The Kohn-Sham Method

Note that the Kohn-Sham Hamiltonian hKS [Eq. (4.1)] is a local operator, uniquely determined by electron density15. This is the main difference with respect to the Hartree-Fock equations which contain a nonlocal operator, namely the exchange part of the potential operator. In addition, the KS equations incorporate the correlation effects through Vxc whereas they are lacking in the Hartree-Fock SCF scheme. Nevertheless, though the latter model cannot be considered a special case of the KS equations, there are some similarities between the Hartree-Fock and the Kohn-Sham methods, as both lead to a set of one-electron equations allowing to describe an n-electron system. 

Density functional theory (DFT) provides an efficient method to include correlation energy in electronic structure calculations, namely the Kohn-Sham method 1 in addition, it constitutes a solid support to reactivity models.2 DFT framework has been used to formalize empirical reactivity descriptors, such as electronegativity,3 hardness4 and electrophilicity index.5 The frontier orbital theory was generalized by the introduction of Fukui function,6 and new reactivity parameters have also been proposed.7,8 Moreover, relationships between those parameters have been found, and general methods to relate new quantities exist.9... 

The purpose of this chapter will be to review the fundamentals of ab initio MD. We will consider here Density Functional Theory based ab initio MD, in particular in its Car-Parrinello version. We will start by introducing the basics of Density Functional Theory and the Kohn-Sham method, as the method chosen to perform electronic structure calculation. This will be followed by a rapid discussion on plane wave basis sets to solve the Kohn-Sham equations, including pseudopotentials for the core electrons. Then we will discuss the critical point of ab initio MD, i.e. coupling the electronic structure calculation to the ionic dynamics, using either the Born-Oppenheimer or the Car-Parrinello schemes. Finally, we will extend this presentation to the calculation of some electronic properties, in particular polarization through the modern theory of polarization in periodic systems. 

The key point of the Kohn-Sham method is to consider an auxiliary system of N non-interacting electrons for estimating the kinetic energy of the real (interacting) system [120-122,125]. 

With the development of GGA functionals, description of molecular systems with the Kohn-Sham method reached a precision similar to other quantum theory methods. It was quickly shown that the GGA s could also well reproduce the hydrogen bond properties. Short after, liquid water at ambient condition was first simulated by Car-Parrinello MD, with a sample of 32 water molecules with periodic boundary conditions [31]. Since then, many simulations of liquid water at different temperatures and pressures and of water solutions have been performed [32-39]. Nowadays, Car-Parrinello MD has become a major tool for the study of aqueous solutions [40-64]. 

To motivate the Kohn-Sham method, we return to molecular Hamiltonian [Eq. (2)] and note that, were it not for the electron-electron repulsion terms coupling the electrons, we could write the Hamiltonian operator as a sum of one-electron operators and solve Schrodinger equation by separation of variables. This motivates the idea of replacing the electron-electron repulsion operator by an average local representation thereof, w(r), which we may term the internal potential. The Hamiltonian operator becomes... 

The Kohn-Sham wave function, KS, is not expected to be a good approximation to the exact wave function indeed, it is a worse approximation to the exact wave function than the Hartree-Fock wave function. Flowever, unlike the electron density obtained from the Flartree-Fock equations, the Kohn-Sham method yields, in principle, the exact electron density. Thus we do not need to use the Kohn-Sham wave function to compute the properties of chemical systems. Rather, motivated by the first Hohenberg-Kohn theorem, we compute properties directly from the Kohn Sham electron density. How one does this, for any given system and for any property of interest, is an active topic of research. 

Because of its critical role in constructing the potential energy surface [cf. Eq. (3)] for a molecule, thence in the prediction of molecular structure and chemical reactivity, we mention how one may compute the electronic energy of a system using the Kohn-Sham method. In particular, one has... 

No useful explicit form for Exc[p] is known, but approximate exchange-correlation energy functionals often provide an excellent approximation to the energetic properties of the molecule. To explain how approximate exchange-correlation functionals work (and when they fail), recall the essence of the Kohn-Sham method the Kohn-Sham method constructs the model system with the energy functional... 

The Kohn-Sham method involves two coupled equations which must be solved sequentially in the first step, one solves the partial differential equations (45) for a... 

The idea of the Kohn-Sham method is best understood as follows. Consider a generalized Hamiltonian of Eq. (2) in which the term 4e is scaled by an electron-electron coupling constant A. We are interested in values of A between 0 and 1. Each value of A corresponds to a distinct universal functional of the density. In Levy s constraint search formulation [36] of the Hohenberg-Kohn principle, this is explicitly stated as... 

By the early 1980s, the first reviews on DFT appeared [107-110], underlining its great potential for applications to atoms and molecules. Also the first publications about analytic gradients for the Kohn-Sham method appeared in the literature [111,112]. This clearly moved the Kohn-Sham method into the computational chemistry arena. Still it took almost a decade more before stable geometry optimization with analytic gradients... 

Thus, some decision has to be taken about the very meaning of kinetic energy density before facing the essentially technical difficulty of approximating this density as a functional of the particle density. This conclusion applies just as forcefully to the Kohn-Sham method of approximating the dominant part of the kinetic energy by the single-determinant form ... 

Fig. 10. Shell-closing cflects in the calculated differences I — A for Na dusters. The calculations used the Kohn-Sham method and a non-local description of exchange and correlations effects (see Ref. [30] for details)...

The Kohn-Sham Method. If we know the ground-state electron density po(r), the Hohenberg-Kohn theorem tells us that it is possible in principle to calculate all the ground-state molecular properties from po, without having to find the molecular wave... 

There is only one problem in using the Kohn-Sham method to find p and Eq. No one knows what the correct functional E Xp] i - Therefore, both in the energy expression (15.120) and in (15.121) and (15.124) are unknown. Various approximations to E ic will be discussed shortly. 

In a true density-functional theory, one would deal with only the electron density (a function of three variables) and not with orbitals, and would search directly for the density that minimizes [p]. Because the functional E is unknown, one instead uses the Kohn-Sham method, which calculates an orbital for each electron. TTius, the KS method represents something of a compromise with the original goals of DFT. 

The exchange-correlation energy Exc in the Kohn-Sham method... 

The solution to the equation having the form (8.1) is at the heart of the computer simulation of solids. However, the modern way of calculations follows the Kohn-Sham method (see Section 8.5), which also involves potentials chosen to incorporate the exchange energy and correlation energy. 

In practical calculations making use of the Kohn-Sham method, the Kohn-Sham equation is used. This equation is a one-electron SCF equation applying the Slater determinant to the wavefunction of the Hartree method, similarly to the Hartree-Fock method. Therefore, in the same manner as the Hartree-Fock equation, this equation is derived to determine the lowest energy by means of the Lagrange multiplier method, subject to the normalization of the wavefunction (Parr and Yang 1994). As a consequence, it gives a similar Fock operator for the nonlinear equation. 

Differently to the Hartree-Fock method, the total electronic energy in the Kohn-Sham method is generally calculated using not the exchange-correlation potential functional but the exchange-correlation energy functional, as... 

Similarly to the Hartree-Fock method, the Kohn-Sham method is solved by the... 

Therefore, as shown in Eq. (4.10), the total electronic energy in the Kohn-Sham method is separately calculated using the exchange-correlation energy functionals. Note, however, that no such difference should be included in a universal functional, and there is a possibility in the future to develop an exchange-correlation functional giving no difference. In this case, process 4 is not necessary. 

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