2.3 Graphene
3.1.5 Plane-waves, pseudopotentials, and projector augmented
When modelling an arrangement of atoms, it is convenient to impose periodic boundary conditions, even if the system does not have a three-dimensional periodicity. The main advantage of this approach is that in a periodic system the electronic wave function can be written as a product of a cell-periodic part and a wavelike part [166]. Symbolically written as:
ψi(r) =exp[ik·r]fi(r). (3.19)
The cell-periodic part, fi(r), can be expanded using a convenient discrete
basis set consisting of plane waves. Therefore the electronic wave function can be written as a sum of plane waves:
ψi(r) =
X
G
ci,k+Gexp[i(k+G)·r], (3.20)
where G are the reciprocal lattice vectors, defined by G·l= 2πm for all l lattice vector of the crystal, and m is an integer. The coefficient ci,k+G for
plane waves with small kinetic energy are more important than those with large kinetic energy, allowing truncation of the basis set of plane waves at some particular cutoff.
Among the many advantages of using a plane-wave basis set, it is worth mentioning that the basis set is spatially unbiased and complete; it does not depend on atomic positions. Moreover plane waves and their derivatives in k-space have simple mathematical expressions. The drawback is that an additional approximation must be introduced: “the pseudopotential approx- imation”.
Pseudopotentials
In general, plane waves are a very poor basis set to expand electronic wave functions because a very large number are needed to expand core orbitals and to replicate the rapid oscillations of the wave function in the core region. Fortunately, chemical properties depend on the valence electrons much more than on core ones. Core electrons, tend to be chemically inert, while valence electrons are available for bonding. For many elements, valence electrons can only occupy the outermost electron shell. A simple example is that of methane molecules (CH4), comprising 4 C-H covalent bonds, each formed
by atoms sharing a pair of electrons. The two electrons in the inner shell of carbon are so tightly bound that they do not directly participate in the bonds.
When modelling atomic systems, a good approximation is replacing the core region with an effective potential, called a pseudopotential. This ap- proximation helps in reducing the number of electrons that must be taken into account for the solution of the quantum many-body problem, but more importantly it allows the complicated effects related to the motion of elec- trons close to the nucleus, to be hidden into the overall pseudopotential effect. Figure 3.1 shows a graphical representation of the pseudopotential concept. The corresponding pseudo-wavefunctions are identical to the real wavefunc- tions outside the core region, but are smoother and node-less within the core region [167]. Thus, they can be expanded using a much smaller basis set of plane waves.
0 r rc ΨP Ψ VP V
Figure 3.1: Schematic illustration of the pseudopotential idea. The dashed lines show the all-electron wavefunc- tion, Ψ, and ionic potential, V; the solid lines show the corresponding pseudo-wavefunction, ΨP, given by
the pseudopotential, VP. All quanti-
ties are shown as a function of dis- tance, r, from the atomic nucleus. All-electron wavefunction and pseudo- wavefunction are identical beyond the cutoff radius rc.
the norm-conversation constraints, typical of the so-called norm-conserving pseudopotential [169, 170], prevent localised electronic orbitals from being represented by very smooth pseudo wavefunctions. For first row elements and transition metals, characterised by strongly localised 2pand 3d orbitals, the resulting pseudopotentials require a large plane-wave basis set. To overcome this limitation the cut-off radius could be increased, but this adversely affects transferability.
In order to obtain maximally smooth pseudo wavefunctions, Vander- bit [171] introduced ultrasoft psuedopotentials (USPP), relaxing the norm- conservation constraint. As with norm-conserving approach, the all-electron and pseudo wave functions are required to be equal outside the cutoff radius rc, but inside rc they are allowed to be as soft as possible. The consequence
is that the pseudo wave functions are not normalised inside rc, resulting in
a charge deficit. This problem can be overcome by introducing localised atom-centered augmentation charges.
USPPs allow multiple references states for each angular momentum chan- nel leading to an improved transferability over an extended region of energy [172]. Consequently the construction of pseudopotentials is rather difficult due to the fact that many parameters must be considered.
Theab initio codeCASTEP[173] implements density functional theory us- ing plane wave basis sets and the pseudopotential approximation. It plays a central role in this work, having been employed to perform most of the calcu- lations presented. Theab initio code VASP, which also uses plane wave basis sets, implements, in addition to pseudopotentials, the projector augmented wave (PAW) method. This has been employed in a limited number of cases. The background theory is briefly introduced in the following section.
Projector augmented wave
The drawback of the plane wave pseudopotential method is that all infor- mation on the full wave function close to the nuclei is lost. Bl¨ochl [174] introduced the projector augmented wave method that allows calculations
to be performed with great computational efficiency whilst working directly with the all-electron valence wave function and all-electron valence poten- tials.
The projector augmented wave method [174] combines the versatility of the linear augmented-plane-wave (LAPW) [175] method and the formal sim- plicity of the plane wave pseudopotential approach. The plane-waves have the flexibility to describe the bonding and tail regions of the wave functions. Atomic orbitals can, on the other hand, describe correctly the nodal structure of the wave function near the nucleus.
The PAW method proves full access to the wave function, dividing it into two parts: partial wave expansions in a sphere around the atom (augmenta- tion region) and envelope functions outside the spheres (interstitial region). The envelope functions and partial wave expansions are then matched at the sphere radius of the augmentation region.
The method is based on a linear transformation of the pseudo wave func- tion, ˜Ψ, to the all-electron wave function Ψ:
|Ψi=T |Ψ˜i, (3.21) ˜
Ψ and Ψ differ near the ion core region, thus the linear transformation is assumed to be a sum of non-overlapping atom-centred contributions:
T = 1 +X
R
ˆ
TR, (3.22)
ˆ
TR is localised to sphere denoted ΩR that encloses the atom R. Within ΩR,
it can be expanded into convenient functions such spherical harmonics:
|Ψ˜i=X
i
|φ˜iici. (3.23)
Despite the complex formalism [176], the expression for the total energy, forces and stress are closely related to the ultrasoft approach, differing in the choice of the auxiliary functions and technical aspects.