Kohn-Sham energy

In this paper we will present a way to improve the evaluation of total energies in LMTO calculations. The Kohn Sham energy functional [1] can be written in the form... 

The starting point is the Kohn-Sham energy functional ... 

A Kohn-Sham calculation is then performed on the anion using the potential defined according to Equations 34.38 and 34.39. The electronic energy of the anion is determined using the conventional Kohn-Sham energy expression with the regular Exc term. 

The KS equations are obtained by differentiating the energy with respect to the KS molecular orbitals, analogously to the derivation of the Hartree-Fock equations, where differentiation is with respect to wavefunction molecular orbitals (Section 5.2.3.4). We use the fact that the electron density distribution of the reference system, which is by decree exactly the same as that of the ground state of our real system (see the definition at the beginning of the discussion of the Kohn-Sham energy), is given by (reference [9])... 

A comprehensive view of the above correlation energy decomposition is presented in Fig. 5. For completeness we have also included the energy functional Pcx ct, KS])i namely, the Kohn-Sham energy. We see from Fig. 5 that the DFT correlation energy, E FT, defined as... 

The orthogonal orbitals, which minimize the Kohn-Sham energy functional are obtained from the following set of one-electron equations21 for / = 1, N ... 

Here the point of view is somewhat different and instead of looking for solving the self-consistency equation (58) itself, we look for minimizing the Kohn-Sham energy functional. With a slight extension with respect to the previous section, (47), we will in fact look for minimizing the total energy ... 

Car and Parrinello in their celebrated 1985 paper [2] proposed an alternative route for molecular simulations of electrons and nuclei altogether, in the framework of density functional theory. Their idea was to reintroduce the expansion coefficients Cj(G) of the Kohn-Sham orbitals in the plane wave basis set, with respect to which the Kohn-Sham energy functional should be minimized, as degrees of freedom of the system. They then proposed an extended Car-Parrinello Lagrangian for the system, which has dependance on the fictitious degrees of freedom Cj(G) and their time derivative Cj (G) ... 

Fig. 3. Total energy and energy components for a system of 32 water molecules (simulations parameters see text). Top fictitious kinetic energy of the electrons (Kei), second from top instantaneous ionic temperature, Tions (proportional to the ions kinetic energy, Kiona), middle instantaneous Kohn-Sham energy Eks, second from bottom classical hamiltonian Eclass = Eks + Kions, bottom CP hamiltonian, Eham = Eclass + Kd- Note the change of scale of the vertical axis from one frame to the other...

We can also evaluate the lowest typical frequency for the electronic dynamics from the gap Egap, energy difference between the Lowest Unoccupied Molecular Orbital (LUMO) and the Highest Occupied Molecular Orbital (HOMO), of the Kohn-Sham non-interacting electron system, which determines the lowest curvature of the E c) Kohn-Sham energy functional ... 

Brown, P., Woods, C.J., Mclntosh-Smith, S., Manby, F.R., A massively multicore parallelization of the Kohn-Sham energy gradients, J. Comput. Chem. 2010, 31(10), 2008-13. 

Note that the extended Kohn-Sham energy functional is dependent on the orbitals and implicitly on through the electron-nucleus attraction terms... 

In Kohn-Sham theory, we assume that the time-dependent density p(r, r) is represented in terms of a time-dependent reference Slater determinant z). The Kohn-Sham energy is then written as a functional of this density in the following manner ... 

N (Kohn-Sham) orbitals can be expanded into atomic orbitals according to (4). Furthermore, the expansion coefficients C, can be determined by requiring that they optimize the total (Kohn-Sham) energy. This results in the (Kohn-Sham) matrix equation similar to (5)... 

The expression for p/(l,l ) makes it next possible to write down the corresponding excited state Kohn-Sham energy to second order as... 

The Kohn-Sham construction is a pragmatic one, justified by computational utility. Of special computational utility is the fact that each Kohn-Sham orbital experiences the same potential and that this potential, in turn, is a functional of the electron density alone. This allows us to rewrite the Kohn-Sham energy in terms of the first-order density matrix,... 

One may be tempted then to find the density matrix by minimizing the Kohn-Sham energy, Eq. (65), with respect to the density matrix subject to the constraint that... 

Direct minimization of the Kohn-Sham energy using Eq. (81) is infeasible direct methods for computing S grow as the third power of the size of the system. However,... 

The matrix, Ymax, that maximizes Tr[5(2Y 2XSX)] is the generalized inverse, allowing the variational procedure for the Kohn-Sham energy to be written as [40]... 

Yet another technique is to minimize the Kohn-Sham energy as a functional of the density matrix, Eq. (65). In the first method of this type, due to Li et al. [47], the ground state of the system was only a local minimum of the energy functional. To surmount this difficulty, one may choose to explicitly impose the idempotency constraint on the density matrix, Eq. (71), with a Lagrange multiplier. For exam-... 

In Table 8 we show the calculated energies of the highest occupied orbitals for a set of isolated atoms together with experimental ionization potentials. The table confirms the conclusion above, i.e., that the Kohn-Sham energies are too high. Furthermore, these tend to have a too weak dependence on the system,... 

---