Structure and Electrochemical Properties of Aqueous Suspensions of Functionalized Single- and Multiwalled Carbon Nanotubes

Abstract

Symmetric and antisymmetric terms have been obtained in the framework of the variational approach for two-dimensional (2𝐷) Coulomb systems of symmetric trions 𝑋𝑋𝑌 . Stability diagrams and certain anomalies arising in the 2𝐷 space are explained qualitatively in the framework of the Born-Oppenheimer adiabatic approximation. The asymptotics of energy terms at large distances obtained for an arbitrary space dimensionality are analyzed, and some approximation formulas for 2𝐷 terms are proposed. An anomalous dependence of multipole moments on the space dimensionality has been found in the case of a spherically symmetric field. The main results obtained for the 2𝐷 and 3𝐷 problems of two Coulomb centers are compared.

The Conditions of Stability of Two-Dimensional Quantum Systems of Three Charged Particles

“…For the most of subsequent calculations, we can put M = 1 without loss of generality. Assume similarly to the previous paper [3] that the conditions of stability of the quantum system in an appropriate state (conditions of existence of bound states) are defined by the fact that the energy of this state is below the ground-state energy (threshold) of the most stable subsystem. In the case of three particles with Hamiltonian (1), the conditions of stability ISSN 2071-0194.…”

“…The investigation of bound states of the threeparticle system (1) is performed, by using the variational Galerkin method with Gaussian basis (notation is similar to that for the three-dimensional problem [3]). We have…”

“…The next practically important step in variational calculations is the selection of optimal schemes to minimize the energy with respect to nonlinear variational parameters {a, b, c}. We systematically adopt certain variational stochastic techniques (see, e.g., [3][4][5]) that allow us to minimize the energy of any given state. It was found that the optimal strategy that keeps the balance of performance and accuracy of calculations is to increase the basis size by one function at every step and to perform a random sampling of variational parameters of the added basis function, keeping all the remaining nonlinear parameters frozen.…”

“…First, we formulate the main result of this work, which consists of general conditions for the existence of bound states (diagram of stability) of 2D systems of three charged particles with Hamiltonian (1) depending on the mass m and the charge Z. We also compare the obtained diagrams of stability with those of 3D problem analyzed in [3].…”

“…Increasing the complexity, the next step is to study the general problem of three charged particles such as trions XXY (see [1]), where it is possible to achieve highly accurate results. In the three-dimensional space, the general problem of three charged particles has been widely studied (see [2,3]).…”