delocalized electrons Q = 1/2 1/2 1/2 1/2 .Naa = Q11 Q11 = 1/4. Nab = Q12 Q21 = 1/4. 2Nab = δab : Wiberg-Mayer bond order. AMP (QCG@UniOvi) M2 2013 Sep 2013 5 / 6
11. Bonding ≡ Overlap ≡ Delocalization −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −4 −2 0 2 4 6 ψ / a0 −3/2 z / a0 Symmetrically orthogonalized atomic basis for the H+ 2 molecule Nab = 0 ⇔ ciµ ciν = 0 en a, b ⇔ delocalized electrons Q = 1/2 1/2 1/2 1/2 .Naa = Q11 Q11 = 1/4. Nab = Q12 Q21 = 1/4. 2Nab = δab : Wiberg-Mayer bond order. AMP (QCG@UniOvi) M2 2013 Sep 2013 5 / 6
12. Bonding ≡ Overlap ≡ Delocalization −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −4 −2 0 2 4 6 ψ / a0 −3/2 z / a0 Symmetrically orthogonalized atomic basis for the H+ 2 molecule Nab = 0 ⇔ ciµ ciν = 0 en a, b ⇔ delocalized electrons Q = 1/2 1/2 1/2 1/2 .Naa = Q11 Q11 = 1/4. Nab = Q12 Q21 = 1/4. 2Nab = δab : Wiberg-Mayer bond order. AMP (QCG@UniOvi) M2 2013 Sep 2013 5 / 6
13. Bonding ≡ Overlap ≡ Delocalization Multicenter delocalization is treated on equal footing. H2+ 3 : φ = 1 √ 3(1+2S) (1sa + 1sb + 1sc ) Q =   1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3   a b c a b c a b c Na = 1/3 Nab = 1/32 = 1/9 Nabc = 2 × 1/33 = 2/27 Three center delocalization (bonding). The number of delocalization channels is important. AMP (QCG@UniOvi) M2 2013 Sep 2013 6 / 6
14. Bonding ≡ Overlap ≡ Delocalization Multicenter delocalization is treated on equal footing. H2+ 3 : φ = 1 √ 3(1+2S) (1sa + 1sb + 1sc ) Q =   1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3   a b c a b c a b c Na = 1/3 Nab = 1/32 = 1/9 Nabc = 2 × 1/33 = 2/27 Three center delocalization (bonding). The number of delocalization channels is important. AMP (QCG@UniOvi) M2 2013 Sep 2013 6 / 6
15. Bonding ≡ Overlap ≡ Delocalization Multicenter delocalization is treated on equal footing. H2+ 3 : φ = 1 √ 3(1+2S) (1sa + 1sb + 1sc ) Q =   1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3   a b c a b c a b c Na = 1/3 Nab = 1/32 = 1/9 Nabc = 2 × 1/33 = 2/27 Three center delocalization (bonding). The number of delocalization channels is important. AMP (QCG@UniOvi) M2 2013 Sep 2013 6 / 6