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C#利⽤Guid实现真随机数C#中的随机数可以利⽤Random类很简单地⽣成随机数，代码如下：Random rdmNum=new Random();//⽣成随机数对象int ans=rdmNum.Next(a,b);//⽣成⼤于⼤于a⼩于b的随机数（int）型但是在不指定种⼦时，默认的种⼦是系统时间。
这样的设置对于⼀般的随机数⽣成过程是没有问题的，但是对于⼀些快速、⼤量⽣成随机数的问题（尤其创建与利⽤循环批量⽣成随机数时），会出现由于运算速度太快导致在⼀个相同或者相近的系统时间内⽣成多个随机数的情况，此时⽣成的这些随机数都是相同的，显然是不符合现实情况要求的。
这种问题可以借助GUID（全局唯⼀标识符）来解决：全局唯⼀标识符（GUID，Globally Unique Identifier）是⼀种由算法⽣成的⼆进制长度为128位的数字标识符。
GUID主要⽤于在拥有多个节点、多台计算机的⽹络或系统中。
在理想情况下，任何计算机和计算机集群都不会⽣成两个相同的GUID。
GUID 的总数达到了2^128（3.4×10^38）个，所以随机⽣成两个相同GUID的可能性⾮常⼩，但并不为0。
GUID⼀词有时也专指微软对UUID标准的实现。
在理想情况下，任何计算机和计算机集群都不会⽣成两个相同的GUID。
随机⽣成两个相同GUID的可能性是⾮常⼩的，但并不为0。
所以，⽤于⽣成GUID的算法通常都加⼊了⾮随机的参数（如时间），以保证这种重复的情况不会发⽣。
在 Windows 平台上，GUID ⼴泛应⽤于微软的产品中，⽤于标识如注册表项、类及接⼝标识、数据库、系统⽬录等对象。
格式GUID 的格式为“xxxxxxxx-xxxx-xxxx-xxxx-xxxxxxxxxxxx”，其中每个 x 是 0-9 或 a-f 范围内的⼀个4位⼗六进制数。
例如：6F9619FF-8B86-D011-B42D-00C04FC964FF 即为有效的 GUID 值。
Experimental Vibrational Zero-Point Energies:Diatomic MoleculesKarl K.Irikura a…Physical and Chemical Properties Division,National Institute of Standards and Technology,Gaithersburg,Maryland20899-8380͑Received23August2006;revised manuscript received18December2006;accepted22December2006;published online18April2007͒Vibrational zero-point energies͑ZPEs͒,as determined from published spectroscopicconstants,are derived for85diatomic molecules.Standard uncertainties are also pro-vided,including estimated contributions from bias as well as the statistical uncertaintiespropagated from those reported in the spectroscopy literature.This compilation will behelpful for validating theoretical procedures for predicting ZPEs,which is a necessarystep in the ab initio prediction of molecular energetics.©2007by the U.S.Secretary ofCommerce on behalf of the United States.All rights reserved..͓DOI:10.1063/1.2436891͔Key words:molecular energetics;uncertainty;vibrational spectroscopy;zero-point energy.CONTENTS1.Introduction (389)2.Definition of Zero-Point Energy (389)3.Selection of Molecules (390)4.Contributions to Uncertainties (390)5.Analyses for Individual Molecules (394)6.Results and Discussion (396)7.Acknowledgments (396)8.References (396)List of Tables1.Spectroscopic constants for selected diatomicmolecules (391)2.Derivatives of ZPE with respect tospectroscopic constants,c (392)3.Estimated standard uncertainties associatedwith some diatomic ZPEs͑cm−1units͒asdetermined using the full͓Eq.͑6͔͒,diagonal͓Eq.͑7͔͒,and pessimistic͓Eq.͑8͔͒approximations (393)4.Truncation bias with respect to sextic diatomicvibrational model (394)5.Experimental vibrational zero-point energies͑cm−1͒for selected diatomic molecules (395)1.IntroductionOne of the most popular uses of computational quantum chemistry 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 structure theory and computational efficiency,the precision of ab initio molecular energetics has improved by a factor of about10every10years.With the advent of basis-set ex-trapolation methods and highly correlated theories,the ener-getics of small molecules can now be computed to high pre-cision almost routinely.1As the uncertainties in electronic energy fall away,other sources of uncertainty become in-creasingly noticeable.Vibrational zero-point energy͑ZPE͒has emerged as one of the principal remaining sources of uncertainty in calculations of molecular energetics.In careful work,it is necessary to go beyond the harmonic approxima-tion to obtain a reliable ZPE.2–5However,the most common practice is simply to scale the computed harmonic ZPE by a multiplicative correction factor.The empirical scaling factor carries uncertainty.We have recently quantified the uncer-tainties associated with scaling factors for fundamental vi-brational frequencies.6The uncertainties associated with the experimental vibrational frequencies were required in the analysis,although in the end their contribution was small enough to be neglected.Similar work is underway to provide scaling factors,with their associated uncertainties,appropri-ate for routine predictions of ZPE.Unfortunately,we have been unable to locate any recent compilations of experimen-tally derived ZPEs,or compilations of any age that include uncertainties.Moreover,existing compilations include few data for polyatomic molecules.We are now working tofill this gap by providing benchmark,experimental ZPEs along with the associated uncertainties.The present list,restricted to diatomic molecules,is not intended to be exhaustive.Nev-ertheless,it is the largest such list yet assembled,thefirst to employ spectroscopic data more recent than those compiled by Huber and Herzberg,7and thefirst to include uncertainties.2.Definition of Zero-Point EnergyIn quantum chemistry,the Born-Oppenheimer approxima-tion͑BOA͒is almost8always accepted.Thus,the most com-mon definition of the molecular ZPE is the energy difference between the vibrational ground state and the lowest point on the Born-Oppenheimer potential energy surface.Unfortu-a͒karl.irikura@©2007by the U.S.Secretary of Commerce on behalf of the United States.All rights reserved..0047-2689/2007/36…2…/389/9/$42.00J.Phys.Chem.Ref.Data,Vol.36,No.2,2007389nately,this definition is not convenient for experimentalZPEs,as the BOA is never adopted by real molecules.Ex-perimental spectra are generally analyzed,however,as if theBOA were followed by real molecules;explicit Born-Oppenheimer corrections are only made when simulta-neouslyfitting different isotopologs to the same effective po-tential.Thus,in the present compilation,the experimentalZPE is defined as the difference between the molecularground state and the lowest point on its isotope-specific ef-fective potential.The small difference between this definitionand that used by quantum chemists must be absorbed by theempirical scaling factor that is typically applied to theoreti-cally determined ZPEs.The ZPE cannot be measured directly since no moleculecan be observed below its ground state.Instead,the term“experimental ZPE”describes a value that is usually͑but notnecessarily9͒derived by combining experimental spectro-scopic constants with standard theoretical or empirical mod-els for anharmonic oscillators.Thus,“experimental”ZPEvalues are actually hybrids of experiment and theoreticalmodeling.Most available experimental ZPEs are for diatomic mol-ecules,because far fewer spectroscopic constants are neededfor diatomic molecules than for polyatomic molecules.Al-though thefield of molecular spectroscopy is home tocrowds of molecular constants,among nonspecialists themost common expression for the vibrational energy levels ofa diatomic molecule,relative to the minimum on the poten-tial energy curve,isG͑v͒=e͑v+12͒−e x e͑v+12͒2.͑1͒In Eq.͑1͒,e ande x e are the harmonic frequency and the first anharmonicity constant,respectively,and v is the vibra-tional quantum number,which can assume nonnegative inte-ger values.10Note that the symbole x e represents a single constant,not a product.Thus,the most popular expression for diatomic ZPE is,to second order in͑v+12͒,ZPE=G͑0͒=12e−14e x e.͑2͒This expression is derived by extrapolating Eq.͑1͒to v i=−1 2,which corresponds to the lowest point on the effectivepotential,to a good approximation.11Contributions from higher-order anharmonicities are generally negligible͑e.g., 0.1cm−1for H2,0.07cm−1for OH,and0.0013cm−1for CO͒.Unfortunately,as has been pointed out recently,2–5the popular expression is incorrect.In addition to the linear and quadratic terms in Eq.͑1͒, there is a constant term that is usually overlooked.This was demonstrated by Dunham in his classic power-series analysis.12The resulting energy levels,to second order,are given byG͑v͒=Y00+Y10͑v+12͒+Y20͑v+12͒2,͑3͒where Y10Ϸe and Y20Ϸ−e x e to good approximations.The constant Y00does not influence the line positions͑i.e.,energy intervals͒in a spectrum but contributes to the ZPE.In this paper we include Y00and even the third-order term͑second anharmonicity constant͒when available.Thus,ZPEs of di-atomic molecules are taken to beZPE=G͑0͒=Y00+12e−14e x e+18e y e,͑4͒where Y30Ϸe y e.To a good approximation,Y00can be ex-pressed in terms of conventional spectroscopic constants as13Y00ϷB e4+␣ee12B e+␣e2e2144B e3−e x e4.͑5͒The Dunham constants that correspond to the conventional rotational constants here are Y01ϷB e and Y11Ϸ−␣e.The value of Y00is largest for H2͑Y00=8.9cm−1͒,smaller for hydrides such as OH͑Y00=3.0cm−1͒,and less than one wavenumber for nonhydrides such as CO͑Y00=0.2cm−1͒.3.Selection of MoleculesAn initial list of molecules and associated ground-state spectroscopic constants was culled from the classic compila-tion by Huber and Herzberg7and from the NIST Diatomic Spectral Database.14For compatibility with the NIST Com-putational Chemistry Comparison and Benchmark Database,15only molecules composed of elements lighter than argon͑i.e.,atomic number ZϽ18͒were included.More recent values of constants were taken from the spectroscopic literature as available.Since our goal is a list of reliable ZPEs,diatomic molecules were excluded if their ZPEs obvi-ously͑upon cursory analysis͒had standard uncertainties of about0.5cm−1or more.Ourfinal list of diatomic molecules, with their spectroscopic constants,is provided in Table1.͑In Table1,and throughout this paper,H refers specifically to protium and D to deuterium.͒Standard uncertainties are listed in the spectroscopic style.For example,the quantity 12.345±0.067would be written12.345͑67͒.When the ground state is classically degenerate͑usually2⌸͒,averaged constants are used,as reported in the experimental sources cited.This choice was made for convenient comparison with conventional,non-relativistic quantum chemistry calcula-tions.For Cl2+,only separate constants for2⌸3/2and1⌸3/2 were reported.74.Contributions to UncertaintiesThe spectroscopic constants have both statistical uncer-tainty͑i.e.,uncertainty from random effects͒and bias͑i.e., systematic error͒.As we are interested in the ZPE,it is nec-essary to determine how the uncertainties in the constants contribute to the uncertainty in the corresponding ZPE.For this purpose,we accept the common linearized propagation of uncertainties16given byy2Ϸ͚iii͑ץy/ץx i͒2+2͚iϽjij͑ץy/ץx i͒͑ץy/ץx j͒,͑6͒for a quantity y=f͑x1,x2,...,x i,...͒,wherey2is the esti-mated variance of y andij=͗͑x i−x¯i͒͑x j−x¯j͒͘is the element of the covariance matrix that corresponds to the pair of vari-390K.K.IRIKURA J.Phys.Chem.Ref.Data,Vol.36,No.2,2007EXPERIMENTAL VIBRATIONAL ZERO-POINT ENERGIES391 T ABLE1.Spectroscopic constants for selected diatomic molecules.Standard uncertainties͑1͒are between parentheses and refer to the least significant digits. Values are in wavenumber units͑cm−1͒.Moleculeee x ee y e B e␣e ReferenceFirst-row elements onlyH24401.213͑18͒121.336͑18͒0.8129͑18͒60.8530͑18͒ 3.0622͑18͒7HD3813.15͑18͒91.65͑9͒0.723͑9͒45.655͑9͒ 1.986͑9͒7D23115.50͑9͒61.82͑9͒0.562͑9͒30.4436͑18͒ 1.0786͑9͒7Second-row elementsBeH2061.235͑15͒37.327͑22͒0.084͑17͒10.31992͑5͒0.3084͑2͒22BeD1529.986͑11͒20.557͑12͒0.035͑7͒ 5.68830͑3͒0.1261͑1͒22Be18O1457.09͑22͒11.311͑74͒0.0143͑83͒ 1.5847͑5͒0.01784͑15͒33BF1402.15865͑26͒11.82106͑15͒0.051595͑35͒ 1.51674399͑21͒0.01904848͑22͒23BH2366.7296͑16͒49.33983͑99͒0.362͑10͒12.025755͑45͒0.421565͑22͒34BO1885.286͑41͒11.694͑11͒−0.00952͑83͒ 1.781110͑31͒0.016516͑17͒35C21855.0663͑63͒13.6007͑54͒−0.116͑2͒ 1.820053͑11͒0.0179143͑44͒36C2−1781.189͑18͒11.6717͑48͒0.009981͑28͒ 1.74666͑32͒0.01651͑46͒37CF1307.93͑37͒11.08͑12͒0.093͑15͒ 1.41626͑48͒0.01844͑17͒38CH2860.7508͑98͒64.4387͑85͒0.3634͑27͒14.45988͑20͒0.53654͑33͒39CD2101.05193͑55͒34.72785͑58͒0.14147͑10͒7.8079823͑55͒0.212240͑11͒40CN2068.648͑11͒13.0971͑68͒−0.0124͑17͒ 1.89978316͑67͒0.0173720͑12͒41CO2169.75589͑8͒13.28803͑2͒0.0104109͑14͒ 1.9316023͑7͒0.01750513͑14͒42CO+2214.127͑35͒15.094͑21͒−0.0117͑34͒ 1.976941͑39͒0.018943͑34͒43and44F2916.929͑10͒11.3221͑10͒−0.10572͑67͒0.889294͑11͒0.0125952͑24͒45HF4138.3850͑7͒89.9432͑7͒0.92449͑31͒20.953712͑2͒0.7933704͑65͒46Li2351.4066͑10͒ 2.58324͑41͒−0.00583͑7͒0.6725297͑50͒0.0070461͑16͒47and48 LiF910.57272͑10͒8.207956͑46͒0.569166͑82͒ 1.34525715͑57͒0.02028749͑19͒49LiH1405.49805͑76͒23.1679͑7͒0.17093͑28͒7.5137315͑9͒0.2163911͑24͒50LiD1054.93973͑32͒13.05777͑21͒0.075478͑50͒ 4.23308131͑46͒0.09149428͑84͒50LiO814.62͑15͒7.78͑15͒NA 1.21282948͑11͒0.0178990͑25͒27N22358.57͑9͒14.324͑9͒−0.00226͑9͒ 1.998241͑18͒0.017318͑9͒7N2+2207.0115͑60͒16.0616͑23͒−0.04289͑25͒ 1.93176͑9͒0.01881͑9͒vib,51rot7 NF1141.37͑9͒8.99͑9͒NA 1.205679͑53͒0.014889͑53͒vib,7rot52 NH3282.72͑10͒79.04͑8͒0.367͑23͒16.66792͑6͒0.65038͑17͒53ND2399.126͑30͒42.106͑21͒0.1203͑54͒8.90867͑15͒0.25457͑19͒54NO1904.1346͑18͒14.08836͑89͒0.01005͑20͒ 1.7048885͑21͒0.0175416͑14͒55NO+2376.72͑11͒16.255͑18͒−0.01562͑92͒ 1.997195͑89͒0.018790͑69͒56O21580.161͑9͒11.95127͑9͒0.0458489͑9͒ 1.44562͑9͒0.0159305͑9͒57O2+1905.892͑82͒16.489͑13͒0.02057͑90͒ 1.689824͑91͒0.019363͑37͒58FO1053.0138͑12͒9.9194͑13͒−0.06096͑59͒ 1.05870763͑81͒0.0132951͑23͒59OH3737.761͑18͒84.8813͑18͒0.5409͑18͒18.9108͑18͒0.7242͑9͒7OD+2271.80͑9͒44.235͑9͒0.4267͑18͒8.9116͑18͒0.2896͑9͒7Third-row elementsAlCl481.77466͑20͒ 2.101811͑88͒0.006638͑15͒0.243930066͑12͒0.001611082͑12͒60AlF802.32447͑11͒ 4.849915͑44͒0.0195738͑68͒0.552480208͑65͒0.004984261͑44͒23AlH1682.37474͑31͒29.05098͑29͒0.24762͑12͒ 6.3937842͑17͒0.1870527͑15͒61AlD1211.77402͑15͒15.06477͑11͒0.09244͑4͒ 3.3183929͑8͒0.0698773͑4͒61AlO979.4852͑50͒7.0121͑32͒−0.00206͑56͒0.6413856͑54͒0.0057796͑8͒62AlS617.1169͑33͒ 3.3310͑19͒−0.00924͑32͒0.2800368͑33͒0.00178225͑55͒63BCl840.29472͑63͒ 5.49170͑33͒0.02995͑7͒0.684282͑12͒0.0068124͑14͒17BeS997.94͑9͒ 6.137͑9͒NA0.79059͑9͒0.00664͑9͒7BS1179.91͑3͒ 6.25͑3͒−0.0083͑58͒0.79478͑5͒0.00578͑4͒64CCl876.89749͑69͒ 5.44698͑54͒0.02607͑15͒0.697137͑34͒0.00685277͑45͒25Cl2559.751͑20͒ 2.69427͑20͒−0.003325͑2͒0.24415͑20͒0.001516͑20͒65Cl2+645.61͑9͒ 3.015͑9͒0.007͑9͒0.26950͑18͒0.00164͑9͒7͑⍀=1/2͒644.77͑9͒ 2.988͑9͒NA0.2697͑9͒0.00167͑9͒7ClF783.4534͑24͒ 4.9487͑6͒−0.0176͑1͒0.5164805͑31͒0.0043385͑8͒66ClO853.64268͑13͒ 5.51828͑6͒−0.01256͑30͒0.62345797͑4͒0.0059357͑1͒28CP1239.79924͑8͒ 6.833769͑46͒−0.001377͑7͒0.79886775͑8͒0.00596933͑19͒67CS1285.15464͑10͒ 6.502605͑53͒0.003887͑9͒0.82004356͑4͒0.00591835͑5͒68HCl2990.9248͑15͒52.8000͑15͒0.21803͑55͒10.5933002͑13͒0.3069985͑41͒69J.Phys.Chem.Ref.Data,Vol.36,No.2,2007ables x i and x j.For the problem at hand,y is the ZPE and the x i are generic spectroscopic constants.The required deriva-tives,summarized in Table2,are straightforward but do not appear to have been compiled previously.Typically,only the diagonal elements,ii,of the covari-ance matrix͑i.e.,the variances of thefitting constants͒are reported and available to us.Omitting the off-diagonal con-tributions in Eq.͑6͒yields the estimatey2Ϸ͚ii2͑ץy/ץx i͒2,͑7͒where the standard uncertainties of the x i arei=ͱii.This diagonal approximation introduces a bias in the estimated uncertainty,which will be different for different molecules. To estimate the magnitude of the bias,we consider the situ-ation for BCl,for which the covariance matrix has been published.17Using the derivatives in Table2and the param-eter values and statistical uncertainties from Table1,the standard uncertainty͑i.e.,the square root of the variance͒for the ZPE is0.00065cm−1.When only the diagonal terms are included,the resulting standard uncertainty is only slightly smaller,0.00057cm−1.Given only the variances,the most pessimistic scenario for the correlation coefficients r ij͑−1Յr ijϵij/ijՅ1͒leads to the upper boundT ABLE1.Spectroscopic constants for selected diatomic molecules.Standard uncertainties͑1͒are between parentheses and refer to the least significant digits. Values are in wavenumber units͑cm−1͒.—ContinuedMoleculeee x ee y e B e␣e ReferenceDCl2145.1326͑11͒27.1593͑7͒0.07993͑20͒ 5.4487838͑6͒0.1132345͑15͒69HCl+2673.69͑9͒52.537͑9͒NA9.95661͑18͒0.32716͑18͒7LiCl642.95453͑93͒ 4.47253͑40͒0.020118͑49͒0.70652247͑20͒0.00801019͑32͒70NaLi256.5412͑19͒ 1.62271͑96͒−0.00495͑22͒0.3758620͑89͒0.0031465͑15͒30Mg251.121͑18͒ 1.645͑9͒0.01624͑18͒0.09287͑9͒0.003776͑18͒7MgH1492.7763͑36͒29.847͑4͒−0.3048͑20͒ 5.8255229͑41͒0.177298͑14͒71MgD1077.2976͑26͒15.521͑2͒−0.1184͑7͒ 3.0343436͑21͒0.066607͑5͒71MgO785.2183͑6͒ 5.1327͑3͒0.01649͑7͒0.5748414͑3͒0.0053223͑3͒72MgS528.74͑9͒ 2.704͑9͒NA0.26797͑9͒0.00176͑9͒7Na2159.08548͑44͒0.70866͑29͒−0.004632͑77͒0.15473537͑29͒0.00086375͑11͒29NaCl364.6842͑4͒ 1.7761͑2͒0.005937͑35͒0.21806302͑8͒0.00162479͑6͒73NaF535.65805͑21͒ 3.57523͑13͒0.018453͑34͒0.43690153͑7͒0.00455918͑7͒74NaH1171.968͑12͒19.703͑10͒0.175͑2͒ 4.90327͑13͒0.1370͑4͒75NCl827.95767͑75͒ 5.30015͑61͒−0.00480͑19͒0.649767390͑85͒0.00641432͑20͒76P2780.77͑9͒ 2.835͑18͒−0.00462͑9͒0.30362͑9͒0.00149͑9͒7P2+672.20͑9͒ 2.74͑9͒NA0.27600͑18͒0.00151͑9͒7PF846.75͑9͒ 4.489͑9͒0.019͑9͒0.5667427͑35͒0.00456͑9͒7but B e77 PH2363.774͑36͒43.907͑27͒0.1059͑73͒8.53904͑19͒0.25339͑28͒54PN1336.948͑20͒ 6.8958͑57͒−0.00605͑48͒0.7864844͑28͒0.0055337͑36͒78PO1233.34͑9͒ 6.56͑9͒NA0.733223657͑22͒0.005466162͑50͒vib,7rot79S2725.7102͑97͒ 2.8582͑25͒NA0.29539516͑30͒0.00159754͑59͒80SF837.6418͑5͒ 4.46953͑18͒NA0.555173͑4͒0.004459͑10͒81SH2696.2475͑58͒48.7420͑28͒0.1124͑6͒9.600247͑51͒0.27990͑10͒82Si2510.98͑9͒ 2.02͑9͒NA0.2390͑9͒0.00135͑18͒7SiCl535.59͑2͒ 2.1757͑50͒0.00604͑36͒0.256103͑14͒0.0015817͑72͒83SiF837.32507͑22͒ 4.83419͑9͒0.019807͑16͒0.58125735͑21͒0.00503859͑39͒84SiH2042.5229͑8͒36.0552͑5͒0.1254͑1͒7.503898͑30͒0.21814͑2͒85SiH+2157.17͑9͒34.24͑9͒NA7.6603͑9͒0.2096͑9͒7SiN1151.284͑43͒ 6.455͑21͒−0.0069͑20͒0.730927͑15͒0.005685͑30͒86bute y e87 SiO1241.54388͑7͒ 5.97437͑2͒0.006090͑3͒0.72675206͑2͒0.00503784͑1͒88SiS749.64559͑7͒ 2.58623͑4͒0.001048͑9͒0.303527856͑6͒0.001473130͑3͒88SO1150.7913͑10͒ 6.4096͑5͒0.01306͑11͒0.72082210͑2͒0.00575080͑2͒vib,89rot90T ABLE2.Derivatives of ZPE with respect to spectroscopic constants,c.cץ͑ZPE͒/ץcZPE=e2−e x e2+e y e8+B e4+␣ee12B e+␣e2e2144B e3e1/2+B ee s͑2s+1͒,where sϵ␣ee12B ee x e−1/2e y e1/8B e1/4−s͑3s+1͒␣e B e␣e s͑2s+1͒392K.K.IRIKURA J.Phys.Chem.Ref.Data,Vol.36,No.2,2007yՅ͚ii͉ץy/ץx i͉,͑8͒oryՅ0.00098cm−1for the BCl example.This is50% larger than the result from Eq.͑6͒and appears inferior to the diagonal approximation.These various estimates are summa-rized in Table3,along with analogous estimates for three other diatomic molecules,based upon unpublishedfitting data generously provided by Dr.F.J.Lovas.Based upon the data for these four molecules,the diagonal approximation appears reasonable and in this paper we use Eq.͑7͒to esti-mate the statistical contribution to the standard uncertainty of the diatomic ZPEs.Most of the standard uncertainties presented in Table1are from the experimental papers in which the constants were reported.In some cases,uncertainties were reported but not described;they have been assumed to represent standard un-certainties͑1͒.When uncertainties have been described as 95%confidence intervals,they have been divided by2to estimate standard uncertainties.Other special cases are de-scribed in Sec.5.The uncertainties associated with spectroscopic constants only reflect the statistical uncertainties resulting fromfitting the observed line positions to a model Hamiltonian.In this context,a Hamiltonian is a physically motivatedfitting func-tion involving selected spectroscopic constants and various quantum numbers.In some cases,the uncertainties were propagated from v-dependentfitting constants or from esti-mated͓type B͑Ref.18͔͒uncertainties in line positions.In addition to the statistical uncertainties,there are sources of bias͑“systematic error”͒which may dominate the com-bined uncertainties.True bias cannot be known,since that requires knowledge of true values.When ancillary informa-tion is available about the possible values of bias,we can make a correction for bias,as recommended in the Guide to the Expression of Uncertainty in Measurement,published by the International Organization for Standardization.19There is uncertainty associated with the correction for bias,corre-sponding to the uncertainty of the ancillary information.If we have no information,we choose the value of the correc-tion to be zero,but there is still an uncertainty associated with this͑null͒value.This uncertainty is combined with the statistical uncertainties to obtain the combined uncertainty. One source of bias derives from the truncation in Taylor-series expansions such as Eq.͑1͒.Such model-dependent uncertainties are seldom discussed.20–22Thefitted values of low-order constants are affected when higher-order constants are included in thefitting procedure.Further,our ZPE com-putation͓Eq.͑4͔͒includes terms only through third order. For diatomic molecules,we estimate the uncertainty due to truncation using the simplified analytical model described in the following paragraph.Assume,for this approximate model,that the diatomic vi-brational energy levels are perfectly described by the sextic polynomialG͑v͒=͚i=06b i͑v+12͒i,͑9͒where v is the quantum number,G͑v͒is the energy of the associated level,and the b i are constants.The exact zero-point energy within this model isZPE=b0+12b1+14b2+18b3+116b4+132b5+164b6.͑10͒Also suppose that n transition frequencies,y j=G͑j͒−G͑0͒, are observed and arefitted to the empirical expressionG͑v͒Ϸ͚i=0na i͑v+12͒i,͑11͒where a i=Y i0or the equivalent conventional constant and n Յ6.This is an exactfit because we have assumed,for math-ematical convenience,that the number of free parameters is equal to the number of data.Then the apparent zero-point energy isZPE app=a0+12a1+14a2+18a3.͑12͒ZPE app depends upon the number,n,offitted constants even though the higher-order constants are not explicit in Eq.͑12͒. The difference͑ZPE app−ZPE͒,determined from Eqs.͑10͒and͑12͒,is the truncation bias in this simple model;its ab-solute value is an estimate for the uncertainty arising from truncation of the spectroscopic Hamiltonian.The biases in the cubic approximation͑12͒,for different orders n of the fitting polynomial͑11͒,are listed in the second column of Table4.The value for n=2uses the more severely truncatedquadratic expression ZPE app=a0+12a1+14a2instead of Eq.͑12͒,because a3is undefined.The differences͑a0−b0͒, which appear in the expressions for estimated bias,require evaluation.If we identify a0and b0with Y00,we can use Eq.͑5͒to estimate that͑a0−b0͒Ϸ١Y00·͑a−b͒=␣e12B eͩ1+␣e6B e2eͪ͑a1−b1͒−14͑a2−b2͒.͑13͒The expressions for͑a1−b1͒and for͑a2−b2͒are given in the last two columns of Table4.To obtain numerical values for the uncertainty arising from truncation for a particular di-atomic molecule,we determine n based upon the order of the experimentalfit,choose b iϷY i0,and combine Eq.͑13͒with the expressions in Table4.However,frequently the higherT ABLE3.Estimated standard uncertainties associated with some diatomicZPEs͑cm−1units͒as determined using the full͓Eq.͑6͔͒,diagonal͓Eq.͑7͔͒,and pessimistic͓Eq.͑8͔͒approximations.Molecule Full Diagonal a PessimisticBCl0.000650.00057͑−12%͒0.00098͑+51%͒CS0.000440.00045͑+2%͒0.00050͑+14%͒SiO0.000330.00052͑+58%͒0.00082͑+148%͒SiS0.000360.00037͑+3%͒0.00041͑+14%͒a Percentage deviation from“full”value given between parentheses.EXPERIMENTAL VIBRATIONAL ZERO-POINT ENERGIES393J.Phys.Chem.Ref.Data,Vol.36,No.2,2007Y i0are unknown.As typically͉Y i+1,0͉Ͻ͉Y i0͉,we estimate the missing Y i0crudely by geometric extrapolation from the two highest measured Y i0.As pointed out by a referee,Y i0and Y i+1,0usually differ in sign.Thus,an alternating sign is as-sumed in the present work.However,for about one-third of the data at hand,the sign does not change,i.e.,Y i0Y i−1,0Ͼ0͑for iϾ3͒.To test the sensitivity of the results to the choice of sign,we consider a positive sign in the extrapolation.This changes the magnitude of͑ZPE app−ZPE͒by a mean factor of 1.8͑standard deviationϭ1.5͒.Thus,we multiply the trunca-tion bias values by1.8to reflect the uncertainty of the ex-trapolation.Among the remaining sources of uncertainty,the most im-portant is probably that Dunham’s canonical treatment is it-self an approximation that leads to bias.In particular,any perturbations from electronically or vibrationally excited states will displace some rovibrational levels,skewing the values of thefitting constants.These effects are specific to individual molecules.We do not attempt to quantify the as-sociated uncertainties.However,they are probably small for the ground states of diatomic molecules.Dunham’s analysis is also for non-degenerate electronic states͑1⌺͒and must be considered more approximate in other cases.To illustrate the current procedure,consider BF as an ex-ample.Constants from Table1are substituted into Eq.͑5͒to obtain Y00=0.3111cm−1.Applying Eq.͑4͒yields ZPE =698.4416cm−1,as listed in Table5.The associated uncer-tainty is computed by propagating the uncertainties in the spectroscopic constants,estimating the uncertainty associ-ated with the͑null͒correction for truncation bias,and com-bining these two quantities to obtain a combined standard uncertainty.To propagate uncertainties,substitute values from Table1into the formulas in Table2to obtain the partial derivativesץ͑ZPE͒/ץ͑e͒=0.50307,ץ͑ZPE͒/ץ͑e x e͒=−0.5,ץ͑ZPE͒/ץ͑e y e͒=0.125,ץ͑ZPE͒/ץ͑B e͒=−3.5257,and ץ͑ZPE͒/ץ͑␣e͒=226.11.The intermediate quantity s =0.96750͑Table2͒.The standard uncertainties for the spec-troscopic constants,from Table1,are combined with thesederivatives according to Eq.͑7͒to obtain the estimated sta-tistical contribution to the variance of the ZPE.Taking thesquare root gives u statϷ0.000159cm−1,as listed in Table5͑rounded to two digits͒.To estimate the uncertainty associ-ated with the͑null correction for͒truncation bias,check theliterature͑Zhang et al.23͒tofind the available Dunham con-stants Y n0.In this case,Y40=0.0003464cm−1and no higher constants are available.Thus,the appropriate row in Table4 is that for n=4.Values of b j in Table4are approximated as b jϷY j0.The missing constants are estimated by geometric extrapolation with sign alternation as Y i+1,0Ϸ−͉Y i,02/Y i−1,0͉͑Y i,0/͉Y i,0͉͒,so Y50Ϸ−2.33ϫ10−6cm−1and Y60Ϸ1.56ϫ10−8cm−1.Then from the third column of Table 4,͑a1−b1͒=0.000225cm−1and from the fourth column of Table4,͑a2−b2͒=−0.000255cm−1.Substituting these val-ues into Eq.͑13͒then gives͑a0−b0͒=6.435ϫ10−5cm−1.Fi-nally the expression in the second column of Table4evalu-ates to͑ZPE app−ZPE͒=0.000107cm−1.Multiplying by1.8 to account for sensitivity to guessed Y i0values,and taking the absolute value,u truncϷ0.000193cm−1.Thefinal,esti-mated,combined standard uncertainty is then=͑u stat2 +u trunc2͒1/2=0.000250cm−1,which is rounded to a single digit in Table5.5.Analyses for Individual MoleculesIn their classic compilation,Huber and Herzberg͑H&H͒did not attempt to estimate the uncertainties associated withthe reported spectroscopic constants.7Instead,they providedthe following guidance.“…we hope that the number of dig-its quoted may serve as a very rough indication of the esti-mated order of magnitude of the error,generally±9units ofthe last decimal place.Where the last digit is given as asubscript,we expect that the uncertainty may considerablyT ABLE4.Truncation bias with respect to sextic diatomic vibrational model͓Eqs.͑9͒–͑13͔͒.The order of thefitting polynomial is n. n ZPE app−ZPE͑a1−b1͒͑a2−b2͒60005͑a0−b0͒−116b4−132b5+6794b648818b6−1213916b64͑a0−b0͒−116b4−48516b5−600516b6−168916b5−1290b64754b5+2206116b63͑a0−b0͒+10516b4+1052b5+892532b622b4+271116b5+17652b6−432b4−150b5−1190916b62͑a0−b0͒−158b3−13516b4−43516b5−247532b6−234b3−24b4−119916b5−210b692b3+292b4+1654b5+177116b6394K.K.IRIKURA J.Phys.Chem.Ref.Data,Vol.36,No.2,2007。
