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Atoms and the Periodic Table of the Elements
Published in Franco Battaglia, Thomas F. George, Understanding Molecules, 2018
Franco Battaglia, Thomas F. George
The hydrogen atom (or, if we wish, a hydrogen-like atom, i.e., a single electron bound to a nucleus with charge Ze) is a system of two particles subjected to electromagnetic interaction of which the electrostatic part is the most important contribution and depends only on the electron–nucleus distance. The latter scenario allows us to reduce the two-particle problem to a one-particle in a central potential problem, as we shall readily see. Using index 1 for the quantities pertaining to the nucleus and index 2 for those pertaining to the electron, the Hamiltonian of the system (kinetic energy plus potential energy) is
Bohr Model
Published in Zbigniew Ficek, Quantum Physics for Beginners, 2017
Show that in the Bohr atom model, the electron’s orbits in a hydrogen-like atom are quantized with the radius r = n2ao/Z, where ao = 4πε0ħ2/me2 is the Bohr radius, n = 1, 2,…, and Z is atomic number. Z = 1 refers to a hydrogen atom, Z = 2 to a Helium He+ ion, and so on.
The Theory of Atom of Hydrogen
Published in Mikhail G. Brik, Chong-Geng Ma, Theoretical Spectroscopy of Transition Metal and Rare Earth Ions, 2019
Mikhail G. Brik, Chong-Geng Ma
The hydrogen-like atom (or an ion) has only one electron, and all others have been removed by various ways of ionization. An essential feature of these systems is the presence of one electron only. The Coulomb interaction between this electron and the nucleus is spherically symmetrical. Later, we shall see that the situation becomes considerably more complicated in the case of multielectron atoms.
Atoms and molecules in soft confinement potentials
Published in Molecular Physics, 2020
L. F. Pašteka, T. Helgaker, T. Saue, D. Sundholm, H.-J. Werner, M. Hasanbulli, J. Major, P. Schwerdtfeger
For a confined hydrogen-like atom with nuclear charge Z, we must solve the radial Schrödinger equation with boundary conditions for and . In the absence of confinement, solutions may be expressed in terms of confluent hypergeometric functions as with x = 2kr, where and the quantum number is introduced to have bound solutions. Under confinement, it is convenient to keep n, but it now serves as a counting number [58, 59], as will be further elaborated in the following.
A simple model for scalar relativistic corrections to molecular total atomisation energies
Published in Molecular Physics, 2019
Jan M. L. Martin, Nitai Sylvetsky
From considering the solutions of the Dirac equation for the hydrogen-like atom, it is clear that relativistic corrections are largest for s orbitals, followed by p1/2 spinors. It has hence been received wisdom in the relativistic quantum chemical community (see, e.g. [39,40] for an early example) that changes in s orbital population drive many relativistic effects. For instance, the following statement in [41] comes to mind:The central atom in each of [BF3, AlF3, and GaF3] has a sizeable change in the s populations from the atom to the molecule, and hence an appreciable scalar relativistic contribution to the atomization energy.(Compare also p. 458 of [5] and Section 16.1 of [6]) It occurred to us that it would be helpful to verify this conjecture for a significantly sized sample of molecular data. We will use here the W4-17 dataset, and we will show that not only is there a clear statistical link with the atomic s populations, but that a simple additive model based on computed changes in s populations can account for over 98% of the variance in the dataset, and over 99% for first-row molecules. By adding correction terms for p populations, 99% of the variance in the dataset can be recovered for both first- and second-row molecules.
Local frame transformation theory for two classes of diatomic molecules
Published in Molecular Physics, 2019
The final step in determining the LFT matrix elements is to evaluate the right-hand side of Equation (51). To lowest order in r, the regular wave function for a hydrogen-like atom, at or away from a bound state, is