Universal Method to Calculate the Stability, Electronegativity, and

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Universal Method to Calculate the Stability, Electronegativity, and Hardness of Dianions La´szlo´ von Szentpa´ly* Institut fu¨r Theoretische Chemie, UniVersita¨t Stuttgart, Pfaffenwaldring 55, D-70569 Stuttgart, Germany ReceiVed: July 30, 2010; ReVised Manuscript ReceiVed: August 31, 2010

The electronic stability of gas-phase dianions of arbitrary size, X2-, is determined by the first universal method to calculate second electron affinities, A2. The model expresses A2,calc ) A1 - (7/6)η0 by the first electron affinity, A1, and chemical hardness, η0, of the neutral “grandparent” species. A comparison with 37 reference data of atoms, molecules, superatoms, and clusters yields A2,ref ) 1.004A2,calc - 0.023 eV, with a mean unsigned deviation of MUD ) 0.095 eV and a correlation coefficient of R ) 0.9987. Predictions of second electron affinities are given for a further 24 species. The universality of the model is apparent from the broad variety of compounds formed by 30 diverse elements. The electronegativity and hardness of dianions are determined for the first time as χ(X2-) ) A2 and η(X2-) ) (7/12)η0, respectively. Pearson and Parr’s operational assumption regarding the hardness of anionic bases for the hard-soft acid-base (HSAB) principle is rationalized, and predictions for hard and soft dianionic bases are presented. For trianions, first criteria and predictions for electronic stability are given and require A1 > (7/4)η0. 1. Introduction Dianions are important and ever-present in solid-state and nanosciences, plasma physics, solution chemistry, organic synthesis, acid-base and redox reactions, and the bioregulation of metabolic and cellular processes. The existence and properties of dianions have received massively growing attention in the last two decades, as documented by several reviews addressing fundamental questions about their intrinsic properties.1-4 The electronic stability of a dianion is characterized by the second electron affinity of the neutral “grandparent” species, X, defined as the energy difference A2(X) ) E(X-) - E(X2-). However, there has been no general method determining their electronic stability by calculating, rationalizing, and predicting the second electron affinity for arbitrary systems. Additionally, for dianions, neither the electronegativity, χ, nor the chemical hardness, η, are known to any useful accuracy, despite their prominence as most frequently used ordering principles of chemistry.5-12 In density functional theory (DFT)13-15 and concepts related to electronegativity6-12 the total energy, E(N,Vex), is a continuous function of the average electron number N and the external potential Vex. Consider a neutral chemical system, X, of electron number N0 and accurately known ionization energy I and first electron affinity values A1. The first and second partial derivatives (at constant Vex) define the electronegativity, χ(X,N) ) -∂E/∂N, and chemical hardness, η(X,N) ) ∂2E/∂N2, respectively.6-13 The energy EDFT(N) consists of straight line segments with slope discontinuities at integer values of N.9,13-17 Therefore, in DFT, the electronegativity is segment-wise constant for noninteger excess negative charges, ∆N ) N - N0. In particular, χDFT(X,∆N) ) I in the cationic range -1 < ∆N < 0 and jumps to a different constant value χDFT(X,∆N) ) A1 for anionic 0 < ∆N < 1.16,17 A direct method to calculate the energy of a dianion could be an extrapolation of the total energy up to N0 + 2 ) Ndianion. However, the slope discontinuity at N0 + 1 ) Nanion is unknown, unless A2 itself is known beforehand. It is a crucial problem of the state-of-the-art DFT that the knowledge of A2 is required as input for finding the continuation of EDFT beyond * E-mail: [email protected].

the anion. Other current limitations of DFT are discussed in refs 9, 15, and 18. Because the research of dianions is extremely demanding, the limits of experimental and computational feasibility in obtaining A2 data are frequently reached. At this state of the art, empirical concepts and approximations estimating dianion properties can be helpful. Large A1 values1-4 or A1/I ratios8,9 provide guidelines in seeking electronically stable dianions. However, qualitative considerations are difficult to convert into quantitative models, and no general criteria have been reported so far. The previously proposed homogeneous charged-sphere1,3,19 and stabilized-jellium models20 are not applicable to atoms and small molecules but are limited to large and nearly spherical clusters. In addition to their A1 input data, these models introduce further species-specific parameters, such as adjustable diameters, Ø, and dielectric constants, ε. Our aim is to generate a simple and efficient method for differentiating between stable and unstable dianions of arbitrary size without explicit computation or measurement. Starting from the Parr-Szentpa´ly-Liu (PSzL) electrophilicity concept,8,9 we develop the first universal method to determine the second electron affinity, the electronegativity, and the hardness of gasphase dianions of arbitrary size. 2. Universal Method to Calculate Second Electron Affinities According to PSzL, a neutral species, X, with N0 electrons may be saturated with excess electrons to the point Nmax, where its electronegativity χ(X,Nmax) becomes 0. Therefore, the anion X- is predicted to be electronically stable if χ(X-) > 0, while the dianion is expected to be electronically stable with

A2(X) > 0

if χ(X2-) > 0

(1)

However, the two models of E(X,N) considered in refs 8 and 9, namely, the ground-state parabola (GSP) and the valencestate parabola (VSP), provide very different values for χ(X-) and χ(X2-). On the GSP scale, χGS(X-) ) 1/2(3A1 - I) and χGS(X2-) ) (1/2)(5A1 - 3I) are obtained by extrapolating

10.1021/jp107177d  2010 American Chemical Society Published on Web 09/16/2010

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Figure 1. Determination of the maximum uptake of electrons, ∆Nmax, by the zero intercept of the electronegativity, χVS(N), for the case I ) 3A1. Density functional theory (DFT) results are in black, and those for the VSP are in magenta. A gas-phase dianion is stable if ∆Nmax g 2. According to eq 3, this is the case for (1/2)(3A1 - I) > 0.

Pritchard’s seminal χ(X,N) function.6-9 To form a stable dianion, the species X has to fulfill the condition (1/2)(5A1 - 3I) > 0. This requires the hardly ever encountered ratio A1/I > 0.600.8,9 According to the GSP model, even the evidently stable monoanions should be unstable, unless fulfilling (1/2)(3A1 - I) > 0.8,9 This result is in striking contradiction to the basic tenet and definition that an anion is electronically stable if and only if A1 > 0. Therefore, the GSP electronegativity cannot be considered for determining the stability of anions and dianions. The VSP model8-10,21-24 gives χVS(X-) ) A1; thus, for monoanions, the PSzL stability criterion χ(X-) > 0 is identical to the principal requirement A1 > 0. Therefore, we turn to this method to find a stability criterion for dianions. Originally developed for atoms-in-molecules10 and functional groups-inmolecules,9 the VSP model has been successfully extended to arbitrary systems.8,9,24 Its applications range from bond polarities,10,22-24 electrophilicity indices,8,9 and transferable force constant increments21 in a universal potential energy curve21-23 to new relationships and predictions for the thermochemistry of diatomic molecules, ionic solids, and metals.24 The valencestate electronegativity, χVS(N) ) -∂EVS/∂N, connects the values for χDFT(Ncation) ) I and χDFT(Nanion) ) A1 by a straight line, reproducing Mulliken’s electronegativity χ0 ) (1/2)(I + A1) at N0 (Figure 1).

χVS(N) ) χ - η ∆N 0

0

(2)

where η0 ) (1/2)(I - A1) is the chemical hardness of the neutral species. The correct DFT value of χVS(N) at Nanion is an important precondition allowing reasonable extrapolations to dianions. In the VSP model, the electronegativity of the dianion is expressed by two properties of the neutral grandparent species, A1 and I, or χ0 and η0

1 χVS(X2-) ) (3A1 - I) ) χ0 - 2η0 ) A1 - η0 2

(3)

Combining eqs 1 and 3 provides a dianion stability criterion, A1 > η0. In analogy to the identity χVS(X-) ) A1, it seems logical to search for simple relationships between χVS(X2-) and reliable literature data, A2,ref. Until the millennium, the body of available experimental A2,ref data for such a test was negligible, but thereafter, modern experimental methods succeeded in generating and measuring an increasing number of stable gas-phase

dianions.4,19,20,25-31 During the last two decades, the computational studies on first and second electron affinities have become more frequent and accurate.1-3,28,32-55 However, in aiming for simple relations, we must beware of possible biases in correlating properties of neutral species with those of dianions. In general, a scaling of η0 ) (1/2)(I - A1) is needed to model the increased electron repulsion (see Supporting Information). In addition, the VSP model is based on fixed external potentials and internuclear distances during the charging process.8-10,21-24 Changes in the molecular geometry between the neutral, anionic, and dianionic species affect the energetics, as manifested by differences between the vertical, V, and adiabatic, ad, values of A2, A1, and I. The question of whether all of these influences may be absorbed by the parameters of a linear regression could be addressed by postulating the identity

A2,VS ) χVS(X2-)

(4)

as a first-order approximation, followed by a linear regression of A2,VS on a larger set of reference data A2,ref, leading to a relationship A2,ref ≈ aA2,VS + b. However, the number of parameters is further reduced by the “ansatz”

A2,calc ) A1 - cη0

(5)

where c is adjusted to a single “anchor” molecule having wellestablished values of A2, A1, and I. For calibration, we take the C70 fullerene with A2 ) +0.02(3) eV,26 A1 ) 2.765(10) eV,26 and I ) 7.48(5) eV,33 giving η0 ) 2.357(26) eV. The result c ) 1.165(10) ≈ 7/6 yields a general equation

7 A2,calc ) χcalc(X2-) ) A1 - η0 6

(6)

Note that other anchor molecules also lead to the same equation. Therefore, eq 6 will be tested for gas-phase dianions of arbitrary size. 3. Dianion Stability and Electronegativity: Results and Discussion Table 1 shows the results of a thorough literature search for atoms, molecules, superatoms, and clusters having reported values for A2, A1, and I. All of the input data are documented with their sources as Supporting Information. Table 1 displays a broad variety of 42 chemical systems, with second electron affinities covering the wide range of -8 < A2/eV < +3. Predictions for 24 further compounds are given in Table 2. The tables include (i) SiF61,28,34,35 and transition-metal complexes,1-4,32,36-38 (ii) “superatomic”4,39,56 fullerenes,4,26,27,33,40 heterofullerenes,41 and endohedral metallofullerenes,39 (iii) metal clusters,19,31,42 (iv) small molecules, mostly having negative second electron affinities,1-4,30,43-52 and (v) gas-phase atoms forming unstable dianionic resonance states only.1-4,53-55 For assessing atoms, the hardness η0VS ) (1/2)(IVS - A1,VS) obtained from valence-state configuration energies is preferred because second-order effects including fine-structure splitting are averaged out.5-7,9,10,57 The differences to (1/2)(I - A1), however, become significant for atoms of groups 15 and higher only (N, P, As, O, and S).5-7,57 Table 1 allows comparison of calculated A2,calc and published A2,ref values. Although it seemed unlikely that the various complex effects in dianion formation might be sufficiently

Stability, Electronegativity, and Hardness of Dianions

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TABLE 1: Calculated Second Electron Affinities, A2,calc, Dianion Electronegativities, χcalc(X2-), and Previously Published A2,ref, Ordered As Related Species in the Sequence of Decreasing Dianion Stabilitya,b species ALn SiF6

A2,calc ) χcalc(X2-) (eV)

TABLE 2: Twenty-Four Predicted Values of the Second Electron Affinity, A2,calc, and Dianion Electronegativity, χcalc(X2-), Ordered in the Sequence of Decreasing Dianion Stabilitya X

A2,calc ) χcalc(X2-) (eV)

X

A2,ref (eV)

PtF6 UO2+ MoF6 WF6

adc: 2.63; vc: 2.27 1.82 1.00 0.56 -2.86

ad: 2.5835 v: 2.334 1.8936 1.1237 1.7 0.79(10)a 0.70 0.31 [anchor: 0.02(4)] -0.26

1.841a 0.82(1)27 0.6027 0.32527 ad: 0.02(3)4,26 -0.34,27

Sc@C82b C59N Cs19 Sc@Cu16b Cs13 Ga3O4

1.11 0.84 0.27 0.15 0.08 0.02(30)

ZnO RuO4 OCN UF4 TiO2 O3

Clusters Au20 (Td) Au12 Au11 Au10 Ag22 Ag11 S8 S7 S3 S2 Al4 B4 Be3

ad: 0.24(3) 0.12 0.13 -0.163 -0.35(33) -0.35(33) 0.41(8) -0.05(8) -2.24(4) v: -3.15(2) -1.22(17) -2.82 -3.02(17)

0.2531 0.1531 >019 -0.231 -0.331 (1/3)A1, which is equivalent to A1 > (7/4)η0 or A1/I > 7/15. Among the predicted stable dianions (Table 2), the fullerenes C1443- and C120H123- are candidates for stable trianions (cf. Supporting Information). The gold cluster Au32 with its experimental A1 ) 3.96(2) eV64 and calculated I ) 6.835 eV65 fulfills A1/I > 7/15; thus, the criterion predicts the trianion Au323- as stable. However, according to Herlert and Schweikhard,19 the minimum size for trianions should be n > 50. The criterion A2 > (1/3)A1 and the extremely high electronic stability of ZrF62- with A2,obsd(ZrF6) ) 2.9(2) eV25 and A1,calc(ZrF6) ≈ 7.1 eV66 allow prediction of ZrF63- as electronically stable, albeit highly reactive and thermodynamically unstable. According to the calculated data, A2(CrF6) ) 2.44 eV32 and A1(CrF6) ) 8.24 eV, the trianion CrF63- is expected to be unstable according to our criteria. The predictions for trianions are, of course, sensitive to the quality of the input data. The electronegativity and hardness are expected to play major roles in the stabilization of multiply charged anions by counterions and solvents and in assessing their redox and acid-base reactions. Detailed applications of the valence-state electronegativity and hardness to such topics will be published in future articles. 6. Conclusions and Broader Context The aim to develop a simple and efficient method for differentiating between stable and unstable dianions of arbitrary size without explicit computation or measurement has been achieved. This is the first general extension of a parabolic E(N) model beyond the PSzL electrophilicity concept.8,9 One cannot

expect to construct a general model of dianions from a series of high-level computations and experiments. The process of setting up universal relationships, such as eq 6, often involves some initial evidence, including “informed guesses” and even hunches, as in ref 8, to be followed by selective model building.67 In our case, the VSP model provides an approximate stability criterion requiring χVS(X2-) > 0 and suggests A2,VS ≈ A1 - η0. The subsequent fine-tuning and anchoring to the observed A2(C70) ≈ 0 proves to be extremely efficient. The energies of unstable dianionic resonance states of atoms are calculated to the same accuracy as the highly positive second electron affinities of large clusters. Amazingly, the electronic stability, electronegativity, and chemical hardness of dianions are quantitatively expressed as functions of two properties, A1 and I, of the neutral grandparent species that are easier to determine. The model widens the basis of the HSAB rule by (i) quantifying dianionic hardness as η(2-) ) (7/12)η0 and (ii) justifying and extending Parr and Pearson’s operational assignment of hardness values η0 to anionic bases. The present work may also draw more attention to the alternative VS electrophilicity index, ω2 ) A1I/(I - A1).8,9 Future work will focus on (i) trianions and more complex species, (ii) reduction potentials in solutions, and (iii) the generation of charge-dependent electronegativity and hardness scales for multiply charged anions. Acknowledgment. I thank Prof. Guntram Rauhut for providing the ionization energy of SiF6 needed for calculating its second electron affinity, Prof. Michael C. Bo¨hm for valuable discussions and critical reading of drafts and manuscripts, and Prof. Hans-Joachim Werner for the kind hospitality in his institute. Supporting Information Available: (i) The methods section, (ii) input data for calculating second electron affinities: first electron affinities, A1, ionization energies, I, and atomic valence-state hardness data, ηVS0, (iii) the calculated second electron affinities, A2,calc and previously published A2,ref, ordered in sequence of decreasing dianion stability, and (iv) literature references for the input data. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Scheller, M. K.; Compton, R. N.; Cederbaum, L. S. Science 1995, 270, 1160–1166.

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