Interpretation of the Kohn-Sham values

The fundamental objects of the KS approach are the KS spin-orbitals and the KS en- ergies, which are obtained as the eigenvectors and the eigenvalues of the system of KS equations (1.21). We recall that the KS system was constructed to reproduce the correct ground state density of an interacting many-body system via Eq. (1.16). For now we will ignore the approximations that are made to construct the XC functional and treat the KS approach as exact. As such the KS eigenvalues are only mathematical tools that allow us to construct the density. In other words, we can not grant physical meaning to the KS spin-orbitals. In practice, this fact is regularly ignored and the KS spin-orbitals are treated as quasi-particle states, which in this case have an infinite lifetime, since they are obtained as the eigenvectors of the KS system. Results, which are obtained in this way, are usually in good agreement with experiment. For example, a practical application of the GW approach relies directly on the KS eigenstates to obtain a correc- tion for the gap (see Ch. A4 in Ref. [51]). In other words, the KS eigenstates are often a good starting point to describe the system quasi-patricle spectrum. An explanation for this fact can be offered using the same line of arguments that are given to justify the independent-particle approach. The expected lifetime of the quasi-particles in, e.g. normal metals, is much larger than the relaxation time due to other sources of scatter- ing (see Ch. 17 in Ref. [65]), especially at higher temperatures. Therefore, the decay of the quasi-particle state, on the relevant timescale, does not play an important role and the KS spin-orbitals can be safely used. Obviously, there will be systems where this approximation breaks down, but the problem is not directly related to DFT.

Similarly, the KS eigenvalues can not be given a direct physical significance, with the exception of the highest occupied energy level. In the latter case Janak’s theorem [66] demonstrates that this eigenvalue corresponds to the first ionization energy of the system. By combining the expressions (1.18) and (1.21), the total energy of the KS system can be expressed as E =X i εi− 1 2EH +EXC − Z drn(r)Vxc[n(r)]. (1.35)

From here it is obvious that the sum of the KS eigenvalues does not yield the energy of the system. The additional terms are known as the double counting correction. From an empirical point of view, one can observe that the variation procedure employed to obtain the ground state usually yields structural parameters of molecules and solids which are within 1 % to 2 % of the experimental results. Similarly, the derivative of the energy, with respect to the position of the atoms in the Born-Oppenheimer approximation, i.e. the forces on the atoms, are also valid. This allows for structural relaxation to be performed, which is crucial for practical applications. For example, the bulk modulus, describing the elasticity of solids, is usually captured with an error smaller than 10 % [67]. Therefore, the energies obtained from DFT are valid, although the KS eigenvalues can not be directly interpreted.

There are however situations when the KS eigenvalues are directly used, e.g. to estimate the bandgap of a material. Most of the cases implicitly treat DTF as an effective “mean- field” theory, which gives a sufficiently good description of the quasi-particle states. Some of these cases are discussed in more detail in section 1.1.3. Here we will shortly discuss the bandgap problem to illustrate some of the limitations of DFT, which have to be kept in mind. We start from the definition of the conduction and the valence band energy, cand v, respectively. These can be expressed as the difference in the total energy of a

true many-body system ofN particles,EN_{, as}

c=EN+1−EN

v =EN−EN−1 . (1.36)

The fundamental bandgap is defined as ∆ =c−v =

=EN+1+EN−1−2EN , (1.37)

where (1.36) was used in the last step. Therefore, the fundamental bandgap can be calculated directly, via Eq. (1.37), by calculating the energies of the systems withN+1, N-1, and N electrons. The former two correspond to the negatively and positively ionized systems hence, the energy differences in Eq. (1.36) can also be expressed in terms of the ionization energy, I, and the chemical affinity, A (cf. figure 1.2). It was demonstrated [68] that within the KS approach the fundamental gap is related to the

vacuum level (N+1) HOMO (N) LUMO (N) HOMO

A

A

Δ

Δ

Δ

I

Figure 1.2: Molecular level diagram illustrating the Kohn-Sham (KS) bandgap prob- lem. The HOMO and the LUMO denote the highest occupied and the lowest unoccupied molecular orbital, respectively. The number of electrons in the system,N, is indicated in the brackets. The true ionization energy, I, and the affinity, A, are shown. Within the KS approach, the affinity,AKS, is determined by the KS bandgap, ∆KS. The latter

represents a lower bound for the fundamental bandgap, ∆.

derivative of the energy functional as

∆ = δE[n] δn N=N+ −δE[n] δn N=N− = = δTKS[n] δn N=N+ − δTKS[n] δn N=N− + δEXC[n] δn N=N+ −δEXC[n] δn N=N+ = = ∆KS+ ∆XC , (1.38) where ∆KS corresponds to the difference between the lowest unoccupied and the highest

occupied energy level in the ground state calculation. Here we have used a shorthand notation: N± ≡N ±δ. Therefore, the KS bandgap will always underestimate ∆. The relation is illustrated in figure 1.2. The calculations based on the local density approxi- mations (cf. 1.1.2.4) show that the ∆KS systematically underestimates the bandgap by

roughly 0.5 eV to 1.0 eV [69].

It is a well known property of the true XC functional that the derivative of the energy functional in Eq. (1.38) has to be discontinuous, since it is determined by the difference in the chemical potential forN andN±1 particle system. For a fractional occupancy the total energy is piecewise linear, since it can only be defined in terms of an ensemble of

systems, which have an integer number of electrons (see Ch. 2 in Ref. [41]). Therefore, the functional derivative has to change discontinuously at the points with integer occupancy. The local density approximations of the XC functional, which are most commonly used in practice, do not describe this feature. Furthermore, from Eq. (1.38) we can infer that any approximation to the XC functional will result in an error when estimating the bandgap, even in the case when the total energies for a system withN,N±1 electrons are directly calculated. This is due to the underestimation of the correlation effects that should be contained in ∆xc. A common way to practically obtain a correction for the

bandgap is to employ the one-shot GW approach [51, Ch. A4], as a post-processing step.