Kohn-Sham wave function

Properties can be computed by finding the expectation value of the property operator with the natural orbitals weighted by the occupation number of each orbital. This is a much faster way to compute properties than trying to use the expectation value of a multiple-determinant wave function. Natural orbitals are not equivalent to HF or Kohn-Sham orbitals, although the same symmetry properties are present. 

The premise behind DFT is that the energy of a molecule can be determined from the electron density instead of a wave function. This theory originated with a theorem by Hoenburg and Kohn that stated this was possible. The original theorem applied only to finding the ground-state electronic energy of a molecule. A practical application of this theory was developed by Kohn and Sham who formulated a method similar in structure to the Hartree-Fock method. 

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

Writing the Euler-Lagrange equations in terms of the single-particle wave functions (tpi) the variation principle finally leads to the effective singleelectron equation, well-known as the Kohn-Sham (KS) equation ... 

The Slater—Condon integrals Ft(ff), Ft(fd), and Gj-(fd), which represent the static electron correlation within the 4f" and 4f 15d1 configurations. They are obtained from the radial wave functions R, of the 4f and 5d Kohn—Sham orbitals of the lanthanide ions.23,31... 

The Wyboume crystal field parameters B (f, f), B (d, d), and Bjj(f, d), which describe the interaction due to the presence of the ligands onto the electrons of the lanthanide center. They are deduced from the ligand field energies and wave functions obtained from Kohn—Sham orbitals of restricted DFT calculations within the average of configuration (AOC) reference by placing evenly n — 1 electrons in the 4f orbitals and one electron in the 5d.33... 

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. 

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. 

To understand how Kohn and Sham tackled this problem, we go back to the discussion of the Hartree-Fock scheme in Chapter 1. There, our wave function was a single Slater determinant SD constructed from N spin orbitals. While the Slater determinant enters the HF method as the approximation to the true N-electron wave function, we showed in Section 1.3 that 4>sd can also be looked upon as the exact wave function of a fictitious system of N non-interacting electrons (that is electrons which behave as uncharged fermions and therefore do not interact with each other via Coulomb repulsion), moving in the elfective potential VHF. For this type of wave function the kinetic energy can be exactly expressed as... 

Just as in the unrestricted Hartree-Fock variant, the Slater determinant constructed from the KS orbitals originating from a spin unrestricted exchange-correlation functional is not a spin eigenfunction. Frequently, the resulting (S2) expectation value is used as a probe for the quality of the UKS scheme, similar to what is usually done within UHF. However, we must be careful not to overstress the apparent parallelism between unrestricted Kohn-Sham and Hartree-Fock in the latter, the Slater determinant is in fact the approximate wave function used. The stronger its spin contamination, the more questionable it certainly gets. In... 

One should therefore expect that the basis set requirements in Kohn-Sham calculations are less severe than in wave function based ones. Indeed, in most applications this is the case (see, e. g., Bauschlicher et al., 1997, and Martin, 2000). 

Morrison, R. C., Zhao, Q., 1995, Solution to the Kohn-Sham Equations Using Reference Densities from Accurate, Correlated Wave Functions for the Neutral Atoms Hehum Through Argon , Phys. Rev. A, 51, 1980. 

The technique used to extract the wave function in this work is conceptually simple the wave function obtained is a single determinant which reproduces the observed experimental data to the desired accuracy, while minimising the Hartree-Fock (HF) energy. The idea is closely related to some interesting recent work by Zhao et al. [1]. These authors have obtained the Kohn-Sham single determinant wave function of density functional theory (DFT) from a theoretical electron density. 

The Kohn-Sham determinant is the single determinant which reproduces the electron density and minimises the kinetic energy [1,9].) They observed that for the Be atom, the Kohn-Sham orbitals were nearly indistinguishable from the HF orbitals, and on this evidence they claim that the problem of finding a physically meaningful wave function from an electron density is solved . Here, we merely note that there are a number of desirable features for our model ... 

Provided the potential t) is local in r, in the limit that X - oo we will have p - p, independent of the choice of t). In this limit then, Equation (5) gives the Kohn-Sham orbitals and eigenvalues. The determinant formed from these orbitals is a wave function obtained from the density p,. 

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. 

Morrison, R. C., Q. Zhao, R. C. Morrison, and R. G. Parr. 1995. Solution of the Kohn-Sham equations using reference densities from accurate, correlated wave functions for the neutral atoms helium through argon. Phys. Rev. A51, 1980. 

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