Force Constants and Bond Orders of Nitrogen Bonds BY PROF. DR. J. GOUBEAU LABORATORIUM FUR AWORGANISCHE CHEMIE DER TECHNISCHEN HOCHSCHULE STUTTGART (GERMANY) The force constants and the corresponding bond orders of nitrogen bonds have been calculated from the vibrational spectra (infrared and Raman spectra) of a great number of nitrogen compounds. Plotting the maximum bond order of stable nitrogen bonds against the sum of Pauling’s electronegativities of the bonding partners ( x x ) leads to one continuows curve for the N - X bonds where X represents elements of the first and the second short period of the periodic table. Furthermore, when the bonds formed between these elements are arranged in a coordinate system in such a way that the position of each bond is determined by the diference between the electronegativities of the bonding partners ( A x along the ordinate) and the sum of the electronegativities of the bonding partners ( r x along the abscissa), the bonding partners capable of forming multiple bonds all lie within a closed domain, where their position can be correlated with their polymerizability and other reactivities of the multiple bonds. Also discussed are the orders of bonds between nitrogen and some transition elements. In an appendix, the present methods used to calculate force constants and bond orders are surveyed. A. Introduction cumulation of high formal charges. The types of bonds thaLnitrogen can form are summarized in Table 1. The conditions for the formation of x-bonds have been previously summarized as follows [ I ] : 1) electron deficiency on both bonding partners; 2) the sum of the Pauling electronegativities of the partners must be at least 5; 3) the difference between the Pauling electronegativities of the bonding partners must be small. We have endeavored in the past few years to verify these conditions on further examples and, in particular, to define more accurately condition 3). Condition 2) has recently been confirmed by Herber and Stockler [*I, who investigated the Mossbauer effect of tin compounds and showed that the isomeric shift is invariably proportional to the sum of the electronegativities of all bonding partners involved. We have chosen nitrogen for our investigations, since it forms a great number of compounds and has the high electronegativity of 3,0, enabling it to form bonds with numerous elements in such a way that the sum of the electronegativities is at least 5 . Unlike the halogens, and like oxygen, nitrogen can form x-bonds without the acTable I . Types of bond f o r m e 3 between nitrogon a n 3 a partnor X . Imines, imides 1 e e E-E-H tt X = N H tt X E N H In addition to the preferred bonds, in which the two atoms involved have a low formal charge (cf. Table l), bonds in which the nitrogen carries a formal positive charge are also important. This formal positive charge is to some extent compensated by the fact that nitrogen is at the same time the negative end of the polar bonds which it forms in most cases. B. Calculation of Bond Order The following discussion is based on vibrational spectra (infrared and Raman spectra) furnishing data for the force constants, which in turn are used for the calculation of the bond orders. In these calculations, the geometry of the molecule and the assignment of the observed frequencies to the fundamental vibrations is also accounted for. Fadini, Sawodny, and Ballein [31 have used a new method of calculation, enabling them to obtain the force constants and all interaction constants directly from the measured frequencies without a trialand-error procedure 141. The resulting valence bond constants, the only ones discussed below, have the advantage of being mutually comparable, since they are obatined by the same method. Deviations of more than 5-10 % correspond to actual changes in the nature of the bonding. Siebert’s rule r4.51 has been used for the calculation of the force constants of single bonds. This rule gives good results, suitable for a general comparison of all force constants. Siebert’s values for single bonds correspond . Nitrides [I] J . Goubeau, Angew. Chem. 69, 77 (1957). [ 2 ] R . H. Herber and A. Stockler, Chem. Engng. News 42, No. 28, 66 (1964). Angew. Chem. internat. Edit. VoI. 5 (1966) 1 No. 6 . [3] A. Fudini, Z . angew. Math. Mechan. 44, 506 (1964); W. Sawodny, A. Fudini, and K . Ballein, Spectrochim. Acta 21, 995 (1965). [4] J. Goubeuu, Angew. Chem. 7 3 , 305 (1961). [5] H. Siebert, Z. anorg. allg. Chem. 273, 170 (1953). 567 to sp3 hybridization. Since differences in the hybridization can lead to large (up to 50%) variations in the force constants of single bonds, it has been found necessary to introduce an empirical correction factor 111. Using these theoretical force constants for single bonds, we can calculate the bond order, b. It is assumed that higher force constants are essentially due to x-bonds. Siebert has given two equations for the bond order. One is based on the assumption that the bond is proportional to the force constant up to about b = 1.5, the other being an empirical formula (giving good results) for b > 1.5 (see Appendix 111). The calculations are generally based on direct proportionality even in the case of higher bond orders. It is further possible to correlate, on a theoretical basis, the bond orders of single and double bonds in certain compounds with the force constants, and then to find the bond orders corresponding to the other force constants, as has been done in Tables 3 and 5 (column btr). and, therefore, it can form multiple bonds (at the most a three-electron bond in NF2). Distributed over two N-F bonds, this leads to a maximum bond order of 1.25, while in N F the bond order may reach 2.0. fexp. ImdyneiAI fcalc. (single bond) lmdyne/Al 4.38 4.26 4.85 4.05 [el b 1.03 1.20 5.90 171 3.87 [81 4.05 IS1 1.45 P a l C . Order of Bonds between Nitrogen and Elements of the First Period 11. The N-B T h e parameters of the N-0, N-N, a n d t h e N-C bonds a r e well known a n d need no further discussion. In all three cases, the b o n d order can b e between 1 a n d 3 according to the force constants. Bond The most important results concerning force constants and bond orders of N-B bonds in various compounds are listed in Table 3 191, together with the possible Table 3. Force constants f and bond orders b of N-B bonds (X H, CH3, F, CI; Y i = H. CH,) Formula X~BNYJ I. The N-F Canonical forms - tl b1 I -2.8 0.7 I .0 4.0 I .0 1.3 6.0 1.5 1.7 6.3 1.6 I .8 RzBNCO 6.6 1.7 1.9 XzBNYz 7.c 1.8 2.0 BN;‘] 7.0 1.8 2.0 BN 8.3 2. I 2.3 Bond Although, on the basis of the sum of electronegativities, the formation of multiple bonds should be possible in nitrogen trifluoride, this compound possesses no electron deficiency (condition 1 in Section A), and can, therefore, form no x-bonds. In the compounds NF2 161 and N F 171, which have recently been described together with their vibrational spectra, the nitrogen does not attain an electronic octet (i.e. it is electron deficient), [6] F. A . Johnson and C . B. Colbuin, Inorg. Chem. I , 431 (1962); M . D . Harmony and R . J. Myers, J . chem. Physics 37,636 (1962). [7] M. E. Jacox and D . E. Milligan, J . chem. Physics 40, 2451 ( 1964). mesomeric limiting structures. The bond orders bI are based on Siebert’s value for the single bond (f = 3.9 mdyne/&. In calculating thebond order bII, it was assumed that compounds of the type of X3BNY3 are characterized by a single bond, while BN23- contains a [8] In such a case, the value for the a-bond represents an intermediate value, since the number of free electron pairs changes in the transition from single bond to double bond. For N F two values are obtained (3.87 and 4.05). [Sa] The bond orders were determined by a simple averaging method, the extreme values being b = 1.39 and b = 1.52. [9] Cf. J. Goubeau, Advances Chem. Ser. 42, 87 (1964). H . Noth, G. Schmid, and Y . Chung (IUPAC Congress, Vienna, 1964) have found another example of a B-N bond with high bond order in the adduct B(NR2)3,TiC14. However, since the force constant has not been calculated, this example could not be included in Table 3. Angew. Chem. internat. Edit. 1 Vol. 5 (1966) No. 6 double bond. The intermediate bond orders were found on the basis of an assumed proportionality between the bond orders and the force constants. The difference between bI and b,, (about 0.2) is of little significance for the subsequent discussion; the bI values are preferred for purposes of comparison. The maximum bond order has thus been found to be 1.8 for the N-B bond in compounds that are stable at room temperature. 111. The N-Be D. Order of Bonds of Nitrogen with Elements of the Second Short Period Bond Beryllium compounds, like boron compounds, form adducts with ammonia and amines [1*1. The reaction between beryllium alkyls and amines leads to substituted beryllium amides 1191 which could not be prepared from adducts of the type BeC12.2HN(CH3)3 by the abstraction of HC11201. Unlike the boron compounds, these beryllium amides are known only in polymeric form. The force constants have been calculated from the spectra of berylliumdi-[bis-(trimethylsilyl)]amide [I31 and the adducts formed of ammonia, dimethylamine, and trimethylamine with BeC12 and BeBr2[21]. In the first case, the value f = 2.96 mdyne/A has been obtained for the N-Be bond, while in the other cases, calculations on the basis of the antisymmetric vibrations of the BeN2 group (600-785 cm-1) led to f = 1.2 to 1.4 mdyne/h;. Using these values and taking, with Siebert, f = 3.1 mdyne/A for the single bond, we obtain b = 0.95 and b = 0.39 to 0.45, respectively. Since the beryllium amides are polymeric, we cannot expect them to have appreciably higher force constants. In fact, there is no indication at the present of higher bond orders in N-Be compounds. IV. The N-1.i Bond Only one very broad infrared band has been observed for each of the simple compounds available, LiNH2, LiNH, and Li3N. These bands are situated, respectively, [lo]J . R. L. Anister and R . C . Taylor, Spectrochim. Acta 20, 1487 (1964);J . Goubeau and W . Sawodns, Z. physik. Chem. N.F. 4 4 , 227 (1965). [Ill J . Goubeau a r d H . K c l l u , Z . a n o r g . allg. Chem. 272, 303 (1953). [I21 H.-J. Eecher, Z.anorg. allg. Chem. 287, 283 (1956). [I31 B. L. Crawford and J . T . Edsall, J. chem. Physics 7 , 223 (1939);H . Watanabe, M . Norisada, T . Nukagawn, and M . Kubo, Spectrochim. Acta 16, 78 (1960). (141 J. Goubeair and H. Grabner, Che:n. Bir. 93, 1379 (1960). [IS] J . Goubeaii and H.-J. Eecher, Z . anorg. allg. Chem. 268, I33 (1952). [I61 J . Goiibeau and W. Anselment, Z. anorg. allg. Chem. 310, 248 (1961). [I71 A . E. Douglas and G. Herzberg, Canad. J. Res., Sect. A 18, I79 ( 1 940). 118Je.g.R.FrirkeandF.RObk~,Z.anorg.allg.Chem.170,25(1928). [I91 See, e.g., G . E. Coates and F. Clocking, J. chem. SOC.(London) 1954, 22. [20] N. Th. Rakintzis, Dissertation, Technische Hochschule Stuttgart, 1957. [211 Ch. Forker, Dissertation, Technische Hochschule Stuttgart, 1964; H . Burger, Ch. Forker, and J . Goitbeau, Mh. Chem. 96, 597 (1965). Aiigew. Chem. intertiat. Edit. at 403, 465, and 430cm-1 and can be attributed to antisymmetric vibrations [221. The resulting force constants (0.44,0.61, and 0.55 mdyne/& respectively) are probably too high. On the assumption that the single bond has a force constant of 2.3 mdyne/A, the average bond order for the N-Li bond is 0.23. 1 Vol. 5 (1966) / No. 6 I. The N-Cl Bond Owing to the presence of vacant d-orbitals in elements of the second short period, electron deficiency exists here even when the electronic octet is reached. This means that double bonds can be formed, e.g. in NC13, unlike the case of NF3 : .. .. ,N-CI: >NZ'CI: H The experimental data for the N-C1 bonds are listed in Table 4. Table 4. Force constants and bond orders of N- CI bonds in various compounds I*]. HzNCl HNClz NCI, N C1 NC10323 3.1 3.0 2.5-3.0 4.0 8.9 1 .& 1.2, 1.04-1.21 1.71 2.2 1231 [231 123,241 [71 [251 In nitrogen trichloride, the calculated force constant varies with the C1-N-C1 angle; the lower value corresponds to a tetrahedral angle, the higher to an angle somewhat smaller than 120 '. However, the geometry of the molecule is not yet known. The bond orders in chloramine and dichloramine confirm the prediction based on the presence of vacant d-xbitals. These are also partly responsible for the fact that chlorine is the negative end in the N-C1 dipole 1261. The bond order in NC1 is seen to be high; it is higher than in NF, probably because of the smaller difference between the electronegaiivities of the bonding partners. The highes&force constant (f = 8.9 mdyne /A) is exhibited by NC10:--, this value being greater than that corresponding to a double bond. 11. The N-S Bond Out of the reliable data for the force constants of N-S bonds, some characteristic examples are listed in Table 5 . [22]V. Dorn, Dissertation,Technische Hochschule Stuttgart, 1964. [ * ] According to Siebert, the sp3 single bond has a force constant of 4.0 mdyne/>f. [23] G. M . Moore and R. Badger, J . Amer. chem. Soc. 7 4 , 6076 (1952). [24] E . Allenstein, Dissertation, Technische Hochschule Stuttgart, 1953. [25] E. Kilcioglu, Diploma Thesis, Technische Hochschule Stuttgart, 1965;together with E. Jacob. [26] E. Allenstein, Z.anorg. allg. Chem. 308, 3 (1961). 569 Table 5. Force constants and bond order of X-S compounds. bonds in various Ref. 3.0 3.1 3.1 4.3 8. I 8.3 10.7 12.5 0.8 0.8 0.8 0.8 1.1 1.3 1.9 2.C 3.0 3.0 1.2 1.5 2.5 2.6 3.4 3.4 IV. The N-Si Bond Only a few examples have been selected from the chemistry of N-P compounds[331 to illustrate the range of the force constants. Discounting the band spectrum of PN in the gaseous state, which would probably give b 3, we find that the bond order for normal compounds varies between 0.8 and 2.0. The force constants in [C13P-N-PC13]+ [*I depend on the P-N-P angle (which is again unknown): Angle f[mdyne/A] I I 120 I 62 I 6 9 140” 1 160 I 7 8 1 f [mdyne/A] 3.1 3.5 4.1 4.3 6.1 6.9 10.6 1 9 1 111 variois b 0.8 I .3 I 1 1.7 2.0 3.4 .z .z O n t h e basis of calculations 1331, t h e most likely value for t h e angle is 140OC; it is associated with the highest force con[27] F. Watari, Z . anorg. allg. Chem. 332, 322 (1964). [28] H . Siebert, Z . anorg. allg. Chem. 292, 167 (1957); J . Nakagawa, S . Mizushirna, A . J . Saraceno, T . J . Lave, and J . V . Quaglinno, Spectrochim. Acta 12, 239 (1958) (calculated by present author). I291 W. Bubeck, Dissertation, Technische Hochschule Stuttgart, 1962. [30] B. v. Cramon, Staatsexamen Thesis, Technische Hochschule Stuttgart, 1958; F. Rauchle, Diploma Thesis, Technischc Hochschule Stuttgart, 1962. [31] H . Richerr and 0 . Gleinser, Angew. Chem. 72, 585 (1960). [32] H . Richert and 0 . Glemser, Z. anorg. allg. Chem. 307, 328 (1961). [33] R . Boumgartner, CV. Saivoch7y, and J . Gorrbeau, 2. anorg. allg. Chem. 340, 246 ( 1 965). [ * ] The author thanks Frau Prof. Dr. M . Brcke-Goeliring [34] (Heidelberg) for these data. 570 Table 7. Force corist2nls and b o n d orders of N-SI bonds In various compounds. I f [mdyne/*,] Single bond HN[Si(CH,)312; 140 HN(SiCl3)?; 140’ WSiH3).1 N[Si(CH3)3]>’3:’;160’ 3.3 3.5 4.0 4.1 4.2 >Us; 7.3 I b I Ref. 1.1 I .2 1.2 1.3 2.2 Discounting molecular SiN which exist only at high temperatures, we find that the maximum bond order is 1.3 for the N-Si bond in compounds stable at ordinary temperatures. The maximum value of f = 4.8 mdyne/A reported for the compound S=C=N-SiH3 [451 is not included in Table 7, since, on recalculation, we have obtained a considerably lower value. However, even for the published value the bond order would be only 1.45. In the case of silicon, compounds of the type >Si=NR, which are monomeric in the case of sulfur and phosphorus, exist here only in polymeric form. Although we have taken b = 1.3 as the maximum value, the bond order may be somewhat higher in certain compounds. V. The N-A1 180 Table 6 Force constants and bond orders of N - P bonds compounds [**I Bond The reported values of the force constants vary, probably because the bond angle exerts a strong influence on the value off. With Siebert’s value of f = 3.7 mdyne/A for the single bond, we obtain bond orders between 0.8 and 3.4, taking into account the hybridization and assuming a direct proportionality between f and b. However, since the bond order cannot exceed 3, it is better to ascribe the values b = 3 and b = 2 to NSF3 and HNSO, respectively, an3 to recalculate the other values on this basis (brl). The corrected Siebert formula gives b = 2.70 and b = 2.2 in these cases. The properties of these compounds confirm the presence of the double and triple bonds deduced from the bond orders. 111. The N-P stant known t o us (f = 6.9). I t should b e added t h a t compounds such a s R N P C 1 3 , R N P F 3 , NPC12, a n d NPF2, for which high b o n d orders are expected, exist only in polymeric form a t r o o m temperature. Bond In this case and in the following ones, spectroscopic data that may be used for the calculations are relatively scarce, except perhaps for aluminum nitride [*21 and the ammonia and amine adducts of aluminum com[**I Sieberr’s value for the sp3 single bond is f = 3.5. [34] M. Becke-Goehring and W. Lehr, Z . anorg. allg. Chem. 327, 128 (1964). (351 E. Steger, Z . anorg. allg. Chem. 309, 304 (1961). [36] N . Lonhof, Dissertation, Technische Hochschule Stuttgart, 1964. [37] J . v. Iriborne and D . G . de Kowalewski, J. chem. Physics 20, 346 (1952). [38] Calculated by the author from data communicated by R . Appel. [39] J. Curry, L. Herzbcrg, and G. Herzberg, Z . Physik 86, 348 (1933). [40] U. a. H. Kriegsnzanri, Z . Elektrochem., Ber. Bunsenges. physik. Chem. 61, 1088 (1957). [41] G. Gndmundsson, Dissertation, Technische Hochschule Stuttgart, 1964. [42] E. A . V. Ebsworth, J . R. Holl, M . J . Mackilon, D . C. McKean, N. Sheppnrd, and L . A . Woodward, Spectrochim. Acta 13, 202 (1958); H . Kriegsmann and W. Fiirsfer, Z . anorg. allg. Chcm. 2Y8, 212 (1959). [43] 13. Biirger, W . Sriwodny, and I/. Wannagat, unpublished work. [44] F. A . Jenkins and H. rle Laszlo, Proc. Roy. SOC.(London), Ser. A 122, 103 ( I 929). [45] K. Sothianandon and J . C . Margrave, J . molecular Spectroscopy 10, 442 (1963). Angew. Chem. internat. Edit. 1 Vol. 5 (1966) / No. 6 pounds [ l o , 461. The calculated force constants are between 1.9 and 2.5 mdyne/& to which correspond bond orders between 0.6 and 0.8 (Siebert’s force constant of the single bond is f = 3.0). VI. The N-Mg and the N-Na Bond Infrared spectra have been recorded for Mg3N2, Mg(NH2)2, and NaNH2 “221, and the force constants for the MgN (1.33 and 2.00) and NaN bonds (0.53) calculated only from antisymmetric vibrations on the basis of the two-body model now appear to represent upper limits. The resulting bond orders, b = 0.47 and 0.70 for MgN and b = 0.21 for NaN, are compatible with the high polarity of these bonds. Basically the same curve is obtained by plotting b against Ex for the nitrogen bonds formed with elements of the first short period and elements of the second short period (Fig. 1 ; x x denotes the sum of the electronegativities of the bonding partners, and the b values represent the maximum bond orders). Whenever S x > 5, double bonds can be formed (except in the case of F). Recalling condition 2 which states that, for the formation of double bonds, the sum of the electronegativities of the bonding partners must be at least 5, we note that, when this condition is not fulfilled (i.e. for the nitrogen bonds to Li, Be, B and Na, Mg, Al, and Si) the bond orders do in fact lie below b - 2 (b = 1 at x x m 4.5). Bond orders below 1 are due to the fact that the polarity of the bonds increases with decreasing value of >:x [31. The bond orders of 0.42 and 0.72 for the “single bond” adducts of Be and R, respectively, fit in well with this series. The coincidence of the curves for the two short periods may be regarded as a confirmationiof the second condition stated in Section A. dimer (probably a four-membered ring) is stable only at temperatures below -10 “C 1471, further polymerization occurring at higher temperatures. The N=Se bond is, therefore, less stable than the analogous N=S band. Approximate calculations based on the assumption that the dimer forms a four-membered ring have given f = 3.4 and b = 1.3 for the N-Se bond. In addition to insufficient experimental data on the nitrogen bonds of the transition elements, calculations are made difficult by the fact that the electronegativities are not accurately known and the electrons can occupy various orbitals. The force constants and the bond orders of the nitrogen bonds of the transition elements are listed in Table 8. None of the elements has been subjected to exhaustive investigations concerning their bond parameters. Instead, ranges of single values are given in Table 8 for the force constants of azides, since the N3 group was treated as a point mass in the simplified approximation. The lower limit corresponds to mass 14, the upper to mass cu. 30. T a b l e 8. Electronegativities (x). stretching frequencies (v), force constants (f), a n d bond orders (b) of the nitrogen bonds of transition elements Ref. Y f acc. t o Siehert v [cm-’1 I .6 406 565 3.2 I .8 555 3.7 560 615 Ill0 1.9 475 470 530 715 815 3.6 1073 2.0 552 2.1 495 565 780 3.8 1045 Na Mo Ai P SI S CI 2.1 2.2 0 LI Be B C N O F E. Order of Bonds between Nitrogen a n d Elements of Higher Periods Including Transition Elements The bond order for NBr is 1.7[71, and the bond is, therefore, similar to that in NCl. The compound RN=Se=O does not exist as a monomer and even its [46] H . Roszitiski, R . Dautel, and W. Zeil, 2. physik. Chem. N.F. 36, 26 (1961). Aiigew. Chem. Iiiternaf. Edit. / Vol. 5 (1966) / No. 6 4.1 3.8 1025 1023 1084 3.9 fex,. b 1.1 2.0-3.1 2.0-3.1 2.0-3.1 0.3 0.6- I .O 0.5-0.8 2.5 0.7 2.2 8.0 I .8 2.1 0.5-0.8 0.5 0.6 4.8 1.3 8.8 2.2-3.1 7.9 1.6-2.5 2.1 4.9 8.0 8.0 9.0 2.4 0.6- I .O 2.2 0.4-0.6 0.5 1.3 2.1 The higher values, which are more probable, lead to bond orders of 0.6 and 1.0 in accordance with expectations. When the values of b,, for Ti, V, Mo, W, Re and 0 s are plotted against c x (Fig. 2), the points for Ti, Re, and 0 s lie on the curve, while Mo, and particularly W and V (the last two in their chloronitrides with b > 2) lie above the curve. The 0 - V bond in oxygencontaining compounds has a similar anomalous position. [47] J . Goubenu and U . Weser, Z . anorg. allg. Chem. 319, 276 (1963). [48] K . Dehnicke, Habilitation Thesis, Technische Mochschule Stuttgart, 1964. [49] J . Strahle, Dissertation, Technische Hochschule Stuttgart, 1964. [50] M . Schober, Dissertation, Technische Hochschule Stuttgart, 1964. [51J J . Clioffand G. A . Rowe, J. chern. SOC.(London) 1962,4019. [52] G. Wilkiiison, Inorganic Syntheses VI, 169. [53] J . Lewis and G . Wilkinson, J. inorg. nuclear Chem. 6, 12 (1958); L . A . Woodwardand J. A . Creighron, Trans. Faraday SOC. 56, 1266 (1960); U. Hiibler, Staatsexamen Thesis, Technische Hochschule Stuttgart, 1962. 57 1 2.1 2.4 Most element pairs lying within the boundaries drawn for the first short period (shaded area) can similarly reach a bond order of 2, the exceptions being the chlorine bonds of P, S, and C [owing the polar bonding (halogen effect)] and the S-S bond. The latter however, is clearly a limiting case, since S=S bonds do exist in S2 mdecules at 800°C[541. These anomalies indicate that, as expected, Puuling's electronegativities represent only a rough approximation. 0L 5" " " "47' " L 19 - 53 51 zx m 55 20 Fig. 2. Maximum bond order hmax of the nitrogen bonds OF transition and other elements as a function of the suni of Pading's electronegativities of the bonding partners (Xx). 15 It cannot yet be decided whether the electronegatitivies [*I and/or force constants belonging to the single bonds are incorrect, or whether the transition metal bonds are in fact different. A more accurate knowledge of the hybridization may throw some light on this problem. 110 Q 05 F. Comparison with Bonds Involving n o Nitrogen 0 In order to compare the polarities of all possible bonds that may be formed within the first short period, we have constructed Fig. 3 by plotting the sum of the electronegativities against the difference of electronegativities of all such combinations. The three apexes of the triangle represent the three types of bond in pure form. rn 15 50 55 E5 70 Fig. 4. Diagram f o r element pairs from the first and second short period, obtained by plotting S x against Ax. The shaded area comprises bonds with b 2 2 in compounds stable at ordinary temperatures (curve I : b = 1; curve 11: h % 1 . 5 ; curve 111: b R+ 3). G . Conclusions Although it may seem like playing with numbers, the above treatment does lead to significant conclusions concerning the stability and the reactivity of multiple bonds. Thus, we can quote again the following examples (cf. Fig. 1): I x Q 3 :NrN: 10 3 HCEN: t \ ' N6 C X =6.0 ti3C3N3 cu= 5.5 H3B3N3H3 Z X =5.0 -0- 3 HB-NH 0 -20 30 50 40 60 ZX .65183, 70 80 Fig. 3. Diagram for element pairs from the first short period, obtained by plotting the sum of the electronegativities (Ex) against the difference of electronegativities (Ax) of the bonding partners. The shaded area 2 in compounds stable at ordinary tempercomprises bonds with b a tures. > The shaded area contains the element-pairs that can form multiple bonds (b 2.0) in compounds stable at ordinary temperatures. The boundary to the top right is determined by the electron deficiency (condition l), that to the top left by the difference, and to the left by the sum of the electronegativities (conditions 3 and 2, respectively) . Fig. 4 has been obtained in the same way for element pairs formed in the first and second short periods, the range being now set by 2lx = 4.5 and 7.0 and by Ax = 0 and 2.0 comprising the more interesting combinations. [*I The literature values of tlizse electronegativities differ by 0.1 to 0.2 eV. We have taken the mean values, but the highest ones would bring V, W, and M o nearer to the curve. 572 -> In the case of nitrogen with >:x = 6.0, the triple bond is so stable that no polymer is formed. In the case of HCN with r x = 5.5, both monomeric and polymeric forms exist, while only the polymer is known for HBNH characterized by x x = 5.0. In terms of the diagram in Fig. 4, the lower right element pairs form very stable multiple bonds with a small tendency to polymerize. The stability of the multiple bonds decreases, and the polymerizability increases, as one proceedes towards the top left, coming to bonds formed by carbon, phosphorus, and boron frequently encountered in polymer chemistry. The element pairs N - 0 , C-0, and B-0 form similar rows, as do C1-0, S - 0 , and P-0. The N - 0 system affords a further example of the polymerizability increasing with decreasing Cx: having 10 valence electrons, NO@ shows no tendency to polymerize. Nitric oxide has 11 electrons, and its y x value must be lower; accordingly, it dimerizes in the liquid ._ [54] R . Mueder, Helv. phys~caActa 2/, 41 1 (1948) ~ Angew. Chem. infernut. Edit. / Vol. 5 (1966) 1 No. 6 state a t a low temperature. The species NO 3, having 12 electrons and being formed in t h e reaction of NO with alkali metals in liquid ammonia, exists only as a dinier[551. This shows t h a t the position of t h e element pairs in Fig. 4 is not fixed, but varies according to t h e type of bond and t h e o t h e r bonding partners. The polymerizability of C=S has recently been confirmed in t h e case o f FzCS which polymerizes in t h e presence o f Insertion of this result into Eq. (4) leads to a set of linear algebraic equations: (F-AB)q == 0; = wz, (6) which have non-vanishing solutions for q only at certain ). values (eigenvalues). These eigenvalues are connected with the frequencies of the fundamental oscillations by the relationship A = 4x*cW, and they satisfy the secular equation: det(F-AB) = 0. (7a) fluorides 1561. Among the interesting bonds lying outside the shaded area in Fig. 4, whose boundary is not yet well defined in the low-Ax region, we may mention the P-C and the P-S bonds, which have been studied with respect to their reactivities but which require further spectroscopic investigation. H. Appendix Calculation of Force Constants and B o n d Orders (by Dr. W. Sawodny) I. Derivation of the Oscillation E q u a t i o n 157-591 The theoretical treatment of molecular vibrations starts with the assumption that the atoms can be considered as points of mass and the bonds between them as weightless springs, i.e. that we are faced with a mechanical system of coupled oscillators. It is further assumed that the very small motions of the mass points about their rest positions are harmonic vibrations and the potential curve can, therefore, be represented approximately by a parabola. The kinetic energy T and the potential energy V of the oscillating system are given by 2V = C fijqiqj = q’Fq (fij = ( 1 b) fji) IJ where the qi,j are displacements of the mass points from their equilibrium positions. The potential energy coefficients f i j are described as force constants, and are a measure of the strength of the springs, i.c. of the forces acting between the atoms. For sufficiently small and slow vibrations, the coefficients bij and fij can be considered independent of qi. By substituting expressions (la) and (lb) into the energy equation T-: V =: const. (2) The substitution G leads to = B-1, carried out for convenience sake, det(GF-~Eh) =: 0 (7 b) where E is the unit matrix. Besides the eigenvalues A (fundamental vibrations) obtained from the vibrational spectra, the determinant contains the matrix G, which is made up of the atomic masses, the bond angles, and the bond lengths of the molecule, and matrix F, which is constructed of the desired force constants[*]. The derivation of the secular equation (7b) is independent of the choice of coordinates for the oscillator displacement, provided that they satisfy conditions (la) and (lb). On the other hand, the coefficients bij and fij are determined by the coordinates used. Instead of Cartesian coordinates, linear transformations of these are generally used. Most convenient are the internal valence coordinates, which embrace only the internal vibrations of the molecule without its translational and rotational motions (these have zero frequency). The movement of the atoms is thus described in terms of changes in either the bond angle or bond length. The force constants thus assume the following physical meaning: the diagonal eiements of the F matrix represent the valence and the deformation constants, while the non-diagonal members denote the coupling constants between the valence and deformation modes. This is known as the General Valence Force Field (GVFF) model. The mathematical effort rapidly increases as the order of the determinant (7b) increases, though it may be simplified by virtue of molecular symmetry. Within a symmetry class, the fundamental molecular vibrations are subdivided into various symmetry species. One can find linear combinations of the inner coordinates that correspond to the symmetry properties of the fundamental vibrations. These are called ”symmetry coordinates”. The overall secular determinant of the molecule can then be factorized into several smaller determinants whose orders are given by the number of fundamental vibrations of the corresponding symmetry species. These smaller determinants can then be solved separately. Nevertheless, electronic computxs generally must be employed in the treatment of the vibration problem, particularly for molecules of low symmetry or containing many atoms 1601. we obtain ‘ / z (i’B6 + q’Fq) = const. (3) Differentation with respect to time gives the equation of motion Bq+Fq=O, (4) a possible solution of which is q ~ = a sin wt; q = ~ -ad sin c d = - d q . (5) .~ [55] J . Goubeau and K . Laitenbergtr, Z. anorg. allg. Chem. 320, 78 (I963); K . Laitenberger, Dissertation, Technische Hochschule Stuttgart, 1964. [56] C. Walter, US-Pat. 3032537 (1962). [57] R . Zurmiihl: Matrizen. Springer, Berlin-Gottingen-Heidelberg 1961. [58] E. B. Wilson Jr., J . C . Decius, and P . C. Cross: Molecular Vibrations. McGraw-Hill, New York 1955. [59] J. M . Mills in M . Davies: Infrared Spectroscopy and Molecular Structure. Elsevier, London 1963. Angew. Chem. internat. Edit. 1 Vol. 5 (1966) / No. 6 11. Solution of the Secular E q u a t i o n Equation (7b) is termed the characteristic equation of the eigenvalue problem, and can be solved by various methods for the eigenvalues A [ 5 @ . This type of calculation presupposes a knowledge of the force constants and can be used to confirm the assignment of the fundamental vibrations for a given [ * ] If there is only one fundamental vibration, the determinant reduces to one single equation gf - A = 0 which, for a diatomic molecule, leads to 4 ~ i v2z = f I:( 3 -t - . [60] D . E. Mann, T . Sliimanouchi, J. H . Meal, and L . Fano, J . chem. Physics 27, 43 (1957); J . Overend and J . R . Scherer, ibid. 32, 1289 (1960); T . Shimanouchi and I . Suzuki, ibid. 42, 296 (1965); J . Aldous and 1. M . Mills, Spectrochim. Acta 18, 1073 (1962); D . A . Long, R . B. Gravenor, and M . Woodger, ibid. 19, 937 (1963); E. W . Schmid, Z . Elektrochem., Ber. Bunsenges. physik. Chem. 64, 533 (1960); D. Papouselc and J . Pliva, Collect. czechoslov. chem. Commun. 28, 755 (1963). 573 molecule by using the constants of compounds having a similar structure 1611. However, the problem is generally the reverse: we know the vibrational assignment and seek the coefficients of the F-matrix. The derivation of Eq. (7b) in the manner specified above involves the basic difficulty of having to calculate for each symmetry species n(n+ 1)/2 GVFF-force constants from n eigenvalues (in the case of a symmetric F-matrix); as a result, the system of equation is insufficiently defined for n > 1 . We can now attempt to find further defining equations by the use of additional experimental data, or to reduce the number of required force constants to n by use of simplifying assumptions concerning the force field, or else to employ mathematical approximation methods. 1. A d d i t i o n a l E x p e r i m e n t a l D a t a T o obtain additional information, fundamental vibrations of isotope-substituted molecules can be used. The force field is considered to be independent of the masses of the system, i.e. the isotope substitution does not change the F-matrix, but the fundamental vibrations are changed and a new G-matrix results. Thus, additional equations for the force constants are obtained. However, the mass of the isotope-substituted molecule must differ sufficiently from that of the reference molecule, since otherwise the additional equations roughly duplicate the original ones, and the indeterminacy of the F-matrix is retained 1621. Furthermore, as the frequency shifts decrease, their measurement involves an increasing error. Therefore, H-D substitution is the most convenient for the calculation of force constants, although owing to tne small masses, the anharmonicity cannot be neglected introducing a further uncertainty. In addition, there are some isotope rules, such as the product rule of Teller and Redlich [631, which represent relationships independent of the force constants, and decrease the number of independent equations available for the determination of the F-matrix. The vibration-rotation interaction constants can also be utilized in the determination of the force constants. Both the centrifugal stretching constants DJ, DJK, and DK (641 and the Coriolis coupling constants <i [651 depend on the F-matrix, Although their relationship is a more complicated one, these constants can still be used successfully. The force constants are also related to the mean vibration amplitudes obtained from electron diffraction studies 1661 as well as t o the intensities of Raman and infrared bands[671, although all these measurements are not yet sufficiently accurate for the application of the data in the calculation of the force constants. Unfortunately, these additional data are at present available only for simple molecules. It may also happen that even the use of such information leads to no unique solution of the G V F F constants, and resort must then be taken to the application of simplified potential fields. 2. S i m p l i f i e d P o t e n t i a l F u n c t i o n s To reduce the number of forcc constants, we can make certain assumptions concerning the magnitudes of some [61] See, e . g . , H.-J. Becher, W. Sawodny, H. Nfith, and W. Meister, Z. anorg. allg. Chem. 314, 226 (1962); H.-J. Becher, ibid. 271, 243 (1953). I I ,molecdar Spectrosc3py 13, 338 1621 See, e.g., J . L. D U ~ I C UJ. (1964); L. Becknlarnt, L . Gutjalir, and R . Mecke, Spectrochim. Acta 21, 141 (1965). [63] 0.Redlich, Z . physik. Chern. B 28, 371 (1935). [64] D . Kivelson and E. B. Wilson J r . , J. chern. Physics 21, 1229 (1953). [65] J . H . Meal and S. R . Polo, J. chern. Physics 24, 1126 (1956). [66j S . J . Cyvin, Acta chern. scand. 14, 959 (1960). 1671 B. L . Crawford, J. chern. Physics 20, 977 (1952); G. W . Clmrirry and L . A . Woodward, Trans. Fardday SOC.56, I l l 0 ( I 960). 574 constants or the relationships between them. One possibility is to borrow some of the constants from other molecules. Particularly suited for this are constants obtained with the aid of the Urey-Bradley field (681. However, this process becomes questionable when the changes in the bonds caused by the modified surroundings are to be investigated. Simplification of the potential functions is best illustrated by means of an example: a symmetric non-linear three-body system ZXz of symmetry Czv executes three fundamental vibrations, two of the type A1 and one of the type B1. The potential energy is given, in accordance with Eq. ( I ) , by hi)+ 2fr, A r l Ar2 + fa Az2 I 2fr, 2~ = f, ( h + (Aria Ar2x) Therefore, the G V F F contains four constants to be determined. The following two simplifying assumptions can be made. In one case, we assume that all interaction constants are negligibly small, so that they can be put equal to zero. The potential function then contains only two constants: This model is known as the Simple Valence Force Field (SVFF). In the second case, we assume that the potential energy is a pure square function of all the internuclear distances, i.e. that the force acting on a n atom is the sum of the attractive and the repulsive forces due to all the other atoms, these forces coinciding with the lines between the nuclei and depending on internuclear distances alone. This is the ionic model of the Simple Central Force Field (SCFF), in which the potential energy is given by: 2V = fr car:+ Ar?$ + fRhR2 Both the deformation constants and the coupling terms have thus been expressed in terms of the interaction between the two X atoms not bonded to each other. This postulate is inadequate in the case of linear and planar molecules, since it leads to a zero frequency for the out-of-plane vibrations. With certain molecular configurations, such as tetrahedral ZX4 molecules, not only the squared, but also the linear potential energy terms must be taken into account. The rest position of the atoms is then described as a state of equilibrium between the attractive and repulsive forces. Both the SVFF and SCFF system are characterized by a redundancy of parameters. Thus, in the given example, three fundamental vibrations are available for the determination of two force constants. The third equation can then be used to verify the results. The results reveal that both postulates involve oversimplifications [@I. It appears, therefore, that we cannot neglect all the coupling constants. The SVFF treatment gives the better result when the central atom is heavy in comparison to the ligands (weak coupling), whereas it is the SCFF treatment that leads to the better approximation when the central atom is light and the ligands are heavy (strong coupling). The results can be improved by the introduction of additional interaction terms, so as to have the same number of force con~ ~~ - [68] See, e . g . , J. Overend and .I. R . Scherer, J. chcm. Physics 32, 1296, 1720 (1960); 33, 446 (1960); 34, 547 (1961). [69] G. Herzberz: Molecular Spectra and Molecular Structure. Vol. 11, Infrared and Iiarnin Spectra of Polyatomic Molecules. Van Nostrand, Toronto 1951. Angew. Chem. internut. Edit. / Vol. 5 (1966) / No. 6 stants to be calculated as the number of the available fundamental vibrations. These modified models have been used in most force-constant dettrminations. The above mentioned equalization may be achieved by the introduction of the required number of G V F F coupling terms into the SVFF model (Modified Valence Force Field, MVFF). Although the choice of the interaction terms is arbitrary, we can often rely o n analogy and past experience. In many cases all interactions between identical internal coordinates are used, since this advantageously leads to a diagonal F-matrix in the derivation of the oscillator equation with symmetry coordinates. The potential energy of the non-linear Z X 2 molecule is then given by 2v - fr + Ari) + frr Arl Ar2 -1- r,l& A model which combines the SVFF and the S C F F concepts is the Urey-Bradley Force Field model (UBFF) [701. It contains the valence force constants and the deformation constants of the SVFF model and describes all couplings in terms of forces acting between the non-bonded atoms: 2V = fr (Art L Ari) + fa Aa2+ fRAR2 This treatment generally affords valence force and deformation constants that are appreciably smaller than those obtained by the valence force fields, but gives relatively large van der W a d s terms. As mentioned above, the force constants can often be transferred from one molecule to another similar one, but in some cases the UBFF model must be further modified to obtain meaningful results [711. For easy comparison with force constants obtained by other methods, the UBFF model is frequently employed to calculate the potential energy with symmetry coordinates, but the resulting constants are transformed into those of the valence force field. Attempts have also been made to derive simplified potential fields by considering the coupling from the point of view of bond theory. Thus, Heath and Linnett 1731 have used the change in hybridization that occurs during the vibration, and developed the Orbital Valence Force Field (OVFF). When an atom is displaced from its equilibrium position by a change of the bond angle, the overlap of the orbitals decreases. In certain deformation vibrations, the orbitals of the central atom can follow this motion either by a partial rotation of the system or a rehybridization with change of the s and p contributions. We thus arrive at a new definition of the deformation constants. The OVFF model resembles the UBFF model in describing the coupling forces by the interactions between the non-bonded atoms. Rehybridization of the orbitals of the central atom can give rise to yet another definition of the coupling between valence and deformation modes : a n increase in s-character shortens the bond, while an increase in the p-component lengthens it. These considerations have been incorporated in a model by Mi//s[731 called the Hybrid Orbital Force Field (HOFF) model. This treatment has led to good results in force constant calculations[74], but is at present applicable only when d orbitals d o not participate in the bond. A similar, though more complicated, model has been developed by KingC751 for the NH3 molecule; it is known as the Hybrid Bond Force Field (HBFF) model. We must also mention a n attempt at using the interactions of a x-electron system to describe relations between some of the G V F F force constants. This is called the Tc-bond Interaction Valence Force Field (x-IVFF) concept, and its use has made it possible to reduce the number of constants to be determined in the case of Ni(C0)4 from 17 to 10[761. Table 9a. Force constants of tetrahedral molecules Z X j calculatcd \\it11 diNcrcnt potential field.. Com pd. Method OVFF UBFF ( V ) MVFF Fudini GVFF (Rot. const., Isotop.) OVFF UBFF UBFF (V) MVFF (I) MVFF (11) Fudini GVFF (Tsotop.) GVFF (Rot. const.) UBFF U B F F (V) Fudini GVFF (El. diffract.) GVFF (Raman int.) MVFF U B F F (V) Fudini GVFF (Isotop.) OVFF UBFF U B F F (V) MVFF Fudini GVFF (Rot. const.) GVFF (Isotop.) MVFF Fudini GVFF (Raman int.) 1701 H . C. Urey and C . H . Brodley, Physic. Rev. 38, 1969 (1931). 1711 T . Shimanouchi, Pure appl. Chem. 7, 131 (1963). [72] D . F. Heath and J . W . Linne/t, Trans. Faraday Sot. 44, 873 ( 1948); J . w.Linnetr and P. J . Wheotley, Trans. Faraday soc. 45, 33 (1949). Angew. Chem. internat. Edit. / Vol. 5 (1966) i Nu. 6 fI 4.95 4.944 5.37 5.391 5.050 4.3s 3.75 ( I < ) 6.15 9.15 6.72 6.876 6.25 6.97 1.76 (I<) 2.98 3.268 3.59 3. I57 2,77 2.743 2.764 2.769 5.9 5.4 (K) 5.95 5.94 6.400 6.16 6.57 2.75 3.077 3.124 frr FIE ~ -- fra ~~ 0.036 0.051 - 0.008 0.142 0.151 0.036 0.6 (F) 1.35 (F) 1 .oo ~~ 0.81 0.788 0.99 0.75 C.656 (F) 0.45 0.3 70 0.24 0.415 - 0.431 0.436 0.420 0.460 0.430 0.844 -0.3 (F') 0.50 1.13 0.567 0.90 0.59 -0.097 ( F 0.24 0.458 0.36 0.406 - 0.034 0.028 0.027 0.15 (F) 0.43 (F) 0.39 0.4 I 0.254 0.33 0.21 0.441 0.436 0.420 0.486 0.469 0.02 I 0 0.016 0.5 (z) I .33 I 0.71 0.15 ( H ) 0.62 0.71 0.71 0.710 0.70 C.71 0.08 ( H ) 0.33 0.317 0.32 0.34 0.207 0.190 0, I88 0.188 1.014 1.12 1.01 -_ 0.26 (%) 0.46 0.417 0.38 0.420 0.207 0.225 0.231 0.231 0.56 -0.31 (F') 0.26 0. I68 0.03 0.19 0.33 - 0.232 0.21 3 0.100 0.1 10 0.02 ( H ) 0.26 0.268 0.3 ( x ) 0.50 3.268 0.454 0.47 0.27 3.26 3.158 3.157 3.082 0.468 0.44 0.25X 0.236 3.235 [73]I . M . Mills, Spectrochim. Acta 19, 1585 (1963). [741 J. Aldous and 1. M. Mills, Spectrochim. Acta 19, 1567 J ' L. D'"cn'l, ibir'. Igo7 1751 14'. T. Kitis, J. chem. Physics 36, 165 ( I 962). [76]L . H . Junes, J. molecular Spectroscopy 5, 133 (1960). 7'' 575 3. M a t h e m a t i c a l A p p r o a c h Mathematical approximation methods may be used to eliminate the indeterminacy of the secular equation (7b) caused by the fact that the number of force constants to be calculated exceeds the number of experimental data available. Thus, in one approach, the matrix product A = G F is reduced to the Jacobian canonical form, i.e. the determinant (A] now consists only of the product of the diagonal elements. The eigenvaiue Ak involving the fundamental vibration vk is considered as characteristic of the coordinate qk, but corrected for the interaction with the rest of the molecule. The fundamental vibrations are first arranged in a series on the basis of their magnitudes, and the individual terms are then successively removed, so that the force constants fij (i > j, will depend only on eigenvalues between Ai and hn[771. A recent method [31 based on the Cayley-Hamilton theorem gives n(n+ 1)/2 equations for n(n+ 1)/2 force constants, but these equations are not mutually independent. However, this mutual dependence can be eliminated by systematic slight alterations of the equations and the use of an approximate solution. In practice, we start with fully uncoupled vibrations as a first approximation. For this case, a fully defined diagonal F-matrix from the diagonal G-matrix can be obtained. Using this F-matrix as a starting point, a complete F-matrix containing all the coupling terms can be calculated by the iterative and stepwise build-up of the known, complete G-matrix from the diagonal matrix. Table 9 b. Force constants of non-linear symmetric Z X ? molecules calculated uith c!'i?'xent potential fields. Compd. Method H2 MVFF UBFF Tcrkiiigron Fadini GVFF (Isotop.) UBFF MVFF (1) MVFF (11) Torkingrou Fudini GVFF (Isotop. and Rot. const.) UBFF MVFF Torkingron Fadini GVFF (Isotop.) GVFF (Rot. const.) MVFF Torkinglon Fadini GVFF (Rot. const.) UBFF IMVFF 0 ClOl SO? 0 3 NO? Torkington FzO frr f, Fadini GVFF (Isotop.) MVFF Torkingron Fadini GVFF (Rot. const.) 8.250 (K) 8.437 8.330 8.422 8.454 7.150(K) 7.16 6.78 7.144 6.8 I5 7.018 10.09 (K) 9.86 10.172 9.887 10.02 10.006 4.64 6.338 5.624 5.701 8.46 (K) 10.022 10.818 10.1 1 1 10.406 3.08 5.320 3.964 3.950 0.083 (F: -0.118 -0.225 -0. 132 -0.100 -0.441 (F: __ -0.22 0.724 -0.164 -0. I70 0.21 (F) -0.12 0.290 -0. I08 0.03 0.024 0.703 2.160 1.440 1.523 3.03 (F) 1.652 2.210 1.741 2.024 - -0.252 0.862 0.806 frcx 0.407 (F' - 0.104 0 0.234 0.014 (F' 0.18 - 0.328 0.022 0.006 0.31 (F') - 0.452 0.044 0.20 0.189 - 0.964 0.297 0.332 - 0.851 0.068 0.535 -0.14 0.680 0. I63 0.137 fi 0.984(H) 0.751 0.772 0.i54 0.761 0.823 (H) 0.63 0.635 0.778 0.618 0.651 0.993 ( l i ) 0.8 I 0.796 0.808 0.793 0.793 1.696 1.131 1.308 1.285 0.395 (H) I .036 0.870 1.021 1.097 1.13 0.788 0.715 0.724 The UBFF force constants have the following meaning: K F stretching force constant = H x repulsive force constant between non bonded atoms repulsive force constant, linear term F' bending force constant ~ internal tension UBFF(V) denotes constants of valence force field calculated from UBFF. ~ [77] P . Torkington, J. chem. Physics 1 7 , 1026 (1949). [77a] T . Shimanourhi, J . Nakagawa, J . Hireishi, and M . isliii, J . molecular Spectroscopy 19, 78 (1966). 576 The results of calculations carried out for some simple molecules using various force fields are listed in Tables 9a and 9b, but their discussion must be preceded by brief mention of the concept of coupling. If, in the triangular ZXz molecule used earlier, the mass of the central atom m, is large in comparison to that of the ligands m,, then the central atom will remain virtually static during the vibrations: In this case, the vibrations are practically pure valence or deformation modes, owing to the lack of any coupling via thc central atom. When, however, the mass relationship is the reverse (i.e. m, < m,), it is predominantly the central atom that oscillates; because of its similar motion in both vibrations, strong coupling between the two modes results. Although the vibrations are no longer pure stretching and deformation vibrations, they still retain some of their original character, namely, the lower frequency is predominantly the deformation mode and the higher frequency, the valence vibration. On the other hand, in the case of strong coupling between two valence vibrations complete mixing can take place. When, for example, a tetrahedral molecule YZX3 is characterized by the mass relationship of m, < m, = my, we can no longer describe the symmetrical valence vibrations as vzy and vzx3 sym., because these motions actually represent an in-phase and out-of-phase vibration vyzxj. In addition to this coupling of masses, the coupling of vibrational energies also plays a part, as for example, in the NO2 molecule. In this case, the relationship m, < m, does hold, but the valence vibration of the N-0 bond of bond order 1.5 and the deformation vibration differ in their energy so much that they can interact only to a small extent. Weak coupling does not mean that the coupling constant will be very small. Calculations carried out on H 2 0 , C H 4 , and NO2 by the GVFF method have given by no means negligible values for the interaction constant frW However, the coupling constant can vary within wide limits without notable deviations in the valence and the deformation constants. As coupling increases, the system becomes more and more sensitive to changes in the interaction constants. It is, therefore, not surprising that the MVFF method gives poor results in the case of strong coupling. Quite often, no real solutions are obtained at all, as is the case with FzO, C 1 2 0 , C F 4 , and CC14; nor can the situation be remedied by neglecting frr,since its value cannot be neglected in most cases (e.g. CF4). By contrast, the UBFF method gives a better approximation for strongly coupled systems, but it leads to erroneous answers in the case of weak coupling. Whereas the constants obtained for C F 4 and CC14 are in good agreement (after transformation of the valence coordinates), the UBFF method Angew. Chem. internut. Edit. 1 Vol. 5 (1966) No. 6 must be modified to result in reasonable values for CH4. Similarly, additional interaction constants must be introduced in the calculations o n silicon compounds and molecules with resonance structures (e.g. NOz) [711. The resulting modified model (MUBFF) gives good results, but generally contains more constants than the available number of vibration frequencies “J7al. The force constants obtained by the O V F F method are very similar to those of the UBFF method. The H O F F method o n the other hand, represents a considerable improvement. In calculations on YZX3 molecules[741, it was shown to be superior to the MVFF and the U B F F methods, although for tetrahedral ZX4 molecules it again involves more constants than there are vibration frequencies available. The mathematical method put forward by Torkington [771 generally leads to excessively high coupling constants. By contrast, the results obtained by the Cayley-Hamilton theorem[31 for intermediately coupled systems (CF4, SiF4, and SiC14) are in good agreement with those obtained by the G V F F method. For weak coupling, the resulting coupling constant is practically zero, but as mentioned earlier, this has only a small effect on the valence and deformation constants. The method is not applicable to strongly coupled valence vibrations, since a certain vibration character is a prerequisite for the validity of the first approximation (uncoupled vibrat ions). Summing up, we may say that unambiguous answers concerning the coupling constants are available at the present only in very few cases, even with an exact knowledge of the vibrational frequencies and possible additional data as well as all the molecular dimensions (for the influence of the latter, see [781). Thus, even the values obtained by the G V F F method, using various additional data, show considerable deviations, as for example in the case of SiF4. As regards the valence and the deformation constants, on the other hand, the appropriate choice of the potential function ensures a n accuracy of about 10%. Higher accuracy presumably is obtained by the application of a unified method. Deformation force constants sometimes cannot been given in an explicit form, owing to the presence of “redundant coordinates” (e.g. in the case of ZX4 tetrahedra, cf. Table 9a). For the time being, it is reasonable, therefore, to confine the discussion to valence force constants, which, incidentally, are also the most important parameters as regards the properties of bonds. Table 10. Comparison of valence force constants calculated from e m pirical rules and from experimental data. Force constants according. t o : Bond CH NH OH BC BN BO BF Budsrr G,,l.dJ Sirbert 4.33 6.33 7.67 2.04 2.55 4.45 5.81 6.94 2.93 3.5.7 4.05 5.40 6.30 7.20 3.3P 3.94 4.50 2.92 3.14 2.93 3.77 4.31 4.78 cc CN co CF CCI SiH P IH SiF Sic1 PCI AlCl SIC PC GeH GeCl 2.78 7.68 3.44 2.33 I ..57 1.99 1.16 1.53 1.90 2.82 I .25 4.49 5.06 3.76 4.54 5.25 5.85 3.54 2.29 2.74 3.28 2.13 2.50 1.82 2.20 2.56 4.05 4 73 5.40 6.08 3.40 3.74 4.00 2.17 2.10 4.20 2.35 2.52 2.18 2.80 3.00 3.60 2.27 5.05 6.00 6.45 3.64 -2.8 4.16 4.87 4.45 4.49 5.22 6.97 3.59 2.77 3.16 6.57 3.12 3.52 1.97 2.93 3.35 2.61 2.66 1 (CHn) (NH?) (HzO) B(CH3)3 BX,--NR> B(0H)y B F ~ CH,-CHI N(CH,)? O(CH3), CF, CClJ SiH, PHP SiF4 SIC14 Pclg AICij Si(CH,)., P(CH,$ GeH, GeCI, where kij is a constant determined by the position of the bonding atoms in the periodic table, k‘ is a correction factor for ions, and ZA and ZB denote the number of valence electrons of the two atoms. Another relationship involving the number of valence electrons z i has been put forward by Guggenheimer[811 in the form: f = R(zAzB)l12/rt where R and t are constants whose values depend on the polarity of the bond. Gordy’s relationship [821 involving the electronegativities Fah(XAXB ;?-) 3’4 c is a special case of Guggenheimer’s formula[831. The constants a and c here generally have the values of 1.67 and 0.30 (other values have to be assumed when one of atoms involved has only one valence electron) ; b denotes the bond order. Siebert 151 has devised the particularly simple formula: 111. Empirical Rules for Valence Force Constants There have been attempts to correlate the valence force constants with other bond properties such as internuclear distance, dissociation energy, and the number of electrons in the bonding partners. The literature contains many such empirical correlations, though they are generally derived and verified only for diatomic molecules (Table 10). The most important empirical rules are as follows: Badger’s ruler791 correlates the force constant f with the internuclear distance r : where dij and Cij are constants whose values are determined by the position of the bonding atoms in the periodic table. A similar correlation is given by Clark’s formula [801: where the constant P invariably has the value of 7.20, Zi is the nuclear charge and ni is the principal quantum number of the valence electrons of atom i. This formula enables us to calculate the single bond values fi without knowing any experimental data such as bond lengths. However, this formula is strictly valid only for bonds of sp3 hybridization. It can be shown that the formation of free electron pairs at the bonding partners (i.e. the setting-up of overlapping p-orbitals) weakens the bond. This can be taken into account by the introduction of an empirical correction factor in Siebert’s formula“]. Furthermore, the single bond value of the force constants increases with increasing s-character of the bond 1841. [81] K. M . Guggenheimer, Proc. physic. SOC. (London) 58, 456 (1946); Discuss. Faraday SOC.9 , 221 (1950). [82] W. Gordy, J. chern. Physics 14, 305 (1946). 1831 R. L . Williams, J. physic. Chem. 60, 1016 (1956) [84] J . Goubeau, unpublished work. [78] R . R . Hart, Developments appl. Spectroscopy 4 , 171 (1965). [79] R. M . Badger, J. chem. Physics 2, 128 (1935). [80] C . H . D . Clark. Philos. Mag. 18, 459 (1934); H . S. Allen u. A. K . Longair, Philos. Mag. 19, 1032 (1935). Angew. Chem. internat. Edit. / VoI. 5 (1966) / No. 6 1851 L. Solem, J. chem. Physics 38, 1227 (1963); L S a l e m , J. chem. Physics 38, 1227 (1963); P . Phillipson, ibid. 39, 3010 (1963); 44, 633, (1966). [86] S. Bratoz u. M . Allavena, J. chem. Physics 37, 2135 (1962); J. Chirn. physique 60, 1199 (1963). 577 Siebert also gives a formula for the determination of the bond order b from the single-bond value f l : b = rbfhiri ri Moreover, we can tibe up to b = 1.5 whenever the distances rl and Tb are not known. However, for bond orders over 1.5, we must use the formula - 0.29 7 0,71 Acknowledgments an even simpler relationship b = fb/fl b cules (H20, NH3) force constants have been calculated from energy values obtained from an approximative wave function [86J. fbbifl containing empirical correction factors; fb always is the experimental force constant. Finally, it should be mentioned, that also attempts where made to find relations between quantum mechanical data, as the electron density, and force constants [851. For small mole- The author wishes to thank his colleagues for their assistance in the syntheses and the spectroscopic investigations, and especially Dr. W. Sawodny for the calculation of numerousforce constants. He also wishes to express his gratitude to the Deutsche Forschungsgemeinschaft, the Ministry of Scientific Research, and the Fonds der Chemischen Industrie for their financial support. Received: November 26, 1964; revised March 14th, 1966 [A 518 IE] German version: Angew. Chem. 78, 565 (1966) Translated by Express Translation Service, London C 0M M U N ICAT I0N S Amination of Hydrazines to Yield Triazanes By Prof. Dr. Ernst Schmitz, Dip].-Chem. S. Schramm, and Dipl.-Chem. Heide Simon Institut fur Organische Chemie der Deutschen Akademie der Wissenschaften, Berlin-Adlershof (Germany) Experiments with l5N have shown that hydrazine can be aminated by hydroxylamine-0-sulfonic acid, the reaction proceeding by way of triazane [I]. We have found 2-acyloxaziridines [21 to be extremely effective aminating agents. The compound ( I b) aminates cyclohexylamine at room temperature within one minute, giving the hydrazine derivative (2) (m.p. and mixed m.p. [31 141 "C). This reaction led to attempts to aminate hydrazines to triazanes. C&11-NH-NH-CO-NH-C&i H, (2). 61% -, y - C O - m - R in which fission of the N-N bond is coupled with dealkylation. Dilute acid liberates ammonia (0.7 mol) and formaldehyde (0.5 mol) from compound (3) within a second [reaction (b)], and because of this competing reaction the yield of the iodide reaction amounts to only 70-75 %. Neutral aqueous solutions of compounds (3) and (4) decompose five minutes at room temperature with formation of tetramethyltetrazene ( 5 ) and urea or phenylurea, respectively [reaction (c)]. Compound (5) was identified by gas chromatography, phenylurea by melting point and mixed melting point, and urea by means of xanthydrol. The same products are formed when the crystalline compounds ( 3 ) and (4) are kept at room temperature for one hour. During a melting-point determination, compound (4) reacts at 80-85 O C and then shows the melting point of phenylurea. Compounds (3) and ( 4 ) are formulated as triazanium betaines, in analogy to the 2,2-dialkyltriazanium salts obtained by R. COsl[41 from N,N-dialkylhydrazines and hydroxylamine-0-sulfonic acid. Final proof of structure was provided for the triazanes prepared analogously from monoalkylhydrazines. A crystalline compound (6) (m.p. 63-65 "C) separates in 30 % yield from a solution of ( l a ) and methylhydrazine in tetrahydrofuran; (6), too, oxidizes iodide to iodine. Its thermal or acid-catalysed decomposition leads to ammonia, formaldehyde, and semicarbazide [reaction (d)] ; the semicarbazide contains the N-N bond obtained on formation of the triazane. 2-Acyloxaziridines, ( l a ) and ( I b ) , suspended in benzene, dissolved immediately on addition of N,N-dimethylhydrazine, and crystalline compounds separated in 70-80% yield in 1-2 minutes at 5-10°C. Elemental analyses showed that products containing four N-atoms were formed with loss of benzaldehyde. These products liberated iodine from acidic iodide solution in the cold [reaction (a)]. As with peroxides, oxaziridines, and diaziridines, acid initiates an intramolecular redox reaction, I z + NH3 5 3 3 H2N-%-NG-CO-NH-R n 578 - P ----) \ CH3 (3) ,R = H I , , z (a) CHlOf NH3 (b) I hydrazine derivatives with unknown f On reaction with cyclohexylhydrazine, compound ( I b ) affords the triazane (7) (yield 70 %). The position of the alkyl group, not rigidly proved for (6), was made certain for this product (7) : the colorless crystals (7), when brought into contact with acid, pass within seconds into yellow crystals (m.p. 89 "C), the azo compound (8) being formed with loss of ammonia. In (8) the cyclohexyl group has the position required by the triazane structure (7). structure u Angew. Chem. internat. Edit. Vol. 5 (1966) J No. 6

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