Polymeric nitrogen

The equilibrium phase boundary between single-bonded, threefold-coordinated polymeric forms of nitrogen, and the observed, triple-bonded diatomic phases, is predicted to occur at relatively low (50 + 15 GPa) pressure. This conclusion is based on extensive local-density-functional total-energy calculations for polymeric structures (including that of black phosphorus, and another with all gauche dihedral angles) and diatomic structures (including that of the observed high-pressure e-N2 phase). We believe the diatomic phase of nitrogen, observed up to 180 Gpa and room temperature, to be metastable at these conditions, and that such hysteresis enhances the prospects for the existence of a metastable polymeric form of nitrogen at ambient conditions. In this regard, we show that the black-phosphorus and cubic gauche polymeric forms of nitrogen would encounter signi6cant barriers along high-symmetry paths to dimerization at atmospheric pressure.


I. INTRODUCTION
All known solid phases of nitrogen assume a diatomic molecular form, ~characterized by strong triple covalent bonds (N-= N) within each Nz molecule, and weak van der Waals interactions between different molecules.s Ni-  trogen is the only group V element that assumes this form in its condensed phases, "in preference to polymerizing to single-bonded systems as in the case of phosphorus and arsenic."2 It is the purpose of this paper to report ab initio pseudopotential total-energy calculations which suggest, however, that such polymeric forms of nitrogen may exist in the bulk, both as thermodynamically stable phases at high pressure, and as metastable phases at atmospheric pressure.
There has been recent interest by chemists in finite counterparts of polymeric nitrogen phases, i.e. , large N" molecules bound by single (N -N) or single and double (N=N) bonds, as potential energetic materials.s s The- oretical calculations have focused on a hypothetical N4 analogue of observed P4, s on a Ns analogue of benzene as well as other N6 isomers, and on a N8 analogue of cubane.There are also studies on NsHs isomers, in which the nitrogens are singly or doubly bound to one another.While the calculations suggest that many of these hypothetical molecules should be metastable, i.e. , exhibit real vibrational frequencies, there has been as yet no conclusive experimental evidence as to their existence.There are, however, indications that N6 may have been observed, and perhaps N3H3 as well.
There is also considerable interest within the high- pressure physics community in the possible existence of bulk polymeric phases of nitrogen.It is generally ex- pected that the application of sufBcient pressure must eventually destroy covalent bonds in solids, and in particular lead to the dissociation of diatomic bonds in molecu- lar solids.Such dissociation in solid iodine has been well documented, and that in solid hydrogen is the object of considerable current interest.~4 Nitrogen offers an addi- tional complication because of the Ns triple bond, which contrasts with the single bonds present in Iz and Hs.The essential conclusion that may be drawn from previ- ous theoretical work on high-pressure nitrogen phases is that the N=-N triple bond will be destabilized at lower pressures than the N -N single bond.~s ~~That is, ni- trogen should transform from its ambient diatomic form to an intermediate, still covalent, polymeric regime be- fore losing its covalency altogether at still higher pres- sures.Two calculations~s ~s have placed the transition to the polymeric regime at pressures below 100 GPa (100 GPa = 1 Mbar), the lower estimate being at 70 GPa.
~s Subsequent diamond-anvil-cell   experiments to 130 and 180 GPa, however, indicate that nitrogen remains di- atomic to these pressures at room temperature, this con- clusion being based on the fact that only modest changes were observed in the measured intramolecular vibron frequency.s' s It may simply be that the diatomic phase is itself metastable at these pressures, since, for exam- ple, the graphite to diamond transition in carbon must be overdriven by about a factor of 7 in pressure even at a temperature of 1000 K. On the other hand, an anomaly has been observed above 30 GPa in hot shock compressed fiuid nitrogen, 21 and has been interpreted as the occurrence of dissociation to a dense atomic phase.
This interpretation is consistent with the theoretically predicted phase transition at low temperatures.
In this paper, we present extensive ab initio pseudopo- tential total-energy results for both polymeric and di- atomic phases of nitrogen, which not only reaKrm the previous theoretical conclusions, but suggest that an even lower pressure of 50 GPa is the more likely phase boundary above which stable nitrogen phases at low tern- perature should be polymeric.This reduction follows from our identification of a polymeric form of nitrogen, characterized by all gauche dihedral angles, which has a significantly lower (0.86 eV/atom) total energy than pre- 46 14 419 1992 The American Physical Society viously considered polymeric forms.At the same time we report ab initio pseudopotential total-energy calculations for the observed high-pressure c'-N~diatomic phase.We also provide estimates of the local-density-functional errors in this diatomic equation of state, based on compar- isons with experimental data.While our local-density- functional predictions indicate a 33-GPa transition from the e-N2 phase to the new polymeric form of nitrogen, these considerations lead to what we believe to be a re- alistic 50 6 15 GPa placement of the diatomic-polymeric phase boundary for solid nitrogen at low temperatures.
The single-bonded group-V elements exhibit structures that are generally related to simple cubic (sc).This struc- ture is itself observed at high pressure in both phosphorus and arsenic, and a rhombohedral distortion (A7) is the ambient phase of arsenic, antimony, and bismuth, and is also seen at pressures below the sc phase in phosphorus.The two previous theoretical upper bounds on the tran- sition pressure from diatomic to polymeric nitrogen as- sumed these two structures (sc and A7) for the poly- meric phase, with the lower 70 GPa result associated with the lower-energy A7 structure.
The ambient black- phosphorus (BP) phase of phosphorus, i an orthorhombic distortion of sc, is another natural candidate structure to consider for a polymeric form of nitrogen.We report here ab initio total-energy calculations for nitrogen in the BP structure, and find indeed a total energy which is 0.28 eV/atom lower at equilibrium (pressure p = 0) than that of A7 nitrogen.Moreover, the relative energy ordering of these A7 and BP curves for nitrogen is also consis- tent with dependence of the total energy on the dihedral angle in NzX4 (i.e., XzN -NXz) molecules.This corre- spondence has led us to consider a "cubic gauche" (cg) distortion of sc, in which we find nitrogen to have an equi- librium total energy a further 0.58 eV/atom below that of the BP structure.It is to this hypothetical cg phase of polymeric nitrogen that we find a transition from the diatomic form at 50 6 15 GPa.
The location of the low-temperature equilibrium phase boundary between diatomic and polymeric forms of ni- trogen also depends crucially on the calculated equation of state for the diatomic phase.While the structure of nitrogen is unknown above about 20 GPa, there exist strong indications from Raman data that all phases of nitrogen so far observed at pressures over 10 GPa are closely related to the 6-Nz (Pm3n) structure.In par- ticular, the e-Nz (R3c) phase observed at cryogenic tem- peratures from 1.9 to about 20 GPa is a slight rhom- bohedral distortion of the 6-N2 structure due in part to orientational ordering of the Ng molecules. 2Raman data suggest a further lowering of symmetry (possibly R3c) at about 20 GPa.z We report in this paper ab initio local-density-functional calculations of the observed z-Nq structure.
For comparison, we also report calculations for diatomic nitrogen in two other structures which are characteristic of slightly prolate molecules, ~s namely, the o.-Nq (Pa3) and P-02 (R3m) structures.While we find the c-N~structure to have the lowest energy of +he three at high pressure, as expected, the more important conclu- sion is that the differences in energy are relatively small in the vicinity of 50 GPa.We therefore conclude that un- certainties in the precise structure of the diatomic phase in this pressure range have relatively small impact on our location of the equilibrium diatomic-polymeric phase boundary.Larger uncertainties for the diatomic phase are due to local-density-functional errors in the treatment of the intermolecular van der Waals interactions.
However, by a comparison to experimental data for both diatomic nitrogen and similarly van der Waals bonded solid argon, we believe we have relatively good estimates of these errors, which are included in our predicted 50+15 GPa location of the diatomic-polymeric phase boundary.
We suggest that the room-temperature diatomic phase of nitrogen, observedis is at pressures approximately three times higher than our calculated equilibrium phase boundary, must be metastable at these conditions.If diatomic nitrogen exhibits such large hysteresis, which is common for covalent solids, zo polymeric forms stable above 50 GPa may well also be metastable at atrno- spheric pressure.To investigate this question, we also consider high-symmetry transformation paths by which the BP and cg polymeric forms might dimerize at at- mospheric pressure (p 0), and also some paths be- tween the various polymeric forms.In all cases we find that bond breaking or at least significant bond bending is required along these paths, and in all cases we find significant barriers as, for example, a 0.9 eV/atom bar- rier inhibiting dimerization of the atmospheric-pressure cg phase.While these results are far from conclusive, they are consistent with the possibility that each of the A7, BP, and cg phases of nitrogen may be mechanically stable at atmospheric pressure.
In the remainder of this paper, Sec.II gives a descrip- tion of the total-energy methods and of the ab initio pseu- dopotential for nitrogen used in the present work.Af- ter characterization of the various structures, our total- energy results as a function of atomic volume are pre- sented in Sec.III for the polymeric and diatomic phases of nitrogen, as well as for a number of higher-coordination metallic phases.Our predicted location of the polymeric- diatomic phase boundary is also given in this section.
Section IV presents our p = 0 metastability analysis for the three (A7, BP, and cg) polymeric structures, and our conclusions are given in Sec.V.

II. THEORETICAL METHOD
The results presented below were obtained using ab initio pseudopotentials implemented with a plane-wave basis for the expansion of the electronic wave functions.Nitrogen is an element for which a pseudopotential de- scription should be adequate because the core 182 states do not overlap significantly with the valence states.How- ever, since the core of nitrogen consists only of 18 states, there is no cancellation for the p and d states in the core region and, consequently, the resulting pseudopotentials are very attractive for these angular momentum chan- nels.Consequently, a large number of plane waves are required to provide a correct description of the nitrogen pseudowave function.
The pseudopotential calculations whose results are presented below are performed within the frame- work of localdensityfunctiona th~ry29 applied in the momentum-space formalism.sos' We use nonlo- cal, norm-conserving, sz ab initio ionic pseudopoten- tials generated in a separable form.ss A procedure re- cently suggested by Trouiller and Martinss4 to generate soft, nonlocal, ab initio pseudopotentials was adopted.The Ceperley-Alderss model is used for the exchangecorrelation potential, as parametrized by Perdew and   Zunger.ss In addition to the total energy E, the pres- sure p and the forces are calculated analytically from the stress theorem.s7 Brillouin zone summations are performed using sets of special k points generated using the Monkhorst-Pack algorithm.ss ss A kinetic energy cutoff of E«t,ft = 80 Ry was used to calculate the total en- ergy for all the structures and transformation paths.To treat the large set of plane waves resulting from this large E«totr, we used the local optimization scheme of Teter, Payne, and Allan. 40served at high pressure in both phosphorus and arsenic.
Inspection of Fig. 1 and Table I reveals that the six- fold coordinated sc phase has substantially lower energy than the high-coordination (8 and 12) bcc, fcc, and hcp metals, and that its equilibrium energy is even slightly lower than fourfold coordinated cd, which is also metal- lic in the case of nitrogen.This behavior anticipates the fact that further significant reductions in total energy arise from distortions of the sc structure which split its six near-neighbor distances into three first-nearest-and three second-nearest-neighbor distances.Such distortedsc structures exhibit the threefold coordination which is ideally suited to the capacity for each nitrogen atom to form three covalent bonds.We refer to these structures as polymeric in contrast to the triple bonds within the N2 molecules of the diatomic phases.

III. TOTAL ENERGIES OF STRUCTURES
The results of our ab initio total-energy calculations are presented below.Given the large number of struc- tures considered, we organize our presentation by dis- cussing, in turn, metallic, polymeric, and diatomic phases.When describing the various structures, we follow standard crystallographic notation4~b y defining vec- tors (z, y, z) in terms of the conventional unit cell vectors a, b, andc: We now consider threefold-coordinated polymeric forms of nitrogen.These include the arsenic (A7), black- phosphorus (BP), and cubic gauche (cg) phases, whose structural characterization is given in Table II.The A7 and BP structures are natural candidates for polymeric forms of nitrogen, as they are observed phases of other group-V elemental solids.The arsenic A7 structure con- sists of a rhombohedral distortion of the sc phase and (z, y, z)-:@a + yb + zc.
We then use this notation to list both the subset of site positions associated with the primitive cell, and also the three primitive cell vectors ao, bo, and co. nitrogen

A. Metallic phases
The structures of the metallic phases considered here are well known and need not be described.Our calcu- lated results for the total energy per atom as a function of atomic volume are shown in Fig. 1 for the face-centered- cubic (fcc), ideal (c/a = g8/3) hexagonal close-packed (hcp), body-centered-cubic (bcc), cubic diamond (cd), and simple cubic (sc) structures.Values of the equilibrium (p = 0) atomic volume Vo, bulk modulus Bo, and total energy Eo for these metallic phases were obtained by fitting the calculated total energies to Murnaghan's   equation of state4~a nd are indicated in Table I.Our re- sults are in excellent correspondence with those obtained from previous linear muffin-tin orbitals~s and ab initio pseudopotential~s analyses which confirm the accuracy of our soft nonlocal pseudopotential.
The half-filled 2p shell of nitrogen tends to favor crystal structures in which bonds are oriented approximately along the lobes of the pz, p", and p, orbitals in orthog- onal directions. 43This is suggestive of the sc structure which may be considered prototypical of the group-V el- emental solids.Indeed, heavier group-V elements exhibit atmospheric-pressure phases which are distortions of the sc structure, while the sc structure itself is actually ob- Atomic Volume (P / atom) FIG. 1. Metallic phases oj nitrogen.Calculated total en- ergy per atom for the face-centered-cubic (fcc), ideal hexag- onal close-packed (hcp, dashed curve) with (c/a) = g8/3, body-centered-cubic (bcc), simple-cubic (sc), and cubic dia- mond (cd) phases of nitrogen as a function of atomic volume.
A kinetic energy cutofF of E,"« = 80 Ry was used to cal- culate the total energy for all the structures.The number of k points used for the calculations was fcc (28), hcp (36), bcc (26), sc (20), and cd (28).The zero of energy corresponds to the minimum energy of the diatomic o-N2 (Pa3) Metallic phases include the face-centered-cubic (fcc), ideal hexagonal close-packed (hcp), body-centered-cubic (bcc), cu- bic diamond (cd), and simple-cubic (sc) structures.Poly- meric phases (see Table II) include the arsenic (A7), black-phosphorus (BP), trans ci-s chain (ch), and cubic gauche (cg) structures.Diatomic phases (see Table III) include the P-02 and cx-Nz structures.All calculations were performed with a kinetic energy cutofF of 80 Ry.The values of the equi- librium (p = 0) atomic volume Vo, bulk modulus Bc, and total energy Ep were obtained by fitting the calculated total energies to Murnaghan's   ( is the atmospheric-pressure form exhibited by antimony, bismuth, and arsenic itself.The pressure sequence A7 ~sc is observed in As at room temperature. 4The BP structure is an orthorhombic distortion of sc, and is the observed atmospheric-pressure form of phosphorus. Phosphorus is known to exhibit the structural transfor- mation sequence BP ~A7 ~sc under pressure.' Af- ter discussing our theoretical results for the A7 and BP phases, we indicate how these two phases exhibit local geometric relationships similar to those found in small N2X4 molecules of the form XzN -NXz.This correspon- dence between our optimized results for nitrogen in the A7 and BP structures and small molecules led us to con- sider a third topology which we refer to as "cubic gauche" (cg), which is then discussed.The cg phase was found to be the lowest-energy polymeric form of nitrogen among the candidate structures considered here.
The arsenic A7 structure belongs to the R3m space group, with a rhombohedral (trigonal) Bravais lattice, and two atoms per primitive cell.It corresponds to a distortion of the sc phase obtained by a strain of the sc unit cell along the [Ill] direction and a simultaneous dis- placement of the atoms of the basis towards each other in pairs in the same direction.The distortion from sc is typically large enough so that the A7 structure is layer- like, as is evident in the schematic illustration indicated in Fig. 2. The hexagonal unit cell vectors defined by The name (abbreviation) is given for each structure, along with the space group, the magnitudes a, b, c of the unit cell vectors, the unit cell multiplicities and Wyckoff site letters, and the corresponding dimensionless internal parameters x, y, z.Hexagonal axes are used for rhombohedral R3m.The bond lengths, dihedral angles, and bond angles are specified for each structure.
In terms of these, the two c positions associated with the primitive cell are k(0, 0, zgr), while the three primitive cell vectors are (3) ao, bo, cp --(1,0, 0), (z, z, 0), (0, 0, 1) . (6) The BP structure is uniquely specified by a knowledge of the five independent structural parameters aBp, bBp, cBp, yBP, and zBp.A sc-like limit for the individual double layers is obtained for (c/a)Bp = 1, (yb/a)Bp = I/v8, and zBp = 0.However, adjacent double layers appear shifted by za relative to one another in this limit, and a monoclinic path is required to actually reach the sc structure, e. g., the 4e sites of P2i/c.
The results of our total-energy ab initio pseudopoten- tial calculations for the A7 and BP threefold coordinated polymeric phases of nitrogen are presented in Fig. 3, where the calculated total energy per atom is shown as a function of atomic volume, and in Tables I and II, where equilibrium bulk and structural properties are given, re- spectively.The c/a ratio was optimized at every vol- ume for the A7 unit cell, with the zero-pressure value (c/a) &7 --3.25, which is somewhat larger than previously found on the basis of pseudopotential calculations.The internal parameter ZA7 was set to ZA7 --0.217 in accor- dance with previous analyses of the A7 phase of ni- It is evident that the A7 structure is completely deter- mined by a specification of the independent structural parameters a~r, c~7, and zg7.The sc structure cor- responds to the special case where (c/a)+7 -+6 and ZA7= 4 1 The BP phase corresponds to the Cmca space group, with an end-or C-centered orthorhombic Bravais lattice, and four atoms per primitive cell.It is also a layered structure as seen in Fig. 2, however, the corrugated dou- ble layers develop perpendicular to the [100] sc directions, in contrast to the [111] directions in the case of A7.The unit cell vectors defined by Table II are a = aBp x, b = bBp y, and c = cBp z.In terms of these, the four f positions associated with the primitive cell are +(0, yBP zBP) +(0 z+yBP z zBP) while one possible choice for the primitive cell vectors is trogen.Full optimization of the internal parameter is expected to result in small second-order effects on the value of the total energy.Both (b/a) and (c/a) ratios were fully optimized for the BP unit cell near zero pres- sure; however, these equilibrium ratios were then used for calculations at all other volumes.Except as speci6- cally indicated, the internal parameters for the BP phase of nitrogen were fixed at their corresponding known val- ues in phosphorus: yBp = 0.10 aild zBp = 0.08.The equilibrium atomic volume Vc, bulk modulus Bo, and to- tal energy Eo for the A7 and BP phases were obtained from Murnaghan fits as in the case of the metallic phases, and are indicated in Table I; and the corresponding op- timized lattice constants, in Table II As a sensitivity analysis, we considered setting all bond angles equal to 102.2', while retaining the same bond lengths as in Table II.The choice of these parameters requires aBp = 2.405 A. , (b/a) Bp = 3.022, (c/a) Bp = 1.236, yBp = 0.099, and zBp = 0.087.We then carried out total-energy calculations as a function of volume with the lattice-constant ratios and internal parameters fixed at these new values.The resulting curve is quite similar to the BP result shown in Fig. 3, although shifted to 10' larger volumes, and reduced slightly (0.03 eV/atom) in equilibrium energy.The corresponding equilibrium quan- tities are Vo = 7.32 As/atom, Bc = 212.35GPa, and Eo = 1.52 eV/atom.The differences between these re- sults and those given in Fig. 3 and Tables I and II are sufficiently small that we have chosen to report the lat- ter results, obtained with yBp = 0.10 and zBp = 0.08, for which complete optimization of the lattice-constant ratios was carried out at equilibrium.The essential result indicated by Fig. 3 is that dis- tortion of the metallic high-coordination sc phase to threefold coordinated polymeric structures causes sub- stantial lowering of the energy with respect to the sc structure: by 0.89 eV/atom for the sc ~A7 distor- tion and 1.17 eV/atom for the sc ~BP distortion.This behavior is not unexpected: Previous calculations have indeed indicated that the A7 was a lower-energy phase than the sc structure (0.56 eV/atom as reported in Ref. 16), and the possibility of polymerization to a threefold-coordinated BP phase has also been previously suggested.5 However, our results constitute a predic- tion, on the basis of ab initio total-energy calculations, of the existence of a nitrogen BP phase exhibiting lower energy than the arseniclike A7 structure.
We now discuss an analogy between small N2X4 molecules and our hypothetical polymeric phases (A7 and BP) of nitrogen which has lead us to consider yet another polymeric form of nitrogen, a "cubic gauche" (cg) struc- ture, which exhibits all-gauche helicity.Isomers of N~X4 stoichiometry arranged in XqN -NXz configurations can be characterized by a dihedral angle P, defined as the angle by which one of the X~N groups must be rotated about the axis of the N -N bond in order to be located precisely above (i.e., eclipse) the other NXq group.It is believed that the juxtaposition of the lone-pair electrons on each of the two nitrogens has significant impact on the preferred value of P, 4 so that the optimal P may, to a first approximation, be viewed as a property of the N -N bond, independent of the attached Xz group.Re- sults of quantum chemistry calculations2 reveal common features of the energy curves E(P) as a function of the dihedral angle P: These curves tend to exhibit an abso- lute maximum at the cis (P = 0') conformation, absolute minima at gauche (P = +67' for NzF4 and P = +91' for N2H4) conformations, and a local minimum at the trans (P = 180') conformation. 24The local bonding topol- ogy associated with the A7 and BP structures can also be described in terms of such dihedral angles.The A7 structure is seen in Table II to be pure trans, while the BP structure has one-third of its bonds trans, and two-thirds gauche (equal numbers P = +76 and -76 ).Since the equilibrium BP energy in Table I is 0.28 eV/atom lower than that of the A7 structure, an obvious possibility, considering the small-molecule trends discussed above, is that an all-gauche structure might have lower energy yet.If one connects a sc array of sites using only three bonds per site and ensures that all such bonds have the same dihedral angle (for example, all -90'), one arrives at the cubic space group I213.
The cg structure belongs to the I213 space group, with a body-centered-cubic Bravais lattice, and four atoms per primitive cell.Unlike the A7 and BP distortions of sc, where the covalent bonds form corrugated double layers which are van der Waals bound to other such double layers, the cg structure is truly a three-dimensional covalent network, in that every atom is linked to every other atom by a continuous path of covalent bonds.A schematic rep- resentation of the cg structure is indicated in Fig. 2. The unit cell vectors in terms of the parameters in Table II are a= a,gx, b = a, gy, and c =a,gz.In terms of these, the four a sites associated with the primitive cell are cos ~cg &cg (&cg 4 ) 1 *!. + (~.g -4)' (10) The structure is completely defined by a specification of the parameters a, and x, z which permit the realization of any bond length d, g and bond angle 8,g.However, the unique dihedral angle P,g, sec(g, g) = sec(8, g) -1, is completely determined by the bond angle for this struc- ture, and vice versa.
The results of our total-energy ab initio pseudopoten- tial calculations for the cg threefold-coordinated polymeric phase of nitrogen are shown in Fig. 3, where the calculated total energy per atom is indicated as a func- tion of atomic volume.Extensive total-energy calculations were performed to determine the lattice constant a, g and internal parameter x, g yielding minimum-energy configurations at vanishing pressure and forces.These structural parameters are provided in Table II I. To give some idea of the sensitivity of the en- ergetics to x, g, the value z, g --0.048 leads to an equilib- rium energy Ep = 1.57eV/atom, bond length d, g --1.54The cg structure reduces to sc for x~z --0.The cg structure exhibits threefold coordination with all bonds of equal length given by FIG.
3. Polymeric phases of nitrogen Calculated t.otal energy per atom for the threefold-coordinated arsenic (A7), black-phosphorus (BP), snd cubic gauche (cg) phases of ni- trogen, snd for s twofold-coordinated chsinlike (ch) phase of nitrogen, as a function of atomic volume.A kinetic energy cutoÃ of Z,"~g = 80 Ry was used to calculate the total en- ergy for all the structures.The number of k points used for the calculations was A7 (28), BP (16), cg (18), snd ch (16).
The zero of energy corresponds to the minimum energy of the diatomic a-N2 (Pa3) phase, which is discussed in Sec.III D.
The simple-cubic (sc, dashed curve) phase is included here for comparison.
A. , and bond angle e, s --103, which may be compared to the optimized values in the tables.Inspection of Fig. 3 and Table I reveals that the fully optimized polymeric cg phase is lower in energy than the BP structure (by 0.58 eV/atom) which is consistent with the trend exhibited by the small N2X4 molecules in regard to the preference of the N -N bond for gauche dihedral angles.
More important, the polymeric cg structure is the lowest- energy atomic phase of nitrogen which has so far been found in any calculation, to our knowledge, and, as we discuss below, it has significant impact on predictions of the pressure-induced dissociation or polymerization of the ambient diatomic form of nitrogen Finally, it is also worth comparing the bond lengths and bond angles of the equilibrium A7, BP, and cg struc- tures in Table II to what is known from molecular sys- tems.Although the calculated N -N bond length in the A7 structure is somewhat large, the BP and cg bond lengths agree well with the 1.45+0.08A range observed for the single N -N bond in small molecules, especially compared to the length of a typical double (1.23 A in NzFz) or triple (1.10 ~in Nz) bond.Similarly, the bond angles in Table II lie within or close to the 100~- 1160 range of XNY bond angles encountered in small molecules where nitrogen forms single bonds.z In this regard, we have noted above that the equilibrium energy of the BP phase is very slightly reduced if the small 96.2' BP bond angle is increased to 102.2~.

C. Polymeric twofold-coordinated chainlike phases
We have also considered partially polymerized twofold coordinated chainlike structures where single (N -N) and double (N=N) bonds alternate.Within the confines of the present work, we restrict our analysis to only one simple example of such a structure.Our results in this case will hopefully provide an indication of the relative energetics of this class of partially polymerized chainlike phases in comparison to the fully polymerized threefold- coordinated phases, such as A7, BP, and cg, discussed above.
The hypothetical chainlike structure we have consid- ered was motivated by the prevalence of planar XN=NX molecules (e.g., where X is H, F, or CHs) of both trans and cis conformations.z We thus take a planar chain (ch) of alternating trans and cis bonds of lengths charac- terized by a single bond angle.One may, in fact, distort (via the lower-symmetry C222& subgroup) thecorrugated double layers of the BP structure so as to arrive at such chains pointing in the c direction.The resulting chainlike structure is described by the Cmcm space group, with a C-centered orthorhombic Bravais lattice, and four atoms per primitive cell.A schematic representation is indi- cated in Fig. 2. The unit cell vectors defined by Table II are a=a, ~x, b=b, "y, and c=c,hz.Thefour f positions associated with the primitive cell are +(0 ya", zch), +(0 ya &z,h), ( 12) while the primitive cell vectors may be taken as co = (1 o o) (g 2 0) (0 0 1) .
It is evident that the chain structure is uniquely spec- ified by a knowledge of the three lattice constants (a,h, b,h, c, h) and two internal parameters (y,h and z~).
The results of our total-energy cb initio pseudopoten- tial calculations for the hypothetical twofold-coordinated chainlike (Cmem) phase of nitrogen are indicated in Fig. 3 where the calculated total energy per atom is given as a function of atomic volume.Only the lattice con- stant a,h was varied, with fixed ratios (b/a)~= 3.16   and (c/a), h = 1.45, and similarly, fixed internal param- eters y,~= 0.072 and z,~= 0.045.These values were chosen so that the c-direction chains would exhibit 106 bond angles (characteristic of FN=NF), and for one value of a,h, the two different bond lengths would equal typ- ical double (1.23 A) and single (1.53 A. ) nitrogen bond lengths, respectively, as seen in small molecules.z The ratio (b/a), ~was taken simply to be that of the equilib- rium BP structure.It is evident from Table II that the minimum-energy structure as a function of a,~does in- deed yield bond lengths quite close to those adopted by small molecules.We also carried out similar calculations for internal parameters (y,h = 0.083, z,h = 0.057) which would allow more nearly equal bond lengths, taking in this case both ratios (c/a)~a nd (b/a), ~to be the same as the BP equilibrium values.These results gave a curve very similar to that in Fig. 3, shifted to slightly smaller volumes.The equilibrium parameters were Vo = 8.99 A.s/atom, Bo --176.35GPa, and Eo = 1.23 eV/atom in this case.
In spite of the lack of optimization, there are impor- tant conclusions to be drawn from the comparison of the twofold-coordinated chainlike structure with the other re- sults shown in Fig. 3. First, while it has a relatively low equilibrium total energy Eo, the ch curve appears shifted to higher volumes in Fig. 3, relative to the threefold coor- dinated polymeric phases.One consequence of this fact is that it never drops below any of the diatomic curves, raising the possibility that low-temperature diatomic nitrogen may transform under pressure directly to one of the threefold coordinated polymeric phases, without any intermediate regime of chainlike behavior.Secondly, at atomic volumes exceeding approximately 8.5 A.s/atom, the chainlike structure has the lowest energy among the candidate polymeric phases considered here.Since this is the volume regime probed by high-temperature shock compression experiments, it is possible that the formation of chainlike nitrogen fragments may play a role in the anomaly observed in that data.

D. Diatomic phases
Nitrogen at low temperature and atmospheric pressure is a diatomic molecular solid or liquid.Below 35.6 K, n- Nz exhibits the cubic Pa3 structure (4Nz molecules/cell) characterized by molecular axes oriented along the body diagonals of the cubic unit cell. 8At low temperature and at a pressure of 0.35 GPa, the o'-N2 phase transforms into the tetragonal p-N2 characterized by the P4q/mnm space group (2 N2 molecules/cell) with the N2 molecules arranged in layers with a common orientation but with a shift of 90 between adjacent layers. 4Raman data indi- cate a structural transition at 1.9 GPa from tetragonal p- Ng (P4g/rnnm) to s-N2 which exhibits the rhombohedral RSc structure (8 Nz molecules per primitive cell).2s The e-Nz (R3c) phase is known to persist up to 20 GPa above which pressure the structure of nitrogen is unknown, al- though it is likely to be a lower-symmetry distortion of the R3c structure.
Because of our interest in the pressure-induced dis- sociation (or polymerization) of the observed diatomic form of nitrogen, the high pressure s-Nz (RSc) phase is of particular importance to our transition-pressure cal- culations of the next subsection.Furthermore, while the diatomic structure is not precisely known above 20 GPa, we consider a selection of other diatomic structures char- acteristic of nearly spherical molecules like Nq in order to get some sense of the uncertainties.In particular, we also carry out calculations here for the cubic n-N2 (Pa3) structure and the rhombohedral P (Rsm) structure (one molecule per primitive cell) observed for solid Oq.The structural data for each of the s-Nq, cr-Nq, and P-02 structures is given in Table III.
The low-temperature and pressure o;-Nq modifica- tion of nitrogen is described by space group Pa3, with a simple-cubic Bravais lattice, and eight atoms (four molecules) per primitive cell.As noted above, the four N2 molecules orient along the four cube body diagonals.4s They undergo orientational librations as well as center-of- mass vibrations about their equilibrium positions.The conventional unit cell described by Table III  Clearly x~N, =0 gives the center-of-mass positions, which are seen to form a face-centered-cubic lattice.The e-Nz (R3c) phase is the observed low-temperature structure of nitrogen between 1.9 and 20 GPa.It is con- veniently described in terms of the higher-temperature disordered cubic b-Nz structure which was first discov- ered at 300 K and 4.9 GPa.so The b-Nz phase has a disordered cubic structure with space group Prn3n and 8 Nq molecules within the simple-cubic cell: There are six molecules with disklike disorder and two with spher- ical disorder.The b-s transition occurs through the ori- entational ordering and small displacement of the Nq molecules, accompanied by a slight extension of the cubic unit cell along the cube diagonal, resulting in a rhombo- hedral cell characterized by a small reduction of the angle between the axes from 90 to approximately 85 .
A detailed description of the  The name is given for each structure, along with the space group, the ratios b/a and c/a of the magnitudes of the unit cell vectors, the unit cell multiplicities and Wyckoff site letters, and the corresponding dimensionless internal parameters 2:, y, z.Hexagonal axes are used for rhombohedral R3m and R3c.The N2 bond length was taken to be d = 1.10 A. (0, 0, z N ), (0, 0) z+z N ),

Structure
(15a) and the 12 (6 molecules) f sites associated with the prim- itive cell are R3c, with a rhombohedral (trigonal) Bravais lattice, and 16 atoms (8 molecules) per primitive cell.The hexagonal unit cell vectors presumed by Table III are the same as in Eq. ( 2), with substitution of the appropriate e-Nz lattice constants, and similarly the primitive vectors are given by Eq. ( 4).In terms of the unit cell vectors, the four (two molecules) c sites associated with the primitive cell are (2 where as before these expressions are to be evaluated using +d in Table III to obtain the two atoms in each molecule.Note that the c and f sites are analogues of the "spherical" and "disklike" sites, respectively, in the high-temperature cubic 6-N2 structure.The e-Nz struc- ture is characterized by the six independent parameters, two lattice constants (a,N"c,N, ), and four internal pa- rameters (z,N, $, N, , y,N. ..N, ).
The P-Oz structure with one N2 molecule per prim- itive cell corresponds to exactly the same R3rn space group and same set of site positions as the arsenic A7 structure described by Eqs. ( 2) -( 4).The important dif- ference is that the internal parameter zpci, = d/(2cpo, ) is small for P-Oz, and is a measure of the diatomic bond length d, in contrast to z~q ~4.The diatomic P-Oz structure can clearly be continuously transformed into the polymeric A7 structure by varying the inter- nal structural parameter z.The P-Oz structure may be viewed as closed-packed hexagonal planes of molecules, with the molecular axis aligned perpendicular to the planes stacked in the . . .[ABC]. . .sequence character- istic of the fcc structure.is In our metastability calcula- tions of Sec.IV, we also consider the hcp analogue, i.e. , an . . .[AB].. .stacking sequence, described in Table III as the "two-layer P-Oz" structure.We emphasize that there is no evidence for the existence of a P-Oz phase for diatomic nitrogen.However, it is useful to perform total- energy calculations for this structure observed in oxygen at low temperatures and pressures, since the Nz and Oz molecules exhibit similar degrees of prolateness, and so the P-Oz structure is not unreasonable to consider for diatomic nitrogen.Moreover, the ab initio total-energy results presented below for the P-Oz phase of nitrogen can serve as the basis of a comparison with theoretical analyses reported earlier on this structure using similar methods.'6 Results of our ab initio pseudopotential total-energy calculations for the a-N2, e'-N2, and P-Oz phases of ni- trogen are indicated in Fig. 4 where the calculated total energy per atom is shown as a function of atomic volume.
For all calculations performed for the diatomic struc- tures, the intramolecular N-=N bond length was fixed at the value d = 1.10 A, extracted from an independent 0- Ry was used to calculate the total energy for all the struc- tures.The number of k points used for the calculations was n-N2 ( 11), e-N2 (2 and 10 in tests), and P-02 (10) The zero of energy corresponds to the minimum energy of the n-Np phase.
set of ab initio pseudopotential total-energy calculations performed for an isolated N2 diatomic molecule.The calculated value of the Nz intramolecular bond length d'~" = 1.10 A. is in perfect agreement with the experi- mental value d'"o' = 1.10 A.. 2 The internal parameters in Table III defining the structure of the diatomic phases shown in Fig. 4 were therefore determined by requiring that d = 1.10 A. for all atomic volumes.For the cubic cs-Nz phase, this determination of d uniquely specifies the internal structural parameter z N, as is indicated in Table III.In accordance with previous theoretical analy- ses of the e-Nz phase, 2~is our ab initio total-energy cal- culations were performed with the rhombohedral angle a = cos i(ao bo) fixed at the value o. ,N, = 85.72, or equivalently (c/a), N, = 1.3649, and the internal param- eters specifying the center of mass and the orientation of the N2 molecules kept constant at the values theoret- ically determined for this structure at 75 GPa.si For the P-Oz phase, a value of (c/a) po, = 3.0 was adopted at all volumes.This value is close to that used in pre- vious theoretical analysesis and we have not felt it nec- essary to perform extensive energy-minimum structural searches for the P-Oz phase since it is not experimentally observed in nitrogen.Moreover, the structural parame- ter which affects the energy most significantly is the bond length d of the diatomic molecule.
Inspection of Fig. 4 reveals that our calculated e-N2 curve drops below that of the other two diatomic results as volume is decreased away from the right side of the fig- ure, in qualitative agreement with experimental observa- tions.The e'-Nz phase continues to be the lowest-energy diatomic form in the range V = 8 -9 As/atom which will be crucial to our calculated dissociation or polymeriza- tion transition.Equally important is the fact that the differences in energy between the three diatomic phases considered here are relatively small (& 0.1 eV/atom) in this region.We take these differences to be a measure of that contribution to the uncertainty in our diatomic equation of state in this range which is due to our lack of knowledge of the true crystal structure.Specifically, we consider the possibility that the correct structure may have an energy ~0.1 eV/atom lower than that of our calculated e-Nz curve in the vicinity V = 8 -9 A. /atom.
Another source of error in our diatomic results at large volume is evident in Table I, where values of the equilibrium (p = 0), atomic volume Vo, bulk modulus Bo, and total energy Eo for the optimized n-Nz and P-Oz structures of Nz are given.It may be seen that our calculated values of Vo and Bo for the u-Nz struc- ture are 15% smaller and 3.5 times larger, respectively, than the experimental values for this phase, 52 Such dis- agreement is characteristic of local-density approxima- tion errors incurred at large volumes for van der Waals bonded materials.For example, in the case of solid fcc argon, local-density-functional calculations yield val- ues of Vo and Bo which are 21% smaller and 2.9 times larger, respectively, than experimental values.i A com- parison to diamond-anvil-cell measurements, however, shows a signi6cant improvement in the theoretical vol- ume (only 8% smaller than experiment) by a pressure of p 5Bo 14 GPa.Thereafter, a more gradual improve- ment is seen, with the theoretical volume about 5% small by 80 GPa, the highest pressure reached in these experi- ments on argon.
We believe this same type of behavior is evident in the present results for the diatonuc phases of nitrogen.
To quantify better the present uncertainties for our Nz phases, a comparison between our calculated pressure for the e-Nz phase (solid curve labeled "e-Nz" ) and the 100 K data for this phase taken by Mills et al. zs is indicated in Fig. 5. Our theoretical volume is about 10% and 9% smaller than experiment at the limits of this data, 6 and 13 GPa, respectively.This also is about a factor of 2 improvement by p 5Bo in comparison to the a-Nz re- sults at p = 0, as was the case with argon, and we expect analogous gradual improvement at higher pressures.The dashed curve in Fig. 5 is a semiempirical e-N2 equation of state from which we may extract a quantitative measure of the consequence of these local-density approximation, van der Waals errors on the dissociation or polymerization transition to be described in the next subsection.It is a Murnaghan fit to the data of Mills et a/.which is constrained to agree with theory at 500 GPa and, with the argon example in mind, to have a volume 5% larger than the local-density result near 80 GPa.Finally, the equation of state of the cg polymerized phase is also pre- sented (solid curve labeled "cg-N") in Fig 5for corn.par- ison.It is to be emphasized that local-density-functional theory generally provides excellent equation of state re- sults (volume errors (2%) for group-IV and -V fourfold of nitrogen.The dashed curve is a semiempirical e-Nz equa- tion of state constrained to agree with the theory at 500 GPa, and with the experimental data of Ref. 26 (data points) below 13 GPa.and threefold-coordinated phases, and so we expect the primary uncertainties in the present numerical results to arise from the diatomic calculations.

E. Polymerization of nitrogen
The calculated equations of state for the polymeric (Fig. 3) and diatomic (Fig. 4) phases of nitrogen allow a determination of the transition pressure at which the Nz molecules in diatomic solid nitrogen should dissoci- ate, or more specifically, polymerize to a single-bonded threefold coordinated network.The transition pressure for the zero-temperature c-Ng -+ cg-N polymerization reaction is given by the slope of the common tangent to the diatomic and polymeric total-energy curves as a function of volume, as shown in Fig. 6, or equivalently by the crossing of the corresponding Gibbs free-energy curves as a function of pressure.Our results suggest the relatively low transition pressure of p = 50 + 15 GPa, above which a polymeric form of nitrogen should be the thermodynamically stable phase of nitrogen.However, because of the large barriers between diatomic and poly- meric forms of nitrogen discussed in the next section, it may be necessary to overdrive this pressure significantly and/or heat the sample in order to observe the transition experimentally.We have noted that in room-temperature experiments, the diatomic form appears to be stable at pressures up to 180 GPa.
Because of this compli- cation, it is important to establish clearly the theoreti- cal uncertainties in order to rule out the possibility that the equilibrium phase boundary might occur at a signif- Atomic Volume (JP / atom) FIG. 6. Polymerization of nitrogen.The solid curves give the calculated total energy per atom for the polymeric cg-N and diatomic e-Na phases of nitrogen as a function of atomic volume.The dashed curve is the semiempirical e-Ng total energy, corresponding to the dashed curve in Fig. 5.The common tangent to the two theoretical curves (dotted line) gives an e-Nq -+ cg-N transition pressure of 33 Gpa, whereas substitution of the semiempirical result for e-N2 increases the pressure to 50 GPa.i5 icantly higher pressure than predicted.We believe our stated uncertainties to be realistic, and the focus of the discussion in this subsection is our determination of these values.
We begin by making a quantitative comparison of the present calculations with other relevant local-density- (LD) functional results for nitrogen.Our LD total en- ergy for the isolated nitrogen pseudoatom ELD is given in Table IV, along with the corresponding spin-polarized The zero of energy is taken to be the LD equilibrium energy of the diatomic n-N2 phase.
energy Er sD ss .Note that both in this table and through- out the present work we take the zero of energy to be our LD calculated equilibrium energy for the n-N2 phase of nitrogen, and we omit all zero-point contributions.
The difFerence E~h --ErNsD -EP = 3.13 eV/atom is our predicted cohesive energy for the sc phase of ni- trogen, which may be compared with the value of 2.88 eV/atom obtained from linear muffin-tin orbital calcula- tions using a different exchange-correlation potential.is A similar 0.2 eV/atom difFerence (8.63 vs 8.43 eV/atom) exists between E«h values obtained in analogous LD calculations for diamond-phase carbon, s~I so that this agreement is consistent with difFerences due to exchangecorrelation potentials and/or shape approximation efFects (in Refs.15 and 58).Another point of comparison is the dissociation energy D, = 2 (EPsD -Er D) of the N2 molecule, where the factor of 2 is required since all of our total energies are given per atom.We also use our LD energy EzDI here for the isolated pseudomolecule, as spin-polarization corrections should be unimportant for the approximately full shell configuration of the N2 molecule.Our result for the dissociation energy of an iso- lated Nz molecule is therefore D, = 11.58 eV/molecule which may be compared to a recent augmented-plane- wave value of 11. 5 eV/molecule.M Since the cohesive energy E, &' --Ei DI -E& ' of the van der Waals bonded o.-N2 solid is quite small, the difFerence 2D, E, "h pro-- vides a measure of the separation between our monatomic and diatomic curves.In this regard, our present results are seen to be in relatively close quantitative agreement with other LD calculations for nitrogen.
We now consider errors associated with the LD approx- imation itself.It is well known that such calculations overestimate the cohesive energies for certain classes of solids.The error is largest by far for second-period covalent nets, 2 about 1 eV/atom for both o;-12 boronsc si and diamond-phase carbon, sr and we expect a similar error for threefold-coordinated nitrogen.It is also gen- erally believed that the error lies mostly with the local- spin-density (LSD) calculation of the atom in the case of such covalent nets, a belief which has been specif- ically confirmed by quantum Monte Carlo calculations in the case of diamond-phase carbon.sr Moreover, the p(V) equations of state and those bulk properties not in- volving the atom (Vo and Bs) are generally in relatively good agreement with experiment for these materials, es- pecially in contrast to van der Waals bonded molecular or rare-gas solids, as, for example, the argon results men- tioned above.ss We therefore believe our LD cg-N results in Fig. 6 to be accurate in compression, and that the significant LD (or LSD) errors occur at large expansion (V )) Vas = 6.67 A. /atom), culminating in an atom en- ergy E&&D which is 1 eV higher than the true atom en- ergy E,"~t.In sharp contrast, the cohesive energy E, h' of the diatomic solid is sufBciently small that LD errors in this quantity are unimportant.We obtain E h' --0.06 eV/atom, which may be compared to the adjusted ex- perimental value of 0.08 eV/atom, i from which we have removed the zero-point contribution.The significant diatomic LD errors are therefore more likely to appear in compression, which is precisely the implication of the LD- experiment p(V) comparison for e-Nq indicated in Fig. 5.
Specifically, one may expect these diatomic errors in the range 3 A.s/atom ( V ( Vo ', since the inter-and in- tramolecular distances become comparable at the lower end of this range, at which point LD theory should be as accurate for the "diatomic" as for the polymeric or metallic phases. There should be, and is, consistency between the monatomic and diatomic LD and LSD errors for nitrogen.Our semiempirical c-Nq equation of state (dashed curves in Figs. 5 and 6) has been constrained to equal our LD s- Nz results near V 3 A. /atom (actually at 3.5 A. /atom or 500 GPa) where we expect negligible LD errors in the difference E's(V) -E'N'(V).The total energy of this semiempirical equation of state is -0.43 eV/atom at equilibrium, which, given the experimental a-Nz cohesive energy, provides the estimate E,"', -0.35 eV/atom.This value in turn suggests EN, 4.60 eV/atom, based on the experimental dissociation energy of 9.90 eV/molecule, sz and therefore a 1.25 eV/atom error in the LSD atom energy.This 1.25 eV/atom error is both consistent with the above-mentioned ~1 eV/atom E«h errors for boronsos and carbon, 7 and follows indica- tions that it should be somewhat larger for the case of nitrogen. 63  Turning to the consequence of these various errors on the transition pressure, we note first that the ID prediction (the common tangent sketched between the two solid curves in Fig. 6) is 33 GPa.If the diatomic result is replaced by our semiempirical e-Np equation of state (dashed curve), the transition pressure is then 50 GPa.If the separation between these two curves (solid cg-N and dashed e'-N2) is increased by a further 0.2 eV/atom, reflecting perhaps a larger error in ELsD and/or a dif- ferent diatomic structure with lower energy in the vicin- ity of V = 8 A /atom in addition to zero-point energy corrections, 56 the transition pressure would increase to 65 GPa.In consideration of the above uncertainty anal- ysis, we believe the most reliable value for the transition pressure to be 50 GPa, with uncertainties of about +15 GPa.
The establishment of the equilibrium phase boundary between polymeric forms of nitrogen and the observed diatomic phases at relatively low pressure is one of the central results of the present theoretical study.The es- sential conclusion that the N=N triple bond will be desta- bilized at lower pressures than the N -N single bond has also been suggested in previous theoretical work on high- pressure phases of nitrogen. 5 6All theoretical analyses, including the present one, indicate that nitrogen should transform from its ambient diatomic form to an interme- diate, yet still covalent, polymeric regime before losing its covalency altogether at still higher pressures.Two earlier calculations have placed the transition to the polymeric regime at pressures below 100 GPa, the lower estimate being at 70 GPa.Our predicted 50 + 15 GPa transformation pressure is based on a polymeric struc- ture which is significantly lower in energy than those previously considered; however, we include for the erst time estimates of errors in the diatomic equation of state, which partially onset the reduction in pressure im- plied by the new lower-energy polymeric phase.Experi- mentally, diamond-anvil-cell experiments up to 130 and 180 GPa have suggested that nitrogen remains diatomic to these pressures at room temperature, however ~8 ~9 A possible explanation for the existence of diatomic nitro- gen well above our predicted diatomic-to-polymeric transition pressure may be that the diatomic phase is itself metastable at these pressures.This explanation is not singular to the case of nitrogen.In fact, the graphite to diamond transition in carbon must be overdriven by about a factor of 7 in pressure even at a temperature of 1000 K. o Moreover, an anomaly has been seen above 30 GPa in hot shock compressed Quid nitrogen, zi which has been interpreted as dissociation to a dense atomic phase, ~2 and which is consistent with the theoretically predicted phase transition at low temperatures. 23  IV.ATMOSPHERIC-PRESSURE METASTABILITY OF POLYMERIC NITROGEN If diatomic nitrogen persists metastably at room temperature to pressures far above (by a factor of 3) the equilibrium diatomic-polymeric phase boundary, such hysteresis raises the possibility that the polymeric form, once synthesized at high pressure, might also be metastable at atmospheric pressure.We now consider the prospects for atmospheric-pressure (p ~0) metasta- bility of the cg, BP, and A7 polymeric phases of nitrogen discussed in the previous section.One may demonstrate mechanical stability of a given phase by showing that all phonon frequencies are real, which is equivalent to demonstrating a local minimum in the total-energy sur- face as a function of the atomic coordinates.For a struc- tures such as cg or BP, with 4 atoms in the primitive cell and 12 modes at each point in the Brillouin zone, such calculations would be rather challenging for direct methods in which atoms are actually displaced, so that a linear-response treatment would be more suitable.ss Neither approach, however, would give barrier heights, a crucial parameter in any investigation of metastable lifetime.
We therefore follow a difFerent approach, motivated by the collective aspect of martensitic transformations.Specifically, we identify transformation paths (described in the Appendix) between a polymeric phase and a lower- energy phase (either diatomic or polymeric), and then carry out ab initio pseudopotential total-energy calcula- tions along such paths in order to determine potential barrier heights inhibiting the transformation.As in our previous work on carbon, 6 we use high-symmetry sub- groups common to the space groups of the initial and final structures to help select the path.In addition, our experience with the carbon calculations has shown a di- rect correspondence with such calculated barrier heights and the number of bonds broken during the course of the transformation.
Therefore, a realistic calculation of the barrier height requires paths which involve no unneces- sary bond breaking.In regard to the potential instability of the atmospheric-pressure threefold-coordinated poly-meric phases towards dimerization, for example, we seek paths which break only two of the bonds, with the third evolving continuously into the diatomic bond of the Nz molecule.We have adopted such paths unless specifically indicated otherwise.In considering the dimerization of the polymeric phases, we have taken the simplest of the diatomic structures, the P-02 structure, in the case of BP and A7 initial structures, and a distortion of its two- layer variant (see Appendix and Table III) for cg, as the target phase.For convenience, we shall refer to the latter distortion of the two-layer variant as the "P'-02" struc- ture.It is to be emphasized that we make no attempt to conclusively prove metastability, but are rather check- ing against the possibility of the kind of generic instabil- ity that is found, for example, in the high-coordination phases of carbon at atmospheric pressure.
We see no evidence for such behavior, and so our results are consis- tent with the possibility that each of the A7, BP, and cg polymeric phases may be metastable at atmospheric pressure.
The results of our stability analyses are indicated in Fig. 7 where the calculated total energy per atom is shown as a function of the reaction path describing the cg ~P'-Oz, BP -+ P-O2, and A7 ~P-Oz transforma- tions.The cg ~P'-02 transformation is mapped within the P2i2i2i space group and involves eight atoms per primitive cell (two sets of a sites).The BP ~P-Oz trans- formation is mapped within the C2/rn space group and (cubic gauche, black phosphorus, A7) ~P-0 FIG. 7. Metastability of polymeric nitrogen.Calculated total energy per atom as a function of the reaction coordi- nate for cg -+ P'-Os (solid line), BP ~P-02 (dashed line), and A7 ~p-02 (dash-dotted line) transformations, where P'-02 designates a distorted two-layer variant of the P-02 structure.Detailed characterization of these transformation paths is given in the Appendix.These are approximately zero- pressure paths, with the starting atomic volumes equal to the equilibrium values for the polymeric phases (6.67, 6.34, and 6.49A /atom for the cg, BP, and A7 phases, respectively), and the final volume the equilibrium value for the P-02 phase (19.67 A /atom).A kinetic energy cutoff of E,"t e = 80 Ry was used for all the calculations.The numbers of special k points used were 18, 32, and 10 for the cg, Bp, and A7 trans- formations, respectively.The zero of energy corresponds to the diatomic P-02 structure.involves four atoms per primitive cell (two sets of i sites).
The A7 ~P-02 transformation is mapped within the 83m space group and involves two atoms per primitive cell (one set of c sites).Along the transformation path, we define a transformation variable ( which continuously evolves the structure from the initial polymeric phase (( = 0) to the final diatomic phase (( = 1).A linear transition path is defined by specifying the continuous z-(() = (1 -()z; + (z.', where the superscripts i and f indicate initial (cg, BP, or A7) and final (P-Oz or P'-02) structures, and a,b, c,zi, . . ., z"are the structural parameters appropriate to the space group of the transformation path (P2i2i2i, C2/rn, or R3rn).As described in the Appendix, our transforma- tions for the cg and BP cases deviate somewhat from such linear paths in order that the three equal bond lengths in the initial polymeric phase evolve into only two unique lengths d(() and d'(() along the path, i.e. , a shorter incip- ient Nz bond length, and two equal longer "broken-bond" lengths.Our A7 path, on the other hand, is strictly lin- ear, and all three bonds are broken during the course of the transformation, in that the final diatomic bond is formed between atoms which were not near neigh- bors at the start.This path is nevertheless of inter- est given that it is achieved without reduction in spacegroup symmetry, and is analogous to the constant volumepath investigated by Martin and Needs at a volume of 5 As/atom.'s Note that all of our transformation paths correspond to approximately constant-pressure transfor- mations.More precisely, the initial and fina structures correspond strictly to p = 0, while the intermediate struc- tures given by Eq. ( 16), with modifications as specified in the Appendix, may be expected to be p ~0. Accord- ingly, the atomic volume varies along the transformation paths from the equilibrium volumes for the polymeric phases V, 's --6.67 A. /atom, VBP = 6.34 As/atom, or V&7 --6.49 A. /atom to the equilibrium volume for the diatomic phase V&o --19.67 A.s/atom.
Inspection of Fig. 7 reveals that all three curves E(() exhibit positive curvature in the polymeric limit ( = 0, indicating mechanical stability insofar as these modes are concerned.We have also verified that the curves rise in energy at small negative values of (, which cor- responds to the formation of chains in the case of the cg and BP paths.These results are therefore consistent with the atmospheric-pressure stability of the cg and BP phases of polymeric nitrogen against either N:-N dimer- ization or the formation of N -N=N chains, albeit demon- strated here only for select, but physically natural, paths. The actual barrier heights are AEb(cg ~P'-02) = 0.86 eV/atom and AEy(BP -+ P-02) -0.59 eV/atom, which may be viewed as the energy required to break two bonds at each N atom followed by their reconstitution as part of the new triple bond in the remaining bond direction.For the A7 path, on the other hand, the three near-neighbor distances change but remain equal for increasing (, until for ( 0.75 a new neighbor becomes the closest.This observation is in good correspondence with the previous constant-volume work indicating that the maximum en- ergy along this path occurs at this point of approximate fourfold coordination.s In our case (p 0) this leads to a barrier AE~(A7 ~P-Oq) -1.65 eV/atom, which reflects three broken bonds, and is therefore expected to be higher than the other two barriers cited.
We have also considered possible paths between the three polymeric phases.The A7 and BP structures are both corrugated double layers, as discussed in Sec.III.
An A7 ~BP transformation requires either bond break- ing or bending.In the first case, one may imagine re- assembling the A7 double layers to form something like the sc parent, which may then be separated into new BP-like double layers parallel to one of the sc [100] di- rections.The barrier would be roughly the sc -A7 energy difFerence of Table I, or about 0.9 eV/atom.In the sec- ond case, one may change the topology of the individ- ual corrugated double layers by bending bonds, as may be accomplished within the C2/m subgroup of the two structures (two sets of 4i sites per unit cell).We have performed calculations along such a path and found a barrier AEb(A7 -+ BP) -0.36 eV/atom (32 special k- point samplings).These numbers are consistent with the expectation that bond breaking should be more energet- ically costly than bond bending.With this in mind, it is to be noted that the bond breaking followed by re- bonding is essential for any path between the A7 or BP structures and the fully three-dimensional covalent net exhibited by the cg phase of nitrogen.We therefore ex- pect significant barriers inhibiting any A7 ~cg or BP -+ cg transformations.The present metastability calculations are clearly lim- ited and certainly not conclusive.Yet it is to be empha- sized that we see no evidence for mechanical instabili- ties in any of the polymeric phases of nitrogen at atmo- spheric pressure, and instead, rather large energy bar- riers inhibiting transformations from one to another as well as to lower-energy diatomic forms.In particular, we note the 0.86 eV/atom barrier inhibiting dimerization of the atmospheric-pressure cg phase.Moreover, it is pos- sible on the basis of these results to rule out the kind of generic instability which is seen in theoretical calcu- lations for high-coordination structures of atmospheric pressure carbon.
V. SUMMARY cation of low-energy polymeric phases of nitrogen, cal- culation of their equations of state, and on an ab initio pseudopotential calculation of the equation of state for the observed high-pressure e-N2 diatomic phase.Most important among these polymeric nitrogen phases are the black-phosphorus structure and a cubic phase character- ized by the occurrence of all-gauche dihedral angles.The latter cubic gauche (cg) structure was found to be the lowest-energy modification among atomic forms of nitro- gen, and we locate its minimum energy 0.86 eV/atom below the previously investigatedis arsenic (A7) phase.
The e-Nq structure was found to have the lowest en- ergy among the diatomic phases considered at modest compression, in agreement with experiment.Moreover, we have estimated the local-density-functional errors in these results based on a comparison of the present and re- lated theoretical results with available experimental data for both nitrogen itself, as well as for other materials whose bonding is similar to either the polymeric or di- atomic phases of nitrogen.These considerations are in- corporated into our predicted e-Nz -+ cg-N transition pressure of 50 + 15 GPa, a modest pressure on the scale of present diamond-anvil-cell capabilities.We believe the diatomic form of nitrogen that has been observed at room temperature and pressures up to 180 GPa, 9 to be metastable at these conditions.Such hysteresis raises the prospect that a polymeric form of nitrogen, if synthesized at high pressure, might also be metastable at atmospheric pressure.With this possibil- ity in mind, we have carried out total-energy calcula- tions along selected transformation paths between vari- ous polymeric phases, and from these to a simple, lower- energy diatomic phase.In all cases we find significant barriers inhibiting these transformations, and in particular, a 0.9 eV/atom barrier inhibiting dimerization of the cg polymeric form of nitrogen at atmospheric pressure.While by no means compelling, these results are certainly consistent with the existence of a metastable polymeric form of nitrogen at atmospheric pressure.
Note added in proof.Recent room-temperature x-ray difFraction measurements on compressed diatomic nitrogen to 44 GPa [H.Olijnyk, J. Chem.Phys.93, 8968   (1990)]yield a p(V) curve within k2 GPa of our semiem- pirical result in Fig. 6, while Raman data [H.Schneider, W. Hafner, A. Wokaun, and H. Olijnyk, J. Chem.Phys.96, 8046 (1992); are consistent with a diatomic phase which is still only a slight distortion of the R3c e'-N2 structure at 54 GPa.Quantum Monte Carlo calculations [L.Mitas and R. M. Martin (private communication)] also suggest a larger LSD error for the total energy of the nitrogen atom than our 1.25-eV estimate of Sec.III E. If there are no compensating LD errors in the equilibrium total energy of the cg phase, this would lead to a larger c-N2 -+cg transition pressure than predicted here.
Results of ab initio pseudopotential total-energy cal- culations indicate that the equilibrium phase bound- ary between single-bonded, threefold-coordinated polymeric forms of nitrogen, and the observed, triple-bonded diatomic phases may occur at relatively low pressure, 50+15 Gpa.This observation is based on the identifi-

ACKNOWLEDGMENTS
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore Na- tional Laboratory under Contract No. W-7405-Eng-48, and has been supported in part by the Joint DoD/DOE 2&" *lg+(~g -4)   and similarly all bond angles of equal value given by FIG.4.Diatomic phases of nitrogen.Calculated total en- ergy per atom for the a-Nz (Pa3), e-Nz (R3c), and P-02 (R3m, dashed curve) diatomic phases of nitrogen as a func- tion of atomic volume.A kinetic energy cutofF of E,"& p --80 FIG. 5. Pressure-volume equations of state.The solid curves give the calculated pressure as a function of atomic volume for the polymeric cg-N and the diatomic e-Nq phases Total energies of selected equilibrium (p=0) nitrogen solids and isolated (V = oo) nitrogen atoms and mpiecules.All energies are in eV/atom, exclude zero-point vi- brational contributions, and are local-density (Eo and Ei,o), local-spin-density (E+$D), or experimental (E«I,I) results

TABLE I .
phase, whichis discussed in Sec.III D, and included here for comparison.Ground-state properties of nitrogen phases.

Table II areTABLE II .
Structural characterization of theoretically calculated polymeric phases of nitrogen at atmospheric pressure, of the BP bonds have lengths equal to 1.54 A.

Table II .
. Our results for the A7 structure of nitrogen are in good correspondence with ab initio pseudopotentialis analyses which have appeared in the literature for this phase.We are not aware of any previous theoretical analyses for comparison in the case of the BP structure, however.One set of test calculations was carried out for the BP structure with internal parameters different from those in The choice was motivated by the 96.2~b ond angle for BP in this table, which is somewhat smaller than N -N bond angles typically found in small molecules.2 , and the corresponding values of the equilibrium atomic volume Vp, bulk modulus Bp, and total energy Ep are indicated in Table is primi- tive, with ao -a = a~N, x, bo -b = a~N, y, and co -c = a N, z.Four of the c sites are given by for this structure just changes the sign of x N, . which

TABLE III .
Structural characterization of theoretically calculated diatomic phases of nitrogen.