Permutaciones y combinaciones

kinetic energ)-, T[n], is defined in terms of a system of non-interacting electrons with the

same densit}^ as the real system. Equation 2.12.

(

y2 A

T \ n { r ) \ = ^ \ y / , { r ) y / . { r ) d r (2.12) \ Z I

T h e electrostatic energy, J[nJ, arises from the classical interaction betw een tw o charge

densities, n { r i ) o r n ( r2) , w hich is sum m ed over all possible pairwise interactions, E q u a tio n 2.13. T his is the same process as em ployed in H F theorj'.

(2.B ) ^ i * J

C o m bining the kinetic energy, the electrostatic energy’, and the electron-nuclear in teraction energ}' leads to a full expression for the energ\- o f an N electron system. W ithin the K oh n -S h am schem e, is therefore defined as containing a co n tribution due to the difference betw een the true kinetic energ}- o f a system and the approxim ated o ne electron kinetic energ)’, as weU as a contribution due to exchange and correlation energies. W hereas H F only includes an exchange potential, the exchange-correlation functional in D F T is a m ore general expression w hich can include term s accounting for b o th the exchange energy and the electron correlation. If the exact exchange-correlation p otential is know n, th en D F T results in the exact gro u n d state energ}'.

2.2.6 E xchange-C orrelation Energy

T h e exchange-correlation functional, E^c, is crucial for the success o f the D F T approach. T h e K ohn-S ham approach allows for an exact description o f m o st o f the co n trib u tio n s to the electronic energ}’ o f a m olecular system. All rem aining parts are coupled in to the Ejjc- In principle an E^c functional exists th at gives the exact g ro u n d state energy, although its form is unk n o w n and only approxim ate solutions are available. H ow ever, one reason that D F T is so appealing is that even relatively sim ple

approxim ations to E^c can give reasonable results in terms o f the optim ized lattice constant and electronic structure.

The majority o f E^c functionals are based upon the idea o f a uniform electron gas modeP'. This is a hypothetical system in which an almost infinite num ber o f electrons are free to move in an electrically neutral environment. Conceptually, this is in many ways similar to the behaviour o f the valence electrons in an ideal metal. The benefit o f using such a model is that highly accurate exchange and correlation energies are know n for this type o f system through a series o f in-depth quantum -M onte-Carlo simulations'*^’.

The simplest way to derive an potential for a real system based on contributions from the uniform electron gas model is the local density approxim ation (LDA). Here £j;c depends only on the value o f the densit}' at a position r. The density taken at this point is then referenced with the contributions that a uniform electron gas o f equal density would have. This process is repeated for each point in space, with the individual contributions summed up to provide the £xc value for the total system. This is shown schematically in Figure 2.1. The largest approximation made in the LDA is that E^c depends only on the density at the position r and not its local environm ent. As such, this approach works well for systems in which the electron density does not vary rapidly and offers results o f comparable accuracy to the H F approach. LDA calculations generally result in exaggerated binding energies and underestimated bond lengths.

The generalised gradient approxim ation (GGA) was developed to overcom e the shortcomings o f LDA and forms the basis for the m ost frequently used functionals. Here a uniform gas is again considered, as in the LDA. However within the G G A , Eye is dependent on both the value o f the densit}' at a position r and the variation close to r

(the gradient). This accounts better for the non-homogeneit\" o f the true electron

density. The GGA approach allows for more flexibiUt}' and accuracy in determining the

electronic structure o f molecules and solids using DFT. In contrast to the LDA, it has a

tendency to overestimate bond lengths. The exchange and correlation functional derived

by Perdew and Wang (PW91)^’ has been one o f the most widely used GGA functionals,

but this has been superseded by the PBE fu n ctio n al which features a simpler form and

offers improved performance for a number of systems. Both of these functionals were

applied in our D FT calculations and are generally considered to be first-principle in

nature as they do not contain parameters other than fundamental constants and those

specified through a quantum mechanical relation.

from real system

-► E J n (r ,)) --- ► KJn(rj))

from hom ogeneous electron gas

Figure 2.1 Schematic of the LDA approximation^’.

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2.3 Sim ulating Solids

2.3.1 Periodic Boundary Conditions

The large number of electrons contained in a macroscopic crystal prohibits a direct