Kohn-Sham positive-energy

Figure 2.7. Simplified energy diagram of copper and copper-oxo sites (BP/DNP), showing relative positions of the triplet and singlet states, along with the relevant Kohn-Sham orbital contours.

Since the principal applications of MST are in solid-state theory, which traditionally uses Rydberg rather than Hartree units of energy, this convention will be followed in the present chapter. Restricting the formalism to local potential functions as in Kohn-Sham theory, the Schrodinger equation for an electron of positive energy is... 

Since DFT calculations are in principle only applicable for the electronic ground state, they cannot be used in order to describe electronic excitations. Still it is possible to treat electronic exciations from first principles by either using quantum chemistry methods [114] or time-dependent density-functional theory (TDDFT) [115,116], First attempts have been done in order to calculate the chemicurrent created by an atom incident on a metal surface based on time-dependent density functional theory [117, 118]. In this approach, three independent steps are preformed. First, a conventional Kohn-Sham DFT calculation is performed in order to evaluate the ground state potential energy surface. Then, the resulting Kohn-Sham states are used in the framework of time-dependent DFT in order to obtain a position dependent friction coefficient. Finally, this friction coefficient is used in a forced oscillator model in which the probability density of electron-hole pair excitations caused by the classical motion of the incident atom is estimated. 

Once the Kohn-Sham equations have been solved, we are in a position to evaluate energies, forces and the electronic structure of a material itself. In particular, with the eigenvalues e, and corresponding wave functions i/f, (r) we can compute the energy of the system explicitly. As can be seen from the discussion given above, and as has been true with each of the total energy methods introduced in this chapter, in the end we are left with a scheme such that once the nuclear... 

Car and Parrinello [97,98] proposed a scheme to combine density functional theory [99] with molecular dynamics in a paper that has stimulated a field of research and provided a means to explore a wide range of physical applications. In this scheme, the energy functional [ (/, , / , ] of the Kohn-Sham orbitals, (/(, nuclear positions, Ri, and external parameters such as volume or strain, is minimized, subject to the ortho-normalization constraint on the orbitals, to determine the Born-Oppenheimer potential energy surface. The Lagrangian,... 

The Kohn-Sham-Dirac equation (27) introduced in the last section is the basis of most relativistic electronic structure calculations in solid state theory. There are certain aspects which make the numerical solution of this four-component equation more involved than its non-relativistic coimterpart The Hamiltonian of the Kohn-Sham-Dirac equation is, unlike its Schrodinger equivalent and unlike the field-theoretical Hamiltonian (7) with the properly chosen normal order, not bounded below. In the limit of free, non-interacting particles the solutions of the Kohn-Sham-Dirac equation are plane waves with energies e(k) = cVk -I- c, where positive energies correspond to electrons and states with negative energy can be interpreted as positrons. For numerical procedures, which preferably use variational techniques to find electronic solutions, this property of the Dirac operator causes a severe problem, which can be circumvented by certain techniques like the application of a squared Dirac operator or a projection onto the properly chosen electronic states according to their above definition after Eq. (19). 

The positive kinetic energy density of the occupied Kohn-Sham orbitals is... 

Only the minimum value of the Kohn-Sham energy fimctional has physical meaning. At the minimum, the Kohn-Sham energy functional is equal to the groimd-state energy of the system of electrons with the ions in positions Ri. 

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