N‐Local energies as accuracy measures on approximate wave functions

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.

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.

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.

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 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.

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.

Some comments on the calculations of oscillator strengths from approximate wave functions

An integral representation of wave functions for the Temkin-Poet model of electron impact on atomic hydrogen is given. Approximate wave functions are evaluated analytically for large hyperradius to extract the ionization S-matrix element. An ionization cross section of the form exp[{minus}aE{sup {minus}1/6}+bE{sup 1/6}], where a and b are positive constants, is derived. The exponential suppression of ionization for small E appears to be the quantum counterpart of the delayed onset of ionization in the classical theory for this model. {copyright} {ital 1997} {ital The American Physical Society}

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.

Four different expressions for the electric dipole transition moment are currently available. In this paper, the Hamiltonians appropiate for these expressions are derived by classical, infinitesimal transformations. The derivations are restricted to the electric dipole approximation and all second and higher order terms in the field strengths are neglected. The equivalence of these Hamiltonians, within the stated approximations, shows that none of the formalisms can be given any formal priority. The differences which arise when approximate wave functions are used are briefly discussed in the light of the present work.

An original method is proposed for searching for the formal solution of the Hylleraas equation. It allows one not only to confirm a majority of results obtained earlier, and scattered in a special literature, but also to develop simple subroutines for further numerical calculations of energy levels and approximate wave functions.

We discuss a generalized dynamic mean-field method combining the advantages of explicit pair correlations and of configuration interaction. The approximate dynamical method, which we call time-dependent self-consistent-field configuration interaction (TDSCF2-CI), is constructed by including N(N−1)/2 TDSCF2 configurations. In each configuration a given pair of N coupled modes is directly (not in the mean-field approach) correlated; the N(N−1)/2 configurations include all such choices of pairs. As such, it has both the usual advantages of TDSCF and improvements due to some inclusion of correlations (exact results for any two-mode problem, improved descriptions of dynamical corrections, and greater accuracy). A three-mode model Hamiltonian is analyzed using five approximate treatments, i.e., the usual TDSCF, the three TDSCF2 forms, and the TDSCF2-CI one. The quantities for comparison with the exact results include the decay P(t) of the initial state, the time dependencies of the energies e(i) of individual modes, and the overlap S(t) of the corresponding approximate wave function with the exact one. We find, indeed, that explicit inclusion of pair correlations improves the description of the quantum dynamics of the system.

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.