Searching for distinct classes of atomic and molecular states using convergence and separability criteria

Abstract

We demonstrated that the solution ψ to the Schrodinger equation (SE) Ĥψ = εψ converges logarithmically in the classically forbidden region, i.e., as ε′ approaches to the correct eigenvalue ε, the approximate solution ψ′ logarithmically converges to ψ. Knowing that this approximate eigenvalue procedure generates a straightforward but inefficient method to solve a general problem of n-bodies, we thereby discuss the main characteristic of the usual methods to obtain a better convergence for atoms and molecules. Such usual methods consider that the solution of SE can be approached through a linear combination of solutions of systems with analytical solutions. Hydrogen-like atom solutions are used to describe atoms and molecules. This solution avoids the convergence problem of ψ′ even for approximate eigenvalues ε′ since all the terms of the linear combination decay asymptotically to zero. We argue that this type of solution works very well for a large class of almost separable atomic and molecular states in which the separation of electronic (and nucleus) movements occurs. We also establish a comparison of this separability and other systems, like gravitational, in which separability is only possible in particular classes of restricted systems. Finally, we consider the existence of distinct atomic and molecular states that may not be described using usual methods that apply this type of solution, which implies the separability of restricted problems. Therefore, usual methods to describe atoms and molecules may be insufficient to reach solutions with different or more general electronic correlations, as discussed in the text. Strategies to achieve general or distinctive solutions, although approximated, should be further studied and developed.

Appendix

1.1 1.1 Definitions

1.1.1 1.1.1 Usual method in quantum chemistry

In this work, we defined a usual method to solve the SE for a quantum many-electron system, based on the independent particle approximation (hydrogen-like model) followed by the incorporation of a mean-field correction or first-order perturbation. A usual method also includes the electronic correlation correction adding more terms in the expansion of functions or perturbation [11, 41, 71]. We include the DFT and semiempirical as a usual method for our purposes.

1.2 1.2 Separability

Strictly, the separability can be defined as the factorization of the SE solution. For example, the solution Ψ(x,y,z) of the harmonic oscillator in 3-D can be trivially separated in Cartesian coordinates as Ψ(x,y,z) = ψ(x)ψ(y)ψ(z) [14]. The hydrogen atom solution can be put as Ψ(x,y,z) = Ψ(r,θ,ϕ) = R(r)Θ(θ)Φ(ϕ) for spherical coordinates [14]. The H₂⁺ molecule ion can show similar separability using the confocal elliptic coordinate for fixed protons [72]. These systems are (integrable and) separable with exact analytical [14] or numerical solutions [29]. However, this strict separability does not represent the two or more electrons systems behavior. Alternatively, a separable problem could also be defined as the system, which obeys the independent particle approach. However, again this is an improbable manifestation for two or more electrons systems, i.e., this approach is only valid when the electrons are far from each other. The discussions concerning independent particles, integrable, and separable can be trivially extended to gravitational systems.

It is better to define for practical purposes the concept of near separability for two or more electrons systems. A system can show different quantum states, in which each state can have the most varied behaviors concerning the separability. We can define that the system or state is nearly separable or quasi-separable if a theory based on the independent particle approach with mean-field correction can describe very well the specific system or state when compared to experimental or theoretical (with correlation corrections) results. We discuss here that, in general, atoms (and molecules) are, in fact, effective hydrogen-like models, where the separability concept can be applied. We can affirm that the systems with two or more electrons can behave from quasi-separable (near separable) to non-separable systems.

The breakdown of separability can be inferred when a usual and accurate theory with correlation corrections cannot find or describe experimental results with chemical accuracy, alternatively when the eigenvalue of the state cannot be extracted from Hamiltonian diagonalization. In this case, the state can be considered as non-separable. The equivalent gravitational three-body system behaves differently; it will be discussed below.

1.3 1.3 Non-usual separability

The usual separability discussed above is defined in close relation to usual methods. Non-separable states can also be defined as a breakdown of separability using the same kind of usual methods. Remarkably, some subsets of these defined non-separable states can show non-usual separability, which is related to specific dynamics and symmetries of these states. Some of these non-usual separable states are discussed in Sect. 3. It is difficult to work without considering separability, so to employ non-usual separations when possible can be an opportunity to reach unknown states. In the case of charge systems, the most evident separability could be those coordinates that describe the repulsion of charges.

1.4 1.4 Well-behaved and badly behaved states

The well-behaved state is a nearly separable state, which can be well described by the usual methods. In fact, the usual methods were developed to solve the well-behaved states. Usually, the ground state of stable atoms and molecules is a well-behaved state.

The badly behaved state is a non-separable or non-usual separable state. This state presents a strong electronic correlation, which cannot be described by usual methods. The badly behaved states can be considered examples where their analogous classical are nonlinear systems with strong correlated regular and chaotic motions [44]. That is, the SE or Hamiltonian operator is linear concerning wavefunction for any system. However, its classical analogous Hamiltonian is nonlinear for two or more-electron system. In general, a classical nonlinear system can show regular and chaotic regions in phase space, depending on the initial conditions. For instance, the quantum system cannot show chaos due to the linear behavior of wavefunction. Still, it is possible to say chaoticity [73], which describes the influence of the grounding chaotic classical structure on the quantum state. In general, a classical backbone must exist for an atomic and molecular binding (quantum) state to manifest when quantization is performed.