J . Phys. Chem. 1989, 93, 5122-5727
5722
Five Stable Points on the N6 Energy Hypersurface: Structures, Energies, Frequencies, and Chemical Shifts' Ray Engelke Los Alamos National Laboratory, M S P952, Los Alamos. New Mexico 87545 (Received: January 31, 1989)
Five stationary points on the N6 energy hypersurface have been located by use of ab initio self-consistent field (SCF) methods. These five points correspond to the N6 analogues of the (CH), structures: (a) benzene (I), (b) Dewar benzene (2), (c) benzvalene (3), (d) triprismane (4), and (e) trans-3,3'-bicyclopropenyl ( 5 ) . The points on the energy hypersurface are calculated in the restricted Hartree-Fock approximation, using 4-3 1G and 4-3 1G* basis sets. At the stationary points, vibrational frequencies have also been calculated at the RHF/4-31G and RHF/4-31G* levels. Within the RHF/4-31G* model, all the structures no corresponding stationary point is found with the RHF/4-31G model. The are stable. For trans-3,3/-bicyclopropenyl, five structures are higher in energy than three N 2 molecules by (1) 243, (2) 289, (3) 303, (4) 382, and (5) 286 kcal/mol with the RHF/4-31G* model. The effect of electron correlation on the energies is examined by use of the MP2=FC/431G*//RHF/4-31G* and MP2=FC/4-31G//RHF/4-31G models. In most cases, the inclusion of correlation stabilizes the structures to dissociation into three N2 molecules by 30-50 kcal/mol (relative to the mean field results). The lowest (nontorsional) RHF/4-3 1G* vibrational frequency for 2-5 is significantly larger than that of 1, indicating that the geometries of 2-5 are more rigidly defined by the energy hypersurface than is 1's. Nitrogen chemical shifts are also presented; these were calculated at the RHF/4-3 1G level using gauge-invariant atomic orbitals and SCF perturbation theory. Most calculations were also performed on diatomic nitrogen (N2),diimide (N2H2),and hydrazine (N2H4);these results are used as an aid in the interpretation of the N, results
I. Introduction An a b initio quantum theoretical study of five stable points on the N6 energy hypersurface is presented. These five stable points correspond to the nitrogen (hexaaza) analogues of the (CH), hydrocarbon structures: (a) benzene ( l ) , (b) Dewar benzene (bicyclo[2.2.0]hexa-2,5-diene, 2), (c) benzvalene (tricyclo[3.1 .0.02!6]hexene, 3 ) , ( d ) triprismane (tetracyclo[2.2.0.0236.0335] hexane, 4), and (e) bicyclopropenyl (trans-3,3'bicyclopropenyl, 5 ) . In the hexaaza analogues, a nitrogen atom substitutes for each C H group of the hydrocarbon structure; each bonded hydrogen atom is replaced by a nitrogen lone pair. The nitrogen atom is isoelectronic with a C H group; thus one might expect that stable structures containing the C H group could have nitrogen analogues. However, there are destabilizing lone pair interactions in the N 6 case that might preclude the existence of stable structures. The fact that nitrogen single and double bonds are much weaker than analogous bonds of the C H group suggests that the nitrogen structures would be much more energetic compounds than their C H analogues, if they were stable. The (CH), analogues of structures 1-4 have been synthesized; structure 5 has been synthesized with the H's replaced by phenyl groups2 Some of the (CH), species have fairly long half-lives. This is attributed to the lack of symmetry-allowed decomposition pathways. To the author's knowledge, none of the N, species have been positively experimentally identified; although a possible observation of 1 in a low-temperature matrix has been r e p ~ r t e d . ~ This report produced a number of calculations on l.4*5The most advanced of these indicate that the D6h structure is a stable specie^.^ This conclusion is only reached a t the restricted Hartree-Fock ( R H F ) level when a basis set of split-valence plus polarization function quality (or better) is used. This indicates that structure calculations on 2-5 should also be done at this level. In a study of P6 valence isomers, Nagase and Ito6 presented comparison a b initio structure and energy calculations for 1, 2, ( I ) This work was supported by the U S . Department of Energy. (2) (a) Scott, L. T.; Maitland Jr., J. Chem. Reu. 1972, 72, 182. (b) Kobayashi, Y.; Kumasaki, I. Top. Curr. Chem. 1984, 123, 103. (3) Vogler, A.; Wright, R. E.; Kunkley, H. Angew. Chem., Inr. Ed. Engl. 1980, 19, 717. (4) Saxe, P.; Schaefer 111, H. E. J . Am. Chem. SOC.1983, 105. 1760 and references therein. ( 5 ) Ha, T. K.; Cimiraglia, R.; Nguyen, M . T. Chem. Phys. Lett. 1981, 83, 317. (6) Nagase, S.; Ito, K. Chem. Phys. Lett. 1986, 126, 43.
0022-3654/89/2093-5722$01.50/0
and 4. T o our knowledge, there exist no published a b initio calculations on 3 and 5, or frequencies for 2-5, or chemical shifts for any of the N, structures. The chemical shift and vibrational frequency results presented here may prove valuable in the identification of the N, molecules, should they be synthesized. Note there have also been numerous recent calculations on the valence isomers of (SiH)6.7 All our structure calculations were carried out in the restricted Hartree-Fock ( R H F ) approximation, with either a 4-31G or a 4-31G* basis set, using the Gaussian-82 program.' The stability of the five stationary points was verified by vibrational frequency calculations a t both the RHF/4-3 I G and RHF/4-3 1G* levels. Some qualitative differences in stationary point character are found when basis set quality is improved. In order to obtain some feeling for the effect of electron correlation on the relative energies, energy calculations were performed a t the MP2/4-3 lG*//RHF/4-3 1G* and MP2/4-3 1G//RHF/4-31G levels, in the frozen-core approximation. Generally, inclusion of correlation results in a 15-20% decrease in the predicted energy release upon dissociation of an N, isomer into three N, molecules. Nitrogen chemical shifts have been obtained by use of Ditchfield's self-consistent field perturbation m e t h ~ d these ; ~ were calculated a t the RHF/4-3 1G level using gauge-invariant atomic orbitals. Most of the calculations were also done on diatomic nitrogen (N,), diimide (N2H2), and hydrazine (N,H,); these results are used as an aid in the interpretation of the N, results. 11. The Calculations
All the R H F structure, energy, and frequency calculations were done (with the Gaussian-82 program) using 4-31G and 4-31G* basis sets.'o," The condition for a self-consistent field (SCF) was that the maximum change in any density matrix element between Integrals were neglected if their magnitude cycles must be (7) (a) Nagase, S.; Teramae, H.; Kudo, T . J. Chem. Phys. 1987,86,4513. (b) Nagase, S.; Nakano, M.; Kudo, T. J . G e m . Sac., Chem. Commun. 1987, 60. (c) Sax, A. Janoschek, R. Angew. Chem., Inf.Ed. Engl. 1986, 25, 651. (d) Nagase, S.; Kudo, T.; Aoki, M. J . Chem. SOC.,Chem. Commun.1985, 1121. (8) Binkley, J. S.; Frisch, M. J.; DeFrees, D. J.; Rahgavachari, K.; Whiteside, R. A,; Schlegel, H. B.; Fluder, E. M.; Pople, J. A . Department of Chemistry, Carnegie-Mellon University, Pittsburgh, PA. (9) Ditchfield, R. Mol. Phys. 1974, 27, 789. ( I O ) Ditchfield, R.; Hehre, W. J.; Pople, J . A. J . Chem. Phys. 1971, 5 4 , 724. (11) Hariharan, P. C ; Pople, J . A. Theor. Chim. Acta 1973, 28, 213.
0 1989 American Chemical Society
Five Stable Points on the N6 Energy Hypersurface was An INDO initial guess was used to begin an S C F calculation. The following four criteria were required at a stationary point on the energy hypersurface: (1) the maximum force (along a spatial or angular displacement) is 1 4 . 5 X lo4 hartrees/bohr or hartrees/radian, (2) the rms force is 1 3 X IO4 hartrees/bohr or hartrees/radian, (3) the maximum displacement bohr or radian, and (4) rms displacement is 51.2 is 51.8 X X IO-' bohr or radian. The frequencies were calculated by using energy second derivatives computed numerically from analytically calculated first derivatives. The step size in the second derivative calculations was 0.0025 A. The same S C F and integral criteria used in the optimizations were maintained in the frequency calculations. The second-order Mdler-Plesset energy calculations used the Fock Hamiltonian and the R H F wave function obtained by using the above listed S C F criteria; the cores were frozen in all the correlated calculations. Note that below when (e.g.) the abbreviated notation RHF/ 4-31G* is used, we mean that the RHF/4-31G*//RHF/4-31G* model was used. To calculate the chemical shifts, a code due to R. Ditchfield, as modified by C . M. Rohlfing, was used. Ditchfield's methodg uses atomic orbitals that depend on the electromagnetic vector potential; these orbitals ensure that the magnetic shielding constants obtained are gauge invariant. Such orbitals are referred to as gauge-invariant atomic orbitals (GIAO). Ditchfield represented the molecular wave function as a single Slater determinant of these atomic orbitals and then obtained a modified set of Hall-Roothaan ( H R ) equations that apply when a nonzero external magnetic field H is present. He expanded the various terms of these HR equations in powers of H and then solved the resultant first-order equations in H for approximate molecularorbital coefficients. These coefficients are used to obtain the chemical shift tensor. 111. Results Note that below, except where noted, the following notation is used: (1) RHF/4-31G, (2) RHF/4-31G*, and (3) experimental values are given, respectively, (1) unbracketed, (2) in [ 1, and (3) in I ) . A. Comparison Molecules-N2, N2H2,and N2H4 Calculations were performed on the ground electronic states of diatomic nitrogen (N2), diimide (N2H2),and hydrazine (N2H4) in the hope that the properties of these molecules would aid in the interpretation of the N6 results. These molecules can be thought of as containing the sp, sp2, and sp3 hybridized forms of the nitrogen atom. The following results were found. ( i ) N2-Diatomic Nitrogen. The N 2 results will be used below as a standard for detecting nitrogens showing triple bond character. N 2 is the nitrogen analogue of acetylene. RHF/4-3 IG, RHF/4-31G*, and experimental values of the N 2 (XIZ,+ state) equilibrium bond length (A), energy (hartrees), vibrational frequency (cm-I), and absolute chemical shift (ppm) are, respectively, 1.085 [ 1.0761 (1.094),12 -108.754219 5 [108.839325 71, 2675 [2765] (2331),12and -101.4 (-101.3).'3 (ii) cis- and trans-N2H2-Diimide. The molecule N2H2,known as diimide or diazene, has a cis and a trans isomer. The cis isomer will be of greater interest here. However, for completeness, properties of the more stable trans structure are also given. The results for this molecule will be used below as a standard for detecting double-bonded nitrogens. Diimide is the nitrogen analogue of ethene. Note that the nitrogens in the cis form of N 2 H 2can be viewed as a subunit of the N, structures in which there are N N double bonds (Le., 2, 3, and 5 ) , provided ring strain effects are not dominant. The ground electronic (IA,) state of the cis [C,] form of N2H2 has the following geometrical parameters (in angstroms and degrees): (RN.N), 1.224 [1.213]; (RN+),1.019 [1.018]; and (LN( I 2) Herzberg, G . Spectra of Diatomic Molecules; Van Nostrand: New York, 1966; p 5 5 3 . (13) Jameson, C. J.; Jameson, A. K.; Oppusunggu, D.; Wille, S . ; Burrel, P. M.;Mason, J. J . Chem. Phys. 1981, 7 4 , 81.
The Journal of Physical Chemistry, Vol. 93, No. 15, 1989 5723 TABLE I: Hexaaza Isomer Structural Parameters" point struct 4-31G* 4-31G electronic N, form group param basis basis state benzene (1) D6h R, 1.284 1.310b IA, Dewar benzene ( 2 ) C, R, 1.218 1.227 IA,
benzvalene ( 3 )
C2,
R2 R3 A, A, R, R2
prismane (4)
D3h
bicyclopropenyl ( 5 )
C2,
R3 R4 Ai A2 A3 R, R2 R, R2
RS A, A2
1.427 1.447 109.6 85.4 1.189 1.452 1.421 1.399 106.9 105.6 60.5 1.466 1.427 1.426 1.463 1.166 47.0 106.0
1.498 1.561 108.7 83.6 1.191 1.549 1.489 1.509 107.4 104.6 59.4 1.544 1.509 c
c
IA,
1A'I
1 4
c c
c
"Lengths are in A and angles in deg. bThis structure is a transition state. c A stationary point was not found within the model.
N-H), 116.0 [ I 13.11. The values of the energy (hartrees), dipole moment (debyes), and rotational constants (A, B, and C in cm-I) are -109.798 8393 [-109.87658981, 3.76 [3.19], ( A ) 10.6751 [10.2141], (B) 1.3220 [1.3671], and ( C ) 1.1763 [1.2057]. The absolute chemical shift (ppm) is -399.2. The vibrational frequencies (cm-I) and symmetries are 1405 [1417] A2, 1480 [1518] A,, 1682 [I7231 B2, 1826 [I9081 A,, 3394 [3452] B2, and 3464 [3528] A,. Similarly, for the ground electronic (1A,) state of trans N2H2, one finds (1) for the [C2J geometry (RN-N), 1.226 [1.214] (1.252);14( R N - H ) , 1.011 [1.015] (1.028};and (LN-N-H), 110.5 [ 107.51 ( 106.5); (2) for the energy and rotational constants, -109.8126870 [-109.887691 51, ( A ) 11.2685 [10.6494] (10.0002],14( B ) 1.3259 [1.3708] (1.3042),(C) 1.1863 [1.2144] (1.1499);(3) for the absolute chemical shift, -519.3; and (4) for the vibrational frequencies and symmetries, 1446 [ 14771 {1286)14 B,, 1464 [1480] (1250-1350) A,,, 1736 [I7641 (1529) A,, 1821 [1908] (1583)A,, 3543 [3550] (3128) A,, 3588 [3586] (3120)B,. (iii) N2H4-Hydrazine. Hydrazine is the nitrogen analogue of ethane. It can be viewed as a subunit of the N6 structures in which there are N N single bonds (Le,, 2-5)-provided changes due to rotation about the hydrazine N N torsional mode and ring strain effects are small. It is known experimentally that there are two distinct types of NH bonds in N2H4;see Kohata, et al.15 (and references therein). The theory mirrors this observation. We adopt Kohata's method of defining the hydrazine geometry in terms of ( I ) the bond lengths, (2) the (two) N N H angles, and (3) the (dihedral) angle between the bisectors of the H N H planes. The molecule has C2symmetry and a 1A ground electronic state. Its geometrical parameters are (RN-N), 1.401 [ 1.4 131 (1.449 i 0.002);'5 (RL-H),0.995 [1.003] (1.021 f 0.002); (RK-H),0.992 [0.999] (1.021 f 0.002);LNNH,, 116.6 [112.1] (112 f 2),LNNH2, 113.0 [107.8] ( 1 0 6 f 2), thedihedralangle, 131 [I121 (91 f 2). Note that while the two N H H angles are well reproduced by theory, the dihedral angle is not; this can probably be traced to the low energies needed to twist the N H H planes about the N N bond. The increase in basis set quality from 4-31G to 4-31G* improves the dihedral angle estimate significantly. The values of the energy (hartrees), dipole moment (debyes), and rotational constants (cm-I) are-11 1.0067526 [-I 11.061 64781, 1.93 [2.22], ( A ) 5.2924 [4.9666], (4.7855},15( B ) 0.8266 [0.8338] (0.80341,and ( C ) 0.8234 [0.8335] (0.8029). The absolute chemical shift is (14) Carlotti. M.; Johns, J. W. C.; Trombetti, A. Can. J . Phys. 1974, 52, 340.
( I 5 ) Kohata, K.; Fukuyama, T.; Kuchitsu, K. J . Phys. Chem. 1982, 86, 602.
5724
The Journal of Physical Chemistry, Vol. 93, No. 15, 1989
Engelke
+238.5. The vibrational frequencies and symmetries are 605 [471] {3711” A, 670 [978] 1896) A, 750 [1116] 1914) B, 1205 [I2271 (1035)A, 1 3 8 8 B [ 1 4 3 8 B ] { 1 2 8 4 A } ,1396A[1470A](1285B1, 1862 B [I854 B] 11579 AI, 1878 A [1871 A] 11598 B1, 3754 B [3691 B] {3274 A), 3759 A [3703 A] 13293 B), 3894 [3803] 13334) A, and 3898 [3810] 13336) B. B. N , Structures and Frequencies. The structural results and frequencies are presented together, since the frequencies define the character of a stationary point (structure) on the energy hypersurface. The structures are illustrated in Figure 1 and numerical results for geometries and frequencies are given in Tables I and 11, respectively. (I‘) N 6 Benzene. The RHF/4-31G calculations for the D6h structure (1) predict that it is unstable; there is an RHF/4-31G D6hstationary point that, is a transition state. It has one imaginary eigenfrequency (4601’cm-’ with B2, symmetry). Calculations at the RHF/STO-3G level give similar results. At the RHF/4-3IG* level, there is a stable D6* structure with the N-N bond length predicted to be 1.284 A. This length is significantly shorter than the RHF/4-3 lG* N-N single bond length for hydrazine but longer than the N=N double bond length in diimide. This mimics the C-C bond length behavior of benzene and suggests that the IT electrons in N, benzene are delocalized over the whole ring. The RHF/4-31G* structure is rather loosely defined by the hypersurface; it has lowest eigenfrequency of 208 cm-I. The lowest frequency normal-mode vector is an in-plane motion and would lead to a dissociation reaction with three N 2 molecules as products, if carried to large distensions. The next vibratonal level is a pair of low-frequency degenerate modes (324 cm-I, E,, symmetry), along which it is also easy to distort the molecule. These results parallel those of Saxe and S ~ h a e f e r . ~ The RHF/4-31G* rotational constants (cm-I) of 1 are ( A ) 0.2434, ( B ) 0.2434, and (C) 0.1217. ( i i ) 1v6 Dewar Benzene. Both the RHF/4-31G and R H F / 4 31G* models predict that 2 is a stable structure. As shown in Table I, the addition of polarization functions to the basis set significantly decreases the N-N bond lengths and increases the bond angles somewhat. The RHF/4-3 1G* geometrical parameters should be taken as superior; they show the following characteristics. The N-N double bond lengths ( R , = 1.218 A) are quite close to the cis-diimide value (1.213 A), suggesting a standard sp2 hybridization. Note, however, that there is strain within the four member rings, as indicated by the value of the angle A,. The angle A , is very close to the tetrahedral value. The bond length R, ( = 1.427 A) is close to the N-N single bond value in hydrazine (1.41 3 A). R, is the most unusual bond in 2, being significantly longer (ca. 0.037 A) than that in hydrazine. This suggests ineffective orbital overlap due to strain effects. The lowest RHF/4-3 IC* and RHF/4-31G vibrational frequencies indicate that the structure of 2 is more rigorusly defined by the hypersurface than is 1’s. In the lowest frequency normal mode (RHF/4-31G*, 526 cm-’. A 2 symmetry), the N , and N 2 nuclei remain essentially fixed, while N3=N4 and IC,=N, undergo similar, but 180’ out-of-phase, rocking motions. This normal mode does not point to any obvious dissociation reaction. The eigenvector for the highest vibrational frequency (RHF/4-31G*, 1874 cm-’, A , symmetry) is a simultaneous shortening of the N N double bonds and the N,-N2 bond as the other bond distances increase (i.e., during half the motion). I f taken to large separations, this mode would lead naturally to a dissociation into three N, molecules with their bond axes parallel to the N1-X2 bond axis in the original molecule. The RHF/4-31G* dipole moment for 2 is negligible. The rotational constants (cm-I) are ( A ) 0.3524, ( B ) 0.1903, and ( C )
1
2
RI
4
0.151 I .
(iii) iY6 Benzualene. Both the RHF/4-31G and RHF/4-31G* models predict that 3 is a stable structure. The addition of polarization functions to the 4-3 l G basis set significantly decreases all the sp3-like bond lengths (Le., R,, R,, and R4),leaves the sp2-like bond ( R , ) essentially unchanged, and does not alter the bond angles to a n > great eutent. The RHF/4-31G* geometry predicts
Figure 1 .
Five Stable Points on the N6 Energy Hypersurface TABLE 11: Hexaaza Isomer Vibrational Frequencies N6 form basis set 1 2 benzene ( 1 ) 4-31G* 208 324 B2u E2u Dewar benzene (2) 4-31G* 526 546" A2 AI 4-31G 475 506 A2 AI benzvalene (3) 4-31G* 593 634" A2 BI 4-31G 415 438 A2 AI prismane (4) 4-31G* 759 814 A," Elf 4-31G 545 545 Et/ E!/ bicyclopropenyl (5) 4-31G* 36" 366" Au Au
The Journal of Physical Chemistry, Vol. 93, No. IS, 1989 5125
(cm-I) 3 324 Ezu 632" B2 539 Bz 807" AI 535 Bl 814 E// 707 AI// 398' Bu
4 819 E2g 924' BI 605 Bl 849 AI 632
5 819
6 970 B2g 1041" AI 804 AI 912"
7 1188
9 1406
10 1406
El"
El,
1198 1308" A2 AI B2 704 1072 1202 A2 A2 Bz 881 1138 1221 A2 Bl B2 AI 670 696 1021 1100 Bz AI BI B2 AI 993" 993" 1082" 1139 1193 E! E/ E' E" E" A,' 746 871 871 995 1025 1025 Et E! E' E" AI' A/ 520 608 871" 923 1017 1116 Ag Bg Au Bg A, Ag " IR-active mode; criteria for being considered active was that the predicted intensity is greater than 0.01 D2/(amu.A2).
that the Nl=N2 bond ( R , = 1.189 8)is 0.024 A shorter (and therefore possibly stronger) than the N N bond in diimide. The bond length Rz is about 0.040 A longer then the N-N bond in hydrazine, probably due to the strain associated with the angle A I . The bond length R3 (=1.421 A) is slightly longer than the N N hydrazine bond (1.413 A), while R3 (= 1.399 A) is slightly shorter. This is surprising considering that these are NN bond lengths in highly strained azacyclopropane rings ( A 3 ) . The lowest RHF/4-31G* vibrational frequency for 3 (593 cm-], A, symmetry) is almost 3 times that of 1, indicating a more rigidly define structure. This made is a very simple motion; N3and N4 do not move. NI=N, and N5-N6 rock about the C2 axis 180' out of phase with each other. This mode does not point to any obvious dissociation reaction; neither do any of the other 1 1 (harmonic approximation) eigenvectors. The RHF/4-31G* dipole moment (debye) of 3 is 0.28 and the rotational constants (cm-I) are ( A ) 0.3212, (B) 0.2178, and (C) 0.1645. (io) N6 Prismane. Both the RHF/4-31G and RHF/4-31G* models predict that the D3,, structure 4 is stable. The addition of d-shell polarization functions to the 4-3 1G basis significantly decreases the two (sp3-like) bond lengths in the molecule. At the RHF/4-3 1G* level, the bonds forming the azacyclopropane rings ( R 2 = 1.427 A) are shorter than the inter-ring bond lengths (R, = 1.466 A) by ca. 0.039 A. Both these bond lengths are substantially larger than the N-N u bond in hydrazine (1.413 A) -indicating weaker bonds due to ring strain. The lowest RHF/4-31G* vibrational frequency for 4 (759 cm-I, AI" symmetry) is nearly 3.5 times larger than that of 1, indicating a more rigidly defined structure. This mode is a simultaneous rocking motion of N1-N2,N3-N4, and N5-N6;it does not lead to any obvious dissociation reaction. Mode 12, the largest frequency normal mode, (RHF/4-31G*, 1446 cm-I, AI' symmetry) is a breathing mode that if carried to large distensions would lead naturally to three N2 molecules oriented parallel to the C, axis of 4. The RHF/4-3 lG* rotational constants (cm-I) are ( A ) 0.2956, (B) 0.2290, and ( C ) 0.2290. ( u ) N6 Bicyclopropenyl. The bicyclopropenyl molecule (5) produced the most interesting dependence of structural properties on the model used. In the RHF/4-31G model, the molecule, when freed to optimize from reasonable starting geometries, spontaneously dissociates. At the last stages of such optimizations two N2 molecules have formed and the other two nitrogen nuclei are still 1.64 A apart. After fairly extensive searches, with both cis and trans starting geometries, it was concluded that there is no stable form of bicyclopropenyl within the RHF/4-3 lG model. The RHF/4-3 lG* calculations predict that trans-bicyclopropenyl (5) is a stable structure, while the cis form is a transition state. The cis form is unstable to torsion about the N,-N2 bond; this torsion transforms the cis form into the trans form. In the trans form, the bonds denoted by R, (= 1.426 A) and R2 ( = 1.463
%21
?$4" '32 902 B2 941 B2 745 A2 1082" E' 970 E' 888" Bu
8 1276 Blu 1143 A2 969 AI 1103" B2 866 B2 1139
11 1708
12 1708
;?l2O
k 4 " AI 1753 AI 1952" A, 1854 AI 1446 AI' 1171 AI' 2122 Ag
Bl
1682 Bl 1450 AI 1137 AI 1315' A/ 1025 E" 2107" Bu
A) (see Figure 1) are longer than the RHF/4-31G* N-N bond in hydrazine (1.413 A). This indicates these bonds are weakened by poor orbital overlap and due to the electron density concentrated in the R3 bonds. Strong ring strain is indicated by the small value of the angle A , (47'). The R3 (1.166 A) bond length is tending toward an N 2 triple bond (1.076 A), being significantly shorter than the N=N double bond in diimide (1.21 3 A). Apparently, the stability sacrificed in producing the long R , and R, bonds is recovered in forming the very short (strong) R3 bonds. Calculations were also done on 5 in which diffuse functions were added to a polarized basis set. It is sometimes said that diffusion functions are helpful in describing molecules containing lone electron pairs.16 We found nearly no change in 5's structure when calculations were done with and without diffusion functions. Bond lengths and angles changed by ca. 0.001 8, or less and 0.2' or less, respectively. These RHF optmizations utilized 6-3 1G* and 6-3 1+G* basis sets. The very small lowest eigenfrequency of 5 is a torsion about the N l - N 2 u bond. The next higher vibrational frequency (366 cm-I, A, symmetry) is still quite small. The eigenvector for this mode is a scissoring motion of an isosceles triangle's edges; the motion of the opposite triangle's edges is 180' out of phase with that of the first. The highest frequency RHF/4-3lG* normal mode is a vibration in which the N3-N4 and N5-N6 bonds shorten rapidly as the Nl-N2 bond shortens more slowly (during half the oscillation). This normal mode, if carried to large distensions, would lead to dissociation into three N,'s. The RHF/4-31G* rotational constants (cm-I) for 5 are ( A ) 0.6285, (B) 0.0960, and (C) 0.0902. C. N6 Energies. Table 111 is a comparison of various energies associated with structures 1-5. Zero-point energies for the RHF/4-31G* model are given and used to correct the relative energies between 1-5 and three N, molecules. The first thing one notes from Table 111 is that structures 1-5 would be highly energetic materials relative to diatomic nitrogen, if they could be synthesized. One mole of 1, 2, 3, or 5 would dissociate to 3 mol of N, with the liberation of ca. 250-300 kcal/mol of energy, while 4 would yield almost 400 kcal/mol. The RHF/4-3 1G model gives dissociation energies for the reactions 1-4 3N2 that are larger by 20-58 kcal/mol (of reactants) than the RHF/4-31G* model. The RHF/4-31G* energies should be taken as more accurate. The zero-point energies (ZPE) of 1-5 are all about 17.5 kcal/mol, with the largest ZPE difference between structures being 3.1 kcal/mol. The zero-point correction (ZPC) of 1-5 relative to 3N2 is larger; the largest correction being 6.9 kcal/mol. This larger ZPC is due to the decrease in the number of vibrational modes when 1-5 dissociate to 3N2.
-
(16) Frisch, M. J.; Pople, J. A,; Binkley, J . S . J . Chem. Phys. 1984, 80,
3265.
5726
The Journal of Physical Chemistry, Vol. 93, No. 15, 1989
Engelke
TABLE 111: Hexaaza Isomer Relative Energies (kcal/mol)'
N, form benzene (1) Dewar benzene (2) benzvalene (3) prismane (4) bicyclopropenyl ( 5 ) 3N2
4-31G*b 243 [O]' 289 [46] 303 [60] 382 [139] 286 [43] 0 [-2431
RHF 4-3lG*/ZPCc 4-31G*/ZPEc [OI 17.4 [471 18.8 [611 17.9 [ 1401 18.3 [411 15.7 [-2491 11.9
4-31G 263 [O]/ 322 [59] 347 [84] 440 [ 1771 g
0 [-2631
MP2=FCd 4-31G* 4-31G 214 [O] 272 [O] 248 [34] 309 [37] 255 [41] 315 [44] 334 [I201 410 [I391 248 [34] g 0 [-2141 0 [-2721
'Absolute energies as a function of model are (a) RHF/4-31G*//RHF/4-31G* -325 hartrees minus (1) 1.1308459, (2) 1.0575673, (3) 1.0347157, (4) 0.909921 0, ( 5 ) 1.0623189, and (3N2) 1.517977 I ; (b) RHF/4-31G//RHF/4-31G -325 hartrees minus (1) 0.8435629, (2) 0.749893 5, (3) 0.709 137 3, (4) 0.561 095 8, (5) -, and (3N2) 1.262685; (c) MP2=FC/4-31G*//RHF/4-31G* -326 hartrees minus (1) 1.088 8206, (2) 1.034211 9, (3) 1.023 1488, (4) 0.897737 I , (5) 1,0344744, and (3N2) 1.4295103; and (d) MP2=FC/4-31G//RHF/4-31G-326 hartrees minus (1) 0.5329588, (2) 0.4735636, (3) 0.4632048, (4) 0.311 811 3, ( 5 ) -, and (3N2) 0.965674. b A better description of the Is orbitals (Le., an RHF/6-31G* calculation) lowers the isomer energies (relative to 3N2)by 3-5 kcal/mol. eZPC zero-point corrected energies and ZPE zero-point energies. dAll the MP2 calculations were done with cores frozen. 'Numbers enclosed in [ 1's are RHF energies relative to hexaazabenzene (1) and unbracketed numbers are energies measure relative to three isolated N2 molecules. /This structure is a transition state. NO stable point of this isomer could be found within this model. The most energetic of the N, structures is clearly prismane (4). It is 80-100 kcal/mol more energetic than 2, 3, and 5 and 140 kcal/mol more energetic than 1. The energy ordering of the N, isomers is very different from that of their (CH), analogues. Schulman and Disch" found with RHF/6-31G* calculations that (CH), 2 and 3 have similar energies (i.e., they are separated by ca. 3.5 kcal/mol) and 4 and 5 have similar energies (Le., they are separated by ca. 5.3 kcal/mol), but that 2 and 3 lie lower in energy than 4 and 5 by ca. 40 kcal/mol. Of the (CH), analogues of 2, 3, 4, and 5, (CH), bicyclopropenyl is least stable. In the N6 case, bicyclopropenyl (5) is the most stable of these isomers. Furthermore, N, 2, 3, and 5 all lie within ca. 20 kcal/mol of each other (at the RHF/4-31G* ZPC level), while 4 lies 99 kcal/mol higher than 5. It is of interest to examine whether the inclusion of electron correlation alters these conclusions markedly. Since it is computationally demanding to treat electron correlation in molecules containing 42 electrons, it was necessary to use the models M P 4 = F C / 4 - 3 l C * / / R H F / 4 - 3 I G * and MP2=FC/4-31G// RHF/4-3 IC; i.e., no geometry optimizations were performed on the correlated hypersurface and the MP2 energies were obtained with the atomic cores frozen. The results of these calculations are given in Table 111. One notes from Table 111 that the N, energies are generally reduced (relative to either three N, molecules or to N6 benzene) when correlation is included a t the MP2=FC level. The energy releases upon dissociation to three N z molecules remains very large; Le., approximately 250 kcal/mol when a 4-3 1G* basis is used. This is about 3C-50 kcal/mol smaller than the R H F / 4 - 3 l G * / / R h F / 4 - 3 l G * values. No significant reordering of the relative energies of the structures occurs at the MP2=FC level. D. N6 Chemical Shifts. W e shall give the computed chemical shifts in ppm; these values apply to both I4N and I5N nuclei.18 As is known, 15N gives sharper N M R lines than I4N due to its lack of a nuclear electric quadrupole moment.19 It is common to reference nitrogen chemical shifts to that found for NH3. The experimental geometry of N H 3 is R N - H = 1.012 A and LHNH = 106.7O; the measured nitrogen chemical shift is +266.8 ppm.,O We use the RHF/4-31G NH, shift as our reference. NH, was optimized a t the RHF/4-31G level and it was found that R N - H = 0.971 A and L H N H = 101.9". The RHF/4-31G(GIAO) chemical shift is C277.8 ppm. The agreement between the experimental and theoretical NH, chemical shifts is satisfactory (e.g., changes, due to solution, phase change, and temperature effects, of 10-20 ppm in nitrogen chemical shifts are not atypical). Below (17) Schulman, J . M.; Disch, R. L. J . Am. Chem. SOC.1985, 107, 5059. ( 1 8) Becker, E. D.; Bradley, R. B. J . Magn. Reson. 1971, 4 , 136. (19) Witanowski, M.;Webb, G. A., Eds.; Nitrogen N M R ; Plenum: New York, 1973; pp 1-39. (20) Carrington, A.; McLachlan, A. D. Introduction to Magnelic Resonance: Chapman-Hail: New York, 1979; p 62.
TABLE IV: Nitroeen RHF/4-31G Chemical Shifts
molecule N, benzene ( l)c N6 Dewar benzene (2)
nucleus" Nl-N6
N,, N2 N3-N6
N, benzvalene (3)
N2 N,, N4 NS, N6
N, prismane (4)
Nl-N6
Nl,
NH3
N
N2
N
N2H2(cis) N2H2(trans) N2H4
N N N
chemical shift; ppm 539.6 (-261.8) 462.5 (-184.7) 597.4 (-319.6) 454.6 (-176.8) 455.2 (-177.4) 235.4 (+42.4) 331.6 (-53.8) 0.0 (+277.8) 379.2 (-101.4) 677.0 (-399.2) 797.1 (-519.3) 39.3 (+238.5)
"See Figure 1 for the correspondence between the nucleus label above and its position in the hexaaza molecule. bChemical shift values in parentheses are absolute, where as unbracketed values are shifts relative to NH, in ppm. 'This point is a transition state. No chemical shift is given for trans-3,3'-bicyclopropenyl ( 5 ) because no corresponding stable point was found with the RHF/4-31G model. and in Table I V a downfield shift relative to N H 3 is defined as positive. It is important to estimate the quality of the computed chemical shifts. T o do this we compare the calculated and experimental values for N, and N,H,. One obtains a shift for N2, relative to NH3, from Jameson et al. of +324.0 ppm;13 this is ca. 50 ppm less than the RHF/4-31G value. There is a second absolute chemical shift in the literature for N 2 of -102 ppm.,' If this value is used, one obtains a relative shift for N, of +366 ppm, which is within about 13 ppm of the calculation. For hydrazine, Schindler,, gives an experimental shift of +45.4 ppm relative to NH, and the calculation gives +39.3 ppm. These comparisons suggest that the errors in the calculated N6 chemical shifts are probably on the order of 10%. Since most of the N6 calculated (relative) shifts are quite large (Le., ca. 400 ppm), these predictions may be a useful diagnostic in an experimental search. No clear patterns appear in the N, shifts listed in Table IV. However, it does appear that x-bonded nitrogen nuclei (usually) have more negative absolute shifts than u-bonded nuclei.
IV. Discussion and Conclusions A uniform set of a b initio calculations has been performed on all the valence isomers of N, using the RHF/4-31G and RHF/4-31G* models. The properties of these structures are interesting in themselves. These properties also represent the limiting case of the structures (CH),_,N, with 0 In I6 and, thus, cast some light on the behavior of such structures, e.g., their possible stability. ( 2 1 ) Gierke, T. D.; Flygare, W. H. J . A m . Chem. SOC.1972, 94, ( 2 2 ) Schindler, M. J . .4m. Chem. SOC.1987, 109, 5950.
7277.
J . Phys. Chem. 1989, 93, 5727-5135 The conclusion from the R H F / 4 - 3 l G * / / R H F / 4 - 3 I G * structure and frequency calculations is that 1-5 are stable. The structures 2-5 are more rigidly defined than 1, as measured by their lowest (nontorsional) vibrational frequency. The calculated N N bond lengths in 1-5 mimic analogous calculated N N bond lengths in smaller unstrained N H molecules in almost all cases. Assuming the calculated RHF/4-31G*//RHF/4-3 1G* energy differences accurately represent reality, all five N6 structures are highly metastable to decomposition into three N, molecules. This metastability is ca. 250-300 kcal/mol for 1-3 and 5 and ca. 380 kcal/mol for 4. For 2, 4, and 5, the highest frequency normalmode eigenvector suggests a route to dissociation into three N 2 molecules, while for 1 the lowest frequency eigenvector does this. The ( I2 harmonic approximation) eigenvectors for 3 do not give any hint as to a dissociation path. Treatment of electron correlation, with the MP2=FC/4-31G*//RHF/4-3lG* model, lowers the energy released upon dissociation of the N6 structures into three N2 molecules by ca. 30-50 kcal/mol.
5727
An increase in basis-set quality from 4-31G to 4-31G* uniformly increases the vibrational frequencies of 1-5; typically by ca. 20% (the range of values is 5-35%). If the RHF/4-31G* frequencies are to be used as an identification tool, they should be calibrated. A simple calibration can be obtained from the N2, N2H2,and N2H4experimental and calculated frequencies given in section IIIA; it is ueXptl= uald/(l + a ) . Here a is 0.16 0.04 for the comparison molecules, is the estimated (experimental) value, and uald is the RHF/4-31G* value. The quantity a is the , ~ the three calibration molmean value of (uCalcd- U , , ~ , ~ ) / U , , ~ for ecules and 0.04 is its standard deviation. The nitrogen chemical shifts are also potentially valuable for identification purposes-although it should be realized that the relative shifts probably contain errors of ca. 10%.
*
Acknowledgment. I thank C. M. Rohlfing for use of the chemical shift program. Registry No. 1, 7616-35-5; 2, 121328-58-3; 3, 116688-86-9; 4, 116688-87-0; 5, 121328-59-4.
Identification of Bifurcations and Centers in Systems with Complex Dynamic Behavior Moshe Sheintuch Department of Chemical Engineering, Technion, Haifa, Israel
and Dan Luss* Department of Chemical Engineering, University of Houston, Houston, Texas 77004 (Received: February 2, 1988; In Final Form: March 9, 1989)
A methodology is presented for organizing experimental data and for qualitative modeling of complex dynamic behavior
that can be described as due to two different periodic motions. When the experimental data are organized in a bifurcation map the nature of the bifurcation and organizing centers may be identified by a comparison with the dynamic features of a qualitative model having these different time scales. Phase space analysis of this three-variable model shows that its behavior can be determined from the bifurcation diagram of the two-variable model when the slowest variable is treated as a parameter. The two-variable bifurcation diagram can be. constructed from the observed time traces. The analysis and modeling presented here are helpful in organizing experimental data, in classifying observed features, and for developing a mathematical model. These points are demonstrated by identifying and organizing observations of dynamic behavior in the chlorite-thiosulfate liquid-phase reaction (Epstein and Orban) and in acetaldehyde gas-phase oxidation (Gray et al.).
1. Introduction A modern approach of studying dynamical systems is to construct a bifurcation map, showing the boundaries between various characteristic behaviors of the mathematical model or experimental observations. When the dynamic behavior is either stationary or simple periodic, the classification of the various regions is simple (either oscillatory or stationary, and their combination), but identifying the nature of the specific transition (bifurcation) is not always simple. A complete classification of such transitions exists for a system described by two state variables (2-v),' but their experimental identification may be ambiguous due to insufficient resolution or slowly changing properties. These problems become more severe when multipeak oscillations are found since the experiments last longer and since the classification is not complete. The classification of states as periodic, quasiperiodic (having two characteristic frequencies), or chaotic requires sophisticated characterization, such as Fourier analysis or fractal dimension of the attractor. Further classification according to the number of peaks in a periodic state or the frequency ratio in a quasiperiodic motion is very useful. The bifurcations in the
nature of the oscillations appear in sequences (scenarios) and a complete mapping of all the transitions may be either too time consuming or impossible. In many cases mapping of experimental data may be accomplished by classifying the oscillatory motion into simple categories, like fast or slow oscillations or large- and small-amplitude oscillations, and their "mixture". These dynamics are found in many chemical systems like the Belousov-Zhabotinski reaction,2 or the chlorite-thiosulphate system studied by Epstein and 0rba11.~ The observed dynamic behavior shown in the four time traces in Figure 1 varies from simple, large-amplitude, slow relaxation oscillations (denoted by 0) to simple, small-amplitude, fast oscillations (0) through a sequence of complex oscillations (C). The latter consist of one large peak and several small ones; the ratio of these cycles varies as a parameter is changed. The bifurcation map (top of Figure 1) describes regions of operating conditions corresponding to these dynamic behaviors as well as a unique stationary state (2) Maselko, J.; Swinney, H. L. Complex Periodic Oscillations and Farey Arithmetic in the Belousov-Zhabotinskii Reaction. J . Chem. Phys. 1986, 85, 6430-6441.
( I ) , Maselko, J. Determination of Bifurcation in Chemical Systems. An Experimental Method. Chem. Phys. 1982, 67, 17-26.
0022-3654/89/2093-5727$01.50/0
(3) Epstein, I . R.; Orban, M . In Oscillations and Travelling Waves in Chemical Sysfems; Field, R. J . , Burger, M., Eds.; Wiley: New York, 1985; p 251.
0 1989 American Chemical Society
