Electronic Structure: Basic Theory and Practical Methods for Computational Physics and Chemistry
Electronic Structure: Basic Theory and Practical Methods
Electronic structure is the study of how electrons are arranged and behave in atoms, molecules and solids. It is one of the most fundamental topics in physics, chemistry and materials science, as it determines many physical and chemical properties of matter, such as energy levels, bonding, magnetism, conductivity, optical spectra and chemical reactions. In this article, we will introduce the basic theory and practical methods for calculating electronic structure, as well as some examples of applications in different fields.
Electronic Structure: Basic Theory and Practical Methods: Basic Theory and Practical Density Functio
What is electronic structure?
Electrons are quantum particles that obey the Schrödinger equation, which describes their wave-like nature and interactions with other particles. The electronic structure of a system is the solution of the Schrödinger equation for all the electrons in that system, given a certain configuration of nuclei (or other external potentials). The solution consists of a set of wave functions (or orbitals) that describe the probability distribution of finding an electron in a certain region of space, and a set of energies (or eigenvalues) that correspond to each wave function.
For example, the electronic structure of a hydrogen atom consists of one electron orbiting a proton. The wave functions are spherical harmonics that depend on the distance from the nucleus (r) and the angles (θ and φ). The energies are inversely proportional to the square of an integer number (n) called the principal quantum number. The electronic structure of a helium atom consists of two electrons orbiting two protons. The wave functions are linear combinations of hydrogen-like orbitals that depend on both r1 and r2, as well as θ1, φ1, θ2 and φ2. The energies depend on both n1 and n2, as well as another integer number (l) called the orbital quantum number.
The electronic structure of a molecule consists of several electrons orbiting several nuclei. The wave functions are linear combinations of atomic orbitals that depend on the coordinates of all the electrons and nuclei. The energies depend on the geometry and bonding of the molecule. The electronic structure of a solid consists of many electrons interacting with many nuclei arranged in a periodic lattice. The wave functions are Bloch functions that depend on the position within a unit cell (r) and a wave vector (k) that characterizes the periodicity. The energies form bands that depend on k.
Why is electronic structure important?
The electronic structure determines many physical and chemical properties of matter, such as:
The energy levels of atoms and molecules, which affect their stability, reactivity, spectroscopy and thermodynamics.
The bonding between atoms and molecules, which affects their structure, shape, polarity and intermolecular forces.
The magnetism of atoms, molecules and solids, which affects their spin, magnetic moment, domain structure and magnetic interactions.
The conductivity of solids, which affects their electrical resistance, current, voltage and power.
The optical spectra of solids, which affect their absorption, emission, reflection and refraction of light.
The chemical reactions of atoms and molecules, which affect their kinetics, mechanisms, products and catalysts.
Understanding and predicting the electronic structure of matter is therefore essential for many scientific and technological applications, such as:
The design of new materials with desired properties, such as superconductors, semiconductors, nanomaterials and biomaterials.
The discovery of new phenomena and phases of matter, such as topological insulators, quantum Hall effect and high-temperature superconductivity.
The development of new devices and systems, such as solar cells, lasers, transistors and quantum computers.
The exploration of new frontiers of science, such as astrochemistry, nanoscience and quantum information.
However, calculating the electronic structure of matter is also very challenging, as it involves solving a complex many-body problem that requires a lot of computational resources and sophisticated methods. In the next section, we will introduce some of the main approaches for tackling this problem.
How is electronic structure calculated?
There are many different methods for calculating electronic structure, each with its own advantages and limitations. Here we will give a brief overview of some of the most common and widely used methods, and classify them into two main categories: independent particle methods and beyond independent particle methods.
Independent particle methods
Independent particle methods are based on the approximation that each electron in the system behaves as if it were independent of the other electrons, except for an average potential that represents the effect of the other electrons. This approximation simplifies the many-body problem into a one-body problem that can be solved more easily. The two most popular independent particle methods are Hartree-Fock theory and density functional theory.
Hartree-Fock theory
Hartree-Fock theory is based on the assumption that the wave function of the system can be written as a single Slater determinant, which is a product of one-electron orbitals that are antisymmetric under exchange of any two electrons. This ensures that the wave function satisfies the Pauli exclusion principle, which states that no two electrons can have the same quantum numbers. The orbitals are obtained by minimizing the total energy of the system with respect to variations in the orbitals, subject to the constraint that they are orthonormal. This leads to a set of self-consistent equations that can be solved iteratively. The average potential in Hartree-Fock theory is called the Fock operator, which consists of two terms: the Coulomb operator, which represents the classical electrostatic repulsion between electrons; and the exchange operator, which represents the quantum mechanical exchange interaction between electrons with parallel spins.
Hartree-Fock theory is able to capture some important features of electronic structure, such as atomic shell structure, molecular orbital theory, chemical bonding and hybridization. However, it also has some serious limitations, such as:
It neglects electron correlation, which is the quantum mechanical effect that electrons tend to avoid each other due to their mutual repulsion. This leads to errors in energy calculations (such as overestimating ionization potentials and underestimating bond dissociation energies) and in density calculations (such as underestimating electron localization and overestimating bond lengths).
It suffers from basis set dependence, which means that the results depend on the choice of basis functions used to expand the orbitals. This leads to errors in energy calculations (such as basis set superposition error) and in density calculations (such as basis set incompleteness error).
It is computationally expensive, which means that it requires a lot of time and memory to solve for large systems. This limits its applicability to small molecules or clusters with up to a few tens of atoms.
Density functional theory
Density functional theory is based on the idea that the ground state energy and density of a system can be obtained from a universal functional of the density alone, without explicitly solving for the wave function. This idea is justified by two fundamental theorems proved by Hohenberg and Kohn: (1) The ground state density uniquely determines the external potential (up to a constant) and hence all the properties of the system; (2) The ground state energy can be written as a functional of the density that attains its minimum value at the true ground state density. The challenge is to find an explicit expression for this functional that can be evaluated for any given density. The most common approach is to use the Kohn-Sham ansatz, which introduces a set of auxiliary orbitals that are used to construct the density and the kinetic energy. The functional is then split into several terms: the kinetic energy of the noninteracting system, the classical electrostatic energy of the density, the exchange-correlation energy that accounts for quantum effects beyond the classical approximation, and an external potential that represents the nuclei or other external fields. The orbitals are obtained by minimizing the total energy of the system with respect to variations in the orbitals, subject to the constraint that they are orthonormal. This leads to a set of self-consistent equations that can be solved iteratively. The average potential in density functional theory is called the Kohn-Sham operator, which consists of three terms: the external potential, the Hartree potential (which is similar to the Coulomb operator in Hartree-Fock theory), and the exchange-correlation potential (which is derived from the exchange-correlation energy).
Density functional theory is able to overcome some of the limitations of Hartree-Fock theory, such as:
It includes electron correlation, at least in principle, by using an appropriate exchange-correlation functional. This improves the accuracy of energy calculations (such as reducing the error in ionization potentials and bond dissociation energies) and density calculations (such as increasing electron localization and reducing bond lengths).
It reduces basis set dependence, by using a more compact and efficient representation of the orbitals. This reduces the errors in energy calculations (such as basis set superposition error) and density calculations (such as basis set incompleteness error).
It lowers computational cost, by using a simpler and faster algorithm to solve for the orbitals. This enables its applicability to large molecules or solids with up to thousands of atoms.
However, density functional theory also has some challenges, such as:
It relies on approximations for the exchange-correlation functional, which are not known exactly and may not be valid for all systems. This leads to errors in energy calculations (such as underestimating band gaps and overestimating cohesive energies) and density calculations (such as underestimating charge transfer and overestimating spin polarization).
It suffers from self-interaction error, which is a spurious interaction of an electron with itself due to the use of an approximate exchange-correlation functional. This leads to errors in energy calculations (such as overestimating ionization potentials and underestimating electron affinities) and density calculations (such as overestimating electron delocalization and underestimating bond angles).
It is limited by the single-determinant ansatz, which assumes that the ground state wave function can be written as a single Slater determinant. This may not be valid for systems with strong correlation effects, such as transition metal complexes, diradicals and Mott insulators.
Beyond independent particle methods
Beyond independent particle methods are based on more sophisticated treatments of electron-electron interactions that go beyond the average potential approximation. These methods typically involve solving a many-body problem that requires more computational resources and advanced techniques. The two most popular beyond independent particle methods are many-body perturbation theory and dynamical mean field theory.
Many-body perturbation theory
Many-body perturbation theory is based on the idea that the exact Hamiltonian of a system can be written as a sum of a reference Hamiltonian (usually chosen to be solvable) and a perturbation Hamiltonian (usually chosen to be small). The exact wave function and energy of a system can then be expanded in terms of a power series of the perturbation Hamiltonian. The coefficients of this series can be calculated using diagrammatic techniques that involve Feynman diagrams and Wick's theorem. The most common approach in electronic structure calculations is to use Green's function methods, which are based on correlation functions that describe the propagation and interaction of particles in time or frequency domains. One of the most widely used Green's function methods is the GW approximation, which is based on two quantities: G, which is the one-electron Green's function, which describes the propagation of an electron with a given energy and momentum; and W, which is the screened Coulomb interaction, which describes the effective interaction between two electrons after accounting for the screening effect of the other electrons. The GW approximation is based on the assumption that the self-energy can be written as a product of G and W, which can be calculated using perturbation theory. The self-energy is then used to correct the energies and wave functions obtained from a reference method, such as Hartree-Fock theory or density functional theory.
Many-body perturbation theory is able to capture some important features of electronic structure that are beyond the reach of independent particle methods, such as:
It includes dynamical effects, such as frequency dependence and lifetime broadening, that are essential for describing excited states and spectral properties.
It improves band gap calculations, by including quasiparticle corrections that account for the renormalization of energies due to electron-electron interactions.
It enables optical spectra calculations, by including excitonic effects that account for the formation of bound electron-hole pairs due to attractive Coulomb interactions.
However, many-body perturbation theory also has some challenges, such as:
It relies on approximations for the Green's function and the screened Coulomb interaction, which may not be accurate or consistent for all systems. This leads to errors in energy calculations (such as underestimating quasiparticle gaps and overestimating exciton binding energies) and spectral calculations (such as underestimating peak intensities and overestimating peak widths).
It suffers from convergence issues, which means that the results depend on the choice of parameters used to perform the calculations, such as the number of bands, k-points, frequencies and diagrams included. This leads to errors in energy calculations (such as convergence error) and spectral calculations (such as spectral weight transfer error).
It is computationally demanding, which means that it requires a lot of time and memory to solve for large systems. This limits its applicability to small molecules or solids with up to a few hundreds of atoms.
Dynamical mean field theory
Dynamical mean field theory is based on the idea that the electronic structure of a system can be mapped onto an effective impurity model, which consists of a single site (or impurity) coupled to a bath of noninteracting electrons. The impurity represents a local region of interest in the system, such as an atom or a cluster. The bath represents the rest of the system, which provides a dynamical mean field for the impurity. The impurity model can be solved using various techniques, such as exact diagonalization, quantum Monte Carlo or numerical renormalization group. The solution provides a local self-energy, which describes the correlation effects on the impurity site. The self-energy is then used to update the bath parameters using a self-consistency condition that ensures that the impurity model reproduces the local properties of the original system.
Dynamical mean field theory is able to capture some important features of electronic structure that are beyond the reach of independent particle methods and many-body perturbation theory, such as:
It includes strong correlation effects, such as Mott transition and Kondo effect, that are essential for describing systems with partially filled d or f orbitals.
It improves density calculations, by including charge fluctuations and localization effects that account for the formation of magnetic moments and insulating gaps.
It enables thermodynamic properties calculations, by including temperature dependence and entropy effects that account for phase transitions and critical phenomena.
However, dynamical mean field theory also has some challenges, such as:
It relies on approximations for the impurity model and the self-consistency condition, which may not be valid or optimal for all systems. This leads to errors in energy calculations (such as overestimating Mott gaps and underestimating Kondo temperatures) and density calculations (such as overestimating charge localization and underestimating orbital polarization).
It suffers from discretization issues, which means that the results depend on the choice of basis functions used to represent the impurity model and the bath. This leads to errors in energy calculations (such as discretization error) and spectral calculations (such as artificial peaks or gaps).
It is computationally challenging, which means that it requires a lot of time and memory to solve for large systems. This limits its applicability to small molecules or solids with up to a few tens of atoms.
How is electronic structure applied?
In this section, we will give some examples of how electronic structure calculations can be applied to different systems and problems, such as atoms, solids, defects, interfaces and nanostructures. We will focus on the results obtained from density functional theory, as it is the most widely used method in practice. However, we will also mention some cases where beyond independent particle methods are necessary or beneficial.
Electronic structure of atoms
The electronic structure of atoms is the simplest and most fundamental case of electronic structure calculations. It provides the basis for understanding the periodic table, the chemical bonding and the spectroscopy of atoms. The electronic structure of atoms can be calculated using density functional theory with a minimal basis set of atomic orbitals. However, in order to reduce the computational cost and improve the accuracy for systems with many electrons, such as transition metals and lanthanides, it is common to use pseudopotentials instead of atomic orbitals.
Pseudopotentials are effective potentials that replace the core electrons and nuclei of an atom by a smooth potential that reproduces the same scattering properties as the original potential. Pseudopotentials allow to use a smaller basis set of valence orbitals that are relevant for chemical bonding and reactivity. Pseudopotentials can be classified into two types: norm-conserving pseudopotentials, which preserve the norm (or charge) of the valence orbitals; and ultrasoft pseudopotentials, which relax this constraint and allow for more flexibility and transferability. Pseudopotentials can be generated using different methods, such as empirical fitting, inversion or projection.
The electronic structure of atoms can be used to calculate various properties, such as:
The ionization potential, which is the energy required to remove an electron from an atom.
The electron affinity, which is the energy released when an electron is added to an atom.
The atomic radius, which is the distance from the nucleus to the outermost electron.
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