9156
J. Phys. Chem. A 2005, 109, 9156-9168
Structures and Thermochemistry of Calcium-Containing Molecules Naomi L. Haworth,† Michael B. Sullivan,‡ Angela K. Wilson,§ Jan M. L. Martin,| and Leo Radom*,†,‡ School of Chemistry, UniVersity of Sydney, Sydney, NSW 2006, Australia, Research School of Chemistry, Australian National UniVersity, Canberra, ACT 0200, Australia, Department of Chemistry, UniVersity of North Texas, Denton, Texas 76203-5070, and Department of Organic Chemistry, Weizmann Institute of Science, 76100 RehoVot, Israel ReceiVed: May 31, 2005; In Final Form: July 27, 2005
A variety of theoretical procedures, including the high-level ab initio methods G3, G3[CC](dir,full), and W2C//ACQ, have been used to predict the structures and heats of formation of several small calcium-containing molecules (CaH, CaH2, CaO, CaOH, Ca(OH)2, CaF, CaF2, CaS, CaCl, and CaCl2). B3-LYP and CCSD(T) with both the (aug-)cc-pWCVQZ and (aug-)cc-pWCVQ+dZ basis sets are found to give molecular geometries that agree well with the experimental results. The CCSD(T)(riv)/(aug-)cc-pWCVQ+dZ results are found to be the most accurate, with a mean absolute deviation from experiment of just 0.008 Å. Zero-point vibrational energies (ZPVEs) and thermochemical corrections are found to be relatively insensitive to the level of theory, except in the case of molecules with highly anharmonic calcium-centered bending modes (CaH2, Ca(OH)2, CaF2, CaCl2), where special procedures need to be employed in order to obtain satisfactory results. Several potential improvements to the W2C method were investigated, most of which do not produce significant changes in the heats of formation. It was observed, however, that for CaO and CaS the scalar relativistic corrections are unexpectedly large and highly basis set dependent. In these cases, Douglas-Kroll CCSD(T)/(aug-)cc-pWCV5Z calculations appear to give a converged result. The G3[CC](dir,full) and best W2Ctype heats of formation are both found generally to agree well with experimental values recommended in recent critical compendia. However, in some cases (CaO, Ca(OH)2, and CaF2), they differ from one another by more than their predicted error margins. The available experimental data are not sufficiently precise to distinguish definitively between the two sets of results although, in general, when discrepancies exist the W2C heats of formation are lower in energy and tend to be in better agreement with experiment. In the case of CaO, the W2C heat of formation (20.7 kJ mol-1) is ∼20 kJ mol-1 lower than the G3[CC](dir, full) result and most of the experimental data. Extensive investigation of possible refinements of the W2C method has failed to reveal any weaknesses that could account for this discrepancy. We therefore believe that the heat of formation of CaO is likely to lie closer to the more recent direct experimental determination of 27 kJ mol-1 than to the value of ∼40 kJ mol-1 recommended in recent thermochemical reviews.
1. Introduction A major goal of modern computational chemistry is the calculation of accurate thermochemical properties for molecules of chemical interest.1 The recent development of composite theoretical methods such as the Gaussian (Gn) procedures of Curtiss, Raghavachari, and Pople,2 the complete basis set (CBS) methods of Petersson et al.,3 and the Weizmann (Wn) methods of Martin et al.,4 has made the calculation of such accurate heats of formation almost a routine task for many molecules. For example, for the standard G2-1 test set of molecules containing atoms from the first and second rows, the G2, G3, CBS-Q, W1, and W2 procedures are found to show mean absolute deviations from reliable experimental heats of formation of 5.0, 4.6, 4.2, 2.5, and 2.1 kJ mol-1, respectively.5 In recent studies,6,7 however, we found that the standard application of some of these procedures to the determination * To whom correspondence should
[email protected]. † University of Sydney. ‡ Australian National Laboratory. § University of North Texas. | Weizmann Institute.
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of the heats of formation of the alkali and alkaline-earth oxides and hydroxides led to unacceptably large errors (more than 100 kJ mol-1) in some cases. For the G2 procedure, we concluded that three aspects of the standard G2 model were contributing to the poor results.6 In the first place, our calculations confirmed earlier indications8a,9 that, for systems containing the third-row atoms K and Ca, it is essential to include the 3s and 3p orbitals in the correlation space and that, more generally, an analogous relaxed-inner-valence (denoted riv) procedure or a procedure that includes all orbitals in the correlation space (denoted full) is beneficial for the remaining systems. Next, we found (as noted earlier8a for CaO) that the QCISD(T) component of the G2 energy is poorly described for CaO, Na2O (in bent structures), and K2O but that this can be rectified through replacement of QCISD(T) with CCSD(T) (denoted G2[CC]). Finally, removal of the additivity approximation of standard G2 theory through direct (denoted dir) large-basis-set CCSD(T) calculations was found to have a large effect for the oxides Na2O, CaO, and K2O. The best heats of formation in our initial study6 were obtained with a procedure denoted G2[CC](dir,full) that includes the modifications to the standard G2 procedure referred to above.
10.1021/jp052889h CCC: $30.25 © 2005 American Chemical Society Published on Web 09/15/2005
Calcium-Containing Molecules Thus, (a) all orbitals are included in the correlation space (rather than freezing the core), (b) CCSD(T) calculations are used in place of QCISD(T), and (c) the additivity approximation is eliminated by carrying out the large-basis-set CCSD(T) calculation directly. MP2(full)/6-311+G(3df,2p) structures were used in these calculations. With these modifications, G2[CC](dir,full) was found to give heats of formation in reasonable agreement with the available experimental data, although a definitive assessment was difficult because of the large uncertainties in much of the experimental information.6 In a subsequent study,7 we examined the performance of the more recently developed G32c and G3X10 procedures. We found that, unlike the situation for G2, the use of a simple relaxedvalence (rv) correlation space for the metals in G3 and G3X does lead to reasonable estimates of the heats of formation for most of the metal oxides and hydroxides. This improvement was attributed to the incorporation of core correlation in G3 and G3X through the MP2(full)/G3large calculation. However, even better results were obtained for G3 and G3X with an riv correlation space, and this was recommended as the standard procedure. On the negative side, it was found that additivity still performs poorly in some cases (e.g., for Na2O), a shortcoming that could be overcome by carrying out direct (dir) calculations. The QCISD(T) calculation is still a problem in cases such as CaO; this is again overcome by using CCSD(T) instead. The best results were obtained with the G3[CC](dir,full) method. We also examined7 the performance of the W1 and W2 procedures.4 We found that standard W1 and W2 give reasonable results for the heats of formation of the metal oxides and hydroxides. However, expansion of the correlation space to riv in the extrapolation calculations leads to spectacular failures in some instances (with errors of more than 100 kJ mol-1). This occurs because the cc-pVnZ basis sets prescribed for the metal atoms in the standard formulation do not have sufficient flexibility to adequately describe these inner-valence orbitals. Inclusion of core-correlation functions, as in the newly developed cc-pWCVnZ basis sets,11 removes most of these problems. In the case of calcium, further improvement is seen in the extrapolated results when one and two additional diffuse d functions are added to the quadruple- and triple-ζ basis sets, respectively. The resulting basis sets are denoted cc-pWCVQ+dZ and cc-pWCVT+2dZ. Our best directly calculated heats of formation corresponded to W2-type calculations using these basis sets, and this method was denoted W2C//ACQ (where the ACQ indicates that these new basis sets were also used to calculate the molecular geometry rather than the standard ccpVQZ+1 sets12). Satisfactory agreement between the heats of formation predicted by the best modified procedures, G3[CC](dir,full) and W2C//ACQ, and experiment was achieved in almost all cases.7 However, notable among the exceptions were the two calciumcontaining molecules, CaO and Ca(OH)2, where the G3[CC](dir,full) and best W2C results differed from one another by 18 and 20 kJ mol-1, respectively. Further improvements beyond W2C did not lead to significant changes in the W2C results. For CaO, these improvements included the use of multireference methods with various active spaces (CASSCF, CAS-ACPF, CAS-AQCC, W2C-ACPF, and W2C-AQCC) and methods based on Brueckner doubles, as well as the investigation of the effects of CCSDT and full CI calculations using smaller basis sets. Although the W2C procedure is, in principle, a higher level of theory than G3[CC](dir,full), the experimental results for CaO
J. Phys. Chem. A, Vol. 109, No. 40, 2005 9157 and Ca(OH)2 were not sufficiently precise to be able to distinguish which of the two procedures is performing better in these cases. We therefore felt it desirable to try to address this question by broadening the study to a larger set of Ca-containing molecules (CaH, CaH2, CaO, CaOH, Ca(OH)2, CaF, CaF2, CaS, CaCl, and CaCl2) and also to see whether further modifications to the procedures can lead to improved agreement. These are the aims of the present investigation. Some of these molecules have been the focus of earlier ab initio studies. In the 1980s, Bauschlicher, Partridge, and Langhoff13 predicted dissociation energies for CaO, CaOH, CaF, CaS, and CaCl using Hartree-Fock (HF) and configuration interaction with single and double excitation (CISD) calculations, coupled with experimental ionization energies and electron affinities for the atoms (with the assumption that these species are largely ionic). The early 1990s saw intense interest in the nature of the geometries of CaH2 and CaF2.14 In particular, it was found that it was necessary to include multiple d functions in the Ca basis set in order to correctly predict CaF2 to be a bent molecule (at the HF level). Most of these calculations were performed using HF or MP2 and in some cases were compared with CISD results. Very recently, detailed investigations of the geometry and vibrational frequencies of CaH2 and CaF2 were reported by Koput and Roszczak15a and by Koput,15b respectively. This work involved the mapping of the potential energy surface at the CCSD(T)/cc-pV5Z level of theory (core-valence basis set used for Ca). The PES was then approximated by a 3D expansion to allow the calculation of vibrational-rotational energy levels and anharmonic frequencies. Apart from our previous work,6,7 and the original third-row G2 and G3 studies,8 a literature search has revealed only a small number of other high-level investigations of thermochemical data that have been reported for these molecules. These include CCSD(T) calculations of Trachtman et al.16a and Koput and Peterson16b for CaOH and a BD(T) investigation of CaF by Yang, Zhang, and Han.16c 2. Theoretical Methods In the present study, the G3[CC](dir,full) and W2C methods have been used to investigate the heats of formation of 10 small calcium-containing molecules: CaH, CaH2, CaO, CaOH, Ca(OH)2, CaF, CaF2, CaS, CaCl, and CaCl2. Standard G3 calculations have also been performed for comparison. The G3[CC](dir,full) method has been described in detail previously.7 In brief, it involves a high-level geometry optimization (here we have used the CCSD(T)(riv)/(aug-)cc-pWCVQZ level of theory, defined below) followed by a CCSD(T)(full)/ G3Large single-point energy calculation. Vibrational frequencies for zero-point energies and thermochemical corrections are calculated using density functional theory (B3-LYP/cc-pVTZ+1, scaled by 0.985). Spin-orbit corrections for atoms and the higher-level correction (as defined by Sullivan et al.7) are also included. The W2C//ACQ method7 employs the same CCSD(T)(riv)/ (aug-)cc-pWCVQZ geometry as used for G3[CC](dir,full). Single-point calculations are performed using CCSD(T)(riv)/ (aug-)cc-pWCVnZ (n ) T, Q, 5), and the resulting HF, CCSD, and triples (T) energies are extrapolated separately to the complete basis set limit. The HF energies are extrapolated using an n-5 extrapolation scheme while an n-3 scheme is used for CCSD and (T).4 In the case of the triples extrapolation, only the TZ and QZ energies are used, removing the need to perform the expensive triples calculation with the large quintuple-ζ basis set. Corrections for core-core and core-valence correlation (CV) and for scalar relativistic effects (Darwin17 and mass-
9158 J. Phys. Chem. A, Vol. 109, No. 40, 2005 velocity18 terms) are estimated at the CCSD(T) and ACPF19 levels of theory, respectively, in both cases using the MTsmall4a basis sets. Spin-orbit effects are included for atomic species where appropriate, and zero-point vibrational energies and thermochemical corrections are determined using B3-LYP/ccpVTZ+1 vibrational frequencies. As noted above, new core-valence basis sets have been developed11 that enable the W1 and W2 methods to give improved results for molecules containing alkali and alkalineearth metal atoms. For calcium, these include the “standard” cc-pWCVnZ sets, along with the augmented cc-pWCVT+2dZ and cc-pWCVQ+dZ sets. For convenience, in this work we refer to the d augmented basis sets as cc-pWCVn+xdZ basis sets (where n ) T, Q, 5 and x ) 2, 1, 0) or n+xdZ for short. W2C//ACQ calculations performed using the standard nZ basis sets are denoted W2C-1, while the term W2C-2 is used when the n+xdZ sets have been employed (with minor modifications from the standard procedure, see below and Table 9). Correlation-consistent basis sets are also used for the other atoms in these molecules; these include cc-pVnZ for H and augcc-pCVnZ for O and F. Unfortunately, aug-cc-pCV5Z basis sets are not yet available for Cl and S, so it has been necessary to employ the aug-cc-pVnZ sets instead. These are improved through the addition of two extra d functions (with exponents 2.5 and 6.25 times that of the largest exponent in the set) and one additional f function (with exponent 2.5 times the largest in the set) in order to partially compensate for the lack of corecorrelation functions (hence aug-cc-pVnZ+2df).5 The effects of using aug-cc-pVnZ+2df basis sets rather than aug-cc-pCVnZ for Cl and S were assessed using {T, Q} extrapolations. We find that the two pairs of basis sets give results that are consistent to within 0.6 kJ mol-1, so we do not expect this approximation to have a significant effect on the final results. For convenience, these combinations of basis sets are denoted (aug-)cc-pWCVnZ and (aug-)cc-pWCVn+xdZ throughout the present work. We also note here that, unless otherwise indicated, a riv correlation space is always used for calcium atoms.6-8 When determining CV and scalar relativistic corrections, the 1s electrons of Ca, S, and Cl are held frozen in both calculations as the basis sets are not believed to have sufficient flexibility to describe these deep-core electrons adequately. As noted earlier, density functional theory (B3-LYP20) is used to determine harmonic vibrational frequencies. In our previous work,7 we found that it was necessary in some cases (such as Ca(OH)2) to use “ultrafine” integration grids in order to obtain reliable vibrational frequencies. We have therefore used these grids in all frequency calculations in the present work. For CaH2, Ca(OH)2, CaF2, and CaCl2, the potential energy surfaces for the calcium-centered bending modes are found to be extremely anharmonic. In these cases, we have therefore plotted out the potential energy surfaces for these modes and solved the nuclear Schro¨dinger equation in order to obtain anharmonic vibrational frequencies. These frequencies were used unscaled in the calculation of zero-point energies and thermochemical corrections, the latter being calculated via a sum over the enthalpic contributions of all accessible energy levels at the appropriate temperature (in this case 298 K). This anharmonic treatment was also applied to the OH stretching modes of CaOH and Ca(OH)2. We have not attempted in the present study to go beyond G3[CC](dir,full) within the G3 framework. However, we have examined several modifications to the standard W2C//ACQ procedure in order to probe for any weaknesses that may account for the discrepancies between the G3[CC](dir,full), W2C, and
Haworth et al. experimental results. These modifications primarily involve increasing the level of theory and/or the size of the basis sets used to calculate the various components of W2C. We have also estimated the potential effects of basis set superposition error (BSSE) on both the extrapolated valence calculations and the core-valence correlation using the counterpoise method of Boys and Bernardi,21 and investigated the use of the DouglasKroll-Hess approximations22 in pursuit of a more accurate description of the scalar relativistic effects. In addition, the extension of the coupled cluster method beyond CCSD(T) was also considered for CaO, and calculations with explicit triple(CCSDT), quadruple-(CCSDTQ), and quintuple-(CCSDTQ5) excitations, along with a full CI computation, were performed. These more demanding calculations naturally required the use of smaller basis sets (double-ζ in most cases) and more restricted valence spaces. The standard G3 calculations and all the B3-LYP calculations reported in the present study were performed using the Gaussian 0323 suite of programs. MOLPRO 200224 was used for all other calculations (G3[CC](dir,full), W2C, etc.) apart from the work involving the calculation of explicit triple and higher excitations, which was performed using a generalized CI/CC code25 interfaced with ACESII.26 3. Results and Discussion 3.1. Geometries. The experimental and theoretical geometric parameters calculated in the present work are reported in Table 1. Also included are the mean deviation (MD), mean absolute deviation (MAD), and largest deviation (LD) from experiment27 for each of the theoretical methods considered. There is generally good agreement between theoretical and experimental bond lengths except in the case of CaF2. We note, however, that this very early experiment27e incorrectly predicts CaF2 to be a linear molecule, which may have an adverse effect on the prediction of the bond length. We have therefore excluded the CaF2 results from our statistical considerations. Comparison of the theoretical and experimental results reveals that the large-basis-set CCSD(T) bond lengths (with MAD ) 0.008 Å) are, in almost all cases, significantly closer to the experimental values than are the MP2/6-31G(d) parameters (employed in the standard G3 procedure) or the HF/6-31G(d) and B3-LYP parameters (used in predicting vibrational frequencies). We also note that MP2 and HF incorrectly predict Ca(OH)2 to be slightly bent (with C2V symmetry). Improvement in the Ca basis sets from cc-pWCVQZ to cc-pWCVQ+dZ results in very little change in the predicted bond lengths at the CCSD(T) level of theory. With B3-LYP, however, there is an appreciable improvement in the agreement between theory and experiment with the addition of the two extra d functions to the triple-ζ basis sets, such that the B3-LYP/(aug-)cc-pWCVT+2dZ results are approaching an accuracy (MAD ) 0.009 Å) comparable to that of the coupled cluster values. As described below, CaF2 is found to be a bent molecule. The saddle point at the linear geometry is, however, only 0.4 kJ mol-1 above the global minimum. Although it would have been desirable to perform the single-point energy calculations for the W2C methods using the bent geometry, the quintuple-ζ calculations are computationally too demanding with our current resources. We therefore used the geometry of the linear molecule for all single-point calculations. This approximation is expected to lead to only a minor overestimation in our final heats of formation (by ∼0.4 kJ mol-1). 3.2. Vibrational Frequencies and Thermochemical Corrections. Vibrational frequencies for the molecules of interest
Calcium-Containing Molecules
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TABLE 1: Geometric Parameters for Calcium Compoundsa,b
parameter CaH CaH2 (bent) CaH2 (linear) CaO CaOH Ca(OH)2
CaF CaF2 (bent) CaF2 (linear) CaS CaCl CaCl2 MDh MADh LDh
r(Ca-H) r(Ca-H)
