Hypervirial theorems and expectation values for helium

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 form of hypervirial theorem which is appropriate in scattering theory is discussed in general terms. It is shown that variational wave functions which are optimized in accordance with Kohn's variational principle do satisfy hypervirial theorems. Thus such theorems may be useful in selecting approximate wave functions to give accurate phase shifts or scattering amplitudes. The situation is analogous to that of energy-optimized wave functions for bound-state systems.

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.

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.

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

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

The direct inversion in the iterative subspace (DIIS) method is applied to several simple SCF wave functions in an effective Fock matrix formulation. The following cases are treated: high-spin-restricted open shell, open-shell singlet, and two-configuration wave functions. Open-shell singlet states are described by a three-determinant 2×2 CAS expansion which is equivalent to Davidson’s nonorthogonal SCF method in the case of the first open-shell singlet. Very sharp convergence is usually obtained in less than 20 cycles. The method is applicable to slowly convergent or even inherently divergent cases, and able to enforce convergence to excited states not the lowest of their symmetry. For these simple wave functions, the present first order method is asymptotically more efficient than second-order methods. Examples are presented for H2O, H2O2, C2H4, F2, several states of NO2, C2H5, formaldehyde, and ketene.

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.

Certain aspects of Rayleigh-Schr\"odinger (RS) perturbation theory and of the variational principle are combined in a new perturbationally oriented variational approach for obtaining approximate solutions to Schr\"odinger's time-independent equation. This perturbational-variational (PV) procedure is applicable for all perturbed Hamiltonian operators which can be expanded as $H(\ensuremath{\lambda})=\ensuremath{\Sigma}{H}_{t}{\ensuremath{\lambda}}^{t}$, where $\ensuremath{\lambda}$ is an external parameter governing the strength of the perturbing terms; in particular, the theory is developed here for the important special form, $H={H}_{0}+{H}_{1}\ensuremath{\lambda}$. It is shown how the application of the PV procedure to an arbitrary analytic approximate wave function yields the variational parameters embedded in the wave function, the wave function itself, the corresponding approximate energy, and other expectation values, all in the form of energetically optimized PV expansions in powers of $\ensuremath{\lambda}$ to any desired order. A variational wave-operator formalism is introduced to deal with the multivariant expansions required to form the PV expansions. The remainder theorem and the variational Hellmann-Feynman theorem are derived and their role in the PV methodology is discussed. The principal advantages of the new method are: (1) Unlike RS perturbation theory, there is no requirement that the exact solution of a simpler unperturbed system be known; (2) the explicit $\ensuremath{\lambda}$ dependence of the optimum quantities is analytically derived; (3) the numerical aspects of a given variational problem are simplified because the variational equations generated by the PV procedure are almost completely linearized in the variational parameters; and (4) the PV expansions form a natural framework for the systematic classification and evaluation of all perturbed variational wave functions in respect to their quality. The PV classification scheme, previously presented in bare outline, is rigorously derived. The insight obtained from PV theory is further illustrated by the derivation and application of the variational integral Hellmann-Feynman theorem, the virial theorem, and a theorem relating to the expectation value of the unperturbed potential. The a priori PV classification of perturbed variational wave functions is extended to the perturbed variational parameters embedded in the wave functions; it is shown that the anomalous PV expansions of some open-shell parameters and orbitals can be simply explained in terms of PV theory.

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}

Compact Jastrow-Slater-type correlated wave functions for three- and four-electron atoms and ions (Li, ${\mathrm{Be}}^{+}$, ${\mathrm{B}}^{2+}$, ${\mathrm{C}}^{3+}$, Be, ${\mathrm{B}}^{+}$, and ${\mathrm{C}}^{2+}$) are proposed. The Jastrow factor we employed consists of one-body and two-body functions including only two variational parameters in total. We found that a one-body Jastrow function with one variational parameter, which is responsible for the screening effect around the nucleus, is effective for reproducing good total energies of the three- and four-electron atomic systems if hydrogenoid orbitals are adopted in the Slater determinant. On the other hand, a determinant composed of Hartree-Fock orbitals multiplied by the same Jastrow factor was found to give rather worse results than that composed of hydrogenoid orbitals. This result clearly indicates that analytic hydrogenoid orbitals coupled with the one-body Jastrow function are useful for describing simple wave functions and understanding the physical properties of these systems.

LCAO SCF calculations are presented for the ground states of helium, beryllium and neon using universal even-tempered, even-tempered and Clementi–Roetti basis sets. Gross uniform scaling of all exponents for the basis sets considered here (with the exception of the Clementi–Roetti single and double-zeta basis sets) leads to the appearance of multiple minima along selected cuts through the energy hypersurface in exponent space. The calculated energies and orbital expectation values 〈rn〉(n=–1, 1, 2, 3), at the respective minimum energy values of the scaling factor, are in very close agreement with the Hartree–Fock values for nearly all choices of basis set. However, the orbital cusp values, and expectation values involving δ(ri) and p4i, show a greater degree of sensitivity to the choice of scale factor. Furthermore, a similar pattern of results for helium and beryllium is obtained when grossly scaled Clementi–Roetti or Raffenetti basis sets for neon are used. In the case of neon, when basis sets with small numbers of primitive functions are used, significant improvements in the calculated expectation values may be obtained by differentially scaling the exponents of the s- and p-type basis atomic orbitals.