Introduction

The discovery of stable N₅⁺ salts,^1,2 the long-known existence of stable N₃^- salts,³ and the recent experimental detection of the N₅^- anion⁴ have stimulated research in polynitrogen chemistry and a search for nitrogen allotropes. The high endothermicity of polynitrogen compounds renders their preparation and handling very difficult, and reliable stability predictions are important for the selection of suitable target compounds. The enthalpies of formation of crucial species, such as N₃, N₃^-, N₅, N₅⁺, and N₅^-, are not accurately known from experiment, because of the difficulty in handling these compounds and ions. There have been a number of theoretical studies on allotropes of nitrogen using ab initio molecular orbital theory without the use of empirical parameters, but these studies have not reliably established the energetics of these compounds. In particular, we note the extensive work of the Bartlett group on these species,^5,6 the work of Nguyen and Ha on decomposition mechanisms for N₅^x, x = -1, 0, +1,⁷ and that of Martin and co-workers on N₃.⁸ Several theoretical studies on the stability of N₅⁺N₃^- and N₅⁺N₅^- have been published.^5,6,9 In the most recent work, Fau, Wilson, and Bartlett performed ab initio MP2 and CCSD(T) molecular orbital theory and B3LYP density functional theory calculations on N₅⁺N₅^- and N₅⁺N₃^- and their individual ions as well as on quasi-periodic clusters of N₅⁺N₅^-. This work suggested that on a closed shell singlet surface the N₅⁺N₅^- ion pair is stable to dissociation into the cation and anion by 114 kcal/mol but is still ~300 kcal/mol above 5N₂ molecules. Thus, the singlet coupled ion pair represents a metastable structure on the potential energy surface. Approximate transition state searches suggested that the singlet coupled ion pair is stable by 15-25 kcal/mol to dissociation for the channel producing N₃ and N₂. These authors also suggested that N₅⁺N₃^- is unlikely to be stable. The authors noted that this is consistent with their calculated vertical electron affinity and ionization potential calculations, suggesting that the N₅⁺N₃^- ion pair is less ionic than the N₅⁺N₅^- ion pair because the high ionization potential of N₅^-, relative to that of N₃^-, causes in N₅⁺N₅^- effective resistance to charge transfer from the anion to the cation.⁶

This interesting concept and our own interests in such compounds led us to perform a different set of calculations on the stability of these compounds. We have also used high level ab initio molecular orbital theory to address the important question of whether the combination of a stable polynitrogen cation, such as N₅⁺, with a stable polynitrogen anion, such as N₃^-, can lead to a stable ionic nitrogen allotrope, such as N₅⁺N₃^-. The stability predictions were made using Born-Haber cycles, a well-established and widely used method¹⁰ for evaluating the stability of ionic salts, if reliable values for the lattice energies of the salts, the first ionization potential of the anion, and the electron affinity of the cation are available. This approach also requires an excellent knowledge of the nature and stability of the most likely decomposition products.

A critical prerequisite for such stability calculations is the use of accurate heats of formation and electron affinities and ionization potentials. We have developed an approach¹¹ to the reliable calculation of molecular thermodynamic properties, notably enthalpies of formation, based on ab initio molecular orbital theory without using empirical parameters. Our approach involves an accurate calculation of the total atomization energy of a molecule and combines this value with known enthalpies of formation of the atoms to calculate the molecular enthalpy of formation at 0 K. This method starts with coupled cluster theory, including a perturbative triples correction (CCSD(T)),^12-14 combined with the correlation-consistent basis sets^15,16 extrapolated to the complete basis set (CBS) limit to treat the correlation energy of the valence electrons. This step is followed by a number of smaller corrections that are presumed to be additive, including core-valence interactions and relativistic effects, both scalar and spin-orbit. Finally, one must include the zero-point energy obtained from experiment, theory, or some combination. The standard enthalpies of formation of compounds at 298 K can then be calculated from the 0 K value and other calculated properties by using standard thermodynamic and statistical mechanics expressions.¹⁷

This paper provides a conclusive answer to the question of whether the lattice energies of these salts are sufficient to stabilize them as solids, thus preventing spontaneous electron transfer from the anion to the cation and spontaneous decomposition. Furthermore, for N₅⁺N₃^-, experimental data are provided in support of our theoretical predictions.

Experimental Section

Caution! Reactions of N₅⁺ salts with azide ions are violent and can result in explosions. Therefore, these materials should be handled only on a small scale with appropriate safety precautions (face shield, leather gloves, and protective clothing).

Materials and Apparatus. All reactions were carried out in a Teflon-PFA double U-tube apparatus that consisted of a reaction U-tube, a porous Teflon filter assembly, and a receiver U-tube. Volatile materials were handled on a stainless steel/Teflon-FEP vacuum line.¹⁸ Nonvolatile solids were handled in the dry nitrogen atmosphere of a glovebox. Infrared spectra were recorded on a Mattson Galaxy FT-IR spectrometer using dry powders pressed between AgCl windows in an Econo press (Barnes Engineering Co.). Raman spectra were recorded on a Bruker Equinox 55 FT-RA spectrometer using a Nd:YAG laser at 1064 nm and Pyrex melting point capillaries as sample containers.

The N₅SbF₆ was prepared using our previously reported procedure.² The CsN₃ (Aldrich Chemical Co.) was pretreated with an excess of (CH₃)₃SiN₃ to remove any traces of moisture. The SO₂ (Air Products) was used as received.

Reactions of N₅SbF₆ with CsN₃. In a typical experiment, N₅SbF₆ (2.285 mmol) and CsN₃ (2.267 mmol) were loaded in the drybox into a Teflon-PFA reaction vessel that was closed by a Teflon valve. On the vacuum line, anhydrous SO₂ (5.0 mL, liquid) was added at -196 C. The reaction mixture was warmed to the melting point of SO₂ (-64 C). After the SO₂ had partially melted and the slurry of reactants was gently agitated, a violent reaction took place producing a bright yellow flame. The reactor was quickly quenched with liquid nitrogen. A check for noncondensable gases at -196 C revealed 9.05 mmol of N₂ as expected for the complete decomposition of 2.267 mmol of N₅N₃ to N₂. Pumping off all volatiles at -64 C produced 0.8890 g of a white solid residue (weight expected for 2.267 mmol of CsSbF₆ = 0.836 g). The extra weight was due to some residual SO₂. Raman and infrared spectra of the white solid, after pumping at room temperature, showed it to be pure CsSbF₆.

This reaction can be moderated by the separate dissolution of the starting materials in SO₂ and the combination of the resulting clear solutions at -64 C. Under these conditions, the decomposition reaction proceeded rapidly but without violence and gave the same results as described above.

Reactions of N₅SbF₆ with NaN₃. To equimolar amounts of NaN₃ and N₅SbF₆ was added CHF₃ as a solvent at -196 C, and the resulting mixture was warmed to -95 C and then to -64 C. No N₂ evolution was observed under these conditions. The CHF₃ was pumped off at -95 C, and SO₂ was added as a solvent at -196 C. The mixture was warmed to -64 C, but again no N₂ evolution was observed. Pumping off the SO₂ at -64 C resulted in a white solid residue that, based on its room-temperature Raman spectrum, consisted of a mixture of unreacted NaN₃ and N₅SbF₆, indicating that no reactions had occurred due to the very low solubility of NaN₃ in these solvents.

Computational Approaches

Enthalpies of Formation. The augmented correlation consistent basis set aug-cc-pVnZ (n = D, T, Q)¹⁹ was used for nitrogen. Only the spherical components (5-d, 7-f, 9-g and 11-h) of the Cartesian basis functions were used. This family of basis sets contains an additional shell (e.g., spd for DZ) of diffuse functions that are necessary for an accurate description of anions. Calculations were performed using the MOLPRO,²⁰ Gaussian,²¹ and GAMESS²² suites of programs. The open shell CCSD(T) calculations were carried out at the R/UCCSD(T) level. In this approach, a restricted open shell Hartree-Fock (ROHF) calculation was initially performed to generate the set of molecular orbitals and the spin constraint was relaxed in the coupled cluster calculation.^23-25 The CCSD(T) total energies were extrapolated to the CBS limit by using a mixed exponential/Gaussian function of the form:

Most geometries were optimized at the frozen core CCSD(T) level with various augmented correlation-consistent basis sets. With the exception of the N₅ radical, all geometries were optimized with the aug-cc-pVTZ or aug-cc-pVQZ basis sets. If the CCSD(T)/aug-cc-pVQZ geometry was not available, the CCSD(T)/aug-cc-pVTZ geometry was used for the CCSD(T)/aug-cc-pVQZ calculation. Geometries for the N₅ radicals were obtained from ZAPT2/6-311+G(2df) and CCSD(T)/aug-cc-pVDZ calculations. Vibrational frequencies were calculated at the CCSD(T) and MP2 levels with the aug-cc-pVDZ basis set. Calculations were performed on SGI, IBM, and Apple G4 computers.

Core-valence corrections, E_CV, were obtained at the CCSD(T)/cc-pCVTZ level of theory.²⁷ Scalar relativistic corrections (E_SR), which account for changes in the relativistic contributions to the total energies of the molecule and the constituent atoms, were included at the CI-SD (configuration interaction singles and doubles) level of theory using the cc-pVTZ basis set. E_SR is taken as the sum of the mass-velocity and 1-electron Darwin (MVD) terms in the Breit-Pauli Hamiltonian.²⁸ Because N has a ⁴S ground state, there are no atomic spin-orbit corrections to the total atomization energies. The spin-orbit coupling constant for N₃ is 71.3 cm^-1 based on experiment,²⁹ leading to a spin-orbit correction of 0.10 kcal/mol.

By combining our computed D₀ values with the known enthalpies of formation³⁰ at 0 K for N of H_f⁰(N) = 112.53 ± 0.02 kcal mol^-1, we can derive H_f⁰ values for the molecules under study in the gas phase. We obtain enthalpies of formation at 298 K by following the procedures outlined by Curtiss and co-workers¹⁶ and by using the stationary electron convention; that is, we do not assume a change in the enthalpy of formation of the electron as a function of temperature.

Lattice Energies. For this study, eq 2³¹

Results and Discussion

Calculated Geometries. The calculated geometries for the molecules, N₂, N₃, N₃^-, N₅^-, and N₅⁺ are summarized in Table 1.

N₂. As expected, the geometry for N₂ at the CCSD(T)/aug-cc-pVQZ level is in excellent agreement with the experimental value.³⁴ Theory overestimates experiment by 0.0028 Å. For comparative purposes, we also optimized the N-N bond length with the larger aug-cc-pV5Z basis set. This produced a small 0.0010 Å decrease in the bond length, indicating that the aug-cc-pVQZ bond lengths should be within approximately 0.001 Å of the basis set limit for the other molecules that could be treated at this level of theory. If the frozen core approximation is removed via calculations with the core/valence (CV) cc-pCVQZ basis set, another 0.0020 Å shortening of the bond length is observed. The final CCSD(T)/aug-cc-pV5Z + CV bond length of 1.0975 Å is almost identical to the experimental value of 1.0977 Å.

N₃. The geometry for N₃ (D_h) is in excellent agreement with the experimental value.³⁵ The calculations predict a value for the NN bond that is slightly shorter than the experimental value. This result is in contrast to N₂ where the calculations predict a longer bond at the CCSD(T)/aug-cc-pVQZ level.

N₃^-. The calculated geometry for N₃^- is also in very good agreement with experiment, with the calculated NN bond length again being slightly shorter than experiment.³⁶

N₅⁺. The geometry for N₅⁺ has been measured in the crystal by X-ray diffraction techniques.² The calculated values are in good agreement with the crystal structure values considering the uncertainty in the experimental values and the different environment. The calculated geometrical parameters for N₅⁺ are the same as those reported by Fau and Bartlett at the CCSD(T)/aug-cc-pVTZ level,⁵ whereas the geometries calculated for N₃^- at the same level differ slightly.

N₅^-. The value of 1.334 Å calculated for the NN bond length in N₅^- at the CCSD(T)/6-311+(G(d,p) level⁷ lies between our values calculated at the CCSD(T) level with the aug-cc-pVDZ and aug-cc-pVTZ basis sets, respectively.

N₅ (Open Isomer). ZAPT2³⁷ calculations with the 6-311+G(2df) basis set followed by CCSD(T)/cc-pVTZ single-point calculations confirm previous conclusions⁷ that the N₅ radical starting from this geometry is vibrationally unstable. Starting from the optimized N₅⁺ structure, the neutral species dissociates without a barrier to N₃ + N₂ as shown in Figure 1.

Figure 1 Schematic showing the decomposition of the N5 radical formed by electron attachment to N5+ at the ZAPT2/6-311G(2d,f) and CCSD(T)/cc-pVTZ levels. The radical smoothly dissociates to products. The x axis corresponds to the calculation number describing the dissociation, and the y axis corresponds to the energy or to the N3-N4 (numbered starting from the right) interaction distance representing the separation between the N2 and N3 molecules. The large squares correspond to the ZAPT/6-311G(2d,1f) potential energy surface, and the circles correspond to the CCSD(T)/cc-pVTZ potential energy surface. The small squares represent the N3-N4 distance which increases.

N₅ (Cyclic Isomer). The N₅ radical cyclic isomer would be generated by detaching an electron from cyclic N₅^-. The HOMO of N₅^- is a degenerate in-plane orbital involving essentially lone pair interactions. Production of the N₅ radical leads to (e)³ occupancy which Jahn-Teller distorts to a C_2v structure. Two states can be derived from the (e)³ occupancy, a ²A₁ state and a ²B₁ state (a ²B₂ state would be a radical). The geometry for the cyclic N₅ (C_2v) ²A₁ radical was found to be a stationary point on the ZAPT2/6-311+G(2df) potential energy surface. The cyclic N₅ radical at the ZAPT2/6-311+G(2df) geometry is 66.8 kcal/mol above the N₃ + N₂ asymptote at the CCSD(T)/cc-pVTZ level. The longest NN ZAPT/6-311+G(2df) bond distance is that in the base of the pentagon, 1.354 Å. The shortest NN bonds are those connected to the apex, 1.292 Å. We took this geometry and initiated an optimization at the R/UCCSD(T)/aug-cc-pVDZ level for both the ²A₁ and the ²B₁ states under the constraint of C_2v symmetry. The ²B₁ state smoothly dissociated to the N₃ + N₂ asymptote, showing that on the CCSD(T)/aug-cc-pVDZ surface, this state is not a minimum. The N₃ distance converges faster than the N₂ distance. The ²A₁ state is difficult to optimize but finally optimizes to a structure with two elongated N₂ molecules complexed weakly to an N atom (see Table 1), as it cannot go to the products N₂ + N₃. At the CCSD(T)/aug-cc-pVTZ level using the optimized CCSD(T)/aug-cc-pVDZ geometry, the ²A₁ state is 60.2 kcal/mol above the N₂ + N₃ asymptote. These calculations strongly suggest that, on electron detachment from N₅^-, the resulting radical has a path on the ²B₁ surface that leads directly to the N₃ + N₂ asymptote and that there is no stable cyclic N₅ radical structure, as has already been noted previously.⁷ As part of a study of Fe(⁵-N₅)₂, Frenking and co-workers reported a structure for the ²A₁ state of the N₅ radical at the B3LYP/6-31G(d) level which could be a minimum, although the structure was not discussed.³⁸ Our result is also consistent with the prediction⁷ of Nguyen and Ha that cyclic N₅ is not stable with respect to dissociation.

Calculated Frequencies. The calculated vibrational frequencies for N₂, N₃, N₃^-, N₅^-, N₅ (cyclic), and N₅⁺ are summarized in Table 2.

N₂. The harmonic frequency at the frozen core CCSD(T)/aug-cc-pVQZ level is within 5 cm^-1 of the experimental value.³⁴ Even at the CCSD(T)/aug-cc-pVTZ level, the N₂ frequency is within 20 cm^-1, and, at the CCSD(T)/aug-cc-pVDZ level, it is within 40 cm^-1. This level of agreement is in part fortuitous. Whereas the aug-cc-pV5Z basis set frequency is identical to the experimental value, introduction of core/valence correlation increases _e by 9.2 cm^-1. Thus, the final CCSD(T)/aug-cc-pV5Z + CV value is 9 cm^-1 larger than experiment.

N₃. For N₃, the frequencies calculated at the MP2/aug-cc-pVDZ level are in qualitative agreement with the experimental values³⁹ for the bend and the symmetric stretch and in good agreement for the antisymmetric stretch. In this case, we have used the experimental frequencies to calculate the zero-point energy.

N₃^-. For N₃^-, the calculated antisymmetric harmonic stretch is in excellent agreement with the gas-phase experimental value.^35,40 The value for the ₂ bending frequency for this ion has been measured in the solid state, and a free space estimate for ₁ has been obtained based on the spectrum measured for the solid.^41,42 For this ion, we used the experimental frequencies for the zero-point energy determination. Use of the theoretical frequencies introduces a difference of 0.19 kcal/mol because the calculated bend and antisymmetric stretch are ~50 cm^-1 lower than the experimental values. The frequencies calculated by us for ₁ and ₃ are in reasonable agreement with those previously calculated at the CASPT2 level¹⁷ and at the CEPA level.⁴²

N₅^-. For this ion, we used the frequencies calculated at the CCSD(T)/aug-cc-pVDZ level for the zero-point energy calculation as there are no experimental values available. Quite good agreement is found for the frequencies of N₅^- calculated at the MP2/aug-cc-pVDZ level and at the CCSD(T)/aug-cc-pVDZ level. The CASPT2 frequencies¹⁷ are similar to our values.

N₅ (Cyclic Isomer). Frequencies were calculated at the ZAPT2/6-311+G(2df)level of theory for the ²A₁ state. No experimental values are available for this species. As noted above, the ²B₁ structure is not a minimum for the cyclic radical at the R/UCCSD(T)/aug-cc-pVDZ level.

N₅⁺. For N₅⁺, there is reasonable agreement between the frequencies calculated at the MP2/aug-cc-pVDZ level and the average of the frequencies taken from measurements in the solid state.² The largest discrepancy is found for the highest frequency b₂ mode with a difference of ~140 cm^-1. The calculated a₂ and b₁ modes deviate by ~50 cm^-1 from the experimental values. The fact that the differences cancel out leads to calculated and experimental zero-point energies that are essentially identical. We used the average experimental values in our zero-point energy calculations. Our calculated frequencies are in qualitative agreement with those calculated at the CCSD(T)/6-311+G(2d) and B3LYP/6-311+G(2d) levels.^1,2 The CCSD(T) values are in general lower than our MP2 values except for the a₂ and b₁ modes which are higher. The CASPT2 frequencies differ from ours more than might be expected, especially for the two highest modes.¹⁷

Calculated Enthalpies of Formation. The total atomization energies are given in Table 3. The H_f⁰ values were converted to 298 K following the procedure of Curtiss and co-workers.¹⁶

N₂. As shown in Table 3, the error in the atomization energy of N₂ is 0.76 kcal mol^-1 when compared to experiment.³⁰ This leads to an error in the calculated enthalpy of formation at 0 K of 0.76 kcal mol^-1. We can thus estimate that the error bars in our heats of formation are likely to be on the order of ±1 kcal/mol.

N₃. The enthalpy of formation of N₃ has been reported by a number of authors. The NIST value³⁰ of 100 ± 5 kcal mol^-1 at 0 K is almost 10 kcal mol^-1 below our calculated value. Our calculated value of 109.2 kcal mol^-1 is in much better agreement with the value of 112 ± 5 based on the experimental measurement of the acidity of HN₃ and the measurement of the electron affinity of N₃^-.^43,44 Our value of 109.25 kcal mol^-1 is in excellent agreement with the scaled value of 109.25 kcal mol^-1 given by Martin et al.⁸ The calculated atomization energies show that reaction 3

N₃^-. The enthalpy of formation of N₃^- has been determined from gas-phase acidity measurements and the enthalpy of formation of HN₃ to be 48 ± 2 kcal mol^-1.⁴³ This value is in excellent agreement with our calculated value of 47.2 kcal mol^-1. Fau and Bartlett calculated the enthalpy of formation of N₃^- by a variety of methods.⁵ Their best estimated enthalpy of formation of 47.8 kcal mol^-1 is in good agreement with our value. The electron affinity of N₃ has been measured to be 2.68 ± 0.01 eV.⁴⁴ Our calculated adiabatic electron affinity of 2.69 eV is in excellent agreement with this result. With an ANO (4s3p2d1f) basis set, the CASPT2 electron affinity was predicted to be 2.64 eV.¹⁷

N₅⁺. The enthalpy of formation of N₅⁺, calculated by us, is 353.4 kcal mol^-1. This result is in reasonable agreement with the value of 347.1 kcal mol^-1, calculated by Fau and Bartlett.⁵ Christe and co-workers^1,2 predicted the enthalpy of formation of N₅⁺ to be 351.1 kcal mol^-1 at the CCSD(T) level with a polarized double- basis set. Nguyen and Ha predict a heat of formation of 351.3 ± 3.6 kcal/mol based on CCSD(T)/6-311+ G(3df) calculations and the experimental energy of N₃.⁷

The adiabatic electron affinity of N₅⁺ is given by the energy difference between its enthalpy of formation and those of the first metastable neutral reaction or decomposition products. Because the intermediate, open-chain N₅ radical, formed by the addition of an electron to N₅⁺, is unstable and spontaneously dissociates to N₃ (g) + N₂ (g), the energy of reaction 4 provides the adiabatic electron affinity of N₅⁺.

N₅^-. The enthalpy of formation of N₅^- is calculated to be 62.2 kcal mol^-1, in excellent agreement with the value of 62.1 ± 3.6 kcal/mol given by Nguyen and Ha.⁷ This value can be used to calculate the adiabatic ionization potential of N₅^- in the same manner as discussed above for N₅⁺. As noted above, loss of an electron from the degenerate HOMO of N₅^- leads to either a ²B₁ or a ²A₁ radical. We have shown that formation of the ²B₁ radical provides a pathway leading to N₃ + N₂, so the following channel (reaction) represents the adiabatic ionization process:

The important dissociation channel for N₅^- is the formation of N₃^- + N₂. We predict this channel to be exothermic by 14.2 kcal mol^-1 at 0 K. This value is in good agreement with the value of 14.3 kcal mol^-1 reported by Nguyen and Ha at the CCSD(T)/aug-cc-pVTZ level.⁷ It is important to know if there is a barrier separating N₅^- from the asymptotic products N₃^- and N₂. Nguyen and Ha calculated the energy barrier to dissociation as 27.7 kcal mol^-1 at the CCSD(T)/aug-cc-pVTZ level including zero-point corrections based on a geometry calculated at the CCSD(T)/6-311+G(d) level.⁷ We have calculated the energy of the transition state at the CCSD(T)/CBS level with Nguyen and Ha's geometry and with the geometry optimized by us at the MP2/aug-cc-pVDZ level. We calculated the geometry at the latter level to get a value of the zero-point energy as well as the value of the imaginary frequency characterizing the transition state. The results are given in Tables 1-3. Our calculated energy barrier using the MP2 geometry is 29.4 kcal mol^-1, and when the zero-point energy difference is included, the barrier is 27.2 kcal mol^-1. The energy barrier using the Nguyen and Ha geometry is 0.61 kcal mol^-1 higher in energy than our value based on the MP2 geometry, showing that the two geometries give essentially identical barrier heights. The barrier calculated⁷ by Nguyen and Ha is very similar to our CCSD(T)/CBS value using their geometry. At the CASPT2/ANO(4s3p2d) level, the barrier is predicted to be 26 kcal mol^-1, and the dissociation energy to N₃^- and N₂ is predicted to be exothermic by 9 kcal mol^-1,¹⁷ 5 kcal mol^-1 lower than our value. A calculation of the imaginary frequency for the transition state of the N₅^- decomposition shows a large value of 862i cm^-1, suggesting that the ion might be able to tunnel through the barrier to reach the products. We can make a crude estimate of the tunneling effect by using the Wigner expression.⁴⁵ This leads to an enhancement of the dissociation rate by 1.7 at 298 K. This suggests that tunneling should be considered when predicting the stability of N₅^-. For lower temperatures, a larger tunneling effect would be expected.

The origin of the difference between the vertical and the adiabatic potentials, of course, is the dissociation energy of neutral N₅ to N₃ and N₂. Vertical electron affinities or ionization potentials for the ionic systems under study correspond to removal or addition of an electron to the ion to form a radical at the geometry of the ion. Only when the neutral radicals are vibrationally stable species with geometries similar to those of the starting materials will their vertical EA or IP values approximate the adiabatic values. If this is not the case, the radicals generated by the neutralization process spontaneously undergo further decomposition. If these decompositions to the first well-defined, vibrationally stable products do not involve any significant activation energy barriers and if they occur on a fast time scale, it is appropriate to add their energies to the vertical potentials.

Lattice Energy Calculations. Prediction of lattice energies, U_L, based on eq 2 requires an estimate of the relevant ion volumes and gives the following results:

N₅⁺N₃^-. V(N₅⁺) is estimated to be 0.051 (±0.020) nm³ based on the reported² crystal structure volume for N₅⁺Sb₂F₁₁^- by subtraction of V(Sb₂F₁₁^-) (Table 6, ref 31). V(N₃^-) is equal to 0.058 (±0.014) nm³ (Table 5, ref 31), and thus V(N₅⁺N₃^-) = 0.109 (±0.024) nm³, leading to an estimate for U_L(N₅⁺N₃^-) = 142 (±7) kcal mol^-1.

N₅⁺N₅^-. An upper limit for V(N₅^-) can be estimated from consideration of V(N₂) and V(N₃^-). The volume of solid dinitrogen, V(N₂), which is dimorphous,⁴⁶ can be obtained from crystal structure data for the -cubic and the hexagonal forms and averages to V(N₂) = 0.046 (±0.001) nm³, leading to V(N₅⁺N₅^-) = 0.155 (±0.014) nm³, giving a value for U_L(N₅⁺N₅^-) = 129 (±5) kcal mol^-1. Alternatively, we can estimate the volume of the ions at the B3LYP/6-31+G* level as described above: V(N₅⁺) = 0.068 nm³; V(N₅^-) = 0.088 nm³, V(N₃^-) = 0.065 nm³. These estimated volumes yield U_L(N₅⁺N₅^-) = 129 kcal mol^-1 and U_L(N₅⁺N₃^-) = 135 kcal mol^-1, well within the above error limits. The corresponding lattice enthalpies, H_L, appropriate for use in the cycles of Figures 2 and 3 are U_L(N₅⁺N₅^-) + 2RT and U_L(N₅⁺N₃^-) + ³/₂RT, respectively.

	Figure 2 Born-Haber cycle in kcal mol-1 for the decomposition reaction of solid N5+N5-. The large negative value for Hr demonstrates the unlikelihood of this salt to exist.
	Figure 3 Born-Haber cycle in kcal mol-1 for the decomposition reaction of solid N5+N3-. The large negative value for Hr demonstrates that this salt is also unlikely to exist.

Stability Predictions Based on Born-Haber Cycles. For a chemical reaction to be thermodynamically favorable, its free energy change must be zero or negative. If the reaction involves a barrier, a sufficient amount of activation energy must be supplied to overcome this barrier. If the reaction does not have an activation energy barrier, a spontaneous reaction occurs when its free energy change becomes negative. An example of such a reaction is the dissociation of an ionic solid into two radicals (reaction 6), assuming that the electron transfer from the cation to the anion proceeds without a barrier.

Figure 4 Potential energy curves in kcal mol-1 for the decomposition reaction of solid N5+N5- to dinitrogen. The left curve is based on the calculated values from this study and shows that a hypothetical N5+N5- would spontaneously decompose to 2N3 (g) + 2N2 (g). The right curve shows that for N5+N5- to be stable, its enthalpy of formation would have to be significantly lower than those of the gaseous species (2N3 + 2N2). The broken line indicates that the barrier might possibly exceed the B value.

The left curve of Figure 4 represents the case of the enthalpy of reaction 7 being negative and having no significant activation energy barrier. It results in spontaneous decomposition of the solid to the first vibrationally stable intermediates, that is, N₃ radicals and N₂. The right curve of Figure 4 represents the scenario of a positive reaction enthalpy, the only condition under which solid N₅⁺N₅^- might exist. This, however, is a hypothetical case, requiring the enthalpy of formation of N₅⁺N₅^- to be about 100 kcal mol^-1 more negative than the calculated value. Because the values for the IP of N₅^- and the EA of N₅⁺ are fixed, the lattice energy of N₅⁺N₅^- would have to be about 100 kcal/mol higher than our calculated value, which is impossible based on the molar volumes of the ions involved. The relatively small uncertainties in our H_r values, -65 ± 7 kcal mol^-1 for N₅⁺N₅^- and -39 ± 9 kcal mol^-1 for N₅⁺N₃^-, together with the large -TS terms, allow us to predict with confidence that both compounds cannot be stabilized in the form of ionic solids.

The potential energy surface for N₅⁺/N₅^- is quite complicated. The asymptotic energy of N₅⁺ + N₅^- is 416 kcal/mol above that of 5N₂ molecules at 0 K. Our approach to the ion pair energy provides an estimate of 297 kcal/mol above 5N₂, in good agreement with the Fau et al.⁶ value of 296 kcal/mol. The highly endoergic heat of formation means that the complex is metastable as are all polynitrogen compounds. Thus, there may be a number of decomposition paths.⁴⁸ Neither the N₅ radical derived from N₅⁺ at the cation geometry nor the N₅ radical derived from N₅^- at the anion geometry are stable species as shown above. These two N₅ species are doublets, and hence there can be both a triplet coupling and a singlet coupling of the two unpaired electrons after the electron transfer process. The singlet coupling, if the geometries remain similar to the parent ion geometries, should generally be considered as an open shell singlet, not necessarily restricted to a closed shell description.⁶ Enforcing singlet closed shell (i.e., RHF) pairing in the ground-state wave function prevents the two N₅ species from coming apart as N₅ radicals (or their dissociative products N₂ and N₃) even though the product formation of 2N₃ + 2N₂ is more than 80 kcal/mol exothermic. On the other hand, the triplet surface could lead to direct dissociation. Including the open shell-singlet paired configuration in a more general multiconfigurational wave function may have the same result. To further study the N₅⁺/N₅^- ion pair potential surface, we optimized the lowest ³A' and ³A' ' states at the ROHF/6-31+G(d) (restricted open shell Hartree-Fock) level, and the lowest ¹A' and ¹A' ' states using TCSCF (two-configuration self-consistent field) wave functions and the same 6-31+G(d) basis set. For both spin states, the lower energy state is A'. Geometry optimization of the ³A' state (lowest energy triplet at this level of theory) leads to a large separation of the anion and cation, with a concomitant decrease in the charges on the two species to nearly zero. So, the lowest energy triplet is clearly on its way to dissociation to two radicals, and subsequently to N₂ and N₃. The other three states (two singlets and one triplet) do not appear to dissociate at these levels of theory. The ³A' state is about 40 kcal/mol above the ¹A' state but is still ~80 kcal/mol below the separated ion pair dissociation asymptote, showing that there is an electron transfer step leading to dissociation that occurs significantly below the energy of the separated ions. Higher levels of theory may alter these results, and we did not further explore the barriers to dissociation. These results and those of other workers do show that the potential energy surface is very complicated and that there are many potential highly exothermic paths leading to formation of N₃ and/or N₂. In addition, we note that all of these comparisons are at 0 K and that at higher temperatures, there is a significant entropic effect favoring formation of N₂ and N₃. These results are all consistent with our conclusion that formation of a stable N₅⁺/N₅^- ion pair is unlikely, so it cannot be isolated.

Experimental Results and their Comparison with the Predictions. All of our attempts to prepare solid N₅⁺N₃^- by the metathesis reaction 9 failed.

These results are consistent with the predictions by us and by Fau and Bartlett⁵ that N₅⁺N₃^- will not be stable. Fau and Bartlett⁵ noted that the reaction of N₅⁺ and N₃^- does not lead to ion pair formation and that N₅N₃ is a weak complex even on the closed shell singlet energy surface. This structure is located at -149 kcal/mol on the way toward the 2N₃ + N₂ final products at -182 kcal/mol (energies with respect to the separated ions). Their results suggest that on the triplet coupled surface, and possibly on the multiconfigurational singlet surface, this species will fully dissociate.

Assessment of the Performance of Solid N₅⁺N₅^- as a Monopropellant. It was recently suggested^6,52 that N₅⁺N₅^-, if it "could be stabilized, would be an ideal monopropellant because its density is nearly twice that of hydrazine while the energy densities (heat of reaction per gram) are comparable." Indeed, the previous energy density estimate of 2.11 kcal/g for N₅⁺N₅^- is close to our estimate of 1.98 kcal/g. However, our energy density estimate for hydrazine is significantly lower. If one assumes its decomposition to proceed according to eq 10,

Conclusions

The following conclusions can be drawn from our studies:

(i) Our results quantitatively examine the stability of N₅⁺N₅^- and N₅⁺N₃^- and show that both compounds are unstable in the solid state.

(ii) Our results show that for Born-Haber cycle based stability estimates for species, such as N₅⁺ and N₅^-, which decompose to vibrationally unstable species upon neutralization, it is important to use adiabatic ionization potentials and electron affinities.

(iii) In view of the instability of the N₅ radicals and their strongly exothermic spontaneous decompositions to N₃ radicals and N₂, both N₅⁺ and N₅^- are not good choices for preparing a stable ionic nitrogen allotrope.

(iv) Our results suggest that N₅⁺N₅^- would be even more unstable than N₅⁺N₃^-, because the former would contain two unstable N₅ species after transfer of the electron.

(v) Stability calculations for individual ions or ion pairs may not provide sufficient information for stability predictions. Rigorous Born-Haber cycles, including realistic lattice energy estimates, and an evaluation of the stability of the decomposition products are required to make meaningful predictions.

(vi) Our theoretical predictions concerning the instability of N₅⁺N₃^- and the use of the adiabatic electron affinities and ionization potentials are confirmed by extensive experimental studies involving the low-temperature metatheses of N₅⁺SbF₆^- with different alkali metal azides in various solvents and the determinations of electron affinities by bracketing methods, respectively.

Acknowledgment

The authors thank Dr. Jerry Boatz for many helpful discussions and Dr. Chang-guo Zhan for help with the volume calculations. The work at Pacific Northwest National Laboratory (PNNL) was supported in part by the U.S. Department of Energy, Offices of Basic Energy Sciences, Division of Chemical Sciences, and Biological and Environmental Research, under Contract No. DE-AC06-76RLO 1830 for PNNL. Part of this research was performed in the William R. Wiley Environmental Molecular Sciences Laboratory (EMSL) at the PNNL using the Molecular Sciences Computing Facility. The EMSL is a national user facility funded by the Office of Biological and Environmental Research in the U.S. Department of Energy. PNNL is a multiprogram national laboratory operated by Battelle Memorial Institute for the U.S. Department of Energy. The work at the Air Force Research Laboratory was supported by the Defense Advanced Projects Agency (DARPA) and the Air Force Office of Scientific Research (AFOSR). The work at the University of Southern California was funded by DARPA, AFOSR, and the National Science Foundation. The work at Iowa State University was funded by an AFOSR high energy density matter (HEDM) grant.