Impact-parameter method for proton-hydrogen-atom scattering variational bounds on the1scharge-exchange amplitudes

The boundary condition (at $t\ensuremath{\rightarrow}\ensuremath{\infty}$) previously shown to be associated with the customary time-dependent Schr\odinger equation of the impact-parameter method places a restriction upon the approximate or trial wave functions which are to be used in the usual coupled-state calculations. For the trial wave function to be physically acceptable, it must belong to the class of functions which have the asymptotic form of the exact wave function. This simply means that one must obtain from the coupled-state calculation, approximate transition amplitudes for which the absolute squares are time independent as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. It is noted that the trial wave function in the six-, eight-, and nine-state Sturmian calculations are not physically acceptable. The trial wave function in the pseudostate calculation of Cheshire, Gallaher, and Taylor is acceptable; however, if one attempted to include an additional pseudostate with the same symmetry as one already in the set, then this trial wave function would not be physically acceptable.

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 estimate of error in an approximate wave function for a stationary state is put forward that does not require any information about the exact state. The measure is sensitive and direct. Parameters embedded in a trial wave function can be varied to minimize this error as well, leading to a variational principle. Such a scheme works nicely for bound states and the more so for Siegert states, for which the standard energy minimization principle does not apply. Pilot calculations on the anharmonic oscillator system and the radial Stark effect in the hydrogen atom reveal the worth of the endeavor. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

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.

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.

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.

We propose error backpropagation using the absolute error function as an objective function. The error backpropagation is the most popular learning algorithm for multi-layered neural networks. In the error backpropagation, the square error function is usually used as the objective function. But a square function has a drawback in which it is enormously large if the data set includes a few anomalous data, which may be observational errors. On the other hand, the absolute error function is less affected by such data. But since the absolute error function is not differentiable, the standard backpropagation's way cannot be applied directly by using an absolute error function as an objective function. Therefore, we first introduce differentiable approximate functions for the absolute value function. The purpose of introducing an approximate function is to construct a differentiable error function which is close to the absolute error function. Then we propose an error backpropagation algorithm minimizing a differentiable approximate error function. Some computational experiments indicate that the proposed method is practically efficient. In particular, it is observed that the method is more robust and learns faster than the backpropagation with the square error function when teacher signals include some incorrect data.

Confined beryllium atom ground and first excited states electronic structures are calculated by the direct variational method, taking into account the system asymmetric nature of the trial wave function, adding a cutoff function to ensure confinement boundary conditions. The trial wave function is built up from hydrogenic functions, which constitute an adequate basis for energies calculation. Physicochemical properties such as kinetic energy, pressure, and polarizability are also calculated from energy results previously obtained to different confined radii. Using different variational parameters in each hydrogenic function, the energy approximation obtained is improved. Electronic configuration changes as we move toward the strong confinement region (small cavity radii) in function of its atomic number using impenetrable walls, this region was obtained for Z = 4. This is a conclusion of this work. Another important result is that this method is computationally simpler and gives values inside the experimental precision. Aforementioned results are compared with other theoretical publications.

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.

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.

We present approximate off-shell wave functions and $T$-matrix elements obtained using a uniform semiclassical propagator. The approximate off-shell wave function is valid near turning points and stationary-phase points for the incoming or outgoing momenta. Our results for the off-shell $T$-matrix elements reduce to the primitive semiclassical results in appropriate limits but extend the range of validity of the available semiclassical approximations.

Approximate wave functions for the 1sσ and 2pπ states of the HeH2+ molecular ion are obtained by the variational method. The values of the electron total energies, potential energies, dipole moments and quadrupole moments computed using the approximate wave functions are compared with the exact values of these quantities.

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