The Quality of a Calculation and Theoretical Error Bars

The idea of the quality of a theoretical calculation is difficult to quantify but can be notionally separated into two parts. First, and from a strictly computational viewpoint, one must discover how sensitive the computed quantities are to the user-defined features of a given computational scheme (e.g. basis set size and electron correlation treatment in ab initio methods, parameter values in empirical and semi-empirical methods). Secondly, how well does the method reproduce actual experiment. 

A classic illustration of the quality of a calculation is provided considering the effect of the basis set of say, a Hartree-Fock (HF) calculation. (See Sect. 3 for further discussion of the HF method). For MO schemes like HF theory, the variational principle states that the lower the computed total energy, the better the result [1]. The energy can be lowered by increasing the basis set size and so large basis set calculations are often described as good quality. However, this need not imply that experiment is also well reproduced. If the basic assumptions of the HF approximation are inappropriate, then it may not be possible to predict experimental data reliably, irrespective of the basis set size. As will be seen later, this is often the case for HF calculations on TM systems. 

This example illustrates the difficulty of assigning error bars to computed quantities. The sensitivity of the computed results to the theoretical procedures loosely parallels the experimental concept of precision while the absolute reproduction of experimental data corresponds more to accuracy. And yet, 

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