The Kohn—Sham Approach

DENSITY FUNCTIONAL THEORY The Kohn-Sham Approach 

Though the Hohenberg-Kohn theorem clearly established that one could, in principle, work directly with the density in ab initio calculations, it was the subsequent work Kohn and Sham (KS) that offered a practical approach to performing DFT calculations. In the KS approach, the unknown Hohenberg-Kohn energy functional, [p(r)], is partitioned in the following manner  

In this partitioning scheme, [7[p(r)] is simply the classical electrostatic energy, the sum of the electron-nucleus attractions and the electron-elearon repulsions 

The next term, T(p(r)], is defined as the kinetic energy of a system of noninteracting electrons with the same density, p(r), as the real system of interacting electrons being studied. This may seem to be introducing a severe error. How- 

Following Kohn and Sham, p(r) of an N-electron system (with N spin up electrons and spin down electrons) is expressed as the sum of the square moduli of singly occupied, orthonormal Kohn-Sham (KS) molecular orbitals. 

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Since the Fock operator is a effective one-electron operator, equation (1-29) describes a system of N electrons which do not interact among themselves but experience an effective potential VHF. In other words, the Slater determinant is the exact wave function of N noninteracting particles moving in the field of the effective potential VHF.5 It will not take long before we will meet again the idea of non-interacting systems in the discussion of the Kohn-Sham approach to density functional theory. 

Before we enter a more detailed discussion of various aspects in the Kohn-Sham approach, let us summarize the main features of this procedure ... 

In the preceding paragraph we have given a detailed survey of the Kohn-Sham approach to density functional theory. Now, we need to discuss some of the relevant properties pertaining to this scheme and how we have to interpret the various quantities it produces. We also will mention some areas connected to Kohn-Sham density functional theory which are still problematic. Before we enter this discussion the reader should be reminded to differentiate carefully between results that apply to the hypothetical situation in which the exact functional ExC and the corresponding potential Vxc are known and the real world in which we have to use approximations to these quantities. 

Is the Kohn-Sham Approach a Single Determinant Method ... 

Recall the central ingredient of the Kohn-Sham approach to density functional theory, i. e., the one-electron KS equations,... 

The developments in the implementation of the Kohn-Sham approach, helped also to formulate alternative formalisms aiming at approaching the linear scaling195 218"227. In particular, the divide-and-conquer approach of Yang218 have attracted much attention. 

The calculation of the induced electron density may be done in the context of the Kohn-Sham approach to density functional theory, because the response of a KS system to a change in the one particle effective potential (r) corresponds to that of a system of non-interacting electrons. 

Alternative Formulation of the Kohn-Sham Approach Generalized... 

The integrals to be summed are readily calculated. Note that DFT per se does not involve wavefunctions, and the Kohn-Sham approach to DFT uses orbitals only as a kind of subterfuge to calculate the noninteracting-system kinetic energy and the electron density function see below. 

The DFT concept of calculating the energy of a system from its electron density seems to have arisen in the 1920s with work by Fermi, Dirac, and Thomas. However, this early work was useless for molecular studies, because it predicted molecules to be unstable toward dissociation. Much better for chemical work, but still used mainly for atoms and in solid-state physics, was the Xa method, introduced by Slater in 1951. Nowadays the standard DFT methodology used by chemists is based on the Hohenberg-Kohn theorems and the Kohn-Sham approach... 

The reason for the delay is that it took the Kohn-Sham approach to initiate practical DFT calculations on molecules, and time was needed to experiment with techniques for improving the accuracy of calculations [7]. As for why the Hohenberg-Kohn theorems and the Kohn-Sham insight came not until 40 years after the wavefunction and electron density concepts, one can only speculate perhaps scientists were mesmerized by the peculiarities of the wavefunction [8], or perhaps it simply took the creativity of specific individuals to usher in the era of widespread density functional calculations. 

In the Kohn-Sham approach to DFT, the energy of the electronic system can be written as... 

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