Complete th emeasaurement of FTIR for given molecules

READ textbook rovibrational analysis

Identify energies and m quantum numbers 
USE equation (10) as the model 
Fit data with parameters from equation (10)
Identify physical observable and uncertainties (all here)

hcla={{-9,2676.05`},{-8,2701.41`},{-7,2726.`},{-6,2750.3`},{-5,2774.02`},{-4,2797.06`},{-3,2819.72`},{-2,2841.71`},{-1,2863.21`},{1,2904.28`},{2,2923.96`},{3,2942.95`},{4,2961.27`},{5,2978.92`},{6,2995.98`},{7,3012.38`},{8,3028.09`},{9,3043.04`}}

{{-9,2676.05`},{-8,2701.41`},{-7,2726.`},{-6,2750.3`},{-5,2774.02`},{-4,2797.06`},{-3,2819.72`},{-2,2841.71`},{-1,2863.21`},{1,2904.28`},{2,2923.96`},{3,2942.95`},{4,2961.27`},{5,2978.92`},{6,2995.98`},{7,3012.38`},{8,3028.09`},{9,3043.04`},{10,3057.}

Dimensions[hcla]

plota=ListPlot[hcla,Ticks->{Automatic,Automatic},PlotRange->All,AxesLabel->{"m","frequency in cm-1"}]

f1=Fit[hcla,{1,x,x^2,x^3},x]

Clear[a,b,c,d]

nlm=NonlinearModelFit[hcla,a +b x+  c x^2+d  x^3,{a,b,c,d},x]

xxx=NonlinearModelFit[hcla,a+b x+c  x^2+d  x^3,{a,b,c,d},x]

nlm[{"ParameterTable","RSquared","AdjustedRSquared","EstimatedVariance","ANOVATable"}]

nlm["FitResiduals"]

nlm["ParameterConfidenceIntervalTable"]

nlm["ParameterConfidenceRegion"]

nlm["SinglePredictionConfidenceIntervalTable"]