How Accurate Are DFT Calculations

Throughout this book, we have focused on using plane-wave DFT calculations. As soon as you start performing these calculations, you should become curious about how accurate these calculations are. After all, the aim of doing electronic structure calculations is to predict material properties that can be compared with experimental measurements that may or may not have already been performed. Any time you present results from DFT calculations in any kind of scientific venue, you should expect someone in the audience to ask How accurate are DFT calculations  

So how accurate are DFT calculations It is extremely important to recognize that despite the apparent simplicity of this question, it is not well posed. The notion of accuracy includes multiple ideas that need to be considered separately. In particular, it is useful to distinguish between physical accuracy and numerical accuracy. When discussing physical accuracy, we aim to understand how precise the predictions of a DFT calculation for a specific physical property are relative to the true value of that property as it would be measured in a (often hypothetical) perfect experimental measurement. In contrast, numerical accuracy assesses whether a calculation provides a well-converged numerical solution to the mathematical problem defined by the Kohn-Sham (KS) equations. If you perform DFT calculations, much of your day-to-day 

Density Functional Theory A Practical Introduction. By David S. Sholl and Janice A. Steckel Copyright 2009 John Wiley Sons, Inc. 

This discussion of numerical accuracy leads to several principles that should be applied when reporting DFT results  

Accurately define the mathematical problem that was solved by specifying the exchange-correlation functional that was used. Many different functionals exist so it is not sufficient to state the kind of functional (LDA, GGA, etc.)—the functional must be identified precisely. 

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We now turn to the issue of physical accuracy. Just as accuracy should not be considered as a single topic, physical accuracy is too vague of an idea to be coherently considered. It is much better to ask how accurate the predictions of DFT are for a specific property of interest. This approach recognizes that DFT results may be quite accurate for some physical properties but relatively inaccurate for other physical properties of the same material. To give just one example, plane-wave DFT calculations accurately predict the geometry of crystalline silicon and correctly classify this material as a semiconductor,... 

Tam et al. [141] attempted to determine how reliable and accurate CC and DFT/ TDDFT calculations are for this conformationally flexible molecule. In addition, they explored the sensitivity of the chiroptical response to two different factors. One was the accuracy of the mole fractions, and another was how different were the ORs of individual rotamers calculated at different levels of theory. It was found that with DFT, at the B3LYP/aug-cc-pVDZ level, the optical rotations were overestimated while CC yielded better agreement with experiment [141, 142], The predicted gas phase optical rotation, averaged by CC or DFT mole fractions, were not in good agreement with either gas or solution phase experimental measurements. The DFT calculated optical rotations differed between 15 and 65% from experiment. 

Other publications, however, report more accurate values of B3LYP gas phase Gibbs free energy calculations on aliphatic amines, diamines, and aminoamines. In 2007 Bryantsev et al. reported that B3LYP calculations with the basis set 6-31-h-G had a mean absolute error of 0.78 kcal/mol from experimental values of the gas phase basicity (AGg s) of the reverse reaction of equation 1 reported in the NIST database [58]. This accuracy is comparable to that of expensive, high level model chemistries, but because the experimental values have uncertainties of 2 kcal/mol, it is difficult to discern exactly how accurate the calculations are in comparison to values in the other publications [81]. The take-home message remains the same always benchmark DFT calculations for the systems you are interested in computing [52]. 

In this chapter we will provide a critical review of the use of 2- and 4-component relativistic Hamiltonians combined with all-electron methods and appropriate basis sets for the study of lanthanide and actinide chemistry. These approaches provide in principle the more rigorous treatment of the electronic structure but typically demand large computational resources due to the large basis sets that are required for accurate energetics. A complication is furthermore the open-shell nature of many systems of practical interest that make black box application of conventional methods impossible. Especially for calculations in which electron correlation is explicitly considered one needs to find a balance between the appropriate treatment of the multi-reference nature of the wave function and the practical limitations encountered in the choice of an active space. For density functional theory (DFT) calculations one needs to select the appropriate density functional approximation (DFA) on basis of assessments for lighter elements because little or no high-precision experimental information on isolated molecules is available for the f elements. This increases the demand for reliable theoretical ( benchmark ) data in which all possible errors due to the inevitable approximations are carefully checked. In order to do so we need to understand how f elements differ from the more commonly encountered main group elements and also from the d elements with which they of course share some characteristics. 

As was the case for XANES, here electronic structure calculations including solvent effects are used to draw conclusions on the most probable coordination isomer and from that structure the interatomic bond distances can be directly compared to the computed EXAFS spectra [150-152]. As EXAFS spectroscopy probes the structure of a statistically averaged system, the most appropriate way of comparing theoretical EXAFS data to experimental ones is to use molecular dynamics trajectories to sample the configuration space, select snapshots and finally compute a statistically average spectrum [89,153] with a direct estimate of the mean square relative disorder (MSRD, also called or the EXAFS Debye-Waller term). However, as discussed in Section 11.2.3, the relevance of MD simulations hinges on how accurate intermolecular interactions are, given that these are usually obtained at DFT level (in Car-Parinello MD simulations) or with force-fields (in classical MD simulations). 

The calculations above allowed the positions of atoms to change within a supercell while holding the size and shape of the supercell constant. But in the calculations we introduced in Chapter 2, we varied the size of the supercell to determine the lattice constant of several bulk solids. Hopefully you can see that the numerical optimization methods that allow us to optimize atomic positions can also be extended to optimize the size of a supercell. We will not delve into the details of these calculations—you should read the documentation of the DFT package you are using to find out how to use your package to do these types of calculations accurately. Instead, we will give an example. In Chapter 2 we attempted to find the lattice constant of Cu in the hep crystal structure by doing individual calculations for many different values of the lattice parameters a and c (you should look back at Fig. 2.4). A much easier way to tackle this task is to create an initial supercell of hep Cu with plausible values of a and c and to optimize the supercell volume and shape to minimize... 

In Kohn-Sham (KS) density functional theory (DFT), the occupied orbital functions of a model state are derived by minimizing the ground-state energy functionals of Hohenberg and Kohn. It has been assumed for some time that effective potentials in the orbital KS equations are always equivalent to local potential functions. When tested by accurate model calculations, this locality assumption is found to fail for more than two electrons. Here this failure is explored in detail. The sources of the locality hypothesis in current DFT thinking are examined, and it is shown how the theory can be extended to an orbital functional theory (OFT) that removes the inconsistencies and paradoxes. 

This could be taken one step further, to evaluating functionals in general, or at least how good they are for a specific system. Since current DFT methods have problems in accurately calculating solid materials in general and solid surfaces in particular, much work has been undertaken to develop more accurate functionals for solids and solid surfaces [29-31]. Since bond valence parameters are empirically derived from crystal structures and are universal, they should provide a good check on these new functionals. A more accurate functional should yield structures with bond valence sum values that are closer to those derived from crystal structures. Similarly, such a functional would be expected to give a lower Sll for surface structures experimentally known to be stable, at least as compared to those which are known from experiment to be unstable. 

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