Benchmark-Quality Atomization Energies for BeH and BeH2

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Benchmark-Quality Atomization Energies for BeH and BeH Monica Vasiliu, Kirk A. Peterson, and David A Dixon J. Chem. Theory Comput., Just Accepted Manuscript • Publication Date (Web): 19 Jan 2017 Downloaded from http://pubs.acs.org on January 20, 2017

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Journal of Chemical Theory and Computation

Benchmark-Quality Atomization Energies for BeH and BeH2 Monica Vasiliu, a Kirk A. Peterson, b and David A. Dixon a,*,† a

Department of Chemistry, The University of Alabama, Shelby Hall, Tuscaloosa, Alabama

35487-0336, USA b

Department of Chemistry, Washington State University, Pullman WA 99164-4630 USA

Abstract The total atomization energies for BeH and BeH2 have been calculated using the Feller-PetersonDixon approach to better than ±1 kcal/mol. The calculations are based on CCSD(T) all electron calculations extrapolated to the complete limit and CCSDT and CCSDTQ corrections are included. A scalar relativistic correction and a diagonal Born-Oppenheimer correction are included. Accurate zero point energies are used. The total atomization energies at 0K are 47.7 kcal/mol for BeH and 140.0 kcal/mol for BeH2 with error bars of at most ±0.3 kcal/mol.

†

Email: [email protected]

1 ACS Paragon Plus Environment


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There is substantial interest in the thermodynamics of small molecules to serve as tests of computational and experimental approaches. Such a simple set is {Be, BeH, BeH2}. This set has the advantage that in the valence space, CCSD is a full configuration interaction (CI) calculation for Be, CCSDT is a full-CI for BeH, and CCSDTQ is a full-CI for BeH2. Although, the diatomic BeH has been known for a long time, gas phase BeH2 was first observed in 2002 1 following its observation in a matrix using infrared spectroscopy in 1995. 2 There is a high quality potential energy surface for the rotational-vibrational spectrum of BeH2 which has yielded an accurate value of the zero point energy. 3 For ν3, there is sub-cm-1 agreement between experiment 4 and theory. Li and Le Roy have reported a potential energy surface from multi-reference configuration interaction (MRCI) calculations with large basis sets as well. 5 The origin of the ν2 band has recently been assigned. 6 In addition, Peterson and co-workers have developed correlation-consistent basis sets 7 including F12 basis sets 8 for Be and have performed an extensive set of CCSD(T) calculations on the atomization energy of BeH2. We focus on the atomization energies and the bond dissociation energies (BDEs) of BeH and BeH2 as the heat of formation of these compounds cannot be predicted to better than ±1.2 kcal/mol (±5 kJ/mol) due to the error in the heat of formation of the gas phase Be atom. 9,10 The BDE, D00 for BeH has been reported by Huber and Herzberg 11 to be 2.034 eV (46.8 ± 0.9 kcal/mol) from a spectroscopic investigation of the predissociation of the B´ 2Π state. 12 The heat of formation of BeH2 reported in the JANAF Tables is only an estimate and is not reliable. In order to provide reliable BDEs for BeH and BeH2, we have performed calculations at the coupled cluster level of theory with single and double excitations and a perturbative triples correction (CCSD(T)) level 13,14,15,16 extrapolated to the complete basis set (CBS) limit 17 using correlation-consistent basis sets. 18 We follow the Feller-Peterson-Dixon (FPD) approach


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developed by us. 19,20,21,22,23 In addition, we have performed CCSDT and CCSDTQ calculations to estimate the effects of higher correlation. Furthermore, we have included additional corrections for relativistic and diagonal Born-Oppenheimer effects. Our approach is similar in form and spirit to the HEAT, 24,25,26 Wn, 27,28,29 and focal point 30 methods and is at a higher level than the composite Gn 31,32 and ccCA 33,34,35 approaches. Computational Methods All electrons are correlated in the CCSD(T) calculations with aug-cc-pwCVnZ basis sets ,

n = T, Q, 5.7 ,36 The electronic structure of the open-shell BeH was calculated with the R/UCCSD(T) approach where a restricted open shell Hartree-Fock (ROHF) calculation was initially performed and the spin constraint was then relaxed in the coupled cluster calculation. 37,38,39 The equilibrium bond distance of BeH for each basis set was obtained from the minimum of a 7-point polynomial fit, 40 where each of these single point energies were calculated for n = T, Q, 5. The equilibrium geometry of D∞h BeH2 was optimized numerically for each of n = T, Q, 5. The CCSD(T) energies at the equilibrium geometries at each basis set with n = T, Q, and 5 were extrapolated to the complete basis set (CBS) limit using a mixed Gaussian/exponential equation (1). 41 E(n) = ECBS + A exp[−(n − 1)] + B exp[−(n − 1)2]

(1)

The energies with n= Q and 5 were also extrapolated to the CBS limit with equation (2) 42 E(lmax) = ECBS + B/(lmax+1/2)4

(2)

with lmax = n. Scalar relativistic corrections (∆ESr) were obtained by single point CCSD(T) calculations using the second-order Douglas-Kroll-Hess Hamiltonian 43,44,45 with the aug-ccpwCVnZ-DK (n = Q, 5) basis sets 46 and extrapolated to the CBS limit using equation (2). The total atomization energy (TAE = ΣD0) is obtained using equation (3).



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ΣD0 = ∆ECBS + ∆ESR + ∆EHOC + ∆EDBOC + ∆EZPE

(3)

where the summation symbol is given for the two Be-H bonds in BeH2. No summation symbol is needed for diatomic Be-H. The contribution ∆ECBS represents the CCSD(T) atomization energy at the extrapolated CBS limit and accounts for the electron correlation of both valence and core electrons. The anharmonic corrected zero-point energy for BeH is calculated as 1/2ωe − 1/4ωexe, where the ωe and ωexe values are obtained from the CCSD(T)/aug-cc-pwCV5Z calculations. The calculated value for ωe is 2064.9 cm-1 and ωexe is 36.7 cm-1 in comparison to the corresponding experimental values of 2060.78 cm-1 and 36.31 cm-1.11 The BeH ZPE is 1023.3 cm-1. We use the calculated value of 2843.6 cm-1 from Koput and Peterson for the ZPE of BeH2. There are no spin 2

orbit corrections to first order as the ground state atoms Be and H have 1S and S ground states, 2 +

respectively, BeH is a Σ state, and BeH2 is a closed shell molecule. Contributions due to correlation effects beyond CCSD(T), ∆EHOC, have been included via CCSDT/cc-pwCVQZ and CCSDTQ/cc-pwCVTZ calculations with all-electrons correlated. Contributions due to the diagonal Born-Oppenheimer correction (∆EDBOC) have been included by calculating the diagonal correction (DBOC) with the cc-pwCVTZ basis sets (cc-pVTZ on H) with all-electrons correlated at the CCSD level of theory 47 with U/UCCSD for the open-shell species. 48 All contributions in Eq. (3) except for ∆ECBS were calculated at the CBS limit equilibrium geometries. The heats of formation at 0 K are derived from their TAE and the experimental heats of formation of the atoms. The thermal corrections at 298 K are obtained using the normal statistical mechanical expression. 49 The heats of formation at 298 K are calculated using the approach of Curtiss et. al. 50 The calculations were carried out with MOLPRO 2012.1,51,52 MRCC, 53

,54,55

and

CFOUR48 on the local Xeon and Opteron based Penguin Computing clusters, the Xeon based 4 ACS Paragon Plus Environment

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Dell Linux cluster at the University of Alabama, the Opteron and Xeon based Dense Memory Cluster (DMC) and Itanium 2 based SGI Altix systems at the Alabama Supercomputer Center, as well as the Intel Xeon-based computing cluster of the Peterson group at Washington State University. Results and Discussion The key results are shown in Table 1. The basis set extrapolation curves are given in the Supporting Information. For BeH, the basis set extrapolation for Q5 is 0.09 kcal/mol larger than for TQ5. The Q5 CBS value is just 0.19 kcal/mol larger than the CCSD(T)/aug-cc-pwCV5Z value for ∆Eelec, so the dissociation energy is well-converged. For BeH2, the extrapolated Q5 CBS limit value for the atomization energy is 0.16 kcal/mol larger than the TQ5 extrapolated CBS value. The Q5 CBS value is just 0.37 kcal/mol larger than the CCSD(T)/aug-cc-pwCV5Z value for ∆ECBS, again showing good convergence in terms of the basis set. The present Q5 CBS and TQ-F12 extrapolated CBS value reported previously8 of 148.07 kcal/mol differ by just 0.06 kcal/mol. The scalar relativistic correction is small as expected. The ZPE corrections should be good to at least 0.05 kcal/mol.

Table 1. Energy components for total atomization energies in kcal/mol Molecule BeH Σ 2

BeH2 Σg+ 1

ΔECBS

ΔECBS

(TQ5)

(Q5)

50.58 147.97

50.67 148.13

ΔErel -0.02 -0.01

ΔT

ΔQ

ΔEZPE

ΔEBO

D00

D00

(TQ5)

(Q5)

a

+0.00

-2.98

-0.06

47.65

47.74

b

+0.01

-8.12

-0.05

139.90

140.06

+0.13 +0.10

b

a

CCSDT valence-only correction = +0.14 kcal/mol. CCSDT valence-only correction = +0.10 kcal/mol



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We note that the full T corrections for BeH in the valence space (full-CI in this space) calculated with the cc-pwCVQZ basis set differ by 0.01 kcal/mol from the full T correction with all electrons correlated. The full T corrections for BeH are not strongly basis set dependent and the difference between the values with the cc-wCVTZ and cc-wCVQZ basis sets is less than 0.01 kcal/mol. The full T corrections for BeH2 for the valence only and all electron calculations with the cc-pwCVQZ basis set are also the same. The full Q correction with the cc-pwCVTZ basis set for BeH2 is very small, 0.01 kcal/mol, so at this level, CCSD(T) is giving results to within better than 0.2 kcal/mol of the full electron correlation limit. The higher level correlation corrections are comparable in size to any basis set extrapolation errors for BeH, and higher level correlation corrections are smaller than the basis set extrapolation errors for BeH2. The total atomization energy reported by Li and LeRoy at the MRCI level with a valence space (4e, 6o) CAS using the aug-cc-pV5Z basis set and additional core correlation corrections with the MR-ACPF method and the cc-pCV5Z basis set gives a value of 148.11 kcal/mol (6.4321 eV). This electronic energy is consistent with our CCSD(T)/[Q5] CBS value of 148.23 kcal/mol and the TQ5 CBS value of 148.07 kcal/mol for the electronic energy component excluding the ZPE and DBOC values. We included a DBOC correction and these values decrease the atomization energy (See Supporting Information for the individual values). The values of -0.06 kcal/mol for BeH and -0.05 kcal/mol for BeH2 are about half of the higher order correlation calculations. The calculated bond dissociation energy for BeH of 47.7 kcal/mol is just within the error limits of the value of 46.8 ± 0.9 kcal/mol from a spectroscopic measurement.11 The atomization energies yield a value of 92.3 kcal/mol for the H-BeH BDE for BeH2, almost double the BDE for BeH. Clearly, adding the second H to complete the simple 4-electron valency expected for BeH2 has a significant impact on the stability of the molecule.


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The heats of formation for Be and BeH2 are given in Table 2. There is an inherent uncertainty of ±1.2 kcal/mol (±5 kJ/mol) due to the uncertainty in the heat of formation of the gas phase Be atom from experiment. Assuming an additional uncertainty of less than ±0.3 kcal/mol for the calculated total atomization energies gives a maximum estimated uncertainty of ±1.5 kcal/mol for the heats of formation. We cannot really reduce the error in the heat of formation of the Be atom unless there is an improvement of experimental heats of formation of either the atom or of other Be-containing molecules such as BeF2 whose heat of formation does not depend on gaseous Be. The experimental error bar for the heat of formation of BeF2, which is derived from vaporization of the solid, is basically ±1 kcal/mol from either the JANAF9 or CODATA10 Tables or from the compilation of Gurvich et al. 56 There is excellent agreement between the experimental and FPD values 57 for the heat of formation of BeF2 so the experimental heat of formation of the Be(g) atom is reliable. We note that Karton and Martin 58 report a lower error bar of ±0.61 kcal/mol for the Be atom using a similar analysis for BeF2 and BeCl2 with weighting by inverse uncertainties based on their W4 approach. The heat of formation of Be(g) remains the same to within 0.1 kcal/mol, just a smaller error bar is given. Using their error bar value, we would obtain errors in the heats of formation of ±0.9 kcal/mol.

Table 2. Heats of formation in kcal/mol ΔHf,0K

ΔHf,298K

BeH

80.3

80.9

BeH2

39.7

39.5

Molecule

Conclusions In summary, we report reliable values for the total atomization energies of BeH and BeH2 which should be accurate to ± 0.3 kcal/mol. These calculated values are based on all



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electron CCSD(T) calculations using the with aug-cc-pwCVnZ correlation-consistent basis sets extrapolated to the complete basis set limit. In addition, essentially full configuration interaction calculations are reported for BeH and BeH2, and these corrections are small, as is the DBOC correction. An accurate ZPE is available for BeH2 and our calculated ZPE for BeH should be good to better than 5 cm-1 based on comparing to the experimental frequency of the diatomic. Thus the only potential source of error in our calculations is the extrapolation to the complete basis set limit. For BeH, our best calculated value at the CCSD(T)/aug-cc-pwCV5Z level is within 0.2 kcal/mol of the CBS limit so there is not a large extrapolation component or error. For BeH2, the CCSD(T)/aug-cc-pwCV5Z is within 0.16 to 0.37 kcal/mol of the limits of the extrapolation methods as well. The actual extrapolation error will be smaller than 0.4 kcal/mol and we believe the ± 0.3 kcal/mol error bar given above to be conservative. The extrapolation error, which is small, is the largest error in the calculations and is larger than the errors due to higher order corrections. Due to the small error bars, the reported dissociation energies for BeH and BeH2 can serve as reliable benchmark values for any further computational studies, for example, using quantum Monte Carlo or density functional theory approaches. 59,60 Acknowledgments This work was supported by the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, U.S. Department of Energy (DOE) under grant no. DE-FG02-03ER15481 (Catalysis Center Program). KAP gratefully acknowledges the support of the U.S. Department of Energy Office of Science, Office of Basic Energy Sciences, Heavy Element Chemistry Program through Grant No. DE-FG02-12ER16329. DAD also thanks the Robert Ramsay Chair Fund of The University of Alabama for support. Supporting Information Additional calculated bond distances, vibrational frequencies and total energies. This material is available free of charge via the internet at http://pubs.acs.org.


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TOC Figure

BeH2

BeH

150

52

145 D00 = 140.0 kcal/mol

140

De(kcal/mol)

De(kcal/mol)

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Page 16 of 16

135

50 48

D00 = 47.7 kcal/mol

46 44

awD

awT

awQ

aw5

CBS

awD

awT

awQ


aw5

CBS

Benchmark-Quality Atomization Energies for BeH and BeH2

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