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Quantum Theory for Chemical Applications最新文献

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Band Structure Theory for Extended Systems 扩展系统的能带结构理论
Quantum Theory for Chemical Applications Pub Date : 2020-12-21 DOI: 10.1093/OSO/9780190920807.003.0013
J. Autschbach
{"title":"Band Structure Theory for Extended Systems","authors":"J. Autschbach","doi":"10.1093/OSO/9780190920807.003.0013","DOIUrl":"https://doi.org/10.1093/OSO/9780190920807.003.0013","url":null,"abstract":"The electronic structure of infinite periodic systems (crystals) is treated with band structure theory, replacing molecular orbitals by crystal orbitals. The chapter starts out by introducing the electron gas and definitions of the Fermi momentum, the Fermi energy, and the density of states (DOS). A periodic linear combination of atomic orbitals (LCAO) type treatment of an infinite periodic system is facilitated by the construction of Bloch functions. The notions of energy band and band gap are discussed with band structure concepts, using the approximations made in Huckel theory (chapter 12). One, two, and three-dimensional crystal lattices and the associated reciprocal lattices are introduced. The band structures of sodium metal, boron nitride, silicon, and graphite, are discussed as examples of metals, insulators, semi-conductors, and semi-metals, respectively. The chapter concludes with a brief discussion of the projected DOS and measures to determine bonding or antibonding interactions between atoms in a crystal.","PeriodicalId":207760,"journal":{"name":"Quantum Theory for Chemical Applications","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122475171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Electron Spin and General Angular Momenta 电子自旋和一般角动量
Quantum Theory for Chemical Applications Pub Date : 2020-12-21 DOI: 10.1093/OSO/9780190920807.003.0019
J. Autschbach
{"title":"Electron Spin and General Angular Momenta","authors":"J. Autschbach","doi":"10.1093/OSO/9780190920807.003.0019","DOIUrl":"https://doi.org/10.1093/OSO/9780190920807.003.0019","url":null,"abstract":"The historical background of the discovery of the electron spin is provided. The Stern-Gerlach and Einstein-de Haas experiments are discussed. The operators for a single electron spin are defined, along with the formulation in terms of the 2x2 Pauli matrices. The discussion then moves on to the definition of the spin for many-electron systems and explains how the famous Hund rule (or Hund’s first rule) arises from considering the energy of an open-shell spin singlet vs. triplet state. Next, the generalized angular momentum, ladder operators, and spherical vector operators are defined, and the rules for the addition of angular momenta are derived. The chapter concludes with a discussion of the total spin, orbital, and total angular momentum for open-shell atoms, term symbols, and Hund’s second and third rule.","PeriodicalId":207760,"journal":{"name":"Quantum Theory for Chemical Applications","volume":"98 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130995795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Vectors and Functions and Operators 向量，函数和算子
Quantum Theory for Chemical Applications Pub Date : 2020-12-21 DOI: 10.1093/OSO/9780190920807.003.0001
J. Autschbach
{"title":"Vectors and Functions and Operators","authors":"J. Autschbach","doi":"10.1093/OSO/9780190920807.003.0001","DOIUrl":"https://doi.org/10.1093/OSO/9780190920807.003.0001","url":null,"abstract":"This chapter introduces – briefly – vectors and functions and the similarities between them, some basic linear algebra concepts, operators (including the del and Laplace operators), eigenvalues and eigenvectors &eigenfunctions, the scalar (dot) and vector (cross) product between two vectors, the scalar product between two functions, the concepts of normalization, orthogonality, and orthonormality. The concept of an operator is first introduced by considering the rotation and stretching or compression of a vector. It is then generalized to a mathematical prescription that changes a function into another function.","PeriodicalId":207760,"journal":{"name":"Quantum Theory for Chemical Applications","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128908465","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A First Example: The “Particle in a Box“ and Quantized Translational Motion 第一个例子:“盒子里的粒子”和量子化的平移运动
Quantum Theory for Chemical Applications Pub Date : 2020-12-21 DOI: 10.1093/OSO/9780190920807.003.0005
J. Autschbach
{"title":"A First Example: The “Particle in a Box“ and Quantized Translational Motion","authors":"J. Autschbach","doi":"10.1093/OSO/9780190920807.003.0005","DOIUrl":"https://doi.org/10.1093/OSO/9780190920807.003.0005","url":null,"abstract":"The simple ‘particle in a box’ (Piab) is introduced in this chapter so that the reader can get familiar with applying the quantum recipe and atomic units. The PiaB is introduced in its one, two, and three dimensional variants, which demonstrates the use of the separation of variables technique as a strategy to solve the Schrodinger equation for a particle with two or three degrees of freedom. It is shown that the confinement of the particle causes the energy to be quantized. The one-dimensional PiaB is then applied to treat the electronic spectra of cyanine dyes and their absorption colors. The chapter then introduces more general setups with finite potential wells, in order to introduce the phenomenon of quantum tunnelling and to discuss more generally with the unintuitive ‘quantum behavior’ of particles such as electrons. Scanning tunnelling and atomic force microscopes are also discussed briefly.","PeriodicalId":207760,"journal":{"name":"Quantum Theory for Chemical Applications","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128443979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Atomic Units 原子单元
Quantum Theory for Chemical Applications Pub Date : 2020-12-21 DOI: 10.1351/goldbook.a00504
J. Autschbach
{"title":"Atomic Units","authors":"J. Autschbach","doi":"10.1351/goldbook.a00504","DOIUrl":"https://doi.org/10.1351/goldbook.a00504","url":null,"abstract":"This short chapter introduces the dimensionless Hartree atomic units that are used widely in theoretical and numerical quantum chemistry.","PeriodicalId":207760,"journal":{"name":"Quantum Theory for Chemical Applications","volume":"34 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126052591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Dynamic Fields and Response Properties 动态字段和响应属性
Quantum Theory for Chemical Applications Pub Date : 2020-12-21 DOI: 10.1093/OSO/9780190920807.003.0023
J. Autschbach
{"title":"Dynamic Fields and Response Properties","authors":"J. Autschbach","doi":"10.1093/OSO/9780190920807.003.0023","DOIUrl":"https://doi.org/10.1093/OSO/9780190920807.003.0023","url":null,"abstract":"It is shown how electronic transitions can be induced by the interaction with an electromagnetic wave of a suitable frequency. The rate of a transition between two electronic states induced by a time-dependent field is derived. The transition rate expression is used to calculate the absorption coefficient due to electronic transitions. The differential absorption coefficient for left and right circular polarized light is specific to chiral molecules and has different signs for a pair of enantiomers. The discussion then shifts to general functions describing the response of an atom or molecule to an external. The ideas developed thus far are then applied to the dynamic polarizability, molecular linear response functions in general, and the optical rotation. Linear response theory is set up within time-dependent molecular orbital theory. The Chapter concludes with a discussion of non-linear response properties and two-photon absorption.","PeriodicalId":207760,"journal":{"name":"Quantum Theory for Chemical Applications","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126295205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
From Atomic Orbitals to Molecular Orbitals and Chemical Bonds 从原子轨道到分子轨道和化学键
Quantum Theory for Chemical Applications Pub Date : 2020-12-21 DOI: 10.1093/OSO/9780190920807.003.0009
J. Autschbach
{"title":"From Atomic Orbitals to Molecular Orbitals and Chemical Bonds","authors":"J. Autschbach","doi":"10.1093/OSO/9780190920807.003.0009","DOIUrl":"https://doi.org/10.1093/OSO/9780190920807.003.0009","url":null,"abstract":"It is shown how an aufbau principle for atoms arises from the Hartree-Fock (HF) treatment with increasing numbers of electrons. The Slater screening rules are introduced. The HF equations for general molecules are not separable in the spatial variables. This requires another approximation, such as the linear combination of atomic orbitals (LCAO) molecular orbital method. The orbitals of molecules are represented in a basis set of known functions, for example atomic orbital (AO)-like functions or plane waves. The HF equation then becomes a generalized matrix pseudo-eigenvalue problem. Solutions are obtained for the hydrogen molecule ion and H2 with a minimal AO basis. The Slater rule for 1s shells is rationalized via the optimal exponent in a minimal 1s basis. The nature of the chemical bond, and specifically the role of the kinetic energy in covalent bonding, are discussed in details with the example of the hydrogen molecule ion.","PeriodicalId":207760,"journal":{"name":"Quantum Theory for Chemical Applications","volume":"2 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120985056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Quantized Vibrational Motion 量化振动运动
Quantum Theory for Chemical Applications Pub Date : 2020-12-21 DOI: 10.1093/OSO/9780190920807.003.0014
J. Autschbach
{"title":"Quantized Vibrational Motion","authors":"J. Autschbach","doi":"10.1093/OSO/9780190920807.003.0014","DOIUrl":"https://doi.org/10.1093/OSO/9780190920807.003.0014","url":null,"abstract":"The harmonic oscillator of chapter 2 is visited again, now in its quantum theoretical version. The solution of the Schrodinger equation (SE) is shown step-by step, as it features steps that are very similar to those used in solving the equations for the angular momentum and hydrogen-like orbitals in later chapters. The Morse oscillator has a potential function that is much more representative of the vibrations of atoms in molecules as the harmonic potential. The solutions of the harmonic and Morse oscillator are compared. It is then shown how nuclear vibrations in poly-atomic molecules are treated at the harmonic level. This requires the separation of internal degrees of freedom from the overall translation and rotation of a molecule, leading to the normal modes. The chapter also discusses basic aspects of vibrational spectroscopy and the selection rules of infrared and Raman vibrational spectroscopy.","PeriodicalId":207760,"journal":{"name":"Quantum Theory for Chemical Applications","volume":"6 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133170996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Angular Momentum and Rotational Motion in Three Dimensions 三维角动量和旋转运动
Quantum Theory for Chemical Applications Pub Date : 2020-12-21 DOI: 10.1093/OSO/9780190920807.003.0016
J. Autschbach
{"title":"Angular Momentum and Rotational Motion in Three Dimensions","authors":"J. Autschbach","doi":"10.1093/OSO/9780190920807.003.0016","DOIUrl":"https://doi.org/10.1093/OSO/9780190920807.003.0016","url":null,"abstract":"This chapter treats rotations in three dimensions and the angular momentum quantum mechanically. The angular momentum is a 3-vector that plays the same role in the kinetic energy of rotating particles as the linear momentum 3-vector plays in the kinetic energy for translational motion. The quantum operator for the three components do not commute, i.e. the full vector cannot be known. Each component commutes with the operator for the square-length of the angular momentum vector, and therefore one opts for calculating the square-length and one of the components, z, referred to as the projection. The angular momentum is quantized. The (azimuthal) quantum number ℓ = 0,1,2,3… quantifies the square-length as ℓ(ℓ+1), and the projection is quantified by the (magnbetic) quantum number mℓ = -ℓ…ℓ in integer steps. The eigenfunctions are called spherical harmonics. A ‘rigid rotor’ model is set up to treat the rotational spectrum of a diatomic molecule.","PeriodicalId":207760,"journal":{"name":"Quantum Theory for Chemical Applications","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114117309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Post-Hartree-Fock Methods and Electron Correlation: A Very Brief Overview Post-Hartree-Fock方法和电子相关:一个非常简单的概述
Quantum Theory for Chemical Applications Pub Date : 2020-12-21 DOI: 10.1093/OSO/9780190920807.003.0020
J. Autschbach
{"title":"Post-Hartree-Fock Methods and Electron Correlation: A Very Brief Overview","authors":"J. Autschbach","doi":"10.1093/OSO/9780190920807.003.0020","DOIUrl":"https://doi.org/10.1093/OSO/9780190920807.003.0020","url":null,"abstract":"‘This chapter sketches how the electron correlation is treated in post-Hartree-Fock (HF) wavefunction methods. The distinction between static and dynamic correlation is explained. A configuration interaction (CI) wavefunction is a linear combination of several or many Slater determinants (SDs). Following a HF calculation, different SDs can be constructed by replacing 1, 2, 3, … occupied orbitals in the HF wavefunction with 1, 2, 3,… unoccupied or virtual orbitals, leading to pseudo-excited electron configurations at the singles, doubles, triples, … (S, D, T, …) level. The virtual orbitals are usually available as a by-product of the HF calculation in a basis set. Full CI (FCI) considers all possible substitutions, up to N-fold for an N-electron system. FCI is impractical for all but the smallest molecules. CI truncated at a lower level, e.g. S and D, suffers from lack of size extensitivity. Truncated coupled-cluster (CC) is size extensive. Open-shell systems generally require a multi-reference treatment. The chapter concludes with a treatment of the static correlation in the bond breaking of H2.","PeriodicalId":207760,"journal":{"name":"Quantum Theory for Chemical Applications","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133074640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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