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A Primer to Electronic Structure Computation

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A Primer to Electronic Structure Computation Nick Schafer Fall 2006 Abstract A brief overview of some literature the author read as a part of an independent study on Electronic Structure Computation is given. The Hartree Fock Approach and Density Functional Theory are emphasized. Only some important aspects of each are presented and results are stated without proof. Some of the diculties involved are discussed along with recently proposed solutions. The treatment is qualitative. The intent is to introduce the problem to someone who has the physics and mathe- matics background of an advanced undergraduate but no prior experience in Electronic Structure Computation. It should be read as an overview before and a guide during a more in depth study. The interested reader should see the references for further reading. Some familiarity with ba- sic Quantum Mechanics and Linear Algebra is assumed, which can be obtained by following the references. Contents 1 Introduction 2 1.1 Purpose . 2 1.2 Denition . 2 2 Hartree Fock Approach 3 2.1 Basic Procedure . 3 2.2 Schrodinger Equation . 4 2.3 Variational Principle . 4 2.4 Born Oppenheimer Approximation . 5 2.5 Antisymmetry Principle . 5 2.6 Hartree Fock Approximation . 6 2.6.1 Spatial and Spin Orbitals . 6 2.6.2 Hartree Products . 6 2.6.3 Slater Determinants . 7 2.6.4 Fock Operator . 7 2.6.5 Hartree Fock Equations . 8 2.7 Computational Procedure . 9 2.7.1 Basis sets . 9 1 2.7.2 Slater Type . 9 2.7.3 Gaussian Type . 10 2.7.4 Procedure . 10 3 Density Functional Theory 11 3.1 Electron Correlation . 11 3.2 Electron Density as the Fundamental Quantity . 12 3.2.1 Electron Density . 12 3.2.2 Pair Density . 12 3.2.3 Fermi and Coulomb Holes . 13 3.3 Hohenberg-Kohn Theorems . 13 3.3.1 Proof of Existence . 13 3.3.2 Variational Principle . 14 3.4 Kohn-Sham Approach . 14 3.4.1 Approximate Functionals . 15 4 Recent Work on Reducing Computational Intensity 15 4.1 Pseudopotentials and Plane Waves . 16 4.2 Real Space Mesh . 16 4.3 Reconstruction . 17 5 Acknowledgements 17 6 References 17 7 Contact 18 1 Introduction 1.1 Purpose The properties of materials are of intense interest to many branches of science and industry. Unfortunately, direct measurements of these properties is not always feasible or reliable. However, these properties are known to be heavily dependent upon the electronic structure of the material. If the electronic struc- ture can somehow be obtained then, in principle, all properties of the material can be calculated (8, p.11). In addition to static properties, dynamic simulations can be carried out using an iterative process to update the electonic structure and nuclear conguration successively (8, p.317). Dynamic simulation allows us to probe the movements of molecules on time and length scales inaccessible by any other means. 1.2 Denition Electronic Structure Computation is the calculation of information about elec- trons near nuclei from which properties of a material can be derived. Quantum 2 mechanics tells us that the description of an electron as a spherical charge with a denite trajectory is not always appropriate. In particular, electrons near nuclei become delocalized and are best described by a probability distribution, which gives the probability of nding an electron at a certain place and time (9, p.18). It is exactly this distribution that we refer to as the electronic structure and from which we derive the properties of a material. In Hartree Fock Theory the information computed is the many electron wave function. Density Functional Theory has taken on several forms since its inception. Very early attempts used relatively simple functionals of the electron density to compute the total energy. Another group of methods computes wave functions of a ctitous noninteracting system of electrons which allows it to resolve most of the energy contribution quickly and accurately. The remaining energy terms are computed using the approximate functionals. Unfortunately, very often it is this remaining energy term which contains most of the interesting information. These methods are known as Kohn-Sham methods. Just recently there have been attempts to revive the direct use of functionals. These methods go under the name Orbital-Free. The primary and most basic property that can be derived from this infor- mation is the ground state energy, as it can be used to obtain the equilibrium conguration of a system. 2 Hartree Fock Approach The Hartree Fock Approach is the starting point for many more sophisticated approaches to Electronic Structure Computation. It has been extended and corrected in many ways, but most of the core ideas and assumptions remain the same and basic knowledge of these is necessary for understanding any of the more modern approaches to Electronic Structure Computation. 2.1 Basic Procedure Anyone who has done any work with numerical methods will recognize the basic procedure for computing electronic structure using the Hartree Fock method. In the broadest sense, there are four steps: 1. Fix the necessary inputs 2. Make an inital guess at the electron density 3. Perform calculations to update the electron density 4. Compare the new electron density to the old electron density If the two electron densities dier by more than a specied tolerance, then the new electron density becomes the old electron density and steps three and four are repeated. In most cases, an average of the new and old electron densities is used to improve the convergence of the solution. This may sound simple and in a sense it is. However, there are many sub- tleties that go into each step and understanding them all takes many years of study. In fact, the details of how each step should be carried out so that the 3 nal electron density is obtained accurately and eciently is still very much an open question. This paper covers only the basics of some well established techniques, as well as some more recent attempts at making the process more ecient. 2.2 Schrodinger Equation Considering the procedure above, it is helpful to imagine that we are dealing with a classical system. If that was the case, step three would almost certainly involve the use of Newton's second law, which governs the dynamics of classical bodies. However, our system is composed of delocalized electrons interacting with each other and with nuclei. This is not a classical situation and we must look for the appropriate dynamic equation. The Schrodinger Equation governs the evolution of non-relativistic quantum mechanical systems. In particular, it governs the evolution of wave functions of the electrons. It is a linear second order partial dierential equation (9, p.31). When treating materials, the sheer number of variables on which it depends precludes the use of analytical solution techniques. Finding approximate so- lutions to the Schrodinger Equation is the primary concern of basic Quantum Chemistry. Approximate solutions are necessary because complete analytic so- lutions are only available for a few simple systems, such as the hydrogen atom (one electron orbiting a nucleus) where the interaction is suciently simple (9, p.215). It seems convenient to note here that all calculations mentioned herein are done in the non-relativistic limit: where the speed of the electrons is much less than the speed of light. This approximation turns out to be appropriate for atomic species lighter than silver, but begins to fail if applied to heavier species. 2.3 Variational Principle The Variational Principle states that any normalized wave function corresponds to an energy, obtained by taking the expectation value of the Hamiltonian, that is greater than or equal to the ground state energy of the system. Also, if this energy is equal to the ground state energy, then that wave function is the ground state wave function of the system. Equivalently, it is not possible to nd a normalized wave function corresponding to an energy lower than the ground state energy. This result is physically reasonable in the sense that the ground state is completely relaxed in the energetic sense (1, p.31). It may not be immediately obvious how this would be useful. Consider a quantum mechanical system for which you only know the Hamiltonian but not the ground state wave function or the ground state energy. Now, you make two guesses at the ground state wave function of the system. The guess which has a lower expectation value for the Hamiltonian is a better guess at the ground state wave function according to the Variational Principle. A common trick for nding a good estimate of the ground state energy of a system is to try a form of the wave function that depends on one or more parameters, calculate the 4 expectation value of the Hamiltonian, then minimize the resulting expression for the energy with respect to the parameters. Inserting these parameters into the original expression gives the best possible guess at the ground state wave function of that particular form (1, p.37). This method works well for systems where you already know approximately what the wave function should look like. This is not always the case, however, and a more general method is necessary. Guessing an arbitrary form for a wave function is often not sucient, as it may not be exible enough to mimic the actual ground state wave function. The situation is particularly bad if the properties which you are trying to derive depend sensitively on the value of the wave function at its tails. However, the wave function, like any other function, can be expanded in terms of basis functions in a certain space.
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