Natural Spin Orbital Analysis of Hydrogen Molecule Wave Functions

Natural orbital functional (NOF) theory in its PNOF3 and PNOF4 implementations is used to investigate the electronic structure and energetics of selected diradicals and diradicaloids: the trimethylenmethane (TMM) diradical and the imino-allyl (IMA) and oxyallyl (OXA) diradicaloids. These compounds are taken as paradigmatic cases of molecules with full and partial diradical character, and are considered as a case study to illustrate the potentiality of Piris NOF (PNOF) to yield the correct treatment of near-degeneracy effects in diradicals. The degree of diradical character is determined through the inspection of the corresponding natural orbital occupation numbers. In addition, the energetics of these compounds with respect to their cyclic isomers (methylenecyclopropane, MCP, iminocyclopropane, ICP, and cyclopropanone, OCP) is investigated. We have found that PNOF4 shows a promising description of these species, and yield the correct trends in occupation numbers as compared with wave-function methods such as CASSCF. Therefore, PNOF4 can give a correct description of near-degeneracy effects, and hence, is a promising method to treat diradicals and diradicaloids in chemistry, a delicate type of molecular systems to simulate by theoretical methods.

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.

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

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.

The natural orbitals and their corresponding occupation numbers are constructed for several interesting problems to demonstrate that the existence of negative natural orbital occupation numbers for single reference correlation methods provides a simple diagnostic for the need for a multiconfigurational description of the wave function.

We propose a procedure that combines multiconfigurational (MC) wave functions with two-body density correlation functionals by transforming the latter into functionals of the MC natural orbitals and occupation numbers. The method is tested with the spectroscopic constants of a set of 11 diatomics, the diradical-involved automerization barrier of cyclobutadiene, the energy difference between triplet and open-shell singlet states in He and the methylene molecule, and the magnetic coupling constants of several systems, such as NiO, KNiF(3), K(2)NiF(4), La(2)CuO(4), alpha-4-dehydrotoluene, 1,1('),5,5(')-tetramethyl-6,6(')-dioxo-3,3(')-biverdazyl, [Cu(2)Cl(6)](-2), copper(II) acetate monohidrate and H-He-H. The procedure is applied to the Colle-Salvetti [Theor. Chim. Acta 37, 329 (1975); 53, 55 (1979)], functional and to a size-consistent functional depending on the on-top pair density (F1-5-N(eff)). On average, the best results are provided by the transformed F1-5-N(eff) [J. Chem. Phys. 114, 2022 (2001)] functional.

Über die kreiseleigenschaften des atomkerns

A trial wave function that is a linear combination of two traveling hydrogenlike basis functions which contain two variable-charge parameters and a polarization parameter has been used to obtain bounds on the second-order error term in the variational principles of Demkov and Storm that are comparable to the magnitudes of approximate $1s$ charge-exchange amplitudes. It is demonstrated that the error function ${\ensuremath{\Delta}}_{1}({X}_{i})$ can be employed to judge various calculations and as an aid in adjusting parameters in the trial wave function to obtain better bounds. Based upon the use of ${\ensuremath{\Delta}}_{1}$ as a measure of error in an approximate trial wave function, we conclude that the Euler-Lagrange variational method is not the optimal approach. Suggestions are made for future work which might improve the calculated bounds.

A procedure is developed to construct improved wave functions for the bound states in various four-electron atoms and ions. The bound state wave functions are improved in the sense that our trial wave functions are compact and accurate. We employ two independent four-electron spin functions ${\ensuremath{\chi}}_{1}=\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}+\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{-}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\beta}\ensuremath{\alpha}$ and ${\ensuremath{\chi}}_{2}=2\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\beta}+2\ensuremath{\beta}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{-}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}$ in each trial wave function. Optimization of the nonlinear parameters is performed for each radial function. Our procedure is applied to obtain the total energies of the ground $^{1}S^{e}$ states in the following four-electron berylliumlike systems: ${\mathrm{Li}}^{\ensuremath{-}}$, Be, ${\mathrm{B}}^{+}$, ${\mathrm{C}}^{2+}$, ${\mathrm{N}}^{3+}$, ${\mathrm{O}}^{4+}$, ${\mathrm{F}}^{5+}$, ${\mathrm{Ne}}^{6+}$, ${\mathrm{Na}}^{7+}$, and ${\mathrm{Mg}}^{8+}$. All our trial wave functions used in this study are very compact. In fact, each of these functions contains only 300 basis (radial) functions which are constructed from ten-dimensional gaussoids of the five-body relative coordinates ${r}_{12}$, ${r}_{13}$, ${r}_{14}$, ${r}_{15}$, ${r}_{23}$, ${r}_{24}$, ${r}_{25}$, ${r}_{34}$, ${r}_{35}$, and ${r}_{45}$.

It has been shown by Kapuy that any alternant molecular orbital wave function is an antisymmetrized product of strongly orthogonal geminals, in which the geminals are of mixed spin. Employing the localized alternant orbitals previously introduced by the author, this fact is used to generate a wave function which is capable of interpolating between the single-parameter AMO wave function and a wave function of the Hurley, Lennard-Jones, and Pople type. The validity of the unprojected form of the least flexible wave function of this type is tested in a model calculation on the hexagonal ring of hydrogen atoms. In the test case, this unprojected wave function is found to do significantly better than the unprojected AMO wave function and almost as well as the projected AMO wave function. It is shown that in the case of an infinite system, this wave function must do better than the projected AMO wave function. Features of the more general wave functions of this type are discussed, and it is mentioned that any of these wave functions may be written as an antisymmetrized geminal powers (AGP) wave function in which the geminal is of mixed spin.

By using the classically solvable minimal massless scalar minisuperspace model we study two examples of wave functions of the universe in the semiclassical limit. The first one consists of a Lorentzian component and a Euclidean component and admits a clear semiclassical interpretation as a superposition of universes. The second one consists of a Lorentzian (or Euclidean) component and another oscillatory component which corresponds to (neither Lorentzian nor Euclidean) complex classical solutions. This example has some resemblance to Hawking's minimal massive scalar minisuperspace model. We suggest a possible way of recovering the classical interpretation in such a case.

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 Pauli exclusion principle is fundamental to understanding electronic quantum systems. It namely constrains the expected occupancies n i of orbitals φ i according to 0 ≤ n i ≤ 2 . In this work, we first refine the underlying one-body N -representability problem by taking into account simultaneously spin symmetries and a potential degree of mixedness w of the N -electron quantum state. We then derive a comprehensive solution to this problem by using basic tools from representation theory, convex analysis and discrete geometry. Specifically, we show that the set of admissible orbital one-body reduced density matrices is fully characterized by linear spectral constraints on the natural orbital occupation numbers, defining a convex polytope Σ N , S ( w ) ⊂ [ 0 , 2 ] d . These constraints are independent of M and the number d of orbitals, while their dependence on N , S is linear, and we can thus calculate them for arbitrary system sizes and spin quantum numbers. Our results provide a crucial missing cornerstone for ensemble density (matrix) functional theory.

Whereas a density functional that incorporates dispersion interaction has remained elusive to date, we demonstrate that in principle the dispersion energy can be obtained from a density matrix functional. In density matrix functional theory one tries to find suitable approximations to the two-particle reduced density matrix (2RDM) in terms of natural orbitals (NOs) and natural orbital occupation numbers (ONs). The total energy is then given as a function(al) of the NOs and ONs, i.e., as an implicit functional of the 1RDM. The left-right correlation in a (dissociating) bond, as well as various types of dynamical correlation, can be described accurately with a NO functional employing only J and K integrals (JK-only functional). We give a detailed analysis of the full CI wavefunction of the He(2) dimer, from which the dispersion part of the two-particle density matrix is obtained. It emerges that the entirely different physics embodied in the dispersion interaction leads to an essentially different type of exchange-correlation orbital functional for the dispersion energy (non-JK). The distinct NO functionals for the different types of correlation imply that they can be used in conjunction without problems of double counting. Requirements on the (primitive) basis set for Van der Waals bonding appear to be more modest than for other types of correlation.

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.