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HEAT: High accuracy extrapolated ab initio thermochemistry
1.See, for example, the special issues of Spectrochim. Acta 58, 599–898 (2002).
2.For a representative and chemically relevant example, see E. R. Jochnowitz, M. R. Nimlos, M. E. Varner, G. B. Ellison, and J. F. Stanton, J. Am. Chem. Soc. (to be published).
3.K. L. Bak, J. Gauss, P. Jørgensen, J. Olsen, T. Helgaker, and J. F. Stanton, J. Chem. Phys. 114, 6548 (2001).
4.For a review, see S. J. Blanksby and G. B. Ellison, Acc. Chem. Res. 36, 255 (2003).
5.B. Ruscic, J. E. Boggs, A. Burcat et al., J. Phys. Chem. Ref. Data (in press).
6.Well-known tabulations of molecular enthalpies of formation include: M. W. Chase, Jr., in NIST-JANAF Thermochemical Tables, 4th ed. Journal of Physical Chemical Reference Data, Monograph Vol. 9 (1998), p. 1;
6.J. D. Cox, D. D. Wagman, and V. A. Medvedev, CODATA Key Values for Thermodynamics (Hemisphere Publishing Corp., New York, 1989);
6.L. V. Gurvich, I. V. Veyts, and C. B. Alcock, Thermodynamic Properties of Individual Substances, 4th ed. (Hemisphere Publishing Company, New York, 1989).
7.B. A. Flowers, P. G. Szalay, J. F. Stanton, M. Kállay, J. Gauss, and A. G. Császár, J. Phys. Chem. A 108, 3195 (2004), and references therein.
8.A. G. Császár, P. G. Szalay, and M. L. Leininger, Mol. Phys. 100, 3879 (2002).
9.A. G. Császár, M. L. Leininger, and V. Szalay, J. Chem. Phys. 118, 10631 (2003).
10.M. S. Schuurman, S. R. Muir, W. D. Allen, and H. F. Schaefer, J. Chem. Phys. 120, 11586 (2004).
11.S. Parthiban and J. M. L. Martin, J. Chem. Phys. 114, 6014 (2001).
12.Before the development of the Active Thermochemical Tables, atomic enthalpies of formation for O and N were also not known precisely. However, uncertainties in these values have been dramatically reduced in the ATcT approach, B. Ruscic (private communication).
13.B. Ruscic, A. F. Wagner, L. B. Harding et al., J. Phys. Chem. A 106, 2727 (2002).
14.A. L. L. East and W. D. Allen, J. Chem. Phys. 99, 4638 (1993).
15.D. A. Dixon, D. Feller, and G. Sandrone, J. Phys. Chem. A 103, 4744 (1999).
16.C. W. Bauschlicher, J. M. L. Martin, and P. R. Taylor, J. Phys. Chem. A 103, 7715 (1999).
17.J. M. L. Martin and G. de Oliveira, J. Chem. Phys. 111, 1843 (1999).
18.T. Helgaker, W. Klopper, A. Halkier, K. L. Bak, P. Jørgensen, and J. Olsen, Quantum Mechanical Prediction of Thermochemical Data (American Chemical Society, Washington DC, 1998).
19.K. L. Bak, P. Jørgensen, J. Olsen, T. Helgaker, and W. Klopper, J. Chem. Phys. 112, 9229 (2000).
20.Quantum Mechanical Prediction of Thermochemical Data, edited J. Cioslowski (Kluwer, Dordrecht, 2001).
21.W. J. Hehre, L. Radom, P. V. R. Schleyer, and J. A. Pople, Molecular Orbital Theory (Wiley, New York, 1986), p. 298.
22.The reader is reminded that isodesmic reactions are those which would be thermoneutral in the limit that additivity of “bond energies” gave a precise picture of chemical thermodynamics.
23.Notable in this regard are the G(aussian) n 2, and 3) family of methods developed by Pople and co-workers [K. Raghavachari and L. A. Curtiss in Ref. 20]; the W(eizmann) n 2, and 3) methods of Martin and co-workers [Ref. 17 (W1); Ref. 11 (W2); Ref. 25 (W3)]; and the CBS methods of Petersson [G. A. Petersson, in Ref. 20]. A related approach is the focal-point strategy advocated by Allen and collaborators (Ref. 24); see Ref. 10 for a recent example.
24.A. G. Császár, W. D. Allen, and H. F. Schaefer III, J. Chem. Phys. 1998, 108 (9751).
25.A. D. Boese, M. Oren, O. Atasolyu, J. M. L. Martin, M. Kállay, and J. Gauss, J. Chem. Phys. 120, 4129 (2004).
26.R. J. Bartlett, Annu. Rev. Phys. Chem. 32, 359 (1981).
27.An exception is our calculation of spin–orbit coupling, which uses a non-size-extensive approach.
28.Throughout the text, the term “enthalpy of formation” refers to the difference in ground-state energies, i.e., the enthalpy of formation at 0 K.
29.R. D. Cowan and M. Griffin, J. Opt. Soc. Am. 66, 1010 (1976);
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30.As is customary in the quantum chemistry world, the second row of the periodic table is taken to be that containing Li, Be, B, C, N, O, F, and Ne.
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35.It should be noted that inclusion of these effects shifts the position of the equilibrium internuclear distance. Nonetheless, we ignore this complication, and base all of our calculations on equilibrium geometries determined with CCSD(T)/cc-pVQZ. The issue of equilibrium geometry is not of numerical significance. In the vicinity of the true minimum, the energy changes only in second order when the nuclei are displaced. Taking a typical (and large) value for a bond stretching force constant (10 aJ Å−2), an error of 0.005 Å in the equilibrium bond length translates to a calculated energy error of 0.08 kJ mol−1.
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38.J. D. Watts and R. J. Bartlett, J. Chem. Phys. 93, 6104 (1993).
39.S. A. Kucharski and R. J. Bartlett, Theor. Chim. Acta 80, 387 (1991);
39.S. A. Kucharski and R. J. Bartlett, J. Chem. Phys. 97, 4282 (1992);
39.N. Oliphant and L. Adamowicz, J. Chem. Phys. 94, 1229 (1991).
40.Optimized all-electron CCSD(T)/cc-pVQZ geometries used in all calculations:
41.T. H. Dunning, J. Chem. Phys. 90, 1007 (1989).
42.K. Kuchitsu, in Accurate Molecular Structures, edited by A. Domenicano and I. Hargittai (Oxford University Press, Oxford, 1992), pp. 14–46.
43.R. A. Kendall and T. H. Dunning, J. Chem. Phys. 96, 6796 (1992).
44.See, for example, J. F. Stanton and J. Gauss, Adv. Chem. Phys. 125, 101 (2003).
45.P. G. Szalay, C. S. Simmons, J. Vázquez, and J. F. Stanton, J. Chem. Phys. 121, 7624 (2004).
46.D. E. Woon and T. H. Dunning, J. Chem. Phys. 103, 4572 (1995).
47.D. Feller, J. Chem. Phys. 96, 6104 (1992).
48.It can be shown that the atomic correlation energy converges to the exact result according to Eq. (6), if X is defined such that it corresponds to a basis set in which sets of orbitals associated with all angular momenta up to and including l (=0 for s, 1, for p, etc.) are saturated. Use of this formula for molecules stands upon a rather flimsy theoretical foundation in the first place, due to the lack of spherical symmetry. It is however an even more extreme (albeit widely used) approximation to treat a basis set such as aug-cc-pCVTZ as though it is saturated with f functions for first-row atoms, since it has but two f functions. Nevertheless, extrapolation schemes based on the formula work well in practice and seem to be as good as any others that have been advocated for this purpose.
49.T. Helgaker, W. Klopper, H. Koch, and J. Noga, J. Chem. Phys. 106, 9639 (1997).
50.To test the significance of the combined treatment of core and valence correlation, the total energy of C, O, CO, and has been calculated at the frozen core CCSD(T) level extrapolated from aug-cc-pVQZ and aug-cc-pV5Z results. The core correlation effects have been calculated by comparing the extrapolated (Q5) frozen-core and all electron energies at the CCSD(T) level, the latter using aug-cc-pCVXZ basis sets. In all cases, this procedure gives lower total energies compared to the direct inclusion of core correlation effect as used in the HEAT protocol (0.23, 0.52, 0.17, and 0.50 kJ mol−1 for C, O, and CO, respectively). From these data the enthalpy of formation of CO changes significantly (by 0.21 kJ mol−1); that of shifts by less than 0.1 kJ mol−1.
51.J. F. Stanton, Chem. Phys. Lett. 281, 130 (1997).
52.D. Feller, J. Chem. Phys. 111, 4373 (1999).
53.D. Feller and J. A. Sordo, J. Chem. Phys. 113, 485 (2000).
54.M. Kállay and J. Gauss, J. Chem. Phys. 120, 6841 (2004).
55.S. Hirata and R. J. Bartlett, Chem. Phys. Lett. 321, 216 (2000).
56.J. Olsen, J. Chem. Phys. 113, 7140 (2000).
57.M. Kállay and P. R. Surján, J. Chem. Phys. 115, 2945 (2001).
58.M. Kállay and P. R. Surján, J. Chem. Phys. 113, 1359 (2000).
59.S. Hirata, J. Phys. Chem. A 107, 9887 (2003).
60.T. A. Ruden, T. Helgaker, P. Jørgensen, and J. Olsen, Chem. Phys. Lett. 371, 62 (2003).
61.Test calculations were carried out for several species, including the CN and CCH radicals that are plagued by significant spin contamination at the UHF level. In no cases did the absolute energies obtained with ROHF and UHF reference functions at the CCSDT level differ by more than 0.1 kJ mol−1.
62.For NO, CCH, CF, CN, and FO, the contribution was obtained at the ROHF/CCSD(T) level with the cc-pVQZ basis set, at the corresponding equilibrium internuclear distances of 1.150 40; 1.208 39 (CH), and 1.062 14 (CO); 1.270 90, 1.171 85, and 1.351 09 Å, respectively]. All other contributions to the HEAT energy were obtained with UHF reference functions.
63.I. M. Mills, in Modern Spectroscopy: Modern Research, edited by K. N. Rao and C. W. Matthews (Academic, New York, 1972), pp. 115–140.
64.J. Gauss and J. F. Stanton, Chem. Phys. Lett. 276, 70 (1997).
65.P. G. Szalay, J. Gauss, and J. F. Stanton, Theor. Chim. Acta 100, 5 (1998).
66.J. F. Stanton and J. Gauss, Int. Rev. Phys. Chem. 19, 61 (2000).
67.H. H. Nielsen, in Handbuch der Physik, edited by S. Flügge (Springer, Berlin, 1959), Vol. 37, Part 1, p. 171.
67.H. H. Nielsen, Rev. Mod. Phys. 23, 90 (1951).
68.D. G. Truhlar, J. Comput. Chem. 12, 266 (1991).
69.J. Vázquez and J. F. Stanton (unpublished). For asymmetric tops, is given in lowest order, by the expression where is the equilibrium rotational constant corresponding to the αth inertial axis, and are cubic and quartic force constants in the dimensionless normal coordinate representation, is the harmonic frequency of normal mode i, is the Coriolis coupling constant between modes k and l with respect to the αth inertial axis, and is defined as This equation has been checked by a direct summation procedure, in which the rovibrational Hamiltonian is constructed explicitly and and are the nth/order Hamiltonian and ground state wave function, respectively) has been evaluated numerically. Our result also agrees with the unpublished result of Allen and co-workers [W. D. Allen (private communication)]. It should be noted that this equation differs from that published in Ref. 70, although the numbers in that paper are consistent with our expression rather than that which was published. We have also derived higher-order terms contributing to but they are of no significance for the hydrogen peroxide example addressed later in this paper.
70.V. Barone, J. Chem. Phys. 120, 3059 (2004).
71.D. W. Schwenke, J. Phys. Chem. A 105, 2352 (2001).
72.E. F. Valeev and C. D. Sherrill, J. Chem. Phys. 118, 3921 (2003).
73.O. L. Polyansky, A. G. Császár, S. V. Shirin, N. F. Zobov, P. Barletta, J. Tennyson, D. W. Schwenke, and P. J. Knowles, Science 299, 539 (2003).
74.P. Pyykkö, Adv. Quantum Chem. 11, 353 (1978);
74.P. Pyykkö, Chem. Rev. (Washington, D.C.) 88, 563 (1988);
74.K. Balasubramanian, Relativistic Effects in Chemistry, Part A: Theory and Techniques and Part B: Applications (Wiley, New York, 1997).
75.J. L. Tilson, W. C. Ermler, and R. M. Pitzer, Comput. Phys. Commun. 128, 128 (2000).
76.R. M. Pitzer (private communication), see also http://www.chemistry.ohio-state.edu/pitzer
77.W. C. Ermler, R. B. Ross, and P. A. Christiansen, Adv. Quantum Chem. 19, 139 (1988), and references therein.
78.E. R. Davidson, Y. Ishikawa, and G. L. Malli, Chem. Phys. Lett. 84, 226 (1981).
79.H. Lischka, R. Shepard, I. Shavitt et al., COLUMBUS, an Ab Initio Electronic Structure Program, release 5.9, 2003.
80.T. D. Crawford, C. D. Sherrill, E. F. Valeev et al., Psi 3.2, 2003. Freely available at http://www.psicode.org/.
81.J. F. Stanton, J. Gauss, J. D. Watts, W. J. Lauderdale, and R. J. Bartlett, Int. J. Quantum Chem. S26, 879 (1992).
82.J. F. Stanton, C. L. Lopreore, and J. Gauss, J. Chem. Phys. 108, 7190 (1998).
83.For a discussion of CCSDT in the context of atomization energies, see: K. L. Bak, P. Jørgensen, J. Olsen, T. Helgaker, and J. Gauss, Chem. Phys. Lett. 317, 116 (2000).
84.Especially when there is strong spin contamination or symmetry breaking (equivalent to unphysical spin localization when appropriate symmetry elements are not present). In this context, it is not surprising to note that CN has by far the largest post-CCSD(T) correction, as it is the most significantly spin-contaminated species in the study. An ROHF-based analysis would be interesting in this context, but is beyond the scope of this initial presentation and study of the HEAT approach.
85.J. Gauss, W. J. Lauderdale, J. F. Stanton, J. D. Watts, and R. J. Bartlett, Chem. Phys. Lett. 182, 207 (1991).
86.J. D. Watts, J. Gauss, and R. J. Bartlett, J. Chem. Phys. 98, 8718 (1993).
87.Atomic enthalpies of formation (0 K) were taken from Ruscic’s Active Thermochemical Tables (ATcT): H (216.034±0.000 kJ mol−1); C (711.79±0.21 kJ mol−1); N (470.592±0.045 kJ mol−1); O (246.844±0.002 kJ mol−1); F (77.21±0.24 kJ mol−1). These values have considerably smaller error bars than those from the CODATA compilation, and have been chosen in this work for that reason.
88.Enthalpies of formation for carbon and fluorine atoms given in the NIST-JANAF database (see Ref. 6) are 711.19±0.46 kJ mol−1 and 77.28±0.30 kJ mol−1). Note that these uncertainties are larger than the mean absolute errors in the enthalpies of formation determined with HEAT. Recent work by some of us (B. Ruscic, A. G. Császár, and J. F. Stanton, to be published) has focused on an improved estimate for Note the relatively large difference between the NIST-JANAF value and that from the active tables (711.79 vs 711.19 kJ mol−1) for carbon.
89.The idea of the Active Thermochemical Tables is presented in detail in: G. von Laszewski, B. Ruscic, P. Wagstrom et al., in Lecture Notes in Computer Science, edited by M. Parashar (Springer, Berlin, 2002), Vol. 2536, pp. 25–38.
90.The ATcT results presented in this paper were privately communicated by B. Ruscic, and were obtained from Active Thermochemical Tables, ver. 1.20 operating on the Core (Argonne) Thermochemical Network ver. 1.032 (2004). A small subset of these values (H, O, F, and HF) will appear in: B. Ruscic, R. E. Pinzon, M. L. Morton, G. V. Laszevski, S. J. Bittner, S. G. Nijsure, K. A. Amin, M. Minkoff, and A. F. Wagner, J. Phys. Chem. A 108, 9979 (2004).
91.A very basic description of the Active Thermochemical Tables is also scheduled to appear as: B. Ruscic, in Active Thermochemical Tables, McGraw-Hill 2005 Yearbook of Science and Technology (McGraw-Hill, New York, 2004).
92.B. Ruscic (private communication).
93.For the constant contribution to the zero-point energy in Eq. (10)] appears to be important. For the elemental formation reaction, the calculated contribution (J. Vázquez and J. F. Stanton, unpublished) of at the CCSD(T)/DZP level of theory is −32.7 cm−1, or about −0.4 kJ mol−1. When this is added to the HEAT value of agreement with experiment is improved significantly. Note also that an independent variational estimate [M. Mladenovic, Spectrochim. Acta 58A, 809 (2002)] of the ZPE is 22.4 cm−1 below the value in Table I.
94.For a discussion concerning CCH, see P. G. Szalay, L. S. Thøgersen, J. Olsen, M. Kállay, and J. Gauss, J. Phys. Chem. A 108, 3030 (2004).
95.One could even question why diffuse functions are used in the HEAT approach. If indeed the extrapolation formulas gave the correct result for any sequence of hierarchical basis sets (cc-pCVXZ or aug-cc-pCVXZ), then the use of diffuse functions in the HEAT method would offer no benefit whatsoever. However, the extrapolation formulas are essentially empirical in nature (see Ref. 48). Also, if diffuse functions are of importance, then the sequence of basis sets containing them should converge to the exact result more rapidly than that which omits them.
96.J. Noga, W. Klopper, and W. Kutzelnigg, in Recent Advances in Coupled-Cluster Methods (World Scientific, Singapore, 1997), p. 1.
97.S. A. Kucharski and R. J. Bartlett, Chem. Phys. Lett. 158, 550 (1989);
97.S. A. Kucharski and R. J. Bartlett, J. Chem. Phys. 108, 9221 (1998).
98.Y. J. Bomble, M. Kállay, J. Gauss, and J. F. Stanton (unpublished).
99.A. Tatji, M. Kállay, P. G. Szalay, J. Gauss, and J. F. Stanton (unpublished).
100.J. Berkowitz, G. B. Ellison, and D. Gutman, J. Phys. Chem. 98, 2744 (1994).
101.It should be noted that calculation of for using the second and third isodesmic reaction in Table V together with ATcT values of for CCH, and CH yields 227.70 and 229.32 kJ mol−1, respectively, which clearly differ appreciably. The discrepancy is removed to some degree by using values for CH and from Császár and co-workers (Refs. 8 and 9) (592.47 and 390.45 kJ mol−1, respectively), with the former notably outside the ATcT range, which gives (second reaction in Table V) and 229.52 kJ mol−1 (third reaction). Error in these calculations arises almost exclusively from uncertainities in the values used in the subtraction procedure; the calculated reaction enthalpies are unlikely to be in error by more than 0.5 kJ mol−1. This example illustrates the pros and cons of using reaction-based approaches, where the theoretically calculated contribution can be excellent but the experimental uncertainty annoying, and the elemental formation reaction where there is absolutely no uncertainty in the “experimental” numbers, but the theoretical calculations are more demanding. Note that pentuple excitations, essential for the calculation of from elemental reactions, is not at all important with the isodesmic reaction.
102.M. Kállay, P. G. Szalay, and J. Gauss, J. Chem. Phys. 119, 2991 (2003).
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