Symmetry Laws Improve Electronegativity ... - ACS Publications

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Symmetry Laws Improve Electronegativity Equalization by Orders of Magnitude and Call for a Paradigm Shift in Conceptual Density Functional Theory László von Szentpály* Institut für Theoretische Chemie, Universität Stuttgart, Pfaffenwaldring 55, D-70569 Stuttgart, Germany ABSTRACT: The strict Wigner−Witmer symmetry constraints on chemical bonding are shown to determine the accuracy of electronegativity equalization (ENE) to a high degree. Bonding models employing the electronic chemical potential, μ, as the negative of the ground-state electronegativity, χGS, frequently collide with the Wigner−Witmer laws in molecule formation. The violations are presented as the root of the substantially disturbing lack of chemical potential equalization (CPE) in diatomic molecules. For the operational chemical potential, μop, the relative deviations from CPE fall between −31% ≤ δμop ≤ +70%. Conceptual density functional theory (cDFT) cannot claim to have operationally (not to mention, rigorously) proven and unified the CPE and ENE principles. The solution to this limitation of cDFT and the symmetry violations is found in substituting μop (i) by Mulliken’s valencestate electronegativity, χM, for atoms and (ii) its new generalization, the valence-pair-affinity, αVP, for diatomic molecules. Mulliken’s χM is equalized into the αVP of the bond, and the accuracy of ENE is orders of magnitude better than that of CPE using μop. A paradigm shift replacing the dominance of ground states by emphasizing valence states seems to be in order for conceptual DFT.

1. INTRODUCTION The Wigner−Witmer symmetry combination “rules” state that the symmetry states of molecules and their dissociation products must be compatible.1 For example, the ground states 3 P3/2 of Br+ (Br+GS) and 1S0 of F− (F−GS) cannot contribute to the 1Σ+ ground state (GS) of the polar molecule Brδ+Fδ− because a triplet and a singlet cannot combine to a singlet state.1,2 For BrF, one of the reference states upon bond breaking has to be an excited singlet state of Br+, either 1D2 or 1 S0, or the averaged valence state (VS), Br+VS.1−7 In fact, the Wigner−Witmer rules (hereafter referred to as Wigner− Witmer “laws”) establish strict symmetry constraints on bond formation and bond breaking processes, and no exceptions have been reported.2 Theoretical concepts, structural principles, and bonding models must be consistent with them. This is here examined for electronegativity (EN), the most accepted ordering concept of chemistry.3−50 The EN concept has been intimately linked with density functional theory (DFT).13,46−50 Parr and Yang characterize DFT as “a theory of ground (equilibrium) electronic states in which the electronegativity of chemistry plays in the basic variational principle just the role that the energy plays in the basic variational principle of wave-function theory.”13 Conceptual DFT (cDFT) is essentially based on the assumption that the exclusive use of ground-state properties is correct and sufficient to offer widely applicable schemes. This basic conception will be hereafter called the “ground-state paradigm” of cDFT. A paradigm is a set of tacit preconceptions and © XXXX American Chemical Society

explicit a priori conceptions that establish conceptual schemes capable of being applied to most varied phenomena.51 The postulate that EN is equalized by bond formation between atoms is most frequently used in simple techniques to estimate net atomic charges in molecules,10−50 including the construction of universal potential energy curves36−41 and reactive force fields for molecular mechanics and molecular dynamics simulations, e.g., see refs 17 and 31−33. It has been accepted that DFT “leads rigorously to the concept of electronegativity and the principle of electronegativity equalization” (ENE).13,46−50 All of the theoretical ENE models based on Mulliken’s valence-state EN scale, e.g., see refs 14−30 and 34−41, supposedly have been retraced to the more general electronic chemical potential equalization (CPE) principle.13,46−50 I here investigate the impact of the Wigner−Witmer laws on the CPE and ENE principles in molecular bonding. It is shown for bond formation between two atoms and corresponding bond breaking processes that (i) The Mulliken EN, χM, is locally equalized in the bond, and the accuracy of ENE is orders of magnitude better Special Issue: 25th Austin Symposium on Molecular Structure and Dynamics Received: August 20, 2014 Revised: October 18, 2014

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dx.doi.org/10.1021/jp5084345 | J. Phys. Chem. A XXXX, XXX, XXX−XXX

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than that of a global CPE using the operational chemical potential, μop; (ii) To reach comparable results by CPE, the operational definition of μop has to be significantly modified by adapting to VS symmetry requirements, which are incompatible with the current ground-state paradigm of conceptual DFT; (iii) Conceptual DFT13,50 cannot claim to have operationally (not to mention, rigorously) proven and unified the CPE and ENE principles.

The VSEN scale has been conceived to model the formation of electron pair bonds. Mulliken3−5 and other authors6−41,54−57 assume that the electron pair bond, X:Y, consists of three interplaying “structures”, viz., the covalent, X−Y, and two ionic, X+ Y:− and X:− Y+. Consequently, the symmetry states of both the atoms-in-molecules and the ions-in-molecules must be compatible with the molecular symmetry state at equilibrium. The VS ions-in-molecules of the proper symmetry X+VSY−VS and X−VSY+VS significantly contribute to the energy balance defining Mulliken’s VSEN in eq 4. Mulliken3 and Pauling8,9 both mention the presence of ionic structures even in “perfect” homonuclear bonds, e.g., in H2 or Br2. For a recent discussion, see ref 57. These ideas are not unknown in DFT. Conceptual DFT invariably refers to Mulliken’s EN concept and accepts the role of ionic structures in bonding for calculating bond polarities and partial atomic charges.13,46−50,58−66 However, the strict Wigner−Witmer symmetry conditions for compatible ionic structures are neglected by insisting on GS ions-in-molecules13,46−50,58−66 despite their frequent incompatibility with the molecular symmetry (e.g., Br +GS F −GS is incompatible with BrF in its GS).

2. DIFFERENCES BETWEEN CHEMICAL POTENTIAL AND MULLIKEN ELECTRONEGATIVITY For neutral atoms or molecules the exact electronic chemical potential, μexact, must involve only ground-state properties at equilibrium, but its value, A0,v ≤ μexact ≤ I0,v, is only imprecisely known.13,52,53 I0,v and A0,v are the GS vertical ionization energy and the GS vertical electron affinity, respectively. Conceptual DFT emphasizes the ground-state paradigm: its operational chemical potential, μop, for arbitrary species13,46−50 is postulated as the negative of the GS electronegativity, χGS, the arithmetic average of I0,v and A0,v μop = −(1/2)(I0, v + A 0, v ) = −χGS

3. THE VALENCE-PAIR-AFFINITY AS BOND ELECTRONEGATIVITY Mulliken’s VSEN is well-defined for atomic species, but what is meant by the EN and specifically VSEN of a bond? It has been pointed out by Ferreira25,26 and supported by Ghosh and Parr49 that the molecular EN in the bond (“bond EN”) has to be perceived as “geminal electronegativity” or “pair electronegativity” because electron pair-density is accumulated in the bond region. However, the Hinze−Whitehead−Jaffé “bond EN”14−17,27 often used in this context fails to account for pair density in a correct way26,30,34,35,55 and does not qualify as “pair electronegativity”. To clarify essential differences and avoid confusions, I have recently changed the previously used name “valence-state (pair) EN”30,34−41,54,55 to the “valence-pairaffinity” scale (VPA, αVP).55 Note that Mulliken initially presented his EN as a “new electroaffinity scale”.3,4 The dependence of αVP on the orbital occupancy (0 ≤ n ≤ 2) is proportional to ηVS(X, i) = (1/2) [IVS(X, i) − AVS(X, i)], the VS orbital hardness30,34−41,54,55

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According to the CPE principle, the molecular value, μop,mol, is an average of all contributing atomic μop,at. For instance, the geometric mean (GM) equalization principle relates the data as μop,mol = − (1/2)(I0, v + A 0, v )mol ≈ −⟨(1/2)(I0, v + A 0, v )at ⟩GM (2)

However, a large-scale empirical investigation of the “geometric mean equalization principle” has disclosed that most molecules dramatically disobey the CPE and the χGS equalization principle.54,55 The root-mean-square relative error for eq 2 amounts to 71% for 210 molecules.54 The results are at strong variance with the CPE principle and its supposed parenthood to ENE. Under which conditions and how far can the principles be saved at least as useful rules? Let us here focus on diatomic molecules XY. As will be demonstrated, the inequality −μop(XY) ≠ [μop(X)μop(Y)]1/2

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αVP(X, i n) = IVS(X, i) − (1/2)[IVS(X, i) − AVS(X, i)]n

is essentially due to the ground-state paradigm of cDFT, which frequently collides with the Wigner−Witmer laws in molecule formation. The key role of the Wigner−Witmer laws in CPE and ENE is first highlighted in ref 55. A way to address the problem is to employ the Mulliken EN scale,3−5 which unlike μop and χGS allows accounting for the correct atomic symmetry states. In contrast to χGS, Mulliken’s atomic EN χM(X, i) is a valence-state electronegativity scale3−7,14−17,30 (VSEN). For a specific valency (v) of the atom X, the VS ionization energy, IVS, and the electron affinity, AVS, are determined for the same singly occupied atomic orbital, i3−7,14−30,34−41,54−56 χM (X, i) = (1/2)[IVS(X, i) + AVS(X, i)]

= χM (X, i) + (1 − n)ηVS(X, i)

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The qualitative characterization of VPA is similar to Pauling’s for EN:8,9 “The valence-pair-affinity is the charge dependent measure for the ability of an atom (or functional group) to attract an electron-pair in the ‘sharing competition’ with another atom-in-the-molecule”.55 For orbitals of neutral VS atoms with n = 1 we have αVP (X, i1) ≡ χM (X, i). Equation 5 also applies to some open shell diatoms, e.g., B2, discussed below. In general, however, the restriction of IVS and AVS to singly occupied orbitals must be abandoned when generalizing Mulliken’s EN to VPA in localized bonds in closed shell molecules. The electron affinity of a closed shell system cannot be attributed to any bond orbital (BO) because the shell limits are crossed by generating the negative ion. Hence, for closed shell diatoms the molecular A0,v values have to be taken by default in agreement with cDFT, viz., eq 1. Fortunately the spectroscopic observable vertical ionization energy of a diatomic bond, IBO,v(X:Y), is mostly assignable to a particular

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Both the VS ionization and electron uptake equally reduce the valency to (v − 1). The VS energy is calculated as the weighted mean of the energies of the relevant spectroscopic states.3−7,14−30,34−41,55,56 Extended lists of atomic VSEN data have been published by Pritchard and Skinner,6 Hinze et al,.14−17 and Bratsch.56 B



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Table 1. Geometric Valence-Pair-Equilibration ⟨αVP,at⟩GM Compared with Operational Chemical Potential Equalization ⟨−μop,at⟩GM (Data in Parentheses) for Diatomic Moleculesa IBO,v (I0,v)

XY B2 Si2 N2 CO NO O2 S2 Br2 HCl HBr

≈ 9.63 (≈ 10.52 v) (≈ 9.14 ad) 8.4473 (7.92)73 16.9868 (15.60)68 16.9168 (14.01)68 17.58(5)76−78 (9.50)68,77 18.80(6)80,81 (12.30)80,81 13.22(10)82 (9.41)82 14.60(1)89 (10.518(5))89 16.668 (12.73)68 15.668 (11.68)68 70,b

70

70

69

AVS,v (A0,v)

αVP (X:Y, i-j) (−μop)

69

≈ 5.53 (≈ 6.23) 5.22 (4.96) 7.31 (6.62) 7.45 (6.00) 8.37(6) (4.33(5)) 9.60(3) (6.38) 7.32(5) (5.43) 8.04 (5.99) 8.10(7) (6.16(5)) 7.75(7) (5.79(5))

1.43 (1.93) 2.0074 (2.00)74 −2.3675 (−2.36)75 −2.0275 (−2.02)75 −0.85(10)79 (−0.85(10))79 0.3983 (0.39)83 1.4484 (1.44)84 ≈ 1.4785−88 (≈ 1.47)85−88 −0.4;91 {−0.62(5)90} (−0.4) −0.1;91 {−0.10(5)90} (−0.1)

⟨αVP,at⟩GM (⟨−μop,at⟩GM) 65

5.64 (4.29) 5.2216 (4.77)65 7.3116 (7.23)65 7.44 (6.87) 8.39 (7.38) 9.6316 (7.54)65 7.3916 (6.22)65 8.3116 (7.59)65 8.19 (7.71) 7.72 (7.38)

δαVP (δμop) 2 (−31) 0 (−4) 0 (9) 0 (15) 0.2(6) (70) 0 (19) 1 (15) 3 (27) 1 (25) 0 (27)

The percent relative errors δαVP and δμop are defined in eqs 10 and 13, respectively. See the text for further symbols and definitions. All energies, electronegativities, and chemical potentials are given in electronvolts. bCalculated Iv values corrected for their likely underestimation by 0.15 eV; see abstract of ref 70. a

we document and discuss the individual examples; the overall comparison and analysis belongs to the next section together with the conclusions. The percentage errors δμop and δαVP are given relative to μop(XY) and αVP(X:Y) as

BO. To test whether the VPA is equalized upon bond formation, I here define the valence-pair-affinity in an electron pair bond formed by the orbitals i of X and j of Y, αVP(X:Y, i-j), as the VS counterpart of −μop(XY) αVP(X:Y, i‐j) = (1/2)[IBO, v(X:Y, i‐j) + A 0, v (XY)]

δαVP = 100[{αVP(X, i1)αVP(Y, j1 )}1/2 − αVP(X: Y)]/αVP(X: Y) (10)

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How do the differences between αVP and −μop affect their equalization in diatomic molecules? I here test the accuracy of the geometric “valence-pair-equilibration” (VPEq) rule in the bond formed between the orbitals i of X and j of Y

Table 1 contains the evidence for the prime result of this work, that the accuracy of VPEq is orders of magnitude better than that of CPE based on μop. The pictorial comparison of the VPeq and CPE in terms of the percent relative errors is shown in Figure 1.

αVP(X:Y, i‐j) ≈ ⟨αVP,at⟩GM = [αVP(X, i1) ·αVP(Y, j1 )]1/2 (7)

For homonuclear electron-pair bonds the VPEq of eq 7 simplifies to αVP(X:Y, i‐j) ≈ χM (X, i)

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We may alternatively define an equilibrated VPA value, ⟨αVP(X:Y, i-j)⟩CT, by energy minimization via charge transfer (CT)55 using the VS orbital hardness, ηVS(X, i) ⟨αVP(X:Y)⟩CT = [αVP(X, i1)ηVS(Y, j) + αVP(Y, j1 )ηVS(X, i)] /[ηVS(X, i) + ηVS(Y, j)]

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The rough proportionality between the αVP(X, i ) ≡ χM(X, i) and ηVS(X, i) values56 renders ⟨αVP(X:Y)⟩CT similar to the geometric mean, ⟨αVP⟩GM, and may justify the latter averaging scheme also for VPEq 1

Figure 1. Comparison of percent errors δαVP of the valence-pairequilibration VPEq (in red) and δμop of the chemical potential equalization CPE (in light blue).

4. COMPARISON OF CPE AND VALENCE-PAIR-EQUILIBRATION FOR DIATOMIC MOLECULES For atoms of groups 1 and 11, we have χM = −μop, and the molecular αVP(X:Y, i-j) and −μop(XY) do not differ either. The results of CPE and VPEq for selected diatoms of said groups shall be discussed in a follow-up article. Let us focus on cases where χM ≠ −μop for at least one atom, such as the molecules B2, Si2, N2, CO, NO, O2, S2, Br2, HCl, and HBr. The input data and the results are displayed in Table 1. The energy unit is electronvolt, unless noted otherwise. Photoelectron (PE) spectroscopy67,68 provides fingerprints of the neutral molecules and their ionized forms. Thus, most of the IBO,v and I0,v and some A0,v input values qualify as experimental. The remaining ones are from high-level theoretical calculations. In this section

The open shell system B2 is one of the few molecules for which eq 5 may be directly tested for VPeq. A genuine electronpair bond is absent in its ground-state X3Σg−. The atoms are bonded by two donor−acceptor type “half-π-bonds” in opposite directions, i.e., dative one-electron π-bonds compensating their charge transfers.69 Adding an electron to one of these bonds generates a true π-bond in the a2Πu state of B2− with the AVS = 1.43, obtained at a high level of theory.69 The larger GS value A0(B2) = 1.93 corresponds to adding an electron to an empty σ-orbital and generates the GS anion with three “half-bonds” in the quartet 4Σg− state.69 Removing a πelectron from B2 needs IBO,v ≈ 9.6370 to form the 2Πu (σg2σu2πu) state of B2+, which is the ionic VS in Mulliken’s C



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(16.91 − 2.02) = 7.45 perfectly. The charge-transfer alternative of eq 9 gives ⟨αVP⟩CT = 7.39. Nitric oxide, NO, is a very important and much discussed paramagnetic radical.67,68,71,76−78] Its ionic states and potential energy curves have been measured to high resolution by various methods.71,76−78 The vertical process for removing the unpaired electron of the 2π orbital and generating the singlet GS of NO+ needs I0,v = 9.50 eV.68 The anion NO− is a borderline case of electronic stability with A0,a = 0.026 eV;71 the vertical A0,v is here estimated from ref 79 as A0,v = −0.85(10). The comparison of −μop(NO) = 4.33 and ⟨−μop,at⟩GM = 7.38 gives the huge difference, δμop(NO) = 70%. On the other hand, a vacancy in the bonding 1π couples to the unpaired 2π orbital in six different ways. The Wigner−Witmer laws attribute to the 1π vacancy of NO+ the following six states: 1Σ+, 1Σ−, 3Δ, 3Σ+, 3 − Σ , and 1Δ.67,68 High-resolution spectroscopy assigns six states between 14.0 ≤ IBO,v(N−O, 1π) ≤ 23.0 eV.71,79,78 The reproduction of the splitting within a given electron configuration is, of course, beyond the scope of VPeq. In the spirit of Mulliken’s averaging state energies into the VSenergy,3−7,14−17,54−56 the molecular states are here averaged according to their J-multiplicity. The J-weighted average ionization energy is ⟨ IBO,v(N:O, 1π)⟩ = 17.58(5) eV. The resulting VPA values, αVP(N:O, 1π) = (1/2)[17.58(5) − 0.85(10)] = 8.37(6) and ⟨ χ M,at⟩GM = (7.31·9.63) 1/2 = 8.39 eV, agree with δα VP(N:O, 1π) = 0.2(6) %. This is more than two orders of improvement over CPE’s δμ op(NO) = 70%. Remarkably, the averaged homonuclear IBO,v and A 0,v of N2 and O2 are very close to those of NO: 17.87 versus 17.58 and ( 1/2)(−2.32 +0.39) = −0.97 versus −0.85 (10). This new piece of information deserves further investigations. For O2 and S2, the ionization from the bonding πu MO leads to the configuration σg2πu3πg2, which according to Wigner and Witmer forms five states of O2+ and S2+: 4Πu, 3 different 2Πu, and one 2Φu. The J-weighted averages of the individual bands are ⟨IBO(O2, πu)⟩ = 18.80 (6)80,81 and ⟨IBO(S2, πu)⟩ = 13.22(10).82 The ground-state ionization energies are I0,v(O2) = 12.3080,81 and I0,v(S2) = 9.41.82 The vertical electron affinities have been recently calculated by the R-matrix method: A0,v(O2) = 0.3983 and A0,v(S2) = 1.42.84 The evaluations in Table 1 yield δαVP(O2, πu) = 0% and δαVP(S2, πu) = 1% compared to δμop(O2) = 19% and δμop(S2) = 15% . Again the VPEq is more accurate than CPE by more than an order of magnitude. As mentioned at the very beginning, the VSEN of a halogen atom, X, is linked to the ionization process X (s2 px2 py2 pz) → X+ (s2 px2 py2) into the 1D2 and 1S0 states of X+. Their barycenter for Br+ is found at IVS(Br, 4p) = 13.11.16 The ionization to the triplet GS 3P3/2 of Br+(s2px2pypz) requires I0(Br) = 11.814 only.56 The chemical potential is μop(Br) = −7.59; the VPA αVP(Br, p1) = 8.31.16 We consider the ground and valence states of Br2+, X 2Πg and B 2Σg+, and the sole stable anion state of Br2−, 2Σu+. Unfortunately, the electron affinities of the halogen dimers, X2, are still not accurately known.71,85−88 For Br2, the prime options for electron detachments are (i) from an antibonding “lone pair” orbital or (ii) from the σ bond. The vertical values I0,v(Br2) = 10.518 (5)89 and A0,v ≈ 1.4785−88 give μop(Br2) = −5.99. The CPE principle requires μop(Br) = μop(Br2), but this is far from being obeyed here

notation. The αVP(B:B) ≈ 5.53 is very close to the geometric VPA average ⟨αVP⟩GM = 5.64 of the single occupied αVP(B, p1) = χM(B, p) = 3.8456 and the unoccupied atomic orbital’s αVP(B, p0) = IVS(B) = 8.298. The relative error is δαVP = 2%. Note the exceptionally large differences between the vertical I0,v ≈ 10.52 and adiabatic I0,ad ≈ 9.14 energies generating the X 2Σg+ (σg2σu2σg) GS of B2+ and Re(B2+) −Re(B2) ≈ 0.53 Å of the equilibrium bond lengths.70 Therefore, I0,v > IBO,v is irregular, while I0,ad < IBO,ad retains the normal sequence.70 In Table 1, −μop(B2) = 6.23 and ⟨−μop,at⟩GM = 4.2965 correspond to the deviation δμop = −31%, which exceeds δαVP by a full order of magnitude. In view of the uncertainties about the bonding57 and the precise ionization energies and electron affinities of C2,71,72 it seems preferable to postpone C2 and present Si2 instead. The GS of Si2 is X 3Σg (4σg24σu25σg22πu2), and the first two states of Si2+ are the 4Σg− (4σg24σu25σg12πu2) GS with I0 = 7.92 and 2Πu (4σg24σu25σg22πu) excited VS with Iπ‑BO = 8.44.73 Two states of Si2− are found just 0.025 eV apart, the 2Πu (4σg24σu25σg22πu3) and the X 2Σg+ (4σg24σu22πu45σg1). The experimental adiabatic A0,a = 2.202(10) is highly accurate, and different DFT methods indicate that the vertical A0,v value is consistently below the adiabatic one by 0.20(1) eV; thus, A0,v = AVS,v = 2.00(1).74 The relatively small δμop = −4% is still outperformed by δαVP = 0%. For N2, the atomic reference values are quite similar, −μop(N) = (1/2)(14.534 − 0.07) = 7.2350,65 and χM(N, 2p) = 7.31.16 The PE spectrum exhibits vertical ionization energies at 15.60, 16.98, and 18.78, corresponding to electron removal from an (σg2p)2, (πu2p)4, and (σu2s)2 MO, respectively.68 The vibrational structure of the first band around I0 =15.60 confirms the nearly nonbonding character of the HOMO, while the (πu2p)2 is the strongly bonding combination of two singly occupied 2pπ AOs.67,68 Thus, the IBO,v = 16.98 provides a suitable check for αVP-equalization. The vertical electron affinity has been calculated as A0,v(N2) = −2.36.75 According to eq 6 αVP(N2) = (1/2)(16.98 − 2.36) = 7.31

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is the perfect match for χM(N, 2p) = 7.31.16 Given δαVP = 0%, the VPA remains constant when forming the localized π-bond. As to the CPE “principle”, the atomic −μop(N) = 7.23 exceeds the molecular value −μop(N2) = (1/2)(15.60−2.36) = 6.62 by δμop (N2) = 9%. We may infer that μop is not well equalized in diatoms with occupied nonbonding or antibonding MOs. A comparison with the isoelectronic CO molecule sheds more light on the situation. The Mulliken VSEN of the oxygen atom is linked to the VS ionization process O(2s22px 2 2py 2pz ) → O+(2s22px 22py ) + e

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into the barycenter of the 2P and 2D states of O+ and requires IVS = 13.618 + 4.171 − 0.499 = 17.290.56 The ionization to the GS 4P3/2 of O+(2s22px2py2pz) requires I0 = 13.6182 only.56 The VS electron affinity AVS(O) = 1.960 exceeds A0(O) = 1.461 by 0.499.56 Therefore, the operational chemical potential is μop(O) = −7.54065 and the Mulliken VSEN χM(O, 2p) = 9.63.16,56 Because of its doubly occupied 2p2 AO, oxygen contributes a lone-pair to the MO scheme of CO, and its σ2p HOMO is more antibonding than that of N2.67,68 The negative chemical potential amounts to −μop(CO) = 6.00, while the ⟨−μop⟩GM = 6.87. Thus, δμop(CO) = 15% is significantly larger than δμop(N2) = 9%. The geometric mean ⟨αVP⟩GM = ⟨χM⟩GM = 7.44 = (5.75·9.63)1/2 for a π-bond in CO matches αVP(C:O) = (1/2)

δμop = 100[μop(Br) − μop(Br2)]/μop(Br2) ≈ 100( −7.59 + 5.99)/(− 5.99) ≈ 27% D

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The difference between μop(Br) and μop(Br2) seems too large to be attributed to the relaxation of electron density after bonding. In case (ii), the ionization removes an electron from the bond orbital, generating the B 2Σg+ excited state of Br2. High-resolution PE spectra89 give IBO,v(Br2) = 14.60(1), which is higher than the atomic IVS(Br, 4p) = 13.11. For Br2, the difference IBO,v − I0,v = 4.08, which is 39% of I0,v. The observed IBO,v and AVS, v yield αVP(Br:Br) ≈ 8.04. For the difference characterizing the VS-ENE we obtain

(iv) The large differences between the geometric mean value ⟨−μop,at⟩GM and −μop, mol have been previously attributed to relaxation effects on the electron density after bonding.13,58−60,65 Such relaxations do not seem to affect the local equalization of αVP in electron pair bonds. (v) The results strongly challenge cDFT’s ground-states paradigm and many of its popular applications. To save the importance of the ENE principle as comparable to the variational principle of energy,13 the absolute EN scale has to be made consistent with the Wigner−Witmer laws. Conceptual DFT should implement old intentions13,58−60 to incorporate the valence-state concept. A paradigm shift replacing the dominance of ground-states by some emphasis on valence states seems in order for conceptual DFT. This will be equivalent to shifting away from the dominance of I0,v and A0,v to accommodate their valence-state counterparts IVS and AVS for atoms and the new valence-pair-affinities (1/2)(IBO,v + A0,v) in bonds. (vi) Parr’s “minimum promotion energy criterion” for VSs13,58−60 ignores important symmetry issues. The lower bound to suitable IVS by minimizing promotion energies is symmetry determined. The best choice to minimize the VS promotion energy (and to avoid Mulliken’s average of spectroscopic states) could be the Wigner−Witmer allowed state of lowest energy. Thus, for Cl+ the 1D2 state is 1.445 eV above its 3P3/2 GS, and the “operational VS chemical potential” is μop* = −χVS,D = −9.015, to be compared with μop = −8.2965 and χM = 9.37.16 Research in this direction is under way. The present statements and conclusions are in principal agreement with Klopman’s “localized equipotential orbital” selfconsistent-field (SCF) model of ENE18−21 and Hamano’s semiempirical MO theory of σ-electron systems.22,23 Klopman and Hamano independently clarified that valence-state electronegativity equalization is linked to the variational treatment of molecules, and the VS electronegativity appears in the diagonal elements of the SCF matrix,18−23 see also ref 26. Klopman’s and Hamano’s models do not claim the global equalization of EN in all MOs of the molecule, nor do they require an additional inclusion of changes in the external potential v(r). Mulliken’s valence-state electronegativity, χM,3−7,14−17,30 with its present extension to valence-pair-affinity; the operational chemical potential, μop = −χGS;13,46−50,58−66 and the exact but yet undetermined expression of the chemical potential, μexact,13,46,47,52,53 have to be regarded as separate and distinct properties characterizing chemical systems.54,55,92,93 Conceptual DFT cannot claim to have operationally (not to mention, rigorously) proven and unified the CPE and ENE principles. I vividly remember the 1983 symposium in Sorrento,58 where Camille Sándorfy suggestively related the two questions “Are there atoms in molecules?” and “When I am shaking hands with you, am I myself?” The present cautiously positive answer is, “Yes, but I am likely promoted to an excited symmetry state, and so should be atomic electronegativity.” In an upcoming publication CPE and VPEq will be further tested on selected polyatomic molecules characterized by “equivalent bonds”. A different status assessment of the VPEq and CPE rules by an entirely DFT-independent thermochemical approach has been introduced in refs 54 and 55 and is under further investigation. The validity and applicability of the maximum hardness and minimum electrophilicity “principles” will be equally tested.

αVP(Br, p1) − αVP(Br: Br, p − p) ≈ 8.31 − 8.04 = 0.27 (14)

The relative error is reduced by an order of magnitude from δμop ≈ 27% to δαVP ≈ 3%. The PE spectra of the halogen acids, HX, are resolved into the 2Π and 2Σ+ bands68 corresponding to removing an electron from a lone-pair (I0) or the hydrogen-halogen σ-bond (IBO), respectively. The anions HX− are unstable, or metastable, but their A0,v(HX) have been obtained from the resonance peaks in dissociative electron attachment experiments90 and at advanced levels of theory.91 It goes almost without saying that the accuracy of VPEq outclasses that of CPE by at least an order of magnitude.

5. CONCLUSIONS AND OUTLOOK We have compared the performance of the proposed valencepair-affinity equilibration with that of the chemical potential equalization for a diatomic molecules with highly diverse electronic characters, for which at least one atomic μop ≠ − χM. Without exception the atomic valence-pair-affinities αVP = χM are equilibrated to αVP(X:Y) in localized bonds to an unprecedented accuracy of ≈1% or better, while the deviations of the operational δμop fall between −31 and +70%. The data in Table 1 and in Figure 1 prompt several statements and conclusions. (i) When the valence-pair-affinity, 1/2(IBO,v + A0,v), of diatomic bonds is compared with ⟨αVP⟩GM, the geometric average of atomic valence-pair-affinities αVP(X, i 1), the equilibration of the latter is highly accurate. (ii) Different values of (1/2)(IBO,v + A0,v) are attributed to different bonds, in accordance with photoelectron spectra, where different peaks relate to specific bond orbitals. However, μexact as a state function must be constant all over the molecule. Thus, the similarity between the “ENE principle” and a constant chemical potential μexact in molecules is highly deceptive. ENE expressed as VPEq is a very useful rule in a limited range of applicability rather than a general “structural principle.” Equating an operational chemical potential to the negative of electronegativity is a gross simplification54,55,92,93 and leads to contradictions and misconceptions. (iii) The large changes of the external potential, v(r), during bond formation13,46,94−96 do not come into play for VPEq; thus, the constancy of v(r) is not needed here. In DFT, a constant v(r) is required to treat the chemical potential as a function of the electron number N alone. Arguably, the vertical Iv and Av at constant v(r) are more required for the CPE13,46 and chemical hardness changes94−96 than for the present extension of Mulliken’s χM to αVP in molecular bonds. E



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AUTHOR INFORMATION

Corresponding Author

*Phone: 0049 711 6856 4408. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS It is a pleasure to thank Professor Michael C. Boehm and Dr. Gerald Knizia for many useful discussions and patiently reading several drafts. I also thank Professor Hans-Joachim Werner for continued kind hospitality at Das Institut für Theoretische Chemie.

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