Natural Bond Orbital Theory

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1. Natural Bond Orbital Theory Ángel Martín Pendás Quantum Chemistry Group Universidad de Oviedo Spain Madrid 2013 AMP (QCG@UniOvi) M2 2013 Sep 2013 1 / 19
2. Outline 1 Analyzing the 1-particle density in Fock space 2 Natural Hybrid Orbitals, Natural Bond Orbitals 3 Perturbation theory: the donor-acceptor view 4 Natural Localized Molecular Orbitals 5 Natural Resonance Theory 6 Natural Steric Analysis 7 Natural Energy Decomposition Analysis (NEDA) 8 Bonding indices, Summary 9 Example AMP (QCG@UniOvi) M2 2013 Sep 2013 2 / 19
3. Analyzing the 1-particle density in Fock space I Much of the MO paradigm is based on minimal basis (MB) thinking. MB ≡ one basis function/electron. Leads to Hückel or extended Hückel, atomic orbitals, atomic hybrids, molecular orbitals. Hybrid Orbitals Orbital hybridization is one of the most useful qualitative valence concepts (Pauling, Slater). It allows immediate extension of the bonding-antibonding idea to polyatomics. h = N(s + λp), σAB = cA hA + cB hB , σ∗ AB = cB hA − cA hB AMP (QCG@UniOvi) M2 2013 Sep 2013 3 / 19
4. Analyzing the 1-particle density in Fock space II Problem Is it possible to restore a minimal basis in an extended calculation? If so, qualitative arguments might cross the quantitative barrier. What to use? How to compact orbitals? Natural Orbitals ρ(1; 1 ) = N d2d3..dN Ψ∗(1, 2.., N)Ψ(1 , 2, .., N) is quadratic in any orbital basis. ρ(1; 1 ) = i,j cij φi (1)φj (1 ) = i ni χi (1)χi (1 ), 0 ≤ ni ≤ 1 (Löwdin) The NO’s series least-square minimizes a given number of terms approximation to ρ. NO’s are maximum occupancy orbitals. AMP (QCG@UniOvi) M2 2013 Sep 2013 4 / 19
5. Analyzing the 1-particle density in Fock space III Weinhold et. al. procedure Derive Natural atomic orbitals of maximal (Löwdin) occupancy from atomic blocks of the 1-particle matrix. ρ(1; 1 ) =   ρAA(1; 1 ) ρAB(1; 1 ) ρAC(1; 1 ) ... ρBA(1; 1 ) ρBB(1; 1 ) ρBC(1; 1 ) ... ... ... ... ...   Extract the A block of ρ(1; 1 ) ≡ ρAA(1; 1 ) Diagonalize ρAθA i = qA i θA i . ⇒ θA i ≡ pre-NAOs. pre-NAOs orthogonal in but not among groups: θA i |θB i = δij only if A = B. Orthogonalize pre-NAOs to NAOs. How? Löwdin, Gram-Schmidt? All of them have problems of stability on increasing the basis set size. Solution: Occupancy-weighted symmetric Orthogonalization: OWSO AMP (QCG@UniOvi) M2 2013 Sep 2013 5 / 19
6. Analyzing the 1-particle density in Fock space IV OWSO JCP 83, 735 (1985) 1 Symmetry average lm blocks in P and S. Diagonalize PANA = SANAWA. Classify by occupancy (wA i ). The minimal set (MS): wA i large. The Rydberg set (R): wA i close to zero. 2 Occupancy Weighted Symmetric Orthogonalization: The set of MSs is transformed such that i wi |φW i − φi |2 is minimum: {φW i } = ˆ W{φi }. wi = spherically averaged qi . For Schmidt, i |φS i − φi |2 is minimum: {φS i } = ˆ S{φi } 3 Flow (i) Spherically average, diagonalize, and classify; (ii) Apply ˆ W to MS; OW = W(WSW)−1/2 (iii) ˆ S orthogonalize R to MS; (iv) Apply ˆ W to R . (v) Repeat (i). AMP (QCG@UniOvi) M2 2013 Sep 2013 6 / 19
7. Natural Hybrid Orbitals, Natural Bond Orbitals The NHO/NBO Algorithm 1 Take the NAO basis and the atomic blocks of ρ: ρAA. Diagonalize them, extracting "core orbitals", and "lone pairs" hA i with occupancies nA i ≥ 1.90. 2 Deplete the two-atom blocks of ρ from these contributions. ρAA(1; 1 ) ρAB(1; 1 ) ρBA(1; 1 ) ρBB(1; 1 ) − i∈A,B nA(B) i hA(B) i hA(B)† i 3 Diagonalize the depleted two-atom blocks and search for nAB i ≈ 2. hAB i = cA hA + cB hB. These are bond and hybrid natural orbitals. 4 Since NBOs will not be orthogonal, reorthogonalize (ˆ S) the hybrids. 5 Complement the basis with antibonds, hAB∗ i = cB hA − cA hB, and with Rydberg residuals all of them ˆ S orthogonalized. 6 If no set of N/2 electron pairs is found, decrease threshold, or search over trios (3-centered bonds) etc. AMP (QCG@UniOvi) M2 2013 Sep 2013 7 / 19
8. Perturbation theory: the donor-acceptor view I Perturbing a closed-shell HF determinant Let Ψ0 = |h1 ..hN | be written in the NBO basis. (The Natural Lewis Structure). ˆ F is not diagonal in the basis: F = diag(F) + F = F0 + F . Then E0 = i i , and ˆ H = i ˆ fi . E1 is a rescaling of the E0 value. E2 = i E2 i . Variation-Perturbation theory with a trial ˜ h results in: E2 i ≤ − hi |ˆ H |˜ h 2 ˜ h|ˆ H0 − i |˜ h , hi |˜ h = 0. ˜ h ≈ h∗ j ⇒ E2 i→j ≈ − hi |ˆ F|h∗ j 2 ∗ j − i . The first order correction to hi , h1 i ≈ N(hi + λh∗ j ), so qh→h∗ ≈ λ2, and |E2 i→j | ≈ qi→j ( ∗ j − i ). AMP (QCG@UniOvi) M2 2013 Sep 2013 8 / 19
9. Perturbation theory: the donor-acceptor view II ∆E ≈ − hi|ˆ F|h∗ j 2 ∗ j − i . ε ε i j * ∆Ε AMP (QCG@UniOvi) M2 2013 Sep 2013 9 / 19
10. Natural Localized Molecular Orbitals ρ(1; 1) almost diagonal in the NBO NLS By a sequence of 2x2 Jacobi rotations the σσ∗ blocks are zeroed The resulting ρ is block diagonal: NLMOs NLMOs have occupancy 1(2) in the occupied blocks, 0 in the unoccupied. NLMOs are similar to Edminston-Ruedenberg or Boys Localized MOs. φNLMO i = N(hi + λh∗ i + · · · ). Related to the perturbation expansion. Methanamide NBO NLMO MO AMP (QCG@UniOvi) M2 2013 Sep 2013 10 / 19
11. Natural Resonance Theory Many times a single NLS is not enough. NRT: ρ(1; 1 ) = r wr ρr(1; 1 ) ⇒ ˆ M = r wr Mr . Incoherent (vs. coherent) superposition of resonance structures. Candidate Lewis structures may be obtained from donor-acceptor interactions. Example: A − B − C − D and σAB → σ∗ CD . (σAB )2(σBC )2(σCD )2 → (σBC )2(nC )2(nD )2 → (σBC )2(πCD )2(nD )2 A − B − C − D → A+B+ − C−D− → A+B = CD− NRT wi ’s are obtained by minimizing |ρ(1; 1 ) − r wr ρr(1; 1 )| AMP (QCG@UniOvi) M2 2013 Sep 2013 11 / 19
12. Natural Steric Analysis Steric interactions are in NBO identified with the kinetic energy pressure introduced by antisymmetry: exchange repulsions. Antisymmetry ⇒ orthogonality ⇒ increased nodal structure ⇒ larger kinetic energy. Use NAOs and NBOs without the last orthogonalization step: Pre-orthogonal PNAOs, PNBOs. Eexch ≡ Est = i (FNBO ii − FPNBO ii ) To analyze contacts from electron pairs i, j, a partially deorthogonalized NBO (PNBO/2) is obtained by an inverse 2x2 OWSO of the i, j NBO 2x2 block. Eij st = (FNBO ii − FPNBO/2 ii ) + (FNBO jj − FPNBO/2 jj ) These are almost pairwise additive NBO steric exchange differences give good descriptions of repulsive potentials between rare-gas atoms. ∆Est (A..B) = Est (A..B) − Est (A) − Est (B) AMP (QCG@UniOvi) M2 2013 Sep 2013 12 / 19
13. Natural Energy Decomposition Analysis (NEDA) Glendening, Streitwieser JCP 100, 2900 (1994) Based on Kitaura-Morokuma partitioning for an A-B complex. ∆E = ECT + EES + EA def + EB def Construct the ΨA def and ΨB def of the monomers using the NBO basis of the complex. EA def = ΨA def |ˆ HA|ΨA def − ΨA ∞ |ˆ HA|ΨA ∞ Construct the localized A-B wavefunction: ΨAB loc = 2−1/2det|ΨA def ΨB def | EES = ΨAB loc |ˆ H|ΨAB loc − ΨA def |ˆ HA|ΨA def − ΨB def |ˆ HB|ΨB def ES includes residual polarization and exchange (antisymmetrized Ψ’s) ECT = ΨAB|ˆ H|ΨAB − ΨAB loc |ˆ H|ΨAB loc Contains charge delocalization effects including all donor-donor, etc interactions. Other Energetic analyses within NBO are based on deleting (DEL) certain orbitals and approximately recalculating the energy AMP (QCG@UniOvi) M2 2013 Sep 2013 13 / 19
14. Bonding indices, Summary Coulson’s (DMNAO) and Wiberg’s (BNDIDX) are available from NAOs. Natural bond order and valencies: Given optimal NRT weights. bAB = r wr bi AB , both for covalent and ionic bond orders. Ionic character iAB = r wr ir AB /bAB ir AB = |(c2 A − c2 B )/(c2 A + c2 B )| Valency VA = B=A bAB . Summary: Pros and cons Provides easy access to MO reasoning: much chemistry It is based on much cooking AMP (QCG@UniOvi) M2 2013 Sep 2013 14 / 19
15. Example: Methanol (Gaussian, pop=nbo) I NATURAL POPULATIONS: Natural atomic orbital occupancies NAO Atom No lang Type(AO) Occupancy Energy ---------------------------------------------------------- 1 C 1 S Cor( 1S) 1.99949 -11.13313 2 C 1 S Val( 2S) 1.07128 -0.32356 3 C 1 S Ryd( 4S) 0.00475 0.76847 8 C 1 px Val( 2p) 1.16841 -0.09602 9 C 1 px Ryd( 4p) 0.00090 0.79950 12 C 1 py Val( 2p) 0.74785 0.00404 15 C 1 py Ryd( 3p) 0.00005 0.34296 16 C 1 pz Val( 2p) 1.14120 -0.08115 17 C 1 pz Ryd( 4p) 0.00027 0.82113 25 O 2 S Cor( 1S) 1.99981 -20.32466 26 O 2 S Val( 2S) 1.66454 -1.14595 27 O 2 S Ryd( 3S) 0.00092 0.76505 32 O 2 px Val( 2p) 1.62543 -0.38114 33 O 2 px Ryd( 3p) 0.00223 0.77001 36 O 2 py Val( 2p) 1.46299 -0.34709 37 O 2 py Ryd( 3p) 0.00188 0.69539 40 O 2 pz Val( 2p) 1.96578 -0.48595 41 O 2 pz Ryd( 3p) 0.00306 0.72913 Summary of Natural Population Analysis: Natural Population Natural ----------------------------------------------- Atom No Charge Core Valence Rydberg Total ----------------------------------------------------------------------- C 1 -0.14681 1.99949 4.12874 0.01858 6.14681 O 2 -0.73256 1.99981 6.71874 0.01401 8.73256 H 3 0.16011 0.00000 0.83790 0.00199 0.83989 H 4 0.13341 0.00000 0.86468 0.00192 0.86659 H 5 0.13341 0.00000 0.86468 0.00192 0.86659 H 6 0.45244 0.00000 0.54389 0.00367 0.54756 ======================================================================= * Total * 0.00000 3.99930 13.95863 0.04208 18.00000 AMP (QCG@UniOvi) M2 2013 Sep 2013 15 / 19
16. Example: Methanol (Gaussian, pop=nbo) II Natural Population -------------------------------------------------------- Core 3.99930 ( 99.9825% of 4) Valence 13.95863 ( 99.7045% of 14) Natural Minimal Basis 17.95792 ( 99.7662% of 18) Natural Rydberg Basis 0.04208 ( 0.2338% of 18) -------------------------------------------------------- Atom No Natural Electron Configuration ---------------------------------------------------------------------------- C 1 [core]2S( 1.07)2p( 3.06)3d( 0.01)4p( 0.01) O 2 [core]2S( 1.66)2p( 5.05)3p( 0.01)3d( 0.01) H 3 1S( 0.84) H 4 1S( 0.86) H 5 1S( 0.86) H 6 2S( 0.54) NATURAL BOND ORBITAL ANALYSIS: Occupancies Lewis Structure Low High Occ. ------------------- ----------------- occ occ Cycle Thresh. Lewis Non-Lewis CR BD 3C LP (L) (NL) Dev ============================================================================= 1(1) 1.90 17.93096 0.06904 2 5 0 2 0 0 0.02 ----------------------------------------------------------------------------- Structure accepted: No low occupancy Lewis orbitals (Occupancy) Bond orbital/ Coefficients/ Hybrids --------------------------------------------------------------------------------- AMP (QCG@UniOvi) M2 2013 Sep 2013 16 / 19
17. Example: Methanol (Gaussian, pop=nbo) III 1. (1.99843) BD ( 1) C 1 - O 2 ( 33.73%) 0.5808* C 1 s( 24.26%)p 3.11( 75.54%)d 0.01( 0.20%) ( 66.27%) 0.8141* O 2 s( 30.80%)p 2.24( 69.06%)d 0.00( 0.13%) 2. (1.99413) BD ( 1) C 1 - H 3 ( 58.27%) 0.7633* C 1 s( 25.39%)p 2.93( 74.50%)d 0.00( 0.11%) ( 41.73%) 0.6460* H 3 s( 99.89%)p 0.00( 0.11%) 3. (1.99746) BD ( 1) C 1 - H 4 ( 57.17%) 0.7561* C 1 s( 25.43%)p 2.93( 74.46%)d 0.00( 0.12%) 4. (1.99746) BD ( 1) C 1 - H 5 ( 57.17%) 0.7561* C 1 s( 25.43%)p 2.93( 74.46%)d 0.00( 0.12%) ( 42.83%) 0.6544* H 5 s( 99.89%)p 0.00( 0.11%) 5. (1.99143) BD ( 1) O 2 - H 6 ( 72.82%) 0.8533* O 2 s( 22.49%)p 3.44( 77.39%)d 0.01( 0.13%) ( 27.18%) 0.5213* H 6 s( 99.85%)p 0.00( 0.15%) 6. (1.99949) CR ( 1) C 1 s(100.00%) 7. (1.99981) CR ( 1) O 2 s(100.00%) 8. (1.98518) LP ( 1) O 2 s( 46.73%)p 1.14( 53.24%)d 0.00( 0.03%) 9. (1.96756) LP ( 2) O 2 s( 0.00%)p 1.00( 99.95%)d 0.00( 0.05%) 10. (0.00314) RY*( 1) C 1 s( 24.40%)p 2.94( 71.81%)d 0.16( 3.79%) 72. (0.00137) BD*( 1) C 1 - O 2 ( 66.27%) 0.8141* C 1 s( 24.26%)p 3.11( 75.54%)d 0.01( 0.20%) ( 33.73%) -0.5808* O 2 s( 30.80%)p 2.24( 69.06%)d 0.00( 0.13%) 73. (0.01146) BD*( 1) C 1 - H 3 ( 41.73%) 0.6460* C 1 s( 25.39%)p 2.93( 74.50%)d 0.00( 0.11%) ( 58.27%) -0.7633* H 3 s( 99.89%)p 0.00( 0.11%) 74. (0.01783) BD*( 1) C 1 - H 4 ( 42.83%) 0.6544* C 1 s( 25.43%)p 2.93( 74.46%)d 0.00( 0.12%) ( 57.17%) -0.7561* H 4 s( 99.89%)p 0.00( 0.11%) 75. (0.01783) BD*( 1) C 1 - H 5 ( 42.83%) 0.6544* C 1 s( 25.43%)p 2.93( 74.46%)d 0.00( 0.12%) ( 57.17%) -0.7561* H 5 s( 99.89%)p 0.00( 0.11%) 76. (0.00474) BD*( 1) O 2 - H 6 ( 27.18%) 0.5213* O 2 s( 22.49%)p 3.44( 77.39%)d 0.01( 0.13%) ( 72.82%) -0.8533* H 6 s( 99.85%)p 0.00( 0.15%) AMP (QCG@UniOvi) M2 2013 Sep 2013 17 / 19
18. Example: Methanol (Gaussian, pop=nbo) IV Second Order Perturbation Theory Analysis of Fock Matrix in NBO Basis Threshold for printing: 0.50 kcal/mol E(2) E(j)-E(i) F(i,j) Donor NBO (i) Acceptor NBO (j) kcal/mol a.u. a.u. =================================================================================================== within unit 1 2. BD ( 1) C 1 - H 3 / 76. BD*( 1) O 2 - H 6 2.98 1.44 0.059 5. BD ( 1) O 2 - H 6 / 10. RY*( 1) C 1 2.86 2.14 0.070 5. BD ( 1) O 2 - H 6 / 73. BD*( 1) C 1 - H 3 2.46 1.57 0.055 6. CR ( 1) C 1 / 72. BD*( 1) C 1 - O 2 2.10 11.69 0.140 7. CR ( 1) O 2 / 10. RY*( 1) C 1 2.38 21.50 0.202 8. LP ( 1) O 2 / 10. RY*( 1) C 1 3.40 2.02 0.074 8. LP ( 1) O 2 / 73. BD*( 1) C 1 - H 3 3.75 1.44 0.066 9. LP ( 2) O 2 / 11. RY*( 2) C 1 2.33 2.31 0.066 9. LP ( 2) O 2 / 74. BD*( 1) C 1 - H 4 9.06 1.07 0.088 9. LP ( 2) O 2 / 75. BD*( 1) C 1 - H 5 9.06 1.07 0.088 AMP (QCG@UniOvi) M2 2013 Sep 2013 18 / 19
19. Bibliography F. Weinhold, C. Landis, Valency and Bonding, Cambridge 2005. J. P. Foster and F. Weinhold, J. Am. Chem. Soc. 102, 7211-7218 (1980). A. E. Reed and F. Weinhold, J. Chem. Phys. 78, 4066-4073 (1983); A. E. Reed, R. B. Weinstock, and F. Weinhold, J. Chem. Phys. 83, 735-746 (1985). A. E. Reed and F. Weinhold, J. Chem. Phys. 83, 1736-1740 (1985). J. E. Carpenter and F. Weinhold, J. Mol. Struct. (Theochem) 169, 41-62 (1988); F. Weinhold, “Natural Bond Orbital Methods,” in, Encyclopedia of Computational Chemistry, P. v.R. Schleyer, N. L. Allinger, T. Clark, J. Gasteiger, P. A. Kollman, H. F. Schaefer III, P. R. Schreiner (Eds.), (John Wiley & Sons, Chichester, UK, 1998), Vol. 3, pp. 1792-1811; E. D. Glendening, C. R. Landis, and F. Weinhold, WIREs Comput. Mol. Sci. 2, 1-42 (2012). A. E. Reed, L. A. Curtiss, and F. Weinhold, Chem. Rev. 88, 899-926 (1988); F. Weinhold and J. E. Carpenter, in, R. Naaman and Z. Vager (eds.), The Structure of Small Molecules and Ions (Plenum, New York, 1988), pp. 227-236; AMP (QCG@UniOvi) M2 2013 Sep 2013 19 / 19