Scaled hydrogenic approximation wavefunctions. [Hartree-Fock approximation

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.

Consequences of the discontinuity at $Z=30$ in the Coster---Kronig transition probability ${f}_{23}$ have been investigated by Auger and photoelectron spectroscopy. The $\frac{{L}_{3}\ensuremath{-}{M}_{45}{M}_{45}}{{L}_{2}\ensuremath{-}{M}_{45}{M}_{45}}$ Auger intensity ratio is found to undergo a sudden decrease at $Z=30$. Relative ${L}_{2}$ and ${L}_{3}$ level widths of Cu and Zn are derived from photoelectron spectra; while the ${L}_{3}$ width increases from Cu to Zn, the ${L}_{2}$ width of Cu is greater than that of Zn, contrary to the general trend. The data agree qualitatively with earlier calculations of radiationless-transition probabilities based on hydrogenic wave functions, but substantial quantitative discrepancies are found. New calculations have been performed with numerical single-particle wave functions based on Green's atomic independent-particle model. With suitable assumptions regarding Cu and Zn ${M}_{4}$ and ${M}_{5}$ electron binding energies in the presence of an ${L}_{2}$ vacancy, reasonable agreement between calculated and observed Auger-electron intensity ratios and $L$-subshell level widths is attained.

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.

Coupled-cluster calculations starting from exchange-only local-density approximation (XLDA), Krieger-Li-Iafrate (KLI), and Kohn-Sham (KS) wave functions are compared with those using the Hartree-Fock (HF) determinant as a reference. The total energies are found to be close, the difference being maximally 2 mhartree in the systems studied (the first terms in the He, Be, Ne, Mg, Ar isoelectronic series). The convergence is, however, sensitive to the choice of the reference: KLI and KS converge, in general, faster than HF in spite of being a worse approximation in the first two orders of perturbation theory. The improvement of convergence due to the use of the KLI or KS references is more pronounced in the systems showing near degeneracy, such as in the Be series. For XLDA, the convergence properties are either comparable to those of KLI or oscillatory, depending on the system. In a second part, the numerical results are analyzed (in the HF and KLI cases) by using first-order developments with respect to nuclear charge Z at large $Z.$

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.

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

With the development of computing technology, numerical exact diagonalization method plays a vital role in modern computational condensed matter physics, especially in the research area of strongly correlated electron systems: it becomes a benchmark for other numerical computational techniques, such as quantum Monte Carlo, numerical renormalization group, density matrix renormalization group, and dynamic mean field theory. In this paper, we first numerically exactly diagonalize the three-dimensional hydrogen atom with the combination of finite-difference method, and find that the numerical wave function of ground state is in good agreement with the analytical calculations. We then turn to discuss the space dimension confinement hydrogen system, two-dimensional hydrogen atom, and notice that the numerical wave function is no longer in agreement with the analytical calculation, where the ground state wave function has a numerical singularity as radius approaches to zero. Compared with the case of the three-dimensional hydrogen atom, this issue mainly comes from the nature of space dimension confinement. To resolve such an issue of numerical singularity in two-dimensional hydrogen atom, we need to construct a new discrete and normalized Bessel function as a basis to study the ground state behavior of dimension confinement system based on the framework of Lanczos-type numerical exact diagonalization. The constructed normalized Bessel basis is orthogonal and discrete, and thus becomes suitable for practical calculation. Besides, these prominent properties of such a Bessel basis greatly reduce the complexity and difficulty in practical calculation, and thus makes computing work efficient. In addition, Lanczos-type numerical exact diagonalization method can extremely speed up the process of solving the eigenvalue equation. As a result, such a high efficient calculation of our method demonstrates the consistence between numerical and analytical ground state energy value, and the corresponding wave function with enough truncated basis number. Since this kind of numerical singularity occurs in many space dimension confinement systems, our finding for constructing a new discrete Bessel basis function may be helpful in studying the quantum systems with numerical singularity behaviors in wavefunctions in future. On the other hand, it should be pointed out that the Bessel basis is incorporated into the linear augment plane wave method in the density functional theory to study the electronic band structure of the condensed material and obtain high accurate results, especially in the theoretical prediction of topological insulators and in experimental realization as well.

Investigations are continuing of the interaction between an F-center electron and the core electrons of the neighboring ions in the crystal. A pseudopotential is obtained for each ion from free ion wave functions, and the superposition of potentials is expanded around the vacancy in Kubic harmonics. Numerical wave functions are obtained for the spherical part of the expansion, and the crystal field effects of the first nonspherical terms are obtained by perturbation theory to first order in wave functions and third order in energy.

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

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

A method for the Gaussian basis set generation for molecular relativistic Dirac-Fock calculations is proposed. The basis set exponents are obtained in the process of stochastic optimization (a hybrid of simplex and simulated annealing optimization techniques has been employed) of a functional defined as the sum of squares of differences between the numerical relativistic atomic wave functions and the wave functions obtained using the Gaussian function expansion. After this pre-optimization step the exponents are refined by ordinary gradient energy-functional based procedure. The present method seems to be very effective and robust. As an example the optimized basis sets of atoms from H (Z=1) to Ar (Z=18) are presented. Results of the Dirac-Fock calculations for all atoms under study are presented and compared with the numerical Dirac-Fock results and results obtained using the Gaussian basis sets according to Okada et al.: J. Chem. Phys.93 (1990) 5013.

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.

This communication deals with the general theory of obtaining numerical electronic wave functions for the stationary states of atoms and molecules. It is shown that by taking Gaussian functions, and functions derived from these by differentiation with respect to the parameters, complete systems of functions can be constructed appropriate to any molecular problem, and that all the necessary integrals can be explicitly evaluated. These can be used in connexion with the molecular orbital method, or localized bond method, or the general method of treating linear combinations of many Slater determinants by the variational procedure. This general method of obtaining a sequence of solutions converging to the accurate solution is examined. It is shown that the only obstacle to the evaluation of wave functions of any required degree of accuracy is the labour of computation. A modification of the general method applicable to atoms is discussed and considered to be extremely practicable.