Theoretical Basis of Quantum-Mechanical Modeling of Functional Nanostructures

Abstract

The paper presents an analytical review of theoretical methods for modeling functional nanostructures. The main evolutionary changes in the approaches of quantum-mechanical modeling are described. The foundations of the first-principal theory are considered, including the stationery and time-dependent Schrödinger equations, wave functions, the form of writing energy operators, and the principles of solving equations. The idea and specifics of describing the motion and interaction of nuclei and electrons in the framework of the theory of the electron density functional are presented. Common approximations and approaches in the methods of quantum mechanics are presented, including the Born–Oppenheimer approximation, the Hartree–Fock approximation, the Thomas–Fermi theory, the Hohenberg–Kohn theorems, and the Kohn–Sham formalism. Various options for describing the exchange–correlation energy in the theory of the electron density functional are considered, such as the local density approximation, generalized and meta-generalized gradient approximations, and hybridization of the generalized gradient method. The development of methods of quantum mechanics to quantum molecular dynamics or the dynamics of Car–Parrinello is shown. The basic idea of combining classical molecular modeling with calculations of the electronic structure, which is reflected in the potentials of the embedded atom, is described.

Modeling of Magnetic Films: A Scientific Perspective

“…In the realm of quantum mechanics, a distinct avenue of inquiry involves the formulation of theoretical frameworks aimed at scrutinizing the characteristics of intricate molecular systems, denominated as quantum molecular dynamics [ 17 ]. This approach relies upon the application of the Born–Oppenheimer approximation, postulating the segregation of variables into substantial nuclei and agile electrons.…”

“…This approach relies upon the application of the Born–Oppenheimer approximation, postulating the segregation of variables into substantial nuclei and agile electrons. The quantum molecular dynamics methodology, as introduced by Car–Parrinello, entails the utilization of function minimization techniques [ 17 ].…”

“…The trajectory of nuclei is dictated by the ensemble of their respective coordinates {R i |i = 1,…,N n }. The electronic degrees of freedom are defined by a collection of quantum-mechanical wave functions {Ψj|j = 1,…,N e } [ 17 ]. In the given context, the symbol Fi represents the resultant force exerted on the atom.…”

“…The electronic degrees of freedom are defined by a collection of quantum-mechanical wave functions {Ψj|j = 1,. . .,N e } [17].…”

“…The selection of the fictitious electron mass is aimed at expediting the convergence of the algorithm. In parallel, the Lagrange multipliers play a pivotal role in generating additional forces that enforce the orthonormality of the wave functions [17] (7,8).…”