Kohn-Sham equations total energy

Once a solution of the Kohn-Sham equation is obtained, the total energy can be computed from... 

In addition to the energy terms for the exchange-correlation contribution (which enables the total energy to be determined) it is necessary to have corresponding terms for the potential, Vxc[p(i )]/ which are used to solve the Kohn-Sham equations. These are obtained as the appropriate first derivatives using Equation (3.52). 

Given in Table 7.1 are the results [24] of the total energy of some atoms obtained by solving the Kohn-Sham equation self-consistently with the exchange potential Wx within the central field approximation. The energy is obtained from Equation 7.10... 

Compare the calculated electron density, nKs(r), with the electron density used in solving the Kohn-Sham equations, (r). If the two densities are the same, then this is the ground-state electron density, and it can be used to compute the total energy. If the two densities are different, then the trial electron density must be updated in some way. Once this is done, the process begins again from step 2. 

To[ (r)] defines the kinetic energy of a non-interaoting electron gas which requires the density />(r). (r) is the classical (direct) Coulomb interaction which is equivalent to the Hartree potential. Using the constraints that the total number of electrons are conserved, Kohn and Sham [20] reduced the many electron Schrodinger equations to a set of one-electron equations known as the Kohn-Sham equations... 

It should be noted that the eigenvalues e,- obtained in the solution of the Kohn-Sham equations axe not equal to the ionization energies as known by the Koopmans theorem in Hartree Fock [51]. Slater found that ionization energies could be obtained by evaluation of the total energy for the neutral and ionized systems [4], which gives... 

The exact energy functional (and the exchange correlation functional) are indeed functionals of the total density, even for open-shell systems [47]. However, for the construction of approximate functionals of closed as well as open-shell systems, it has been advantageous to consider functionals with more flexibility, where the a- and j8-densities can be varied separately, i.e. E[p, p ]. The variational search for a minimum of tire E[p, p ] functional can be carried out by unrestricted and spin-restricted approaches. The two methods differ only by the conditions of constraint imposed in minimization and lead to different sets of Kohn-Sham equations for the spin orbitals. The unrestricted Kohn-Sham approach is the one most commonly used and is implemented in various standard quantum chemistry software packages. However, this method has a major disadvantage, namely a spin contamination problem, and in recent years the alternative spin-restricted Kohn-Sham approach has become a popular contester [48-50]. 

Since we now have a one-electron problem, the Kohn-Sham equations (2.4) can be solved in a self-consistent manner. We obtain a set of orbitals and their energies, much as in HF theory. The density function, p(r, can be found as the sum of the squares of the w/, for the occupied orbitals. From p(r) the expectation value of the energy can be found, as well as other one-electron properties. Just as in the HF method, the total electronic energy is equal to the sum of the energies of the occupied orbitals, minus a correction because the electron-electron interactions have been counted twice. 

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... 

The last term in Equation (6.1) represents the exchange-correlation energy Exc. The total energy can then be obtained by the solution of the relativistic Kohn-Sham equations (RKS)... 

Similar to the spin-compensated case, the solution of the unrestricted Kohn-Sham equations starts with the external potential and the number of spin a and spin ji electrons in the state of interest (denoted Na and Np, respectively) then Eqs. (48) and (49) are solved until consistency is achieved. Using the Kohn-Sham orbitals and orbital energies, one then computes the total energy of the system using the spin-dependent generalization of Eq. (47),... 

The only unknown functional in eq. (29) is Exc[p]. All of the others are known or can be easily calculated from the wavefunction of the noninteracting system. Once Exc is known, the total energy E can be minimized with respect to the density p, yielding the Kohn-Sham equations that can be solved self-consistently ... 

Whether one uses an all-electron or a pseudopotential technique, the fundamental aims of a DFT calculation are to calculate the total energy and the charge/spin density of a given configuration of atoms. The Kohn-Sham equation... 

The central equation of modem DFT is the Kohn-Sham equation for die total energy (22) ... 

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