A Variational Framework for Spectral Discretization of the Density Matrix in Kohn-Sham Density Functional Theory

I have been very blessed to have two of the most knowledgeable and supportive advisors at Caltech, Prof. Finally and most importantly, I want to thank God for the joy of studying the laws of nature as written in Psalm 111:2, “Great are the works of the Lord; They are studied by all who enjoy them". REKS0 Ground state energy for the transformed extended Kohn-Sham functional 0 Ground state energy of a molecular system - with electron relaxation and.

Sj,kj(u, φ) The spatially and spectrally discretized functional for Column energy T The kinetic energy of the homogeneous electron gas. We prove convergence of the approximate solutions with respect to spectral binning and with respect to an additional spatial discretization of the domain. It is said that in experiments we have a partial understanding of the whole truth; and in calculation we have a full understanding of the partial truth.

The development of DFT in 1964 by Walter Kohn was a major advance in physics because it related the ground state energy of a molecular system to the ground state electron density. Almost all of the information in this background section comes from the following two references, the first of which places more emphasis on explaining the physics [55] and the second which places more emphasis on the mathematics [12].

Figure 1.1: Number of publications with DFT as topic.

Many-body Schr¨ odinger equation

Born-Oppenheimer Approximation

A key assumption for the Born-Oppenheimer approximation is that the motion of the nuclei is slow relative to the motion of the electrons, so that with each motion of the nuclei the electrons reach the ground state configuration. Therefore, we can treat the spatial coordinates of the nuclei {R1,· · · ,RM} as a parameter and find the wave function of the electron ground state for a given set of nuclei coordinates. We replace this approximation with the Rayleigh coefficient in equation (2.7) and get with the electronic Hamiltonian he defined. 2.11).

We will refer to the potential due to the nuclei in the electron problem (2.9) as an external potential; potential due to electrons are intrinsic to the problem. In the limit that the mass of the nuclei goes to infinity, the kinetic energy of the nuclei can be neglected and the wave function of the nuclei is concentrated at the points {RI,· · ·RM}, since the deBroglie wavelength of a nucleus is infinitely small in comparison with the deBroglie wavelength of an electron. The solution of the Schr¨odinger equation can be solved in two steps: first it is solved for the electron ground states, finding the lowest eigenvalue and its corresponding eigenfunction of the He electronic Hamiltonian; then solve a geometry optimization problem to obtain the ground state energy of the molecular system.

The most important consequence of the Born-Oppenheimer approximation is that the electronic Hamiltonian of He has a purely discrete spectrum, i.e. in many cases the number of eigenstates counted. The infimum in the electronic problem in equation (2.9) can be reached with respect to the external potential.

Precursors to density functional theory

The corresponding eigenvalues are also quantized according to wavenumber quantization, but since the energy levels are proportional to|k|2, there will be degenerate eigenstates, i.e., wavefunctions that differ in wavenumber but have the same energy. To find the total energy of the system, which is purely kinetic, we can add up the energy of each electron. In the case where the box is large, i.e., l is very large, and the number of electrons N is also large, we can make an approximation that allows the calculation of the fermi level and the total energy with much less effort.

The allowed wavenumbers can be drawn as follows: we see that the limit l is very large, the spacing between consecutive grid points in k-space 2πl decreases. The force force method would be used to calculate the total (kinetic) energy of the system. If we want to use the approximation that the field is large, the summation k in equation (2.16) can be written as an integral in three dimensions.

Since we know that the energy is only a function of |k|, we can integrate equation (2.17) using spherical coordinates,. With Thomas-Fermi models as a precursor, Kohn and Hohenberg set out to rigorously prove in 1964 the assumption that the ground-state energy of a molecular system can only be written as a function of the electron density.

Figure 2.1: k-points in two dimensional k-space.

Electron density and Hohenberg-Kohn Theorem

Electron density

From the definition of the electron density, we see that the ground state wave function contains more information about the electronic system than the electron density alone. Given a wave function, we can always find its corresponding electron density through integration; but given only the electron density, we cannot recover the wave function.

Hohenberg-Kohn theorem

1If the two potentials differ by only a constant, then the variational problem (2.9) would yield the same ground state wave function, with the ground state energy differing by exactly the same constant. Therefore, there cannot be two external potentials that differ by more than a constant that has the same ground state electron density. From the Hohenberg-Kohn theorem, depending on the density of electrons in the ground state, the number of electrons can be determined by integration, and the external potential is determined up to a constant, so that the Hamiltonian is completely determined and, consequently, the energy in the ground state is completely determined.

The functional FHK is a universal functional, i.e., independent of the external potential of the system; depends only on the number of electrons in the system N. These two open questions make the Hohenberg-Kohn energy functional a theoretical result; however, it illuminated a very promising direction for quantum mechanical calculations.

Kohn-Sham density functional theory

The density matrix, written as a matrix function of the Hamiltonian matrix, is the power function defined in equation (3.5). Now let {ξi} denote the orthonormal eigenvectors of the matrix H(Vmax,∆r) and the eigenvector H(VN,∆r) can be expanded: ψmax. Similarly, let {ξi} denote the orthonormal eigenvectors of the matrix H(Vmin,∆r), the eigenvector H(VN,∆r) can be expanded: ψmin.

If ξ1Hj( ˆφ,u) denotes the corresponding normalized eigenvector of Hj( ˆφ, u), we can derive a lower bound of λH1 j( ˆφ,u) from the ellipticity of the underlying variational problem.