Development of a Correction Term for the Kinetic Energy Density Functional

ABSTRACT  

functional theory (DFT) is a useful and computational tool for electronic structure , which form the basis for the classification of materials into conductors, or insulators.

DFT started with a crude by Thomas and Fermi (TF theory) which calculated the kinetic energy of electrons using the so-called local density approximation (LDA).

Although TF is computationally inexpensive, it provides a poor numerical result due to a lack of understanding of the density dependence of the kinetic energy. Another approximation to the kinetic energy is the von-Weizsacker (vW) term, which greatly improves the TF theory,

yet the full functional form of the kinetic energy remains unknown. We seek to develop a supplemental term to the kinetic energy density functional and compute corrections to the Thomas-Fermi-von-Weizsacker kinetic energy of closed shell atoms in order to improve its accuracy. 

 

TABLE OF CONTENTS

1 Introduction . . . . . . . . . . . . . . . . . . 1
1.1 What is Density Functional Theory (DFT)? . . . . . . . . . . . . . . . . . . 2
1.2 Why DFT? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Uses of DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Focus of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 The Background: The Schr¨odinger Equation . . . . . . . . . . . . . . . . . . 4
1.5.1 One Particle TISE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5.2 System of Several Particles . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5.3 A Real System and The Born-Oppenheimer Approximation . . . . . 6
1.6 The Hartree-Fock Approximation . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Hohenberg-Kohn Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7.1 Represent-ability of Density . . . . . . . . . . . . . . . . . . . . . . . 12

2 Basic Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
2.1 Density Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 Uniform Electron Gas (UEG) Model . . . . . . . . . . . . . . . . . . 14
2.1.2 Local Density Approximation (LDA) . . . . . . . . . . . . . . . . . . 14
2.1.3 Gradient Expansion Approximation (GEA) . . . . . . . . . . . . . . . 15
2.2 Thomas-Fermi Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 The von Weizs¨acker Functional (vW) . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Conceptualization of Density Matrices . . . . . . . . . . . . . . . . . . . . . 18
2.4.1 Electron Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.2 Pair Density: n2(1, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.3 Reduced Density Matrix (RDM) . . . . . . . . . . . . . . . . . . . . . 20

3 Calculation of Corrections for KEDF of closed shell systems . . . . . . . . . . . .21
3.1 Kinetic Energy Expectation Value . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Expression for First Order RDM . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 The Kinetic Energy Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Kinetic Energy Corrections for some closed shell Systems . . . . . . . . . . . 26
4 Development of a Correction term and Discussion of Results 28
4.1 Development of a correction term . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Results of the Correction Terms . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46

INTRODUCTION  

The quantum mechanics of many-electron systems which have descriptions from time dependent and time-independent Schr¨odinger and Liouville equations, is to a good approximation ostensibly a well-understood subject.

The Schr¨odinger equations present the theoretical bases for the description of both the time evolution and pure stationary states properties of atoms and molecules.

In treating some quantum mechanical systems such as biological molecules and liquids where the individuality of molecules ceases to exist, rather collective effects becomes predominant,

it is immaterial to talk of pure states but paramount to consider ensemble of states describable with time-dependent and time-independent Liouville equations in lieu of Schr¨odinger equations.

However, in each case of pure states and ensemble of non-trivial many-electron systems, the equations involved are not without complicated and complex mathematical parameters with little or no analytical or numerical solutions. equations.

Although non-relativistic Hamiltonian operators for systems interacting Coulombically can be written explicitly for these equations, understanding a priori, the subtleties of the manybody behavior that ensues from these interactions remains a challenge.

Thus, this calls for the formulation of a rigorous quantum mechanical approach entirely equivalent to the Schr¨odinger or Liouville equations which certainly opened ways for important developments in atomic,

molecular and condensed matter physics as well as in quantum chemistry, particu1 larly, to avoid the particle-number dependency.

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StudentsandScholarship Team.