{"title":"态、构型和轨道的正交性","authors":"Balakrishnan Viswanathan, Mohamed Shajahan Gulam Razul","doi":"10.1007/s10698-022-09417-y","DOIUrl":null,"url":null,"abstract":"<div><p>This manuscript explores the orthogonality constraints on configurations and orbitals subject to the property that states are mutually orthogonal. The orthogonality constraints lead to properties that affect the description of chemical systems. When states are described as linear combinations of (orthogonal or non-orthogonal) configurations, the coefficient matrix (mapping configurations to states) diagonalises <b><i>S</i></b><sup>−1</sup><b><i>H</i></b>. Therefore, single-configuration states are <i>only</i> possible in one-electron systems: non-orthogonal configurations yield single-configuration states only if <b><i>S</i></b><sup>−1</sup><b><i>H</i></b> is diagonal, but this would violate the orthonormalisation constraint. Further, the coefficient matrix is not constrained to be square (the number of configurations may differ from the number of configurations). Similarly, the orbitals used to construct configurations may also be orthogonal or non-orthogonal; orbitals are only required to be mutually orthogonal at the one-electron limit. Orthogonal orbitals are generally preferred due to their mathematical and conceptual simplicity, leading to fictitious unoccupied orbitals. Since the Fock operator is orthogonality agnostic, non-orthogonal (occupied) orbitals can be generated by solving the Fock equation independently for each electron; the virtual orbitals produced by this conception are true excitation orbitals as they are eigensolutions of the Fock operator. Additionally, we show that the number of molecular orbitals generated is <i>not</i> restricted to the number of atomic orbitals (or basis functions) employed in the computation. This manuscript explores the mathematical relationships that need to be satisfied under the various orthogonality regimes. We also present mathematical relationships that provide results that are independent of the orthogonality approximation within a particular computational method.</p></div>","PeriodicalId":568,"journal":{"name":"Foundations of Chemistry","volume":null,"pages":null},"PeriodicalIF":1.8000,"publicationDate":"2022-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Orthogonality properties of states, configurations, and orbitals\",\"authors\":\"Balakrishnan Viswanathan, Mohamed Shajahan Gulam Razul\",\"doi\":\"10.1007/s10698-022-09417-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>This manuscript explores the orthogonality constraints on configurations and orbitals subject to the property that states are mutually orthogonal. The orthogonality constraints lead to properties that affect the description of chemical systems. When states are described as linear combinations of (orthogonal or non-orthogonal) configurations, the coefficient matrix (mapping configurations to states) diagonalises <b><i>S</i></b><sup>−1</sup><b><i>H</i></b>. Therefore, single-configuration states are <i>only</i> possible in one-electron systems: non-orthogonal configurations yield single-configuration states only if <b><i>S</i></b><sup>−1</sup><b><i>H</i></b> is diagonal, but this would violate the orthonormalisation constraint. Further, the coefficient matrix is not constrained to be square (the number of configurations may differ from the number of configurations). Similarly, the orbitals used to construct configurations may also be orthogonal or non-orthogonal; orbitals are only required to be mutually orthogonal at the one-electron limit. Orthogonal orbitals are generally preferred due to their mathematical and conceptual simplicity, leading to fictitious unoccupied orbitals. Since the Fock operator is orthogonality agnostic, non-orthogonal (occupied) orbitals can be generated by solving the Fock equation independently for each electron; the virtual orbitals produced by this conception are true excitation orbitals as they are eigensolutions of the Fock operator. Additionally, we show that the number of molecular orbitals generated is <i>not</i> restricted to the number of atomic orbitals (or basis functions) employed in the computation. This manuscript explores the mathematical relationships that need to be satisfied under the various orthogonality regimes. We also present mathematical relationships that provide results that are independent of the orthogonality approximation within a particular computational method.</p></div>\",\"PeriodicalId\":568,\"journal\":{\"name\":\"Foundations of Chemistry\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2022-02-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Foundations of Chemistry\",\"FirstCategoryId\":\"92\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10698-022-09417-y\",\"RegionNum\":3,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"HISTORY & PHILOSOPHY OF SCIENCE\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Foundations of Chemistry","FirstCategoryId":"92","ListUrlMain":"https://link.springer.com/article/10.1007/s10698-022-09417-y","RegionNum":3,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"HISTORY & PHILOSOPHY OF SCIENCE","Score":null,"Total":0}
Orthogonality properties of states, configurations, and orbitals
This manuscript explores the orthogonality constraints on configurations and orbitals subject to the property that states are mutually orthogonal. The orthogonality constraints lead to properties that affect the description of chemical systems. When states are described as linear combinations of (orthogonal or non-orthogonal) configurations, the coefficient matrix (mapping configurations to states) diagonalises S−1H. Therefore, single-configuration states are only possible in one-electron systems: non-orthogonal configurations yield single-configuration states only if S−1H is diagonal, but this would violate the orthonormalisation constraint. Further, the coefficient matrix is not constrained to be square (the number of configurations may differ from the number of configurations). Similarly, the orbitals used to construct configurations may also be orthogonal or non-orthogonal; orbitals are only required to be mutually orthogonal at the one-electron limit. Orthogonal orbitals are generally preferred due to their mathematical and conceptual simplicity, leading to fictitious unoccupied orbitals. Since the Fock operator is orthogonality agnostic, non-orthogonal (occupied) orbitals can be generated by solving the Fock equation independently for each electron; the virtual orbitals produced by this conception are true excitation orbitals as they are eigensolutions of the Fock operator. Additionally, we show that the number of molecular orbitals generated is not restricted to the number of atomic orbitals (or basis functions) employed in the computation. This manuscript explores the mathematical relationships that need to be satisfied under the various orthogonality regimes. We also present mathematical relationships that provide results that are independent of the orthogonality approximation within a particular computational method.
期刊介绍:
Foundations of Chemistry is an international journal which seeks to provide an interdisciplinary forum where chemists, biochemists, philosophers, historians, educators and sociologists with an interest in foundational issues can discuss conceptual and fundamental issues which relate to the `central science'' of chemistry. Such issues include the autonomous role of chemistry between physics and biology and the question of the reduction of chemistry to quantum mechanics. The journal will publish peer-reviewed academic articles on a wide range of subdisciplines, among others: chemical models, chemical language, metaphors, and theoretical terms; chemical evolution and artificial self-replication; industrial application, environmental concern, and the social and ethical aspects of chemistry''s professionalism; the nature of modeling and the role of instrumentation in chemistry; institutional studies and the nature of explanation in the chemical sciences; theoretical chemistry, molecular structure and chaos; the issue of realism; molecular biology, bio-inorganic chemistry; historical studies on ancient chemistry, medieval chemistry and alchemy; philosophical and historical articles; and material of a didactic nature relating to all topics in the chemical sciences. Foundations of Chemistry plans to feature special issues devoted to particular themes, and will contain book reviews and discussion notes. Audience: chemists, biochemists, philosophers, historians, chemical educators, sociologists, and other scientists with an interest in the foundational issues of science.