Benchmark Theoretical Study on the Dissociation Energy of Chlorine

ARTICLE pubs.acs.org/JPCA

Benchmark Theoretical Study on the Dissociation Energy of Chlorine J
ozsef Csontos* and Mih
aly K
allay Department of Physical Chemistry and Materials Science, Budapest University of Technology and Economics, Budapest P.O. Box 91, H-1521 Hungary

bS Supporting Information ABSTRACT: The currently accepted D0(35Cl2) is 239.221 ( 0.001 kJ/mol, whereas popular theoretical model chemistries provide values in the range of 233
247 kJ/mol, and even the so-called high-accuracy protocols can yield values as low as 237.9 kJ/mol and as high as 240.1 kJ/mol for D0(35Cl2). The aim of this study was to uncover the sources of error inherent in the theoretical approaches. Therefore, a coupled-cluster-based composite model chemistry was utilized that included contributions of up to pentuple excitations, as well as corrections beyond the nonrelativistic and Born
Oppenheimer approximations. In our calculations, correlation consistent basis set families were used up to octuple-ζ basis sets. It was found that the following factors, in order of significance, can be identified as the most important error sources: (i) the considerably large relativistic contributions carrying large uncertainties, (ii) the very slow convergence of the Møller
Plesset (MP2) correlation energy (with the octuple-ζ basis set, it still contains an error of a few tenth of a kJ/mol), (iii) the slow convergence of the coupled-cluster singles and doubles (CCSD) contribution (it needs a octuple-ζ basis set to converge within 0.1 kJ/mol), and (iv) the relatively large basis set (quadruple-ζ) needed in the calculation of an accurate perturbative quadruples contribution. It is also notable that, for chlorine, the use of a quintuple-ζ basis set for the Hartree
Fock energy, the MP2 correlation energy, and for the CCSD and perturbative triples contributions, which is the usual treatment in almost every high-accuracy model chemistry, resulted in the overestimation of all of these contributions (altogether about by 1.8 kJ/mol). However, this overestimation is accidentally compensated by (i) using an inappropriate, small basis set for the valence electron contribution due to quadruple excitations (∼1.2 kJ/mol), (ii) neglecting the effects of core electron contributions due to quadruple excitations (∼0.2 kJ/mol), and (iii) neglecting relativistic effects beyond the scalar relativistic treatment (∼0.3 kJ/mol). The most reliable theoretical estimate for D0(35Cl2) obtained in this study, 239.27 ( 1.30 kJ/mol, differs by only 0.05 kJ/mol from the most accurate experimental result. This study also underpins the effect of relativistic contributions, which precludes current model chemistries to enter the range of sub-kJ/mol accuracy for second-row systems.

’ INTRODUCTION Recently, the heats of formation for atmospherically important fluorinated and chlorinated methane derivatives were calculated with the help of accurate quantum chemical methods,1 and considerable deviation was observed between theory and experiment. Most of the discrepancy was the consequence of the difference, 1.04 kJ/mol, between the theoretically1 and experimentally2 predicted heats of formation of the chlorine atom Because the theoretical value for ΔfH°(Cl) can be [ΔfH°(Cl)]. 0 0 obtained as half of the dissociation energy of the chlorine molecule [D0(Cl2)], the purpose of this paper is to analyze the factors contributing to the theoretical D0(Cl2) and to reveal the sources of significant errors inherent in the theoretical approaches. The accurate determination of D0(Cl2) has been the topic of several studies, and the experimental findings that apply to the 35 Cl2 isotope are summarized in Table 1 and discussed briefly in the following. One of the earliest spectroscopic studies on diatomic halogen molecules was conducted by Kuhn in 1926,3 who determined r 2011 American Chemical Society

244.76 kJ/mol for D0(Cl2). A couple of years later, Elliott4,5 reinvestigated the absorption spectrum of chlorine at higher resolution and obtained 238.07 kJ/mol for D0(Cl2). Kuhn3 and Elliott4,5 did not publish any statistical measure related to the uncertainty of their values. On the basis of the work of Kuhn3 Gaydon,6,7 assuming that the convergence limit of visible absorption bands lies at 20850 cm
1, 238.80 ( 0.24 kJ/mol was recommended for D0(Cl2). In 1962, Rao and Venkateswarlu8 studied the ground-state vacuum ultraviolet resonance spectrum of chlorine and determined D0(Cl2) = 240.00 ( 0.12 kJ/mol using the Birge
Sponer extrapolation. One year later, Douglas and associates9 measured and analyzed the absorption spectrum of chlorine and found D0(Cl2) = 239.24 ( 0.02 kJ/mol. To resolve the discrepancy between the measurement of Douglas and associates9 and that of Rao and Venkateswarlu,8 Clyne and Received: March 4, 2011 Revised: May 16, 2011 Published: May 23, 2011 7765

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Table 1. Dissociation Energy of Chlorine (kJ/mol) Obtained from Experimental Data. k D0

investigation a

Kuhn (1926)

244.76

Elliott (1930)b Gaydon (1947)c

238.07 238.80 ( 0.24

Rao (1962 d

Table 2. Dissociation Energy of Chlorine Calculated by Theoretical Model Chemistries; the Deviations (δ) from the Most Accurate Experimental Value (239.221 ( 0.001 kJ/mol) Are Also Shown (All Values Are in kJ/mol) D0

δ

a

242.7


3.5

W3K

239.1

0.2

240.00 ( 0.12

G2b

233.5

5.8

W4ai

239.0

0.2

Douglas (1963)e

239.24 ( 0.02

G3c

234.4

4.8

W4bi

239.0

0.2

Clyne (1970)f

239.73 ( 0.24

G4d

236.9

2.4

W2.2j

238.8

0.4

Le Roy (1971)g

239.220 ( 0.004

CBS-4e

236.0

3.2

W3.2j

238.4

0.9

Douglas (1975)h

239.221 ( 0.001

CBS-Qe

246.4


7.2

W4litej

237.9

1.4

Samartzis (1997)i

238.71 ( 1.93

Li (2007)j

239.13þ0.10
0.02

CBS-QB3c Feller (1999)f

245.9 239.3


6.7
0.1

W4j W4.2j

238.6 238.5

0.6 0.7

a

Feller (2003)g

239.7


0.5

W4.3j

238.8

0.4

f

h

Feller (2008)

239.5


0.2

W4.4ak

238.8

0.4

W1c

236.2

3.0

W4.4bk

238.8

0.4

W2c

240.1


0.9

HEATl

238.7

0.5

W3i

239.8


0.6

HEATm,l

240.1


0.8

W3Ai

239.9


0.6

model chemistry G1

Reference 3. b Reference 5. c Reference 7. d Reference 8. e Reference 9. Reference 10. g Reference 11. h Reference 14. i Reference 15. j Reference 16. k The error bars, where available, are also given.

Coxon10 reinterpreted some of the data of ref 8, and 239.73 ( 0.24 kJ/mol was obtained for D0(Cl2). LeRoy and Bernstein11 refined the value given by Douglas and associates9 using the data in ref 9 and their newly derived formula12 for determining the dissociation energy of diatomic molecules.13 They estimated D0(Cl2) to be 239.220 ( 0.004 kJ/mol.11 To reconcile the two D0(Cl2) values derived by Rao and Venkateswarlu8 and Douglas and associates,9 Douglas and Hoy14 reinvestigated the resonance fluorescence spectrum of chlorine at higher resolution than that of Rao and Venkateswarlu, and their study resulted in 239.221 ( 0.001 kJ/mol, confirming the work of LeRoy and Bernstein.11 The above result, 239.221 ( 0.001 kJ/mol, is nowadays the accepted experimental value for D0(Cl2). Recently, D0(Cl2) has also been determined by other spectroscopic techniques. Samartzis and co-workers15 studied the photolysis of the chlorine molecule and obtained D0(Cl2) = 238.71 ( 1.93 kJ/mol from photofragment translational energy distributions, whereas Li and associates,16 using zero kinetic energy photoelectron spectroscopy and ion-pair formation imaging techniques, determined 239.13þ0.10
0.02 kJ/mol for D0(Cl2). Besides the experimental studies, D0(Cl2) has been calculated in numerous theoretical work. In the majority of the theoretical studies, usually it is not clear whether or not the isotope effect was taken into account. Nevertheless, the inaccuracy of the theoretical methods discussed is considerably larger than the effect, 0.02 kJ/mol,2 in D0(Cl2) caused by the presence of the 37Cl isotope. The results obtained by the most popular as well as most accurate model chemistries are summarized in Table 2 and detailed below. Gaussian-n (Gn) theories G1,17,18 G2,19 G3,20 and G421 provide 242.73,18,22 233.46,19 234.43,23 and 236.85 kJ/mol,21 respectively, for D0(Cl2). The representative members of the complete basis set (CBS) model chemistry family, developed by Petersson and associates,24
28 CBS-4, CBS-Q, and CBS-QB3, give, respectively, 235.98,28 246.44,28 and 245.94 kJ/mol23 for D0(Cl2). Although the aimed and claimed accuracy range of the above methods is about (4 kJ/mol, they can hardly reach this regime. Nevertheless, D0(Cl2) has also been calculated by model chemistries that pursue more ambitious and challenging accuracy ranges. Feller and Peterson29 used such a composite approach and obtained 239.33 kJ/mol for D0(Cl2). Their method was based on frozen-core (FC) coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] calculations, which were further

model chemistry i

D0

δ

a

Reference 17. b Reference 19. c Referencef 23. d Reference 21. e Reference 28. f Reference 29. g Reference 30. h Reference 31. i Reference 32. j Reference 33. k Reference 34. l Reference 1. m An improved HEAT version (see text).

corrected by core
valence and scalar relativistic calculations, as well as experimental spin
orbit (SO) contributions. The FC CCSD(T) total energy was extrapolated to the complete basis set limit using the aug-cc-pV(Q,5,6)Z basis sets and an exponential formula. The core
valence correction was obtained from CCSD(T)/cc-pwCVQZ computations, and the scalar relativistic correction was calculated at the FC CISD/cc-pVTZ level. In an extensive study, Feller and associates30 investigated the performance of their composite approach on small halogenated compounds. The approach was similar to that used previously in ref 29; however, in the extrapolation, a two-point formula was utilized using the CCSD(T)/aug-cc-pV(5,6)Z total energies, and the scalar relativistic corrections were obtained from Douglas
Kroll
Hess (DKH) FC CCSD(T)/cc-pVQZ calculations. The computations resulted in 239.74 kJ/mol for D0(Cl2). Recently, Feller and co-workers31 surveyed several factors that play an important role in the accurate determination of atomization energies. For D0(Cl2), 239.45 ( 1.26 kJ/mol was obtained. Their approach included contributions beyond the CCSD(T) level, and the core
valence correlation was extrapolated to the basis set limit. The use of an aug-cc-pV7Z basis set in the extrapolations of the CCSD(T) correlation and SCF energy is also notable. The Weizmann-n (Wn) family of theories has been developed by Martin and his associates,23,32
34 and in the first paper of the series, 236.19 and 240.08 kJ/mol were reported for D0(Cl2) using the W1 and W2 methods, respectively.23 Nevertheless, in ref 23, D0(Cl2) was also evaluated with even higher-level methods, and the best estimate was 240.25 kJ/mol. The different variants of the W3 and W4 theories, W3, W3A, W3K, W4a, and W4b, provided 239.83, 239.87, 239.07, 239.03, and 238.99 kJ/mol, respectively, for D0(Cl2).32 In the further development of the W4 theory, several versions of W4 were defined, and more accurate variants of W2 and W3 appeared.33 The application of W2.2, W3.2, W4lite, W4, W4.2, and W4.3 theories for D0(Cl2) resulted, respectively, in 238.78, 238.36, 237.86, 7766

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238.61, 238.53, and 238.82 kJ/mol.33 The most recent versions of the Weizmann theories, which slightly differ in the extrapolation of the valence (T) contribution, W4.4a and W4.4b, provide 238.82 and 238.78 kJ/mol, respectively, for D0(Cl2).34 One notable and shared feature of the above-mentioned theoretical model chemistries is that the contributions due to core
valence and core
core correlations are separated from that originated from the valence
valence correlation. Unlike the above models in the high-accuracy extrapolated ab initio thermochemistry (HEAT) approach, these contributions are not separated.35
37 For D0(Cl2), the HEAT-345(Q) protocol gave 238.70 kJ/mol.1 An improved version of the HEAT-345(Q) protocol was also used in ref 1, where the CCSDT and CCSDT(Q) contributions were extrapolated to the CBS limit using cc-pVXZ (X = 4 and 5) and cc-pVXZ (X = 3 and 4) basis set series, respectively. These calculations resulted in 240.05 kJ/mol for D0(Cl2), indicating that at least some of the considered contributions are far from convergence. This finding may also indicate that, as expected, some error cancellation occurs in the other theoretical model chemistries as well. At this point, the need for a more elaborate theoretical study is evident in order to pinpoint the sources of error in the theoretical approaches.

’ METHODS Basis Sets. Because correlation consistent polarized core
valence 6-ζ (cc-pCV6Z), 7-ζ (cc-pCV7Z), and 8-ζ-quality (ccpCV8Z) basis sets were previously not available, slightly modified versions of the correlation consistent basis set series38 were generated, as described in refs 39
42, except that the electron correlation was calculated with the CCSD(T) method43 and the 1s2 electrons were not frozen while the CV-functions were being optimized. (We note here that, during the course of this study, Peterson and associates44 published a cc-pCV6Z basis set for second-row elements.) To be consistent, the appropriate DZ, TZ, QZ, 5Z, and 6Z basis set series were regenerated with the above modifications. The initial (sp) set used in the optimization of polarization functions was (22s13p) for the DZ, TZ, and QZ basis sets. For the 5Z, 6Z, 7Z, and 8Z basis sets, the initial (sp) set was (32s23p). The exponents in the initial (sp) sets were fully optimized in restricted open-shell Hartree
Fock (HF) calculations using the downhill simplex method of Nelder and Mead.45 At the optimizations of the polarization functions, the exponents were restricted to be even-tempered expansions; similarly, the s- and p-correlation sets were restricted to be even-tempered. During the determination of the augmenting functions, no constraint was forced. The TURBOMOLE package46 was used for the optimization of the (sp) sets, and all other calculations related to basis set development were carried out by the PSI3 package.47 Total Energies. Total energies were calculated invoking CC theory48 and the family of the modified (see above) cc-pVXZ,39,49 aug-cc-pVXZ,41 cc-pCVXZ, and aug-cc-pCVXZ42 basis sets. In the energy calculations, additivity of the various contributions was assumed according to the following scheme

E ¼ EHF þ ΔEMP2 þ ΔECCSD þ ΔEðTÞ þ ΔET þ ΔEðQ Þ þ ΔEQ þ ΔEP þ ΔEDBOC þ ΔESR þ ΔEDC þ ΔEGaunt ð1Þ In eq 1, EHF is the HF self-consistent field energy. To determine the basis-set limit, the extrapolation of the HF energies is a widely

applied procedure, and several extrapolation formulas have been devised.50
53 However, because of the lack of theoretical arguments, it is still not clear which extrapolation formula provides the best estimates for the HF basis set limit.29,51
53 In this study, the extrapolation formula of Feller,50 E(X) = ECBS þ b 3 e
cX,√and that of Karton and Martin,53 E(X) = ECBS þ A 3 (X þ 1) 3 e
9 X, were tested (see the Supporting Information). Because the two-point extrapolation formula of Karton and Martin provided a bit more consistent data, it was used in this study. ΔEMP2 is the correlation energy evaluated by the second-order Møller
Plesset (MP2)54 method and extrapolated to the basis set limit. As in the HF case, several extrapolation formulas have been advised to accelerate the convergence of the correlation energy.24
29,55
64 The E(X) = ECBS þ B 3 X
3, E(X) = ECBS þ B 3 X
3 þ C 3 X
4, and E(X) = ECBS þ B 3 (X þ 0.5)
4 formulas are used to determine the basis set limit for the correlation component of the total energy, because these functions are more or less backed by theory.65 The average of the obtained values was used as the most reliable estimate for the basis set limit.29
31,66,67 ΔECCSD and ΔE(T) are the correlation contributions defined as ΔECCSD = ECCSD
EMP2 and ΔE(T) = ECCSD(T)
ECCSD, where EMP2, ECCSD, and ECCSD(T) are the total energies obtained with the MP2, CCSD,68 and CCSD(T)43 methods, respectively. The ΔECCSD and ΔE(T) contributions were also extrapolated to the basis set limit using the aforementioned formulas. The correlation energy was also investigated beyond the CCSD(T) level using higher excitations in CC theory. The iterative triples [CCSDT]69 and the perturbative quadruples [CCSDT(Q)]70,71 contributions were also extrapolated to the basis set limit. The effect of the iterative quadruple [CCSDTQ]72,73 and pentuple [CCSDTQP]74 excitations were calculated with the cc-pVTZ and cc-pVDZ basis sets, respectively. The above contributions were defined as follows: ΔET = ECCSDT
ECCSD(T), ΔE(Q) = ECCSDT(Q)
ECCSDT, ΔEQ = ECCSDTQ
ECCSDT(Q), and ΔEP = ECCSDTQP
ECCSDTQ, where EX is the total energy obtained with the corresponding method X. In the CCSDTQ and CCSDTQP calculations, the effects of the core electrons were not studied. The deficiencies of the Born
Oppenheimer (BO) approximation were corrected by adjusting the energy with the diagonal BO correction (DBOC) calculated at the CCSD/cc-pCVXZ (X = D, T, Q) level (ΔEDBOC), as described in ref 75. The scalar relativistic contributions (ΔESR) were taken into account using the fourth-order Douglas
Kroll
Hess (DKH) Hamiltonian in CCSD(T) calculations. ΔEDC is defined as the difference of energies between the four-component Dirac
Coulomb76,77 and DKH treatment; in these calculations, the CCSD(T) method with uncontracted aug-cc-pCVXZ basis sets was used. The Gaunt correction, ΔEGaunt, was obtained with the HF method using uncontracted aug-cc-pCVXZ basis sets. The zero-point vibrational energy (ZPE), 3.34 kJ/mol,78 and the equilibrium bond length, 1.9880 Å,79 of 35Cl2 were taken from experiment. We note here that it is usual to determine the equilibrium structure and the ZPE for a molecule at the CCSD(T)/cc-pVQZ level. For Cl2, this approach resulted in 1.9998 Å and 3.33 kJ/mol for the equilibrium bond length and the ZPE, respectively; and based on our test calculations, these deviations would decrease the value of D0(Cl2) by about 0.1 kJ/mol. In the evaluation of the DKH contribution, restricted openshell HF orbitals were employed. In all other calculations, unrestricted and restricted HF orbitals were used for Cl and Cl2, respectively. 7767

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Table 3. Contributions (in kJ/mol) to the Dissociation Energy of Chlorine, D0(35Cl2), at Various Levels of Theory higher-order terms valence Xa

EHFb

ΔEMP2b ΔECCSDb ΔE(T)b

ΔETc

ΔE(Q)d

ΔEQd

ΔEPd 0.08

2

56.28

145.46


29.45

13.84


0.30

1.08


0.09

3

78.85

172.48


40.33

19.20


1.40

1.76


0.08

4

80.84

182.84


42.32

20.71


1.62

2.01

5

81.21

187.46


43.14

21.29


1.72

2.13


1.78

6

81.21

189.30


43.76

21.53

7

81.21

190.29


44.16

21.63

8

81.21

190.75

3i 4i

80.72 81.09

182.93 190.63


44.53
43.55

21.27 21.78


1.87
1.75

2.04 2.20

5i

81.27

192.30


43.98

21.87


1.82

2.25

6i

81.20

191.50


44.69

21.81


1.87

7i

81.21

191.88


44.87

21.79

8i

81.21

191.51

ΔESRe

core ΔETe

ΔE(Q)e ΔEDBOCe

CCSD(T)g

HFf

CCSD(T)g ΔEGaunth


0.05

0.09

0.00


0.93

0.52


7.35

0.45

0.24


0.05

0.15

0.01


1.38

0.60


7.40

0.42

0.28

0.36


0.02


0.08

HFf

ΔEDCh


0.05 0.00

0.01


1.28

0.58


7.42


1.44

0.56


7.43


1.41

0.56

0.28 0.28

0.17

a Cardinal number of the basis set used. b Results obtained with aug-cc-pCVXZ basis sets. c Results obtained with aug-cc-pVXZ basis sets. d Results obtained with cc-pVXZ basis sets. e Results obtained with cc-pCVXZ basis sets. f Relativistic correction obtained at the HF level. g Relativistic correction due to the correlation treatment with the CCSD(T) method. h Results obtained with uncontracted aug-cc-pCVXZ basis sets. i Basis sets with cardinal numbers (X
1, X) were used to extrapolate the HF energies and correlation contributions to the basis set limit. The average of the E(X) = ECBS þ B 3 X
3, E(X) = ECBS þ B 3 X
3 þ C 3 X
4, and E(X) = ECBS þ B 3 (X þ 0.5)
4 extrapolations is shown for the correlation contributions.

The CCSDT(Q), CCSDTQ, and CCSDTQP calculations were carried out with the MRCC suite of quantum chemical programs81 interfaced to the CFOUR package.82 In the DKH calculations, the MOLPRO package83 was utilized; the four-component Dirac
Coulomb and Dirac
Coulomb
Gaunt calculations were performed with the DIRAC code.84 Calculations that involved basis sets with angular momentums k and l were performed with the PSI3 package.47 All other results were obtained with 82 CFOUR.

’ RESULTS AND DISCUSSION The Hartree
Fock Approximation. Table 3 summarizes the calculations performed during the course of this study. When considering the HF contributions, it can be seen that it reaches convergence, 81.21 kJ/mol, with the aug-cc-pCV5Z basis set. The (Q,5)- and (5,6)-based extrapolations, respectively, over- and underestimate the basis set limit, yielding 81.27 and 81.20 kJ/mol. Although the difference is small, the oscillating behavior is notable. Halkier and his co-workers51 also noted that the reliability of the HFextrapolation schemes is questionable, and the result obtained with the largest basis set might provide the best estimates for the basis set limit. Harding and associates37 also found that “it is better to use HF-SCF energies obtained in a quadruple- or quintuple-zeta basis set than extrapolated HF-SCF energies that include triple-ζ or smaller basis-set contributions”. However, for chlorine, it is clear that even the quadruple-ζ basis set results should be avoided in the extrapolation procedure when very high accuracy is sought. Therefore, no extrapolation is recommended for the HF energies, and the use of the largest basis set results is encouraged. This might be true for other second-row elements as well. Nevertheless, because the HF term is converged, the uncertainty that originates from the use of finite basis sets in the HF calculations is negligible in D0(Cl2).

Electron Correlation up to the CCSD(T) Level. When studying the convergence behavior of correlated calculations, it can be observed in Table 3 that the convergence of the nonextrapolated MP2, CCSD, and (T) terms is monotonic, and the perturbative triples correction converges considerably faster than the MP2 and CCSD terms. The quadruple-ζ (T) contribution differs from the septuple-ζ value by less than 1 kJ/mol, whereas the corresponding differences are close to 2 and 8 kJ/mol for CCSD and MP2, respectively. The fast convergence of the (T) contribution is retained for the extrapolated values as well. The convergence of the extrapolated CCSD part is not as smooth as that of the (T) contribution, but less troublesome than that for MP2. The extrapolated MP2 energies oscillate, although with decreasing amplitude. Consequently, the worst pathological behavior is expressed by the MP2 contribution with the (T,Q)-and (Q,5)based extrapolations, yielding values, 190.63 and 192.30 kJ/mol, below and above the best estimate, 191.51 kJ/mol, respectively, by 0.88 and 0.79 kJ/mol. Nevertheless, the extrapolation always improves on the pure basis set results. Similar conclusions can be drawn from the CCSD and (T) results. Due to the considerable savings in computational time, most of the composite methods describe the electron correlation in the core region at a less-advanced level than they do in the valence region, and it is assumed that the high-level all-electron correlation can be accurately obtained as the sum of the high-level valence and low-level core correlations. Of course, this distinction can be further justified by chemical reasoning; that is, the behavior of electrons is more vital in the valence region than in the relatively rigid, chemically less-relevant core region. Nevertheless, it was found in recent studies31,37 that the calculation of the core (core
core, core
valence) and valence (valence
valence) electron correlations at different levels can introduce an error of about 0.5 kJ/mol for the atomization energies of small first-row systems. Although it can be expected that the effects of the different treatments are amplified in second-row systems, it is 7768

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Table 4. Convergence of the Core and Valence Electron Correlations Calculated at Different Levels of Theory (All Values in kJ/ mol). The HF Results with the Corresponding Basis Sets Are Also Shown ΔEMP2

EHF a

X

b

CV

c

V

d

diff

CV

e

f

V

ΔECCSD g

h

e

core

diff

CV

V

f

ΔE(T) g

core

h

diff

CV

e

f

V

coreg

diffh

5

81.27

81.16

0.11

192.30

189.86

2.63


0.18


43.98


41.03


3.32

0.37

21.87

20.17

1.76


0.06

6

81.20

81.20

0.01

191.50

188.30

3.03

0.18


44.69


41.15


3.32


0.22

21.81

20.03

1.76

0.01

7 8

81.21 81.21

81.16 81.18

0.05 0.02

191.88 191.51

189.34 188.93

2.36 2.69

0.18
0.11


44.87
44.78i


41.59
41.50


3.29
3.28

0.02

21.79 21.76i

20.02 20.00

1.76 1.76

0.00

a Cardinal number of the basis set used. b Results extrapolated from the aug-cc-pCV(X
1,X)Z basis set energies. c Results extrapolated from the aug-ccpV(X
1,X)Z basis set energies; the corresponding aug-cc-pV(Xþd)Z basis sets were used for X = 5,6. d The difference between the CV and V results. e Results extrapolated from the aug-cc-pCV(X
1,X)Z basis set energies; all electrons are correlated. f Results extrapolated from the aug-cc-pV(X
1,X)Z basis set energies; core electrons are frozen. The corresponding aug-cc-pV(Xþd)Z basis sets were used for X = 5,6. g The difference of the all-electron and frozen-core energies extrapolated to the basis set limit using aug-cc-pCV(X
2,X
1)Z data. h The difference between the CV and the sum of the V and core results. i The sum of the V and core results.

also reasonable to think that this effect can be alleviated by using sufficiently large basis sets. Because the treatment of electron correlation beyond the MP2 level using octuple-ζ basis sets started to become very demanding, the additivity of the core and valence electron correlations calculated at different levels of theory was also investigated. The results are shown in Table 4. Because this also affects the calculation of the HF energy, the HF results with the corresponding basis sets are also shown in Table 4. We note here that valence quintuple-, and sextuple-ζ results were obtained with the aug-ccpV(Xþd)Z basis sets,85 which is developed for second-row atoms to correct the deficiencies of the standard aug-cc-pVXZ basis sets.86
91 Although the aug-cc-pV(7þd)Z and aug-ccpV(8þd)Z basis sets are not available, the effects of tight dfunctions on the EHF, ΔEMP2, ΔECCSD, and ΔE(T) contributions is not expected to be relevant beyond the sextuple-ζ level (see the Supporting Information). In Table 4, the core electron correlation was calculated as the difference between the all-electron and frozen-core results obtained with the aug-cc-pCVXZ family of basis sets. The valence electron correlation was obtained using the frozen-core approximation with the aug-cc-pVXZ basis set family, and it is also assumed that the all-electron correlation energy extrapolated from aug-cc-pCV(X,Xþ1)Z basis sets can be approximated as the sum of the extrapolated valence, and core electron correlations using the aug-cc-pV(X,Xþ1)Z and aug-ccpCV(X
1,X)Z basis sets, respectively. It can be seen that the deviation caused by the above approximation always decreases with increasing basis set size. It is also encouraging that this deviation converges rapidly for the ΔECCSD and ΔE(T) contributions. The ΔECCSD =
44.87 kJ/mol and ΔE(T) = 21.79 kJ/mol contributions calculated from all-electron aug-cc-pCV(6,7)Z extrapolation only slightly differ from the corresponding values,
44.88 and 21.78 kJ/mol, obtained as a sum of valence and core electron correlations extrapolated from aug-cc-pV(6,7)Z and aug-cc-pCV(5,6)Z basis sets, respectively. It is worth noting that the core ΔE(T) contribution series is already converged with the aug-cc-pCV(T,Q)Z extrapolation (see Table 4). Similarly, the core ΔECCSD contribution obtained from aug-cc-pCV(T, Q)Z extrapolation differs by only 0.04 kJ/mol from that calculated from aug-cc-pCV(6,7)Z extrapolation. The variations in the valence ΔECCSD and ΔE(T) contributions are considerably larger. For instance, the valence contribution for ΔECCSD and ΔE(T) changes by 0.47 and 0.17 kJ/mol when going from quintuple- to octuple-ζ basis set. The corresponding core variations are 10 times smaller, 0.04 kJ/mol and less than 0.01 kJ/mol, but the relative

errors are on the same scale. Nevertheless, these suggest that the separated treatment of the core and valence regions is a good approximation with practical benefits, and the accuracy of the CCSD and (T) parts of the protocol is determined to a large extent by the description of the valence region. It can be seen that the errors caused by the approximation in the CCSD and (T) terms are smaller than 0.02 kJ/mol. Therefore, on the basis of the above, it is possible to estimate the all-electron aug-cc-pCV(7,8)Z ΔECCSD and ΔE(T) contributions accurately as the sum of the already converged aug-cc-pCV(6,7)Z core and aug-cc-pV(7,8)Z valence values. Thus, our best estimate for the ΔECCSD and ΔE(T) contributions are, respectively,
44.78 and 21.76 kJ/mol. Beyond the CCSD(T) Approximation. Regarding the effects of higher-order terms, our observations are in line with those of Martin and associates,34 who thoroughly investigated the basis set convergence of post-CCSD contributions up to pentuple excitations. The use of double-ζ basis sets to assess the effects of the iterative T term is strongly discouraged; at least a triple-ζ basis set is required. Nevertheless, to reach the (0.1 kJ/mol accuracy range, extrapolation from triple- and quadruple-ζ basis set calculations is needed. The above observations are valid for the (Q) term as well, as was also pointed out in refs 34 and 37. To achieve an accuracy of (0.1 kJ/mol in the (Q) contribution, the exploitation of extrapolation and the use of a quadruple-ζ basis set are desirable. However, because the use of quadruple-ζ basis sets in CCSDT and CCSDT(Q) calculation for larger molecules is prohibitive, the extrapolation from (D,T) basis sets was also tested. Surprisingly, this worked out very well. The (D,T) extrapolation of the valence T contribution resulted in
1.87 kJ/mol, which matches the value obtained in the (5,6) extrapolation. A similar conclusion can be drawn from the valence (Q) results; the (D,T) extrapolation yielded 2.04 kJ/mol, which is (i) relatively close to the (Q,5)-extrapolated value (2.25 kJ/mol), and (ii) slightly better than the pure quadruple-ζ result (2.01 kJ/mol). Furthermore, the fortuitous cancellation of the demanding CCSDTQ and CCSDTQP terms can also be observed, which can further encourage the utilization of the CCSDT(Q) approximation in thermochemical applications, especially for larger systems where the calculations of higher-order terms are nontractable. The effects of the triple and quadruple excitations on the core correlation are remarkably different. Practically, the triple excitations have no impact on D0(Cl2), whereas the perturbative quadruple excitations contribute a significant amount to the dissociation energy, 0.15 kJ/mol with the cc-pCVTZ basis set, suggesting the possible importance of quadruple excitations in 7769

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The Journal of Physical Chemistry A the core region of second-row atoms and molecules. On the basis of the facts that (i) the cc-pV(D,T)Z basis set extrapolation worked well for the valence (Q) contribution and (ii) the ccpCVDZ and cc-pCVTZ basis sets are considerably larger than the corresponding valence basis sets, the cc-pCV(D,T)Z extrapolated contribution, 0.17 kJ/mol, is taken as the best estimate for the core ΔE(Q) contribution. It can be observed in Table 3 that each higher-order correction to D0(Cl2) is converged within 0.05 kJ/mol. The error in the Q and P corrections is probably less than 0.01 kJ/mol, and as previously mentioned, the two terms cancel each other. The error in the core T and (Q) terms altogether is expected to be less than 0.02 kJ/mol. Furthermore, it can be seen that D0(Cl2) is decreased and increased, respectively, by the valence T and (Q) contributions, and the errors in the T,
0.04 kJ/mol, and (Q), 0.05 kJ/mol, terms are about to cancel. On the basis of the above, it is fair to say that the uncertainty in D0(Cl2) due to the higherorder corrections is less than 0.05 kJ/mol. Beyond the Born
Oppenheimer and Nonrelativistic Approximations. The DBOC term converges rapidly, and its overall effect on D0(Cl2) is almost negligible, 0.01 kJ/mol. Unlike the DBOC term, the relativistic effects have a substantial impact on D0(Cl2). Scalar relativistic effects account for
0.85 kJ/mol with the sextuple-ζ basis set. It can be observed that the HF part of ΔESR is more sensitive to the quality of the basis set; the difference between the double- and sextuple-ζ results are 0.04 and 0.48 kJ/mol for the correlation and HF terms, respectively. It can also be seen that the quintuple-ζ basis set is needed to achieve convergence within 0.1 kJ/mol, and with the sextuple-ζ basis set, the error in ΔESR is less than 0.03 kJ/mol. We note here that it might be possible to achieve faster convergence using basis sets specifically recontracted for DKH calculations; however, this was not tested here. The effects of higher excitations on ΔESR was also studied using the CCSDT(Q)/cc-pCVDZ level of theory, and it was found that the neglect of higher excitation in the DKH treatment can introduce an error of about (0.01 kJ/mol in D0(Cl2). The HF part of ΔEDC is converged within 0.01 kJ/mol at the quintuple-ζ level; however, the correlation part of the contribution still contains an error of 0.06 kJ/mol with the quadruple-ζ basis set. Because the effect of higher-order excitations beyond the CCSD(T) level is expected to be smaller by an order of magnitude,
7.07 ( 0.10 kJ/mol can be estimated for ΔEDC. The Gaunt correction calculated with the HF method quickly converges with increasing basis set size; it is converged with the triple-ζ basis set, ΔEGaunt = 0.28 kJ/mol. It is clear that the introduction of electron correlation would have a considerable impact on ΔEGaunt; however, this effect is expected to be not larger than ΔEGaunt itself calculated at the HF level. Therefore, an error bar of (0.30 kJ/mol attached to ΔEGaunt seems a reasonable estimate. The neglect of the retardation term in the Breit operator and that of the quantum electrodynamic (QED) effects introduces additional uncertainties; however, the calculation of the corresponding contributions is severely limited due to the lack of implementations. Nevertheless, on the basis of previous investigations,92 the errors can be assessed. The correction due to the residual term in the Dirac
Coulomb
Breit Hamiltonian can be as large as the Gaunt correction; thus, the uncertainty should be increased by (0.30 kJ/mol. Similarly, the QED effects can contribute in the order of ΔEGaunt, and this further increases the error bar by (0.30 kJ/mol. Altogether, the error bar, which originates from the approximate relativistic treatment, is (1.00 kJ/mol for D0(Cl2).

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Table 5. D0(35Cl2) and the EHF Energy as Well as the ΔEMP2, ΔECCSD, ΔE(T), and ΔET Contributions Calculated Using Quintuple-, Sextuple-, Septuple-, and Octuple-ζ Basis Sets Xa

EHF

ΔEMP2 ΔECCSD ΔE(T)

ΔET

D0b

5

81.27 192.30
43.98

21.87


1.82

241.07

6

81.20 191.50
44.69

21.81


1.87

239.39

7

81.21 191.88
44.87

21.79


1.87c 239.58

8

81.21 191.51
44.78

d

experiment (ref 14)

d

21.76


1.87c 239.27 239.22

a

Basis sets with cardinal numbers (X
1, X) were used to extrapolate the HF energies and correlation contributions to the basis set limit. b Theoretical results also include experimental ZPE =
3.34 kJ/mol, ΔEDBOC = 0.01 kJ/mol, ΔEDC =
7.07 kJ/mol, ΔESR =
0.85 kJ/mol, ΔEGaunt = 0.28 kJ/mol, as well as higher-order contributions due to valence
valence electron correlation [ΔECCSDT(Q) = 2.25 kJ/mol, ΔECCSDTQ =
0.08 kJ/mol, and ΔECCSDTQP = 0.08 kJ/mol] and core
core and core
valence electron correlations [ΔECCSDT = 0.00 kJ/mol and ΔECCSDT(Q) = 0.17 kJ/mol]. c Results obtained from (5,6)-ζ extrapolation. d The core
core and core
valence electron correlations are estimated separately from the valence
valence contribution (see text).

Best Theoretical Estimate for D0(Cl2) and for ΔfH0°(Cl), as Well as the Most Troublesome Contributions. Table 5 shows

the theoretical estimates obtained in this study for D0(Cl2) and those nonrelativistic contributions that still vary while expanding the basis set from the quintuple-ζ to the octuple-ζ level. The first conclusion that can be drawn is that the (T) and T contributions hardly change and are practically converged and only affects the value of D0 slightly. It is important to notice that the errors are amplified with the quintuple-ζ basis, because all contributions are overestimated, and this causes an error of 1.85 kJ/mol relative to the best estimate. Most of this error comes from the MP2, 0.79 kJ/mol, and from the CCSD, 0.71 kJ/mol, terms. We note here that this overestimation accidentally does not appear in most model chemistries because (i) the effects of quadruple excitations on the core and valence electron correlations are underestimated by about 1.4 kJ/mol and (ii) the Gaunt correction (∼0.3 kJ/mol) and the second-order spin
orbit coupling are not considered. The situation is considerably better reaching the sextuple-ζ level; the cumulative error reduces to 0.17 kJ/mol. However, at the septuple-ζ level, the oscillating behavior of the MP2 and CCSD contributions causes a non-negligible increment in D0(Cl2). Nevertheless, when considering the error estimates given in the previous sections for the various contributions, a conservative error bar of (0.30 kJ/mol can be deduced for the nonrelativistic part of the calculated dissociation energy. A very similar error bar can be determined, if one assumes that the error predominantly originates from the error of the extrapolation, and 2 times the standard deviation of the basis set limit values yielded by the three extrapolation formulas is used as an error estimate. In this way, for (7,8) extrapolations, an error estimate of (0.26 kJ/mol can be obtained. Although a fairly tight error bar can be given for the nonrelativistic contributions, the error inherent in the relativistic treatment of D0(Cl2) ruins the quality of the theoretical estimate, increasing the total error bar to (1.30 kJ/mol. Thus, our best theoretical value for D0(Cl2) is 239.27 ( 1.30 kJ/mol, and consequently, 119.64 ( 0.65 kJ/mol can be derived for ΔfH0°(Cl). Both values are in excellent agreement with the corresponding most accurate experimental values, D0(Cl2) = 239.221 ( 0.001 kJ/mol14 and = 119.62 ( 0.01 kJ/mol.2 ΔfH°(Cl) 0 7770

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The Journal of Physical Chemistry A

’ CONCLUDING REMARKS This study presents the most detailed theoretical investigation for D0(Cl2) and, consequently, for ΔfH0°(Cl). It also reveals several issues that can be responsible for the inaccuracies inherent in the theoretical calculation of these quantities for systems containing second-row elements, that is, (i) the considerably large relativistic contributions carrying large uncertainties, which cannot be resolved due to the lack of available theoretical tools, (ii) the very slow basis set convergence of the MP2 correlation energy, (iii) the slow convergence of the CCSD contribution, and (iv) the relatively large basis set (quadruple-ζ quality) that is required for an accurate calculation of the (Q) contribution. It is probable that the same issues are responsible for the relatively large errors associated with the heats of formation of chlorine-containing species calculated by highaccuracy model chemistries. These issues also indicate that it is not yet possible to reach the range of sub-kJ/mol accuracy for species containing second-row elements. ’ ASSOCIATED CONTENT

bS

Supporting Information. Total energies and basis sets. This material is available free of charge via the Internet at http:// pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT Professor Lucas Visscher’s help with the Dirac code and his fruitful comments are gratefully acknowledged. We are also thankful to the reviewers for the constructive suggestions and comments. Financial support has been provided by the European Research Council (ERC) under the European Community’s Seventh Framework Programme (FP7/2007-2013), ERC Grant Agreement No. 200639, and by the Hungarian Scientific Research Fund (OTKA), Grant No. NF72194. ’ REFERENCES (1) Csontos, J.; Rolik, Z.; Das, S.; K
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