J . Phys. Chem. 1985, 89, 2198-2203
2198
Theoretical Thermochemistry. 1. Heats of Formation of Neutral AH, Molecules (A = Li to CI) John A. Pople,* Brian T. Luke, Department of Chemistry, Carnegie- Mellon University, Pittsburg, Pennsylvania 1521 3
Michael J. Frisch, Department of Chemistry, University of California, Berkeley, Berkeley, California 94720
and J. Stephen Binkley Scientific Computing Division 8233, Sandia National Laboratories, Livermore, California 94550 (Received: August 8, 1984)
Ab initio molecular orbital theory (Mdler-Plesset theory to full fourth order and a series of extended basis sets) is used to compute total atomization energies for the complete series of AH, hydrides (A = Li to Cl). These data are used in combination with generally accepted experimental data on isolated atoms to predict ideal gas heats of formation at 298.15 K for these molecules. Comparison with high-quality experimental data indicates that, for hydrides, this level of theory has an error level of about +=2kcal mol-I. For several of these hydrides, there are no experimentally determined heats of formation. We predict AHf029s values (in kcal mol-’) for BeHz ( 3 9 4 , MgH2 (38.6), A1H2 (64.0), A1H3 (29.5), PH (59.2), and PH2 (34.8). Our computed heats of formation call into question the experimental heats of formation of BH2 and SiH2.
Introduction The techniques of ab initio molecular quantum mechanics have recently achieved considerable success in the computation of bond energies,’ equilibrium structures,2 and vibrational frequencies3 These data constitute the raw material needed for the calculation of ideal gas thermodynamic functions at any temperature, using standard methods of statistical mechanics. The time now seems ripe to exploit this new capability for systematic prediction of thermodynamic properties. Introduction of such a theoretical thermochemistry is the aim of this paper. Ab initio methods are, of course, still subject to considerable uncertainties and are not yet competitive with the best experimental techniques when applicable. The heat of formation of water as an ideal gas, for example, is known with high precision by calorimetric mea~urement.~ On the other hand, experimental techniques are less successful in examining many transient species, for which heats of formation may be subject to uncertainties of several kcal mol-I. Here theory should be competitive and may frequently provide better results. For those species for which there are no experimental data, theoretical heats of formation should replace speculative guesswork often quoted in current tabulations. In this paper, we consider only the set of neutral AH,, molecules, where A is a non-hydrogen atom (Li to Cl). These molecules are sufficiently small for the theory to be taken to a relatively high level. A uniform approach is taken for all systems to predict equilibrium structures, harmonic frequencies, and bond dissociation energies. These are then used to compute thermodynamic functions for the AH, molecular ground states, using generally accepted data for the monatomic species and the elemental standard state^.^ The reliability of the theory is tested by comparison with the best available experimental values and the whole (1) (a) J. A. Pople, M. J. Frisch, B. T. Luke, and J. S. Binkley, In?. J . Quantum Chem. Symp., 517 307 (1983); (b) J. S.Binkley and M. J. Frisch, In?.J. Quantum Chem. Symp., S17, 331 (1983); (c) M.J. Frisch, J. A. Pople, and J. S.Binkley, J. Chem. Phys., 80, 3265 (1984); (d) M.J. Frisch and J. S.Binkley, J . Chem. Phys., accepted for publication. (2) (a) D. J. DeFrees, B. A. Levi, S.K. Pollack, W. J. Hehre, J. S.Binkley, and J. A. Pople, J . Am. Chem. Soc., 101.4085 (1979); 102,2513 (1980); (b) D. J. DeFrees, K. Raghavachari, H. B. Schlegel, and J. A. Pople, J . Am. Chem. SOC.,104 5576 (1982). (3) J. A. Pople, H. B. Schlegel, R. Krishnan, D. J. DeFrew, J. S. Binkley, M. J. Frisch, R. A. Whiteside, R. J. Hout, and W. J. Hehre, Int. J . Quantum Chem. Symp., S15, 269 (1981). (4) M. W. Chase, J. L. Curnutt, J. R. Downey, R. A. McDonald, A. N. Syverud, and E. A. Valenzuela, J . Phys. Chem. Ref. Data, 11, 695 (1982),
and references therein. (5) CODATA Task Group, J . Chem. Thermodyn., 10, 903 (1978).
0022-3654/85/2089-2198$01.50/0
set of data is then critically reviewed. Theoretical Methods and Results Initial studies are made at the Hartree-Fock level using the 6-31G(d) b a s k 6 This is a split-valence basis with polarization functions on non-hydrogen atoms. Spin-restricted Hartree-Fock (RHF) theory is used for singlet states and the unrestricted form of the theory (UHF) elsewhere. This HF/6-31G(d) model is used to locate the global minima on the AH, potential energy surfaces, leading to theoretical values for bond lengths and angles at the equilibrium geometries. Some of these have been published previously and compare satisfactorily with structures known experimentally.2 The complete set of structural data is listed in Table I. These geometrical parameters are used, without further modifications, to compute theoretical moments of inertia by using the principal isotopes. It should be noted that HF/6-31G(d) geometries are less satisfactory for many larger molecules with two or more non-hydrogen atoms. The same HF/6-31G(d) model was next used to determine harmonic force constants at the potential minima. The masses of principal isotopes were then used to compute harmonic vibrational frequencies, which are also listed in Table I. Many Hartree-Fock frequencies were compared with observed fundamentals in a previous publication3and were found to systematically overestimate the experimental values. Following the recommendation made in that study, all frequencies listed in Table I are multiplied by a scale factor 0.89 in all subsequent computations. In particular, the scaled values are used to compute the zero-point vibrational energy as 1/2Cvi. The principal task of the ab initio theory is to calculate the depth of the AH,, potential minima relative to the separated atoms A + n H . This gives the total atomization energy for fixed nuclei and may be equated to EDe, the sum of the sequential equilibrium dissociation energies. This atomization energy is found by single-point calculations, which include electron correlation and use larger basis sets, at the HF/6-3 lG(d) equilibrium geometries. Electron correlation is included only for valence electrons (the frozen core approximation) and is computed by using fourth-order Maller-Plesset perturbation theory (MP4).’ We now turn to the basis sets used in the single-point calculations. For molecules with first-row A atoms, the principal basis (6) P. C. Hariharan and J. A. Pople, Theor. Chim.Acta, 28, 213 (1973). (7) R. Krishnan, M. J. Frisch, and J. A. Pople, J . Chem. Phys., 72, 4244 (1980).
0 1985 American Chemical Society
Heats of Formation of Neutral AH,, Molecules
TABLE I: HF/6-31G(d) Equilibrium Geometries and Harmonic Frequencies"
bond bond molecule symmetry length, A angle, deg freq, cm-I 0.729 96 4646 (2) H2 LiH 1.635 78 1416 (2) BeH 1.348 39 1.334 15 180.00 BeH2 BH BH2
1.225 39 1.18548
126.53
BH3
1.18833
120.00
CH CH2
1.108 17 1.07 1 06
130.72
CH3
1.072 57
120.00
CH4
1.083 66
109.47
NH 2"
1.023 76 1.01256
104.37
3"
1.002 49
107.21
OH OH2
0.958 46 0.947 32
105.50
FH NaH MOH MgH2
0.91095 1.91435 1.748 59 1.71768
180.00
AIH AlH2
1.652 27 1.595 27
118.02
AlH3
1.58407
120.00
SiH SiH2
1.51521 1SO8 60
93.32
SiH3
1.475 87
110.88
SiH4
1.475 25
109.47
PH PH2
1.41053 1.407 16
93.43
pH3
1.403 14
95.41
SH SH2
1.33009 1.326 37
94.37
CIH
1.266 24
1128 (AI), 2728 (AI), 2867 (B2) 1225 (Az''), 1305 (E'), 2693 (AI'), 2813
(E') 3058 (2) 1244 (AI), 3323 (AI), 3522 (B2) 308 (A;), 1540 (E'), 3285 (AI'), 3461
(E')
1488 (Tz), 1703 (E), 3197 (AI), 3302 (T2) 3528 (2) 1711 (Ai), 3608 (AI), 3708 (B2) 1207 (AI), 1849 (E), 3690 (AI), 3823 (E) 3995 (2) 1827 (AI), 4070 (AI), 4189 (B2) 4358 (2) 1198 (2) 1532 (2) 475 1641 @J, 1667 (2,) 1771 (2) 820 (AI), 1954 (AI), 1976 (B2) 760 (A;), 846 (E'), 2029 (E'), 2813
(W,
(AI')
1949 (2) 1131 (Ai), 2207 (AI), 2218 (B2) 875 (AI), 1015 (E), 2364 (AI), 2383 (E) 1016 (T2), 1051 (E), 2386 (T2), 2393 (Ai) 2557 (2) 1258 (AI), 2577 (BZ), 2579 (AI) 1140 (AI), 1271 (E), 2601 (E), 2606 (AI) 2900 (2) 1368 (AI), 2920 (AI), 2931 (B2) 3186 (2)
Geometrical parameters were not optimized to five significant figures; these quantities are tabulated to this precision for the purpose of reproducibility. set used is 6-31 lG(d,p), a triple-split valence basis with polarization functions on all atoms.* This is extended in three distinct ways, as described in ref IC. First, a standard set of diffuse functions is added (one sp set on each heavy atom) to give the basis set denoted 6-31 l+G(d,p). Second, the number of polarization functions on heavy atoms is doubled by replacing the original set of functions of exponent by two sets having exponents 1/2a and 2a, producing the 6-31 1G(2d,p) basis set. Finally, a single set of second polarization functions (f functions) with standard exponents are added, producing the 6-31 lG(df,p) basis set. The basis set incorporating all three of these improvements is denoted 6-3 1l+G(Zdf,p). For practical reasons, calculations were not (8) R. Krishnan, J. S. Binkley, R. Seeger, and J. A. Pople, J. Chem. Phys., 72, 650 (1980).
The Journal of Physical Chemistry, Vol. 89, No. 11, 1985 2199 carried out with the full 6-31 1+G(2df,p) basis set; instead, separate calculations were carried out with the original 6-3 1lG(d,p) basis and with each of the basis sets produced by the separate enhancements. The energy for each molecule from the full 631 1+G(2df,p) basis was then approximated by assuming that the three enhancements are additive, and hence E[6-311+G(2df,p)] = E[6-311+G(d,p)] E[6-311G(2d,p)] + E[6-311G(df,p)] - 2E[6-311G(d,p)] (1)
+
This approximation is known to slightly overestimate the energy calculated with the full basis set.' Total energies calculated by this method for the first-row systems are listed in Table IIa. The 6-31 lG(d,p) basis set is not yet available for second-row atoms, so the calculations on the AH,, systems involving second-row atoms were carried out with the smaller 6-31G(d,p) basis set. The same extensions are applied to this basis set and the same additivity approximation is made. Total energies of the second-row molecules at this level are found in Table IIb. The total energies from these single-point calculations are now used to determine total atomization energies, CD,. The simplest procedure would be direct calculation of the energy of the dissociation reaction AH,-A+nH
(2)
where free atoms are in their electronic ground state. It is known that extensions to the hydrogen 6-31 lG(d,p) basis set analogous to those described above for heavy atoms contribute about 2 kcal mol-' to the energy of each A-H bond.lcSd Since the explicit inclusion of these basis functions would increase the cost of the computations substantially, a preferable procedure is to determine the energy of each bond relatiue to that of the hydrogen molecule and then to convert to atomization energies by using the exact dissociation energy of H2. This approach assumes that the contribution of the missing hydrogen basis functions is constant for all bonds, an approximation which is known to be good for the few c a m for which it has been tested so far.'" There is also some smaller cancellation of missing correlation energy comparing the A-H and H-H bonds. Thus for lithium hydride, we consider the reaction LiH
+H
-
Li
+ H2
(3)
According to Table 11, the energy of this process (for fixed nuclei) is -0.08175 hartree (or 51.30 kcal mol-'). Using the exact dissociation energy De for the hydrogen molecule of 109.48 kcal mol-', we deduce that D, for lithium hydride is 58.32 kcal mol-'. Reaction 3 is a particular example of an isogyric process, in which the number of unpaired electron spins is left unchanged. We may, in fact, write unique isogyric comparisons with H2 for the ground states of all of the AH,, systems. For methane, the appropriate reaction is
CH4
+ 2H
-
C
+ 3H2
(4)
since carbon has a triplet ground state (two unpaired electrons). For the amino radical, it is NH2 + 2H N 2H2 (5)
+
with three unpaired spins on both sides. Note that a process which conserves the number of unpaired electrons also conserves the number of electron pairs. This isogyric energy correction is approximately 4 kcal mol-' per hydrogen molecule. Such isogyric comparisons give the total atomization energies ED,. Subtraction of the AH,, zero-point energies (from HF/63 1G(d) frequencies scaled by 0.89) then gives the vibrationally corrected atomization energies EDo, listed in Tables I11 and IV. These are the theoretical predictions for the enthalpy of the atomization reaction 2 at zero temperature. To obtain AH,, heats of formation at zero temperature, the atomization energies E D o are subtracted from the known heats of formation of the isolated atoms AHOf(AH,,,O K) = AHoI(A,O K) + nAHof(H,O K) - E D , (6)
2200
Pople et al.
The Journal of Physical Chemistry, Vol. 89, No. 11, 1985
TABLE II: MP4 Total Energies (bartree) at HF/6-31G(d) Geometries A
molecule H H2
0 OH OH2 F FH
6-31 lG(d,p) -0.499 8 1 -1.167 63 -7.432 03 -8.015 52 -14.61307 -15.189 31 -15.838 60 -24.588 12 -25.21498 -25.846 98 -26.518 20 -37.764 30 -38.385 88 -39.05291 -39.730 60 -40.404 8 1 -54.490 89 -55.107 55 -55.75262 -56.427 54 -74.933 33 -75.588 22 -76.275 90 -99.565 34 -100.274 21
6-31 l+G(d,p) -0.499 8 1 -1.1 67 63 -7.43203 -8.01 5 6 1 -14.61308 -1 5.189 38 -15.838 67 -24.588 52 -25.21 5 7 1 -25.847 33 -26.5 18 34 -37.765 20 -38.387 59 -39.05405 -39.731 87 -40.405 11 -54.492 5 1 -55.11091 -55.757 51 -56.433 89 -74.937 24 -75.595 23 -76.286 62 -99.571 12 -100.286 08
molecule
6-3 lG(d,D)
H H2 Na NaH Mg MOH MgH2 AI AIH AlH2 AIH3
-0.498 23 -1.16453 -161.841 44 -162.39925 -199.627 38 -200.168 09 -200.773 53 -241.899 50 -242.504 80 -243.078 49 -243.709 71 -288.887 65 -289.490 94 -290.10728 -290.721 01 -291.36560 -340.763 33 -341.36297 -341.977 72 -342.605 92 -397.571 47 -398.194 13 -398.833 71 -459.569 83 -460.223 92
6-31+G(d,p) -0.498 23 -1.164 53 -161.841 44 -162.399 34 -199.627 46 -200.168 22 -200.773 75 -241.900 11 -242.50601 -243.079 82 -243.71 1 39 -288.888 50 -289.492 41 -290.109 17 -290.722 75 -29 1.367 55 -340.764 23 -341.36426 -341.979 30 -342.607 66 -397.573 07 -398.19591 -398.835 67 -459.571 62 -460.226 23
Li
LiH Be BeH BeH2 B BH BH2 BH3 C CH CH2 CH3 CH4 N NH 2" 3"
6-31 1G(2d,p) -0.499 8 1 -1.167 63 -7.432 03 -8.016 22 -1 4.61 3 03 -1 5.190 27 -1 5.840 11 -24.590 64 -25.218 71 -25.851 10 -26.522 99 -37.77029 -38.393 26 -39.059 62 -39.738 56 -40.413 88 -54.500 76 -55.1 18 54 -55.76477 -56.439 70 -74.948 44 -75.60423 -76.291 92 -99.585 80 -100.295 68
6-31 lG(df,p) -0.499 8 1 -1.167 63 -7.432 03 -8.017 31 -14.613 09 -15.191 01 -15.841 35 -24.589 93 -25.215 75 -25.85202 -26.524 41 -37.768 98 -38.393 89 -39.062 16 -39.742 32 -40.41762 -54.499 22 -55.120 10 -55.768 44 -56.445 85 -74.949 90 -75.607 95 -76.297 91 -99.587 67 -100.29861
est +(2df,p) -0.499 8 1 -1.167 63 -7.43203 -8.018 10 -14.61306 -1 5.192 04 -15.84293 -24.592 85 -25.220 21 -25.85649 -26.529 34 -37.775 87 -38.40298 -39.07001 -39.751 55 -40.426 99 -54.5 10 7 1 -55.134 45 -55.785 48 -56.464 36 -74.968 92 -75.630 97 -76.324 65 -99.61391 -100.331 95
6-31G(2d,p) -0.498 23 -1.164 53 -161.841 44 -162.401 09 -199.627 16 -200.168 76 -200.774 36 -241.905 87 -242.51080 -243.083 89 -243.715 35 -288.90029 -289.502 75 -290.11938 -290.730 86 -291.375 48 -340.778 37 -341.377 86 -341.993 44 -342.622 73 -397.589 15 -398.213 78 -398.85493 -459.590 11 -460.247 5 1
6-31G(df,p) -0.498 23 -1.164 53 -161.841 44 -162.40221 -199.627 40 -200.172 35 -200.781 99 -241.901 41 -242.51036 -243.087 51 -243.722 38 -288.892 83 -289.500 59 -290.12005 -290.736 17 -291.383 35 -340.773 73 -341.378 38 -341.99692 -342.627 68 -397.594 22 -398.22032 -398.862 22 -459.602 34 -460.258 56
est +(2df,p) -0.498 23 -1.16453 -161.841 44 -162.404 14 -199.627 26 -200.173 15 -200.783 04 -241.908 39 -242.51757 -243.094 24 -243.729 70 -288.906 32 -289.513 87 -290.134 04 -290.747 76 -291.395 18 -340.789 67 -341.394 56 -342.01422 -342.646 23 -397.613 50 -398.241 75 -398.885 40 -459.62441 -460.28446
B
Si SiH SiHz SiH3 SiH, P PH PH2 PH3 S SH SH2
c1
C1H
TABLE 111 Total Atomization Energies Dn (kcal mol-') molecule theory expt"
CH4
394.4 276.5 218.7 135.4 171.7 101.5
3"
0% FH SH2
CIH
392.5 276.7 219.3 135.3 173.1 102.2
"Based on 0 K heats of formation from CODATAS for atoms and JANAF4 for molecules.
CODATA values5 are used for the atomic MHO and f lead to the molecular A H O f ( 0 K) listed in Table IV. Finally, corrections to 298.15 K are made AH0f(AH,,,298.15 K) = Mof(AH,,,O K) [H0(AH,,,298.15 K) - Ho(AH,,,O K)] - [H0(A,298.15 K) Ho(A,O K)Ist n[H0(H,298.15 K) - Ho(H,O K)Ist
+
+
The three heat capacity corrections (in square brackets) are treated
differently. The correction for the AH,, gas is made by using the scaled theoretical frequencies for vibrations and the classical approximation for translation and rotation, while the elemental heat corrections for A and H refer to the standard state and are taken from the CODATA tablese5 The resulting heats of formation WdAH,,,298.15 K) are listed in Table IV a s theoretical values, although, as noted, they are based on some experimental data for monatomic and standard species. Experimental AH, heats of formation at 298.15 K are also listed in this table; comparisons will be made in detail in the next section. Discussion Before considering the full set of results for all AH,, species, it is helpful to test the reliability of the theory for the molecules with the best experimental data. We do this by comparing theory and experiment for the total atomization energies E D o of the saturated molecules CH4, NH3, OHz, FH, SHz,and ClH. This is shown in Table 111. The level of agreement is good, with the mean absolute difference between theory and experiment being
Heats of Formation of Neutral AH,, Molecules
The Journal of Physical Chemistry, Vol. 89, No. 11, 1985 2201
TABLE I V Total Binding Energies and Heats of Formation (kcal mol-')
AHi'290 molecule LiH BeH BeH2 BH BH2 BH3 CH CH2 CH3 CH4 NH 2" 3"
OH OH2 FH NaH MgH MgH2 A1H AIH2 AlH3 SiH SiH2 SiH3 SiH4 PH PH2 PH3 SH SH2 C1H
D
e
58.3 49.7 148.5 84.1 169.7 282.4 83.9 188.9 306.9 421.2 81.8 180.8 297.2 105.9 231.6 141.0 44.5 29.9 104.0 73.6 122.9 213.0 72.6 153.1 225.6 323.3 70.9 151.2 239.1 85.6 180.9 105.6
ED0 56.5 46.9 140.9 80.9 161.1 266.9 80.0 178.6 289.8 394.4 77.3 169.3 276.5 100.8 218.7 135.4 43.0 28.0 98.6 71.3 116.9 202.1 71.1 146.0 212.8 304.5 67.6 143.0 224.4 81.9 171.7 101.5
Mroo
33.2 81.1 39.8 103.3 74.7 20.6 141.6 94.6 35.2 -17.9 86.9 46.5 -9.1 9.8 -56.5 -65.3 34.5 58.6 39.4 58.5 64.7 31.1 88.0 63.8 48.6 8.6 59.4 35.7 5.9 35.5 -2.8 -21.3
H'29a.15-
2.1 2.1 2.2 2.1 2.4 2.4 2.1 2.4 2.6 2.4 2.1 2.4 2.4 2.1 2.4 2.1 2.1 2.1 2.4 2.1 2.4 2.6 2.1 2.4 2.5 2.5 2.1 2.4 2.4 2.1 2.4 2.1
H'o
theory 33.1 81.7 39.5 104.1 74.8 19.7 142.4 94.7 34.5 -19.8 86.9 45.8 -10.8 9.8 -57.2 -65.3 34.0 58.4 38.6 58.5 64.0 29.5 88.3 63.4 47.3 6.3 59.2 34.8 4.0 35.4 -3.5 -21.4
expt 33.6" 77 f 7.a 81.7b 105.8 f 2,O
