A linear combination of atomic orbitals or LCAO is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry.[1] In quantum mechanics, electron configurations of atoms are described as wavefunctions. In a mathematical sense, these wave functions are the basis set of functions, the basis functions, which describe the electrons of a given atom. In chemical reactions, orbital wavefunctions are modified, i.e. the electron cloud shape is changed, according to the type of atoms participating in the chemical bond.
It was introduced in 1929 by Sir John Lennard-Jones with the description of bonding in the diatomic molecules of the first main row of the periodic table, but had been used earlier by Linus Pauling for H2+.[2][3]
https://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals
The generalized valence bond (GVB) method is one of the simplest and oldest valence bond method that uses flexible orbitals in the general way used by modern valence bond theory. The method was developed by the group of William A. Goddard, III around 1970.[1][2]
https://en.wikipedia.org/wiki/Generalized_valence_bond
In chemistry, resonance, also called mesomerism, is a way of describing bonding in certain molecules or ions by the combination of several contributing structures (or forms,[1] also variously known as resonance structures or canonical structures) into a resonance hybrid (or hybrid structure) in valence bond theory. It has particular value for describing delocalized electronswithin certain molecules or polyatomic ions where the bonding cannot be expressed by one single Lewis structure.
https://en.wikipedia.org/wiki/Resonance_(chemistry)
In quantum mechanics, orbital magnetization, Morb, refers to the magnetization induced by orbital motion of charged particles, usually electrons in solids. The term "orbital" distinguishes it from the contribution of spin degrees of freedom, Mspin, to the total magnetization. A nonzero orbital magnetization requires broken time-reversal symmetry, which can occur spontaneously in ferromagnetic and ferrimagnetic materials, or can be induced in a non-magnetic material by an applied magnetic field.
https://en.wikipedia.org/wiki/Orbital_magnetization
Modern valence bond theory is the application of valence bond theory, with computer programs that are competitive in accuracy and economy with programs for the Hartree–Fock method and other molecular orbital based methods. The latter methods dominated quantum chemistry from the advent of digital computers because they were easier to program. The early popularity of valence bond methods thus declined. It is only recently that the programming of valence bond methods has improved. These developments are due to and described by Gerratt, Cooper, Karadakov and Raimondi (1997); Li and McWeeny (2002); Joop H. van Lenthe and co-workers (2002);[1] Song, Mo, Zhang and Wu (2005); and Shaik and Hiberty (2004)[2]
https://en.wikipedia.org/wiki/Modern_valence_bond_theory
In chemistry, molecular orbital theory (MO theory or MOT) is a method for describing the electronic structure of molecules using quantum mechanics. It was proposed early in the 20th century.
In molecular orbital theory, electrons in a molecule are not assigned to individual chemical bonds between atoms, but are treated as moving under the influence of the atomic nuclei in the whole molecule.[1] Quantum mechanics describes the spatial and energetic properties of electrons as molecular orbitals that surround two or more atoms in a molecule and contain valence electrons between atoms.
Molecular orbital theory revolutionized the study of chemical bonding by approximating the states of bonded electrons—the molecular orbitals—as linear combinations of atomic orbitals (LCAO). These approximations are made by applying the density functional theory (DFT) or Hartree–Fock (HF) models to the Schrödinger equation.
Molecular orbital theory and valence bond theory are the foundational theories of quantum chemistry.
https://en.wikipedia.org/wiki/Molecular_orbital_theory
In chemistry, valence bond (VB) theory is one of the two basic theories, along with molecular orbital (MO) theory, that were developed to use the methods of quantum mechanics to explain chemical bonding. It focuses on how the atomic orbitals of the dissociated atoms combine to give individual chemical bonds when a molecule is formed. In contrast, molecular orbital theory has orbitals that cover the whole molecule.[1]
https://en.wikipedia.org/wiki/Valence_bond_theory
https://en.wikipedia.org/wiki/Category:Electronic_structure_methods
In solid-state physics, the tight-binding model (or TB model) is an approach to the calculation of electronic band structure using an approximate set of wave functionsbased upon superposition of wave functions for isolated atoms located at each atomic site. The method is closely related to the LCAO method (linear combination of atomic orbitals method) used in chemistry. Tight-binding models are applied to a wide variety of solids. The model gives good qualitative results in many cases and can be combined with other models that give better results where the tight-binding model fails. Though the tight-binding model is a one-electron model, the model also provides a basis for more advanced calculations like the calculation of surface states and application to various kinds of many-body problem and quasiparticle calculations.
https://en.wikipedia.org/wiki/Tight_binding
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called band gaps or forbidden bands).
Band theory derives these bands and band gaps by examining the allowed quantum mechanical wave functions for an electron in a large, periodic lattice of atoms or molecules. Band theory has been successfully used to explain many physical properties of solids, such as electrical resistivity and optical absorption, and forms the foundation of the understanding of all solid-state devices (transistors, solar cells, etc.).
https://en.wikipedia.org/wiki/Electronic_band_structure
In solid-state physics, the k·p perturbation theory is an approximated semi-empirical approach for calculating the band structure (particularly effective mass) and optical properties of crystalline solids.[1][2][3] It is pronounced "k dot p", and is also called the "k·p method". This theory has been applied specifically in the framework of the Luttinger–Kohn model (after Joaquin Mazdak Luttinger and Walter Kohn), and of the Kane model (after Evan O. Kane).
https://en.wikipedia.org/wiki/K·p_perturbation_theory
Complete active space perturbation theory (CASPTn) is a multireference electron correlation method for computational investigation of molecular systems, especially for those with heavy atoms such as transition metals, lanthanides, and actinides. It can be used, for instance, to describe electronic states of a system, when single reference methods and density functional theory cannot be used, and for heavy atom systems for which quasi-relativistic approaches are not appropriate.[1]
https://en.wikipedia.org/wiki/Complete_active_space_perturbation_theory
The Peierls substitution method, named after the original work by Rudolf Peierls[1] is a widely employed approximation for describing tightly-bound electrons in the presence of a slowly varying magnetic vector potential.[2]
In the presence of an external magnetic vector potential the translation operators, which form the kinetic part of the Hamiltonian in the tight-binding framework, are simply
and in the second quantization formulation
The phases are defined as
https://en.wikipedia.org/wiki/Peierls_substitution
In physics, a pseudopotential or effective potential is used as an approximation for the simplified description of complex systems. Applications include atomic physics and neutron scattering. The pseudopotential approximation was first introduced by Hans Hellmann in 1934.[1]
https://en.wikipedia.org/wiki/Pseudopotential
The projector augmented wave method (PAW) is a technique used in ab initio electronic structure calculations. It is a generalization of the pseudopotential and linear augmented-plane-wave methods, and allows for density functional theorycalculations to be performed with greater computational efficiency.[1]
Valence wavefunctions tend to have rapid oscillations near ion cores due to the requirement that they be orthogonal to core states; this situation is problematic because it requires many Fourier components (or in the case of grid-based methods, a very fine mesh) to describe the wavefunctions accurately. The PAW approach addresses this issue by transforming these rapidly oscillating wavefunctions into smooth wavefunctions which are more computationally convenient, and provides a way to calculate all-electron properties from these smooth wavefunctions. This approach is somewhat reminiscent of a change from the Schrödinger picture to the Heisenberg picture.
https://en.wikipedia.org/wiki/Projector_augmented_wave_method
Ab initio quantum chemistry methods are computational chemistry methods based on quantum chemistry.[1] The term ab initiowas first used in quantum chemistry by Robert Parr and coworkers, including David Craig in a semiempirical study on the excited states of benzene.[2][3] The background is described by Parr.[4] Ab initio means "from first principles" or "from the beginning", implying that the only inputs into an ab initio calculation are physical constants.[5] Ab initio quantum chemistry methods attempt to solve the electronic Schrödinger equation given the positions of the nuclei and the number of electrons in order to yield useful information such as electron densities, energies and other properties of the system. The ability to run these calculations has enabled theoretical chemists to solve a range of problems and their importance is highlighted by the awarding of the Nobel prize to John Pople and Walter Kohn.[6]
https://en.wikipedia.org/wiki/Ab_initio_quantum_chemistry_methods
Symmetry-adapted perturbation theory or SAPT[1][2] is a methodology in electronic structure theory developed to describe non-covalent interactions between atoms and/or molecules. SAPT is a member of the family of methods known as energy decomposition analysis (EDA). Most EDA methods decompose a total interaction energy that is computed via a supermolecularapproach, such that:
where is the total interaction energy obtained via subtracting isolated monomer energies and from the dimer energy . A key deficiency of the supermolecular interaction energy is that it is susceptible to basis set superposition error (BSSE).
https://en.wikipedia.org/wiki/Symmetry-adapted_perturbation_theory
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