Electronic structure theory: Fundamentals to frontiers. VI. Analysis and more. MARTIN HEAD-GORDON Department of Chemistry, University of California, Berkeley, and, Chemical Sciences Division, Lawrence Berkeley National Laboratory
Outline 1. Densities and density changes 2. Populations, localized orbitals, oxidation states... 3. Intermolecular interactions 4. Wrap-up
One particle density matrix and orbitals Hartree-Fock and DFT: 1PDM is made from the occupied orbitals eigenvalues, nq, are 1, 0 eigenvectors are therefore, in general,... a) the same as those from the SCF? b) or are they different? Pu q = n q u q Electron correlation methods 1PDM now has non-integer eigenvalues, nq eigenvectors are the natural orbitals some will be strongly occupied (bonding, lone pair) some will be weakly occupied (virtual)
Attachment-detachment density analysis A way of examining difference densities... Decompose the change in the one-particle density matrix between ground and excited states into the difference of A particle density (the attachment density, A) And a hole density (detachment density, D) A, D integrate to the number of electrons rearranged, and can be visualized. Eigenvectors, U, are difference natural orbitals... sometimes only 2 are significant... Δ = P 1 P 2 = A D Δ = Uδ U A = UaU a q = max( δ q,0) D = UdU d q = min( δ q,0)
Example: CIS for 2 1 A g state of butadiene Can give a very clear picture of the excitation CIS finds 2 1 A g to be Rydberg; no valence states nearby. CIS: 7.23 ev
Transition densities The transition density, yields the transition moment via: µ z = drρ T r ( ) The transition density must look like a dipole to yield a large transition moment. It can be plotted & decomposed... In a HOMO to LUMO transition, it is the product of the HOMO and LUMO. z
Outline 1. Densities and density changes 2. Populations, local orbitals, oxidation states Localized orbital bonding analysis (LOBA) Dr. Alex Thom, Eric Sundstrom PCCP 11, 11297-11304 (2009) 3. Intermolecular interactions 4. Wrap-up
Oxidation states from quantum chemistry: Atomic charges? Like atomic charges, oxidation states are not observables Little or no correlation with atomic populations Octahedral 3d complexes... d 3 d 5 d 4 d 6 d 5 d 8 d 10
Oxidation states from quantum chemistry: Atomic spin densities? Much more useful than total charges, but still ambiguous Higher oxidation states of Mn... d 3 d 5 d 4 d 6 d 5 d 8 d 10 d 3 d 2 d 1 d 0
Localized orbital bonding analysis (LOBA) (1) Transform from canonical orbitals to localized orbitals. a canonical occupied orbital: a localized occupied orbital: (2) Do population analysis on localized orbitals. An orbital is (loc)alized on an atom if its population > threshold. (3) Form the metal oxidation state as Z nloc
Testing LOBA: Is there a threshold that reproduces accepted oxidation states? Thresh > 0.75 Oxid. states too positive. Thresh < 0.4 Oxid. states too negative.
A model water oxidation catalyst oxygen-evolving center (OEC): 4 oxo-bridged Mn atoms O2 evolution via 4 proton-coupled electron transfers many aspects are still unknown -- consider instead a synthetic (Mn)2 analog... which we further simplify... actual computed
Possible catalytic cycle IV IV IV V V V
LOBA plots -- input for oxidation states O A1
Character of the assumed resting state (O) 3 α d electrons on each Mn atom. Bridging oxygens are 2 1 dative bond from each O to each Mn Mn(d α ) O Mn
Location of the first oxidation (to A1) H + and α e have been removed β Oxidized α e taken from bridging O Terminal O donates β e to Mn Net effect: no change in Mn oxidation states Differs from expectation that Mn adjacent to OH would be oxidized from IV to V.
Location of the second oxidation (to A) 2nd H + and β e have been removed Bridging O have excess α e β e is shared by bridging O atoms Mn atoms are (still) not oxidized IV IV π donation from O to Mn Cross-ring O-O interaction
Outline 1. Densities and density changes 2. Populations, localized orbitals, oxidation states... 3. Intermolecular interactions Energy/charge decomposition; applications Dr. Rustam Khaliullin (Prof. Alex Bell) JPC A 111, 8753 (2007); JCP 128, 184112 (2008); Inorg. Chem. 47, 4032 (2008); Chem. Eur. J. 15, 851 (2009) 4. Wrap-up
Interaction energy analysis: an incomplete history K. Kitaura and K. Morokuma, Int. J. Quantum Chem. 10, 325 (1976). Kitaura-Morokuma decomposition P.S. Bagus, K. Hermann, C.W. Bauschlicher, J. Chem. Phys. 80, 4378 (1984) CSOV analysis E.D. Glendening, A. Streitwieser, J. Chem. Phys. 100, 2900 (1994). natural energy decomposition analysis Y.R. Mo, J.L. Gao, S.D. Peyerimhoff, J. Chem. Phys. 112, 5530 (2000). BLW EDA Alternative development is symmetry-adapated perturbation theory (SAPT). See e.g. B. Jeziorski, R. Moszynski & K. Szalewicz, Chem. Rev. 94, 1887 (1994); A.J. Misquitta, B. Jeziorski & K. Szalewicz, Phys. Rev. Lett. 91 (2003)
Analysis of binding energies Geometric distortion Non-interacting fragment densities p(0) Frozen electrostatics: Interaction energy using p(0) Coupling of permanent moments & exchange repulsion Polarization: treat by ALMO-SCF (new) Induction effects treated fully self-consistently Donor-acceptor interactions: charge-transfer correction (new). Can be decomposed into forward/back donation Higher-order charge transfer: full SCF Not decomposable ΔE = Tr ( P - p)f = Tr X VO F OV R.Z Khaliullin, R.Lochan, E.Cobar, A.T.Bell, MHG, J. Phys. Chem A 111, 8753 (2007)
Absolutely localized molecular orbitals (ALMO s) For molecular clusters and complexes, molecular fragments are well-defined. Define absolutely localized molecular orbitals as: φ xi = ω xµ T xµ,xi µ x Transformation T is molecule-blocked ALMO s are non-orthogonal No charge transfer between fragments T = T A 0 0 0 T B 0 0 0 T C No borrowing your neighbor s basis functions for your own selfish variational purposes!
Water dimer (HF/aug-cc-pVDZ basis) R.Z Khaliullin, MHG, A.T.Bell, J. Chem Phys. 124, 204105 (2006)
Example: alkane binding affinities (CH 4 )Re(CO) 2 Cp (E.A.Cobar, R. Khaliullin, R.G.Bergman, MHG) Cp back-bonding ΔE = -25 kj/mol Re CO CO Term kj/mol ---------------------------- FRZ +66 POL -27 CT -96 TOT -57 forward donation ΔE = -71 kj/mol
Water - metal ion interactions: M + H 2 O Frozen density Polarization Charge transfer Electrostatics dominates CT is not significant Size-dependence
Complementary occupied-virtual orbital pairs X VO is decomposed into (x,y) intermolecular pairs: ΔE = Tr X VO F OV ( x, y) y,x = Tr ( ) X FOV VO x y The principal orbitals for charge transfer are obtained by singular value decomposition of ( ) ( x, y) X VO = Vx x ( x, y) U y X VO x, y Principal virtual (acceptor) orbitals Weight of each donor-acceptor pair Principal occupied (donor) orbitals R.Z Khaliullin, A.T.Bell, MHG, J. Chem. Phys. 128, 184112 (2008)
Forward donation in (CO) 5 W-CO (Rustam Khaliullin) ΔE(L M) = 101 kj/mol ΔQ(L M) = 0.04 e 1st complementary orbital pair: 98% of ΔE 97% of ΔQ Bold: donor orbital Faint: acceptor orbital R.Z Khaliullin, A.T.Bell, MHG, J. Chem. Phys. 128, 184112 (2008)
Back-donation in (CO) 5 W-CO (Rustam Khaliullin) ΔE(M L) = 142 kj/mol ΔQ(M L) = 0.25 e 1st complementary orbital pair: 50% of ΔE 50% of ΔQ Bold: donor orbital Faint: acceptor orbital R.Z Khaliullin, A.T.Bell, MHG, J. Chem. Phys. 128, 184112 (2008)
Charge-transfer in the water-water hydrogen bond Analysis of modern ab initio wave functions in terms of natural bond orbitals strongly suggests the resonancetype "charge transfer" character of H-bonding, contrary to the widely held classical-electrostatic viewpoint F. Weinhold, Adv. Protein Chem. 72, 121-155 (2006) Water dimer binding energy = 18.9 kj/mol Charge transfer energy = 37.3 kj/mol Electrons transferred = 15-20 me E. Glendening, J. Phys. Chem. A 109, 11936 (2005)
Water dimer (Rustam Khaliullin) ΔE = -6.6 kj/mol ΔE = -0.3 kj/mol Term kj/mol ---------------------------- FRZ -5.2 POL -6.5 CT -7.2 TOT -18.9 NBO: -37.3 BSSE-corrected B3LYP/aug-cc-pVQZ R.Z Khaliullin, A.T.Bell, MHG, Chem Eur. J.15, 851 (2009)
Electrons transferred (Rustam Khaliullin) ΔQ = 2.8 me Term ΔQ/me ΔQ = 0.1 me ---------------------------- D A 2.8 A D 0.1 POL -0.4 TOT 2.5 NBO: 15.0 BSSE-corrected B3LYP/aug-cc-pVQZ R.Z Khaliullin, A.T.Bell, MHG, Chem Eur. J.15, 851 (2009)
Form of the donor & acceptor orbitals Donor orbital does not rotate with rotation of the donor molecule! R.Z Khaliullin, A.T.Bell, MHG, Chem Eur. J.15, 851 (2009)
CT and the water dimer: B3LYP/aug-cc-pVTZ Effect of charge transfer: on structure / binding? optimize with ALMO ALMO full ΔE / kj mol 1 13.7 18.8 ΔEfrz / kj mol 1 10.5 6.4 ΔEpol / kj mol 1 3.2 5.8 ΔECT / kj mol 1 0.0 6.8 R(OH) / Å 2.15 1.96
CT and the red shift of the OH stretch? ALMO Full ω9 (I9) 3791 (85) 3683 (330) ω10 (I10) 3803 (7) 3800 (10) ω11 (I11) 3895 (121) 3877 (84) ω12 (I12) 3902 (79) 3899 (85)
Outline 1. Densities and density changes 2. Populations, localized orbitals, oxidation states... 3. Intermolecular interactions 4. Wrap-up
Topics not mentioned... Molecular properties as responses (energy derivatives) Relativistic effects Born-Oppenheimer breakdown Quantum Monte Carlo Hybrid (QM-MM) methods Walking on potential energy surfaces