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Experimental Vibrational Zero-Point Energies: Diatomic Molecules

Karl K. Irikuraa… Physical and Chemical Properties Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8380

͑Received 23 August 2006; revised manuscript received 18 December 2006; accepted 22 December 2006; published online 18 April 2007͒

Vibrational zero-point energies ͑ZPEs͒, as determined from published spectroscopic constants, are derived for 85 diatomic molecules. Standard uncertainties are also pro- vided, including estimated contributions from bias as well as the statistical uncertainties propagated from those reported in the literature. This compilation will be helpful for validating theoretical procedures for predicting ZPEs, which is a necessary step in the ab initio prediction of molecular energetics. © 2007 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.. ͓DOI: 10.1063/1.2436891͔

Key words: molecular energetics; uncertainty; vibrational spectroscopy; zero-point energy.

CONTENTS about 10 every 10 years. With the advent of -set ex- trapolation methods and highly correlated theories, the ener- getics of small molecules can now be computed to high pre- 1. Introduction...... 389 cision almost routinely.1 As the uncertainties in electronic 2. Definition of Zero-Point Energy...... 389 energy fall away, other sources of uncertainty become in- 3. Selection of Molecules...... 390 creasingly noticeable. Vibrational zero-point energy ͑ZPE͒ 4. Contributions to Uncertainties...... 390 has emerged as one of the principal remaining sources of 5. Analyses for Individual Molecules...... 394 uncertainty in calculations of molecular energetics. In careful 6. Results and Discussion...... 396 work, it is necessary to go beyond the harmonic approxima- 7. Acknowledgments...... 396 tion to obtain a reliable ZPE.2–5 However, the most common 8. References...... 396 practice is simply to scale the computed harmonic ZPE by a multiplicative correction factor. The empirical scaling factor List of Tables carries uncertainty. We have recently quantified the uncer- tainties associated with scaling factors for fundamental vi- 1. Spectroscopic constants for selected diatomic brational frequencies.6 The uncertainties associated with the molecules...... 391 experimental vibrational frequencies were required in the 2. Derivatives of ZPE with respect to analysis, although in the end their contribution was small spectroscopic constants, c...... 392 enough to be neglected. Similar work is underway to provide 3. Estimated standard uncertainties associated scaling factors, with their associated uncertainties, appropri- −1 with some diatomic ZPEs ͑cm units͒ as ate for routine predictions of ZPE. Unfortunately, we have determined using the full ͓Eq. ͑6͔͒, diagonal been unable to locate any recent compilations of experimen- ͓Eq. ͑7͔͒, and pessimistic ͓Eq. ͑8͔͒ tally derived ZPEs, or compilations of any age that include approximations...... 393 uncertainties. Moreover, existing compilations include few 4. Truncation bias with respect to sextic diatomic data for polyatomic molecules. We are now working to fill vibrational model...... 394 this gap by providing benchmark, experimental ZPEs along 5. Experimental vibrational zero-point energies with the associated uncertainties. The present list, restricted −1 ͑cm ͒ for selected diatomic molecules...... 395 to diatomic molecules, is not intended to be exhaustive. Nev- ertheless, it is the largest such list yet assembled, the first to 1. Introduction employ spectroscopic data more recent than those compiled by Huber and Herzberg,7 and the first to include One of the most popular uses of computational uncertainties. models is to predict molecular energetics, which is required for applications such as thermochemistry and reac- tion kinetics. As a result of steady progress in electronic 2. Definition of Zero-Point Energy structure theory and computational efficiency, the precision of ab initio molecular energetics has improved by a factor of In , the Born-Oppenheimer approxima- tion ͑BOA͒ is almost8 always accepted. Thus, the most com-

a͒ mon definition of the molecular ZPE is the energy difference [email protected] © 2007 by the U.S. Secretary of Commerce on behalf of the United States. between the vibrational and the lowest point on All rights reserved.. the Born-Oppenheimer potential energy surface. Unfortu-

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͑ nately, this definition is not convenient for experimental paper we include Y00 and even the third-order term second ZPEs, as the BOA is never adopted by real molecules. Ex- anharmonicity constant͒ when available. Thus, ZPEs of di- perimental spectra are generally analyzed, however, as if the atomic molecules are taken to be BOA were followed by real molecules; explicit Born- ZPE = G͑0͒ = Y + 1 ␻ − 1 ␻ + 1 ␻ y , ͑4͒ Oppenheimer corrections are only made when simulta- 00 2 e 4 e e 8 e e Ϸ␻ neously fitting different isotopologs to the same effective po- where Y30 eye. To a good approximation, Y00 can be ex- tential. Thus, in the present compilation, the experimental pressed in terms of conventional spectroscopic constants as13 ZPE is defined as the difference between the molecular B ␣ ␻ ␣2␻2 ␻ x ground state and the lowest point on its isotope-specific ef- Ϸ e e e e e e e ͑ ͒ Y00 + + 3 − . 5 fective potential. The small difference between this definition 4 12Be 144Be 4 and that used by quantum chemists must be absorbed by the The Dunham constants that correspond to the conventional empirical scaling factor that is typically applied to theoreti- Ϸ Ϸ ␣ rotational constants here are Y01 Be and Y11 − e. The cally determined ZPEs. ͑ −1͒ value of Y00 is largest for H2 Y00=8.9 cm , smaller for The ZPE cannot be measured directly since no molecule ͑ −1͒ hydrides such as OH Y00=3.0 cm , and less than one can be observed below its ground state. Instead, the term wavenumber for nonhydrides such as CO ͑Y =0.2 cm−1͒. “experimental ZPE” describes a value that is usually ͑but not 00 necessarily9͒ derived by combining experimental spectro- scopic constants with standard theoretical or empirical mod- 3. Selection of Molecules els for anharmonic oscillators. Thus, “experimental” ZPE An initial list of molecules and associated ground-state values are actually hybrids of experiment and theoretical spectroscopic constants was culled from the classic compila- modeling. tion by Huber and Herzberg7 and from the NIST Diatomic Most available experimental ZPEs are for diatomic mol- Spectral Database.14 For compatibility with the NIST Com- ecules, because far fewer spectroscopic constants are needed putational Chemistry Comparison and Benchmark for diatomic molecules than for polyatomic molecules. Al- Database,15 only molecules composed of elements lighter though the field of molecular spectroscopy is home to than argon ͑i.e., atomic number ZϽ18͒ were included. More crowds of molecular constants, among nonspecialists the recent values of constants were taken from the spectroscopic most common expression for the vibrational energy levels of literature as available. Since our goal is a list of reliable a diatomic molecule, relative to the minimum on the poten- ZPEs, diatomic molecules were excluded if their ZPEs obvi- tial energy curve, is ously ͑upon cursory analysis͒ had standard uncertainties of −1 G͑v͒ = ␻ ͑v + 1 ͒ − ␻ x ͑v + 1 ͒2. ͑1͒ about 0.5 cm or more. Our final list of diatomic molecules, e 2 e e 2 with their spectroscopic constants, is provided in Table 1. ͑In ͑ ͒ ␻ ␻ In Eq. 1 , e and exe are the harmonic frequency and the Table 1, and throughout this paper, H refers specifically to first anharmonicity constant, respectively, and v is the vibra- protium and D to .͒ Standard uncertainties are tional quantum number, which can assume nonnegative inte- listed in the spectroscopic style. For example, the quantity 10 ␻ ͑ ͒ ger values. Note that the symbol exe represents a single 12.345±0.067 would be written 12.345 67 . When the constant, not a product. Thus, the most popular expression ground state is classically degenerate ͑usually 2⌸͒, averaged ͑ 1 ͒ constants are used, as reported in the experimental sources for diatomic ZPE is, to second order in v+ 2 , cited. This choice was made for convenient comparison with ͑ ͒ 1 ␻ 1 ␻ ͑ ͒ ZPE = G 0 = 2 e − 4 exe. 2 conventional, non-relativistic quantum chemistry calcula- + 2⌸ 1⌸ ͑ ͒ tions. For Cl2, only separate constants for 3/2 and 3/2 This expression is derived by extrapolating Eq. 1 to vi = 7 1 were reported. − 2 , which corresponds to the lowest point on the effective potential, to a good approximation.11 Contributions from higher-order anharmonicities are generally negligible ͑e.g., 4. Contributions to Uncertainties −1 −1 −1 0.1 cm for H2, 0.07 cm for OH, and 0.0013 cm for The spectroscopic constants have both statistical uncer- ͒ 2–5 CO . Unfortunately, as has been pointed out recently, the tainty ͑i.e., uncertainty from random effects͒ and bias ͑i.e., popular expression is incorrect. systematic error͒. As we are interested in the ZPE, it is nec- ͑ ͒ In addition to the linear and quadratic terms in Eq. 1 , essary to determine how the uncertainties in the constants there is a constant term that is usually overlooked. This was contribute to the uncertainty in the corresponding ZPE. For demonstrated by Dunham in his classic power-series 12 this purpose, we accept the common linearized propagation analysis. The resulting energy levels, to second order, are of uncertainties16 given by given by x ͒, ͑6͒ץ/yץ͑͒ xץ/yץ͑ x ͒2 +2͚ ␴ץ/yץ͑ ␴2 Ϸ ͚ ␴ ͑ ͒ ͑ 1 ͒ ͑ 1 ͒2 ͑ ͒ y ii i ij i j G v = Y00 + Y10 v + 2 + Y20 v + 2 , 3 i iϽj Ϸ␻ Ϸ ␻ ͑ ͒ ␴2 where Y10 e and Y20 − exe to good approximations. The for a quantity y= f x1 ,x2 ,...,xi ,... , where y is the esti- ͑ ␴ ͗͑ ͒͑ ͒͘ constant Y00 does not influence the line positions i.e., energy mated variance of y and ij= xi −¯xi xj −¯xj is the element intervals͒ in a spectrum but contributes to the ZPE. In this of the covariance matrix that corresponds to the pair of vari-

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TABLE 1. Spectroscopic constants for selected diatomic molecules. Standard uncertainties ͑1␴͒ are between parentheses and refer to the least significant digits. Values are in wavenumber units ͑cm−1͒.

␻ ␻ ␻ ␣ Molecule e exe eye Be e Reference

First-row elements only ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ H2 4401.213 18 121.336 18 0.8129 18 60.8530 18 3.0622 18 7 HD 3813.15͑18͒ 91.65͑9͒ 0.723͑9͒ 45.655͑9͒ 1.986͑9͒ 7 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ D2 3115.50 9 61.82 9 0.562 9 30.4436 18 1.0786 9 7 Second-row elements BeH 2061.235͑15͒ 37.327͑22͒ 0.084͑17͒ 10.31992͑5͒ 0.3084͑2͒ 22 BeD 1529.986͑11͒ 20.557͑12͒ 0.035͑7͒ 5.68830͑3͒ 0.1261͑1͒ 22 Be18O 1457.09͑22͒ 11.311͑74͒ 0.0143͑83͒ 1.5847͑5͒ 0.01784͑15͒ 33 BF 1402.15865͑26͒ 11.82106͑15͒ 0.051595͑35͒ 1.51674399͑21͒ 0.01904848͑22͒ 23 BH 2366.7296͑16͒ 49.33983͑99͒ 0.362͑10͒ 12.025755͑45͒ 0.421565͑22͒ 34 BO 1885.286͑41͒ 11.694͑11͒ −0.00952͑83͒ 1.781110͑31͒ 0.016516͑17͒ 35 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ C2 1855.0663 63 13.6007 54 −0.116 2 1.820053 11 0.0179143 44 36 − ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ C2 1781.189 18 11.6717 48 0.009981 28 1.74666 32 0.01651 46 37 CF 1307.93͑37͒ 11.08͑12͒ 0.093͑15͒ 1.41626͑48͒ 0.01844͑17͒ 38 CH 2860.7508͑98͒ 64.4387͑85͒ 0.3634͑27͒ 14.45988͑20͒ 0.53654͑33͒ 39 CD 2101.05193͑55͒ 34.72785͑58͒ 0.14147͑10͒ 7.8079823͑55͒ 0.212240͑11͒ 40 CN 2068.648͑11͒ 13.0971͑68͒ −0.0124͑17͒ 1.89978316͑67͒ 0.0173720͑12͒ 41 CO 2169.75589͑8͒ 13.28803͑2͒ 0.0104109͑14͒ 1.9316023͑7͒ 0.01750513͑14͒ 42 CO+ 2214.127͑35͒ 15.094͑21͒ −0.0117͑34͒ 1.976941͑39͒ 0.018943͑34͒ 43 and 44 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ F2 916.929 10 11.3221 10 −0.10572 67 0.889294 11 0.0125952 24 45 HF 4138.3850͑7͒ 89.9432͑7͒ 0.92449͑31͒ 20.953712͑2͒ 0.7933704͑65͒ 46 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ Li2 351.4066 10 2.58324 41 −0.00583 7 0.6725297 50 0.0070461 16 47 and 48 LiF 910.57272͑10͒ 8.207956͑46͒ 0.569166͑82͒ 1.34525715͑57͒ 0.02028749͑19͒ 49 LiH 1405.49805͑76͒ 23.1679͑7͒ 0.17093͑28͒ 7.5137315͑9͒ 0.2163911͑24͒ 50 LiD 1054.93973͑32͒ 13.05777͑21͒ 0.075478͑50͒ 4.23308131͑46͒ 0.09149428͑84͒ 50 LiO 814.62͑15͒ 7.78͑15͒ NA 1.21282948͑11͒ 0.0178990͑25͒ 27 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ N2 2358.57 9 14.324 9 −0.00226 9 1.998241 18 0.017318 9 7 + ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ 51 7 N2 2207.0115 60 16.0616 23 −0.04289 25 1.93176 9 0.01881 9 vib, rot NF 1141.37͑9͒ 8.99͑9͒ NA 1.205679͑53͒ 0.014889͑53͒ vib,7 rot52 NH 3282.72͑10͒ 79.04͑8͒ 0.367͑23͒ 16.66792͑6͒ 0.65038͑17͒ 53 ND 2399.126͑30͒ 42.106͑21͒ 0.1203͑54͒ 8.90867͑15͒ 0.25457͑19͒ 54 NO 1904.1346͑18͒ 14.08836͑89͒ 0.01005͑20͒ 1.7048885͑21͒ 0.0175416͑14͒ 55 NO+ 2376.72͑11͒ 16.255͑18͒ −0.01562͑92͒ 1.997195͑89͒ 0.018790͑69͒ 56 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ O2 1580.161 9 11.95127 9 0.0458489 9 1.44562 9 0.0159305 9 57 + ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ O2 1905.892 82 16.489 13 0.02057 90 1.689824 91 0.019363 37 58 FO 1053.0138͑12͒ 9.9194͑13͒ −0.06096͑59͒ 1.05870763͑81͒ 0.0132951͑23͒ 59 OH 3737.761͑18͒ 84.8813͑18͒ 0.5409͑18͒ 18.9108͑18͒ 0.7242͑9͒ 7 OD+ 2271.80͑9͒ 44.235͑9͒ 0.4267͑18͒ 8.9116͑18͒ 0.2896͑9͒ 7 Third-row elements AlCl 481.77466͑20͒ 2.101811͑88͒ 0.006638͑15͒ 0.243930066͑12͒ 0.001611082͑12͒ 60 AlF 802.32447͑11͒ 4.849915͑44͒ 0.0195738͑68͒ 0.552480208͑65͒ 0.004984261͑44͒ 23 AlH 1682.37474͑31͒ 29.05098͑29͒ 0.24762͑12͒ 6.3937842͑17͒ 0.1870527͑15͒ 61 AlD 1211.77402͑15͒ 15.06477͑11͒ 0.09244͑4͒ 3.3183929͑8͒ 0.0698773͑4͒ 61 AlO 979.4852͑50͒ 7.0121͑32͒ −0.00206͑56͒ 0.6413856͑54͒ 0.0057796͑8͒ 62 AlS 617.1169͑33͒ 3.3310͑19͒ −0.00924͑32͒ 0.2800368͑33͒ 0.00178225͑55͒ 63 BCl 840.29472͑63͒ 5.49170͑33͒ 0.02995͑7͒ 0.684282͑12͒ 0.0068124͑14͒ 17 BeS 997.94͑9͒ 6.137͑9͒ NA 0.79059͑9͒ 0.00664͑9͒ 7 BS 1179.91͑3͒ 6.25͑3͒ −0.0083͑58͒ 0.79478͑5͒ 0.00578͑4͒ 64 CCl 876.89749͑69͒ 5.44698͑54͒ 0.02607͑15͒ 0.697137͑34͒ 0.00685277͑45͒ 25 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ Cl2 559.751 20 2.69427 20 −0.003325 2 0.24415 20 0.001516 20 65 + ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ Cl2 645.61 9 3.015 9 0.007 9 0.26950 18 0.00164 9 7 ͑⍀=1/2͒ 644.77͑9͒ 2.988͑9͒ NA 0.2697͑9͒ 0.00167͑9͒ 7 ClF 783.4534͑24͒ 4.9487͑6͒ −0.0176͑1͒ 0.5164805͑31͒ 0.0043385͑8͒ 66 ClO 853.64268͑13͒ 5.51828͑6͒ −0.01256͑30͒ 0.62345797͑4͒ 0.0059357͑1͒ 28 CP 1239.79924͑8͒ 6.833769͑46͒ −0.001377͑7͒ 0.79886775͑8͒ 0.00596933͑19͒ 67 CS 1285.15464͑10͒ 6.502605͑53͒ 0.003887͑9͒ 0.82004356͑4͒ 0.00591835͑5͒ 68 HCl 2990.9248͑15͒ 52.8000͑15͒ 0.21803͑55͒ 10.5933002͑13͒ 0.3069985͑41͒ 69

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TABLE 1. Spectroscopic constants for selected diatomic molecules. Standard uncertainties ͑1␴͒ are between parentheses and refer to the least significant digits. Values are in wavenumber units ͑cm−1͒.—Continued

␻ ␻ ␻ ␣ Molecule e exe eye Be e Reference

DCl 2145.1326͑11͒ 27.1593͑7͒ 0.07993͑20͒ 5.4487838͑6͒ 0.1132345͑15͒ 69 HCl+ 2673.69͑9͒ 52.537͑9͒ NA 9.95661͑18͒ 0.32716͑18͒ 7 LiCl 642.95453͑93͒ 4.47253͑40͒ 0.020118͑49͒ 0.70652247͑20͒ 0.00801019͑32͒ 70 NaLi 256.5412͑19͒ 1.62271͑96͒ −0.00495͑22͒ 0.3758620͑89͒ 0.0031465͑15͒ 30 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ Mg2 51.121 18 1.645 9 0.01624 18 0.09287 9 0.003776 18 7 MgH 1492.7763͑36͒ 29.847͑4͒ −0.3048͑20͒ 5.8255229͑41͒ 0.177298͑14͒ 71 MgD 1077.2976͑26͒ 15.521͑2͒ −0.1184͑7͒ 3.0343436͑21͒ 0.066607͑5͒ 71 MgO 785.2183͑6͒ 5.1327͑3͒ 0.01649͑7͒ 0.5748414͑3͒ 0.0053223͑3͒ 72 MgS 528.74͑9͒ 2.704͑9͒ NA 0.26797͑9͒ 0.00176͑9͒ 7 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ Na2 159.08548 44 0.70866 29 −0.004632 77 0.15473537 29 0.00086375 11 29 NaCl 364.6842͑4͒ 1.7761͑2͒ 0.005937͑35͒ 0.21806302͑8͒ 0.00162479͑6͒ 73 NaF 535.65805͑21͒ 3.57523͑13͒ 0.018453͑34͒ 0.43690153͑7͒ 0.00455918͑7͒ 74 NaH 1171.968͑12͒ 19.703͑10͒ 0.175͑2͒ 4.90327͑13͒ 0.1370͑4͒ 75 NCl 827.95767͑75͒ 5.30015͑61͒ −0.00480͑19͒ 0.649767390͑85͒ 0.00641432͑20͒ 76 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ P2 780.77 9 2.835 18 −0.00462 9 0.30362 9 0.00149 9 7 + ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ P2 672.20 9 2.74 9 NA 0.27600 18 0.00151 9 7 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ 77 PF 846.75 9 4.489 9 0.019 9 0.5667427 35 0.00456 9 7butBe PH 2363.774͑36͒ 43.907͑27͒ 0.1059͑73͒ 8.53904͑19͒ 0.25339͑28͒ 54 PN 1336.948͑20͒ 6.8958͑57͒ −0.00605͑48͒ 0.7864844͑28͒ 0.0055337͑36͒ 78 PO 1233.34͑9͒ 6.56͑9͒ NA 0.733223657͑22͒ 0.005466162͑50͒ vib,7 rot79 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ S2 725.7102 97 2.8582 25 NA 0.29539516 30 0.00159754 59 80 SF 837.6418͑5͒ 4.46953͑18͒ NA 0.555173͑4͒ 0.004459͑10͒ 81 SH 2696.2475͑58͒ 48.7420͑28͒ 0.1124͑6͒ 9.600247͑51͒ 0.27990͑10͒ 82 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ Si2 510.98 9 2.02 9 NA 0.2390 9 0.00135 18 7 SiCl 535.59͑2͒ 2.1757͑50͒ 0.00604͑36͒ 0.256103͑14͒ 0.0015817͑72͒ 83 SiF 837.32507͑22͒ 4.83419͑9͒ 0.019807͑16͒ 0.58125735͑21͒ 0.00503859͑39͒ 84 SiH 2042.5229͑8͒ 36.0552͑5͒ 0.1254͑1͒ 7.503898͑30͒ 0.21814͑2͒ 85 SiH+ 2157.17͑9͒ 34.24͑9͒ NA 7.6603͑9͒ 0.2096͑9͒ 7 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ␻ 87 SiN 1151.284 43 6.455 21 −0.0069 20 0.730927 15 0.005685 30 86 but eye SiO 1241.54388͑7͒ 5.97437͑2͒ 0.006090͑3͒ 0.72675206͑2͒ 0.00503784͑1͒ 88 SiS 749.64559͑7͒ 2.58623͑4͒ 0.001048͑9͒ 0.303527856͑6͒ 0.001473130͑3͒ 88 SO 1150.7913͑10͒ 6.4096͑5͒ 0.01306͑11͒ 0.72082210͑2͒ 0.00575080͑2͒ vib,89 rot90

ables xi and xj. For the problem at hand, y is the ZPE and the appear to have been compiled previously. ␴ xi are generic spectroscopic constants. The required deriva- Typically, only the diagonal elements, ii, of the covari- tives, summarized in Table 2, are straightforward but do not ance matrix ͑i.e., the variances of the fitting constants͒ are reported and available to us. Omitting the off-diagonal con-

TABLE 2. Derivatives of ZPE with respect to spectroscopic constants, c. tributions in Eq. ͑6͒ yields the estimate

͒ ͑ 2͒ ץ ץc ␴2 Ϸ ␴2͑ץ/ZPE͒͑ץ c y ͚ i y/ xi , 7 i ␻ ␻ x ␻ y B ␣ ␻ ␣2␻2 ZPE = e − e e + e e + e + e e + e e 3 ␴ ͱ␴ 2 2 8 4 12Be 144Be where the standard uncertainties of the xi are i = ii. This diagonal approximation introduces a bias in the estimated uncertainty, which will be different for different molecules. B ␣ ␻ 1/2 + e s͑2s +1͒, where s ϵ e e To estimate the magnitude of the bias, we consider the situ- ␻ ␻ B2 e e 12 e ation for BCl, for which the covariance matrix has been published.17 Using the derivatives in Table 2 and the param- ␻ exe −1/2 eter values and statistical uncertainties from Table 1, the ␻ eye 1/8 standard uncertainty ͑i.e., the square root of the variance͒ for ͑ ͒ Be 1/4−s 3s+1 the ZPE is 0.000 65 cm−1. When only the diagonal terms are B included, the resulting standard uncertainty is only slightly e s͑2s +1͒ ␣ ␣ −1 e e smaller, 0.000 57 cm . Given only the variances, the most ͑ pessimistic scenario for the correlation coefficients rij −1 Յ ϵ␴ ␴ ␴ Յ ͒ rij ij/ i j 1 leads to the upper bound

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TABLE 3. Estimated standard uncertainties associated with some diatomic low-order constants are affected when higher-order constants ͑ −1 ͒ ͓ ͑ ͔͒ ͓ ͑ ͔͒ ZPEs cm units as determined using the full Eq. 6 , diagonal Eq. 7 , are included in the fitting procedure. Further, our ZPE com- and pessimistic ͓Eq. ͑8͔͒ approximations. putation ͓Eq. ͑4͔͒ includes terms only through third order. Molecule Full Diagonala Pessimistic For diatomic molecules, we estimate the uncertainty due to truncation using the simplified analytical model described in BCl 0.00065 0.00057 ͑−12%͒ 0.00098 ͑+51%͒ the following paragraph. ͑ ͒ ͑ ͒ CS 0.00044 0.00045 +2% 0.00050 +14% Assume, for this approximate model, that the diatomic vi- ͑ ͒ ͑ ͒ SiO 0.00033 0.00052 +58% 0.00082 +148% brational energy levels are perfectly described by the sextic SiS 0.00036 0.00037 ͑+3%͒ 0.00041 ͑+14%͒ polynomial aPercentage deviation from “full” value given between parentheses. 6 ͑ ͒ ͑ 1 ͒i ͑ ͒ G v = ͚ bi v + 2 , 9 i=0 ͒ ͑ ͉ ץ ץ͉ ␴ Յ ␴ y ͚ i y/ xi , 8 i where v is the quantum number, G͑v͒ is the energy of the associated level, and the b are constants. The exact zero- or ␴ Յ0.000 98 cm−1 for the BCl example. This is 50% i y point energy within this model is larger than the result from Eq. ͑6͒ and appears inferior to the diagonal approximation. These various estimates are summa- 1 1 1 1 1 1 ͑ ͒ ZPE = b0 + 2 b1 + 4 b2 + 8 b3 + 16 b4 + 32 b5 + 64 b6. 10 rized in Table 3, along with analogous estimates for three ͑ ͒ ͑ ͒ other diatomic molecules, based upon unpublished fitting Also suppose that n transition frequencies, yj =G j −G 0 , data generously provided by Dr. F. J. Lovas. Based upon the are observed and are fitted to the empirical expression data for these four molecules, the diagonal approximation n appears reasonable and in this paper we use Eq. ͑7͒ to esti- ͑ ͒Ϸ ͑ 1 ͒i ͑ ͒ G v ͚ ai v + 2 , 11 mate the statistical contribution to the standard uncertainty of i=0 the diatomic ZPEs. where a =Y or the equivalent conventional constant and n Most of the standard uncertainties presented in Table 1 are i i0 Յ6. This is an exact fit because we have assumed, for math- from the experimental papers in which the constants were ematical convenience, that the number of free parameters is reported. In some cases, uncertainties were reported but not equal to the number of data. Then the apparent zero-point described; they have been assumed to represent standard un- energy is certainties ͑1␴͒. When uncertainties have been described as 1 1 1 ͑ ͒ 95% confidence intervals, they have been divided by 2 to ZPEapp = a0 + 2 a1 + 4 a2 + 8 a3. 12 estimate standard uncertainties. Other special cases are de- scribed in Sec. 5. ZPEapp depends upon the number, n, of fitted constants even ͑ ͒ The uncertainties associated with spectroscopic constants though the higher-order constants are not explicit in Eq. 12 . ͑ ͒ ͑ ͒ only reflect the statistical uncertainties resulting from fitting The difference ZPEapp−ZPE , determined from Eqs. 10 ͑ ͒ the observed line positions to a model Hamiltonian. In this and 12 , is the truncation bias in this simple model; its ab- context, a Hamiltonian is a physically motivated fitting func- solute value is an estimate for the uncertainty arising from tion involving selected spectroscopic constants and various truncation of the spectroscopic Hamiltonian. The biases in ͑ ͒ quantum numbers. In some cases, the uncertainties were the cubic approximation 12 , for different orders n of the ͑ ͒ propagated from v-dependent fitting constants or from esti- fitting polynomial 11 , are listed in the second column of mated ͓type B ͑Ref. 18͔͒ uncertainties in line positions. Table 4. The value for n=2 uses the more severely truncated 1 1 In addition to the statistical uncertainties, there are sources quadratic expression ZPEapp=a0 + 2 a1 + 4 a2 instead of Eq. ͑ ͒ ͑ ͒ of bias ͑“systematic error”͒ which may dominate the com- 12 , because a3 is undefined. The differences a0 −b0 , bined uncertainties. True bias cannot be known, since that which appear in the expressions for estimated bias, require requires knowledge of true values. When ancillary informa- evaluation. If we identify a0 and b0 with Y00, we can use Eq. tion is available about the possible values of bias, we can ͑5͒ to estimate that ͒ ͑ make a correction for bias, as recommended in the Guide to ͑ ͒Ϸ١ a0 − b0 Y00 · a − b the Expression of Uncertainty in Measurement, published by 19 ␣ ␣ 1 the International Organization for Standardization. There is e ͩ e ␻ ͪ͑ ͒ ͑ ͒ = 1+ 2 e a1 − b1 − a2 − b2 . uncertainty associated with the correction for bias, corre- 12Be 6Be 4 sponding to the uncertainty of the ancillary information. If ͑13͒ we have no information, we choose the value of the correc- ͑ ͒ ͑ ͒ tion to be zero, but there is still an uncertainty associated The expressions for a1 −b1 and for a2 −b2 are given in the with this ͑null͒ value. This uncertainty is combined with the last two columns of Table 4. To obtain numerical values for statistical uncertainties to obtain the combined uncertainty. the uncertainty arising from truncation for a particular di- One source of bias derives from the truncation in Taylor- atomic molecule, we determine n based upon the order of the ͑ ͒ Ϸ ͑ ͒ series expansions such as Eq. 1 . Such model-dependent experimental fit, choose bi Yi0, and combine Eq. 13 with uncertainties are seldom discussed.20–22 The fitted values of the expressions in Table 4. However, frequently the higher

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TABLE 4. Truncation bias with respect to sextic diatomic vibrational model ͓Eqs. ͑9͒–͑13͔͒. The order of the fitting polynomial is n.

͑ ͒͑͒ n ZPEapp −ZPE a1 −b1 a2 −b2

60 0 0

1 1 679 4881 12139 ͑a − b ͒ − b − b + b b − b 5 0 0 16 4 32 5 4 6 8 6 16 6

1 485 6005 1689 475 22061 ͑a − b ͒ − b − b − b − b − 1290b b + b 4 0 0 16 4 16 5 16 6 16 5 6 4 5 16 6

105 105 8925 2711 1765 43 11909 ͑a − b ͒ + b + b + b 22b + b + b − b − 150b − b 3 0 0 16 4 2 5 32 6 4 16 5 2 6 2 4 5 16 6

15 135 435 2475 23 1199 9 29 165 1771 ͑a − b ͒ − b − b − b − b − b −24b − b −210b b + b + b + b 2 0 0 8 3 16 4 16 5 32 6 4 3 4 16 5 6 2 3 2 4 4 5 16 6

͉ ͉ Ͻ͉ ͉ ͑ ͒ Yi0 are unknown. As typically Yi+1,0 Yi0 , we estimate the =0.967 50 Table 2 . The standard uncertainties for the spec- missing Yi0 crudely by geometric extrapolation from the two troscopic constants, from Table 1, are combined with these ͑ ͒ highest measured Yi0. As pointed out by a referee, Yi0 and derivatives according to Eq. 7 to obtain the estimated sta- Yi+1,0 usually differ in sign. Thus, an alternating sign is as- tistical contribution to the variance of the ZPE. Taking the Ϸ −1 sumed in the present work. However, for about one-third of square root gives ustat 0.000 159 cm , as listed in Table 5 Ͼ ͑ ͒ the data at hand, the sign does not change, i.e., Yi0Yi−1,0 0 rounded to two digits . To estimate the uncertainty associ- ͑for iϾ3͒. To test the sensitivity of the results to the choice ated with the ͑null correction for͒ truncation bias, check the of sign, we consider a positive sign in the extrapolation. This literature ͑Zhang et al.23͒ to find the available Dunham con- ͑ ͒ −1 changes the magnitude of ZPEapp−ZPE by a mean factor of stants Yn0. In this case, Y40=0.000 346 4 cm and no higher 1.8 ͑standard deviationϭ1.5͒. Thus, we multiply the trunca- constants are available. Thus, the appropriate row in Table 4 tion bias values by 1.8 to reflect the uncertainty of the ex- is that for n=4. Values of bj in Table 4 are approximated as Ϸ trapolation. bj Y j0. The missing constants are estimated by geometric Among the remaining sources of uncertainty, the most im- extrapolation with sign alternation as Yi+1,0 Ϸ ͉ 2 ͉͑ ͉ ͉͒ Ϸ ϫ −6 −1 portant is probably that Dunham’s canonical treatment is it- − Yi,0/Yi−1,0 Yi,0/ Yi,0 ,soY50 −2.33 10 cm and Ϸ ϫ −8 −1 self an approximation that leads to bias. In particular, any Y60 1.56 10 cm . Then from the third column of Table ͑ ͒ −1 perturbations from electronically or vibrationally excited 4, a1 −b1 =0.000 225 cm and from the fourth column of ͑ ͒ −1 states will displace some rovibrational levels, skewing the Table 4, a2 −b2 =−0.000 255 cm . Substituting these val- ͑ ͒ ͑ ͒ ϫ −5 −1 values of the fitting constants. These effects are specific to ues into Eq. 13 then gives a0 −b0 =6.435 10 cm . Fi- individual molecules. We do not attempt to quantify the as- nally the expression in the second column of Table 4 evalu- ͑ ͒ −1 sociated uncertainties. However, they are probably small for ates to ZPEapp−ZPE =0.000 107 cm . Multiplying by 1.8 the ground states of diatomic molecules. Dunham’s analysis to account for sensitivity to guessed Yi0 values, and taking 1 Ϸ −1 is also for non-degenerate electronic states ͑ ⌺͒ and must be the absolute value, utrunc 0.000 193 cm . The final, esti- ␴ ͑ 2 considered more approximate in other cases. mated, combined standard uncertainty is then = ustat 2 ͒1/2 −1 To illustrate the current procedure, consider BF as an ex- +utrunc =0.000 250 cm , which is rounded to a single ample. Constants from Table 1 are substituted into Eq. ͑5͒ to digit in Table 5. −1 ͑ ͒ obtain Y00=0.3111 cm . Applying Eq. 4 yields ZPE −1 =698.4416 cm , as listed in Table 5. The associated uncer- 5. Analyses for Individual Molecules tainty is computed by propagating the uncertainties in the spectroscopic constants, estimating the uncertainty associ- In their classic compilation, Huber and Herzberg ͑H&H͒ ated with the ͑null͒ correction for truncation bias, and com- did not attempt to estimate the uncertainties associated with bining these two quantities to obtain a combined standard the reported spectroscopic constants.7 Instead, they provided uncertainty. To propagate uncertainties, substitute values the following guidance. “…we hope that the number of dig- from Table 1 into the formulas in Table 2 to obtain the partial its quoted may serve as a very rough indication of the esti- ͒ ␻͑ץ ͒ ͑ץ ͒ ␻͑ץ ͒ ͑ץ derivatives ZPE / e =0.503 07, ZPE / exe =−0.5, mated order of magnitude of the error, generally ±9 units of ͒ ͑ץ ͒ ͑ץ ͒ ␻͑ץ ͒ ͑ץ ZPE / eye =0.125, ZPE / Be =−3.5257, and the last decimal place. Where the last digit is given as a ͒ ␣͑ץ ͒ ͑ץ ZPE / e =226.11. The intermediate quantity s subscript, we expect that the uncertainty may considerably

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TABLE 5. Experimental vibrational zero-point energies ͑cm−1͒ for selected TABLE 5. Experimental vibrational zero-point energies ͑cm−1͒ for selected diatomic molecules. Combined standard uncertainties ͑1␴͒ are between pa- diatomic molecules. Combined standard uncertainties ͑1␴͒ are between pa- rentheses and refer to the least significant digit. The combined uncertainties rentheses and refer to the least significant digit. The combined uncertainties ͑ ͒ ͑ ͒ include both the estimated statistical uncertainties ustat and the estimated include both the estimated statistical uncertainties ustat and the estimated ͑ ͒ ͑ ͒ uncertainties arising from truncation bias utrunc . uncertainties arising from truncation bias utrunc .—Continued

Molecule State designation ZPE ustat utrunc Molecule State designation ZPE ustat utrunc

1⌺+ ͑ ͒ 35 1⌺+ ͑ ͒ H2 g 2179.3 1 0.022 0.11 D Cl 1066.607 7 0.00067 0.0065 HD 1⌺+ 1890.3͑2͒ 0.14 0.13 H35Cl+ 2⌸ 1326.͑5͒ 0.047 5.2 1⌺+ ͑ ͒ 7 35 1⌺+ ͑ ͒ D2 g 1546.50 8 0.065 0.049 Li Cl 320.550 2 0.00051 0.0019 9BeH 2⌺+ 1022.23͑1͒ 0.015 0 23Na7Li 1⌺+ 127.817͑1͒ 0.0011 0 9 2⌺+ ͑ ͒ 24 1⌺+ ͑ ͒ BeD 760.372 9 0.009 0 Mg2 g 25.26 1 0.011 0.0013 9Be18O 1⌺+ 725.8͑1͒ 0.12 0.00039 24MgH 2⌺+ 739.1͑9͒ 0.0028 0.89 11B19F 1⌺+ 698.4416͑3͒ 0.00016 0.00019 24MgD 2⌺+ 534.9͑1͒ 0.0017 0.14 11BH 1⌺+ 1172.64͑5͒ 0.0018 0.055 24Mg16O 1⌺+ 391.433͑1͒ 0.00035 0.0011 11B16O 2⌺+ 939.89͑2͒ 0.022 0.00017 24Mg32S 1⌺+ 263.7͑1͒ 0.065 0.073 12 1⌺+ ͑ ͒ 23 1⌺+ ͑ ͒ C2 g 924.0 5 0.0043 0.52 Na2 g 79.3359 3 0.00026 0 12 −2⌺+ ͑ ͒ 23 35 1⌺+ ͑ ͒ C2 g 887.7 1 0.10 0.00018 Na Cl 181.9709 2 0.00022 0.0000013 12C19F 2⌸ 651.6͑2͒ 0.20 0.0011 23Na19F 1⌺+ 267.1154͑1͒ 0.00013 0.000018 12CH 2⌸ 1416.07͑4͒ 0.014 0.033 23NaH 1⌺+ 581.63͑2͒ 0.017 0.014 12CD 2⌸ 1042.792͑9͒ 0.00068 0.0091 14N35Cl 3⌺− 412.886͑2͒ 0.00049 0.0021 12 14 2⌺+ ͑ ͒ 31 1⌺+ ͑ ͒ C N 1031.133 7 0.0065 0.00089 P2 g 389.70 8 0.075 0.00016 12 16 1⌺+ ͑ ͒ 31 +2⌸ ͑ ͒ C O 1081.74682 5 0.000054 0 P2 u 335.4 1 0.087 0.060 12C16O+ 2⌺+ 1103.36͑2͒ 0.022 0.00019 31P19F 3⌺− 422.41͑6͒ 0.057 0.0017 19 1⌺+ ͑ ͒ 31 3⌺− ͑ ͒ F2 g 455.41 2 0.0051 0.020 PH 1171.9 4 0.028 0.40 H19F 1⌺+ 2050.77͑1͒ 0.00055 0.010 31P14N 1⌺+ 666.79͑1͒ 0.011 0.00011 7 1⌺+ ͑ ͒ 31 16 2⌸ ͑ ͒ Li2 g 175.0259 6 0.00056 0.00021 P O 615.1 2 0.064 0.19 7 19 1⌺+ ͑ ͒ 32 3⌺− ͑ ͒ Li F 453.70762 7 0.000063 0.000017 S2 g 362.19 6 0.0050 0.060 7LiH 1⌺+ 697.952͑5͒ 0.00053 0.0050 32S19F 2⌸ 417.9͑1͒ 0.0038 0.13 7LiD 1⌺+ 524.762͑1͒ 0.00019 0.0012 32SH 2⌸ 1337.2͑2͒ 0.0064 0.19 7 16 2⌸ ͑ ͒ 28 3⌺− ͑ ͒ Li O 405.6 4 0.11 0.39 Si2 g 255.0 1 0.12 0.043 14 1⌺+ ͑ ͒ 28 35 2⌸ ͑ ͒ N2 g 1175.78 5 0.045 0.00079 Si Cl 267.34 1 0.011 0.00035 14 +2⌺+ ͑ ͒ 28 19 2⌸ ͑ ͒ N2 g 1099.40 2 0.025 0.0024 Si F 417.6275 2 0.00019 0.000021 14N19F 3⌺− 568.8͑4͒ 0.065 0.37 28SiH 2⌸ 1013.336͑9͒ 0.0012 0.0092 14NH 3⌺− 1623.6͑6͒ 0.065 0.55 28SiH+1⌺+ 1071.͑3͒ 0.080 2.8 14ND 3⌺− 1190.13͑5͒ 0.021 0.046 28Si14N 2⌺+ 574.10͑3͒ 0.027 0.00016 14N16O 2⌸ 948.647͑1͒ 0.0011 0 28Si16O 1⌺+ 619.39217͑4͒ 0.000036 0.00000090 14N16O+ 1⌺+ 1184.33͑6͒ 0.059 0.00032 28Si32S 1⌺+ 374.21174͑4͒ 0.000038 0.0000051 16 3⌺− ͑ ͒ 32 16 3⌺− ͑ ͒ O2 g 787.380 6 0.0045 0.0040 S O 573.9499 6 0.00057 0.00019 16 +2⌸ ͑ ͒ O2 g 948.91 4 0.043 0.0016 19F16O 2⌸ 524.053͑1͒ 0.0011 0 16OH 2⌸ 1850.69͑5͒ 0.035 0.042 16OD+3⌺− 1126.5͑1͒ 0.065 0.083 exceed ±10 units of that last decimal place.” Although rarely 27Al35Cl 1⌺+ 240.4516͑1͒ 0.00011 0.0000042 explicit, the implicit assumption of normal distributions per- 27Al19F 1⌺+ 400.13958͑6͒ 0.000062 0.0000024 vades the literature of molecular spectroscopy, with coverage 18,19 27AlH 1⌺+ 835.024͑8͒ 0.00023 0.0081 factors ͑i.e., multipliers of the standard uncertainty͒ rang- 27AlD 1⌺+ 602.685͑1͒ 0.00010 0.0011 ing from 1 to 3. To accommodate the “very rough” quality of 27Al16O 2⌺+ 487.976͑3͒ 0.0030 0.000013 the estimated uncertainties, we assume a coverage factor of 1 27Al32S 2⌺+ 307.672͑2͒ 0.0019 0.00054 for the uncertainty estimates from H&H. Where we have 11 35 1 + B Cl ⌺ 418.984͑3͒ 0.00057 0.0034 taken spectroscopic constants from H&H, we have thus as- 9 32 1⌺+ ͑ ͒ Be S 497.4 2 0.052 0.20 sumed the associated standard uncertainties to be 9 in the last 11B32S 2⌺+ 588.39͑3͒ 0.025 0.00024 digit except when printed as a subscript, in which cases we 12C35Cl 2⌸ 437.3613͑6͒ 0.00048 0.00035 35 1⌺+ ͑ ͒ assumed 18 in that last, subscripted digit. Cl2 g 279.22 2 0.016 0 35 +2⌸ ͑ ͒ A few spectroscopic constants were reported without as- Cl2 3/2g 322.09 8 0.077 0.00034 35 +2⌸ ͑ ͒ sociated uncertainties. In such cases we estimated the uncer- Cl2 1/2g 321.7 1 0.077 0.074 35 19 1 + 18 ͑␻ ͒ ͑␻ Cl F ⌺ 390.510͑2͒ 0.0013 0.0013 tainties. This was done for Be O e ,BH eye, taken 35Cl16O 2⌸ 425.6295͑6͒ 0.000090 0.00061 equal to that estimated for the corresponding constant in the 12 31 2⌺+ ͑ ͒ 24͒ ͑ ͒ ͓ ͑ ͒ C P 618.20033 9 0.000086 0.0000060 A state ,O2 all constants ,BS all deperturbed constants, 12C32S 1⌺+ 641.03295͑6͒ 0.000060 0.0000010 ͔ ͑ estimated from effect of deperturbation , CCl Be, based 1 + H35Cl ⌺ 1483.89͑2͒ 0.0011 0.023 upon the discrepancy between Jin et al.25 and Endo et al.26͒, ͑ ͒ ͑␻ ͒ Cl2 all constants , and ClO eye . For LiO, we derived a

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␣ 27 value for e from published values of B0 and B1. In some atomic molecules. The reference data were not reported but ͑ ͒ reports of isotopic studies, uncertainties u Yl,m are reported literature references were provided. The earlier, classic paper only for the Dunham constants of the most common isoto- by Grev et al.32 included 12 diatomic and 12 polyatomic polog. In such cases, uncertainties for the Dunham constants molecules. Reference data and citations were both provided. of the rarer isotopologs are estimated by using Eq. ͑14͒, The present work is the first to include uncertainty analyses where ␮ is the reduced nuclear mass and the primed quanti- and to include copious, recent spectroscopic data. ties refer to the rarer isotopolog:12 The principal utility of the present compilation is for de- veloping empirical corrections for ZPEs obtained from ab u͑YЈ ͒Ϸu͑Y ͒͑␮/␮Ј͒m+l/2. ͑14͒ l,m l,m initio calculations. However, only stretching vibrations are For ClO, there is an apparent transcription error; Y11 represented by diatomic molecules, so reference ZPEs for ͑Ϸ ␣ ͒ 28 − e should be negative. For BeO, note that the tabu- polyatomic molecules are still needed. Data evaluation and lated values are for the rare isotopolog 9Be18O. uncertainty analysis are substantially more complicated for polyatomic molecules than for diatomic. For example, vibra- ͑ ͒ 6. Results and Discussion tional resonances i.e., perturbations are common, the ana- log of Y00 is not readily determined, and far more spectro- Table 5 lists our recommended values for ZPE as derived scopic constants are required for characterization. from experimental spectroscopic constants. The uncertainties Furthermore, adequate spectroscopic data are available for listed are combined standard uncertainties ͑i.e., coverage fac- fewer than 20 polyatomic molecules. Despite these chal- ͒ ͑ ͒ lenges, such an evaluation is currently underway at NIST. tor k=1 that include both statistical uncertainties ustat and ͑ ͒ ͑ ͒ uncertainties arising from null corrections for bias utrunc . Ͼ For nearly half the molecules, utrunc ustat. As expected, 7. Acknowledgments ͑ cases truncated most severely, at n=2 i.e., for which Y20 is ͒ the highest Yn0 measured , tend to have high values of the The author thanks Dr. Frank J. Lovas for sharing detailed ͑ ϭ ϭ ratio utrunc/ustat mean 18 and standard deviation 32, for results from spectroscopic analyses and for reviewing this 12 values͒. In contrast, this mean ratio equals 3 ͑for 26 val- manuscript, and Dr. Russell D. Johnson III and Dr. Raghu N. ues; standard deviationϭ6͒ and 4 ͑for 28 values; standard Kacker for many helpful discussions and suggestions. deviationϭ7͒ for n=3 and 4, respectively. Surprisingly, the highest mean ratio ͑50, for 12 values; standard deviation 8. References ϭ ͒ 93 is for the most lightly truncated nontrivial case, n=5. 1 T. H. Dunning, Jr., J. Phys. Chem. A 104, 9062 ͑2000͒. Upon inspection, we find that the highest values of the ratio 2 V. Barone, J. Chem. Phys. 120, 3059 ͑2004͒. 3 utrunc/ustat are associated with low values of the ratio M. S. Schuurman, S. R. Muir, W. D. Allen, and H. F. Schaefer III, J. Chem. Phys. 120, 11586 ͑2004͒. Yn−1,0/Yn,0. That is, truncation bias is expected to be large 4 ͓ ͑ ͔͒ A. Tajti, P. G. Szalay, A. G. Császár, M. Kállay, J. Gauss, E. F. Valeev, B. when the polynomial expansion compare Eq. 11 con- A. Flowers, J. Vázquez, and J. F. Stanton, J. Chem. Phys. 121, 11599 verges slowly. Log-log linear regression yields a correlation ͑2004͒. coefficient of −0.64 with slopeϭ−1.12 and interceptϭ4.75. 5 G. I. Csonka, A. Ruzsinszky, and J. P. Perdew, J. Phys. Chem. A 109, Ϸ 6779 ͑2005͒. This corresponds crudely to utrunc/ustat 100Yn,0/Yn−1,0, with 6 ͑ K. K. Irikura, R. D. Johnson III, and R. N. Kacker, J. Phys. Chem. A substantial scatter. Note that our estimate of utrunc second 109, 8430 ͑2005͒. column of Table 4͒ is identically zero for n=6 only because 7 K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Struc- the model is based upon a sextic polynomial approximation. ture: IV. Constants of Diatomic Molecules ͑van Nostrand Reinhold, New York, 1979͒. This may underestimate the final, combined uncertainties 8 M. Cafiero, S. Bubin, and L. Adamowicz, Phys. Chem. Chem. Phys. 5, significantly when n=6 and convergence is slow. For ex- 1491 ͑2003͒. 9 ample, BeH, BeD, FO, Na2, and NaLi have rather low values G. Michalski, R. Jost, D. Sugny, M. Joyeux, and M. Thiemens, J. Chem. ͑ ͒ Phys. 121, 7153 ͑2004͒. of Y50/Y60 6, 8, 16, 16, and 19 respectively . BeH, BeD, 10 22 22 29 G. Herzberg, Spectra of Diatomic Molecules, 2nd ed. ͑van Nostrand Re- Na2, and NaLi have been measured to n=8, 8, 12, and ͒ 30 inhold, New York, 1950 . 9, respectively, but FO could have significant uncertainty 11 E. W. Kaiser, J. Chem. Phys. 53, 1686 ͑1970͒. arising from the ͑null correction for͒ polynomial truncation 12 J. L. Dunham, Phys. Rev. 41, 721 ͑1932͒. 13 bias. G. Herzberg, Spectra of Diatomic Molecules, 2nd ͑corrected͒ ed. ͑van Nostrand Reinhold, New York, 1989͒, p. 109. This equation was incorrect The present study is limited to 85 diatomic molecules, in earlier editions. some of them isotopologs. In contrast, earlier work has in- 14 F. J. Lovas, E. Tiemann, J. S. Coursey, S. A. Kotochigova, J. Chang, K. cluded polyatomic molecules, which is clearly essential if Olsen, and R. A. Dragoset, Diatomic Spectral Database NIST Standard ͑ bending vibrations are to be represented. This was important Reference Database 114 National Institute of Standards and Technology, Washington, D.C., 2003͒; http://physics.nist.gov/PhysRefData/MolSpec/ because the focus of the earlier work was to determine em- Diatomic/index.html. pirical scaling factors for ZPEs obtained from ab initio cal- 15 R. D. Johnson, III, NIST Computational Chemistry Comparison and culations. Among the many previous studies, the most Benchmark Database (version 12) ͑National Institute of Standards and 31 Technology, Washington, D.C., 2005͒ http://srdata.nist.gov/cccbdb/. heavily cited is that by Scott and Radom. In that study, 16 ͑ ͒͑ P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis experimental ZPEs were computed by using Eq. 2 or its for the Physical Sciences, 3rd ed. ͑McGraw-Hill, Boston, 2003͒. polyatomic analog͒ for a set of 29 diatomic and 10 poly- 17 A. G. Maki, F. J. Lovas, and R. D. Suenram, J. Mol. Spectrosc. 91,424

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