Understanding the Hydrogenic orbital expressions (original) (raw)
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Molecular orbital theory is a conceptual extension of the orbital model, which was so successfully applied to atomic structure. As was once playfully remarked, "a molecule is nothing more than an atom with more nuclei." This may be overly simplistic, but we do attempt, as far as possible, to exploit analogies with atomic structure. Our understanding of atomic orbitals began with the exact solutions of a prototype problem – the hydrogen atom. We will begin our study of homonuclear diatomic molecules beginning with another exactly solvable prototype, the hydrogen molecule-ion \(H_{2}^{+}\).
On the problem of the exact shape of orbitals
The Chemical Educator
To explain the inconsistency in the representation of the orbital in many textbooks, we considered the contour and polar representation of orbitals. Most introductory textbooks use the polar representation in their discussions, but the orbitals seems to be the most misrepresented orbital. In particular it is represented with a "lifesaver'' middle section. We argue that this shape belongs to the contour representation and is inconsistent with the polar representation. 2 d z 2 d z
Angular dependence of Gaussian‐Lobe orbitals. I. Analysis of standard p‐ and d‐orbitals
International Journal of Quantum Chemistry, 1978
Multipole expansions of Gaussian‐lobe atomic orbitals around their centers are theoretically investigated in order to study the exact angular dependence of such functions. Analytical expressions of the multipole coefficients are derived for standard lobe orbitals. It is shown that the average‐square values of multipole components are related to a unique orbital parameter λ. The numerical values of p‐ and d‐components are given for selected λ and the choice of this parameter is discussed on the basis of symmetry and computational arguments. The transferability of optimized atomic exponents from harmonic (or Cartesian) functions to lobe functions is established so that the possibility of applying the Gaussian‐lobe orbital approach in chemical studies is greatly extended.
Atomic Orbitals: Explained and Derived by Energy Wave Equations
2019
The electron’s orbital distance, ionization energy and shape can be modeled based on classical mechanics when the recently-discovered pentaquark structure is used as the model of the proton. This paper accurately models atomic orbital distances based on this five-quark structure of the proton, in which the orbiting electron is both attracted by an anti-quark and repelled by quarks in the proton. The orbital distance is classically defined as the point where the sum of the forces is zero, removing the need for a separate set of laws in physics, known as quantum mechanics, to describe the electron’s position in an atom.
Atomic orbitals revisited: generalized hydrogen-like basis sets for 2nd-row elements
Theoretical Chemistry Accounts, 2018
In the present work, we revisit the problem of atomic orbitals from the positions mostly dictated by semiempirical approaches in quantum chemistry. To construct basis set, having proper nodal structure and simple functional form of orbitals and representing atomic properties with reasonable accuracy, authors propose an Ansatz based on gradual improvement of hydrogen atomic orbitals. According to it, several basis sets with different numbers of variable parameters are considered and forms of orbitals are obtained for the 2nd-row elements either by minimization of their ground state energy (direct problem) or by extracting from atomic spectra (inverse problem). It is shown that so-derived three-and four-parametric basis sets provide accurate description of atomic properties, being, however, substantially provident for computational requirements and, what is more important, simple to handle in analytic models of quantum chemistry. Since the discussed Ansatz allows a generalization for heavier atoms, our results may be considered not only as a solution for light elements, but also as a proof of concept with possible further extension to a wider range of elements.
Physical meaning of the natural orbitals: Analysis of exactly solvable models
2010
We investigate the suitability of natural orbitals as a basis for describing many-body excitations. We analyze to which extent the natural orbitals describe both bound as well as ionized excited states and show that depending on the specifics of the excited state the ground-state natural orbitals may yield a good approximation. We show that the success of reduced density-matrix functional theory in describing molecular dissociation lies in the flexibility provided by fractional occupation numbers while the role of the natural orbitals is minor.
Atomic Model Theory: Shapes of Electron Orbitals
We explore the different proposed theories in quantum mechanics for the shapes of atomic orbitals; Ultimately arriving at Schrodinger's wave equation. We examine the solution to Schrodinger's equation and then run simulations in MATLAB to demonstrate the consistency in his solution.