{"title":"莫尔斯振子的阶梯算子及摄动振动问题","authors":"S. V. Krasnoshchekov, Xuanhao Chang","doi":"10.1080/0144235X.2019.1593583","DOIUrl":null,"url":null,"abstract":"ABSTRACT Quantum-mechanical methods of solving the polyatomic vibrational Schrödinger equation need higher quality zero-order approximations than ones originating from the harmonic oscillator (HO). Ladder operators built on the HO have a number of unique features simplifying both the operator perturbation theory and practical implementations of matrix-elements-based methods. Therefore, finding suitable ladder operators for solvable anharmonic oscillators and mainly the Morse oscillator remain one of the major challenges of nuclear vibrational dynamics. In this work, we review the problem of building Morse oscillator ladder operators (MLOs) and the prospects of their use in various methods of solving the many-dimensional anharmonic vibrational problem. The features of several existing approaches for building MLOs are explored and analysed. The native MLOs obtained by the factorisation method are not quite suitable for expressing a perturbed potential energy operator. Supersymmetric quantum mechanics (SUSYQM) does not solve the problem either since corresponding ladder operators only connect states from related potentials. The SU(2) vibron model provides an approximate solution based on a formal isomorphism of energy states. We have found that for the present the only useful model for MLOs is based on the so-called quasi-number states basis set (QNSB) built on modified Laguerre polynomials. QNSB yields a finite tridiagonal matrix representation of the Morse Hamiltonian corresponding to the exact solution. The convenience and accuracy of QNSB approach in comparison to second/fourth-order perturbation theory is illustrated with the HF molecule. The general conclusion is that QNSB-based MLOs are suitable for building many-body treatments, for instance, with the VSCF/VCI approach.","PeriodicalId":54932,"journal":{"name":"International Reviews in Physical Chemistry","volume":null,"pages":null},"PeriodicalIF":2.5000,"publicationDate":"2019-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"Ladder operators for Morse oscillator and a perturbed vibrational problem\",\"authors\":\"S. V. Krasnoshchekov, Xuanhao Chang\",\"doi\":\"10.1080/0144235X.2019.1593583\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"ABSTRACT Quantum-mechanical methods of solving the polyatomic vibrational Schrödinger equation need higher quality zero-order approximations than ones originating from the harmonic oscillator (HO). Ladder operators built on the HO have a number of unique features simplifying both the operator perturbation theory and practical implementations of matrix-elements-based methods. Therefore, finding suitable ladder operators for solvable anharmonic oscillators and mainly the Morse oscillator remain one of the major challenges of nuclear vibrational dynamics. In this work, we review the problem of building Morse oscillator ladder operators (MLOs) and the prospects of their use in various methods of solving the many-dimensional anharmonic vibrational problem. The features of several existing approaches for building MLOs are explored and analysed. The native MLOs obtained by the factorisation method are not quite suitable for expressing a perturbed potential energy operator. Supersymmetric quantum mechanics (SUSYQM) does not solve the problem either since corresponding ladder operators only connect states from related potentials. The SU(2) vibron model provides an approximate solution based on a formal isomorphism of energy states. We have found that for the present the only useful model for MLOs is based on the so-called quasi-number states basis set (QNSB) built on modified Laguerre polynomials. QNSB yields a finite tridiagonal matrix representation of the Morse Hamiltonian corresponding to the exact solution. The convenience and accuracy of QNSB approach in comparison to second/fourth-order perturbation theory is illustrated with the HF molecule. The general conclusion is that QNSB-based MLOs are suitable for building many-body treatments, for instance, with the VSCF/VCI approach.\",\"PeriodicalId\":54932,\"journal\":{\"name\":\"International Reviews in Physical Chemistry\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.5000,\"publicationDate\":\"2019-01-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Reviews in Physical Chemistry\",\"FirstCategoryId\":\"92\",\"ListUrlMain\":\"https://doi.org/10.1080/0144235X.2019.1593583\",\"RegionNum\":2,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"CHEMISTRY, PHYSICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Reviews in Physical Chemistry","FirstCategoryId":"92","ListUrlMain":"https://doi.org/10.1080/0144235X.2019.1593583","RegionNum":2,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"CHEMISTRY, PHYSICAL","Score":null,"Total":0}
Ladder operators for Morse oscillator and a perturbed vibrational problem
ABSTRACT Quantum-mechanical methods of solving the polyatomic vibrational Schrödinger equation need higher quality zero-order approximations than ones originating from the harmonic oscillator (HO). Ladder operators built on the HO have a number of unique features simplifying both the operator perturbation theory and practical implementations of matrix-elements-based methods. Therefore, finding suitable ladder operators for solvable anharmonic oscillators and mainly the Morse oscillator remain one of the major challenges of nuclear vibrational dynamics. In this work, we review the problem of building Morse oscillator ladder operators (MLOs) and the prospects of their use in various methods of solving the many-dimensional anharmonic vibrational problem. The features of several existing approaches for building MLOs are explored and analysed. The native MLOs obtained by the factorisation method are not quite suitable for expressing a perturbed potential energy operator. Supersymmetric quantum mechanics (SUSYQM) does not solve the problem either since corresponding ladder operators only connect states from related potentials. The SU(2) vibron model provides an approximate solution based on a formal isomorphism of energy states. We have found that for the present the only useful model for MLOs is based on the so-called quasi-number states basis set (QNSB) built on modified Laguerre polynomials. QNSB yields a finite tridiagonal matrix representation of the Morse Hamiltonian corresponding to the exact solution. The convenience and accuracy of QNSB approach in comparison to second/fourth-order perturbation theory is illustrated with the HF molecule. The general conclusion is that QNSB-based MLOs are suitable for building many-body treatments, for instance, with the VSCF/VCI approach.
期刊介绍:
International Reviews in Physical Chemistry publishes review articles describing frontier research areas in physical chemistry. Internationally renowned scientists describe their own research in the wider context of the field. The articles are of interest not only to specialists but also to those wishing to read general and authoritative accounts of recent developments in physical chemistry, chemical physics and theoretical chemistry. The journal appeals to research workers, lecturers and research students alike.