Orbits, atomic elliptical

Pauli s Exclusion Principle The principle that no two identical elementary particles having half-integer spin (fermion) in any system can be in the same quantum state (i.e., have the same set of quantum numbers). In order to account for the various spectral characteristics of the different elements, one must assume that no two electrons in a given atom can have all four quantum numbers identical. This means that, in any orbit (circular, elliptical, or tilted), two electrons at most may be present and of these two, one must spin clockwise and the other must spin counterclockwise. Thus, the presence of two electrons of opposite spin in a given orbit excludes other electrons. 

The Hartree-Fock orbitals are expanded in an infinite series of known basis functions. For instance, in diatomic molecules, certain two-center functions of elliptic coordinates are employed. In practice, a limited number of appropriate atomic orbitals (AO) is adopted as the basis. Such an approach has been developed by Roothaan 10>. In this case the Hartree-Fock differential equations are replaced by a set of nonlinear simultaneous equations in which the limited number of AO coefficients in the linear combinations are unknown variables. The orbital energies and the AO coefficients are obtained by solving the Fock-Roothaan secular equations by an iterative method. This is the procedure of the Roothaan LCAO (linear-combination-of-atomic-orbitals) SCF (self-consistent-field) method. 

In general, if a particle is bound (E < 0) it will oscillate (classically) between some limits r = a, and r = b. For example, in an elliptic orbit of a hydrogen atom, the radius oscillates periodically between inner and outer limits. Only for a circular orbit is there no oscillation. Among the eigenvalues which have the same n, the one with lowest l has the largest amplitude in the vicinity of the nucleus. 

Following Sommerfeld s proposal of elliptical electron orbits in 1915, Bohr amended his original theory, which had included only circular orbits. 14 A 1922 paper in Zeitschriftfur Physik outlined the "Aufbauprinzip" by which electrons are fed into atomic subshells. There was a neat correlation between periodic groups containing 2, 8, 8, 18,... 

Three quantum numbers had been proposed, based on spectral lines and inferences about electron energy levels a principal quantum number to specify energy level of the atom an azimuthal quantum number to specify the angular momentum of electrons moving elliptically and an inner or magnetic quantum number to express the orientation of the plane of the electron s orbit in a magnetic field. 20... 

To a first approximation, we may assume that the electrons follow elliptical orbits with the nucleus at one focus of the ellipse. This model of the atom thus resembles the Solar System, with the electrons in the role of planets and the nucleus standing in for the Sun. 

Which is all very well, but the Bohr atom is wrong. The picture of a dense nucleus surrounded by electrons is accurate enough, but they do not follow nice elliptical orbits like those of the planets. Venus and Mars follow Newton s laws, but electrons are governed by the... 

While the theory of Bohr was a major step forward, and it helped to rmderstand the observed hydrogen spectrum, it left many other observations in the dark. New light was shed on the subject of atomic structure and the line spectra by Arnold Sormnerfeld (1868-1951) (27). He elaborated the basic theory of Bohr by observing that the orbits eould also be elliptical, and that for each principal energy level, there eotrld be a specific number of elliptical orbits of different... 

Atomic. From spectroscopic studies, it is known lliat when an electron is bound to a positively charged nucleus only certain fixed energy levels are accessible to the electron. Before 1926, the old quantum theory considered that the motion of the electrons could be described by classical Newtonian mechanics in which the electrons move in well defined circular or elliptical orbits around the nucleus. However, the theory encountered numerous difficulties and in many instances there arose serious discrepancies between its predictions and experimental fact. 

A useful feature of the molecular orbital approach is that the eigenvalue equation of Eq. (23.22) can be separated in confocal elliptic coordinates,23 and, equally important, these eigenfunctions are apparently somewhat similar to the final atomic eigenfunctions.22 The coordinates are given by22... 

Bohr s theory was extended in various ways, especially by Somerfeld, who showed how to deal with elliptical orbits. There was a certain amount of qualitative success in applying the theory to atoms with several electrons. These developments in what is now called the old quantum theory were important as they laid much of the groundwork necessary for a correct theory. Ultimately, they were unsuccessful. Bohr s theory did not really explain what is going on why should only some orbits be allowed Where does the quantization condition (eqn 4.12) come from Following the developments of... 

The only quantum number that flows naturally from the Bohr approach is the principal quantum number, n the azimuthal quantum number Z (a modified k), the spin quantum number ms and the magnetic quantum number mm are all ad hoc, improvised to meet an experimental reality. Why should electrons move in elliptical orbits that depend on the principal quantum number n Why should electrons spin, with only two values for this spin Why should the orbital plane of the electron take up with respect to an external magnetic field only certain orientations, which depend on the azimuthal quantum number All four quantum numbers should follow naturally from a satisfying theory of the behaviour of electrons in atoms. 

Circular orbits are defined by n = 0. The principal quantum number specifies energy shells. For n = 1 the only solution is n = 0, k = 1, which specifies two orbits with angular momentum vectors in opposite directions. The solutions n = 0, k = 2 and n = 1, k = 1 define 8 possible orbits, 4 circular and 4 elliptic. The angular momentum vectors of each set are directed in four tetrahedral directions to define zero angular momentum when fully occupied. Taken together, these tetrahedra define a cubic arrangement, closely related to the Lewis model for the Ne atom. 

The Sommerfeld model for Ne is shown in figure 2.8. The He atom presented a special problem as the quantum numbers restrict the two electrons to the same circular orbit, on a collision course. One way to overcome this dilemma was by assuming an azimuthal quantum number k = for each electron, confining them to coplanar elliptic orbits with a common focal point. To avoid interference they need to stay precisely out of phase. This postulate, which antedates the discovery of electron spin was never seen as an acceptable solution to the problem which eventually led to the demise of the Sommerfeld model. 

The vehement opposition generated by van t Hoff s proposals was evidently directed against the notion that atoms possessed non-spherical three-dimensional structures. The equivalent, more careful, formulation of Le Bel referred explicitly to the symmetry of molecules, such as methane, and is free of this criticism. It is unfortunate that it was the van t Hoff picture which became established, first in terms of Sommerfeld s elliptic orbits and later on in Pauling s hybrid orbitals. 

The remarkable accord between the postulates of van t Hoff and Sommer-feld s elliptic orbits must, no doubt have convinced many sceptics of a more fundamental basis of both phenomena to be found in atomic shape. The new quantum theory that developed in the late 1920 s seemed to define such a basis in terms of the magnetic quantum number mi. 

In contrast to the four tetrahedrally oriented elliptic orbits of the Sommer-feld model, the new theory leads to only three, mutually orthogonal orbitals, at variance with the known structure of methane. A further new theory that developed to overcome this problem is known as the theory of orbital hybridization. In order to simulate the carbon atom s basicity of four an additional orbital is clearly required. The only possible candidate is the 2s orbital, but because it lies at a much lower energy and has no angular momentum to match, it cannot possibly mix with the eigenfunctions on an equal footing. The precise manoeuvre to overcome this dilemma is never fully disclosed and appears to rely on the process of chemical resonance, invented by Pauling to address this, and other, problems. With resonance, it is assumed that, linear combinations of an s and three p eigenfunctions produce a set of hybrid orbitals with the required tetrahedral properties. 

The quantum content of current theories of chemical cohesion is, in reality, close to nil. The conceptual model of covalent bonding still amounts to one or more pairs of electrons, situated between two atomic nuclei, with paired spins, and confined to the region in which hybrid orbitals of the two atoms overlap. The bond strength depends on the degree of overlap. This model is simply a paraphrase of the 19th century concept of atomic valencies, with the incorporation of the electron-pair conjectures of Lewis and Langmuir. Hybrid orbitals came to be introduced to substitute for spatially oriented elliptic orbits, but in fact, these one-electron orbits are spin-free. The orbitals are next interpreted as if they were atomic wave functions with non-radial nodes at the nuclear position. Both assumptions are misleading. 

The term atom and the names of its constituents—protons, neutrons, and electrons—have become almost household words during the last decade. The schematic sketch of the solar-system atom with a nucleus at the center and electrons revolving about it in circular or elliptical orbits has become as familiar as many of the commercial trademarks. This picture of the atom is that which is often presented to the student during his early chemistry courses while such a picture fits a number of important properties of atoms, it is no longer regarded as a good approximation. 

These were Sommerfeld s orbits for the hydrogen atom. When the atom is in the n = 2 state, the electron can move in either the circular orbit k = 2) or the elliptical orbit k = 1) in the = 3 state, it can move in three different orbits. 

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