Phenomenological two-nucleon potentials and the deuteron

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTVariational Principle for a Particle in a BoxJuan I. Casaubon and Graham Doggett View Author Information Facultad de Ciencias Exactas y Naturales, Universidad de Belgrano y CONICET, Villanueva 1324, 1426 Buenos Aires, Argentina Chemistry Department, University of York, Heslington, York YO10 5DD, UKCite this: J. Chem. Educ. 2000, 77, 9, 1221Publication Date (Web):September 1, 2000Publication History Received3 August 2009Published online1 September 2000Published inissue 1 September 2000https://pubs.acs.org/doi/10.1021/ed077p1221https://doi.org/10.1021/ed077p1221research-articleACS PublicationsRequest reuse permissionsArticle Views1245Altmetric-Citations8LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InRedditEmail Other access optionsGet e-Alertsclose SUBJECTS:Chemical calculations,Energy,Excited states,Wave function Get e-Alerts

The method of convolutions is used to form smoothed functions from approximate wave functions which are discontinuous or have discontinuous first derivatives. As a parameter ε in the smoothed function decreases, the smoothed function approaches the approximate wave function. The expectation values for physical properties corresponding to the approximate wave function are defined to be the limit as ε approaches zero of the expectation values corresponding to the smoothed function. It is found that if the approximate wave function is discontinuous, the corresponding expectation value for the kinetic energy is infinite. Therefore, it seems unlikely that discontinuous approximate wave functions can ever be useful. However, if the approximate wave function is continuous but has a discontinuity in its first derivative, then, as a result of the discontinuity, there is a contribution δT̄12 to the expectation value of the kinetic energy. For a one-dimensional problem δT̄12=−(ℏ2/2m)ψ*(0) [ψ2′(0)−ψ1′(0)].Here ψ(0) is the value of the approximate wave function at the point zero where the discontinuity in its first derivative occurs, and ψ1′(0) and ψ2′(0) are the first derivatives of ψ taken from the left and from the right, respectively, at this point. Similarly, for an N-dimensional problem having a surface S12=0 over which the first derivatives of the approximate wave function are discontinuous, δT̄12= ∫ surfaceψ*[(∂/∂n) (ψ2−ψ1)]dS12.Here the ∂/∂n is the normal derivative with the normal pointed from region 1 towards region 2.

An approximate light-cone wave function for the pion effective quark-antiquark Fock sector corresponding to a small value of the renormalization group parameter is presented. The approximate wave function is motivated by the LF-holography and the quadratic confinement potential in the front form of Hamiltonian dynamics, which is in harmony with the linear confining potential in the instant form. The pion radius, decay constant and form-factor are also presented.

The effect of hydrostatic pressure on the binding energy and diamagnetic susceptibility of a laser dressed donor impurity in a GaAs/GaAlAs nanowire superlattice

Investigations of the fractional quantum Hall effect are extended to spatially varying magnetic fields. Approximate single-particle wave functions are proposed and compared with ones obtained by numerical integration. As in the uniform field case, the interacting many-electron system forms an incompressible fluid and has fractionally charged excitations. Field inhomogeneities can trap collective excitations. © 1990 The American Physical Society.

Taking the ground state of helium as an example, evidence is presented in support of the theory that when an approximate wave function satisfies a hypervirial theorem, then the expectation values of certain operators calculated using that wave function will be quite accurate. Firstly, it is discovered which hypervirial theorems are satisfied by various simple approximate wave functions when their disposable parameters are chosen to optimize the energy. Appropriate expectation values are tested, and found to be reasonably accurate. Secondly, the optimum energy criterion is abandoned in certain cases, and parameters are chosen to satisfy alternative hypervirial theorems. This leads to improvement in expectation values related to the hypervirial operators. Some advantages and drawbacks of the hypervirial method of fixing parameters in approximate wave functions are briefly discussed.

Quantum-selected configuration interaction (QSCI) is an approach for quantum chemical calculations using current quantum computers. In conventional QSCI, Slater determinants used for the wave function expansion are sampled by iteratively performing approximate wave function preparation and subsequent measurement in the computational basis, and then the subspace Hamiltonian matrix is diagonalized on a classical computer. In this approach, preparation of a high-quality approximate wave function is necessary to accurately compute total energies. Here we propose a Hamiltonian simulation-based QSCI (HSB-QSCI) to avoid this difficulty, by sampling the Slater determinants from quantum states generated by the real-time evolution of approximate wave functions. We provide numerical simulations for the lowest spin-singlet and triplet states of oligoacenes (benzene, naphthalene, and anthracene), phenylene-1,4-dinitrene, and hexa-1,2,3,4,5-pentaene. We found that the HSB-QSCI is applicable not only to molecules where the Hartree-Fock provides a good approximation of the ground state, but also to strongly correlated systems where preparing a high-quality approximate wave function is hard. Hardware demonstrations of the HSB-QSCI are also reported for carbyne molecules expressed by up to 36 qubits, using an IBM quantum processor. The HSB-QSCI captures more than 99.18% of the correlation energies in the active space by considering about 1% of all the Slater determinants in 36 qubit systems, illustrating the ability of the proposed method to efficiently consider important electronic configurations.

We present the results obtained from spectroscopic investigations and quantum chemical calculations of the interaction of anisole (methoxybenzene) with small water clusters. The experiments have been carried out using resonant two-photon ionization (R2PI) and IR-UV double-resonance vibrational spectroscopy (IR/R2PI) in the region of the OH stretches. Apart from the vibrational spectra of the water moiety in the clusters, their intermolecular vibrations in the electronically excited S1 state are identified by IR/R2PI hole burning spectroscopy and assigned according to the vibrations calculated for the S1 state and compared with the vibrations calculated for the S0 state. The calculations for the S0 state were carried out at the second order Møller-Plesset level of theory using both the 6-31+G* and aug-cc-pVDZ basis sets and for the S1 state at the configuration interaction singles (CIS) level with the 6-31+G* basis set. In the electronic ground state (S0), the interaction of a water monomer to anisole is mediated through its oxygen atom, and that of a water dimer both through the oxygen atom (σ type of interaction) and the arene ring (π type of interaction). Thus in contrast to the interaction of fluorinated benzenes with water clusters, wherein a conformational transition from an in-plane σ to a on-top π bonding emerges starting with a water trimer, this conformational transition appears in case of anisole already with a water dimer. In the excited state (S1) of the investigated systems, there is a pronounced weakening of the interaction of the water cluster with the aromatic chromophore, which is also responsible for the blue shift of the electronic transitions. Consequently, the structures of the complexes of anisole with a water monomer or dimer are very different in both states. The weakening results from a diminished electron density of the oxygen atom and of the π system of anisole in the excited state. The calculated binding energies of the ground-state conformers indicate that these small water clusters are bound more strongly to anisole than to other π systems like benzene, toluene, fluorobenzene, and p-difluorobenzene. The many-body decomposition of the binding energy (S0) reveals that a progressive increase in the size of the water clusters results in a weakening of the σ O⋯H interaction and a concurrent strengthening of the π-H interaction. The complex containing a cyclic water trimer exhibits in the excited state also a π-type H-bonding interaction, but its stability emerges from a delocalization of the electron density from the water trimer to the anisole oxygen. Excepting the water dimer complexes, there is a good agreement between the calculated OH vibrational frequencies and the experimental IR spectra recorded for the ground state. The surprisingly good agreement of the calculated and the experimentally observed intermolecular modes in the excited state of these complexes provides convincing evidence that the experimental spectra emerge from structures similar to those predicted at the CIS/6-31+G* level.

A class of multilocal functions called n‐local energies, which can serve as accuracy criteria on approximate wave functions, are defined. Their interpretation is discussed and their multilocal form is shown to be a result of the holistic character of the quantum description. The analogy with the reduced local energy of Thomas, Javor, and Rothstein is demonstrated. The explicit formula for the simplest n‐local energy capable of measuring the deviation from the exact solutions of the Schrödinger equation is obtained in the one‐determinantal case.

The time-dependent variational principle in the stationary action principle form is formulated with constraint conditions for parametrized wave functions. The constraint conditions are classified into the first and the second classes as in Dirac's constrained classical mechanics based on the commutability of operators. If the local bases in the parametrized wave functions construct a complete basis set, the classification of constraints with the commutator becomes equivalent to that with the complex generalized Poisson brackets (CGPB). However, in approximate wave functions with limited variational parameters, the equivalence between the CGPB and the commutator of operators is lost. The first-class constraints and the constants of motion classified with the commutator should be reconsidered as the pseudo-second-class constraints in the approximate wave functions.

Perturbation theory is used to obtain approximate energies and wavefunctions for the hydrogen molecule ground state. A function of simple form is chosen as zero-order function, the zero-order Hamiltonian is constructed, and the variational equation for the first-order wavefunction is solved approximately by expansion in James—Coolidge basis functions, yielding the second- and third-order energies. All integrals are calculated numerically. The binding energy and equilibrium internuclear distance are fair, but not as good as what one obtains from a comparable variational calculation.

In nature every undisturbed quantum system tends to its lowest allowed energy. This state is known as the ground state. An excited state is any state with energy greater than the ground state. It is not possible to make a direct measurement of the quantum wave function. A quality measure of an approximate wave function is how well it reproduces the ground state energy. In this chapter I plan to focus on the computation of the ground state energies of multi-electron atoms/ions with special reference to the application of the variational methods developed in the previous chapter.

The electronic structure of GaAs and InAs quantum dots (QDs) containing a single substitutional Mn impurity is investigated in the envelope-function formalism. The Mn impurity in these compounds is known to be a shallow acceptor in the configuration ${d}^{5}+h$ and characterized by a strong antiferromagnetic $sp\text{\ensuremath{-}}d$ exchange interaction between the hole $(h)$ and the Mn ion. Our model for the hole states is based on the Luttinger Hamiltonian and the Coulomb potential with a central-cell correction that accounts for the observed binding energy and the effective $g$ factor in the bulk. The binding energy as well as the exchange contribution is found to increase with decreasing QD size. However, in contrast with the case of spherical nanocrystals (NCs), the binding energy in lens-shaped self-assembled QDs in the low-confinement limit is lower than that in the bulk because of their highly anisotropic shape. With an on-center impurity, NCs retain the bulk ${T}_{d}$ symmetry and the ground state is a $j=3/2$-like ${\ensuremath{\Gamma}}_{8}({T}_{d})$ level. In self-assembled QDs it splits into two doublets: ${\ensuremath{\Gamma}}_{6}$ $(|{j}_{z}|=1/2)$ and ${\ensuremath{\Gamma}}_{7}$ $(|{j}_{z}|=3/2)$ of ${D}_{2d}$, which mix in the presence of in-plane asymmetry, both belonging to ${\ensuremath{\Gamma}}_{5}$ of the reduced symmetry ${C}_{2v}$. The order and the splitting between the doublets depend on the degree of confinement and the strain-induced separation between the light- and heavy-hole valence bands. In lattice-matched GaAs/(Ga,Al)As QDs the ground-state doublet is $|{j}_{z}|=3/2$-like in the low-confinement limit. As the lateral size decreases there is a rapid crossover to a $|{j}_{z}|=1/2$-like ground state in QDs of typical sizes. On the other hand, in strained InAs/GaAs QDs the ground state is always $|{j}_{z}|=3/2$-like and the splitting relatively large. The $sp\text{\ensuremath{-}}d$ coupling with the Mn spin $S=5/2$ finally leads to a splitting of the ground-state doublet into six doubly-degenerate levels. The components are close to one another as the effective exchange parameters are an order of magnitude smaller than in the bulk. Our results thoroughly contradict the previously adopted picture based on treating the confinement potential as a small perturbation to the bulk impurity levels. We also consider the lowest two-hole states: the ground state in InAs/GaAs QDs is a singlet almost uncoupled to the Mn spin. We deduce the zero-field fine structure of the excitonic transitions and compare the results with the recently reported photoluminescence spectra.

The system of a quarkonium confined by an impenetrable spherical cavity filled with a hot quantum chromodynamics (QCD) medium is studied by solving the Schr\"{o}dinger equation. This is the first time this issue has been raised for discussion. The Schr\"{o}dinger equation with an appropriate boundary condition of a quarkonium in an impenetrable cavity filled with a hot medium is derived. The numerical results are obtained with the help of Gaussian Expansion Method. Binding energies and radii of the ground and low-excited states are obtained as a function of the medium temperature and the cavity radius. We find the behaviour of quarkonium in this cavity is different from that in infinite space. Our results show that the quarkonium dissociation temperature decreases as the cavity radius decreases and the finite volume effects on the ground state are more obvious than on the excited states. We also find that the less mass of the constituents and the bigger radius of the quarkonium lead the finite volume effects to become more obvious.

A formal proof is given that for $N$-boson systems in which two-body interaction potentials are described by a single Gaussian the ratio of higher-order to the lowest-order hyper-radial potentials decreases at $N\ensuremath{\rightarrow}\ensuremath{\infty}$ as ${N}^{\ensuremath{-}1/2}$ or faster. As a result, for such potentials, the convergence of binding energies for ground and several lowest excited states, obtained in expansion of the $N$-body wave function over the hyperspherical-harmonics basis, improves with increasing number of bosons. For a phenomenological three-body repulsive potential, introduced to account for the missing hard core, the ratio of higher-order to lowest-order hyper-radial potential, corresponding to this three-body potential, also decreases as ${N}^{\ensuremath{-}1/2}$ or faster when $N\ensuremath{\rightarrow}\ensuremath{\infty}$. Although adding the three-body contributions leads to increased influence from the total nondiagonal couplings around the node of the lowest hyper-radial potential, the arguments are given that this should not dramatically deteriorate the convergence if the range of repulsion is properly chosen. This means that the hyperspherical-harmonics expansion with soft two-body and repulsive three-body effective forces may become an attractive tool for studying the spectra of many-body systems. It is suggested that fine tuning of the three-body repulsion to reproduce the binding energies over a large region of $N$ is possible. In particular, it has been shown that an $N$-independent choice for the three-body repulsion exists for which the ground-state binding energies of $N\ensuremath{\le}112$ atoms of helium, obtained in the lowest-order approximation of the hyperspherical-harmonics expansion, are close to the prediction of the Green's function Monte Carlo method with a hard-core He-He potential.