Semi-empirical correction of ab initio harmonic properties by scaling factors: a validated uncertainty model for calibration and prediction
aa r X i v : . [ phy s i c s . c h e m - ph ] O c t Semi-empirical correction of ab initio harmonic propertiesby scaling factors: a validated uncertainty modelfor calibration and prediction
Pascal Pernot
Laboratoire de Chimie Physique, Univ Paris-Sud, Orsay,F-91405 andCNRS, UMR8000, Orsay, F-91405 a) Fabien Cailliez
Laboratoire de Chimie Physique, Univ Paris-Sud, Orsay,F-91405
Bayesian Model Calibration is used to revisit the problem of scaling factor calibrationfor semi-empirical correction of ab initio harmonic properties ( e.g. vibrational fre-quencies and zero-point energies). A particular attention is devoted to the evaluationof scaling factor uncertainty, and to its effect on the accuracy of scaled properties.We argue that in most cases of interest the standard calibration model is not statis-tically valid, in the sense that it is not able to fit a set of experimental calibrationdata within their uncertainty limits . This impairs any attempt to use the resultsof the standard model for uncertainty analysis and/or uncertainty propagation. Wepropose to include a stochastic term in the calibration model to account for modelinadequacy. This new model is validated in the Bayesian Model Calibration frame-work. We provide explicit formulae for prediction uncertainty in typical limit cases:large and small calibration sets of data with negligible measurement uncertainty, anddatasets with large measurement uncertainties.
Keywords: Bayesian data analysis; Model calibration; Uncertainty propagation; Scal-ing factor; Vibrational frequency; Zero point energy. a) Electronic mail: [email protected] . INTRODUCTION One considers generally two types of uncertainty, arising either from random errors orfrom systematic errors. In quantum computational chemistry, random uncertainties, suchas those arising from non-zero convergence threshold, have been shown by Irikura et al. to be negligible. The major uncertainty sources are biases due to basis-set and/or theorylimitations. For quantum chemistry to be predictive, i.e. to be able to predict observableswith confidence intervals , these biases have to be corrected. A common way to do this is bysemi-empirical corrections, i.e. corrections by additive or multiplicative factors calibratedon sets of experimental data. Semi-empirical corrections of ab initio results by linear scaling are efficient for many ob-servables. It is often overlooked that semi-empirical corrections are statistical operations,and as such, accompanied by an uncertainty which has to be considered in the uncertaintybudget of model predictions, of which it is liable to be a major contribution. A sound uncer-tainty budget for these corrections is important in many circumstances. For instance, it isacknowledged that ZPE is a major source of uncertainty in thermochemistry with chemicalaccuracy.
A good evaluation of ZPE prediction uncertainty is therefore essential for theassessment of the accuracy of computed thermochemical properties. In another field, in-frared spectral fingerprinting, confidence intervals on corrected vibrational frequencies couldhelp to ascertain the identification of spectral features.
Estimation of uncertainty oncomputational chemistry results is also of paramount importance for their transfer in multi-scale chemical modeling.
As quantum computational chemistry is at the lowest scale ofchemical simulation, uncertainty on its results has to be carefully propagated to the higherscales in order to get quantified predictions. An example is the use of computational ther-mochemistry to predict the rates of reactions that could have a direct impact on macroscopicobservables in combustion simulations. The concept of
Virtual Measurement has been introduced by Irikura et al. , with the aimto recast model outputs in the standardized uncertainty management framework establishedfor experimental measurements in the Guide to the Expression of Uncertainty in Measure-ment (also known as ”the GUM”). To be a Virtual Measurement, a model output has tobe qualified by a standard uncertainty or confidence interval.In a recent article (hereafter IJKK09), Irikura et al. address the problem of uncertainty2valuation for scaled zero-point energies (ZPE), in the continuity of their 2005 paper (here-after IJK05) on vibrational frequencies. Scaling of harmonic vibrational frequencies is animportant example of semi-empirical correction method in computational chemistry, whereestimation of a vibrational frequency ν is obtained by multiplying the corresponding har-monic vibrational frequency ω , routinely calculated by computational chemistry codes, byan empirical scaling factor s (Fig. 1) ν = ω s. (1)Optimal scaling factors have been computed for numerous sets of theory/basis-set combinations. More sophisticated scaling schemes have been designed to increase the precision of semi-empirical corrections. They make use of frequency-range or mode adapted scaling factorsfor frequencies, or internal coordinate adapted scaling factors for force constants.
Inall cases, the scaling factors are optimized to reproduce at best a set of experimental data,and are affected by a calibration uncertainty, which depends on a few factors, as the sizeof the calibrartion set and the precision of the data within. We focus in the following onthe importance of this calibration uncertainty and concentrate on the widely used uniformscaling factors ( i.e. a single scaling factor for all frequencies), without loss of generality.In the majority of publications about scaling factors, two summary statistics are providedfor each theory/basis-set combination: the optimal scaling factor and the root mean squaresdeviation, characterizing the average distance between experimental and corrected valuesestimated on the calibration dataset. From a reference dataset of experimental { ν exp,i } Ni =1 and calculated { ω i } N vibrational frequencies, the optimal scaling factor obtained by theleast-squares procedure is ˆ s = X ω i ν exp,i / X ω i (2)and the quality of the correction is estimated by the root mean squares (rms) value γ = (cid:18) N X ( ν exp,i − ˆ sω i ) (cid:19) / . (3)To our knowledge, these values have not been explicitly used for uncertainty propagation,but the rms γ provides an estimate of the residual uncertainty resulting from the scalingcorrection (”something like the target accuracy”, or ”a surrogate for uncertainty” accordingto Irikura et al. ), and is used as a criterion for theory/basis-set selection.Acknowledging that scaling factors obtained by calibration on experimental datasets areuncertain, Irikura et al. proposed that (i) this uncertainty is the major contribution to3rediction uncertainty using the scaling model; and (ii) prediction uncertainty is propor-tional to the calculated harmonic property (frequency or ZPE). These authors argue alsothat scaling factors are accurate to only two significant figures, and that all other stud-ies overstate their precision by reporting them with four figures. This approach has beenadopted by the National Institute of Standards and Technology (NIST) and put into prac-tice in the Computational Chemistry Comparison and Benchmark DataBase (CCCBDB), section XIII.C.2, where scaling factors are provided with uncertainties derived according tothe procedure of IJK05/IJKK09. These results can also have a direct impact on the criteriato define the best basis/method level of theory for a given observable.In the present paper, we revisit the problem of scaling factor calibration and propertiesprediction through the Bayesian Model Calibration framework, reputed for providing con-sistent uncertainty evaluation and propagation. Section II presents the methodologicalelements used for calibration and validation procedures, which are applied to a few repre-sentative vibrational frequency and zero point vibrational energy datasets and compared tothe approach by IJK05/IJKK09 in Section III. We point out a statistical inconsistency inthis approach, the main consequence being a much too large scaling factor uncertainty, fromwhich misleading conclusions can be derived. A set of recommendations for reliable uncer-tainty estimation of scaled properties is provided in the Conclusion. Bayesian calculationsused in this study are fairly standard and straightforward, but for the sake of completenessand for readers unfamiliar with statistical modeling, detailed derivations are provided in theAppendix.
II. METHODS
In the following sections, we present the calibration procedure for uniform scaling factorsof vibrational frequencies, but it can be easily transposed to any other property usuallycomputed at the harmonic level and corrected by a multiplicative scaling factor (ZPE, en-tropy...). It is also straightforward to transpose this procedure to semi-empirical correctionschemes involving multiple frequency-adapted scaling factors.4
Harmonic frequency (cm -1 ) E xp e r i m e n t a l fr e qu e n c y ( c m - ) Figure 1. Correlation plot between calculated harmonic frequencies ω i and measured frequen-cies ν expi for a set of vibrations extracted from the CCCBDB for the HF/6-31G* combination oftheory/basis-set (dots). The full line is the regression line ν exp = sω ; the dashed line is a visual aidto appreciate the bias. A. Scaling factor calibration
Considering a measured frequency ν exp , one can assume that it is related to the true orexact value ν true by ν exp = ν true + ǫ exp (4)where ǫ exp ∼ N (0 , u exp ) is a normal random variable, centered at zero with standard uncer-tainty u exp , which represents the measurement error.Calculated harmonic vibrational frequencies ω are also affected by random errors, relatedto numerical convergence defined by non-zero thresholds and the choice of starting pointin geometry optimization, and to non-zero thresholds in wave-function optimization. Ithas been shown that these uncertainties are negligible when compared to the measurement5ncertainty u exp . In the following, one can thus assume that, for one choice of theory/basis-set, the harmonic vibrational frequencies are computed without significant uncertainty.
1. The standard calibration model
If one makes the hypothesis of a linear relationship between ν true and ω , as popularizedby Pople at al. , the standard calibration model is ν exp,i = sω i + ǫ exp,i , (5)where one considers a set of i = 1 , N frequencies. For a single frequency, there is an optimalscaling parameter s i = ν exp,i /ω i . As ν exp,i is uncertain, with standard uncertainty u exp,i ,the value of s i cannot be known exactly and has a standard uncertainty u s i = u exp,i /ω i .For a calibration dataset with uniform measurement uncertainty u exp , it can be shown thatthe optimal value for s is given by the least squares solution ˆ s , Eq. 2, and its standarduncertainty by u s = u exp / qP Ni =1 ω i ( cf. Section II C 3).Applicability of this formula is subject to one condition: the model (Eq. 5) has to bestatistically valid, which means that the residuals { ν exp,i − ˆ sω i } Ni =1 should have a normaldistribution centered on zero, with variance u exp . Normality is not always verified, butmost important, the variance condition is violated in most cases where precise data are usedfor calibration: the linear model (Eq. 5) is typically unable to reproduce a given set ofmeasured frequencies within their measurement uncertainty . Consequently, the width of thedistribution of residuals is dominated by model misfit instead of measurement uncertainty( γ ≫ u exp ), which invalidates the distributional hypothesis of the standard calibration model(Eq. 5). In these conditions, this model should not be used to infer u s , the uncertainty of s .Note that this is the key point to explain statistical inconsistencies in IJK05/IJKK09, as will be discussed later.
2. An improved calibration model
An option to solve this problem would be to search for better ab initio methods, able toreproduce experimental properties within their measurement uncertainties. This is an activeresearch area which is out of the scope of the present study . Considering the practical6nterest of correction by scaling factors, we rather focus on restoring statistical consistencyby improving the calibration model.Observing the apparent randomness of the residuals { ν exp,i − ˆ sω i } Ni =1 (Fig. 2), we con-sider that the model misfit is not deterministically predictable. A solution to preserve astatistically valid linear scaling model is to introduce an additional stochastic variable ǫ mod to represent the discrepancy between model and observations ν exp,i = sω i + ǫ mod + ǫ exp,i . (6)This equation is similar to the basic statistical model introduced by Kennedy and O’Hagan for Bayesian Calibration of Model Outputs. The discrepancy variable ǫ mod could formallydepend on ω , but we observed on representative datasets that the residuals between modeledand observed frequencies are not markedly frequency dependent (Fig. 2). Therefore ǫ mod is considered null in average, with unknown variance u mod : ǫ mod ∼ N (0 , u mod ) . (7)The new calibration model (Eq. 6) depends on two unknown parameters, s and u mod . B. Model predictions and uncertainty propagation
The new stochastic prediction model used within the calibration model (Eq. 6), ν = sω + ǫ mod , (8)is linear with respect to uncertain variables s and ǫ mod , and one can use standard uncertaintypropagation rules to estimate the average value and variance of predicted frequencies: ν = s ω (9) u ν = (cid:18) ∂ν∂s (cid:19) s = s u s + (cid:18) ∂ν∂ǫ mod (cid:19) ǫ mod =0 u mod (10) = ω u s + u mod , (11)where s denotes the average value of the scaling factor, and u s its variance.In order to provide evaluated predictions of vibrational frequencies, we need therefore toestimate s , u s and u mod from a calibration dataset. This is done in the next section, usingBayesian Model Calibration. 7 Harmonic frequency (cm -1 ) -300-200-1000100200300 R e s i du a l s ( c m - ) Index in the reference set -300-200-1000100200300 R e s i du a l s ( c m - ) Figure 2. Residuals between calculated harmonic frequencies ω i and measured frequencies ν i for aset of vibrations extracted from the CCCBDB for the HF/6-31G* combination of theory/basis-set(dots). Bottom: residuals as a function of ω . In order to suppress the grouping effect linked withfrequencies, the residuals were also plotted as a function of their order in the reference set (top). C. Bayesian Model Calibration (BMC)
1. General case
Starting from the calibration model (Eq. 6), one derives the expression for the poste-rior probability density function (pdf) of the parameters, given a set of N measured and8 Sample Cumulative Density Function N o r m a l C u m u l a ti v e D e n s it y F un c ti on Figure 3. Plot of the cumulative density function (CDF) for the residuals (same as in Fig. 2) againsta normal CDF shows that globally there is very little deviation from normality in this dataset. calculated frequencies (details of derivation are provided in Appendix A 1) p (cid:16) s, u mod | { ν exp,i , u exp,i , ω i } Ni =1 (cid:17) ∝ u mod Q Ni =1 q u mod + u exp,i × exp − N X i =1 ( ν exp,i − sω i ) (cid:0) u mod + u exp,i (cid:1) ! . (12)Estimates of s , u s and u mod are obtained from this pdf. In the general case, this has to bedone numerically. Two limit cases of interest ( i.e. negligible and very large measurementuncertainties), amenable to analytical results, are presented in the next sections.9 . The case of negligible measurement uncertainties
In the commonly met situation where the amplitude of the discrepancy between cali-bration model and experimental data is much larger than any other sources of uncertainty( u mod ≫ u exp ), we can consider the approximate calibration model ν exp,i = sω i + ǫ mod , (13)for which the posterior pdf (Eq. 12) can be simplified and rearranged to (see Appendix A 2) p (cid:16) s, u mod | { ν exp,i , ω i } Ni =1 (cid:17) ∝ u N +1 mod exp (cid:18) − N γ u mod (cid:19) exp − ( s − ˆ s ) P Ni =1 ω i u mod ! , (14)from which one can analytically derive estimates of the parameters: • s = ˆ s : the average value for s is identical to the optimal value of least-squares analysis(Eq. 2); • u s , the standard uncertainty on s , is related to the rms γ by u s = γ r N/ h ( N − X ω i i ; (15) • and the estimate of u mod is related to γ according to u mod = γ N/ ( N − . (16)Inserting these values in Eq. 11, we obtain the standard uncertainty of a predicted frequency: u ν = γ s NN − (cid:18) ω P i ω i + 1 (cid:19) . (17)It can be seen that for large calibration sets of few hundreds of frequencies p N/ ( N − ≃ and ω / P i ω i ≪ , and thus u ν ≃ γ. (18)In such conditions, it is possible to derive directly confidence intervals on scaled propertiesfrom the summary calibration statistics ˆ s and γ typically provided in the literature. Assuming the normality of uncertainty distributions, confidence intervals can be defined forprediction purpose, e.g. the 95% confidence interval for ν is given by CI ( ν ) = [ˆ sω − . u ν , ˆ sω + 1 . u ν ] . (19)10 . The case of very large measurement uncertainties When model discrepancy is negligible compared to measurement uncertainties ( u mod ≪ u exp ), the standard linear model is statistically valid, and one recovers the Bayesian versionof weighted least squares. The posterior pdf for s is then p ( s | { ν exp,i , u exp,i , ω i } Ni =1 ) ∝ N Y i =1 u − exp,i exp − N X i =1 ( ν exp,i − sω i ) u exp,i ! , (20)from which one obtains ˆ s = N X i =1 (cid:0) ω i ν exp,i /u exp,i (cid:1) / N X i =1 (cid:0) ω i /u exp,i (cid:1) , (21) u s = 1 / N X i =1 (cid:0) ω i /u exp,i (cid:1) . (22)For uniform experimental uncertainty over the dataset, the scaling factor uncertainty is u s = u exp / vuut N X i =1 ω i . (23) D. The Multiplicative Uncertainty (MU) method
Irikura et al. , after a thorough analysis of the uncertainty sources in the ab initio calculation of harmonic vibrational frequencies, proposed that the major contribution toprediction uncertainty would be the uncertainty on the scaling factor ˆ s . They estimate u s from the weighted variance of s with weights a i = ω i . This weighting scheme is derived intwo steps: (1) they propose that the probability density function (pdf) for the scaling factoris a linear combination of pdf’s for individual scaling factors in the reference set; and (2)from the comparison of the expression of the average value derived from this propositionwith the least-squares solution Eq. 2. This way, they obtain a standard uncertainty u ∗ s ≃ (cid:18) P ω i X ω i ( s i − ˆ s ) (cid:19) / , (24)which can be related to the rms γ by u ∗ s ≃ γ p N/ P ω i .More recently, Irikura et al. derived another expression by standard uncertainty propa-gation from the least-squares solution Eq. 2, adding a new term to their previous expression u ∗ s ≃ P ω i X ω i ( s i − ˆ s ) + 1( P ω i ) X ω i u exp,i ! / . (25)11hey showed that the contribution of the latter term is negligible, validating the use oftheir former expression. Note that, unless all frequencies ω i are equal, this uncertainty u ∗ s isdifferent from the dispersion of s values within the calibration set δ s = (cid:18) N X ( s i − ˆ s ) (cid:19) / (26)and attributes larger weights to the high frequencies.Using either of Irikura et al. expressions for u ∗ s , uncertainty on a scaled frequency isapproximated by u ν ≃ ωu ∗ s , (27)hence the name of ”Multiplicative Uncertainty” (MU) used hereafter.The salient feature of Eq. 27 is that prediction uncertainty is always proportional tothe calculated harmonic frequency, ignoring the additive term present in Eq. 11. Simplestatistical validation test of the MU method have apparently not been published and areperformed in the next sections. III. APPLICATIONS AND DISCUSSION
In the following, we validate the BMC approach and compare it to the MU approach onrepresentative test cases of vibrational frequencies and zero point energies.
A. Vibrational frequencies
The reference dataset of 2737 frequencies for the HF/6-31G* combination of theory/basis-set has been downloaded from the NIST/CCCBDB in July 2008. Correlation betweenexperimental and harmonic frequencies is plotted in Fig. 1.
1. Calibration
In absence of detailed information on the measurement uncertainties for this dataset,and considering the typical high accuracy of spectroscopic data, we assume that they arenegligible and apply the corresponding equations for the BMC model. Using Eqs. 15 and16, we obtain ˆ s = 0 . ± . , and u mod = 45 . ± . cm − (Table I). The latter12 ummary stat. MU BMC ˆ s γ (cm − ) u ∗ s %CI u s u mod (cm − ) %CI All frequencies ( N = 2737 )Full set 0.89843 45.33 0.025 - 0.00046 45.35 -Calibration set 0.89860 45.27 0.024 - 0.00065 45.31 -Validation set - - - 83.0 - - 94.6 High frequencies, between 3180 and 3500 cm − ( N = 479 )Full set 0.90502 28.71 0.00869 - 0.00040 28.78 -Calibration set 0.90517 23.32 0.01005 - 0.00046 23.44 -Validation set - - - 97.4 - - 95.4Table I. Statistical estimates and validation for MU and BMC models for vibrational frequenciesextracted from the CCCBDB for the HF/6-31G* combination of theory/basis-set. value is very close to the rms value γ = 45 . cm − , which validates the use of Eq. 18 forlarge calibration datasets.For this same dataset, the CCCBDB proposes ˆ s = 0 . ± . , which can be recov-ered using Eq. 24 (Table I). The standard uncertainties on ˆ s evaluated by both methodsdiffer thus by a factor 50, which can be expected to have noticeable effect on predictionuncertainty (see Section III A 5). In order to visualize the difference, we plotted the 95 %confidence intervals on predicted frequencies obtained from both methods (Fig. 4). It isimmediately visible that the the MU approach has a tendency to underestimate uncertaintyat low frequencies and to overestimate it at high frequencies.
2. Validation
To better quantify this inconsistency, we performed a standard test in statistical cali-bration/prediction: the dataset is split randomly in two subsets, one for calibration, theother one for validation. Both sets are taken here of equal size (plus or minus one unit). Inthis case, one gets slightly different values of the parameters, as reported in Table I. Usingthese values, we generate 95 % confidence intervals (Eq. 19; the residuals of this datasethave a nearly normal distribution) and calculate the percentage of inclusion of the exper-13
Harmonic frequency (cm -1 ) -300-200-1000100200300 R e s i du a l s ( c m - ) Figure 4. Residuals of the linear scaling model for a set of 2737 vibrational frequencies and the HF/6-31G* combination of theory/basis-set (dots). Model 95 % confidence intervals for residuals: dashed(green) lines for the Bayesian Model Calibration method; solid (red) lines for the MultiplicativeUncertainty model of IJK05/IJKK09. imental values of the validation subset within these prediction intervals (Fig. 4). For aconsistent predictor, one should find a frequency close to 95 %. BMC succeeds for 94.7 %of the frequencies in the validation set, whereas the MU model succeeds for only 83 % (Ta-ble I). Considering the size of the samples, the difference is significant, and the statisticalconsistency of the MU approach can be questioned. When contrasted with the BMC, oneunderstands that the MU method, which does not consider model inadequacy explicitly,”absorbs” it at least partially in u ∗ s .
3. Test on a restricted frequency scale
As stated in IJK05, ”to apply the fractional bias correction, it is important to select aclass of frequencies similar to the ones to be estimated”. For instance, if one selects in thereference set only those frequencies between 3180 and 3500 cm − , one gets a much moreuniform picture than with the full reference set.The MU calibration procedure was done with this limited set of 479 frequencies, providing ˆ s = 0 . ± . (Table I). In this case, the uncertainty factor for s is practically identical14o the standard deviation calculated from the sample (0.00869 vs. 0.00871): u ∗ s ≃ δ s . Dueto the restricted frequency range, one has indeed ω i / P i ω i ≃ /N , hence the identitybetween evaluations by Eqs 24 and 26.This set has been split in two, as before. The scaling factor obtained by MU from thecalibration subset is now ˆ s = 0 . ± . , and 97.4 % of the validation frequencies fallwithin the 95 % confidence interval. This result is quite close to the one obtained with BMC(Table I).It appears thus that in restrictive conditions, the MU method can be valid for referencesets where the individual scaling factors are uniformly distributed with regard to the har-monic frequencies. In such cases however, the uncertainty is recovered as the conventionalunweighted standard deviation of the sample of individual scaling factors (Eq. 26). Notealso that the MU method is used in the CCCBDB out of these favorable conditions.
4. Significant figures and uncertainty reporting
Good practice in uncertainty reporting is to provide one or two significant figures for theuncertainty and to truncate the average/optimal value at the same level. If the reportednumber is to be used in further calculations (which is the case for uncertainty propagation),two digits is better. The common practice is to publish scaling factors for vibrationalfrequencies with four significant digits.
At the risk of being pedantic, one could arguethat they should be reported with five significant digits, e.g. ˆ s = 0 . ± . , in sharpcontrast with the two digits recommendation of Irikura et al. , based on their biased scalingfactor uncertainty evaluation.
5. Prediction and uncertainty propagation
The relative importance of both factors u s and u mod in Eq. 11 can be evaluated on theexample of a calculated harmonic frequency in the higher frequency range ω = 3000 cm − (Table II).In this case, the uncertainty on the scaling factor contributes only to one thousandthof the total prediction variance. When dealing with large datasets of accurate vibrationalfrequencies, the uncertainty on the scaling factor can thus be neglected. The uncertainty15 roperty Theory/Basis set ω Method ω u s u mod ν ± u ν Frequency HF/6-31G* 3000 cm − BMC 2.25 2052.09 2695 ±
45 cm − MU 5625.0 - 2695 ±
75 cm − ZPE HF/6-31G* 100 kJ mol − BMC 0.073 0.53 91.35 ± − MU 2.592 - 91.35 ± − ZPE B3LYP/6-31G* 100 kJ mol − BMC 0.029 0.19 98.12 ± − MU 1.061 - 98.12 ± − Table II. Compared prediction uncertainty with the BMC and MU methods for a set of 2737 vibra-tional frequencies extracted from the CCCBDB for the HF/6-31G* combination of theory/basis-setand for a set of 39 ZPE of the Z1 set for the HF/6-31G* and B3LYP/6-31G* combinations. on u mod is also much too small to be relevant for confidence intervals calculation. One cantherefore safely apply the uncertainty propagation formula (Eq. 18), using the rms providedby most reference articles dealing with scaling factors calibration. For smaller calibrationsets, the rms can be seen as an inferior limit to prediction uncertainty, and Eq. 17 wouldprovide more reliable confidence intervals (see next Section).Comparing the prediction uncertainties for the BMC (45 cm − ) and MU (75 cm − ) meth-ods, one sees that the factor 50 between u s and u ∗ s observed at the calibration stage ispartially damped at the prediction level by the fact that the BMC uncertainty is stronglydominated by the model inadequacy parameter u mod . B. Zero Point Vibrational Energies
We consider ZPE as an additional test because the reference datasets are considerablysmaller than for the vibrational frequencies (e.g. 39 molecules in the Z1 set of Merrick etal. ), which is expected to emphasize the role of u s , the uncertainty on the scaling factor.The uncertainties reported by Irikura for diatomic molecules are typically very small (onthe order of 0.01 cm − ), but transposition to larger molecules is not straightforward. Inthe absence of a systematic review of measurement errors for ZPE of polyatomic molecules,we consider here that they can be neglected. The effect of non-negligible measurement16ncertainties is addressed at the end of this section.
1. Calibration - Validation
Using BMC with the Z1 reference set, one gets ˆ s = 0 . ± . and u mod = 0 . ± . kJ mol − (Table III), which is consistent with the rms obtained by Merrick et al. forthe HF/6-31G* theory/basis-set combination. Relative uncertainties on these parametershave been increased by one order of magnitude, when compared to the vibrational frequenciescase, a direct effect of the smaller sample size. The validation test shows once more thatthe MU model fails to provide correct confidence intervals, with a score of only 0.63 for CI (Table III).
2. Uncertainty propagation
For such a small reference dataset, it is interesting to check if the approximate formula(Eq. 18) for uncertainty propagation, which was validated for large sets of vibrationalfrequencies still holds, i.e. if the contribution of the multiplicative term involving u s staysnegligible or not for the larger ZPE values. If one considers a calculated ZPE of 100 kJ mol − (HF/6-31G*), one has u ν = p (100 ∗ . + 0 . = 0 . kJ mol − , to be compared to γ = 0 . kJ mol − (Table III). It is to be noted also that the uncertainty on u mod might alsocontribute, with u u mod = 0 . kJ mol − . Taking all uncertainty sources into account troughEq. A27 by Monte Carlo Uncertainty Propagation (MCUP), one gets u ν = 0 . kJ mol − .The uncertainty on u mod can therefore be neglected.In the same conditions, for the combination B3LYP/6-31G*, one gets γ = 0 . kJ mol − and u ν = 0 . kJ mol − , to be compared with a reference value obtained by MCUP of u ν = 0 . kJ mol − (Table III).There is globally only a 10% increase compared to the rms γ . In this range of ZPEs, γ still provides a good approximation of the prediction uncertainty (Table II). However,the amplitude of the discrepancy between γ and u ν will probably increase with the size ofthe molecule. In consequence, for uncertainty propagation with ZPEs, notably for largemolecules, it would be safer to use the full UP formula (Eq. 17), involving the multiplicativeuncertainty factor. Compilations of scaling factors for ZPE should thus report the easily17 ummary stat. MU BMC ˆ s γ (kJ mol − ) u ∗ s %CI u s u mod (kJ mol − ) %CI HF/6-31G*
Full set 0.9135 0.707 0.0161 - 0.0027 0.731 ± ± B3LYP/6-31G*
Full set 0.9812 0.423 0.0103 - 0.0017 0.437 ± ± calculated value of u s = γ p N/ (( N − P ω i ) , in addition to the rms γ .
3. The case of non-negligible experimental uncertainties
When the measurement uncertainty becomes comparable to the rms, model inadequacyshould be small, and confidence intervals for prediction should account for the measure-ment uncertainty (Eq. 11). In the absence of an exhaustive compilation of experimentaluncertainties on measured ZPE, we performed simulations assuming a uniform uncertaintydistribution over the full dataset. In order to test the sensitivity of the model parametersto this uncertainty, we repeated the estimations of the previous section, using Eq. A6, forvalues of u exp between 0.1 and 1.0 kJ mol − . The results are reported in Fig. 5.As expected from the properties of the posterior pdf (Eq. 12), the average/optimal valueof the scaling factor is not sensitive to the amplitude of u exp . Moreover, we observe only aslight absolute increase of u s from 0.002 to 0.004. A transition from a constant u s , definedby the u exp = 0 limit, to a linear increase consistent with the weighted least squares limit(Eq. 23) is observed around u exp = γ , where both limit equations intersect. A closer lookshows that the transition occurs indeed at values of u exp slightly smaller than γ , in a region18 u exp ≃ . where u s displays a minimum.The evolution of the model inadequacy factor u mod is more dramatic: it displays a sharpdecrease and falls down to zero as soon as the measurement uncertainty reaches the value ofthe rms γ . For values of u exp below . , u mod follows the u mod + u exp = γ law (representedas a dashed line in Fig. 5), but the calculated decrease becomes much faster in the transitionzone. The uncertainty on u mod increases notably in the transition region.In the limit of large experimental uncertainties, using Eq. 23, the uncertainty propagationformula can be written as u ν = ω u s = u exp ω / X ω i . (28)In this case, the model inadequacy variable ǫ mod becomes useless, as the standard calibrationmodel is statistically valid.This test shows that the BMC model is able to adapt nicely to various conditionsof measurement uncertainty, with an automatic and smooth transition from the ”modelinadequacy”- to the ”measurement uncertainty”-dominated regimes. IV. CONCLUSIONS AND RECOMMENDATIONS
A reanalysis of the scaling factor calibration problem as stated by Irikura et al. iden-tified two uncertainty components, besides the experimental one: a parametric uncertainty u s attached to the optimal scaling factor, and a model inadequacy factor u mod accountingfor the inability of the linear scaling correction model to reproduce sets of calibration datawithin their experimental uncertainties. A general estimation framework, based on BayesianModel Calibration, has been defined and validated in cases of interest.The general formula for prediction of a scaled property ν from a harmonic value ω is ν = ˆ sω ± u ν , (29)where ˆ s is the optimal value of the scaling factor provided by the least squares formula (Eq.2) for negligible or uniform measurement uncertainty, or more generally by the weighted leastsquares formula (Eq. 21), and u ν is a standard uncertainty, for which explicit expressionshave been derived in limit cases, depending on the size and precison of the calibration set: • large calibration sets of precise data ( u exp ≪ γ ): u ν ( ω ) = γ ;19 .0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.9780.9800.9820.9840.986 S ca li ng f ac t o r u exp ( kJ mol -1 ) u m od ( k J m o l - ) Figure 5. Evolution of measurement model parameters with the amplitude of an hypotheti-cal uniform experimental measurement uncertainty u exp on ZPE; B3LYP/6-31G* combination oftheory/basis-set (green squares with error bars). The brown vertical dashed line indicates the valueof the rms γ . Top panel: the red dashed lines represent the 1 σ confidence interval in the limitof null experimental uncertainty; the blue dashed line represent the 1 σ confidence interval in theweighted least squares limit. Bottom panel: the red dashed line represents the u mod + u exp = γ law, truncated to positive values of u mod . • small calibration sets of precise data ( u exp ≪ γ ): u ν ( ω ) = γ q NN − (1 + ω / P i ω i ) ; • calibration sets with large measurement uncertainties ( u exp ≥ γ ): u ν ( ω ) = ω/ qP i ω i /u exp,i ,simplified to u ν ( ω ) = u exp ω/ pP i ω i for uniform measurement uncertainty.The Multiplicative Uncertainty method proposed by Irikura et al. has been shown hereto be statistically inconsistent when large frequency ranges are considered. It is only valid inparticular situations, either when the dataset spans a restricted frequency range (in which20ase the uncertainty is reduced to a trivial unweighted standard deviation), or in the extremecase of large uniform measurement uncertainty in the calibration dataset. For vibrationalfrequencies, the MU method underestimates prediction uncertainty for small values of ω andoverestimate it (up to a factor 2) at the high end of the ω scale.We would like to stress out that the validity of the formulas proposed above for uncer-tainty propagation depends to some extent on the normality of the residuals { ν i − ˆ sω i } Ni =1 ofthe linear regression. Inspection of histograms of residuals (see e.g. Fig. 1 in Ref. ) showsthat this is not always the case. The usual approach of choosing an optimal theory/basis-setcombination is to assess their performance by the rms alone, maybe weighted by computa-tional cost considerations. Researchers concerned by prediction uncertainty might alsoconsider an additional ”normality criterion” in order to reject theory/basis-set combinationsproviding non-normal residuals and from which the summary statistics cannot be used reli-ably for uncertainty propagation. Analysis of restricted ranges of data as presently done bysome authors for vibrational frequencies is one way to improve the normality of residuals,but as demonstrated above, prediction from small calibration sets calls for more informationthan the rms.
A. Recommendations to calibrators of scaling factors
1. For large calibration sets of accurate data, as the ones used for calibration of uniformscaling factors for vibrational frequencies, reliable prediction uncertainty can be simplybased on the rms γ (Eq. 3). In this case, prediction uncertainty is purely additive.2. For much smaller datasets of a few dozens of data or less, as in the case of ZPEsor mode-specific frequencies, one has a combination of additive and multiplica-tive uncertainty (or rather, variance). Ideally, uncertainty on the scaling factor u s = γ p N/ (( N − P ω i ) should be reported along with the additive term u mod = γ p N/ ( N − , for use in the general uncertainty propagation equation u ν = ω u s + u mod (Eq. 11). It certainly would be a large step towards the general applicability ofthe Virtual Measurement concept, if statistically pertinent estimators were systemat-ically reported in the literature devoted to the calibration of semi-empirical correctionparameters. 21. An indicator of the normality of the residuals in the calibration dataset would also bewelcomed. B. Recommendations to users of scaling factors
1. For the end user of scaling factors, it is important to remind, as pertinently statedby Irikura et al. , that semi-empirical correction of a property by scaling is not adeterministic procedure: a scaled property has an attached uncertainty, which dependson the level of theory/basis-set used for the calculation of harmonic properties (itdepends also on the quantity and quality of the calibration dataset, but this is out ofreach of the end user).2. In the present state of affairs, the best estimate of the prediction uncertainty availablefor most levels of theory/basis-set is the rms γ , i.e. one has to assume u ν ≃ γ . The use of the multiplicative scaling factor uncertainty as reported presently (March2010) in the CCCBDB cannot be recommended for the estimation of uncertainty ofscaled properties.3. Users are encouraged to(a) report the uncertainty along with the scaled properties, i.e. ν = ˆ sω ± u ν , and(b) account for uncertainty when scaled properties are used as inputs to a model ,or for comparison with experimental data.4. One has to be conscious that γ provides only a lower limit of the uncertainty forproperties with small calibration data sets ( e.g. ZPE). For numerical examples, seeTable II.
ACKNOWLEDGMENTS
The authors would like to thank Prof. Leo Radom for providing the Z1 ZPE dataset. B.Lévy is warmly acknowledged for helpful discussions.22 ppendix A: Appendix1. Bayesian analysis of scaling factor calibration model
We consider the calibration model ν exp,i = sω i + ǫ mod + ǫ exp,i , (A1)where ǫ exp,i ∼ N (0 , u exp,i ) is the measurement uncertainty of ν exp,i , and ǫ mod ∼ N (0 , u mod ) is a variable accounting for the discrepancy between the linear model and the observations.This model has two unknown parameters, s and u mod , to be estimated on a calibrationdataset consisting of N calculated harmonic frequencies { ω i } Ni =1 , and their correspondingexperimental frequencies { ν exp,i , u exp,i } Ni =1 .In the Bayesian data analysis framework, all information about parameters can beobtained from the joint posterior pdf p (cid:16) s, u mod | { ν exp,i , u exp,i , ω i } Ni =1 (cid:17) . In order to simplifythe notations, we will omit in the following the list indices when they are not necessary.This pdf is obtained through Bayes theorem p ( s, u mod | { ν exp , u exp , ω } ) ∝ p ( { ν exp } | s, u mod , { u exp , ω } ) p ( s, u mod ) , (A2)where p ( { ν exp } | s, u mod , { u exp , ω } ) is the likelihood function and p ( s, u mod ) is the prior pdf.In the hypothesis where the difference between observation and corrected frequency isexpected to arise from a normal distribution ν exp,i − sω i ∼ N (0 , u mod + u exp,i ) , (A3)the likelihood function for a single observed frequency is p ( ν exp,i | s, u mod , u exp,i , ω i ) = (cid:0) π (cid:0) u mod + u exp,i (cid:1)(cid:1) − / exp −
12 ( ν exp,i − sω i ) u mod + u exp,i ! . (A4)Considering that all frequencies are measured independently (with uncorrelated uncertainty)the joint likelihood is the product of the individual ones, i.e. p ( { ν exp } | s, u mod , { u exp , ω } ) = N Y i =1 (cid:0) π (cid:0) u mod + u exp,i (cid:1)(cid:1) − / × exp − N X i =1 ( ν exp,i − sω i ) u mod + u exp,i ! . (A5)23s there is a priori no correlation between s and u mod , we use a factorized prior pdf p ( s, u mod ) = p ( s ) p ( u mod ) . In the absence of a priori quantified information on s , a uni-form pdf p ( s ) = cte is used. For u mod , we enforce a positivity constraint through a Jeffrey’sprior, p ( u mod ) ∝ u − mod . The posterior pdf is finally defined up to a norm factor which isirrelevant for the following developments p ( s, u mod | { ν exp , ω, u exp } ) ∝ u − mod N Y i =1 (cid:0) u mod + u exp,i (cid:1) − / × exp − N X i =1 ( ν exp,i − sω i ) u mod + u exp,i ! . (A6)
2. Case of negligible measurement uncertainties
For the analysis of vibrational frequencies, it is generally considered that experimentaluncertainties are negligible when compared to model inadequacy ( u exp,i ≪ u mod ). Thegeneral expression for the posterior pdf (Eq. A6) can then be simplified accordingly: p ( s, u mod | { ν exp , ω } ) ∝ u − N − mod exp − u mod N X i =1 ( ν exp,i − sω i ) ! . (A7)Using Eq. 2 and 3 we derive the identity (see e.g. Ref. (, Eq. 9.4, p. 214)) N X i =1 ( ν exp,i − sω i ) = ( s − ˆ s ) X ω i + N γ , (A8)which enables to write the posterior pdf in a convenient factorized form p ( s, u mod | { ν exp , ω } ) ∝ u − N − mod exp (cid:18) − N γ u mod (cid:19) exp (cid:18) − ( s − ˆ s ) P ω i u mod (cid:19) (A9)from which we can derive analytical estimates for the parameters and their uncertainties. a. Estimation of s The marginal density for s is obtained by integration over u mod ( s | { ν exp , ω } ) = Z ∞ du mod p ( s, u mod | { ν exp , ω } ) (A10) ∝ Z ∞ du mod u − N − mod exp − u mod N X i =1 ( ν exp,i − sω i ) ! (A11) ∝ N X i =1 ( ν exp,i − sω i ) ! − N/ , (A12)which, using Eq. A8, can be rewritten as p ( s | { ν exp , ω } ) ∝ (cid:18) s − ˆ s ) P ω i N γ (cid:19) − N/ , (A13)and has the shape of a Student’s distribution Stt( x ) ∝ (cid:18) x n (cid:19) − ( n +1) / . (A14)Posing n = N − and x = ( N − /N ( s − ˆ s ) P ω i /γ , we can directly use the propertiesof the Student’s distribution E[ x ] = 0; Var[ x ] = n/ ( n − , (A15)to derive E[ s ] ≡ s = ˆ s (A16) Var[ s ] ≡ u s = γ N ( N − P ω i Var[ x ] (A17) = γ N ( N − P ω i (A18) b. Estimation of u mod The marginal density for the standard uncertainty of the stochastic variable ǫ mod is p ( u mod | { ν exp , ω } ) = Z ∞−∞ ds p ( s, u mod | { ν exp , ω } ) (A19) ∝ u N +1 mod exp (cid:18) − N γ u mod (cid:19) Z ∞−∞ ds exp (cid:18) − ( s − ˆ s ) P ω i u mod (cid:19) (A20) ∝ u Nmod exp (cid:18) − N γ u mod (cid:19) . (A21)25sing the formula Z ∞ dx x − n e − a/x = 12 Γ (cid:18) n − (cid:19) /a ( n − / (A22)to recover the normalization constant of p ( u mod | { ν exp , ω } ) and to calculate mean values of u mod and u mod , one obtains readily the following estimates ˆ u mod = γ (A23) u mod = γ r N N − / N − / (A24) u mod = NN − γ (A25) u u mod = γ s NN − − N (cid:18) Γ [( N − / N − / (cid:19) . (A26)
3. Prediction and uncertainty propagation
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