J. Phys. Chem. 1995, 99, 17130-17133
17130
Relative Bond Formation Energies of Isoelectronic Reactants D. W. Davis Monitoring and Laboratoly Division, Califomia Air Resources Board, Sacramento, Califomia 95814 Received: April 17, 1995; In Final Form: August 15, 1995@
An electrostatic potential model for changes in energy associated with addition of localized positive charge is extended to transfer of nuclear charge among nuclei. This model is applied to relative bond formation energies (RBFEs) of isoelectronic reactants. Predicted RBFEs are approximately additive functions of productreactant or “chemical” shifts in electrostatic potentials at corresponding nuclei in isoelectronic reactants.
Properties of isoelectronic molecules often are compared in and reactants. It is shown below for the addition of a species the scientific literature. Many subgroups of isoelectronic R to N2 and CO, forming R N 2 and RCO. molecules have closely corresponding valence-electron orbitals and similar nuclear geometries,’s2 so their properties may be expected to be related. The chemical behavior of isoelectronic molecules may be similar;’ however, the associated reaction energies may be significantly different. A well-known example is the (spin-allowed) addition of atomic oxygen (ID) to Nz and CO, forming NNO and C02. The reaction energies (exothermic) From the above cycle, are 84 and 171 kcaymol, respectively, using tabulated3 enthalpies of formation of the reactants and products at 0 K and the AE2 - AE, = AE, - AE3 (1) ’D - 3P energy difference for the oxygen atom., The first model of reaction energies among isoelectronic The quantity A E 2 - AEl is the RBFE, while AE, and A E 3 are species related them to variations in repulsion energies among the respective changes in energy accompanying the transfer of atomic cores or “kemels” (nuclei plus inner-shell electrons). nuclear charge between two nuclear sites in pairs of isoelectronic This model correctly predicted the relative stabilities of isomers reactants and products. There is no restriction on the added such as NCO- and CNO-, but it often overestimated relative species R. It may be a surface as well as atomic or molecular reaction energies. Other workers have used first-order perturbaspecies, so this approach should be applicable to relative tion theory to estimate energies of molecules from the wave adsorption energies of isoelectronic species. Also, this approach functions of their isoelectronic counterparts. The first-order is not restricted to addition reactions. It can be applied to estimates differed by 2-7% from the directly calculated SCF reactions where bonds are broken as well as formed, such as energies. It was concluded that this approach is not sufficiently substitution reactions and hydrogenation, as long as there are accurate for the prediction of bond energie~.’.~ pairs of corresponding isoelectronic reactants and products. Relative bond formation energies (RBFEs) of isoelectronic The molecular energy will be treated as usual according to reactants will be discussed here using an extension of an the Bom-Oppenheimer approximation as a sum of electronic, electrostatic potential energy model9-I5 for addition (or subtracvibrational, rotational, and translational components. Only the tion) of highly localized charge to a system. This model electronic component will be analyzed here. It is known accounts for the effects of changes in both the nuclear and generally to make the main contribution to reaction energies. electronic charge distributions on the total energy by use of Further, the other components of the total energy should tend electrostaticpotentials at nuclei, whereas previous to make smaller contributions to RBFEs than to absolute bond to reaction energies of isoelectronic molecules neglected changes formation energies, especially if the corresponding isoelectronic in the distribution of valence electrons. The electrostatic reactants and products have similar structures and chemical potential model to be extended and used here was first applied bonds. Therefore in what follows, the symbol E will refer to to inner-shell electron-binding energies9*I0and to Auger enerE is assumed to the electronic component of the total energy. gies.” Later, it was used to interpret structural effects on gasbe a function of the nuclear charges, which are allowed to be phase acidities and basicities. 12.16-19 Also, applications of this continuous variables. model to solvation energies of ions have been The Changes in E upon transfer of nuclear charge as in steps 3 core of the model is integration of the generalized Hellmannand 4 above will be related to electrostatic potentials at sites of Feynman theorem**with respect to nuclear charge, which makes nuclei (VJ)using eq 2a, which may be obtained directly from it closely related to the Wilson charging process23 and its the generalized Hellmann-Feynman theorem:22 developments for molecular energies and bond energies, especially in the work of Parr, Politzer, Fliszar, and c o - ~ o r k e r s . * ~ - ~ ~ Method
The starting point for the electrostatic potential model of RBFEs is a thermochemical cycle among isoelectronic products @
Abstract published in Advance ACS Abstracts, November 1, 1995.
0022-365419512099-17130$09.0010
The index i in eq 2b refers to electrons, and the index K refers to nuclei. The terms (llrJ,) on the right-hand side of eq 2b
0 1995 American Chemical Society
Bond Formation Energies of Isoelectronic Reactants
J. Phys. Chem., Vol. 99, No. 47, 1995 17131
depend on the nuclear charges ZK through the dependence of the wave function on the ZK. The number of electrons is constant during steps 3 and 4, and within the Born-Oppenheimer approximation the internuclear distances are fixed. Therefore, using eq 2a, the total differential of E for steps 3 and 4 is
J
J
Equation 3 will be integrated by the introduction of a “charging” parameter t, as done in the Wilson charging processz3 and in classical electrostatics to obtain the energy of a charge distribution: a j
=h
z j
(4)
dt
The factor AZJ in eq 4 is the total change in nuclear charge at the Jth nuclear site; the parameter t varies from 0 to 1 over the integration. Equation 4 therefore defines a path of integration along which at any point each nuclear charge ZJ has changed by the same fraction of AZJ. From elementary calculus, the integral is independent of the path of integration if E is a function of the zj’s. Integration of eq 3 while using eq 4 gives the change in energy associated with transfer of nuclear charge as
Equation 5 is exact if the wave functions employed to calculate E and the VJ satisfy the generalized HellmannFeynman theorem. This category includes Hartree-Fock, certain multiconfigurational, and exact wave functions.22 the integral for each index J Following previous in eq 5 will be approximated here as OS(C Vf), where the superscripts o and f refer to the initial and final states of the integration. This approximation gives what has been called the linear potential model (LPM), because it is equivalent to linearization of the VJ’S with respect to t (or the ZJ’S: if VJ is a linear function of t, then it is easy to show, using eq 4, that it also varies linearly with the ZJ’S).An alternative to the LPM is to approximate each of the integrals in eq 5 by the value of VJ at t equal to the intermediate value of 0.5. This approximation was found to be slightly more accurate than linearization of potentials for addition of a unit of positive charge to nuclei of diatomic molecule^.'^ The result of the multicenter LPM for the transition N2 CO is given below as an example. In this transition, one unit of nuclear charge is transferred between two nuclei:
+
-
AE = ~ / ~ ( V N ( N-I-J Vo(CO) - VN(N2) - Vc(CO)) = e/2(Vo(CO> - VC(C0)) ( 6 ) Finally, RFBEs are predicted from the multicenter LPM by using eq 5 in eq 1 and linearizing the electrostatic potentials:
+
AEz - AEl = 0.5cA,ZJ(6V,0 SV;)
(74
J
dVJ = VJ(products) - VXreactants)
(7b)
In eq 7a the summation is over only those sites of nuclei which gain or lose nuclear charge. The “chemical shifts” SVJ” and SV? contribute additively to the relative bond formation energy (more exactly, from eq 5, the SVJ at each point in the integration are cumulative), and they were weighted by AZJ. These are
consequences of the VJ being derivatives of the total energy instead of direct components of it. Also, the contributions of the SVJ to the RBFE depend on the signs of their corresponding
MJ.
Results and Discussion Equation 5 and the approximation of linear potentials were used to estimate AE for transfer of nuclear charge within a molecule. The predictions are given in Table 1 along with the directly calculated SCF values of AE and the percentage difference between them. Most of the values of E and electrostatic potentials used for Table 1 are taken from the work of Snyder and Baschz (electrostatic potentials are labeled in ref 2). They used double-5 basis sets of Gaussian orbitals in their calculations. Additionally, the near Hartree-Fock calculations of Dingle, Huzinaga, and Klobukowski28were used for six transitions involving diatomic molecules; in five transitions in Table 1, a proton is lost. For these transitions, the electrostatic potential energy of a unit positive charge at the vacant site of the proton is required in the LPM and was obtained from the calculations of Kollman and R ~ t h e n b e r g , ~ ~ who used a double-5 Gaussian basis set slightly smaller than that of ref 2. The multicenter LPM predicts AE to within 0.5% for 21 out of the 25 entries in Table 1. The LPM predictions using the near Hartree-Fock calculations of ref 28 for transitions involving transfer of one unit of charge in diatomics are more accurate than the predictions for these transitions using the double-5 calculations of ref 2. It seems likely that this result is due to the near Hartree-Fock wave functions more closely satisfying the generalized Hellmann-Feynman theorem. The multicenter LPM gives only qualitative agreement for two of the transitions in Table 1, which are both isomerizations: CH3NC CH3CN and CHZNZ NH2CN. Part of the high-percentage errors for these transitions may be ascribed to their comparatively very low energies; however, even the absolute errors are larger than for many of the other transitions in Table 1. The relatively poor results for these two reactions suggest that the LPM is not adequate for isomerizations. Unusually large errors in the LPM prediction were found also for the transitions O3 CF:! and CH3NC BH3CO. In both these transitions, one nuclear site loses or gains two units of nuclear charge, while two other nuclear sites gain or lose one unit of charge each. These are the only transitions in Table 1 where unequal amounts of charge are gained or lost among nuclear sites. Because of this, the deviation of the electrostatic potentials from linearity probably affects the LPM prediction more than for the other transitions in Table 1. This was suggested by examination of the dependence of the SCFz*28 electrostatic potentials at the nuclei of BF, CO, and NZon their nuclear charges. Their electrostatic potentials were least-squares fitted using both linear and quadratic functions of the nuclear charges. The electrostatic potentials were fit significantly better by the quadratic function; the difference between integrating the quadratic function versus integrating the linear function increased with the range of integration. However, this difference had opposite signs for nuclear sites gaining and losing nuclear charge, as linearization of the potentials overestimated the magnitudes of the contributions from both these nuclear sites to the change in energy. These results indicated that the linear errors for the nuclear sites participating in transfer of nuclear charge would cancel to some extent in the LPM prediction, especially when there are equal numbers of nuclear sites, each gaining and losing the same amount of nuclear charge. When unequal amounts of nuclear charge are transferred, as in the
-
-
-
-
Davis
17132 J. Phys. Chem., Vol. 99, No. 47, 1995 above two transitions, it is likely that there will be less cancellation of the linear errors in the LPM prediction. The internuclear distances should be held constant for the transitions in Table 1 because the method is carried out within the Born-Oppenheimer approximation. They actually change somewhat, usually by about 5%. However, for a few transitions in Table 1, including LiF B e 0 and FNO 0 3 , the internuclear distances change considerably more, by as much as 20%. Nevertheless, the LPM predictions for these transitions are about as accurate as for the others, which suggests a cancellation of errors due to using the fixed-nuclei approximation for these transitions. If this approximation were dropped, the ~ / R J K terms in eq 2b for the electrostatic potential (and in the total energy) would be replaced by the expectation values (~/RJK).With this modification, the above method could in principle handle transitions which involve changes in both nuclear charges and internuclear distances. The electrostatic potentials and energies calculated using this modification should be different (and more accurate than those calculated using fixed nuclei. The results in Table 1 suggest that these differences tend to cancel each other in the LPM.
-
+
Applications and Conclusions The multicenter LPM will next be applied to several addition reactions of isoelectronic molecules, including the addition of atomic oxygen and CH2 to N2 and CO. These two pairs of reactions are represented by the above thermochemical cycle. The RBFE from eq 1 and eqs 7a-7b is
+
AE2 - AE, = e/2(SVN,j SV, - SV,
- SV,)
(8a)
TABLE 1: SCF and LPM Energies for Transfer of Nuclear Charge -A€"
transition
SCF
LPM
(A%)b
ref
CO CO BF BF B e 0 LiF NNO CO2 FNO CF2 0 3 FNO COF2 BF3 CHzNi CHrCO NH2CN CHiCO CH3CN BH3CO H20 HF CH3OH CH3F NH3 H20 CHI NH3 CzH2 HCN CH3NC CH3CN CH2N2 NHzCN Nz-BF N? BF CZ B e 0 0 3 CFz CHiNC BH3CO CZ LiF
3.796 3.807 1 1.377 11.396 17.540 3.962 8.059 4.362 1 1.604 3.889 3.816 7,200 24.012 24.014 19.832 15.989 16.037 0.027 0.074 15.174 15.202 14.050 12.421 7.227 31.589
3.796 3.816 1 1.372 1 1.426 17.532 3.947 8.057 4.347 11.631 3.886 3.800 7.195 23.943 23.952 19.765 15.918 15.960 0.007 0.043 15.152 15.220 14.050 12.200 7.389 3 1.523
0.01 -0.24 0.04 -0.27 0.04 0.37 0.02 0.33 -0.23 0.08 0.41 0.06 0.28 0.26 0.34 0.45 0.48 72.8 41.4 0.14 0.12 0 1.78 -2.24 0.21
28 2 28 2 28 2 2 2 2 2 2 2 2, 29 2, 29 2, 29 2, 29 2, 29 2 2 28 2 28 2 2 28
-
N2 N2 CO CO
----.-. +
-
+
+
-.
-+
--
--
--- -
' Atomic units. ' 100(A€(LPM) - A€(SCF)/IAE(SCF)( TABLE 2: Predicted Relative Bond Formation Energies and Chemical Shifts in Electrostatic Potentials at Nuclei and in Atomic Charges RBFE" reactants products expth,d SCFh' LPM" 6V,'f
SV, = VJRCO) - V,(CO)
Nz.0
c0,o Nz, CH2
NNO
co1
CHiNz C0,CHz CHzCO CO,Fz CFzO BF. Fz BF?
SV, = V,(RCO) - V,(CO)
87
97
82
39
52
44
118
131
129
101 80 32 17 137 95
d V p f dqll(.Y dq,/
20 -5 -15 -24 -16
-9
0.26 0.39 0.03 0.21 0.55 0.54
l'
0.03 0.09 -0.03 -0.05 -0.05 0.01
p; the
Lower entry in each reaction pair has the more exothermic bond formation energy. kcal/mol. Atomic units. Experimental RBFEs obtained from AH:, in refs 3 and 30 (AH:, (CH2N2) estimated from AH;,,, (CH2N2)). e From ref 2. f kcal/(mol unit charge).
The SCF RBFEs for the three pairs of reactions in Table 2 are predicted semiquantitatively by the multicenter LPM. Both the LPM and the SCF predictions reproduce the increase in the experimental RBFE with the electronegativity of the added substituent. The LPM predictions could perhaps be improved by use of energies and electrostatic potentials calculated with
basis sets larger than double-5. This is suggested by the more accurate results, shown in Table 1, obtained when near HartreeFock calculations were used to estimate the energy for transfer of a unit of nuclear charge in diatomics. The LPM predictions could probably be improved further by using correlated wave functions (which satisfy the generalized Hellmann-Feynman theorem). For addition of atomic oxygen to N2 and CO and also for addition of F2 to CO and BF, the 6Vu's shown in Table 2 are all positive and they are much larger in magnitude than the corresponding dVp's. Therefore, from eq 8a (or its equivalent for addition to CO and BF), the RBFEs for these two reactions are due primarily to the shifts in the electrostatic potentials at the a nuclear sites. The positive 6V,'s can be attributed mainly to the large and positive shifts in atomic charges of the a atoms, 6q,'s, because the dqp's are much smaller and the atomic charges calculated in ref 2 for the 0 and F atoms in the reaction products are negative. It should be mentioned that the contribution of dq, to 6Vu depends inversely on the sizes of the valence orbitals of the a atom. This is a major reason that for each pair of reactions in
In eqs 8a-8e the nitrogen atoms in R" are labeled a and nitrogen atom labeled a is bonded directly (or most closely) to the added species R. This notation will be used to distinguish the atoms or nuclear sites which are involved in transfer of nuclear charge. The carbonyl carbon in CO and RCO is an a atom, and the carbonyl oxygen is a p atom. Table 2 gives experimental and predicted values of RBFEs and "chemical shifts" in V and in Mulliken net charges q of the a and p atoms for addition of atomic oxygen and CH2 to N2 and CO. The dqa's and dqp's are included because they are measures of changes in electronic density in the atomic neighborhoods of the a and p nuclei, which may be expected to have a significant effect on the 6 V s . Results for the addition of F2 to CO and BF, forming carbonyl fluoride and boron trifluoride, are also included in Table 2. Equations of the same form as eqs 8a-8e apply to the latter reactions, except that the a atoms for these reactions are C and B, while the p atoms are 0 and F. The SCF energies, electrostatic potentials, and Mulliken populations used in Table 2 were obtained from ref
Bond Formation Energies of Isoelectronic Reactants Table 2 the 6V of the a atom with the higher nuclear charge (and smaller valence orbitals) is larger, even when its 6qa is smaller. The LPM indicates that the 64’s at the two a (or /3) atoms have cumulative effects on the RBFE of isoelectronicmolecules. This is apparent from eq 8a and the approximately linear dependence of 6Va on 6qa. It implies that as more charge is withdrawn from (or donated to) the a atoms, the magnitude of the RBFE will tend to increase. It also implies that sizable RBFEs may occur even if the bonds created are similar in terms of changes in atomic charges (or more precisely, changes in the spatial distributions of the valence electrons). The addition of FZto CO and BF is an example of this, as the dqa’s, which are dominant for this pair of reactions, are nearly the same. Likewise, the LPM indicates that large RBFEs for isoelectronic molecules don’t necessarily imply large differences in the chemical bonds created. Finally, the LPM prediction of the relative reaction energy for the addition of CH2 to N2 and CO will be discussed briefly. In this case, all the 6Va’s and 6Vg’s contribute significantly to the predicted RBFE for addition of CHz. The GVg’s have opposite signs to the 6Va’s, so they reinforce each other in eq 8a (this holds for five of the six reactions in Table 2 ) . Also, it may be noted that 6 q N a and ~ V Nfor , addition of CHZto NZare positive, despite the lower electronegativity of C relative to N.
References and Notes (1) Bent, H. A. J . Chem. Ed. 1966, 43, 170. (2) Snyder, L. C.; Basch, H. Molecular Wave Functions and Properties; John Wiley & Sons: New York, 1972.
J. Phys. Chem., Vol. 99,No. 47, 1995 17133 (3) Chase, M. W., Jr.; Davies, C. A.; Downey, J. R., Jr.; Frurip, D. J.; McDonald, R. A.; Syverud, A. N. JANAF Thermochemical Tables, 3rd ed.; J . Phys. Chem. Ret Data 1985, 14, Suppl. 1. (4) Moore, C. E. Tables of Spectra of Hydrogen, Carbon, Nitrogen and Oxygen Atoms and Ions; CRC: Boca Raton, 1993. ( 5 ) Pauling, L.; Hendricks, S. B. J. Am. Chem. SOC.1926, 48, 641. (6) Latimer, W. M.J. Am. Chem. SOC.1929, 51, 3185. (7) Gilson, B. R.; Arents, J. J . Chem. Phys. 1963, 38, 1572. (8) Chang, T. Y.; Brown, W. B. Theor. Chim. Acta 1966, 4, 393. (9) Hedin, L.; Johansson, A. J . Phys. B 1969, 2, 1336. (10) Davis, D. W.; Shirley, D. A. Chem. Phys. Lett. 1972, 15, 185. (11) Shirley, D. A. Chem. Phys. Lett. 1972, 16, 220. (12) Davis, D. W.; Rabalais, J. W. J. Am. Chem. SOC.1974, 96, 5305. (13) Levy, M. J . Chem.Phys. 1978, 68, 5298. (14) Politzer, P.; Sjoberg, J. J . Chem. Phys. 1983, 78, 7008. (15) Sen, K. D.; Seminario, J. M.; Politzer, P. J . Chem. Phys. 1989, 90, 4373. (16) Davis, D. W.; Shirley, D. A. J . Am. Chem. SOC.1976, 98, 7898. (17) Smith, S. R.; Thomas, T. D. J. Am. Chem. SOC.1978. 100, 5459. (18) Davis, D. W. J . Mol. Struct. 1985, 127, 337. (19) Siggel, M. R.; Thomas, T. D. J . Am. Chem. SOC.1986, 108,4360. (20) Davis, D. W. Chem. Phys. Lett. 1982, 91, 459. (21) Contreras, R. R.; Mendizabal, F.; Aizman, A. J. Phys. Rev. A 1994 49, 3439. (22) Yurtsever, E.; Hinze, J. J . Chem. Phys. 1979, 71, 1511. (23) Wilson, E. B. J . Chem. Phys. 1962, 36, 2232. (24) Politzer, P.; Parr, R. G. J. Chem. Phys. 1974, 61, 4258. (25) Politzer, P. J . Chem. Phys. 1976, 64, 4239. (26) Fliszar, S. J . Am. Chem. SOC.1980, 102, 6946. (27) March, N. H. Electron Density Theory of Atoms and Molecules; Academic Press: London, 1992; Appendix 6.1. (28) Dingle, T. W.; Huzinaga, S.; Klobukowski, M. J . Comput. Chem. 1989, 10, 753. (29) Kollman, P.; Rothenberg, S. J. Am. Chem. SOC.1977, 99, 1333. (30) Lias, S. G.; Bartmess, J. E.; Liebman, J. F.; Holmes, J. L.; Levin, R. D.; Mallard, W. G. J . Phys. Chem. Ref. Data Suppl. 1988, 17.
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