Chapter 4 Effects of microscopic density variations on stopping power
4.2 Introducing Density Functional Theory (DFT)
It has been stated previously that single-particle systems can be described math- ematically by wavefunctions that are solutions to the single-particle Schr¨odinger equation. Although, when considering more complex systems of many particles, the system can still be described by a composite wavefunction. In this scenario, the wavefunction will be a solution to the n-particle Schr¨odinger equation. This equation can be written as
" X i − ~ 2 2me∇ 2 ri+Vext(ri) +Ve−e(ri) # Ψ(r) =EΨ(r), (4.1) whereVext(ri) is the external Coulomb potential,Ve−e(ri) is an electron interaction potential, E is the energy eigenvalue of the system and Ψ(r) is the many-particle wavefunction. Solving for the wavefunctions of complex many-particle systems can be complicated and so certain assumptions may simplify this process. DFT simpli- fies the system by solving for a set of single-particle wavefunctions in an effective potential. This effective potential is tuned to represent the local effects of the n
single-particle potentials of the surrounding medium. In systems where an assump- tion of periodicity at some scale can be made, the full DFT calculation is often greatly simplified. Naturally, from the full description of a periodic system element, a description of a medium of arbitrary size can be produced.
DFT can provide an accurate representation of the electronic structure of a dense plasma and so is useful in modelling the systems that we are attempting to describe. DFT is built on the Hohenberg-Kohn theorem which states that the ground state electron density completely describes a physical system [Hohenberg and Kohn [1964]]. Once the ground state electron density profile is known, every
physical property of the system can be calculated from functionals of this ground state density, where the density can be determined from the wavefunction by the relation,
ρ=|ψ|2 =ψψ∗. (4.2)
The density is therefore independent of the phase of the wavefunction and so no ef- fects of phase dependence are observed in the physical quantities of the system. The Kohn-Sham ansatz can be employed to connect the original many-particle system to a secondary, non-interacting system under the premise that the original ground state density must be preserved [Kohn and Sham [1965]]. Using atomic units, the Hamiltonian of the non-interacting system can be written as [Martin [2004]]
HKSσ = 1 2∇
2+Vσ
KS(r), (4.3)
whereVKSσ is the Kohn-Sham effective potential seen by the electrons. This potential will reproduce the original ground state density and consists of several contributions, which are given as
VKSσ (r) =Vext(r) +VHartree[n] +Vxcσ[n]. (4.4)
Vext(r) is the real potential from ions in the system and external fields, VHartree[n]
is the Hartree potential, determined by the local density andVxcσ[n] is the exchange correlation potential which is also determined by the local density. Determining the exchange correlation term is often difficult and in many correlated systems this term will need to be approximated. These approximations introduce some uncertainty into this method of calculation [Martin [2004]]. Much work has been done on refining the values of this exchange correlation term. Two effective methods are the local density approximation (LDA) and the generalised gradient approximation (GGA). LDA assumes that for any given particle, the exchange correlation energy is the same as the energy of a homogeneous electron gas with equal density to the system being investigated. This method can be refined, incorporating the effects of spin, which is known as the local spin density approximation (LSDA). GGA makes a local approximation to the exchange correlation energy, taking the gradient of the density into account. For many systems this consideration results in an improvement in the accuracy of the final calculated ground state density. To investigate a system using DFT, the Kohn-Sham equations are solved in a self-consistent manner. An initial ground state density can be estimated and then, also inserting an estimation of the total potential, the generalised Schr¨odinger equation must be solved for the electron
wavefunctions.
−12∇2+VKSσ (r)
ψσi(r) =εiσψσi(r), (4.5) where εσi and ψσi(r) are the particle energy eigenvalues and wavefunctions respec- tively. These updated wavefunction solutions give a new value for the electron den- sity. From this density, a new potential can be calculated and again the Schr¨odinger equation can be solved, resulting in a new electron density. Repeating this process until convergence is reached, the system will quickly relax into its true ground state density.
Originally DFT assumed a system was in its absolute ground state, with zero temperature. Later work included effects of temperature such as the work of Mermin [Mermin [1965]]. This temperature modelling is important for the appli- cation of DFT in the work presented in this thesis. Since their conception, DFT calculations have also been successfully combined with molecular dynamics (MD) simulations, which was achieved in 1985 by Car and Parrinello [Car and Parrinello [1985]]. Originally, the work of Car and Parinello assumed that the global electron structure would be constant for many MD steps and so to lower the computational intensity, their approach performed many MD steps for each complex DFT calcu- lation. However, this method was reversed in the Born-Oppenheimer approach in which it is assumed that the lightweight electrons respond instantly to the motion of the heavier ions and so a DFT convergence of the electron wavefunctions is cal- culated for each ion MD step. This approach gives improved results over the Car and Parrinello technique.
As one might expect, the unification of the DFT modelling with the dynamic processes of the ions increases the level of accuracy in the resulting calculated elec- tron structure. A side-effect of employing this technique is that the system must now also be allowed to reach a thermodynamic equilibrium, i.e. the ion structure must be allowed to relax into a self-consistent configuration, before physical prop- erties can be calculated. All DFT-MD simulations undertaken in the course of the work presented in this thesis were performed by J. Vorberger using the simulation package VASP [Kresse and Hafner [1993]]. These calculations are employed to a considerable extent in the treatment of microscopic density fluctuations described in this chapter.