Natural Bond Orbital Theory

Transcript

1. Natural Bond Orbital Theory
   Ángel Martín Pendás
   Quantum Chemistry Group
   Universidad de Oviedo
   Spain
   Madrid 2013
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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
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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
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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.
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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
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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).
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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.
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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
   ).
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9. Perturbation theory: the donor-acceptor view II
   ∆E ≈ − hi|ˆ
   F|h∗
   j
   2
   ∗
   j
   − i
   .
   ε
   ε
   i
   j
   *
   ∆Ε
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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
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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 )|
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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)
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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
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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
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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
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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
    ---------------------------------------------------------------------------------
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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%)
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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
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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).
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    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;
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