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DRAFT Energy Decomposition Analysis of Single Bonds Within Kohn-Sham Density Functional Theory

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DRAFT Energy Decomposition Analysis of Single Bonds Within Kohn-Sham Density Functional Theory UC Berkeley UC Berkeley Previously Published Works Title Energy decomposition analysis of single bonds within Kohn-Sham density functional theory. Permalink https://escholarship.org/uc/item/9vt5381f Journal Proceedings of the National Academy of Sciences of the United States of America, 114(48) ISSN 0027-8424 Authors Levine, Daniel S Head-Gordon, Martin Publication Date 2017-11-20 DOI 10.1073/pnas.1715763114 Peer reviewed eScholarship.org Powered by the California Digital Library University of California 1 63 2 Energy Decomposition Analysis of Single Bonds 64 3 65 4 Within Kohn-Sham Density Functional Theory 66 5 67 6 Daniel S. Levinea and Martin Head-Gordona,* 68 7 69 aKenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California, Berkeley, California 94720, USA and Chemical Sciences Division, 8 Lawrence Berkeley National Laboratory Berkeley, California 94720, USA 70 9 71 10 This manuscript was compiled on October 3, 2017 72 11 73 An energy decomposition analysis (EDA) for single chemical bonds Analyzing chemical bonds in more complex molecules has 12 74 is presented within the framework of Kohn-Sham density functional also attracted great attention. Ruedenberg and co-workers 13 75 theory, based on spin-projection equations that are exact within have been developing generalizations of their classic analysis 14 76 wavefunction theory. Chemical bond energies can then be under- procedures with this objective(8, 9). Valence bond theory, 15 77 stood in terms of stabilization due to spin-coupling, augmented by while uncompetitive for routine computational purposes, in- 16 78 dispersion, polarization, and charge transfer in competition with volves conceptually simple wave functions that are suitable 17 79 destabilizing Pauli repulsions. The EDA reveals distinguishing fea- for extracting qualitative chemical bonding concepts(10). The 18 80 tures of chemical bonds ranging across non-polar, polar, ionic and emergence of the “charge-shift bond” paradigm, exemplified 19 81 charge-shift bonds. The effect of electron correlation is assessed by the F molecule, is a specific example of its value(11). The 20 2 82 by comparison with Hartree-Fock results. Substituent effects are widely used Natural Bond Orbital (NBO) approach (12), pro- 21 83 illustrated by comparing the C-C bond in ethane against that in vides localized orbitals, predominant Lewis structures, and 22 84 bis(diamantane), and dispersion stabilization in the latter is quanti- information on hybridization and chemical bonds. The quan- 23 85 fied. Finally, three metal–metal bonds in experimentally character- tum theory of atoms-in-molecules (QTAIM) (13), describes 24 86 ized compounds are examined: a MgI−MgI dimer, the ZnI−ZnI bond the presence of bonds by so-called bond critical points in the 25 87 in dizincocene, and the Mn−Mn bond in dimanganese decacarbonyl. electron density, as well as partitioning an energy into intra- 26 88 atomic and inter-atomic terms. Another topological approach 27 89 is the electron localization function (ELF), which is a function 28 90 Energy Decomposition Analysis | Chemical Bonding of the density and the kinetic energy density. Many other 29 91 methods also exist for partitioning a bond energy into sums 30 92 of terms that are physically interpretable (4). 31 nderstanding the chemical bond is central to both syn- 93 32 Uthetic and theoretical chemists. The approach of the EDA schemes have been very successful at elucidating the 94 33 synthetic chemists is based on qualitative, empirical features nature of non-covalent interactions(2, 14, 15). These methods 95 34 (electronegativity, polarizability, etc.) gleaned over the past typically separate the interaction energy by either perturba- 96 35 150 years of research and investigation. These features are no- tive approaches or constrained variational optimization. Per- 97 36 tably absent from the toolbox of the theoretical chemist, who turbative methods include the popular Symmetry Adapted 98 37 relies on a quantum mechanical wavefunction to holistically Perturbation Theory (SAPT)(16, 17) method and the Natural 99 38 describe the electronic structure of a molecule; in essence, a Energy Decomposition Analysis (NEDA)(18), based on NBOs. 100 39 numerical experiment. Bridging this gap is the purview of Variational methods include Kitaura and Morokuma (KM) 101 40 bonding analysis and energy decomposition analysis (EDA), EDA(19), the Ziegler-Rauk method(20), the Block-Localized 102 41 which seeks to separate the quantum mechanical energy into Wavefunction (BLW-EDA)(14) and the Absolutely Localized 103 42 physically meaningful terms. Bonding analysisDRAFT and EDA ap- Molecular Orbital (ALMO-EDA) of Head-Gordon et al(21–24). 104 43 proaches are necessarily non-unique, but different well-designed A number of non-covalent EDA methods have been applied to 105 44 approaches provide complementary perspectives on the nature 106 45 of the chemical bond. This task is not yet complete, despite 107 46 intensive effort and substantial progress(1–5). 108 47 Significance Statement 109 48 The chemical bond was originally viewed(6) as being electro- While theoretical chemists today can calculate the energies 110 49 static in origin, based on the virial theorem, and supported of molecules with high accuracy, the results are not readily 111 50 by an accumulation of electron density in the bonding re- interpretable by synthetic chemists who use a different lan- 112 51 gion relative to superposition of free atom densities. The guage to understand and improve chemical syntheses. Energy 113 52 chemical bond is still often taught this way in introductory decomposition analysis (EDA) provides a bridge between theo- 114 53 classes. However, the quantum mechanical origin of the chem- retical calculation and the practical insight. Most existing EDA 115 54 + methods are not designed for studying covalent bonds. We 116 ical bond in H2 and H2 (classical mechanics does not explain 55 bonding) lies in lowering the kinetic energy by delocalization, developed an EDA to characterize single bonds, providing an 117 56 that is, via constructive wavefunction interference. This was interpretable chemical fingerprint in the language of synthetic 118 57 first established(1) 55 years ago by Ruedenberg for H+.A chemists from the quantum mechanical language of theorists. 119 58 2 120 secondary effect, in some cases, such as in H2, is orbital con- 59 traction, which is most easily seen by optimizing the form of MHG designed research; DSL performed research; Both analyzed data and wrote the paper. 121 60 a spherical 1s function as a function of bond-length(7). Polar- The authors declare no conflicts of interest. 122 61 ization and charge-transfer contribute to further stabilization. 123 62 *To whom correspondence should be addressed. E-mail: [email protected] 124 www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX PNAS | October 3, 2017 | vol. XXX | no. XX | 1–8 125 bonds(2, 20, 25), although the single-determinant nature of a lone pair in the p-orbital in a molecule. Rearranging the odd 187 126 these methods leads to spin-symmetry broken wavefunctions, electron of each radical fragment to be in the hybrid orbital 188 127 which contaminates the EDA terms with effects from the other that is appropriate for spin-coupling will incur an energy cost, 189 128 terms. ∆EHYBRID, that completes the preparation energy: 190 129 191 130 To address this challenge, we recently reported a spin-pure ∆EPREP = ∆EGEOM + ∆EHYBRID [2] 192 131 extension of the ALMO-EDA scheme to the variational analysis 193 132 of single covalent bonds(26). The method, which reduces to the 194 We define ∆EHYBRID as the energy change due to rotations of 133 ALMO-EDA scheme for non-covalent interactions(24), includes the β hole in the span of the α occupied space from the isolated 195 134 modified versions of the usual non-bonded frozen orbital (FRZ), radical fragment to the correct arrangement in the bond. This 196 135 polarization (POL) and charge transfer (CT) terms, as well as is accomplished by variational optimization of the fragments’ 197 136 a new spin-coupling (SC) term describing the energy lowering RO orbitals (in the spin-coupled state) only allowing doubly 198 137 due to electron pairing. The final energy corresponds to occupied-singly occupied mixings. Afterwards the modified 199 138 the CAS(2,2) (equivalently, 1-pair perfect-pairing or TCSCF) 200 fragment orbitals are used to evaluate ∆EHYBRID. As limited 139 wavefunction. While this is a fully ab initio model, it lacks 201 orbital relaxation is involved, ∆EHYBRID may also be viewed 140 the dynamic correlation necessary for reasonable accuracy. as a kind of polarization and indeed it was previously placed 202 141 in the POL term.(26) 203 142 By far the most widely used treatment of dynamic corre- 204 lation in quantum chemistry today is Kohn-Sham density 143 However, ∆EHYBRID is also partially present here in that the 205 144 functional theory (DFT)(27). DFT methods yield RMS errors geometry of the radical fragment is fixed to be that of the 206 145 in chemical bond strengths on the order of a few kcal/mol, interacting fragment. For instance, free methyl radical is an 207 146 which approaches chemical accuracy at vastly lower compu- sp2-hybridized planar molecule while a methyl group in a bond 208 147 tational effort than wavefunction methods. The purpose of is a pyramidalized sp3 fragment, and it is the latter that is 209 this paper is to recast our bonded ALMO-EDA method into 148 employed in this EDA scheme. We have moved ∆EHYBRID 210 149 a single-determinant formalism that allows the computation here for that reason, and because it can be much larger than 211 150 of a dynamically-correlated bonded EDA with any existing the other contributions to POL and therefore its presence in 212 151 density functional.
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