J . Phys. Chem. 1989, 93, 5419-5426
5419
New Potential Energy Function for Bond Extensions Andreas A. Zavitsas*it and Athelstan L. J. Beckwith* Research School of Chemistry, The Australian National University, G.P.O. Box 4, Canberra, A C T 2601, Australia, and the Department of Chemistry, Long Island University, University Plaza, Brooklyn, New York I 1 201 (Received: December 22, 1988)
A new potential energy function, V(r) = De[u
+ (1 + u)“ exp(-@x)],where x = r - re,@ = 8.486(k,/D,)1/2,and u = exp(-2@x)
- 2 exp(-ox), predicts the energy-distance relationship for bond dissociation more accurately than all available alternatives.
The exponent n can be calculated from the normalized force constant (kN = ke/De),the normalized bond length (rN= re/De), and the electronegativity difference (Ax), by n = 1.95 + (1.5 X 104)/(kNrN- 3.05 X lo4) + (4 X 1 0 4 ) ( A ~ ) 2 / ( k N r NFor ). ground-state molecules with experimental (RKR) potentials known beyond 50% of dissociation, the overall average deviation between calculated and RKR points is f0.39 kcal/mol. The new function is both accurate and consistent: for bond extensions ( r > re),all ground-state potentials tested were reproduced with an average deviation of less than f l kcal/mol, the worst case being 02,for which the average deviation is f0.85 kcal/mol.
A potential energy curve describes quantitatively the most central phenomenon in chemistry: the making or breaking of a chemical bond. Energy is expressed as a function of distance between two atoms. Our continuing interest in enhancing understanding through modeling of reacting systems1 has led us to seek an analytical function capable of describing bond breaking and making with an accuracy sufficient for realistic descriptions. This accuracy is generally of the order of f l kcal/mol, the socalled “chemical accuracy”. What is required can be described as an a priori calculation, i.e., one that utilizes information not derived from the process it describes. The classic example for potential energy is the Morse function,2 eq 1: V ( r ) = D,u (la) 2e-8x = e-sx(e-sx + c ) u = (1b)
0 = 8.486(k,/De)1/2
(IC) where V ( r ) is the potential energy at distance r, x = r - re, C = -2, and the other symbols have their usual meaning and are described in the Appendix. The units used here are kilocalories per mole for bond energies, angstroms for distance, and millidynes per angstrom for the force constant, which is obtained from the IR stretching frequency and the atomic masses (see Appendix). The bond strengths, bond lengths, and IR stretching frequencies required for the construction of Morse curves are generally available for common species. Thus, the function correlates some of the most fundamental properties of chemical bonds. The shape of the actual potential energy curve can be deduced from spectroscopic measurements involving the various vibrational levels of the bond, from v = 0 up to the dissociation limit, through the RKR c a l ~ u l a t i o nwhich , ~ ~ ~ gives the energy of each level from the bottom of the potential curve and two “turning points” for each level, Le., the width of the potential curve in terms of two distances, rminand rmax. Thus a set of RKR points provides information on the diminution of binding energy with distance at various points on either side of re. There are few bonds for which the detailed spectroscopic information required for the RKR calculation is available to the dissociation limit, but there are several for which RKR pote4ials cover a reasonable fraction of the dissociation. For all cases, the Morse function matches the available RKR points quite accurately near re but fails to provide the “chemical accuracy” of f l kcal/mol at significant bond extensions. So defined, failure occurs at different extents of bond breaking with different bonds; e.g., with Hz failure occurs when bonding has declined to approximately 60% of its value at re;with I2 as early as 30%; and with C O as late as 70%. Consequently, many potential energy functions have been proposed as alternatives Long Island University. *The Australian National University.
0022-3654/89/2093-5419$01.50/0
or improvements to the Morse function. Among some of the better known attempts for a priori calculations of potential curves are those of R ~ d b e r g Rosen ,~ and Morse,5 Poschl and Teller,6 Linnett,’ Varshni,8 and L i p p i n ~ o t t . ~ Functions utilizing spectroscopic information from vibrational levels above v = 0 have also been proposed, the best known and most accurate being the Hulbert-Hirschfelder function.1° Such functions, in essence, require prior knowledge of at least part of the potential curve. As a result, the Hulbert-Hirschfelder and similar expressions have been considered curve-fitting functions rather than functions designed for a priori predictions.” A recent report of a new modification of the Morse function by Biirgi and Dunitz also gives several examples of the power of simple functions to correlate a large variety of experimental observations pertaining to both molecular structure and chemical reactivity.’2 The performance of the best known functions has been compared for various application^,^,'^ and a thorough review has appeared that also examines the theoretical implications of the various proposed p0tentia1s.l~ The most common criterion for the existing comparisons has been the accuracy of predicted spectroscopic potential constants. However, Steele et al.15 have pointed out that a much more stringent criterion is the ability of a function to predict the potential energy curve, as judged by agreement with known RKR points. This is also the most crucial (1) (a) Zavitsas, A. A. J . Am. Chem. SOC.1972, 94, 2779-2789. (b) Zavitsas, A. A,; Melikian, A. A. J . Am. Chem. SOC.1975, 97, 2757-2763. (c) Beckwith, A. L. J.; Schiesser, C. H. Tetrahedron 1985, 41, 3925-3941. (d) Saebo, S.; Beckwith, A. L. J.; Radom, L. J . Am. Chem. SOC.1984,106, 5119-5122.
(2) Morse, P. M. Phys. Rev. 1929, 34, 57-64. (3) Rydberg, R. 2.Phys. 1931, 73, 376; Ibid. 1933, 80, 514. (4) (a) Klein, 0. Z . Phys. 1932, 76, 221. (b) Rees, A. L. G. Proc. Phys. Soc. (London) 1947, 59, 998. ( 5 ) Rosen, N.; Morse, P. M. Phys. Rev. 1932, 42, 210. (6) Poschl, G.; Teller, E. Z . Phys. 1933, 83, 143. (7) Linnett, J. W. Trans. Faraday Soc. 1940, 36, 1123; Ibid. 1942, 38, 1. (8) Varshni, Y . P. Rev. Mod. Phys. 1957, 29, 664-682. (9) (a) Lippincott, E. R. J . Chem. Phys. 1953, 21, 2070; Ibid. 1955, 23, 603. (b) Lippincott, E. R.; Schroeder, R. J. Chem. Phys. 1955,23, 1099; Ibid. 1955,23, 1131; J . Am. Chem. SOC.1956,78,5171; J . Phys. Chem. 1957,61, 921. (c) Lippincott, E. R.; Dayhoff, M. 0. Spectrochim. Acta 1960, 16, 807. (d) Lippincott, E. R. J . Chem. Phys. 1957, 26, 1678. ( e ) Lippincott, E. R.; Steele, D.; Caldwell, P. J . Chem. Phys. 1961, 35, 123. ( f ) Steele, D.; Lippincott, E. R. J . Chem. Phys. 1961, 35, 2065. (10) Hulburt, H. M.; Hirschfelder, J. 0. J . Chem. Phys. 1941, 9, 61-69. (11) Wright, J. S. J . Chem. Soc., Faraday Trans. 2 1988,84, 219-226. (12) Biirgi, H. B.; Dunitz, J. D. J . Am. Chem. SOC.1987,109,2924-2926. (13) Jain, D. C. In?. J . Quantum Chem. 1970,6, 579-586. (14) Goodisman, J. Diatomic Interaction Potential Theory; Physical Chemistry Vol. 31-1, 31-2; Loebl, E. M., Ed.; Academic Press: New York, 1973. (15) Steele, D.; Lippincott, E. R.; Vanderslice, J. T. Rev. Mod.Phys. 1961, 34. 239-251.
0 1989 American Chemical Society
5420
The Journal of Physical Chemistry, Vol. 93, No. 14, 1989
Zavitsas and Beckwith
requirement for the description of bond breaking and bond making or for the construction of potential energy surfaces based on such functions. On this basis, the Lippincott function has been reported to be the most accurate available a priori function,15 and our experience confirms this. We repeat it here because ref 15 has a misprint: V(r) = D e [ ] - exp(-P2x2r,/r)] x { l - 0.8[(1.065@re- l ) ~ / ( r r , ) ~ exp(-1.065@x)J /~] The symbols have the same meaning as in eq 1; the above expression reproduces published value^'^ for r > re. For bond stretching (r > re), the Lippincott function has been found to be as accurate as the Hulbert-Hirschfelder function in a large variety of ground- and excited-state diatomics. The measure of performance was the average of the fraction IV(RKR) - V(calcd)l/D, expressed as a percent, where V(RKR) denotes the RKR energy value at each reported distance and V(ca1cd) is the corresponding value calculated with each function. The conclusions from these extensive comparison^'^ were that the best a priori functions requiring knowledge of three physical properties of the bond (generally De,re, and we in addition to the masses) should give an average error of 2-3% in the fraction, while the best of the Hulbert-Hirschfelder type (sometimes called five-parameter functions) should give 1-2% average error. The Lippincott function and the Hulbert-Hirschfelder function each showed such an error of 1.44% in the region of r > re; this level of error is not adequate for chemical accuracy, especially since for some bonds the resulting error is much greater. An additional shortcoming of existing a priori functions is that they do not appear to have an adequate functional form for describing potential energy curves. In general, existing functions cannot be fitted to one RKR point or accurate ab initio value about midway in the potential curve and then be capable of emulating the entire energy-distance range. For example, Brown and Truhlar16 reported recently that, when such fitting was attempted in the region of 65-70% remaining bond energy, the functions of Morse, Rydberg, Varshni, and Lippincott all deviated at other regions to various extents with the last two functions showing the best performance. As a result, even when accurate RKR or ab initio points are available and a function is sought to describe the potential curve, one must often resort to high-order polynomial fittings; e.g., an eighth-order polynomial has been used to fit the H2 curve in the limited region 0.40-1.53 A.17 Alternatively, fl of the Morse function has been treated as an adjustable variable and expressed by a polynomial in r fitted to produce a potential curve matching known points, with De and re simultaneously also being treated as adjustable curve-fitting It is unlikely that there exists a universal potential energy function capable of describing exactly the energy-distance relationship for the dissociation of a bond in terms of a few generally known physical properties of the molecule. Nevertheless the search for better analytical functions continue^'^^^^ despite the numerous efforts that have already been made.14 Recent publications of several RKR results covering almost the entire range of dissociation (see Tables I and 11) and the ever improving accuracy of ab initio calculations prompted us to search for a new potential function. The New Function The Morse function, eq 1, can be solved for C, the bonding or attractive term of the potential, when RKR values are available: C = [ V(RKR) - D e ] / [ D eexp(-@x)] - exp(-px), where V(RKR) - De = V(r). In the region r > re,we find that Cremains constant, at -2, for various extents of bond stretching but ultimately it begins to increase toward -1. Plots of C so obtained vs u of eq 1 are shown in Figure 1 for H2 and 12, as typical examples. With HZ, (16) Brown, F. B.;Truhlar, D.G. Chem. Phys. Left. 1985,113,441-446. (17) Daudel, R.; Leroy, G.; Peeters, D.; Sana, M. Quantum Chemistry; Wiley-Interscience: New York, 1983; Chapter 7. (18) Hase, W. L.; Mondro, S. L.; Duchovic, R. J.; Hirst, D. M. J . Am. Chem. SOC.1987, 109, 2916-2922. (19) JenE, F.; Brandt, B. A. Theor. Chim. Acta 1987, 72, 411-432.
-U
Figure 1. Plot of C vs u of eq 1. C and u are calculated from reported V(RKR),., (rmaJ2pairs by C = exp(-pxi) - [V(RKR),. - D,]/[De X exp(-(3xi)] and u = exp(-2pxi) - 2 exp(-Oxi), where x i = (rmaJi - re. Circles, H,;squares, I1.
C remains essentially at -2 for almost 50% of the dissociation; with 12, C is constant for less than 20%. The dependence of C on u can be approximated by C = -2 + (1 + u)". For Hz, the curve in Figure 1 is drawn with n = 5.25; for 12, with n = 2.08. The behavior of different molecules is characterized by different values of the exponent n. Hence, for the region r > re we propose the following function:
V(r) = Dee-@[e-OX- 2
+ (1 + u)"] = De[u + e-bX(l + u)"] (2)
where the symbols have the same meaning as in eq 1 and n is a constant for each bond. For the region r < re, the Morse function may be retained, as in this steeply rising portion of the curve extremely small changes in r cause large changes in V(r) and none of the existing a priori functions duplicate RKR potentials very accurately. In the following sections we will show that eq 2 is of a functional form appropriate for potential energy curves and that a value of the exponent n exists that will reproduce known RKR-derived potential points with an average accuracy of f l kcal/mol, usually much less; that n can be estimated adequately by a simple a priori calculation; that this function is more accurate than the best a priori method presently avaiable; that it emulates the results of accurate ab initio calculations; and that it appears to be appropriate for describing the breaking of one bond in polyatomic molecules. Functional Form Table I lists representative ground-state molecules for which (20) Weissman, S.; Vanderslice, J. T.; Battino, R. J. Chem. Phys. 1963, 39, 2226-2228. (21) (a) Giroud, M.; Nedelec, 0. J . Chem. Phys. 1980, 73, 4151-4155. (b) Verma, K. K.; Stwalley, W. C. J. Chem. Phys. 1982, 77, 2350-2355. (c) Hsieh, Y. K.; Yang, S.C.; Verma, K. K.; Stwalley, W. C. J. Mol.Spectrosc. 1980,83, 311-316. (d) Yang, S.C. J . Chem. Phys. 1982, 7 7 , 2884-2894. (22) Krupenie, P. H.; Mason, E. A,; Vanderslice, J. T. J. Chem. Phys. 1963, 39, 2399-2401. (23) DiLonardo, G.; Douglas, A. E. Can. J . Phys. 1973, 51, 434. (24) Kirschner, S. M.; Watson, K. G. J. Mol. Spectrosc. 1974, 51,
321-333. (25) Lofthus, A.; Krupenie, P. H. J . Phys. Chem. ReJ Data 1972, 1, 113-307. (26) Krupenie, P. H. J . Phys. Chem. Re$ Data 1972, I , 423-534. (27) Coxon, J. A. J . Mol.Spectrosc. 1986, 117, 361-387. (28) (a) Kusch, P.; Hessel, M. M. J. Chem. Phys. 1978,68, 2591-2606. (b) Ross, A. J.; Effantin, C.; D'Incan, J.; Barrow, R. F. Mol.Phys. 1985, 56, 903-912. (29) Diemer, U.;Weickenmeir, H.; Wahl, M.; Demtroder, W. Chem. Phys. Lett. 1984, 104, 489-495. (30) Field, R. W.; Capelle, G. A,; Revelli, M. A. J. Chem. Phys. 1975, 63, 3228-3237. (31) Verma, R. D. J . Chem. Phys. 1960, 32, 738-149. (32) Brand, J. C. D.; Hey, A. R. J. Mol.Spectrosc. 1985, 114, 197-209. (33) Weickenmeir, W.; Diemer, U.; Wahl, M.; Raab, M.; Demtriider, W.; Muller, W. J . Chem. Phys. 1985, 82, 5354-5363.
The Journal of Physical Chemistry, Vol. 93, No. 14, 1989 5421
Potential Energy Function for Bond Extensions TABLE I: Optimum Value of n in Eq 2 for Fitting RKR-Derived Energies for Bond Extensions (r > re)’ n, for eg 2
avb dev maxc dev RKR range R K R ref
Ground States
’
H2(X ’Zgt) H O ( X 211,) HF(X ‘Et) N a H ( X IZt) CO(X ‘Zt) N2(X ’Zgt) N O ( X ’II1p) 02(X ’Zi) HCI(X ‘Z’) Na2(X IZgt) NaCs(X ‘Zt) BaO(X IZt) 12(X ’Zgt) ICI(X ‘Et) C S ~ ( X‘Zgt)
5.22 3.35 5.94 3.55 9.61 2.72 2.55 2.45 3.15 1.89 1.79 4.19 2.08 2.17 1.78
0.36 0.20 0.76 0.10 0.46 0.21 0.44 0.43 0.63 0.19 0.19 0.51 0.14 0.51 0.09
Hz(C In,) HO(A 2Zt) CO(e ’27 CO(a’ ,Zt) N2(B 311g) N2(A ’2,’) NO(B 211) 02(B Z)’;
4.42 2.44 2.63 4.10 2.23 2.34 1.87 2.01
Excited States 0.34 -1.02 0.50 -1.26 0.76 -2.46 0.57 -2.30 0.41 -1.51 0.10 -0.27 0.67 -1.48 +0.20 0.07
-0.90 +0.46 -1.65 -0.38 +0.86 -0.45 -1.32 -1.96 -2.20 -0.54 -0.45 +1.05 -0.23 -1.46 -0.29
103.3-0.4 101.3-26.4 135.3-1.0 44.9-7.5 255.8-77.8 225.1-102.6 149.7-46.7 118.0-38.4 102.2-4.3 16.9-1.6 14.0-0.9 133.5-67.6 35.6-0.4 49.7-0.2 10.4-0.1
20 15 23 21a 24 25 15 26 27d 28 29 30 31 32 3 3e
55.1-1.5 53.8-3.1 73.9-9.0 98.0-6.2 110.5-39.0 82.8-35.9 74.4-25.5 22.2-3.0
34
15 35 35 25 25 15 26
‘The RKR range exceeds 50% of dissociation for all species. Deviation and range values in kcal/mol. bAverage of IV(calcd) - [V(RKR) - Dell. ‘V(calcd) - [V(RKR) - De] for the most poorly fitted point. d N o t RKR; least-squares fitting. < N o t RKR; IPA points (inverted perturbation approach).
RKR points are available for 50% of dissociation or more. In addition to ground states (X designation), representative excited states have been included. The performance of eq 2 was measured by the deviation of each calculated potential, V(calcd), from the RKR-derived potential, V(RKR) - De, for each RKR distance for bond breaking ( r > re). Table I also lists the value of the exponent n of eq 2 that minimizes the sum of the absolute value of the deviations in each species; also listed are the average deviation, the maximum deviation from any one RKR-derived point, and the range of energy of the RKR-derived potential. It should be noted that reactive intermediates are included and that singlets, doublets, and triplets are represented in both the ground and excited states, with the latter including species that dissociate to ground-state atoms or to atoms in excited states. Table I shows that, in every instance, the form of eq 2 is capable of reproducing RKR-derived energies with an average deviation well under f l kcal/mol, the worst case being f0.76 for an excited state of carbon monoxide, CO(e 3Z-).The overall average deviation for all the ground states is f0.37kcal/mol. For the excited states, the overall performance is somewhat poorer in terms of both average and maximum deviation, possibly reflecting greater uncertainty in the required physical constants of the species. The overall average deviation for all species in Table I is f0.40 kcal/mol. These levels of discrepancies must be evaluated in the context of the fact that RKR values measure the falloff of energy from r = re, not the actual potential from V ( m ) = 0. The RKR-derived potentials used in the comparisons, V(r) = V(RKR) - De, depend on the dissociation energy that is used. Thus, for a bond of 100 f 1 kcal/mol, the RKR-derived value at 99% dissociation will be 1 f 1 kcal/mol. Nevertheless, Table I shows that the maximum deviation at any RKR point for ground states is of the order of f l kcal/mol and never exceeds 2.2 kcal/mol, the worst case being HC1. This level of performance indicates that the form of eq 2 is generally adequate for representing potential energy curves within “chemical accuracy”. (34) Namioka, T. J . Chem. Phys. 1965, 43, 1636-1644. (35) Tilford, S. G.; Simmons, J. D. J . Phys. Chem. Ref.Data 1972, I , 147-1 87.
Potential energy curves that exhibit one or more local maxima cannot be fitted by the form of eq 2 or the Morse, the Lippincott, and similar functions. Such curves have been described36as having a “hump” and can be viewed as having a significant energy of activation for bond formation between the two atoms. Among the ground states for which RKR points are available to more than 50% of dissociation, we have encountered one such example, CuH. The reported3’ energy of its highest observed vibrational level is 4.7 kcal/mol greater than its reported% dissociation energy. Evidently such species, although known or suspected, are not common among ground states, but they appear to be somewhat more prevalent among excited states, one example being CO(A lrI).36
A Priori Estimation of Potential Energy Curves The best value of n for ground-state homonuclear diatomics from Table I was found to be a function of the normalized force constant, kN = ke/De,38and of the normalized bond length, rN = re/De;for these molecules, a plot of n vs kNrNgives a smooth curve with little scatter. The functional relationship can be expressed by a hyperbola39of the general form, y i - a = b / ( x i - c), where a is the asymptote parallel to the x axis, c is the asymptote parallel to the y axis, and b is the proportionality constant that also accommodates the units: n - 1.95 = (1.5 x 1 0 - 4 ) / ( k ~-r 3.05 ~ x (3) The heteronuclear species deviate from this pattern. The most obvious distinction between the nature of bonds in homonuclear and heteronuclear species is the electronegativity difference present in the latter, Ax. We sought a correlation between Ax and the difference between the “best” n of Table I and n as given by eq 3. The difference in n is a function of ( A x ) 2and the electronegativity effect becomes less pronounced as kNrN increases. Thus n can be estimated from eq 4:
The electronegativity values used are from Pauling’s “Complete Electronegativity Scale”.40 Equation 2 was used to calculate the potential energy curves with the values of n calculated by eq 4 for the species shown in Table 11. This table lists the n value calculated by eq 4, the average deviation between the RKR-derived potential points and the corresponding V(r)calculated by eq 2 with n from eq 4, and the maximum deviation for any one point. Also listed are the corresponding values calculated by the Morse function, as a familar reference point, and the values calculated by the full (36) Huber, K. P.; Herzberg, G. Molecular Spectra and Molecular Strucfure;Van Nostrand Reinhold: New York, 1979; Vol. 4. (37) Castaiio, F.; De Juan, J.; Martinez, E. Spectrochim. Acfa 1982, 38A, 545-548. (38) It should be noted that the “spectroscopic constant” of the Morse equation is p = 8.486(kN)1/2. (39) The relationship can also be approximated by an exponential, n = 2 + 100 exp(-kNrN x io4); then eq 4 becomes: n = 2 + 100 exp(-kNrN x 104) + (4 x l o - 4 ) ( ~ ~ ) 2 / ( k N r NThe ) . overall accuracy of the exponential relationship is approximately the same as that of the hyperbola, but we were tired of exponentials. (40) Pauling, L. The Nature of the Chemical Bond, 3rd ed.; Cornell University Press: Ithaca, NY, 1960; p 93. The only element omitted from this scale is hydrogen, for which x = 2.2 was used, the commonly accepted value (Bergmann, D.; Hinze, J. In Structure and Bonding, Clarke, J. M., et al., Eds.; Springer-Verlag: Berlin, 1987, Vol. 66, p 152). (41) Luh, W.-T.; Stwalley, W. C. J. Mol. Spectrosc. 1983, 102, 212-223. (42) Coxon, J. A. J . Mol. Spectrosc. 1975, 58, 1-28. (43) Fallon, R. J.; Vanderslice, J. T.; Cloney, R. D. J . Chem. Phys. 1962, 37, 1097-1100. (44) Ram, R. S.; Bernath, P. F. J . Mol. Spectrosc. 1987, 122, 275-281. (45) Nair, K. P. R.; Singh, R. B.; Rai, D. K. J . Chem. Phys. 1965, 43, 3570-3574. (46) Coxon, J. A. J . Mol. Spectrosc. 1980, 82, 264-282. (47) Ross, A. J.; Crozet, P.; D’Incan, J.; Effantin, C. J . Phys. B 1986, L 145-L148. (48) Barrow, R. F.; Clark, T. C.; Coxon, J. A,; Yee, K. K. J . Mol. Specrrosc. 1974, 51, 428-449. (49) Ginter, M. L.; Battino, R. J . Chem. Phys. 1965, 42, 3222-3229. (50) Gerber, G.; Broida, H. P. J . Chem. Phys. 1976, 64, 3423-3437.
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TABLE 11: A Priori Calculation of Potential Energy Curves by Eq 2 and 4 for Bond Extensions; Comparison with RKR-Derived Potentials and the Lippincott and Morse Potentialsa this work Lippincott Morse n, from eq 4 av dev* max devc av dev* max d e 9 av devb max defl RKR range RKR ref Ground States. RKR > 50% 4.94 3.97 2.13 3.38 5.95 3.34 13.14 2.79 2.38 2.17 2.94 2.05 2.04 2.02 3.43 3.77 2.04 5.10 2.00 2.07 2.01
f0.39 f0.55 10.04 f0.20 10.76 f0.14 f0.55 f0.43 f0.60 f0.85 f0.73 f0.25 f0.25 f0.20 f0.05 f0.50 f 0 . 15 f0.55 f0.16 f0.52 fO.15
-0.84 -1.62 -0.15 +0.39 -1.65 +0.26 -1.10 -0.64 +1.29 + 1.57 -2.12 -0.57 -0.49 -0.42 -0.28 -1.75 -0.52 -1.08 +0.32 -1.40 -0.35
2.58 3.38 3.37 2.78 2.20 3.04 3.62 2.04 2.01 2.25 2.34 2.03
f0.43 20.06 f0.07 10.53 f0.34 f0.27 f0.07 f0.34 f0.02 f0.34 f0.11 f0.82
+0.86 -0.08 -0.10 -0.84 +0.59 +1.01 +0.22 +0.57 -0.03 +0.65 -0.19 +1.72
2.60 2.46 2.44 2.71 2.53 2.12 2.07 2.21 1.97 1.96
f 1.09
+2.16 -1.27 -2.46 +2.94 + 1.22 -1.01 +1.13 -2.63 +0.21 -1.32 +OX6
f0.83 f0.68 f0.33 f0.88 f2.76 f0.35 f8.71 f0.78 f0.48 f0.55 f0.95 f0.5 1 f0.44 f0.39 f0.36 50.56 f0.46 f2.10 f0.32 f0.80 f0.27
-1.47 -1.63 -0.75 +1.61 +5.44 +0.66 14.66 +1.31 -1.25 -2.27 -2.10 -0.80 -0.69 -0.61 +0.60 -1.48 -0.70 +5.20 -0.58 -1.88 -0.52
+
f1.96 f1.76 10.72 f2.41 f2.38 f1.16 f0.60 f4.29 f4.84 f4.12 f3.88 f1.13 f0.97 f0.83 f0.69 f1.73 fl.O1 f0.86 f1.51 f2.68 f0.57
-4.28 -3.95 -1.65 -5.59 -5.48 -2.62 -1.45 -10.39 -10.91 -9.71 -6.85 -1.85 -1.65 -1.42 -1.99 -3.62 -1.61 -1.35 -3.06 -4.84 -1.20
103.3-0.4 56.0-0.46 24.1-11.7 101.3-26.4 135.3-1.0 44.9-7.5 255.8-77.8 225.1-102.6 149.7-46.7 118.0-38.4 102.2-4.3 16.9-1.6 14.9-0.1 12.6-2.1 40.1-13.9 41.1-0.4 14.0-0.9 133.5-67.6 35.6-0.4 49.7-0.2 10.4-0.1
20 21bd 22 15 23 21a 24 25 15 26 27c 28 28b 47 21c 21d 29 30 31 32 29'
Ground States, 20% < RKR < 50% f0.63 f0.38 f0.37 f0.38 f0.28 fl.10 f0.87 f 0.12 f 0 . 13 f0.09 fO.08 f0.39
+1.29 +0.77 +0.75 f0.69 -0.67 +2.72 +2.09 -0.30 -0.32 +0.15 -0.14 +0.97
f0.12 f0.22 f0.23 f3.34 f1.48 f0.34 f0.11 fl.01 f0.34 f0.41 f0.59 f0.61
+0.21 -0.49 -0.52 -8.1 1 -3.24 -0.74 -0.33 -2.26 -0.87 -1.01 -1.29 -1.27
78.9-60.2 102.8-81.0 102.4-80.6 185.1-9 1.6 66.4-36.4 169.6-122.0 148.1-1 12.0 57.0-35.0 45.2-36.5 45.4-30.7 80.5-61.8 47 .O-25.4
41 42 42 438 44 45 45 46 48 49 45 50
fl.39 f2.82 f3.38 f2.30 f2.81 f4.44 f2.29 f3.62 f1.18 f2.00 f0.13
-2.80 -3.92 -6.99 -5.99 -6.48 -9.42 -5.33 -7.93 -1.68 -2.84 +0.32
55.1-1.5 53.8-3.14 73.9-9.0 98.0-6.2 82.8-23.1 110.4-39.0 82.8-35.9 74.4-25.5 22.2-2.98 17.9-0.3 9.2-0.0
34 15 35 35 35 25 25 15 26 26 46
Excited States, RKR > 50% f 0 . 51 f0.90 f1.50 f0.63 f0.47 f0.60 f0.85 f0.09 f0.91 f0.38
f0.55 f0.56 f0.88 f1.75 f0.68 f0.98 f0.23 f1.48 f0.21 fI.16 f0.29
-1.21 -1.39 -2.44 +3.29 +1.21 -2.58 -0.69 -3.65 -0.38 -1.56 +0.74
"Values for deviations and dissociation range are in kcal/mol. bAverageof IV(ca1cd) - [ V(RKR) - DJl. V(calcd) - [V(RKR) - De] for the most poorly fitted point. "Not RKR; "rotationless" potential, where rminand rmr pairs for each level are reported at different energies. For CsH, the reported RKR values for J" = 15 (re = 2.516) are reproduced significantly better by our calculation: av dev = f0.27; max dev = C0.65. 'Not RKR; least-squares fitting. /Not RKR; IPA points. gFor value of Do = 185.1 see Data and Methods. Lippincott function, the best available a priori function; the range of the known RKR potential is noted for each species. Ground states for which RKR values are available for a large extent of dissociation allow the most discerning evaluation of the performance of potential energy functions; such species constitute the first group in Table 11: the physical constants are fairly accurately known and the knonw region of energy extends beyond the immediate neighborhood of re where many functions perform adequately. For this group, the overall average of the deviations between our calculated potentials and RKR-derived potentials is 10.39 kcal/mol, well within chemical accuracy. This can be compared to 11.2 kcal/mol for the Lippincott function and 1 2 . 4 kcal/mol for the Morse function. In addition to this 3-fold improvement in average performance over the best available a priori function, Table I1 also shows that eq 2 and 4 are consistently reliable: in the first group of 21 molecules, the average deviation is always less than f l kcal/mol, the worst fitting being to 02, f0.85kcal/mol. In contrast, both the Morse and the Lippincott functions show average deviations greater than f 2 kcal/mol for
several molecules, the worst being the fitting to CO by the Lippincott function, f8.7 1 kcal/mol. Even more important is the fact that our calculated potentials never deviate by more than 2.2 kcal/mol from any RKR-derived point for any ground-state species examined, the worst fit being to the point corresponding to the highest reported vibrational level of HCI, where we obtain V(2.88 A) = -6.42 kcal/mol instead of the recommended*' value of -4.30 kcal/mol.51 On the basis of the overall performance of our calculation with all the ground state species of Table 11, it may be expected that its use will lead to maximum errors of the order of fl-2 kcal/mol at any one point of a potential energy curve of a ground-state species. In contrast, both the Morse and Lippincott functions occasionally show maximum deviations greater than 5 kcal/mol, the worst case being CO with 14.7 kcal/mol with the Lippincott function. (51) The values for HCI are not RKR points, but it is not is the cause of the larger deviation.
clear that this
The Journal of Physical Chemistry, Vol. 93, No. 14, 1989 5423
Potential Energy Function for Bond Extensions
TABLE 111: Comparison of RKR-Derived Potential Points with Those Given by the Lippincott Function and Eq 2 with n from Eq 4" (in kcal/mol below V ( m ) = 0) H ~b 02' Cod Do = 135.14 Do = 117.97 Do 255.78 re = 1.207 54 A re = 1.12832 A re = 0.91681 A we = 1580.19 cm-' we = 2169.81 cm-' we = 4138.3 cm-' eq 2 RKR-De Lippincott eq 2 RKR-D, Lippincott eq 2 RKR-De Lippincott 135.3 123.6 112.5 102.0 92.1 82.6 73.7 65.3 57.3 49.8 42.7 36.2 30.1 24.6 19.5 15.0 11.0 7.5 4.6 2.1
135.3 124.0 113.2 102.6 92.9 83.5 74.5 66.0 57.5 50.2 43.0 36.2 29.8 23.9 18.6 13.7 9.5 5.9 3.0 1 .o
135.2 123.1 111.4 100.1 89.3 79.1 69.5 60.6 52.4 44.9 38.1 31.9 26.3 21.4 16.9 13.0 9.6 6.6 4.1 1.9
118.0 113.5 109.0 104.5 100.1 95.7 91.3 87.1 82.9 78.9 74.9 71.1 67.3 63.7 60.2 56.9 53.6 50.5 47.5 44.6 41.8 39.1
118.0 113.5 109.1 104.8 100.6 96.4 92.3 88.2 84.2 80.3 76.4 72.7 68.9 65.3 61.7 58.1 54.7 51.3 47.9 44.7 41.7 38.4
118.0 113.5 109.1 104.7 100.4 96.2 92.0 88.0 84.0 80.1 76.3 72.5 68.9 65.4 61.9 58.6 55.3 52.2 49.1 46.2 43.4 40.6
255.8 243.5 231.5 219.7 208.3 197.2 186.4 176.0 165.8 156.0 146.6 137.4 128.6 120.1 111.9 104.1 96.5 89.3 82.3 78.9
255.8 243.6 23 1.7 220.1 208.9 197.9 187.2 176.8 166.7 156.9 147.4 138.1 129.2 120.5 112.1 104.0 96.2 88.6 81.3 77.8
255.7 243.1 230.6 218.1 205.8 193.6 181.7 170.1 158.7 147.7 137.0 126.7 116.8 107.3 98.3 89.7 81.6 73.9 66.8 63.4
"Oxygen demonstrates the worst performance for eq 2 with n from eq 4 encountered for any ground-state species. bVibrational levels 0-19. 'Vibrational levels 0-21. dAlternate vibrational levels 0-36 and the highest available level, v = 37. The second group in Table I1 is comprised of ground-state molecules whose RKR potentials are known for more than 20% but less than 50% of dissociation. Within this group all three compared functions perform better, as expected for a smaller energy range nearer the bottom of the potential well. The Morse function shows an overall average deviation of f0.70 kcal/mol, the Lippincott function f0.38 kcal/mol, and our proposed function of f0.26 kcal/mol. The maximum deviations are 3.3, 2.7, and 1.7 kcal/mol, respectively. The greater accuracy of eq 2 and 4 is again demonstrated but with the expected smaller margin. For all ground states in Table 11, the average deviation from all RKR-derived potential points is f0.34 kcal/mol. Figure 2 shows a plot of RKR-derived points and the curves generated by eq 2 with n from eq 4; it illustrates the ability of our function to reproduce ground-state potentials adequately, despite the great variety of possible shapes. It appears that, within its limitations for accuracy, one of the most favorable aspects of our proposed function is its consistency. Other available a priori functions occasionally fail very badly in an unpredictable fashion. The third group in Table I1 is a representative sample of excited states whose RKR-derived potential is known to extents greater than 50% of dissociation. The overall average deviations for these excited states are f2.40 kcal/mol for the Morse function, f0.80 kcal/mol for the Lippincott function and f0.72 kcal/mol for eq 2 and 4. Our function is still marginally better and overall more consistent, in that no individual species is fitted with an average deviation greater than f 1.5 kcal/mol; nevertheless, this level of performance does not allow predictions to be made with confidence of being within chemical accuracy as with ground states. It is interesting that the performance of the Morse and Lippincott functions is not drastically different for ground and for excited states, whereas the accuracy of our proposed function deteriorates. The difficulty does not lie with the form of eq 2, which is adequate for fitting excited states with the proper value of the exponent n (Table I). The problem is that eq 4 fails to give a very accurate estimate of n. In some cases, one contributing reason for this failure is that we used the electronegativities of the ground-state atoms for calculating A x . However, this is not always appropriate. For instance, homonuclear diatomics dissociating to atoms in different electronic states will not have A x = 0; the ionization potentials, the major component of electronegativity, of the product
250
v
251
i
I ~ ~ ~ ~ I , , ~ ~ l , I , ~ , I , , 1.0
1.5
2.0
2.5
30
r/re
Figure 2. Plot of reported points [V(RKR),- De] vs [(r,,,&/re] for (a) 12, (b) LiH, (c) H2, (d) 02,(e) HF, and (f) CO. Inset: 1OX magnification for H2.The curves are given by eq 2, with n from eq 4. The use of r / r e is for plotting convenience only. Not all RKR points are shown. For 02,the curve drawn extends beyond the known RKR points.
atoms are different. For such cases, therefore, eq 4 would require the electronegativity of the excited-state atoms. Nevertheless, for simplicity, all calculated values in Table I1 were obtained by using the electronegativities of the ground-state atoms as given by P a ~ l i n g . ~ ~ Table 111 shows RKR-derived potential energies and values calculated with the Lippincott and our function for three examples. Hydrogen fluoride is shown as the molecule with the largest A x ; our function fits it worse than average. Oxygen serves as an example of species with unpaired electrons and was also chosen to demonstrate the poorest performance of our function encountered with a ground state. CO illustrates the poorest performance
5424
The Journal of Physical Chemistry, Vol. 93, No. 14, 1989
of the Lippincott function; our function fits it worse than average. Comparison with Other Functions The fraction lOOlV(RKR) - V(calcd)l/D, has been employed as a measure of performance of the better known potential energy functions. The conclusion was reached that the best a priori functions would be expected to produce overall errors between 2% and 3% and that “it does not seem likely that any substantial improvement (errors less than 1%) can be made by suggesting new functions ...n.15 Our function gives a value of 0.54% by this criterion when applied to bond extensions of all ground states in Table 11. In order to establish the relative performance of different potential functions vs our proposed function in a manner not biased by sample selection, we examined the sample of species used by Steele et al.I5 for their similar comparison and we retained the data they used at the time. This sample consists of 8 ground and 1 1 excited states. The compared functions gave the following reported average overall error in the fraction lOOlV(RKR) V(calcd)l/D, in the region r > re: Morse, 3.20; HulbertHirschfelder, 1.44; Rosen-Morse; 2.80; Rydberg, 2.27; PoschlTeller, 3.28; Linnett, 5.07; Frost-Musulin, 3.30; Varshni 111, 1.68; Lippincott, 1 .44.15 Equation 2 with n from eq 4 gives 0.96 with this sample which contains a majority of excited states, including CO(A In),subsequently found not to be a “well-behaved” species, as it exhibits a local m a x i ~ n u m . ~ ~ , ~ ~ Comparison of a Priori with ab Initio Potential Energy Points When accurate ab initio potential energy points are available, we find that our function reproduces them quite closely. The ab initio results of Kolos et a1.s2.53for H2, which have been called “exact”, are reproduced by eq 2 with n calculated by eq 4 with an average deviation of f0.34 kcal/mol and a maximum deviation of -0.91 kcal/mol at 2.54 A. For the extended energy region from Do = 103.3 kcal/mol down to 18 kcal/mol, the agreement is considerably better: average deviation, fO. 19 kcal/mol; maximum deviation +0.37 kcal/mol. For the hydroxyl radical, RKR points are available from Do = 101.3 to 26.4 kcal/mol. Good-quality ab initio calculations to the dissociation limit54give De = 104.3 kcal/mol, 2.34 kcal/mol lower than the experimental value. With the a b initio value of De, our calculation reproduces the ab initio values with an average deviation of f0.34 kcal/mol and a maximum deviation of -0.98 kcal/mol for the entire range of r > re. Older ab initio valuess5 for H O are also similarly reproduced with the ab initio value of De = 102.15 kcal/mol, with an average deviation of f0.34 kcal/mol and a maximum deviation of -1.2 kcal/mol. Other examples are the ab initio results for H F and HCl where the values of D,obtained are smaller than experimental by 6.5 and 4.2 kcal/mol, r e s p e c t i ~ e l y . ~Equations ~ 2 and 4 with the ab initio De reproduce the a b initio potential energies with average deviations of f 0 . 5 6 and f0.69 kcal/mol, respectively, while the maximum deviations are 1.6 and 1.2 kcal/mol. The ab initio values given for the HBr potential in the same works6 would be expected to be less accurate-too many electrons-and indeed the same procedure with our function gives an average deviation of k3.9 kcal/mol and a maximum deviation of 9.9 kcal/mol. Unlike a b initio calculations, the accuracy of the a priori calculation does not seem to be affected by the atomic number of the bonded species (Table 11). Where a b initio calculations are indispensible is in approximating potential energy curves for breaking one bond in a polyatomic molecule. No RKR potentials or experimental equivalents are available for this type of dissociation process, which is important for describing bond breaking in a chemical reaction, e.g., (52) Kolos, W.; Wolniewicz, L. J. Chem. Phys. 1965, 43, 2429-2441. (53) Kolos, W.; Szalewicz, K.;Monkhorst, H. J. J. Chem. Phys. 1986, 84, 3278-3283. (54) Langhoff, S . R.; Werner, H.-J.; Rosmus, P. J. Mol. Spectrosc. 1986, 118, 507-529. (55) Chu, S.-I.;Yoshimine, M.; Liu, B. J. Chem. Phys. 1974, 61, 5392. (56) Werner, H.-J.; Rosmus, P. J . Chem. Phys. 1980, 73, 2319-2328.
Zavitsas and Beckwith TABLE I V Comparison of V ( r ) / D , Derived from ab Initio and from a Priori Calculations H3C-H: V ( r ) / D , H2N-H: V ( r ) / D , r.
A
1.086 1.136 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 4.00
ai.“
a.o.b
r. a,,
a.i?
anb
1.000 0.992 0.929 0.702 0.475 0.296 0.173 0.096 0.051 0.026
1.000 0.993 0.934 0.705 0.463 0.279 0.162 0.094 0.055 0.032 0.004
2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.2 3.5 4.0 4.5
0.936 0.883 0.823 0.759 0.693 0.629 0.572 0.518 0.464 0.368 0.247 0.113 0.046
0.929 0.878 0.820 0.758 0.694 0.630 0.567 0.507 0.482 0.351 0.236 0.119 0.060
0.001
a Ab initio values calculated from ref 58. b A priori values calculated from eq 2, with n given by eq 4. ‘Ab initio values calculated from the 30° values of ref 62. a. = 0.529 177 A.
the abstraction of one atom from a polyatomic molecule by a radical. Polyatomic Molecules Potential energy functions derived for and tested against diatomic molecules have been used to approximate the potential energy of one bond in polyatomic molecules, for example, in calculating activation energies in atom-transfer reaction^.'^*^' Whether the approximation was valid or not could only be inferred from the accuracy of the calculated activation energy, which is a poor criterion. Recent ab initio calculations can shed some light on this point. A series of publications have appeared, focusing on the ab initio calculation of the potential energy curve for the dissociation of one bond in methane,16*’8*s8 in order to construct the potential energy surface for the simplest alkyl radical/atom association reaction. The surface has been used in classical trajectory and variational transition state theory calculations of the rate constant for CH3’ H’ and for a Monte Carlo transition-state study of the d i s s o c i a t i ~ n . ~In~ Table IV we compare the predictions of our function with those of H i r ~ twhich , ~ ~ included optimization of all angles at each calculated H3C-H distance. The table compares the ab initio values of V ( r ) / D ,and our values obtained with the ab initio value of De and the experimental values of re and u , . ~A x was obtained from the group electronegativity values of Boyd and Edgecombe.61 The agreement between the ab initio and the a priori calculation is good, with an average deviation of only f0.0064 kcal/mol in V ( r ) / D ,and a maximum deviation in energy of only 1.9 kcal/mol at 2.0 A. This agreement appears surprising since the ab initio energy values are quite dependent on proper geometry optimization at every calculated distance, while our function treats the H3C-H bond as if it were a diatomic with masses of 12 and 1 amu; the possibility of course remains that the agreement is fortuitous, but the agreement with Brown and Truhlar’s ab initio potential energy points for methaneI6 is also adequate, with an average deviation of f0.87 kcal/mol (maximum deviation +1.6 kcal/mol) for their four points at r > r,. It should be noted that, although the ab initio energies are dependent on geometry optimization at each distance, the ratio V ( r ) / D , that results is much less sensitive to it. When Hirst’s valuesS8of V(r) for constant tetrahedral geometry are expressed as V ( r ) / D eand compared to our corresponding ratios, the average deviation is changed little, to f0.0078 kcal/mol. The dissociation of one N-H bond in ammonia has also been studied recently by ab initio calculations by McCarthy et a1.62
+
(57) Johnston, H. S.; Parr, C. J . Am. Chem. SOC.1963.85, 2544. (58) Hirst, D. M. Chem. Phys. Lett. 1985, 122, 225-229. (59) References to pertinent work can be found in ref 18. (60) Gray, D. L.; Robiette, A. G. Mol. Phys. 1979, 37, 1901-1920. (61) Boyd, R. J.; Edgecombe, K. E. J . Am. Chem. SOC.1988, 110, 4182-4186. A value of x = 2.2 was used for hydrogen.
The Journal of Physical Chemistry, Vol. 93, No. 14, 1989 5425
Potential Energy Function for Bond Extensions The reported total energies for various extensions of the H2N-H bond at various angles of deviation from planarity were compared to our calculated values obtained with the a b initio values of re and De = 90.6 kcal/mol, which appears to be the a b initio value at 30° geometry. Dissociation appears to occur with an angle between 20° and 30' deviation from planarity. Table IV shows that our calculation mimics the 30' ab initio results; the average deviation in V(r)/De is f0.0075 kcal/mol. A difficulty arises in attempting to use a priori functions for a bond in a polyatomic molecule; usually the value of we is not known and therefore fl, kN, and De must be approximated from the observed stretching frequency, u, in cm-' (see Appendix): kN' N ( ~ / 1 3 0 3 ) ~ ~ p' / D=~ 8.486(kN')1/2; ; and D,' = Do 0.00143~. The results with methane and ammonia allow a test of the adequacy of these approximations. When u = 2993.8 and 3388.4 cm-I are used for H3C-H and H2N-H,63 respectively, the agreement with the ab initio values of V(r)/De does not change dramatically: it improves marginally with methane to f0.005 and deteriorates somewhat with ammonia to f0.014. An additional problem with polyatomics is that the stretching frequency may be coupled with other vibrations in the spectrum so that it is not clear what value should be used to calculate the force constant. This problem does not usually arise with bonds to hydrogen. For carbon-carbon, carbon-oxygen, and carbon-halogen bonds, the force constant may be obtained directly from the bond dissociation energy following our recently published procedure.64 The results of Table IV appear reassuring as to the adequacy of our function for realistic descriptions of single-bond formation in polyatomic molecules, even when we is not available.
+
Discussion The value of u is negative for bond extensions and changes from -1 at re to zero at r = m; therefore, (1 + u ) is~ always positive. As a result, eq 2 can cause only a narrowing of the Morse potential well, i.e., produce weaker bonding at a given distance. The smaller the value of the exponent n, the greater this narrowing. Values of n greater than about 15 essentially duplicate the Morse function for up to 80% dissociation. With C = -2 (1 + u ) and ~ n given by eq 4,attractive forces are described adequately for ground states. This signifies that the attractive term of the Morse function, -2e-BX, and the exponential dependence on distance that it implies are adequate to a certain point, but then break down. Figure 1 illustrates that this breakdown occurs early but gradually with 12,while with H2 it occurs much later and is more abrupt. With CO, the breakdown occurs very late and is very precipitous as shown by the high value of n > 9. It appears reasonable to ascribe the cause of the breakdown of the exponential energy-distance relationship to some repulsive forces, not present at re, coming into play later during bond breaking. A possible rationalization for the breakdown of the exponential dependence of bonding energy on distance may be constructed in terms of some type of bonding-antibonding mixing. Singlet-triplet mixing has been proposed on theoretical grounds for hyperfine states of H 2 at long distances (r > 4.75 A).65 Experimental evidence supporting such proposals has been suggested from the spectrum of Cs2.33 For H2, however, we find that the onset of the breakdown of the exponential energy-distance relationship becomes apparent much earlier, at r N 2re N 1.5 A. The effect is reminiscent of the (undesirable) "spin contamination" of unrestricted Hartree-Fock calculations. In . UHF, the calculated spin-orbit annular momentum.. (. S ). tends -
+
(62) McCarthy, I . M.; Rosmus, P.; Werner, H.-J.; Botschwina, P.; Vaida, V. J . Chem. Phys. 1987, 86, 6693-6700. (63) For H,C-H, u = ((1/4 [(2917)2 3(3019)2])1/2;for H2N-H, = l(1/3)[(3336.7)2 + 2(3414)211' '. The values inside the parentheses are the observed frequencies for the symmetric and degenerate stretches, appropriately weighted (seeref 64). Frequencies from: Nakamoto, K. Infrared and Raman Spectra of Inorganic and Coordination Compounds, 3rd ed.; Wiley: New York, 1977. The values of we are 3091.85 cm-I for H3C-H and 3408 for H2N-H. (64) Zavitsas, A. A. J . Phys. Chem. 1987, 91, 5573-5577. (65) Hirschfelder, J. 0.;Meath, W. J. Adu. Chem. Phys. 1967, 12, 92.
+
to go from zero to unity during bond breaking;66 ( S 2 )is 0 for a pure singlet state and +2 for a pure triplet. Finally, we note that the excited state H2(B is the only state we encountered where the value of C appears to definitely become more negative than -2 during bond breaking; the excited state C12(B311(0,+)) (Table 11) is marginally wider that the Morse function, as are weakly bound or van der Waals type species (HgH or Mg2, respectively). Such species may illustrate the reverse of the usual case: electronic repulsions present at re are diminished during bond breaking faster than the exponential energy-distance relation would dictate. In the normal ground states examined, C = -2 (1 u)", with n obtained from eq 4, describes the breakdown in the exponential energy-distance dependence within chemical accuracy; this includes the case of 02,a ground-state triplet dissociating to product atoms that are also triplets.
+ +
Conclusions The functional form of eq 2 with the proper n is adequate for fitting known potential energy points; an overall average deviation of f0.36 kcal/mol with representative ground and excited states has been obtained. The exponent n of eq 2 can be calculated by eq 4 with an accuracy sufficient to describe a large variety of ground-state potentials with an overall average deviation of f 0 . 4 0 kcal/mol, a substantial improvement over other available a priori functions. The performance of our function is remarkably consistent; known ground-state-potential points are always reproduced with an average deviation well below f l kcal/mol. The new function appears adequate for describing the breaking of one bond in polyatomic molecules, as demonstrated by agreement with recent ab initio calculations for H3C-H and H2N-H. One of the distinctive features of the new function, compared to the other better known f ~ n c t i o n s , ~is~ the ~ - ' inclusion ~ of the electronegativity factor. In retrospect it is not surprising that a successful attemp to describe bond making and breaking would have to take into consideration electronegativity differences, so central to describing the nature of the chemical bond. Earlier attempts to consider ionization potentialsgbwere a step in this direction.
Data and Methods The values used for Do, re, and we are those compiled by Huber and H e r ~ b e r or g ~those ~ given in the RKR reference, except for Do of CN. The dependence of the RKR-derived potential on the choice of bond dissociation energy is illustrated by CN, for which the RKR potential is available to about 50%of dissociation. With the value of Do = 178.94 kcal/mol of ref 36, our function fits the RKR-derived potential with an average deviation of f 0 . 7 4 kcal/mol and a maximum deviation of -1.6 kcal/mol at the highest reported vibrational level, u = 18.43 With the more recent value of Do= 185.1 kcal/m01,~' the average deviation decreases considerably to f0.45 kcal/mol and the maximum error is only -0.56 kcal/mol, at u = 8. The more recent value was used in Table 11. The effect of an incorrect value for Do becomes more pronounced as the extent of RKR range approaches complete dissociation. When the average deviation between our calculated potential energies and those derived from RKR points is of the order of f0.3 kcal/mol or less, we note that the specific details of the RKR calculation can be equally as important as the accuracy of our function. One example is the excited state 0 2 ( B 32;) for which there exist several RKR calculations. Two alternative sets of data are given in ref 49; with one, our average deviation is f0.304 kcal/mol, while the other it is f0.166 kcal/mol. Comuared to the RKR potential for the same species from ref 15, we obtain f0.107 kcal/mol; from ref 26, we obtain f 0 . 0 9 0 kcal/mol. The level of uncertainty in some of the literature RKR values is also illustrated by the ground state of co: Our function deviates from the highest RKR level ( u = 28) reported by Mantz et aL6* (66) Gordon, M. S.; Truhlar, D. G . Theor. Chim. Acta 1987, 71, 1-5. (67) Griller, D.; Kanabus-Kaminska, J. M.; Maccoll, A. J . Mol. Struct. (Theochem) 1988, 163, 125-131. (68) Mantz, A. W.; Watson, J. K. G.; Rao, K. N.; Albritton, D. L.; Schmeltekopf, A. L.; Zare, R. N. J. Mol. Spectrosc. 1971, 39, 180-184.
5426
J. Phys. Chem. 1989, 93, 5426-5431
by +0.334 kcal/mol; however, our deviation at u = 28 from ref 24 is only +0.163 kcal/mol (see Table 111). Variations in different reported RKR values of rmaxdo not appear significant but can be substantial when expressed in terms of corresponding energy. For instance, for u = 18 of HF, ref 15 reports r,,, = 2.555 8, (re = 0.917 17 A), and ref 23 reports rmax= 2.563 (re = 0.91681 A); conversion of these distances to energy values gives -4.55 and -4.07 kcal/mol, respectively. This discrepancy of 0.5 kcal/mol is greater than our average deviations in most cases. Similar extreme examples are provided by comparing the RKR data collected in ref 15 with the more recent values from the references given in Table I1 for u = 19 of N 2 (energy difference 0.6), u = 14 of N 2 (energy difference 1.6), u = 22 of O2 (energy difference 0.2), etc. While we do not doubt that the accuracy of a priori functions will be improved further, the present function appears to be approaching the limits of reliability of the RKR points in some instances. At large r, high-order polynomials in 1/ r are superior to exponential functions for curve fitting to known RKR points. We find that our function also shows increasing values of percent error beyond 95-98% of dissociation, overestimating stability in that region. Absolute errors are small in the region of the long-range potential. Computer evaluation of the energy at exactly re with fractional powers of n can lead to an invalid argument in the log routine. Negative powers of n have not been examined. Equation 2 can be made to produce potential wells wider that those of the Morse function by using C = -2 - (1 + u ) ~ .
Acknowledgment. We thank L. R. Zavitsas for her assistance in the establishment of the RKR data base and her advice on properties of various mathematical functions. Partial support of this work by the Committee on Research of the Brooklyn Campus of Long Island University and by the Research School of Chemistry of the Australian National University is gratefully acknowledged.
Appendix The zero of energy is defined as V ( a ) ,the energy of the two separated atoms. The thermodynamic bond dissociation energy, Do, is the energy difference between the lowest vibrational level and V(m), expressed as a positive number. Similarly, De is the depth of the potential well. De and Do differ by the zero-point energy; the approximation De = Do + 0 . 0 0 1 4 3 ~is~generally accurate to better than 0.07 kcal/mol. The equilibrium vibrational frequency we, is related to the observed vibrational frequency, v, by we = v 2 (w,xe) cm-’, where ( w & ~ ) is the “anharmonicity”. The force constant is given by k, = (we/ 1303)2p, where p is the reduced mass in atomic mass units. De, we, and k, are theoretical constructs. In terms of the experimentally obtainable molecular parameters Do and v, the following approximations may be used if necessary: D,’ = Do + 0.00143~,k’= ( ~ / 1 3 0 3 ) ~ p ,= 8.486(k’/00)’f2, kN = kf/Do, and rN = re/Dd. The use of k’together with Do in calculating p leads to partial cancellation of errors.
+
Infrared Matrix Isolation Study of Hydrogen Bonds Involving C-H Bonds: Substituent Effects Mei-Lee H.Jeng and Bruce S. Ault* Department of Chemistry, University of Cincinnati, Cincinnati, Ohio 45221 (Received: January 20, 1989)
The matrix isolation technique combined with infrared spectroscopy has been employed to isolate and characterize hydrogen-bonded complexes between a series of substituted alkynes and several oxygen and nitrogen bases. Distinct evidence for hydrogen bond formation was observed in each case, with a characteristic red shift of the hydrogen stretching motion us. Shifts between 100 and 300 cm-’ were observed, the largest being for the complex of CF3CCH with (CH3)3N. The perturbed carbon-carbon triple bond stretching vibration was observed for most complexes, as was the alkynic hydrogen bending motion. Attempts were made to correlate the magnitude of the red shift of us with substituent constants for the different substituted alkynes; a roughly linear correlation was found with the Hammett u parameter. Lack of correlation Aus with either u1 or C T ~alone suggests that both inductive and resonance contributions to the strength of the hydrogen-bonding interaction are important.
Introduction Numerous studies have been carried out to study hydrogen bonding as a consequence of the very significant role this interaction plays in chemical, physical, and biological processes.’-3 Infrared spectroscopy has emerged as one of the most effective experimental tools for the study of hydrogen-bonding interactions, since hydrogen bond formation gives rise to distinct, readily identifiable spectral features.’ The matrix isolation technique4“ combined with infrared spectroscopy has been applied very effectively by a number of researchers to characterize hydrogenbonded The most frequent participants in hydrogen bonding are the highly electronegative elements nitrogen, oxygen, and fluorine. Although the electronegativities of carbon and hydrogen are similar, the possibility that a C-H group may serve as a proton donor has generated substantial experimental and theoretical It is generally accepted that the proton-donating ability, the acidity of a C-H group, is dependent on the hybrid-
* Author to whom correspondence should be addressed. 0022-3654/89/2093-5426$01.50/0
ization of the carbon as well as the substituent groups in the molecule. In previous reports from this l a b o r a t ~ r y , hydro~~.~~ ( I ) , Pimentel, G. C.; McClellan, A. L. The Hydrogen Bond; Freeman: San Francisco, 1960. (2) Vinogradov, S. N.; Linnell, R. H. The Hydrogen Bonding; Van Nostrand-Reinhold: New York, 1971. (3) Kollman, P. J . Am. Chem. SOC.1977, 99, 4875. (4) Craddcck, S.; Hinchliffe, A. Matrix Isolation; Cambridge University Press: New York, 1975. (5) Andrews, L. Annu. Rev. Phys. Chem. 1971, 22, 109. (6) Hallam, H., Ed. Vibrational Spectroscopy of Trapped Species; Wiley: New York, 1973. (7) Ault, B. S. Acc. Chem. Res. 1982, 15, 103. (8) Truscott, C. E.; Ault, B. S. J . Phys. Chem. 1985, 89, 1741. (9) Ault, B. S.; Pimentel, G . C. J . Phys. Chem. 1973, 77, 57, 1649. (10) Andrews, L. J . Mol. Struct. 1983, 100, 281. (11) Johnson, G. L.; Andrews, L. J . Am. Chem. Soc. 1982, 104, 3043. (12) Barnes, A. J . J . Mol. Strucf. 1983, 100, 259. (13) Sapse, A. M.; Jain, D. C. Chem. Phys. Lett. 1966, 124. 517. (14) Frisch, M. J.; Pople, J. A.; Del Bene, J. E. J . Chem. Phys. 1983, 78, 4063. ( 15) Truscott, C. E.; Ault, G. S. J . Phys. Chem. 1984, 88, 2323. (16) Manceron, L.; Andrews, L. J . Phys. Chem. 1985, 89, 4094.
0 1989 American Chemical Society
