Energy decomposition analysis of single bonds within INAUGURAL ARTICLE Kohn–Sham density functional theory Daniel S. Levinea,b and Martin Head-Gordona,b,1 aKenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California, Berkeley, CA 94720; and bChemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected in 2015. Contributed by Martin Head-Gordon, October 10, 2017 (sent for review September 7, 2017; reviewed by Gernot Frenking and Roald Hoffmann) An energy decomposition analysis (EDA) for single chemical bonds ally simple wave functions that are suitable for extracting qual- is presented within the framework of Kohn–Sham density func- itative chemical bonding concepts (10). The emergence of the tional theory based on spin projection equations that are exact “charge-shift bond” paradigm, exemplified by the F2 molecule, within wave function theory. Chemical bond energies can then is a specific example of its value (11). The widely used natural be understood in terms of stabilization caused by spin-coupling bond orbital (NBO) approach (12) provides localized orbitals, augmented by dispersion, polarization, and charge transfer in predominant Lewis structures, and information on hybridization competition with destabilizing Pauli repulsions. The EDA reveals and chemical bonds. The quantum theory of atoms-in-molecules distinguishing features of chemical bonds ranging across nonpo- (QTAIM) (13) describes the presence of bonds by so-called bond lar, polar, ionic, and charge-shift bonds. The effect of electron critical points in the electron density as well as partitioning an correlation is assessed by comparison with Hartree–Fock results. energy into intraatomic and interatomic terms. Another topo- Substituent effects are illustrated by comparing the C–C bond in logical approach is the electron localization function, which is ethane against that in bis(diamantane), and dispersion stabiliza- a function of the density and the kinetic energy density. Many tion in the latter is quantified. Finally, three metal–metal bonds other methods also exist for partitioning a bond energy into sums in experimentally characterized compounds are examined: a MgI– of terms that are physically interpretable (4). MgI dimer, the ZnI–ZnI bond in dizincocene, and the Mn–Mn bond EDA schemes have been very successful at elucidating the CHEMISTRY in dimanganese decacarbonyl. nature of noncovalent interactions (2, 14, 15). These methods typically separate the interaction energy by either perturbative energy decomposition analysis j chemical bonding j density functional approaches or constrained variational optimization. Perturba- theory j covalent bonds j charge-shift bonds tive methods include the popular Symmetry Adapted Perturba- tion Theory (16, 17) method and the Natural EDA (18) based on NBOs. Variational methods include Kitaura and Morokuma nderstanding the chemical bond is central to both synthetic EDA (19), the Ziegler–Rauk method (ZR-EDA) (20), the Block- Uand theoretical chemists. The approach of the synthetic Localized Wave function EDA (14), and the absolutely localized chemists is based on qualitative, empirical features (electronega- molecular orbital energy decomposition analysis (ALMO-EDA) tivity, polarizability, etc.) gleaned over the past 150 y of research of Head-Gordon and coworkers (21–24). A number of nonco- and investigation. These features are notably absent from the valent EDA methods have been applied to bonds (2, 20, 25), toolbox of the theoretical chemist, who relies on a quantum although the single-determinant nature of these methods leads mechanical wave function to holistically describe the electronic to spin-symmetry broken wave functions, which contaminate the structure of a molecule: in essence, a numerical experiment. EDA terms with effects from the other terms. Bridging this gap is the purview of bonding analysis and energy To address this challenge, we recently reported a spin-pure decomposition analysis (EDA), which seek to separate the quan- extension of the ALMO-EDA scheme to the variational analysis tum mechanical energy into physically meaningful terms. Bond- of single covalent bonds (26). The method, which reduces to the ing analysis and EDA approaches are necessarily nonunique, but ALMO-EDA scheme for noncovalent interactions (24), includes different well-designed approaches provide complementary per- spectives on the nature of the chemical bond. This task is not yet Significance complete, despite intensive effort and substantial progress (1–5). The chemical bond was originally viewed (6) as being electro- While theoretical chemists today can calculate the energies static in origin, based on the virial theorem, and supported by of molecules with high accuracy, the results are not readily an accumulation of electron density in the bonding region rela- interpretable by synthetic chemists who use a different lan- tive to superposition of free atom densities. The chemical bond guage to understand and improve chemical syntheses. Energy is still often taught this way in introductory classes. However, the decomposition analysis (EDA) provides a bridge between the- + quantum mechanical origin of the chemical bond in H2 and H2 oretical calculation and practical insight. Most existing EDA (classical mechanics does not explain bonding) is in lowering the methods are not designed for studying covalent bonds. We kinetic energy by delocalization (that is, via constructive wave developed an EDA to characterize single bonds, providing an function interference). This was first established 55 y ago by Rue- interpretable chemical fingerprint in the language of synthetic + denberg (1) for H2 . A secondary effect in some cases, such as in chemists from the quantum mechanical language of theorists. H2, is orbital contraction, which is most easily seen by optimizing the form of a spherical 1s function as a function of bond length Author contributions: M.H.-G. designed research; D.S.L. performed research; D.S.L. and (7). Polarization (POL) and charge transfer (CT) contribute to M.H.-G. analyzed data; and D.S.L. and M.H.-G. wrote the paper. additional stabilization. Reviewers: G.F., University of Marburg; and R.H., Cornell University. Analyzing chemical bonds in more complex molecules has also The authors declare no conflict of interest. attracted great attention. Ruedenberg and coworkers (8) have Published under the PNAS license. been developing generalizations of their classic analysis proce- 1To whom correspondence should be addressed. Email: [email protected]. dures with this objective (9). Valence bond theory, while uncom- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. petitive for routine computational purposes, involves conceptu- 1073/pnas.1715763114/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1715763114 PNAS Early Edition j 1 of 8 Downloaded by guest on October 3, 2021 frags X ∆Eint = Emolecule − EZ Z = ∆EPREP + ∆EFRZ + ∆ESC + ∆EPOL + ∆ECT: [1] Each term is described in a corresponding subsection below. Preparation Energy. We begin from two doublet radical frag- ments, each of which is described by a restricted open shell (RO) Hartree–Fock (HF) or Kohn–Sham DFT single determi- nant with orbitals that are obtained in isolation from the other. ∆EPREP includes the energy required to distort each radical fragment to the geometry that it adopts in the bonded state: ∆EGEOM. This “geometric distortion” arises in most EDAs. There is another distortion energy that may also be incorpo- rated into ∆EPREP. Many radicals have a different hybridization than in the corresponding bond. For example, an F atom has an unpaired electron in a p orbital, while an F atom in a bond will 2 be sp-hybridized. The amine radical, NH2, is sp -hybridized with an unpaired electron in a p orbital, while an amine group is often sp3-hybridized or sp2-hybridized with a lone pair in the p orbital in a molecule. Rearranging the odd electron of each radical frag- ment to be in the hybrid orbital that is appropriate for SC will incur an energy cost, ∆EHYBRID, that completes the preparation energy (PREP): ∆EPREP = ∆EGEOM + ∆EHYBRID: [2] Fig. 1. (A) EDA of representative single bonds. (B) EDA of a few repre- sentative bonds with the FRZ and SC terms summed into a single FRZ + SC We define ∆EHYBRID as the energy change caused by rotations term. of the β hole in the span of the α occupied space from the iso- lated radical fragment to the correct arrangement in the bond. modified versions of the usual nonbonded frozen orbital (FRZ), This is accomplished by variational optimization of the frag- POL, and CT terms as well as a spin-coupling (SC) term describ- ments’ RO orbitals (in the spin-coupled state) only allowing dou- ing the energy lowering caused by electron pairing. The final bly occupied–singly occupied mixings. Afterward, the modified energy corresponds to the complete active-space(2,2) [CAS(2,2)] fragment orbitals are used to evaluate ∆EHYBRID. As limited (equivalently, one-pair perfect pairing (1PP) or two configura- orbital relaxation is involved, ∆EHYBRID may also be viewed as tion self-consistent field (TCSCF)) wave function. While this is a a kind of POL, and indeed, it was previously placed in the POL fully ab initio model, it lacks the dynamic correlation necessary term (26). for reasonable accuracy. However, ∆EHYBRID is also partially present here in that the By far, the most widely used treatment of dynamic correla- geometry of the radical fragment is fixed to be that of the inter- tion in quantum chemistry today is Kohn–Sham density func- acting fragment. For instance, free methyl radical is an sp2- tional theory (DFT) (27). DFT methods yield rms errors in hybridized planar molecule, while a methyl group in a bond is a chemical bond strengths on the order of a few kilocalories per pyramidalized sp3 fragment, and it is the latter that is used in this mole, which approach chemical accuracy at vastly lower com- EDA scheme.