4.4 Kohn–Sham equations
4.4.2 Self–consistent Iterative scheme
Computationally, the Kohn–Sham method for DFT is implemented in an iterative scheme, as the Kohn–Sham equations are a nonlinear set of differential equations. Roughly, withimarking the iteration step counter, the scheme may be summarized as follows:
1. An existing distribution ρ(i) (initial guess distribution for the first step i = 0) is used to construct the potential terms: the external potential V by means of an Ewald summation over the ions, the Coulomb potentialJ solving the Poisson equation for the charge distribution, and the exchange–correlation potentialVxc determined point–to–point from tabulated values. The single terms are then summed to obtain the effective potential
Vef f(i) =V(i)+J(i)+Vxc(i). (4.81) 2. The corresponding Kohn–Sham equation
−~ 2 2m∇ 2+V(i) ef f(r)
φ(i)λ)(r) =(i)λ φ(i)λ,(r) (4.82) is solved either by matrix inversion or some other method such as a predictor– corrector scheme.
3. The total energy is calculated as a sum over the occupied space and the correc- tions:
E(i)= occ X
λ
(i)λ −J(i)+Exc(i)−
Z
Vxc(i)(r)ρ(i)(r)dr. (4.83) 4. The energy thus obtained is compared with the previous one. If they are equal
within a predetermined accuracy tresholdδ, the iteration is stopped:
if E(i)−E(i−1)< δ Stop; (4.84)
if not, the calculation proceeds to the next step.
5. All the occupied states are summed up to obtain a new density
ρ(i)new= occ X
λ
6. Part of this new charge density is mixed with the previous charge density:
ρ(i+1)=M[ρ(i), ρ(i)new] (4.86) where the operatorMrepresents the mixing algorithm of choice among the many available. This procedure ensures numerical stability.
7. Go back to step 1.
The Kohn–Sham scheme is illustrated as a flow chart in Figure 4.1.
This scheme provides a single–point calculation of the electronic ground state of the system. This means that the external potential is determined by the ionic configuration and is not, in general, the minimum of the Born–Oppenheimer potential energy surface. By virtue of the Hellmann–Feynman theorem [19, 20] (see Appendix A), the scheme above is usually embedded in a cycle of ionic relaxation, that is, after a geometry is electronically converged, the forces between the ions are calculated, then the ions are moved by a small displacement and a new self–consistent cycle is started, until the maximum force acting on the ions is smaller than a predetermined threshold. Both the speed and the reliability of a ionic relaxation depend strongly on the initial guess geometry, which must be wisely chosen, as the minimization algorithms are local and will converge to the closest local minimum instead of the global minimum. A relaxation calculation typically consists of few to few hundreds ionic steps.
START
Guess densityρi
Calculate Vef f(i) =V(i)+J(i)+Vxc(i)
Solve Kohn–Sham equation −~ 2 2m∇ 2+V(i) ef f(r)
φ(i)λ)(r) =(i)λ φ(i)λ,(r)
Calculate total energyE(i)=Pocc λ (i) λ −J(i)+E (i) xc − R Vxc(i)(r)ρ(i)(r)dr E(i)−E(i−1) < δ ? STOP
New densityρ(i)new=Poccλ |φ (i) λ |2
Mix densityρ(i+1) =M[ρ(i), ρ(i)new] yes
no
Figure 4.1: Flow chart summarizing the passages of the iterative Kohn–Sham scheme for Density Functional Theory.
REFERENCES
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136(3B):B864–B871, 1964.
[4] M. Levy. Universal variational functionals of electron densities, first–order density matrices, and natural spin–orbitals and solution of the v-representability problem.
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[5] M. Levy and J. P. Perdew. In defense of the Hohenberg–Kohn theorem and density functional theory. International Journal of Quantum Chemistry, 21(2):511–513, 1982.
[6] E. H. Lieb. Thomas–Fermi and related theories of atoms and molecules. Reviews of Modern Physics, 53:603–641, 1981.
[7] E. H Lieb. Density functionals for Coulomb systems. International Journal of Quantum Chemistry, 24(3):243–277, 1983.
[8] J. Harriman. Orthonormal orbitals for the representation of an arbitrary density.
Physical Review A, 24(2):680–682, 1981.
[9] L. H. Thomas. The calculation of atomic fields. Mathematical Proceedings of the Cambridge Philosophical Society, 23(05):542, 1926.
[10] E. Fermi. Eine statistische methode zur bestimmung einiger eigenschaften des atoms und ihre anwendung auf die theorie des periodischen systems der elemente.
Zeitschrift f¨ur Physik, 48(1-2):73–79, 1928.
[11] C. F. V. Weizs¨acker. Zur theorie der kernmassen. Zeitschrift f¨ur Physik, 96(7- 8):431–458, 1935.
[12] W. Kohn and L. J. Sham. Self–consistent equations including exchange and cor- relation effects. Physical Review, 140(4A):A1133–A1138, 1965.
[13] E. Wigner. On the interaction of electrons in metals.Physical Review, 46(11):1002– 1011, 1934.
[14] M. Gell-Mann and K. Brueckner. Correlation energy of an electron gas at high density. Physical Review, 106(2):364–368, 1957.
[15] J. P. Perdew. Self–interaction correction to density–functional approximations for many–electron systems. Physical Review B, 23(10):5048–5079, 1981.
[16] S. H. Vosko, L. Wilk, and M. Nusair. Accurate spin–dependent electron liquid correlation energies for local spin density calculations: a critical analysis.Canadian Journal of Physics, 58(8):1200–1211, 1980.
[17] M. Levy and J. P. Perdew. Hellmann–Feynman, virial, and scaling requisites for the exact universal density functionals. Shape of the correlation potential and diamagnetic susceptibility for atoms. Physical Review A, 32(4):2010–2021, 1985. [18] E. H. Lieb and S. Oxford. Improved lower bound on the indirect Coulomb energy.
International Journal of Quantum Chemistry, 19(3):427–439, 1981.
[19] H. Hellmann. Einf¨uhrung in die Quantenchemie, page 285. F. Deuticke, 1937. [20] R. P. Feynman. Forces in molecules. Physical Review, 56:340–343, 1939.
Chapter 5
Theory of STM
5.1
Electron transport in the low–conductance regime
In Scanning Tunneling Microscopy the main physical obstacle to the transport of elec- trons is the vacuum barrier between the sample and the probe tip. In this case variations of the conductance across the tunneling barrier due to electron–electron interactions can be considered small enough to be treated with perturbation theory. Therefore, the main task is a suitable description of the transport across the barrier. Additional effects, such as electron–phonon excitations, can be incorporated as extensions of the basic model. Together with the variation of the tunneling current due to the magnetic properties of the system, they account for the bulk of experimental observations. At present, the following four theoretical models of electron tunneling are used in nearly all simulations of STM processes:
• the Tersoff–Hamann approach [1, 2]: isocurrent contours are derived from the electronic structure of the sample alone;
• the Bardeen, or transfer Hamiltonian approach [3]: the electronic structure of the tip is included;
• the Landauer–B¨uttiker approach [4]: equivalent to the Bardeen’s method, but it includes multiple tunneling pathways between the tip and the surface;
• the Keldysh [5] approach, or non–equilibrium Green’s function approach: it in- corporates inelastic effects.
The models are listed in order of increasing complexity. For the purpose of the present Thesis, the Tersoff–Hamann approach has proved sufficiently accurate and therefore will be presented in more detail. We will present Bardeen’s tunneling derivation first, as it provides the basis upon which the Tersoff–Hamann model is built. A thorough treatment can be found in Ref. [6].