Quantum in physics - Coursework Example

(Belloni et al, 2005) Local density Approximation (LDA) Exchange correlation (XC) is the relationship between different electrons and the exchange of energy in the electronic setup of a particular quantum system. Further density functional theory (DFT) is study of an electronic structure when it is at its least excited state called the ground state or the zero-point energy of the system. Electron density is defined as the probability of an electron filling up a miniscule space around any particular point.

It is denoted by n(r). (Computational Materials Science Group,1998) Local Density approximation can now be defined as an approximation of the exchange correlation in the Density functional theory or in other words the energy relationship between different electrons in an electronic structure at ground state. This can be found out using a function of the electron density at each spatial point. Further Homogeneous electron gas (HEG) is the interaction of positive atomic nuclei that are uniformly distributed in space with the negatively charged electrons that have a uniform density in the same space.

Local Density approximations are thus most accurately derived when functional integrals are made on the HEG approximation. (Computational Materials Science Group,1998) For a unpolarized system the LDA can be written as ELDAXC=??xc(n(r)n(r)dr Where n(r) is the electron density and ?xc is the exchange –correlation energy density. Exc can further be split up as Exc=Ex + Ec where Ex are the exchange functions and Ec is the correlation function.( Computational Materials Science Group,1998) The Perdew-Burke-Ernzerhof (PBE) The interpretation and evaluation of Density functional Theory (DFT) has made the calculation of systems at ground state very effective albeit with several drawbacks.

The exchange correlation energy obtained was an approximate value using the LDA method. Further improvement resulted in the introduction of GGA’s or generalized gradient approximation to fine tune the LDA. Creating a functional without any empirical parameters the GGA were made to follow certain basic constraints. Since true electron density was actually non-homogenous, it was proposed in 1980 to enhance the density n(r) at a particular point r with inputs regarding the gradient of electron density.

To derive an accurate value of the DFT a higher functional satisfying several parameters is ideally chosen. The PBE functional is an ideal functional proposed by Perdew, Burke, Ernzerhof in 1996. Experiments conducted thereafter have proved that the values obtained using these GGA’s were in accordance with those obtained using numerical tests.( Evarestiv R.A, 2007) This PBC functional can be defined as a summation of two derivatives, the XC hole and the energy derivative. This functional is constructed on the premise that the constraints of a particular hole are known and the exchange correlation hole is defined per these constraints.

ELSDXC[na, nb]=? d3rn(r)[?x(n(r))f(?, r) + ?c(rs(r), ?(r))] Where ?=(na-nb)/ (na+nb) is the relative spin polarization and f(?)=1/2[(1+?)4/3 +(1-?)4/3] The exchange energy ?x per electron depends on rs=[3n/4л]1/3 and correlation energy ?c depends on rs and ?.( Evarestiv R.A, 2007) The exchange PBE functional is written as a combination of Ex and Ec. Here the exchange PBE functional EPBEX(n)=?d3rn?x(n)Fx(s) With Fx(s) =1+k-k/(1+µs2/k) , here k=0.804 and µ=021951 EPBEC[na, nb]=? d3rn[?C(rs,?)+ H(rs,?,t] where H=? ?3 ln