Wave-function functionals

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.

We scrutinize the behavior of eigenvalues of an electron in a helium (He) atom as it interacts with electric field directed along the z-axis and is exposed to linearly polarized intense laser field radiation. To achieve this, we freeze one electron of the He atom at its ionic ground state and the motion of the second electron in the ion core is treated via a more general case of screened Coulomb potential model. Using the Kramers–Henneberger (KH) unitary transformation, which is the semiclassical counterpart of the Block–Nordsieck transformation in the quantized field formalism, the squared vector potential that appears in the equation of motion is eliminated and the resultant equation is expressed in the KH frame. Within this frame, the resulting potential and the corresponding wave function are expanded in Fourier series and using Ehlotzky’s approximation, we obtain a laser-dressed potential to simulate intense laser field. By fitting the more general case of screened Coulomb potential model into the laser-dressed potential, and then expanding it in Taylor series up to , we obtain the solution (eigenvalues and wave function) of an electron in a He atom under the influence of external electric field and high-intensity laser field, within the framework of perturbation theory formalism. We found that the variation in frequency of laser radiation has no effect on the eigenvalues of a He electron for a particular electric field intensity directed along z-axis. Also, for a very strong external electric field and an infinitesimal screening parameter, the system is strongly bound. This work has potential application in the areas of atomic and molecular processes in external fields including interactions with strong fields and short pulses.

The compressed helium atom variationally treated via a correlated Hylleraas wave function

A variational wave function is used to describe the binding energy of atomic negative ions using a two-electron system in the 2p2-state. Each electron is described by a modified screened hydrogenic wave function involving two free screening parameters denoted by c and a. The model is applied to hydrogen, helium, lithium, and boron anions, where the optimum values of the screening parameters are deduced through fitting the optimized energy to available experimental and theoretical values. The behavior of the optimum wave function for each anion is also investigated as a function of electronic radial distance and compared with its counterpart in the screened hydrogenic model and the Hartree–Fock method.

Involutory transformations and variational principles with multi-varoables in thin plate bending problems

In this paper we execute the solution procedure for second order linear fuzzy difference equation by Lagrange's multiplier method. In crisp sense the difference equation are easy to solve, but when we take in fuzzy sense it forms a system of difference equation which is not so easy to solve. By the help of Lagrange's multiplier we can solved it easily. The results are illustrated by two different numerical examples and followed by two applications.

A variational theory called free-complement (FC) ${s}_{ij}$ theory for solving the Schr\odinger equation of atoms and molecules using only one- and two-electron integrals over Slater or Gaussian functions is proposed. It is derived from the scaled Schr\odinger equation [Phys Rev. Lett. 93, 030403 (2004)] by replacing the two-electron part ${r}_{ij}$ of the scaling $g$ operator with ${s}_{ij}={r}_{ij}^{2}$, both of which solve the divergence difficulty inherent to the original Schr\odinger equation by avoiding the interelectron collisions. The ${s}_{ij}$ function can be rewritten with only one-electron functions so that when the initial wave function of the FC theory is composed of only one-electron functions, the FC wave function including ${s}_{ij}$ can be rewritten with only one-electron functions. Therefore, the variational calculations of the FC ${s}_{ij}$ theory can be performed with only one- and two-electron integrals. However, in comparison with the ${r}_{ij}$ function, the ${s}_{ij}$ function behaves less efficiently when two electrons come close to each other: the electron-electron cusp condition is not satisfied with the FC ${s}_{ij}$ theory, though the wave function has explicit ${r}_{ij}$ dependence. On the other hand, the electron-nuclear cusp condition is satisfied, even with the Gaussian functions, for the presence of the electron-nuclear function ${r}_{iA}$ in the $g$ operator. Test applications of the FC ${s}_{ij}$ theory were done to He, Li, and the $^{5}S^{\mathrm{o}} {\mathit{sp}}^{3}$ state of carbon atom and to hydrogen molecule using the local-molecular orbital (MO)- and valence bond (VB)-type initial functions. We examined both Slater and Gaussian functions. As the order of the FC theory increased, the wave functions were improved and the energies approached from a few kcal/mol to less than 1 kcal/mol accuracies relative to the known exact values. Thus, with the FC ${s}_{ij}$ theory, the Schr\odinger equation was solved to less than 1 kcal/mol accuracy for all the systems examined. The costs for the ${s}_{ij}$ calculations were much smaller than those of the ${r}_{ij}$ theory. However, for the $^{5}S^{\mathrm{o}} {\mathit{sp}}^{3}$ state of the carbon atom, we observed that the convergence rate became slow when the calculation came close to $\ensuremath{\sim}1$ kcal/mol accuracy. This suggests that for the FC ${s}_{ij}$ theory, the calculations would become easier if the required accuracy is within a few kcal/mol for the absolute energy. Actually, what we need in chemical studies are mostly related to the difference energy, whose accuracy could be better than that of the absolute energy with the robust variational method. Two lines of future studies were suggested.

The single-reference coupled-cluster method has proven very effective in the ab initio description of atomic and molecular systems, but its successful application is limited to states dominated by a single Slater determinant, which is used as the reference. In cases where several determinants are important in the wave function expansion, i.e., we have to deal with nondynamic correlation effects, a multi-reference version of the coupled-cluster method is required. The multi-reference coupled-cluster approaches are based on the effective Hamiltonian formulation providing a two-step procedure, in which dynamic correlation effects can be efficiently evaluated by the wave operator, while nondynamic correlation contributions are given by diagonalization of the effective Hamiltonian in the final step. There are two classical multi-reference coupled-cluster formulations. In this paper, the focus is on the so-called Fock-space coupled-cluster method in its basic version with one- and two-particle operators in the exponent. Computational schemes using this truncation of the cluster operator have been successfully applied in calculations in one- and two-valence sectors of the Fock space. In this paper, we show that the approach can be easily extended and effectively employed in the three-valence sector calculations.

Some comments on the coupled cluster with generalized singles and doubles (CCGSD) ansatz

Generally "exact" quantum Monte Carlo computations for the ground state of many bosons make use of importance sampling. The importance sampling is based either on a guiding function or on an initial variational wave function. Here we investigate the need of importance sampling in the case of path integral ground state (PIGS) Monte Carlo. PIGS is based on a discrete imaginary time evolution of an initial wave function with a nonzero overlap with the ground state, which gives rise to a discrete path which is sampled via a Metropolis-like algorithm. In principle the exact ground state is reached in the limit of an infinite imaginary time evolution, but actual computations are based on finite time evolutions and the question is whether such computations give unbiased exact results. We have studied bulk liquid and solid (4)He with PIGS by considering as initial wave function a constant, i.e., the ground state of an ideal Bose gas. This implies that the evolution toward the ground state is driven only by the imaginary time propagator, i.e., there is no importance sampling. For both phases we obtain results converging to those obtained by considering the best available variational wave function (the shadow wave function) as initial wave function. Moreover we obtain the same results even by considering wave functions with the wrong correlations, for instance, a wave function of a strongly localized Einstein crystal for the liquid phase. This convergence is true not only for diagonal properties such as the energy, the radial distribution function, and the static structure factor, but also for off-diagonal ones, such as the one-body density matrix. This robustness of PIGS can be traced back to the fact that the chosen initial wave function acts only at the beginning of the path without affecting the imaginary time propagator. From this analysis we conclude that zero temperature PIGS calculations can be as unbiased as those of finite temperature path integral Monte Carlo. On the other hand, a judicious choice of the initial wave function greatly improves the rate of convergence to the exact results.

A computational procedure for calculating correlated wave functions is proposed for three-particle systems interacting through Coulomb forces. Calculations are carried out for the muonic helium atom. Variational wave functions which explicitly contain interparticle coordinates are presented for the ground and excited states. General Hylleraas-type trial functions are used as the basis for the correlated wave functions. Excited-state energies of the muonic helium atom computed from 1- and 35-term wave functions are listed for four states.

This paper is about the problem of optimization of bracing columns--height of the parallel chords, cross-sectional areas of the chord and bracing and area moments of inertia of the chord and bracing. Optimization parameters have been determined by Lagrange's multipliers method--inequality constraint functions. Total mass of the construction has been selected as the objective function. Criterions of buckling (general buckling of the bracing column, local buckling of the chord and local buckling of the bracing diagonal) and criterion of deformation are used as the constraint functions. Numerical examples are performed by usage of the obtained theoretical solutions.

We address the issue of the relation between the canonical functional Schr\"odinger representation in quantum field theory and the approach of precanonical field quantization proposed by the author, which requires neither a distinguished time variable nor infinite-dimensional spaces of field configurations. We argue that the standard functional derivative Schr\"odinger equation can be derived from the precanonical Dirac-like covariant generalization of the Schr\"odinger equation under the formal limiting transition $\gamma^0\varkappa\rightarrow\delta(\mathbf{0})$, where the constant $\varkappa$ naturally appears within the precanonical quantization as the inverse of a small "elementary volume" of space. We obtain a formal explicit expression of the Schr\"odinger wave functional as a continuous product of the Dirac algebra valued precanonical wave functions, which are defined on the finite-dimensional covariant configuration space of the field variables and space-time variables.

SchrÖdinger Wave Functional in Quantum Yang–Mills Theory from Precanonical Quantization

The conventional tensor-network states employ real-space product states as reference wave functions. Here, we propose a many-variable variational Monte Carlo (mVMC) method combined with tensor networks by taking advantages of both to study fermionic models. The variational wave function is composed of a pair product wave function operated by real space correlation factors and tensor networks. Moreover, we can apply quantum number projections, such as spin, momentum and lattice symmetry projections, to recover the symmetry of the wave function to further improve the accuracy. We benchmark our method for one- and two-dimensional Hubbard models, which show significant improvement over the results obtained individually either by mVMC or by tensor network. We have applied the present method to hole doped Hubbard model on the square lattice, which indicates the stripe charge/spin order coexisting with a weak $d$-wave superconducting order in the ground state for the doping concentration less than 0.3, where the stripe oscillation period gets longer with increasing hole concentration. The charge homogeneous and highly superconducting state also exists as a metastable excited state for the doping concentration less than 0.25.