Ground-state energy for periodic Anderson model

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 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.

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.

Multi-electron systems in strong magnetic fields II: A fixed-phase diffusion quantum Monte Carlo application based on trial functions from a Hartree–Fock–Roothaan method

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.

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.

Approximate wave functions for the hydrogen molecule ground state previously available in the literature are analyzed quantitatively into approximate natural spin orbital functions with particular attention to the corresponding occupation numbers. The analysis demonstrates the very great similarity of all such trial wave functions, and especially the largely molecular orbital nature of the wave function. In addition it shows a close relationship between the molecular orbital and valence bond functions, and the importance for allowing for angular correlation of the electrons by including terms dependent upon the azimuthal coordinate. The analysis particularly demonstrates that approximate natural spin orbital occupation numbers are nearly invariant under a wide variety of choices of basis functions, and therefore are particularly suitable for comparison of different approximate functions and for discussion of their respective properties.

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.

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

Models of carrier dynamics in quantum dots rely strongly on adequate descriptions of the carrier wave functions. In this work we numerically solve the one-band effective mass Schr\odinger equation to calculate the capture times of phonon-mediated carrier capture into self-assembled quantum dots. Comparing with results obtained using approximate carrier wave functions, we demonstrate that the capture times are strongly influenced by properties of the wetting layer wave functions not accounted for by earlier theoretical analyses.

We propose a new variational wave function for the periodic Anderson model in the case of antiferromagnetically ordered f-electrons. Expectation values in terms of this wave function can be calculated analytically in arbitrary dimensions d. The wave function automatically produces the exact limiting behavior of the ground state energy for weak hybridization and strong f-electron interaction in the symmetric case, which the Gutzwiller wave function fails to do. Several additional improvements are discussed. Thereby one obtains a wave function that leads to extremely good ground state energies for the periodic Anderson model for arbitrary U in all dimensions d.

Although widespread use of computer codes for the solution of Schrodinger equations makes available numerical Hartree-Fock model radial wave functions, there remains persistant interest in simple analytic expressions for atomic wave functions. One such frequency favored approach employs hydrogenic functions, suitably scaled, as approximate wave functions. The following note displays typical inaccuracies to be expected from such approximations. 13 references.