How to Write an Electronic Wave Function for Electron Configuration + Hund's Law

Atomic structure and bonding

To understand bond formation, it is necessary to know the general features of the electronic structure of atoms—that is, the arrangement of electrons around the central nucleus. For background information about this subject and further details, see atom.

Atomic structure

The modern version of atomic structure begins with Ernest Rutherford's recognition that an atom consists of a single, central, massive, positively charged nucleus surrounded by electrons. The number of protons in the nucleus is the atomic number, Z, of the element. (For hydrogen Z = 1, and for carbon Z = 6.) A proton is positively charged, and an electron carries an equal but opposite negative charge. For an atom to be electrically neutral, it must contain the same number of extranuclear electrons as there are protons in the nucleus. Hence, the number of electrons in a neutral atom of atomic number Z is also Z. A hydrogen atom consequently has one electron, and a carbon atom has six electrons.

Rutherford atomic model

Physicist Ernest Rutherford envisioned the atom as a miniature solar system, with electrons orbiting around a massive nucleus, and as mostly empty space, with the nucleus occupying only a very small part of the atom. The neutron had not been discovered when Rutherford proposed his model, which had a nucleus consisting only of protons.

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The Bohr model

The first attempt to introduce quantum theory to account for the structure of atoms was made by the Danish physicist Niels Bohr in 1913. He asserted that the electron in a hydrogen atom occupies one of an array of discrete (but infinite in number) orbits, each orbit being progressively farther from the nucleus and labeled with an integer n = 1, 2,…. This integer is an example of a quantum number, which in general is an integer (in some cases, a half-integer) that labels the state of a system and which, through an appropriate formula, determines the values of certain physical properties of the system. By matching the centrifugal effect of the electron's motion in its orbit to the electrostatic attraction of the nucleus for the electron, Bohr was able to find a relation between the energy of the electron and the quantum number of its orbit. The result he obtained was in almost perfect agreement with the observed values of the energy levels of a hydrogen atom that had previously been obtained by spectroscopic methods.

Bohr's triumph was the first apparently successful incorporation of quantum theoretical ideas into the description of a mechanical system. The numerical success of the model has turned out to be coincidental, however, and Bohr's model is now regarded as no more than a historically important step in the evolution of quantum mechanics. The cracks in its validity were noted quite soon after its introduction. Thus, it was remarked that Bohr had not really derived the existence of discrete orbits from more fundamental principles but had merely imposed them on the model. Furthermore, all attempts to extend his theory to atoms that consisted of more than one electron (helium, with two electrons, for instance) utterly failed. Although the model was augmented by more elaborate specifications of the orbits (most notably, first, by allowing for elliptical orbits and introducing a second quantum number to specify the elongation of the ellipse and, second, by allowing for the effects of relativity), the failure to generalize to many-electron atoms remained a fatal flaw.

The quantum mechanical model

Current understanding of atomic structure had to await the introduction of quantum mechanics by the scientists Werner Heisenberg of Germany and Erwin Schrödinger of Austria in the mid-1920s. Indeed, the structure of the hydrogen atom that is still employed today was developed by Schrödinger in the four papers with which he introduced his version of quantum mechanics—wave mechanics—to the world. The quantum mechanical model of the hydrogen atom has the same numerical agreement with experiment that proved so coincidental with the Bohr model, but it is more fundamentally founded (i.e., the discreteness of the allowed energy states emerges from more general aspects and is not imposed), and the model can be extended (albeit with difficulty) to many-electron atoms. Moreover, unlike Bohr's theory, it is consistent with the fundamental principles of quantum mechanics—specifically the wave character of the electron and the requirements of the uncertainty principle, which states that the position and momentum (mass times velocity) of a particle cannot be specified simultaneously.

The location of the electron

In the quantum mechanical model of the hydrogen atom, the location of the electron is expressed in terms of a probability distribution, so one speaks of the probability that an electron will be found at a particular location near a nucleus. The probability distribution, in turn, is determined by a mathematical function known as a wave function, denoted ψ. Wave functions for the distribution of particles are a general feature of quantum mechanics, and for electrons in atoms they are known as atomic orbitals. The name orbital is intended to express a distribution that is less precise than the explicit orbits of the Bohr model. The probability of finding an electron at a specified location is proportional to the square of the amplitude of the wave function at that point. Hence, the sign (positive or negative) of the orbital is not relevant to the location of the electron, because taking the square of ψ eliminates any negative sign it may have. However, as explained below in Molecular orbital theory, the sign is of crucial importance in the discussion of bonding between atoms and so cannot be ignored.

Quantum numbers

Three quantum numbers are needed to specify each orbital in an atom, the most important of these being the principal quantum number, n, the same quantum number that Bohr introduced. The principal quantum number specifies the energy of the electron in the orbital, and, as n increases from its lowest value 1 through its allowed values 2, 3,…, the energies of the corresponding orbitals increase. The ground state, or lowest energy state of the hydrogen atom, is the state in which it is normally found and has n = 1, it consists of a single electron in the orbital closest to the nucleus. As n increases, so does the average distance of the electron from the nucleus, and, as n approaches infinity, the average distance also approaches infinity. The energy required to elevate the electron from the orbital with n = 1 to the orbital with n = ∞ is called the ionization energy of the hydrogen atom; this is the energy required to remove the electron completely from the atom.

The quantum number n labels the shell of the atom. Each shell consists of n ² individual orbitals with the same principal quantum number and hence (in the hydrogen atom) the same energy. Broadly speaking, each shell consists of orbitals that lie at approximately the same distance from the nucleus. The shells resemble the layers of an onion, with successive shells surrounding the inner shells.

The next quantum number needed to specify an orbital is denoted l and called the orbital angular momentum quantum number. This quantum number has no role in determining the energy in a hydrogen atom. It represents the magnitude of the orbital angular momentum of the electron around the nucleus. In classical terms, as l increases, the rate at which the electron circulates around the nucleus increases. The values of l in a shell of principal quantum number n are limited to the n values 0, 1, 2,…, n − 1, and the value of l of an orbital in a given shell determines the subshell to which that orbital belongs. It follows from the allowed values of l that there are n subshells in a shell of principal quantum number n. As will be explained, there are 2l + 1 orbitals in a given subshell.

Although subshells are uniquely specified by the values of n and l, it is conventional to label them in a slightly different manner. A subshell with l = 0 is called an s subshell, one with l = 1 is called a p subshell, and one with l = 2 is called a d subshell. Other subshells are encountered, but these three are the only ones that need to be considered here. The three subshells of the shell with n = 3, for example, are called the 3s, 3p, and 3d subshells.

As noted above, a subshell with quantum number l consists of 2l + 1 individual orbitals. Thus, an s subshell (l = 0) consists of a single orbital, which is called an s orbital; a p subshell (l = 1) consists of three orbitals, called p orbitals; and a d subshell (l = 2) consists of five orbitals, called d orbitals. The individual orbitals are labeled with the magnetic quantum number, m _l , which can take the 2l + 1 values l, l − 1,…, −l. The orbital occupied in the lowest energy state of the hydrogen atom is called a 1s orbital, signifying that it belongs to (and is in fact the only member of) the shell with n = 1 and subshell with l = 0.

How to Write an Electronic Wave Function for Electron Configuration + Hund's Law

Source: https://www.britannica.com/science/chemical-bonding/Atomic-structure-and-bonding

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