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Energy decomposition analysis of single bonds withinKohn–Sham density functional theoryDaniel 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 SciencesDivision, 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 bondsis presented within the framework of Kohn–Sham density func-tional theory based on spin projection equations that are exactwithin wave function theory. Chemical bond energies can thenbe understood in terms of stabilization caused by spin-couplingaugmented by dispersion, polarization, and charge transfer incompetition with destabilizing Pauli repulsions. The EDA revealsdistinguishing features of chemical bonds ranging across nonpo-lar, polar, ionic, and charge-shift bonds. The effect of electroncorrelation is assessed by comparison with Hartree–Fock results.Substituent effects are illustrated by comparing the C–C bond inethane against that in bis(diamantane), and dispersion stabiliza-tion in the latter is quantified. Finally, three metal–metal bondsin experimentally characterized compounds are examined: a MgI–MgI dimer, the ZnI–ZnI bond in dizincocene, and the Mn–Mn bondin dimanganese decacarbonyl.

energy decomposition analysis | chemical bonding | density functionaltheory | covalent bonds | charge-shift bonds

Understanding the chemical bond is central to both syntheticand theoretical chemists. The approach of the synthetic

chemists is based on qualitative, empirical features (electronega-tivity, polarizability, etc.) gleaned over the past 150 y of researchand investigation. These features are notably absent from thetoolbox of the theoretical chemist, who relies on a quantummechanical wave function to holistically describe the electronicstructure of a molecule: in essence, a numerical experiment.Bridging this gap is the purview of bonding analysis and energydecomposition analysis (EDA), which seek to separate the quan-tum mechanical energy into physically meaningful terms. Bond-ing analysis and EDA approaches are necessarily nonunique, butdifferent well-designed approaches provide complementary per-spectives on the nature of the chemical bond. This task is not yetcomplete, despite intensive effort and substantial progress (1–5).

The chemical bond was originally viewed (6) as being electro-static in origin, based on the virial theorem, and supported byan accumulation of electron density in the bonding region rela-tive to superposition of free atom densities. The chemical bondis still often taught this way in introductory classes. However, thequantum mechanical origin of the chemical bond in H+

2 and H2

(classical mechanics does not explain bonding) is in lowering thekinetic energy by delocalization (that is, via constructive wavefunction interference). This was first established 55 y ago by Rue-denberg (1) for H+

2 . A secondary effect in some cases, such as inH2, is orbital contraction, which is most easily seen by optimizingthe form of a spherical 1s function as a function of bond length(7). Polarization (POL) and charge transfer (CT) contribute toadditional stabilization.

Analyzing chemical bonds in more complex molecules has alsoattracted great attention. Ruedenberg and coworkers (8) havebeen developing generalizations of their classic analysis proce-dures with this objective (9). Valence bond theory, while uncom-petitive for routine computational purposes, involves conceptu-

ally simple wave functions that are suitable for extracting qual-itative chemical bonding concepts (10). The emergence of the“charge-shift bond” paradigm, exemplified by the F2 molecule,is a specific example of its value (11). The widely used naturalbond orbital (NBO) approach (12) provides localized orbitals,predominant Lewis structures, and information on hybridizationand chemical bonds. The quantum theory of atoms-in-molecules(QTAIM) (13) describes the presence of bonds by so-called bondcritical points in the electron density as well as partitioning anenergy into intraatomic and interatomic terms. Another topo-logical approach is the electron localization function, which isa function of the density and the kinetic energy density. Manyother methods also exist for partitioning a bond energy into sumsof terms that are physically interpretable (4).

EDA schemes have been very successful at elucidating thenature of noncovalent interactions (2, 14, 15). These methodstypically separate the interaction energy by either perturbativeapproaches or constrained variational optimization. Perturba-tive methods include the popular Symmetry Adapted Perturba-tion Theory (16, 17) method and the Natural EDA (18) basedon NBOs. Variational methods include Kitaura and MorokumaEDA (19), the Ziegler–Rauk method (ZR-EDA) (20), the Block-Localized Wave function EDA (14), and the absolutely localizedmolecular orbital energy decomposition analysis (ALMO-EDA)of Head-Gordon and coworkers (21–24). A number of nonco-valent EDA methods have been applied to bonds (2, 20, 25),although the single-determinant nature of these methods leadsto spin-symmetry broken wave functions, which contaminate theEDA terms with effects from the other terms.

To address this challenge, we recently reported a spin-pureextension of the ALMO-EDA scheme to the variational analysisof single covalent bonds (26). The method, which reduces to theALMO-EDA scheme for noncovalent interactions (24), includes

Significance

While theoretical chemists today can calculate the energiesof molecules with high accuracy, the results are not readilyinterpretable by synthetic chemists who use a different lan-guage to understand and improve chemical syntheses. Energydecomposition analysis (EDA) provides a bridge between the-oretical calculation and practical insight. Most existing EDAmethods are not designed for studying covalent bonds. Wedeveloped an EDA to characterize single bonds, providing aninterpretable chemical fingerprint in the language of syntheticchemists from the quantum mechanical language of theorists.

Author contributions: M.H.-G. designed research; D.S.L. performed research; D.S.L. andM.H.-G. analyzed data; and D.S.L. and M.H.-G. wrote the paper.

Reviewers: G.F., University of Marburg; and R.H., Cornell University.

The authors declare no conflict of interest.

Published under the PNAS license.1To whom correspondence should be addressed. Email: mhg@cchem.berkeley.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1715763114/-/DCSupplemental.

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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 + SCterm.

modified versions of the usual nonbonded frozen orbital (FRZ),POL, and CT terms as well as a spin-coupling (SC) term describ-ing the energy lowering caused by electron pairing. The finalenergy corresponds to the complete active-space(2,2) [CAS(2,2)](equivalently, one-pair perfect pairing (1PP) or two configura-tion self-consistent field (TCSCF)) wave function. While this is afully ab initio model, it lacks the dynamic correlation necessaryfor reasonable accuracy.

By far, the most widely used treatment of dynamic correla-tion in quantum chemistry today is Kohn–Sham density func-tional theory (DFT) (27). DFT methods yield rms errors inchemical bond strengths on the order of a few kilocalories permole, which approach chemical accuracy at vastly lower com-putational effort than wave function methods. The purpose ofthis paper is to recast our bonded ALMO-EDA method into asingle-determinant formalism that allows the computation of adynamically correlated bonded EDA with any existing densityfunctional. After outlining our approach, we turn to the charac-terization of a variety of chemical bonds, ranging from familiarsystems to less familiar dispersion stabilized bonds, and severalsingle metal–metal bonds.

Variational EDAThe single bond of interest is, by definition, the differencebetween the DFT calculation on the molecule and the sum ofDFT calculations on the separately optimized, isolated frag-ments. This interaction will be separated into five terms:

Table 1. EDA of representative bonds (in kilocalories per mole)

Bond PREP FRZ SC POL CT Sum

H3C–CH3 36.9 279.7 −344.6 (82.7) −11.8 (2.8) −60.1 (14.4) −99.8H–Cl 13.1 219.0 −253.7 (74.7) −49.3 (14.5) −36.7 (10.8) −107.5F–SiF3 41.5 259.1 −227.3 (48.9) −119.0 (25.6) −118.8 (25.5) −164.4F–F 9.2 186.3 −124.1 (52.7) −37.2 (15.8) −74.3 (31.5) −40.1Li–F 0.0 30.5 2.7 −7.8 (4.5) −164.6 (95.5) −139.2

Values in parentheses are the percentages of the total stabilizing interaction energy.

∆Eint = Emolecule −frags∑Z

EZ

= ∆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 radicalfragment 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 hybridizationthan in the corresponding bond. For example, an F atom has anunpaired electron in a p orbital, while an F atom in a bond willbe sp-hybridized. The amine radical, NH2, is sp2-hybridized withan unpaired electron in a p orbital, while an amine group is oftensp3-hybridized or sp2-hybridized with a lone pair in the p orbitalin a molecule. Rearranging the odd electron of each radical frag-ment to be in the hybrid orbital that is appropriate for SC willincur an energy cost, ∆EHYBRID, that completes the preparationenergy (PREP):

∆EPREP = ∆EGEOM + ∆EHYBRID. [2]

We define ∆EHYBRID as the energy change caused by rotationsof the β hole in the span of the α occupied space from the iso-lated radical fragment to the correct arrangement in the bond.This is accomplished by variational optimization of the frag-ments’ RO orbitals (in the spin-coupled state) only allowing dou-bly occupied–singly occupied mixings. Afterward, the modifiedfragment orbitals are used to evaluate ∆EHYBRID. As limitedorbital relaxation is involved, ∆EHYBRID may also be viewed asa kind of POL, and indeed, it was previously placed in the POLterm (26).

However, ∆EHYBRID is also partially present here in that thegeometry of the radical fragment is fixed to be that of the inter-acting fragment. For instance, free methyl radical is an sp2-hybridized planar molecule, while a methyl group in a bond is apyramidalized sp3 fragment, and it is the latter that is used in thisEDA scheme. We have moved ∆EHYBRID here for that reasonand because it can be much larger than the other contributionsto POL; therefore, its presence in POL can obscure trends inPOL and SC across rows of the periodic table as the hybridizationof the radical fragment changes. Even with ∆EPREP so defined,one must still reorient the frozen orbitals to resolve degenera-cies and obtain the correctly oriented α density as previouslydescribed (26).

Nevertheless, regarding orbital rehybridization as part ofpreparation (as we do in this paper) and regarding orbital rehy-bridization as part of POL [as was done previously (26)] areboth defensible choices. Also, in cases where ∆EHYBRID is large,the consequences of where it is placed can be considerable, as∆EPREP, ∆EFRZ, ∆ESC, and ∆EPOL all change as a result. There-fore, the reader can compare and decide whether they agree

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Fig. 2. (A) EDA of first row element–H bonds. (B) EDA of first row element–Hbonds with the FRZ and SC terms summed into one FRZ + SC term.

with our choice; corresponding data tables with rehybridizationas part of POL (and ∆EPREP = ∆EGEOM) are included in SIAppendix.

Frozen Energy. The second term in Eq. 1, ∆EFRZ, is the energychange associated with the two radical fragments interactingwithout permitting SC, POL, or CT. For simplicity (but with-out loss of generality), let us assume that both radicals haveS = 1/2;MS = +1/2. In the FRZ energy, the fragment wave func-tions are combined to form a spin-pure triplet single-determinantwave function (S = 1;MS = +1) without allowing the orbitals torelax. This term is entirely a nonbonded interaction and will typ-ically be repulsive for a chemical bond because of Pauli repul-sion. It includes contributions from interfragment electrostatics,Pauli repulsion, exchange–correlation, and dispersion. The addi-tion of dispersive effects, a dynamic correlation property, usingDFT should result in smaller frozen energy terms than whatis calculated in the original CAS(2,2) ALMO-EDA method.

A set of orbitals is said to be “absolutely localized” if themolecular orbital (MO) coefficient matrix is block-diagonal inthe fragments. Since the atoms partition into fragments, so dotheir corresponding atomic orbitals (AOs), and hence, the iso-lated fragment MOs, T , automatically satisfy the absolutelylocalized molecular orbital (ALMO) constraint. The “frozen”occupied ALMOs constructed by block-diagonally concatenat-

Table 2. EDA of first row E–H bonds (in kilocalories per mole)

Bond PREP FRZ SC POL CT Sum

H–H 0.0 245.4 −311.4 (88.0) −30.6 (8.6) −11.9 (3.4) −108.4H–Li 0.0 34.8 −51.2 (53.7) −18.9 (19.8) −25.3 (26.5) −60.6H–BeH 0.1 188.8 −252.3 (87.1) −14.6 (5.1) −22.7 (7.8) −100.6H–BH2 3.7 276.1 −349.0 (89.0) −21.4 (5.5) −21.6 (5.5) −112.2H–CH3 15.4 284.6 −356.3 (86.2) −26.0 (6.3) −30.9 (7.5) −113.2H–NH2 32.0 293.3 −344.5 (78.1) −40.8 (9.2) −55.9 (12.7) −115.9H–OH 32.2 284.6 −310.9 (70.3) −36.7 (8.3) −94.5 (21.4) −125.2H–F 27.9 279.4 −268.7 (60.0) −55.1 (12.3) −124.2 (27.7) −140.8

Numbers in parentheses are the percentages of the total stabilizing interaction energy.

ing isolated RO fragments need not be orthogonal (ALMOsare generally nonorthogonal), and therefore, they have an over-lap matrix, σ. Therefore, as a function of interfragment separa-tion, the frozen density matrix, PFRZ = Tσ−1T†, undergoes Paulideformation, although T is constant.

The energy associated with this density (EFRZ) may then becomputed by HF or DFT. The frozen interaction energy is thedifference relative to noninteracting, prepared fragments:

∆EFRZ = EFRZ −frags∑Z

EZ. [3]

This ALMO-EDA FRZ term may be further separated into con-tributions corresponding to permanent electrostatic interactions,Pauli repulsion, and dispersion (23).

Spin-Coupling Energy. The third term in Eq. 1, ∆ESC, is the energydifference caused by electron pairing: that is, changing the SCof the two radical electrons from high-spin triplet to low-spin(LS) singlet. The SC energy accounts for the difference betweendestructive wave function interference [in the high-spin (HS)case] and constructive wave function interference (in the LScase). It is worth pointing out that delocalization does not requiretwo spins and is present even in H2

+. Like FRZ, SC will be eval-uated with frozen orbitals, but while FRZ is typically stronglyrepulsive (dominated by Pauli repulsion), SC is typically stronglyattractive in the overlapping regime associated with covalentbond formation. For this reason and because we are primarilyinterested in the singlet surface (as opposed to the triplet surfaceof the initial supersystem), FRZ and SC may be grouped togetherinto a total frozen orbital term (FRZ + SC).

From the HS frozen determinant (S = 1;MS = +1), flippingthe spin (α→β) of one of the two half-occupied orbitals reducesthe MS value by one (i.e., MS = +1 → MS = 0). The objective isto change the SC from (not bonding) triplet to (bonding) sin-glet. However, in the single-determinant formalism, the resultof the spin flip is a broken symmetry (BS) ALMO determinant.Its energy, EBS, contains the desired spin-coupled LS energy,ELS, but also a single contaminant, which is HS: EHS =EFRZ andSHS =SLS + 1. We may write

EBS = (1− c)ELS + cEHS. [4]

To obtain c, we examine the value of 〈S2〉BS, which is contami-nated in exactly the same way as the energy:

〈S2〉BS = (1− c)〈S2〉LS + c〈S2〉HS

= 〈Sz〉LS(〈Sz〉LS + 1) + 2c(〈Sz〉LS + 1). [5]

Using the calculated 〈S2〉BS value (a derivation of the 〈S2〉 valuefor a BS single determinant with nonorthogonal orbitals is givenin SI Appendix), we solve for c via Eq. 5:

c =〈S2〉BS − 〈Sz〉LS(〈Sz〉LS + 1)

2(〈Sz〉LS + 1). [6]

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Fig. 3. Comparison of EDA terms computed at HF, PBE0-D3, ωB97X-D, andωB97M-V/aug-cc-pvtz of (A) ethane and (B) F2.

In turn, this permits us to solve Eq. 4 for the spin-pure energy,ELS =αEBS+(1−α)EHS, where α= (1− c)−1. This result corre-sponds to Yamaguchi’s spin-projection scheme (28, 29). Finally,with ELS in hand, the SC term is given by

∆ESC = ELS − EFRZ, [7]

where the orbitals are still the frozen fragment ones.The above derivation is exact, because a spin-pure ELS is

obtained if 〈S2〉BS and 〈E〉BS are evaluated consistently fromthe same one-particle density matrices and two-particle densitymatrices (2PDMs). An example is the case of HF wave func-tions. Unfortunately, this condition is not strictly satisfied forKohn–Sham DFT, because the interacting 2PDM is not available(30), and thus, the value of 〈S2〉BS corresponding to 〈E〉BS is notavailable. This dilemma arises because the fundamental theo-rems of DFT allow construction of the exact ground-state energywithout knowledge of the 2PDM. The best that can be straight-forwardly accomplished is to use the noninteracting 2PDM(i.e., from the Kohn–Sham determinant) to evaluate 〈S2〉BS inDFT. For any functional but HF, this choice leads to a smallinconsistency in the final energy, the remedy for which is des-cribed below.

Regarding comparison of this approach with other EDAs,this SC term is only partly contained in the frozen orbital termin EDA schemes such as the ZR-EDA approach (20) usedextensively for bonding analysis (2). Such EDAs form only onefrozen supersystem on the LS surface rather than separate FRZand SC terms. The resulting LS frozen energy is exactly EBSabove. From the analysis above, because of spin contamination,∆EFRZ(BS) >∆EFRZ + ∆ESC.

Polarization Energy. The fourth term in Eq. 1, ∆EPOL, arisespartly from the orbitals (with low spin coupling) relaxing becauseof the presence of the field of the other fragment. ∆EPOL is theterm that includes contributions from polarization in the bond,but the ALMO constraint prevents CT contributions. To providea well-defined basis set limit, fragment electric response func-tions (FERFs) are used as the ALMO virtual basis (22, 24). The

FERFs are the subset of virtual orbitals that exactly describe thelinear response of each fragment to an applied electric field. Fol-lowing previous work (22, 24), the dipole and quadrupole FERFswill be used to define the fragment virtual spaces for electricalpolarization. For a hydrogen atom, the three-dipole functions arep-like, and the five-quadrupole functions are d-like.

In addition to electrical polarization, there is another contribu-tion to POL that we have discussed in detail elsewhere (31). Thefrozen orbitals may contract toward the nucleus to lower theirenergy without any induced electrical moments. This contractioneffect was first identified by Ruedenberg (1) as part of his clas-sic analysis of the one-electron chemical bond in H2

+. We haveshown (31) that orbital contraction can be accurately modeled byadding a so-called monopole function to the FERF virtual spacefor each occupied orbital. On the H atom, the monopole FERFis a 2s-type function.

The overall FERF basis is thus of monopole, dipole, quadru-pole (MDQ) type, and thus,

∆EPOL = ∆ECONPOL + ∆EELE

POL , [8]

where (31) ∆ECONPOL = EALMO/ M − EFRZ and ∆EELE

POL =EALMO/ MDQ−EALMO/ M. Our results showed that orbital con-traction was very important in bonds to hydrogen but ratherinsignificant in bonds only involving heavier elements (31). Thisdecreased energy lowering in heavy element bonds can be viewedas arising from diminished violation of the virial theorem on SCwith frozen orbitals relative to bonds to hydrogen.

The additional mathematics necessary to implement POL inconjunction with the approximate DFT spin-projection methodis described in SI Appendix.

Charge-Transfer Energy. The fifth term, ∆ECT, contains CT con-tributions, allowing electrons to move between the fragments.It is the dominant term in ionic bonds and an important partof charge-shift bonds (11). Mathematically, we will releasethe ALMO constraint and reoptimize the orbitals to obtainan unconstrained spin-projected energy. Implemented with HFdeterminants, this gives the CAS(2,2)/1PP energy. With DFT, weobtain the approximately spin-projected DFT analog ESP-DFT.

However, as already mentioned, in DFT, the 〈S2〉 value usedin the optimization is only approximate. Moreover, the approxi-mate exchange–correlation functional accounts for some amountof static correlation. Hence, ESP-DFT obtained at this final stepis generally lower than the DFT energy of a single determinant,EDFT. Since DFT functionals are typically developed (or fitted) toproduce accurate results only as a single determinant, this over-counting of correlation leads to molecules being slightly over-bound (on the order of 1–15 kcal/mol). To address this issue,we simply rescale the terms calculated on the approximately

Fig. 4. Comparison of the EDA of the E–E bond (E = F, Cl, Br) computed atHF and ωB97M-V/aug-cc-pvtz (Table 3).

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Table 3. EDA of E–E bonds (E = halogen)

Bond PREP FRZ SC POL CT Sum

F–F 9.2 186.3 −124.1 (52.7) −37.2 (15.8) −74.3(31.5) −40.1Cl–Cl 6.8 154.8 −121.5 (54.7) −52.8 (23.8) −47.9 (21.6) −60.6Br–Br 4.4 116.4 −98.9 (57.7) −40.7 (23.7) −32.0 (18.6) −50.8

All energies are in kilocalories per mole. Numbers in parentheses arethe percentages of the total stabilizing interaction energy that each termrepresents.

spin-projected LS surface by a factor cR, such that ∆ESC←cR∆ESC, ∆EPOL← cR∆EPOL, and ∆ECT← cR∆ECT. cR isdefined as

cR =(EDFT − EFRZ)

(ESP-DFT − EFRZ), [9]

so that the final interaction energy exactly satisfies Eq. 1. cR istabulated in SI Appendix.

For DFT, this rescaling scheme will be inadequate for ana-lyzing bonds whenever a single DFT determinant is itself inad-equate to describe the bond, such as in strongly diradicaloidmolecules. In such cases, it may be preferable to use the approx-imately spin-projected result directly, such as is quite often donein BS DFT calculations (32, 33). In contrast, no rescaling wouldbe needed for second-order Møller–Plesset perturbation theory(MP2) or coupled cluster theory, because the spin projectionwould be exact, and such methods could additionally describestrongly diradicaloid systems.

Especially for symmetrical systems, one may inquire about thenature of the CT term. The SC term includes some amount ofionic-like contribution because of the nonorthogonality of theorbitals. In the limiting cases, when the singly occupied spaces donot overlap at all, the spin-coupled state is purely covalent, andwhen they overlap fully, for symmetrical systems, the HF result isobtained. The CT term then measures the fraction of ionic con-tribution that was unavailable to the fragments with the FERFand ALMO Hilbert space constraints.

Computational DetailsA development version of Q-Chem 4.4 was used for all calcula-tions (34). All calculations were performed using the ωB97M-Vdensity functional (35) [a range-separated hybrid (RSH) meta-generalized gradient approximation (meta-GGA)] and the aug-cc-pVTZ basis (36) unless otherwise specified. Numerical testssuggest that ωB97M-V is among the most accurate available forchemical bond energies (35). For decomposition of the frozenterm (23), the dispersion-free functional used with ωB97M-Vwas HF. Decomposition of the frozen term was carried out usingunrestricted fragments forming an unrestricted supersystem.While the preparation energy [2] has two components, we reportonly their sum here (the breakdown is in SI Appendix). Likewise,since the contraction energy has been discussed elsewhere (31),here we report only the total POL energy as defined in Eq. 8.Finally, when HF is used, identical EDA results (apart from thereclassification of the hybridization energy) are obtained as withthe previously described multideterminant [CAS(2,2) or 1PP]

Table 4. EDA of bis(diamantane) and ethane (ωB97M-V/6-31+G**)

Bond PREP FRZ* DISP SC POL CT Sum

Diamantane 46.5 274.1 −60.4 (15.3) −264.2 (66.8) −23.1 (5.8) −47.9 (12.1) −75.1H3C–CH3 37.6 285.9 −8.3 (2.0) −346.5 (81.9) −22.1 (5.2) −46.4 (11.0) −99.7

All energies are in kilocalories per mole. DISP is the dispersion energy as determined by the frozendecomposition (23).*FRZ is the frozen energy less the dispersion energy.

method, which is hereafter referred to as Hartree–Fock energydecomposition analysis (HF-EDA) (26).

Results and DiscussionRepresentative Bonds. We first verify the behavior of the termsof the EDA by investigating some representative bonds with theωB97M-V functional: the C–C bond in ethane (a nonpolar cova-lent bond), the H–Cl bond in HCl (a polar covalent bond), theF–Si bond in SiF4 (a polar bond with ionic character), the F–Fbond in F2 (a nonpolar, charge-shift bond), and the Li–F bondin LiF (an ionic bond) (Fig. 1 and Table 1). The EDA gives a“fingerprint” for different classes of bonds: covalent and charge-shift bonds have relatively high SC energies, polar bonds haverelatively high POL energies, and charge-shift and ionic bondshave relatively high CT energies. The EDA thus recovers classi-cal bonding concepts from quantum mechanical methods (5).

First Row Element–H Bonds. This method allows us to investigatetrends across periods and down groups of the periodic table. Toillustrate, first row element–H bonds were investigated (Fig. 2and Table 2). Moving right across the first row, the elementsbecome more electronegative, and the E–H bonds switch frombeing nonpolar covalent bonds to polar covalent bonds withincreasing CT. This change is most obvious when the FRZ andSC terms are summed into a total FRZ + SC term (Fig. 2B).For the nonpolar covalent bonds, total FRZ + SC interactionsaccount for most of the bond energy. By contrast, in the moder-ately polar covalent bond in ammonia, POL becomes significant,and with increasing ionic character in water and HF, CT is a largesource of binding.

A variety of density functionals were investigated to deter-mine how the HF-EDA terms are altered by the inclusion ofdynamic correlation as well as to check that this method is nottoo sensitive to choice of functional. Full data tables compar-ing HF, BLYP-D3 (a dispersion-corrected GGA), PBE0-D3 [adispersion-corrected hybrid GGA (37)], ωB97X-D [a dispersion-corrected RSH GGA (38)], B97M-V [a meta-GGA (39) withVV10 nonlocal correlation], and ωB97M-V [an RSH meta-GGA(35) with VV10] for all of the bonds described above are avail-able in SI Appendix. Generally speaking, addition of dynamic cor-relation decreases the frozen energy (owing to the inclusion ofdispersion) and increases CT and POL stabilization relative toHF. Fig. 3 shows these trends for the C–C bond in ethane andthe F–F bond in fluorine. Some differences between functionalsare inevitable, as none are exact. Overall, the small discrepanciesevident in Fig. 3 and SI Appendix appear acceptable.

Bonding in Halogens. Dynamic correlation effects in bonds aremost pronounced when the local electron density is high, such asin molecules with many lone pairs near each other (40). Hence,dynamic correlation is necessary for studying bonds, such asthose in halogens (Fig. 4). Comparing the HF-EDA and ωB97M-V-EDA of the homoatomic halogens F2, Cl2, and Br2, we seethat dynamic correlation is indeed very important to obtain cor-rect bond energies. Moreover, in these molecules, inclusion ofdynamic correlation mostly increases the CT term, and therefore,it is dynamic correlation associated with greater electron delo-calization that contributes most strongly to the stability of bonds

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N

NN

Ar

Ar

MgN

NN

Ar

Ar

Mg

Ar = 2,6-Me2C6H3

Zn

ZnMn Mn COOC

C

C

C

CC

C C

C

O O

O OO

OO

O

Fig. 5. Metal–metal bonds probed by the ALMO-EDA.

between halogen atoms as might be expected from a charge-shiftbond. Dispersion plays only a modest role (≈9 kcal/mol) in thesemolecules. This charge-shift bonding, in which ionic structurecontributes significantly to the ground state, is manifest in thechemistry of halogens [for example, in the stabilization of charge-separated species during the formation of halonium ions in thehalogenation of alkenes (41)].

Dispersion-Assisted Bonds. In molecules with bulky side groups,dispersion can play a significant role in the stabilization of bonds(42, 43). An example is the elongated 1.65 A C–C bond inbis(diamantane), which, based on bond length, is expected tohave a bond strength of ≈40 kcal/mol (44). Experimentally, it isconsiderably stronger (showing no decomposition up to 300 ◦C),which has been attributed to many stabilizing dispersive interac-tions between the interfacial C–H bonds (45). Comparison of theEDA for this bond vs. ethane (at the ωB97M-V/6-31+G** level)allows quantification of the forces that stabilize it as shown inTable 4.

As seen in Table 4, while the POL and CT terms are fairly simi-lar for ethane and bis(diamantane), the SC term is 82 kcal/mol lessfor bis(diamantane) following expectations based on bond length.However, the bis(diamantane) bond is not that much weakerthan ethane, because it also has a less destabilizing FRZ energy.Applying the ALMO frozen decomposition, it was found that dis-persion accounts for this large increase in bond strength [60.4vs. 8.3 kcal/mol of stabilization for bis(diamantane) and ethane,respectively]. Enhanced dispersion is the key factor in account-ing for the unusual stability of bis(diamantane), even as the bondelongation is a result of partially relieving the close contacts.

Metal–Metal Bonds. We next consider some single bonds, whichare less well-studied: main group and transition metal metal–metal bonds. We investigated a slightly truncated version of theMg–Mg dimer of Jones and coworkers (46), the Zn–Zn bond indizincocene from Carmona and coworkers (47), and the classicMn–Mn bond (48, 49) in the dimanganese decacarbonyl complex(Fig. 5). The relatively new Mg–Mg and Zn–Zn bonds are inter-esting for their novelty and have proven to be important chem-ical synthons (50, 51). However, the nature of these symmetri-cally bonded complexes is difficult to guess on first inspectionbecause of our unfamiliarity with the chemistry of Mg and Zn inthe formal +1 oxidation state. Will these be conventional cova-lent bonds, or will they have charge-shift character? These sys-tems are, therefore, good candidates for use of the EDA, becausewe can compare their EDA results against the well-understoodsystems presented earlier.

Table 5. EDA of metal–metal bonds [ωB97X-D/6-31+G** withbasis set superposition error (BSSE) correction]

Bond PREP FRZ SC POL CT Sum

Mg–Mg 0.1 69.7 −118.3 (97.3) −3.1 (2.5) −0.2 (0.2) −51.8Zn–Zn 4.1 103.5 −150.0 (82.7) −7.6 (4.2) −23.7 (13.1) −73.7Mn–Mn 3.5 33.2 −37.4 (51.6) −8.0 (11.1) −27.0 (37.3) −35.7

All energies in kilocalories per mole.

Table 6. EDA of select bonds (ωB97M-V/aug-cc-pvtz) with BSEDA methods

Bond H–H H3C–CH3 F–F Li–F Li+–F−

Prep 0.0 17.9 0.0 0.0 45.6Elec 3.9 −135.1 −41.9 −17.0 −202.2Pauli −10.9 194.1 165.2 55.1 42.9Orb −101.4 −175.5 −161.2 −176.1 −24.3

All energies in kilocalories per mole.

The total bond energies obtained for the Mn–Mn bond arein close agreement with experimental measures (40.9 vs. 38±5 kcal/mol) (52) and previous calculations (53, 54). There areno direct experimental measures for the given Mg–Mg and Zn–Zn bonds. The Mg–Mg bond in ClMgMgCl was extrapolatedfrom experimental measurements to be 47.1 kcal/mol, in closeagreement with the results obtained here (55). ExperimentalZn–Zn bond dissociation energy (BDE) measurements wereobtained from the homoatomic dimer (which is, in principle,doubly bonded), but the measured BDE is relatively close tothe Zn–Zn bond calculated here (82.2 vs. 93.7 kcal/mol) (56).Both classes of bonds have previously been studied by theoreti-cal methods (refs. 50 and 57 and references therein).

The EDA results are given in Table 5. The Mg–Mg bond turnsout to be a classic nonpolar covalent bond analogous to H2:the bond strength is mainly caused by SC (i.e., electron pair-ing) between the unpaired electrons on Mg(I) centers. Thereis almost no CT: consistent with the high reduction potentialof Mg(0), Mg(0)–Mg(II)/Mg(II)–Mg(0) contributions are notimportant in this bond. This is consistent with NBO calculationscarried out in the initial disclosure of this molecule, which foundthe bond to be a covalent single-bond dominated by s-orbitalcontributions (46).

However, the less reducing Zn in the Zn–Zn bond, whichis principally covalent, does exhibit some ionic Zn(0)–Zn(II)/Zn(II)–Zn(0) resonance contributions, much like in ethane.These ionic contributions account for most of why the Zn–Znbond is stronger than the Mg–Mg bond. Our method providesan accurate dissection of the metal–metal bond in multimetal-locenes, which was not possible before (58, 59). It also gives insightinto the origins of the relative bond strengths in metal–metalbonds: although both Mg–Mg and Zn–Zn bonds have strong cova-lent stabilization, the more easily oxidized Zn is further stabi-lized by ionic resonances, making it a much stronger bond. Thiswas hinted at in a recent QTAIM study, which showed that maingroup–main group bonds in M2Cp2 had more “covalent char-acteristics”, while transition metal–transition metal bonds had“closed shell ionic characteristics” (60).

By contrast, the bond in dimanganese decacarbonyl is acharge-shift bond much like in F2, with CT playing a majorrole in stabilizing the bond. Previous studies using QTAIM havealso implicated “closed shell interactions” and indicated that thebond is intermediate to a covalent and ionic bond (61), whileother studies favor a more covalent picture (62). This hybridcovalent–CT stabilization is quantified here and appears anal-ogous to the charge-shift bonding picture (11).

Comparison with Other EDA Methods. The main advantage of ourEDA over alternatives for studying covalent bonds is its fulluse of valid, spin-pure intermediate wave functions. By con-trast, the Morokuma EDA (19), the ZR-EDA (2, 25), and theextended transition state-natural orbitals for chemical valence(ETS-NOCV) (20) method use BS spin-contaminated inter-mediate wave functions to determine the energy components,which are likewise nonphysically spin-contaminated. This prob-lem does not arise for intermolecular interactions, which are typ-ically between closed shell fragments. In such cases, BS solutions

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do not enter, and the ALMO-EDA frozen energy correspondsdirectly to that in these earlier EDAs.

The degree of spin contamination changes during the stepwisevariational optimization of the wave function (e.g., the final stateis typically spin-pure and closed shell), and hence, the effect ofthe contamination is inconsistently distributed among the energyterms for a molecule. The SC energy of the method presentedhere is distributed between the Elec, Pauli, and Orb terms in asystem-specific manner in these BS-based methods.

Another advantage of our EDA is that it can resolve differentclasses of chemical bonds. For comparison, the ZR-EDA/ETS-NOCV energy terms for H−H, H3C–CH3, F−F, and Li−F aregiven in Table 6. Note that no chemical fingerprint is evidentfrom these data, which significantly decrease the utility of BSmethods for understanding bonded interactions. It is unclearhow much of the unphysical negative value of the H2 Pauli repul-sion is because of spin contamination vs. self-interaction error ofthe functional. The ionic picture of LiF can be recovered withthe ZR-EDA by first ionizing the fragments and noting a rela-tively small Orb term. However, this requires knowing how thefragments should be prepared, whereas in our method, the resultfalls out naturally.

Conclusionsi) An EDA method for single bonds has been developed in a

single-determinant formalism. This EDA allows use of DFT

with approximate spin projection, thereby allowing for theefficient inclusion of dynamic correlation effects, such as dis-persion.

ii) Numerical tests show that the DFT-based EDA is not toosensitive to the choice of functional, and relative to theuse of uncorrelated HF determinants, it has improved thetreatment of a variety of single bonds, including the firstrow E–H bonds and the bonds in halogens. Comparisonsbetween different molecules should all be made with a singlefunctional.

iii) The inclusion of dispersion effects allows for meaningfulstudy of bonds that rely heavily on dispersion as illustratedby the example of bis(diamantane).

iv) Analysis of single metal–metal bonds with this method haspermitted characterization of Mg(I)-Mg(I), Zn(I)-Zn(I), andMn(0)-Mn(0) bonds that have been synthesized and suggestsphysical reasons for the range of bond strengths seen in maingroup metal and transition metal bonds.

v) The main limitation of this EDA is its restriction to sin-gle chemical bonds. Only technical challenges inhibit theextension of this single-bond EDA approach to correlatedab initio methods—we are currently working on addressingthose issues.

ACKNOWLEDGMENTS. This work was supported by Grants CHE-1665315 andCHE-1363342 from the US National Science Foundation.

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