A Model for the Hydrogen Bond LELAND C

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Proc. Nat. Acad. Sci. USA Vol. 72, No. 12, pp. 4701-4705, December 1975 Chemistry A model for the hydrogen bond LELAND C. ALLEN Department of Chemistry, Princeton University, Princeton, New Jersey 08540 Communicated by R. S. Mulliken, September 2, 1975

ABSTRACT A model of the linear hydrogen bond has 1 is defined as that radial distance which encloses 98% of been devised which is characterized by three physical pa- the charge in the highest occupied monomer molecular or- rameters of the isolated electron donor and proton donor bital (Table 2), and it bears an inverse relationship to ioniza- molecules: MA-H, the bond dipole moment; AI, the difference 1 in ionization potential between that of the electron donor tion potential. Thus for a given row in the periodic table, is and the corresponding noble gas atom; and 1, the length of almost a linear function of AW. the electron donor lone-pair of electrons. The model is able The proton donor parameter is MAA-H, and its relationship to explain dimerization energy, charge transfer, internuclear to the monomer dipole moment is shown by the diagrams of separation, directionality, stretching force constants, dimer Fig. 2. The monomer dipole moment is made up of the pro- dipole moment, and infrared intensity enhancement. jections of the bond dipoles plus the other dipole moment components of the molecular charge distribution. The prin- The normal hydrogen bond is the relatively weak (1-10 cipal other contributing components are the symmetry axis kcal/mole, approximately one-twentieth that of a typical lone-pairs, and there is one of these for each monomer (in ionic or covalent bond) link that connects a covalently general these lone-pairs are more tightly bound than the hy- bound, slightly positive hydrogen atom to an atom in the drogen bonding lone-pairs; in NH3 and PH3 they are identi- upper right-hand corner of the periodic table (N, 0, F, P, S, cal). Analysis of the monomer charge density distribution Cl, As, Se, Br, and sometimes C). In spite of its central im- axis lone- portance to much of chemistry and biology, a model relating shows that the dipole moment of the symmetry the properties of the hydrogen bond to the physical quan- pairs, AsIp, is small for FH (and CIH), medium for OH2 tities characterizing the participating molecules has not been available. Although a large and significant body of quantita- H tive experimental measurements exists, as well as six books, almost all of the observations have been made on molecules H) in solution rather than on the simple isolated species that are H traditionally studied when a basic understanding of force H laws are sought. Recently, a moderate sized literature of ab initio quantum mechanical calculations has arisen along H: with a few high resolution beam spectroscopy measurements for selected dimers, and this provides a data base for devel- H- oping mechanistic models. These results show that the hydrogen atom and the two atoms attached to it lie almost H 3 along a line. The model proposed here has been designed to represent linear hydrogen bonds between all pair combina- tions of the molecules, NH3, OH2, FH, PH3, SH2, and ClH. Covalent H Bond (The geometrical arrangement for the water dimer is shown radii radii (extents) on the upper left in Fig. 1: electron donor on left, proton donor on right.) It can be extended also to strong hydrogen .8 bonds and to electron donors of biological interest. .6 Psro) PARAMETERS OF THE MODEL .4 I H2 The model uses the conventions of the molecular orbital rep- .2 resentation because this has proved to be the simplest frame- o.ov I work that can quantitatively account for a wide range of 0 .4 .8 4.2 4.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 chemical bonding phenomena. The highest occupied mono- .8 - C r mer molecular orbital forms the hydrogen bonding lone-pair .6 - of the electron donor (BHn), and its angular and radial de- P(r. pendence is displayed in Fig. 1. The hydrogen bond to a ).4 / \ I1sH2 noble gas atom is taken as zero, and AI, the ionization potential of a hydride electron donor, BHn, referenced to the .2 - noble gas atom in the row of B, is the relative energy for hy- 0 /4L 8 2 4.6 2.0= 2.4 2.8 3.2 3.6 4.0 4.4 4.8 drogen bonding (Table 1). The hydrides are formed, hydro- I rs gen bonding long-pairs are created, and the symmetry is lowered as protons are pulled from the noble gas atom; Al is analogous to the splitting parameter in crystal field theory. FIG. 1. Electron distribution of electron donor lone-pairs. A principal source of hydrogen bonding arises from charge Polar plots for the six monomers (top right). Radial amplitude for. rearrangements in the monomers, and Al is also a measure OH2 and SH2 (bottom). Definition of coordinate system for (H20)2 of the relative polarizability of the electron donor. in upper left-hand corner. 4701 Downloaded by guest on September 30, 2021 4702 Chemistry: Allen Proc. Nat. Acad. Sci. USA 72 (1975)

Table 1. Ionization potentials, I, and H-F-Hal.82 D ionization potential differences, AI (electron volts) 0 i=1.82D ,-H=10a.13 D I Al H Monomer BHn Exper.* Calc.t Exper. Calc. t1.85D NH3 10.88 11.23 10.68 11.77 H/ OH2 12.62 13.59 8.94 9.41 FH 16.05 17.08 5.51 5.92 H \ALN-H=0.66D Neon 21.56 23.0 PH3 10.60 10.38 5.16 5.42 --aX . /LSlp SH2 10.47 10.43 5.29 5.37 HES /1 =4.47 D ClH 12.74 12.77 3.02 3.03 H Argon 15.76 15.8 Monomer Dipole Bond Dipole Symmetry Axis * First ionization potentials (1, 2). Moment, fu Moment; P-A-H Lone - Pair t First ionization potentials obtained by Koopmans' theorem from FIG. 2. Schematic diagram showing relationship between mo- wavefunctions constructed with the 4-31 G basis (3). nomer dipole moment, bond dipole moment, AA-H, and the symmetry axis lone-pair dipole moment, ,Ip. The sum of the AA-H compo- (SH2), and large for NH3, (PH3). Their contributions to the nents along the direction of the monomer dipole equals tM minus monomer dipole moment are therefore approximated as 0, /slp. 25, and 50%, respectively (Table 3). The bond dipole moment may be thought of as a quantitative measure of the erally unknown R values are taken from computed results electronegativity of the proton donor, and- as such it obeys since relative internuclear separations are in error by less the well-known diagonal relationship which is a similarity in than 3% in all sets of computed data. values between oxygen and chlorine and between nitfiogen and sulfur. Although not a condition on the validity of the model, it simplifies the analysis and helps smooth the data Charge redistribution base if diagonal averages (0-H with Cl-H and N-H with In hydrogen bonding there is appreciable charge transfer S-H) are used (Table 3). from the electron donor to the proton donor but considera- bly less than occurs in some conventional donor-acceptor RESULTS complexes. The pattern of this charge transfer is ordered by the lone-pair radial extent, 1. A large I implies a relatively Energy formula large overlap of the proton donor, resulting in a relatively The energy of the hydrogen bond complex can be expressed large charge transfer. Charge transfer is ordered according as: to the electron donor sequences NH3 > OH2 > FH and PH3 > SH2 > ClH, and transfers for the third row are compara- ED = K/A-HAIIR [1] ble to those of the second even though the average dissociation energies are in the ratio of 1 to 2.25. where K is an energy scale factor and R is the A...B internu- The driving force for charge redistribution is the proton clear separation. Because there is a nearly linear relationship donor bond dipole. Charge transfer and total charge change between I and AI for each row, the binding energy formula on the electron donor are roughly linear with MIA-H. The pat- can also be expressed in terms of 1-1 where lo = 0.01 and tern of charge change on the proton donor parallels that of 1.44 A for the second and third row, respectively. I-lo aids in ED. extending the model to complicated electron donors where an ionization potential reference is not readily apparent. Table 3. Monomer dipole moments, p, and Fig. 3 shows the match achieved by Eq. [1] for the three bond dipole moments, 1A -H (Debyes) most accurate sets of computed data available in the literature (diagonal averaging was used in the left graph). Eq. [1] Diagonal with experimentally derived AA-H and Al compared to the Average five available dimerization energy measurements [experi- M_____ H'A Ht 11A-H mental values in parentheses with K fit to (FH)2] are (FH)2, Monomer 7.0 (7.0 1 1); (OH2)2, 6.7 (5.2 4 1.5); (NH3)2, 4.1 (4.5 4 0.4); Hm A-H Exper. * Calc. Exper. Calc. Exper. Calc. (CIH)2, 1.5 (2.1 4 0.2); and (SH2)2, 1.2 (1.7 1 0.3). The gen- F-H 1.82 2.28 1.82 2.28 1.82 2.28 HO-H 1.85 2.61 1.13 1.60 1.11 1.73 Table 2. Radial extents, I (A), and H2N-H 1.47 2.28 0.66 1.02 1 1 internuclear separation index Cl-H 1.08 1.86 1.08 1.86 0.59 1.00 0.97 1.78 0.53 I 11 12I HS-H 0.97 .'BHn H2P-H 0.58 1.05 0.18 0.33 0.18 0.33 NH3 1.77 19.9 35.2 * From ref. 4. OH2 1.58 21.5 33.9 t Sum of AA-H projections along molecular axis equated to 1.0,0.75, FH 1.35 23.1 31.1 and 0.50 times the monomer dipole moment of FH, OH2, and PH3 2.16 22.4 48.4 NH3, respectively, and of ClH, SH2, and PH3, respectively. The SHI 2.13 22.2 47.3 column designated Exper. applies this prescription to measured CIH 1.95 24.9 48.6 monomer dipole moments; that designated Calc. uses calculated monomer dipole moments. Downloaded by guest on September 30, 2021 Chemistry: Allen Proc. Nat. Acad. Sci. USA 72(1975) 4703

46.0k F-H

6-31 G* HARTREE - FOCK AO 12 0 12.0r F-H ED IOH) (kcal/mol)

8 0 _ 8.0F ED HO-H _N H2N-H)' (kcal/Inol) HS-H 4.0 4.0 _ N-H H2N-H _

H2P-H C 2 - ~~~~~~~~-- --,- o 1, 1 ,1 OH2 F NH3 OH2 FH PH3 SH2 CIH NH3 H NH3 OH2 PROTON ACCEPTORS PROTON ACCEPTORS PROTON ACCEPTORS FIG. 3. Dissociation energy versus electron donors for specified proton donor from Eq. [1] (solid lines) compared to ab initio quantum mechanical calculations (dashed lines). Left graph is 36 dimers referenced to ab initio wavefunctions using the 4-31G basis (refs. 3, 6) with K obtained by matching HO-H.-FH dimer. Middle graph is nine dimers referenced to ab initio calculations using a 6-31G* basis (J. D. Dill, L. C. Allen, W. C. Topp, and J. A. Pople, J. Am. Chem. Soc., in press) with K from (H20)2 match. Right graph is nine dimers referenced to ab initio calculations using a Hartree-Fock atomic orbital basis (ref. 5) with K from (H20)2 match.

Internuclear separation IB2IB = constant. The approximate constancy of these two internuclear separation indices is shown in Table 3. Internuclear separation is inversely proportional to ALA-H and From Fig. 4, we can see also that the difference in inter- practically constant for all electron donors in a given row. nuclear separation between electron donors of the second These relationships are displayed for the 36 systems in Fig. 4 and third rows for a specified proton donor is a constant [r(H...B) is used here instead of R(A-..B) to eliminate the de- (;z0.8 A). This follows because the separation between rows pendence on the A-H covalent bond distance; diagonal aver- for given A-H is determined by the lone-pair extents,.1. The aging is also used]. The fact that r (and R) is almost constant ratio of average I for the second to third row is 0.75 com- for all electron donors in a given row follows from an ap- ratio of second to third proximate scaling relation between atomic potential energy pared to 0.73 for the corresponding curves. Around atom B the effective potential is known to row average r(H...B). have an analytical form between Coulombic and inverse square; thus Directionality The energy involved in changing the angle, 0, between the VB - KB/r or VB - K' r2. electron donor and proton donor is an order of magnitude less than that required to change the internuclear separation and since VB = IB when r = 'B, KB = IBIB, KB' = WB2IB, and by the same percent. For this reason there is considerable' the criteria for constant r (or R) is that 'BIB = constant or uncertainty in both experimental and computed values. Directionality is determined by a competition between HCI - H3P the electron donor dipole-proton donor dipole interaction on H2S the one hand and the more favorable interaction achieved 3.2 _ along the lone-pair angular maxima on the other hand. This implies that for a specified electron donor lower angles will be realized for larger AA-H- For a specified proton donor, 2.8 lower angles are expected for larger electron donor dipole r(H *--B) moments, and since third row moments are notably smaller, (A) angles will be larger than for second row electron donors. 2.4_ Currently available computational and experimental data support these trends.

2.0 Force constants Badger's rule governs stretching force constants in ordinary covalent bonds. Thus it is noteworthy that the same relation- 1C F.- H-H 0.4 08 ( H4-H 12 1.6 (Ho-H) 2.0 F-H 2.4 ship holds for hydrogen bonds: AHSH \CI-H PA-H (Debyes) KAB(R - dAB)1 = 1.86 FIG. 4. Internuclear separation, r(H...B), versus bond dipole moment for 36 hydride dimers (ref. 3). Diagonal average values where the dAB parameter for hydrogen bonds is indepen- used. The two off-graph points arise from a slight imbalance in the dent of the row and equal to 1.00, 0.80, and 0.55 for groups PH3 basis set. V, VI, and VII, respectively. The reason for this simple re- Downloaded by guest on September 30, 2021 4704 Chemistry: Allen Proc. Nat. Acad. Sci. USA 72 (1975) sult is that KAB is governed by 1 and average l as a percent of interpeptide hydrogen bond in an aqueous environment. average R is nearly the same for the second and third row. The result, 5.8 kcal/mole, may then be compared with the The proton donor stretching force constant, KAH, is also or- denatured state represented by the average dimerization en- dered by the parameters of the model. Relative to the isolat- ergy of the two N-methylacetamide-water complexes: (7.6 ed monomer force constant, KAH is inversely proportional to + 6.0)/2 = 6.8 kcal/mole. The thermodynamic estimate for AI because large AI produces large mixing between the the hydrogen bond contribution to this interpeptide bond electron donor and proton donor potential energy surfaces, disruption process obtained by Klotz et al. (7, 8) also was 1.0 yielding low KAH. Similarly, relative KAH is inversely pro- kcal/mole. portional to AA-H because large MA-H produces a large charge Strong Hydrogen Bonds. When hydrogen bonds formed redistribution on the proton donor, resulting in a low KAH. with ions are included in the categorization, dissociation energy magnitudes vary by a factor of fifty, a range that is Dipole moment and intensity enhancement equal to or greater than that of any other type of chemical For proton donors in a given row, the dipole moments of the bond. Using the same noble gas atom ionization potential complex are proportional to AA-H because AA-H controls reference as for neutral hydrogen bonds, Eq. [1] leads to the charge redistribution. For third row electron donors the di- following ratio: pole moments of the complex are smaller than for those of the second row because Os are larger for the third than for EJ)[F HOH]/ E1)[C HOH] the second row. = (A\I[F ]//\1[C:1]) Infrared intensity enhancement is a characterizing feature of the hydrogen bond, and it arises because of the charge re- X (R[Cl- HOHI/R[F HOHI) distribution attendant to bond formation. Although there is or computational data currently Experimental electron affinities for F and Cl are 3.50 and insufficient experimental 3.62 e.v., respectively, yielding AI[F-] = 18.06 and AI[Cl-] available for test, it is to be expected that for a given row, in- = 12.13. Extended basis calculations for these two complexes tensity enhancement will be ordered according to l for a give internuclear separations and dissociation energies of specified proton donor and ordered according to MA-H for a 2.51 and 3.31 A, and 23.54 and 11.86 kcal/mole for the fluo- specified electron donor. rine and chlorine complexes, respectively (12). The left side EXTENSIONS AND APPLICATIONS of the above expression is therefore (23.54)/(11.86) = 1.98 compared to the right side (18.06)(3.31)/(12.13)(2.51) = A Biologically Important, Multiply Bonded Electron 1.96, thus demonstrating that AI successfully represents Donor. A problem of long-standing interest has been the ionic electron donors. However, when Eq. [1] is used to ob- question of whether or not interpeptide hydrogen bonds tain a direct prediction of the dissociation energy of F-.--H- contribute to the stabilization of protein configuration in OH (K is evaluated from calculations on H20-.-H-OH with aqueous solution. Klotz and coworkers (7, 8) carried out near the same basis), a value approximating 50% of that comput- infrared studies and heat of solution measurements, showed ed is obtained. Similar results occur for (FHF)- and other that the model peptide group, N-methylacetamide, did not representative examples. The origin of this discrepancy is aggregate in water, and obtained enthalpies for amide trans- the failure to take into account the Coulomb interaction be- fer from an apolar to an aqueous solution. The dissociation tween F- and the slightly positive hydrogen and the addi- energy formula, Eq. [1], can be used with existing calcula- tional charge polarization of the proton donor also produced tions to confirm their results by giving dissociation energy by F-. An appropriate Coulomb term can be added to Eq. estimates for a solvated model peptide and for the intramo- [1] and, since the added polarization will be proportional to lecular hydrogen bonding (model peptide dimer).Geometry MA-H, an extension of the dissociation energy formula to optimized quantum mechanical calculations with a good bonds can be written as: basis set are available for: N-methylacetamide and form- strong hydrogen amide as N-H proton donors with water as electron donor, E[) = KMA-H(AI/R + 10/r) N-methylacetamide and formamide as electron donors with water as proton donor, the formamide dimer, and the water Calculations using extended bases are available for four ions. dimer (9-11). Since these calculations give the same internu- Values from the above equation compared to the ab initio clear separation for complexes with N-methylacetamide and calculations (in parentheses) are: F---H-OH, 22.5 kcal/mole with formamide, an ED estimate for the N-methylacetamide (23.54); Cl----H-OH, 12.8 (11.86); (FHF)-, 39.4 (40.24); and dimer can be obtained from Eq. [1] as (HOHOH)-, 27.3 (31.8). A corresponding formula for strong positive ion hydrogen bonds can be written as ED = K(A-H E jOCHNHI,]2En[OC (CH;,)N (CHO)H OH.] + D)AI/R, where D is the dipole moment lying along the X E,[HOH OC(CH;3)N(CH3)H]/ same line as uA-H that results from the positive charge. Although not a strong hydrogen bond, we include here EJ[OCHNH,- OHJ] the hydrogen bond between the free fluorine atom and the X En[HOH OCHNH.2] hydrogen fluoride molecule because, like the strong hydrogen bonds themselves, this tests the ability of the model to The corresponding energy of solvated N-methylacetamide is encompass the full range of the phenomena. AI[F] = 21.56 - 17.42 = 4.14 e.v. Using this in Eq. [1] with K again ob- (Ej[OC(CH,)N(CH.;)H OH2] tained from the normal hydrogen bond reference calcula- + E,,[HOH-OC(CH,)N(CH,)H])/2 tions, ED(F--.H-F) is found to be 3.2 kcal/mole compared to the ab initio ED of 3.0 (13). Using the values from references 9-11 yields ED- Hydrogen Bonded PFplymers. Although derived from [OC(CH3)N(CH3)H]2 = 5.5 kcal/mole, and this number av- data on pairs of interacting monomers, the model leads to an, eraged with ED[OH2]2 = 6.1 yields an estimate for the intact understanding of the well-known nonadditivity in the heat Downloaded by guest on September 30, 2021 Chemistry: Allen Proc. Nat. Acad. Sci. USA 72 (1975) 4705 of formation bound in trimers and higher polymers. Mono- matic rings) can be obtained from ab initio calculations with mer and dimer quantities can be used to predict trimer a single proton donor (6*). A chart may be constructed for properties. Eq. [1] shows that EDIMER --A(of monomer) converting the covalent bonding pattern of the electron IATOM, -IMONOMER. If R is the same for dimer and trimer donor sequence into the corresponding hydrogen bonding (a single type of monomer is assumed), then: electron donor parameters by ordering the electron donors along the abscissa according to their electronic structure and EWIIMER - EDIMER 'zAI(of dimer) - by calculating sufficient points to define the conversion AI(of monomer) = 'MONOMER - IDIMER curve. As discussed previously, R will generally vary slowly From the first relationship we can obtain the scale factor be- for most electron donor sequences; this leads to curves that tween EDIMER and AJ(of monomer), and from the second have simple shapes. Thus it will be possible to obtain a fine- the resulting nonadditivity can be predicted for an optimally grained characterization of hydrogen bond strengths and op- positioned sequential trimer. The formation of a dimer is ac- timum internuclear separations for collections of biologically companied by charge transfer from electron donor to proton interesting electron donors. donor. This additional charge raises the potential around A, A fuller account of the present model along with other chemical slightly lowering the ionization potential and raising A! of applications is to appear in the Journal of the American Chemical the A lone-pair which will become the hydrogen bonding Society (November 12, 1975). I thank the NSF, Molecular Biology lone-pair for the new bond in the trimer. Because Al is in- Section, for financial support of this research. creased, an optimally positioned sequential trimer will be 1. Lempka, H. J., Passmore, T. R. & Price, W. C. (1968) Proc. R. more stable than two isolated dimers. Values for (HF)2 from Soc. London Ser. A 304,53-64. the parent set of 36 dimer wavefunctions are EDIMER = 7.87 2. Potts, A. W. & Price, W. C. (1972) Proc. R. Soc. London Ser. kcal/mole, AI(of monomer) = 5.92 e.v., and IMONOMER - A 326, 181-197. IDIMER = 0.975 e.v. Therefore the two above equations pre- 3. Topp, W. C. & Allen, L. C. (1974) J. Am. Chem. Soc. 96, dict a nonadditivity ETRIMER - EDIMER = 1.3 kcal/bond. 5291-5293. This result may be compared with an ab initio molecular or- 4. Nelson, R. D., Lide, D. R. & Maryott, A. A. (1967) U.S. Dept. bital calculation (14) for the hydrogen fluorine trimer (using of Commerce, Nat. Bur. Stand., NSRDS-NBS 10, 1-49. a slightly inferior basis), which yielded 1.15 kcal/bond. A 5. Kollman, P. A. & Allen, L. C. (1971), J. Am. Chem. Soc. 93, similar prediction for the water trimer is 0.95 kcal/bond 4991-5000. 6. Kollman, P. A. , McKelvery, J., Johnanson, A. & Rothenberg, nonadditivity compared to direct computation of the water S. (1975) J. Am. Chem. Soc. 97,955-965. trimer with two somewhat different orbital bases that gave 7. Klotz, I. M. & Franzen, J. S. (1962) J. Am. Chem. Soc. 84, 1.05 and 0.71 kcal/bond (refs. 15, 16, respectively). 3461-3466. The model also provides a conceptual picture for under- 8. Kresheck, G. C. & Klotz, I. M. (1969) Biochemistry 8, 8-12. standing other cases of nonadditivity. An electron donor hy- 9. Pullman, A., Alagona, G. & Tomasi, J. (1974) Theor. chim. drogen bonded to two proton donors is less stable (negative Acta 33, 87-90. cooperativity) than two isolated dimers because the in- 10. Dreyfus, M. & Pullman, A. (1970) Theor. chim. Acta 19, 20- creased charge transfer lowers the potential around the elec- 37. tron donor, lowering AI beyond that for a single proton 11. Alagona, G., Pullman, A., Scrocco, E. & Tomasi, J. (1974) Int. donor. Hydrogen bonds involving two hydrogens attached to J. Peptide Protein Res. 5, 251-259. A also result in coo- 12. Kistenmacher, H., Popkie, H. & Clementi, E. (1973) J. Chem. a common proton donor atom negative Phys. 58,5627-5638. perativity. Charge transfer to one A-H bond reduces AA-H of 13. Noble, P. N. & Kortzeborn, R. N. (1970) J. Chem. Phys. 52, the other bond. Negativity cooperativity for double proton 5375-5387. donors is greater than for double electron donors because 14. Del Bene, J. E. & Pople, J. A. (1971) J. Chem. Phys. 55, hydrogen bond energies change by a greater amount for a 2296-2299. given charge change on A-H than on B. These double elec- 15. Del Bene, J. E. & Pople, J. A. (1970) J. Chem. Phys. 52, tron donor and double proton donor trends have been found 4858-4866. in ab initio molecular orbital calculations on water trimers 16. Hankins, D., Moskowitz, J. W. & Stillinger, F. H. (1970), J. (16). Chem. Phys. 53,4544-4554. Generalization to Any Electron Donor. Because the obtained from leads to a (ED) (R) product Eq. [1] separable * These authors have predicted hydrogen bond energies by using function of the proton donor and electron donor, representa- an energy expression based on the assumption of statistical inde- tion of a given class of electron donors (e.g., a sequence of pendence, ED = f(A-H)g(B). This generally yields reasonable ED substituents at carbon for 0 = C, or a series of single-, dou- values because of the weak dependence on R for B in a given row ble-, and triple-bonded nitrogen compounds, or a set of aro- discussed in the section on internuclear separation. Downloaded by guest on September 30, 2021