Uniform semiclassical off-shell wave functions andT-matrix elements

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.

We show that in relativistic heavy-ion collisions the independent-electron model can be used to predict cross sections for multiple inner-shell ionization in a single collision. Charge distributions of 430- and 955-MeV/amu ${\mathrm{U}}^{90+}$, ${\mathrm{U}}^{89+}$, ${\mathrm{U}}^{83+}$, and ${\mathrm{U}}^{68+}$ beams emerging from thin solid targets were used to obtain single- and multiple-electron stripping cross sections. The probabilities of stripping electrons from the K, L, or M shells were calculated using the semiclassical approximation and Dirac hydrogenic wave functions. The data generally agree with theory. An influence of the Auger effect is seen in ${\mathrm{U}}^{68+}$ collisions.

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.

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.

Semiuniform off-shell wave functions and {ital T}-matrix elements are presented that remain valid even in the limit of equal incoming and outgoing momenta. Numerical comparisons are made with both primitive semiclassical and fully quantal predictions for scattering described by a Hulthen potential.

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.

The accuracy of the semiclassical theory of forward glory scattering for a state-to-state chemical reaction is investigated using a J-shifted Eckart parameterization for the scattering matrix element. The parameters are chosen initially to model the angular scattering of the H+D2→HD(vf=3)+D reaction, following D. Sokolovski (Chem. Phys. Lett., 370, 805 (2003)). Then the parameters are systematically varied to generate different scattering patterns. The theory assumes that the scattering amplitude can be expanded in a Legendre partial wave series (PWS). The theoretical techniques applied to the PWS include: a nearside–farside decomposition, and a local angular momentum (LAM) analysis; both techniques include resummations of the PWS. The semiclassical techniques used include: a uniform semiclassical approximation (USA), a primitive semiclassical approximation, a classical semiclassical approximation, an integral transitional approximation, a semiclassical transitional approximation and a semiclassical LAM. The LAM for the classical collision of two hard spheres is also employed. It is demonstrated that the USA can accurately describe glory oscillations for scattering angles on and near the forward direction (as defined by the axial caustic associated with the glory); in favourable cases the USA is accurate for sideward scattering angles and even for angles close to the backward direction.

A discussion is given of the relations which exist among vibrational matrix elements for diatomic molecules represented as Morse oscillators. This is done according to the wave function approximation of Trischka and Salwen. It is shown how the matrix element of the dipole moment Mvi′ is related to the matrix elements M0i where v′—v≤i≤v′+v with v′>v. This relation is examined for the case of the 1–2 band of HCl. It is also shown how the matrix element M01 for DCl can be related to the matrix elements of HCl. Finally, the influence of the vibration-rotation interaction is incorporated in the wave function approximation. In an example, it is shown that the matrix element {M0,Ji,J+1 can be expressed in terms of the set of matrix elements{M0,0i,1},i=0,1,··· .

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.

Uniform WKB wave functions are developed for prolate spheroidal wave functions which arise when the Schrödinger equation is given in prolate spheroidal coordinates. A detailed numerical test is presented for angular spheroidal wave functions. It shows that the approximate uniform WKB wave functions are excellent in accuracy.

Ground-state energy for periodic Anderson model

The energy-variance-based optimization procedures have proven to be useful tools to describe N-electron spectra. However, the resulting wave functions usually present spin-contaminant contributions. The goal of this work is to reduce the spin contamination of the results arising from the unrestricted doubly occupied configuration interaction method in its energy variance minimization version [Alcoba et al., J. Chem. Phys. 160, 164107 (2024)]. We propose to incorporate the half-projection technique, which allows removing the spin components with even or odd spin quantum number of an approximate N-electron wave function, into the framework of the unrestricted doubly occupied configuration interaction treatment. This implementation can be carried out following several possible ways, whose results are analyzed in detail, in order to show the behavior of each procedure. Numerical determinations performed on selected strongly correlated N-electron systems, in ground and excited states, allow us to assess the most suitable procedure.

The time-independent energy sudden (ES) representation is defined through application of the energy shift operator S=exp[−(h−ωn)∂/∂ε], where h is the internal (molecular) Hamiltonian. Our introduction of S follows from an earlier study by Chang, Eno, and Rabitz where exp[−iht], which ‘‘factors out’’ internal motion, was used to define the time-dependent ES representation. Exact integral equations for the scattering wave function within the ES representation are derived, the leading terms being the approximate ES wave function. Corrections to the ES wave function are nonsingular and involve the generalized potential increment V=S−1VS−V, where V is the interaction potential. Boundary conditions and transition amplitudes are discussed, as is the connection between wave functions in the ES and the original representations.

Comparison of several equivalent local potentials for microscopic nonlocal Nα potentials