Density function theory Kohn-Sham orbitals

Key words Density functional theory - Kohn-Sham -Electron correlation - Molecular orbital theory... 

Density Functional Theory (DFT) has become the method of choice for the study of the electronic structure of solids. Advances in computer technology have made possible the development of DFT-based codes providing a detailed ab initio description of the electronic structme of complex materials. Following the two celebrated papers by Hohenberg and Kohn and Kohn and Sham, a wide variety of approaches have been developed and turned into very efficient computational tools. These approaches differ in the way they represent the density, potential, and Kohn-Sham orbitals. Essentially DFT approaches can be classified in two main groups all electron methods and pseudopotential methods. 

In the pursuit of quantitative accuracy from density-functional theory, one typically abandons the idea of expressing the kinetic energy directly in terms of the electron density. Instead, one introduces a set of auxiliary functions, the Kohn-Sham orbitals, ij/ p r] / i, themselves functionals of the ground-state electron density, and computes an accurate approximation to the kinetic energy from these orbitals according to the formula [15],... 

Finally, we consider density functional theory (DFT) computations of p-space properties. A naive way of calculating p-space properties is to use the Kohn-Sham orbitals obtained from a DFT computation to form a one-electron, r-space density matrix Fourier transform / according to Eq. (14), and proceed further. This approach is incorrect because the Kohn-Sham density matrix F is not the true one and, in fact, corresponds to a fictitious non-interacting system with the same p(r) as the true system. On the other hand, Hamel and coworkers [112] have shown that if the exact Kohn-Sham exchange potential is used, then the spherically averaged momentum densities of the Kohn-Sham orbitals should be very close to those of the Hartree-Fock orbitals. Of course, in practical computations the exact Kohn-Sham exchange potential is not used since it is generally not known. 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

Kohn-Sham orbitals functions for describing the electron density in density functional theory calculations... 

It is a truism that in the past decade density functional theory has made its way from a peripheral position in quantum chemistry to center stage. Of course the often excellent accuracy of the DFT based methods has provided the primary driving force of this development. When one adds to this the computational economy of the calculations, the choice for DFT appears natural and practical. So DFT has conquered the rational minds of the quantum chemists and computational chemists, but has it also won their hearts To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations. There has been misunderstanding concerning the status of the one-determinantal approach of Kohn and Sham, which superficially appeared to preclude the incorporation of correlation effects. There has been uneasiness about the molecular orbitals of the Kohn-Sham model, which chemists used qualitatively as they always have used orbitals but which in the physics literature were sometimes denoted as mathematical constructs devoid of physical (let alone chemical) meaning. 

Goddart, J. D., Orlova, G., 1999, Density Functional Theory with Fractionally Occupied Frontier Orbitals and the Instabilities of the Kohn-Sham Solutions for Defining Diradical Transition States Ring-Opening Reactions ,... 

However, one feature of the HF potential is that it is not a local potential. In the case of perfect data (i.e. zero experimental error), the fitted orbitals obtained are no longer Kohn-Sham orbitals, as they would have been if a local potential (for example, the local exchange approximation [27]) had been used. Since the fitted orbitals can be described as orbitals which minimise the HF energy and are constrained produce the real density , they are obviously quite closely related to the Kohn-Sham orbitals, which are orbitals which minimise the kinetic energy and produce the real density . In fact, Levy [16] has already considered these kind of orbitals within the context of hybrid density functional theories. 

A new and accurate quantum mechanical model for charge densities obtained from X-ray experiments has been proposed. This model yields an approximate experimental single determinant wave function. The orbitals for this wave function are best described as HF orbitals constrained to give the experimental density to a prescribed accuracy, and they are closely related to the Kohn-Sham orbitals of density functional theory. The model has been demonstrated with calculations on the beryllium crystal. 

Applying the variational principle to the energy given by Eq. 1, Kohn and Sham reformulated the density functional theory by deriving a set of one-electron Hartree-like equations leading to the Kohn-Sham orbitals v().(r) involved in the calculation of p(r)15. The Kohn-Sham (KS) equations are written as follows ... 

All calculations presented here are based on density-functional theory [37] (DFT) within the LDA and LSD approximations. The Kohn-Sham orbitals [38] are expanded in a plane wave (PW) basis set, with a kinetic energy cutoff of 70 Ry. The Ceperley-Alder expression for correlation and gradient corrections of the Becke-Perdew type are used [39]. We employ ah initio pseudopotentials, generated by use of the Troullier-Martins scheme [40], The coreradii used, in au, were 1.23 for the s, p atomic orbitals of carbon, 1.12 for s, p of N, 0.5 for the s of H, and 1.9, 2.0, 1.5, 1.97,... 

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