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
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
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
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
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
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
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
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
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
(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
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
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
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