Beyond the Local Density Approximation

It was a major breakthrough when it was realized, in the early 1980s, that instead of power-series-hke systematic gradient expansions one could experiment with more general functions of p r) and Vp r), which need not proceed order by order. Such functionals, of the general form 

The semiempirical functionals are fitted to selected data from experiment or from the ab-initio calculations. The higher the rung of the functional the larger is the number of parameters (functionals with as many as 21 fit parameters are popular in chemistry). Is DFT ab-initio or semiempirical As was suggested in [357] it can fall in between as a nonempirical theory when the functionals are constructed without empirical fitting. 

The best nonempirical functional for a given rung is constructed to satisfy as many exact theoretical constraints as possible while providing satisfactory numerical predictions for real systems [357]. Once a rung has been selected, there remains little choice about which constraints to satisfy (but greater freedom in how to satisfy them), [357]. Accuracy is expected to increase up the ladder of rungs as additional local ingredients enable the satisfaction of additional constraints. A short summary of 

We close this subsection with more detailed consideration of the PBE functional 

The exchange energy per electron ex (p) of the unpolarized uniform electron gas of density p r) depends on = p as sx p) = the correlation energy 

According to many authors (e.g. Handy, 1993), the local density approximation (LDA) is not adequate for useful predictions in computational chemistry. 

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Perdew, J. P., 1991, Unified Theory of Exchange and Correlation Beyond the Local Density Approximation , in Electronic Structure of Solids, P. Ziesche, H. Eschrig (eds.), Akademie Verlag, Berlin. 

Beyond the local-density approximation in calculations of Compton profiles... 

Anisimov VI (ed.) (1999) Strong Coulomb Correlatins in Electronic Structure Calculations Beyond the Local Density Approximation. Gordon Breach, New York... 

T. Grabo, T. Kreibich, S. Kurth and E.K.U. Gross, in V.I. Anisimov (Ed.), Strong coulomb correlations in electronic structure calculations Beyond the local density approximation, Gordon and Breach, Amsterdam, 2000. 

Beyond the Local Density Approximation Gradient-corrected Functionals... 

J.R Perdew in Unified Theory of Exchange and Correlation Beyond The Local Density Approximation, Nova Science, New York, 1991. 

Perdew JP (1991) Unified theory of exchange and correlation beyond the local density approximation. In Ziesche P, Eschrig H (eds) Electronic structure of solids 91. Akademie-Verlag, Berlin,... 

G. Scalmani, M. J. Frisch. A New Approach to Noncollinear Spin Density Functional Theory beyond the Local Density Approximation. /. Chem. Theory Corn-put., 8 (2012) 2193-2196. 

Strong Coulomb Correlations in Electronic Structure Calculations. Beyond the Local Density Approximation, ed. by Vladimir I.Anisimov (Gordon and Breach Sci. PubL, Amsterdam, 2000)... 

Electronic Structure Beyond the Local Density Approximation,. I. Anisimov, Ed., Gordon and Breach, Tokyo, 2000, pp. 203-311. Orbital Eunctionals in Density Eunctional Theory The Optimized Effective Potential Method. 

Functional Theory beyond the Local-Density Approximation. 

Barone, V., Scuseria, G. E. (2004). Theoretical study of the electronic properties of narrow single-walled carbon nanotubes Beyond the local density approximation. The Journal of Chemical Physics, 121,10376-10379. 

A very difficult challenge is the increase in accuracy. The melting of silicon discussed above shows rather dramatically the enormous demands on a theory to predict the temperatures of phase transitions. It is not clear at the moment how such a high accuracy could be achieved. Improvements beyond the local density approximations such as gradient corrections have been a step forward. However, the lack of a systematic approach towards the exact limit is discouraging. Possibly quantum Monte Carlo holds the promise for such a systematic approach. 

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