Downloaded by guest on October 3, 2021 www.pnas.org/cgi/doi/10.1073/pnas.1715763114 a Levine S. Daniel theory functional within density bonds Kohn–Sham single of analysis decomposition Energy ini h atri unie.Fnly he ea–ea bonds Mg metal–metal a three examined: Mg Finally, are compounds quantified. characterized is experimentally in stabiliza- latter dispersion the in and in bond bis(diamantane), tion C–C in the that electron comparing against by results. ethane of illustrated Hartree–Fock effect are with effects The comparison by bonds. assessed charge-shift is and correlation nonpo- ionic, across ranging polar, bonds lar, reveals in chemical EDA of transfer The features repulsions. charge distinguishing Pauli and destabilizing spin-coupling with polarization, by competition caused dispersion, then stabilization by can of augmented energies terms exact bond in are understood Chemical that theory. be equations function projection func- wave spin density within on Kohn–Sham based of theory framework Hoffmann) tional Roald the and within Frenking bonds Gernot presented chemical by single is reviewed 2015. for 2017; in (EDA) 7, elected analysis September decomposition Sciences review energy of An for Academy (sent National 2017 the 10, of October members Head-Gordon, by Martin Articles by Inaugural Contributed of series special the of part is contribution This 94720 CA Berkeley, Laboratory, National Berkeley Lawrence Division, eiiefrruiecmuainlproe,ivle conceptu- involves purposes, uncom- computational while routine theory, proce- for bond Valence petitive analysis (9). classic objective this have their with (8) dures of coworkers generalizations and developing Ruedenberg been attention. great attracted to length contribute bond (CT) of transfer function stabilization. charge a additional as and function (POL) 1s Polarization spherical (7). a of form the analysis decomposition energy decacarbonyl. dimanganese in H wave Rue- H by for constructive ago (1) y via denberg 55 established is, first was (that This interference). delocalization function the lowering by in energy is bonding) H kinetic explain in not bond does chemical mechanics the the (classical However, of classes. origin bond introductory in mechanical chemical way quantum The this densities. taught rela- often atom by region still free supported is bonding of and the superposition in theorem, to density virial tive electron the of on accumulation based an origin, in static yet (1–5). not progress is substantial task and This effort bond. intensive chemical despite the complete, of nature the on per- spectives complementary provide but approaches nonunique, well-designed necessarily different are Bond- approaches EDA terms. and meaningful analysis physically ing energy into quan- energy and experiment. the mechanical analysis separate numerical to tum bonding seek of which a (EDA), purview analysis essence, the decomposition quantum is in gap a molecule: this electronic Bridging on a the relies describe of holistically who structure the to chemist, from function wave theoretical absent mechanical the notably are of research features of toolbox y These 150 investigation. past the and over gleaned etc.) (electronega- features polarizability, empirical tivity, qualitative, on based is chemists U theory ent .Pte etrfrTertclCeity eateto hmsr,Uiest fClfri,Bree,C 42;and 94720; CA Berkeley, California, of University Chemistry, of Department Chemistry, Theoretical for Center Pitzer S. Kenneth 2 nlzn hmclbnsi oecmlxmlclshsalso has molecules complex more in bonds chemical Analyzing h hmclbn a rgnlyvee 6 sbigelectro- being as (6) viewed originally was bond chemical The sobtlcnrcin hc sms aiyse yoptimizing by seen easily most is which contraction, orbital is , I ie,teZn the dimer, n hoeia hmss h praho h synthetic the of approach The synthetic chemists. both to theoretical central and is bond chemical the nderstanding | oaetbonds covalent I a,b –Zn 2 + eodr feti oecss uha in as such cases, some in effect secondary A . n atnHead-Gordon Martin and | I hresitbonds charge-shift odi iiccn,adteM–nbond Mn–Mn the and dizincocene, in bond | hmclbonding chemical | est functional density a,b,1 2 + n H and I 2 – 1073/pnas.1715763114/-/DCSupplemental at online information supporting contains article This 1 the under Published interest. of conflict no declare authors University.The Cornell R.H., and Marburg; of paper. University the G.F., wrote Reviewers: M.H.-G. and D.S.L. and data; analyzed M.H.-G. fsnl oaetbns(6.Temto,wihrdcst the includes to (24), reduces interactions which noncovalent method, for scheme The ALMO-EDA (26). analysis bonds variational covalent the single to of scheme ALMO-EDA the of extension terms. other the from 25), effects the with contaminate 20, terms leads which EDA (2, functions, methods wave bonds these broken to of spin-symmetry to nature applied nonco- single-determinant been of the have number although methods A EDA (21–24). coworkers valent and Head-Gordon (ALMO-EDA) of analysis localized decomposition absolutely energy the orbital and molecular (14), EDA function Block- Wave based the Localized (20), (18) Morokuma (ZR-EDA) method and EDA Ziegler–Rauk the Kitaura Natural (19), include EDA the methods and Variational method NBOs. on Perturba- 17) Adapted (16, Symmetry Theory popular Perturba- tion the optimization. include perturbative methods variational tive either methods constrained by These or energy 15). approaches interaction 14, the (2, separate interactions typically noncovalent of nature (4). Many interpretable density. sums physically into are energy energy that bond kinetic terms a of the partitioning is for and exist which also density methods function, other the localization of topo- electron function an Another the a partitioning terms. is as interatomic approach well and logical as intraatomic density into electron energy the in bond so-called points by bonds critical of presence the atoms-in-molecules describes of (13) theory (QTAIM) quantum The natural bonds. orbitals, hybridization chemical used on localized and information widely provides and The structures, (12) (11). Lewis approach predominant value (NBO) its orbital of bond example F the specific the of a by emergence is exemplified The qual- paradigm, (10). extracting bond” for concepts “charge-shift suitable bonding are chemical that itative functions wave simple ally uhrcnrbtos ..G eindrsac;DSL efre eerh ...and D.S.L. research; performed D.S.L. research; designed M.H.-G. contributions: Author owo orsodnesol eadesd mi:[email protected]. Email: addressed. be should correspondence whom To hmssfo h unu ehncllnug ftheorists. of language mechanical quantum the synthetic from of an chemists language providing the We bonds, in fingerprint bonds. single chemical characterize interpretable covalent to studying EDA EDA for an existing developed designed Most not insight. are practical methods and the- calculation between bridge oretical a provides lan- (EDA) Energy analysis syntheses. different readily chemical decomposition a improve not and use understand are who to results guage chemists the synthetic energies accuracy, by the high interpretable calculate with can molecules today of chemists theoretical While Significance oadesti hleg,w eetyrpre spin-pure a reported recently we challenge, this address To the elucidating at successful very been have schemes EDA NSlicense. PNAS . www.pnas.org/lookup/suppl/doi:10. b NSEryEdition Early PNAS hmclSciences Chemical 2 molecule, | f8 of 1

CHEMISTRY INAUGURAL ARTICLE 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 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 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. We have moved ∆EHYBRID here for that reason putational effort than wave function methods. The purpose of and because it can be much larger than the other contributions this paper is to recast our bonded ALMO-EDA method into a to POL; therefore, its presence in POL can obscure trends in single-determinant formalism that allows the computation of a POL and SC across rows of the as the hybridization dynamically correlated bonded EDA with any existing density of the radical fragment changes. Even with ∆EPREP so defined, functional. After outlining our approach, we turn to the charac- one must still reorient the frozen orbitals to resolve degenera- terization of a variety of chemical bonds, ranging from familiar cies and obtain the correctly oriented α density as previously systems to less familiar dispersion stabilized bonds, and several described (26). single metal–metal bonds. Nevertheless, regarding orbital rehybridization as part of preparation (as we do in this paper) and regarding orbital rehy- Variational EDA bridization as part of POL [as was done previously (26)] are The single bond of interest is, by definition, the difference both defensible choices. Also, in cases where ∆EHYBRID is large, between the DFT calculation on the molecule and the sum of the consequences of where it is placed can be considerable, as DFT calculations on the separately optimized, isolated frag- ∆EPREP, ∆EFRZ, ∆ESC, and ∆EPOL all change as a result. There- ments. This interaction will be separated into five terms: fore, the reader can compare and decide whether they agree

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) −99.8 H–Cl 13.1 219.0 −253.7 (74.7) −49.3 (14.5) −36.7 (10.8) −107.5

F–SiF3 41.5 259.1 −227.3 (48.9) −119.0 (25.6) −118.8 (25.5) −164.4 F–F 9.2 186.3 −124.1 (52.7) −37.2 (15.8) −74.3 (31.5) −40.1 Li–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.

2 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.1715763114 Levine and Head-Gordon Downloaded by guest on October 3, 2021 Downloaded by guest on October 3, 2021 u oso eeaiy,ltu sueta ohrdcl have radicals both that assume with- us (but let simplicity generality), For interacting S CT. of fragments or loss radical POL, out two SC, permitting the without with associated change eieadHead-Gordon and Levine concatenat- block-diagonally “frozen” by The constructed constraint. ALMOs (ALMO) occupied orbital iso- molecular the localized MOs, do hence, so and fragment fragments, (AOs), lated into orbitals atomic partition in corresponding atoms block-diagonal their the is Since matrix fragments. coefficient the (MO) what orbital than molecular terms method. energy ALMO-EDA CAS(2,2) frozen original the smaller in using in calculated property, is result correlation dynamic should a DFT effects, dispersive of addi- repul- The tion Pauli dispersion. and of electrostatics, exchange–correlation, interfragment because repulsion, Pauli from bond contributions chemical includes a It typ- will sion. for and repulsive interaction nonbonded be a ically entirely is term This relax. (S single-determinant function triplet spin-pure wave a form to combined are tions Energy. Frozen (and POL Appendix. rehybridization of with part tables data as corresponding choice; our with term. SC + FRZ one into summed terms SC and FRZ the with bonds 2. Fig. 1 = e fobtl ssi ob asltl oaie”i the if localized” “absolutely be to said is orbitals of set A /2; D ffis o element–H row first of EDA (B) bonds. element–H row first of EDA (A) M S +1 = h eodtr nEq. in term second The 1; = nteFZeeg,tefamn aefunc- wave fragment the energy, FRZ the In /2. 284.6 32.2 188.8 H–F H–OH 0.1 H–NH H–CH H–BH H–BeH H–Li odPE FRZ PREP mole) H–H per kilocalories (in bonds E–H row first Bond of EDA 2. Table ubr nprnhssaetepretgso h oa tblzn neato energy. interaction stabilizing total the of percentages the are parentheses in Numbers M ∆E T 3 2 2 S uoaial aif h absolutely the satisfy automatically , +1 = PREP ihu loigteobtl to orbitals the allowing without ) ∆E = 279.4 27.9 293.3 284.6 32.0 15.4 276.1 3.7 34.8 0. 245.4 0. GEOM 1, ∆ r nlddin included are ) E FRZ steenergy the is , (60.0) −268.7 (70.3) −310.9 (87.1) −252.3 (78.1) −344.5 (86.2) −356.3 (89.0) −349.0 (88.0) −311.4 (53.7) −51.2 SC SI ftesi i saboe ymty(S LOdeterminant. ALMO (BS) E symmetry broken result sin- energy, a the (bonding) is Its formalism, flip to spin single-determinant triplet the the bonding) of in (not from However, SC glet. the change to the (α spin the SC). + (FRZ term orbital together frozen grouped total be a may SC surface into primarily and triplet FRZ are the supersystem), to initial we opposed the covalent (as of because surface and singlet with the reason in associated interested this regime For formation. overlapping strongly bond the typically strongly is typically in is FRZ SC attractive repulsion), while Pauli but by (dominated orbitals, repulsive frozen H LS with in even the uated present require is (in not and does spins delocalization interference that two (HS) out function pointing worth high-spin is wave It the case). constructive [in and low-spin interference between case] to difference function the triplet wave for high-spin accounts destructive SC energy from SC the electrons The changing singlet. radical (LS) is, two that the pairing: of electron by caused difference Energy. Spin-Coupling (23). dispersion and repulsion, Pauli interactions, con- electrostatic into permanent separated to further corresponding be tributions may term FRZ ALMO-EDA This the is energy interaction fragments: prepared frozen noninteracting, The to DFT. relative difference or HF by computed although deformation, matrix, density frozen the (ALMOs tion, over- orthogonal an have be matrix, they not therefore, lap and need nonorthogonal), generally fragments are RO isolated ing S ae neatytesm a steenergy: the as way same the exactly in nated obtain To o Ssnl eemnn ihnnrhgnlobtl sgiven is orbitals nonorthogonal with in determinant single BS a for calculated the Using HS LS rmteH rzndtriat(S determinant frozen HS the From (E density this with associated energy The ,w ov for solve we Appendix), SI u loasnl otmnn,wihi HS: is which contaminant, single a also but , M = (12.3) −55.1 (8.3) −36.7 (5.1) −14.6 (19.8) −18.9 (9.2) −40.8 (6.3) −26.0 (5.5) −21.4 (8.6) −30.6 S S LS au yoe(i.e., one by value POL hS 1 + c → σ 2 eeaietevleof value the examine we , E i hrfr,a ucino nefamn separa- interfragment of function a as Therefore, . BS emywrite may We . BS β c foeo h w afocpe riasreduces orbitals half-occupied two the of one of ) otistedsrdsi-ope Senergy, LS spin-coupled desired the contains , (1 = = = E hS ∆E hS BS hS z (27.7) −124.2 h hr emi Eq. in term third The − i (21.4) −94.5 (7.8) −22.7 (26.5) −25.3 (12.7) −55.9 (7.5) −30.9 (5.5) −21.6 (3.4) −11.9 2 FRZ LS T (1 = 2 i c i BS (hS sconstant. is )hS BS CT M = hS − 2(hS au adrvto fthe of derivation (a value z S − 2 c i E +1 = i LS i Eq. via LS c FRZ P z z )E )+2c + 1) + i FRZ i + LS LS 2 LS + − c (hS → 1) + ieFZ Cwl eeval- be will SC FRZ, Like . = hS + X frags 5: hS −125.2 −100.6 −140.8 −115.9 −113.2 −112.2 −108.4 Z Tσ z M 2 Sum −60.6 cE i i 1; = LS 2 S E NSEryEdition Early PNAS HS −1 i HS (hS 0 = 1, Z BS 1) + . T . ∆E hc scontami- is which , M z † FRZ .Teojcieis objective The ). i nege Pauli undergoes , S LS . E SC +1 = a hnbe then may ) HS 1). + steenergy the is , = hS ,flipping ), E 2 FRZ i | value f8 of 3 and [6] [3] [4] [5]

CHEMISTRY INAUGURAL ARTICLE FERFs are the subset of virtual orbitals that exactly describe the linear response of each fragment to an applied electric field. Fol- lowing previous work (22, 24), the dipole and quadrupole FERFs will be used to define the fragment virtual spaces for electrical polarization. For a hydrogen atom, the three-dipole functions are p-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). The frozen orbitals may contract toward the nucleus to lower their energy without any induced electrical moments. This contraction effect was first identified by Ruedenberg (1) as part of his clas- + sic analysis of the one-electron chemical bond in H2 . We have shown (31) that orbital contraction can be accurately modeled by adding a so-called monopole function to the FERF virtual space for each occupied orbital. On the H atom, the monopole FERF is a 2s-type function. The overall FERF basis is thus of monopole, dipole, quadru- pole (MDQ) type, and thus, CON ELE ∆EPOL = ∆EPOL + ∆EPOL , [8] CON ELE where (31) ∆EPOL = EALMO/ M − EFRZ and ∆EPOL = EALMO/ MDQ − EALMO/ M. Our results showed that orbital con- traction was very important in bonds to hydrogen but rather insignificant in bonds only involving heavier elements (31). This decreased energy lowering in heavy element bonds can be viewed Fig. 3. Comparison of EDA terms computed at HF, PBE0-D3, ωB97X-D, and as arising from diminished violation of the virial theorem on SC ωB97M-V/aug-cc-pvtz of (A) ethane and (B)F2. with frozen orbitals relative to bonds to hydrogen. The additional mathematics necessary to implement POL in conjunction with the approximate DFT spin-projection method In turn, this permits us to solve Eq. 4 for the spin-pure energy, is described in SI Appendix. −1 ELS = αEBS +(1−α)EHS, where α = (1 − c) . This result corre- Charge-Transfer Energy. The fifth term, ∆ECT, contains CT con- sponds to Yamaguchi’s spin-projection scheme (28, 29). Finally, tributions, allowing electrons to move between the fragments. E with LS in hand, the SC term is given by It is the dominant term in ionic bonds and an important part

∆ESC = ELS − EFRZ, [7] of charge-shift bonds (11). Mathematically, we will release the ALMO constraint and reoptimize the orbitals to obtain where the orbitals are still the frozen fragment ones. an unconstrained spin-projected energy. Implemented with HF The above derivation is exact, because a spin-pure ELS is determinants, this gives the CAS(2,2)/1PP energy. With DFT, we 2 obtained if hS iBS and hEiBS are evaluated consistently from obtain the approximately spin-projected DFT analog ESP-DFT. the same one-particle density matrices and two-particle density However, as already mentioned, in DFT, the hS 2i value used matrices (2PDMs). An example is the case of HF wave func- in the optimization is only approximate. Moreover, the approxi- tions. Unfortunately, this condition is not strictly satisfied for mate exchange–correlation functional accounts for some amount Kohn–Sham DFT, because the interacting 2PDM is not available of static correlation. Hence, E obtained at this final step 2 SP-DFT (30), and thus, the value of hS iBS corresponding to hEiBS is not is generally lower than the DFT energy of a single determinant, available. This dilemma arises because the fundamental theo- EDFT. Since DFT functionals are typically developed (or fitted) to rems of DFT allow construction of the exact ground-state energy produce accurate results only as a single determinant, this over- without knowledge of the 2PDM. The best that can be straight- counting of correlation leads to molecules being slightly over- forwardly accomplished is to use the noninteracting 2PDM bound (on the order of 1–15 kcal/mol). To address this issue, 2 (i.e., from the Kohn–Sham determinant) to evaluate hS iBS in we simply rescale the terms calculated on the approximately DFT. For any functional but HF, this choice leads to a small inconsistency 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 term in EDA schemes such as the ZR-EDA approach (20) used extensively for bonding analysis (2). Such EDAs form only one frozen supersystem on the LS surface rather than separate FRZ and SC terms. The resulting LS frozen energy is exactly EBS above. From the analysis above, because of spin contamination, ∆EFRZ(BS) > ∆EFRZ + ∆ESC.

Polarization Energy. The fourth term in Eq. 1, ∆EPOL, arises partly from the orbitals (with low spin coupling) relaxing because of the presence of the field of the other fragment. ∆EPOL is the term that includes contributions from polarization in the bond, but the ALMO constraint prevents CT contributions. To provide a well-defined basis set limit, fragment electric response func- Fig. 4. Comparison of the EDA of the E–E bond (E = F, Cl, Br) computed at tions (FERFs) are used as the ALMO virtual basis (22, 24). The HF and ωB97M-V/aug-cc-pvtz (Table 3).

4 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.1715763114 Levine and Head-Gordon Downloaded by guest on October 3, 2021 Downloaded by guest on October 3, 2021 eieadHead-Gordon and Levine 1PP] or [CAS(2,2) multideterminant with described as Eq. the obtained previously in from are the (apart defined energy) results hybridization as EDA the energy identical of reclassification used, POL is total HF the when (31), Finally, only elsewhere report discussed been we has here energy contraction in is the breakdown since report (the we here supersystem. components, sum two their unrestricted has only [2] energy an preparation the using forming While out carried fragments was with term frozen unrestricted used the frozen of functional Decomposition the HF. dispersion-free of was the decomposition (23), For (35). term energies bond tests chemical Numerical specified. otherwise that suggest unless aug- (36) the basis meta- and cc-pVTZ (RSH) (meta-GGA)] approximation hybrid gradient range-separated generalized [a the calcula- (35) using all functional performed for were density used calculations was All 4.4 (34). Q-Chem tions of version development A Details Computational FERF the with constraints. fragments space the con- Hilbert ionic ALMO to of and unavailable fraction was the that measures then tribution is result term and HF CT covalent, the The systems, purely obtained. symmetrical is for state fully, overlap spin-coupled they the when do all, spaces occupied the at of singly overlap of the amount not when nonorthogonality some cases, limiting includes the the In term of orbitals. SC because The contribution term. ionic-like CT the of nature describe additionally could projection methods spin systems. such the diradicaloid and strongly because exact, theory, be cluster would coupled theory would or perturbation rescaling Møller–Plesset no (MP2) second-order contrast, for In 33). needed (32, done be often calculations diradicaloid quite DFT approx- is strongly the BS as use in in such to directly, preferable as result be spin-projected may such imately it inad- cases, bond, such itself the In is molecules. describe determinant DFT to single equate a whenever bonds lyzing Eq. satisfies exactly energy interaction in tabulated final the that so as factor defined a by c surface LS spin-projected term each that energy interaction stabilizing total represents. the of percentages the 116.4 .8 154 SC 4. 186.3 6.8 Br–Br 9.2 Cl–Cl FRZ F–F halogen) = PREP (E bonds E–E Bond of EDA 3. Table R seilyfrsmerclsses n a nur bu the about inquire may one systems, symmetrical for Especially ana- for inadequate be will scheme rescaling this DFT, For are parentheses in Numbers mole. per kilocalories in are energies All ∆ E SC , ∆ IAppendix. SI ω E 9MVi mn h otacrt vial for available accurate most the among is B97M-V POL ← FZi h rzneeg estedseso energy. dispersion the less energy frozen the is *FRZ (23). decomposition 274.1 46.5 Diamantane (ω ethane and bis(diamantane) Bond of EDA 4. Table H 2. (54.7) −121.5 (52.7) −124.1 c 89(57.7) −98.9 3 c R C–CH l nrisaei ioaoisprml.DS stedseso nrya eemndb h frozen the by determined as energy dispersion the is DISP mole. per kilocalories in are energies All R = ∆ E (E 3 POL (E SP-DFT DFT and , 28(23.8) −52.8 (15.8) −37.2 07(23.7) −40.7 RPFZ DISP FRZ* PREP − 76285.9 37.6 − E POL E FRZ ∆E FRZ c .Likewise, Appendix). SI ) R CT ) uhthat such , , ← −74.3(31.5) 79(21.6) −47.9 20(18.6) −32.0 c R 04(15.3) −60.4 ∆ TSum CT . (2.0) −8.3 E CT ω ω B97M-V B97M-V ∆ . 1. E c −50.8 −60.6 −40.1 SC c R R [9] ← 8. is is 4. (81.9) −346.5 (66.8) −264.2 B97M-V/6-31+G**) aiainta otiue otsrnl otesaiiyo bonds of stability delo- the electron to greater strongly most with contributes associated that calization correlation dynamic of therefore, is inclusion and term, it molecules, CT the these increases mostly in cor- correlation dynamic Moreover, obtain to energies. important bond very rect indeed is F as correlation halogens dynamic such that homoatomic the bonds, and of HF-EDA studying the V-EDA Comparing for 4). Hence, (Fig. halogens (40). necessary in other those is each near correlation pairs as lone dynamic such many high, with is density molecules electron in local the when pronounced most Halogens. in Bonding and discrepancies 3 and small Fig. the ethane in Overall, in functionals evident exact. are between bond none differences C–C as Some inevitable, the fluorine. are for in to trends bond F–F relative these the stabilization shows of 3 POL inclusion Fig. and the HF. CT to increases (owing and energy avail- dispersion) frozen are the above decreases described bonds relation the of in all with able for (39) VV10] meta-GGA with (35) [a and B97M-V correlation], nonlocal (38)], VV10 GGA RSH compar- [a corrected tables PBE0-D3 (37)], GGA), data GGA not dispersion-corrected hybrid Full is dispersion-corrected (a functional. of method BLYP-D3 of this inclusion HF, that choice the ing check to by to sensitive altered as too well are as terms correlation HF-EDA dynamic the how mine large a is CT binding. HF, and of water source in character significant, moder- ionic becomes the increasing POL in with ammonia, 2B). and contrast, in (Fig. By interactions bond energy. covalent term SC bond polar + SC ately the FRZ of + most total FRZ for bonds, total account covalent and a nonpolar FRZ with into the the bonds summed when For obvious are covalent most terms polar is SC change elements to This from the bonds CT. switch row, increasing covalent bonds first E–H nonpolar the the being and across 2 electronegative, right (Fig. more Moving investigated become 2). were To bonds Table table. element–H and periodic the row of first groups illustrate, down and periods across trends Bonds. Element–H Row First (5). methods classi- mechanical recovers quantum thus from bonds EDA concepts ionic The bonding cal energies. have and CT bonds charge-shift high polar and relatively a have energies, energies, gives SC POL EDA high high The relatively relatively 1). charge- have Table and covalent bonds and bonds: shift 1 of classes (Fig. different bond) for “fingerprint” ionic (an LiF the in bond), covalent F polar in (a bond SiF HCl in in bond bond F–Si H–Cl the bond), the lent with bonds representative some ω investigating by EDA the of Bonds. Representative Discussion and Results energy Hartree–Fock as (26). to (HF-EDA) analysis referred decomposition hereafter is which method, SC 9MVfntoa:teCCbn nehn annoa cova- nonpolar (a ethane in bond C–C the functional: B97M-V ait fdniyfntoaswr netgtdt deter- to investigated were functionals density of variety A eeal paig diino yai cor- dynamic of addition speaking, Generally Appendix. SI 2 annoa,cag-hf od,adteL– bond Li–F the and bond), charge-shift nonpolar, (a 21(5.2) −22.1 (5.8) −23.1 POL 4 aplrbn ihinccaatr,teF–F the character), ionic with bond polar (a IAppendix SI yai orlto fet nbnsare bonds in effects correlation Dynamic efis eiytebhvo fteterms the of behavior the verify first We 64(11.0) −46.4 (12.1) −47.9 hsmto losu oinvestigate to us allows method This TSum CT ω peracceptable. appear 9MV[nRHmeta-GGA RSH [an B97M-V −99.7 −75.1 ω 2 NSEryEdition Early PNAS Cl , 9XD[ dispersion- [a B97X-D 2 n Br and , 2 ω esee we , B97M- | f8 of 5

CHEMISTRY INAUGURAL ARTICLE Ar Ar Table 6. EDA of select bonds (ωB97M-V/aug-cc-pvtz) with BS OOO N N C C C O EDA methods N Mg Mg N C + − N N Zn Bond H–H H3C–CH3 F–F Li–F Li –F OC Mn Mn CO Ar Ar Zn Prep 0.0 17.9 0.0 0.0 45.6 C C C O C Elec 3.9 −135.1 −41.9 −17.0 −202.2 O O O Ar = 2,6-Me2C6H3 Pauli −10.9 194.1 165.2 55.1 42.9 Orb −101.4 −175.5 −161.2 −176.1 −24.3 Fig. 5. Metal–metal bonds probed by the ALMO-EDA. All energies in kilocalories per mole.

between halogen atoms as might be expected from a charge-shift The total bond energies obtained for the Mn–Mn bond are ≈ bond. Dispersion plays only a modest role ( 9 kcal/mol) in these in close agreement with experimental measures (40.9 vs. 38 ± molecules. This charge-shift bonding, in which ionic structure 5 kcal/mol) (52) and previous calculations (53, 54). There are contributes significantly to the ground state, is manifest in the no direct experimental measures for the given Mg–Mg and Zn– chemistry of halogens [for example, in the stabilization of charge- Zn bonds. The Mg–Mg bond in ClMgMgCl was extrapolated separated species during the formation of halonium ions in the from experimental measurements to be 47.1 kcal/mol, in close halogenation of alkenes (41)]. agreement with the results obtained here (55). Experimental Zn–Zn bond dissociation energy (BDE) measurements were Dispersion-Assisted Bonds. In molecules with bulky side groups, obtained from the homoatomic dimer (which is, in principle, dispersion can play a significant role in the stabilization of bonds doubly bonded), but the measured BDE is relatively close to ˚ (42, 43). An example is the elongated 1.65 A C–C bond in the Zn–Zn bond calculated here (82.2 vs. 93.7 kcal/mol) (56). bis(diamantane), which, based on , is expected to Both classes of bonds have previously been studied by theoreti- have a bond strength of ≈40 kcal/mol (44). Experimentally, it is ◦ cal methods (refs. 50 and 57 and references therein). considerably stronger (showing no decomposition up to 300 C), The EDA results are given in Table 5. The Mg–Mg bond turns which has been attributed to many stabilizing dispersive interac- out to be a classic nonpolar covalent bond analogous to H2: tions between the interfacial C–H bonds (45). Comparison of the the bond strength is mainly caused by SC (i.e., electron pair- EDA for this bond vs. ethane (at the ωB97M-V/6-31+G** level) ing) between the unpaired electrons on Mg(I) centers. There allows quantification of the forces that stabilize it as shown in is almost no CT: consistent with the high reduction potential Table 4. of Mg(0), Mg(0)–Mg(II)/Mg(II)–Mg(0) contributions are not As seen in Table 4, while the POL and CT terms are fairly simi- important in this bond. This is consistent with NBO calculations lar for ethane and bis(diamantane), the SC term is 82 kcal/mol less carried out in the initial disclosure of this molecule, which found for bis(diamantane) following expectations based on bond length. the bond to be a covalent single-bond dominated by s-orbital However, the bis(diamantane) bond is not that much weaker contributions (46). than ethane, because it also has a less destabilizing FRZ energy. However, the less reducing Zn in the Zn–Zn bond, which Applying the ALMO frozen decomposition, it was found that dis- is principally covalent, does exhibit some ionic Zn(0)–Zn(II)/ persion accounts for this large increase in bond strength [60.4 Zn(II)–Zn(0) resonance contributions, much like in ethane. vs. 8.3 kcal/mol of stabilization for bis(diamantane) and ethane, These ionic contributions account for most of why the Zn–Zn respectively]. Enhanced dispersion is the key factor in account- bond is stronger than the Mg–Mg bond. Our method provides ing for the unusual stability of bis(diamantane), even as the bond an accurate dissection of the metal–metal bond in multimetal- elongation is a result of partially relieving the close contacts. locenes, which was not possible before (58, 59). It also gives insight into the origins of the relative bond strengths in metal–metal Metal–Metal Bonds. We next consider some single bonds, which bonds: although both Mg–Mg and Zn–Zn bonds have strong cova- are less well-studied: main group and transition metal metal– lent stabilization, the more easily oxidized Zn is further stabi- metal bonds. We investigated a slightly truncated version of the lized by ionic resonances, making it a much stronger bond. This Mg–Mg dimer of Jones and coworkers (46), the Zn–Zn bond in was hinted at in a recent QTAIM study, which showed that main dizincocene from Carmona and coworkers (47), and the classic group–main group bonds in M2Cp2 had more “covalent char- Mn–Mn bond (48, 49) in the dimanganese decacarbonyl complex acteristics”, while transition metal–transition metal bonds had (Fig. 5). The relatively new Mg–Mg and Zn–Zn bonds are inter- “closed shell ionic characteristics” (60). esting for their novelty and have proven to be important chem- By contrast, the bond in dimanganese decacarbonyl is a ical synthons (50, 51). However, the nature of these symmetri- charge-shift bond much like in F2, with CT playing a major cally bonded complexes is difficult to guess on first inspection role in stabilizing the bond. Previous studies using QTAIM have because of our unfamiliarity with the chemistry of Mg and Zn in also implicated “closed shell interactions” and indicated that the the formal +1 . Will these be conventional cova- bond is intermediate to a covalent and ionic bond (61), while lent bonds, or will they have charge-shift character? These sys- other studies favor a more covalent picture (62). This hybrid tems are, therefore, good candidates for use of the EDA, because covalent–CT stabilization is quantified here and appears anal- we can compare their EDA results against the well-understood ogous to the charge-shift bonding picture (11). systems presented earlier. Comparison with Other EDA Methods. The main advantage of our EDA over alternatives for studying covalent bonds is its full Table 5. EDA of metal–metal bonds [ωB97X-D/6-31+G** with use of valid, spin-pure intermediate wave functions. By con- basis set superposition error (BSSE) correction] trast, the Morokuma EDA (19), the ZR-EDA (2, 25), and the Bond PREP FRZ SC POL CT Sum extended transition state-natural orbitals for chemical valence (ETS-NOCV) (20) method use BS spin-contaminated inter- Mg–Mg 0.1 69.7 118.3 (97.3) 3.1 (2.5) 0.2 (0.2) −51.8 − − − mediate wave functions to determine the energy components, Zn–Zn 4.1 103.5 150.0 (82.7) 7.6 (4.2) 23.7 (13.1) −73.7 − − − which are likewise nonphysically spin-contaminated. This prob- Mn–Mn 3.5 33.2 37.4 (51.6) 8.0 (11.1) 27.0 (37.3) −35.7 − − − lem does not arise for intermolecular interactions, which are typ- All energies in kilocalories per mole. ically between closed shell fragments. In such cases, BS solutions

6 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.1715763114 Levine and Head-Gordon Downloaded by guest on October 3, 2021 Downloaded by guest on October 3, 2021 1 hi ,Dnvc ,W ,HbryP 20)Cag-hf odn n t manifes- its and bonding Charge-shift (2009) PC Hiberty W, Wu D, Danovich S, Shaik (2007) 11. PC Hiberty S, Shaik 10. 2 edA,CrisL,Wihl 18)Itroeua neatosfo natural a from interactions Intermolecular (1988) F Weinhold LA, Curtiss AE, Reed 12. 7 oesenE,Serl D(02 aeucinmtosfrnnoaetinterac- noncovalent for methods Wavefunction (2012) CD Sherrill EG, Hohenstein inter- to 17. approach theory Perturbation (1994) K Szalewicz R, Moszynski B, Jeziorski 16. analysis decomposition Energy (2015) CK Skylaris CS, Tautermann T, Fox MJS, block-localized Phipps a 15. on based analysis decomposition Energy (2011) J Gao P, Bao Y, molecules. Mo in Atoms 14. (1985) RFW Bader 13. 1 hlulnR,CbrE,Lca C elA,Ha-odnM(07 naeln the Unravelling (2007) M Head-Gordon AT, Bell RC, Lochan EA, decomposi- Cobar energy RZ, and Khaliullin charge combined 21. A (2009) T Ziegler molecu- A, Michalak for MP, scheme Mitoraj decomposition 20. energy new A (1976) K Morokuma K, Kitaura density 19. to Extension analysis: decomposition energy Natural (2005) ED Glendening 18. 2 onP,Ha-odnM(05 oaiaincnrbtost nemlclrinter- intermolecular to contributions Polarization (2015) M Head-Gordon PR, Horn 22. ,H H−H, ZR-EDA/ETS- for the terms comparison, energy For NOCV bonds. chemical of a classes in terms Orb presented methods. and BS-based method Pauli, these the Elec, in manner of the system-specific energy between distributed SC is The here molecule. of a effect energy the for the hence, among terms distributed and inconsistently shell), is contamination closed the and state final spin-pure the typically (e.g., function is wave the of optimization variational corresponds EDAs. energy earlier frozen these in ALMO-EDA that the to directly and enter, not do Conclusions result the the naturally. method, how out our rela- knowing falls in whereas a requires prepared, noting this be with should However, and fragments recovered term. fragments be Orb the small can ionizing tively LiF first of by ZR-EDA picture the ionic of error The self-interaction functional. vs. contamination the unclear spin H of the is because of BS is value It sion of negative unphysical interactions. evident utility the of bonded is the much how fingerprint decrease understanding chemical significantly for no which methods that data, Note these 6. from Table in given eieadHead-Gordon and Levine .Reebr 16)Tepyia aueo h hmclbond. chemical the of nature physical The (1962) K Ruedenberg 1. .CrooT,Nsiet A 20)Eeg attoigfrgnrlzdproduct generalized for partitioning Energy (2009) MAC Nascimento TM, Cardozo drive the 9. by created the are of bonds Covalent role (2014) K fundamental Ruedenberg The J, Ivanic bonding: MW, Schmidt Covalent 8. 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Waals der van of surfaces 1887–1930. energy potential molecular patterns. interaction protein–drug Rev prototypical Soc on Chem evaluation their and approaches theory. functional density 13:6760–6775. multistate and wavefunction rgno nemlclritrcin sn boueylclzdmlclrorbitals. molecular localized A absolutely Chem using Phys interactions intermolecular of origin analysis. bond for scheme tion approximation. Hartree-Fock the 340. within interactions lar clusters. water in effects A cooperative of analysis and methods functional cin eiie ihfamn lcrcfil epnefunctions. response electric-field fragment with 143:114111. revisited actions nEAmto o igebnshsbe eeoe na DFT in of developed use been allows has EDA bonds This formalism. single single-determinant for method EDA An 109:11936–11940. IE optMlSci Mol Comput WIREs 2:43–62. WlyVH enem Germany). 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Comput bonding. chemical of analysis 1–86. pp NJ), Hoboken, (Wiley, DB Boyd chemistry. understanding and molecular localized absolutely using analysis orbitals. decomposition energy generation ond calculations energies. theory interaction functional intermolecular density of in dispersion and repulsion, Pauli electrostatics, rdetapoiaindniyfntoas h erhfrB97M-V. for search The functionals: 142:074111. density approximation gradient corrections. dispersion 6620. atom–atom damped with correction. dispersion range hydrogen. and neon through boron atoms The correla- nonlocal VV10 with functional tion. density meta-GGA hybrid, range-separated package. program 4 Q-Chem formulations. new and Phys insight Chem Deeper calculations: theory functional sity groups. prosthetic bioinorganic Biol of Mol properties magnetic Methods and electronic to applied ogy bonding. ical B systems. di-chromium to application 450. its and method and diradicals polyradicals. singlet for wavefunctions Møller–Plesset and theory. 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Waals der van eosdrn trceffects. steric reconsidering biii ehd—eaecretywrigo addressing on working correlated issues. currently sin- to the those are approach to inhibit methods—we EDA restriction initio challenges single-bond ab its technical this is of Only EDA extension bonds. this chemical of gle limitation main bonds. main The metal in transition seen and strengths metal bond of group range the suggests for and reasons synthesized physical been have has that method bonds Mn(0)-Mn(0) this and Zn(I)-Zn(I), with Mg(I)-Mg(I), of bonds characterization permitted metal–metal illustrated single as of dispersion meaningful Analysis bis(diamantane). on of for heavily example rely the allows by that effects bonds dispersion of study of inclusion Comparisons The single halogens. first a with in the made be functional. bonds including all should the bonds, molecules different and the single between the bonds to of improved E–H variety relative has row a and it of determinants, functional, too HF treatment of not uncorrelated is choice of EDA use the DFT-based to the that sensitive show tests Numerical dis- as the such for effects, persion. correlation allowing dynamic of thereby inclusion projection, efficient spin approximate with 12:2111–2120. ,Guih N, e ´ hmPhys Chem J ,McaldsA(02 esetv:Avne n hlegsi treating in challenges and Advances Perspective: (2012) A Michaelides J, s B d(1991) ed ZB, c ˇ ´ 12:4812–4820. hsCe Phys Chem Phys 17:14375–14382. hmPy Lett Phys Chem r ,MliuJ 21)Si eotmnto fboe-ymtyden- broken-symmetry of decontamination Spin (2015) JP Malrieu N, ery ´ hsCe Lett Chem Phys J 144:214110. 1122:269–296. rai Chemistry Organic hoeia oeso hmclBonding Chemical of Models Theoretical hswr a upre yGat H-651 and CHE-1665315 Grants by supported was work This 18:23067–23079. ne hmItEd Int Chem Angew optChem Comput J 149:537–542. o Phys Mol eiw nCmuainlChemistry Computational in Reviews 8:1967–1972. 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