Electronic states multielectron atoms

The Electronic States of Atoms. II. The Zero-Order Approximation for Multielectron Atoms... 

Briefly, XANES is associated with the excitation process of a core electron to bound and quasibound states, where the bound states interacting with the continuum are located below the ionization threshold (vacuum level) and the quasibound states interacting with the continuum are located above or near the threshold. Thus, XANES contains information about the electronic state of the x-ray absorbing atom and the local surrounding structure. However, as stated above, unhke EXAES, since the excitation process essentially involves multielectron and multiple scattering interactions, interpretation of XANES data is substantially more complicated than that of EXAFS data. 

The ground-state electron configuration of a multielectron atom is arrived at by following a series of rules called the aufbau principle. 

A multielectron atom can exist in several electronic states, called microstates, which are characterized by the way the electrons are distributed among the atomic orbitals. The number of microstates for a free atom with a valence shell consisting of a set of degenerate orbitals with orbital angular momentum quantum number I housing n electrons is given by ... 

This normal sequence of events accords with the convexity principle (7), which states that the energy of stabilization upon addition of the first electron to an atomic or molecular system (-AGO is greater than the stabilization energy attending addition of the second (-AG2). The converse set of conditions (-AG2 > -AGi, 2° > 1°, AGdisp < 0) may arise, however, when a change in structure or composition, e.g., ligation, protonation, ion-pair formation, accompanies electron transfer. This circumstance results in a multielectron event via an inversion of potentials and the destabilization of A". Numerous examples of such behavior have been identified and discussed 8-13),... 

In atoms with more than one electron, wave functions should include the coordinates of each particle, and a new term representing the electrostatic interactions between electrons. Even for the case of only two electrons, such a wave equation is so complex that it has never been solved exactly. To analyse multielectron atoms some approximations have to be made. The most practical one is to assume that the electron considered moves in an electrical potential that is a combination of all other electrons and the nucleus, and that this potential has spherical symmetry. This approximation has proven very useful, as it allows a description of energy states in a similar manner to that employed for the H atom by using a comparable set of four quantum numbers. An important, additional condition appears no two electrons can have the same set of quantum numbers in other words, no more than one electron can occupy the same energy state. This is Pauli s exclusion principle. 

We begin with the assumption that the electrons in a multielectron atom can in fact be assigned to approximate hydrogen-like orbitals, and that the wavefunction of the complete atom is the product of the wavefunctions of each occupied orbital. These orbitals can be labeled with the quantum number labels Is, 2s, 2p, 3s, 3p, and so on. Each s,p,d,f,... subshell can also be labeled by an quantum number, where ranges from — to T (2T + 1 possible values). But it can also be labeled with a spin quantum number m either -f or —The spin part of the wavefunction is labeled with either a or p, depending on the value of for each electron. Therefore, there are several simple possibilities for the approximate wavefunction for, say, the lowest-energy state (the ground state) of the helium atom ... 

In addition to the conditions for the electronic structures of multielectron atoms established by the monoelectronic wave functions and their relative energies mentioned above, other restrictions should also be considered. One of them is the Pauli principle stating that no two electrons can have the same quantum numbers. Thus one orbital can contain a maximum of two electrons provided they have different spin quantum numbers. Other practical rules or restrictions refer to the influence of interelectronic interactions on the electronic structures established by Hund s rules. The electrons with the same n and / values will occupy first orbitals with different nti and the same rris (paired spins). 

There are several commonly used approximation schemes that can be applied to the electronic states of multielectron atoms. The first approximation scheme was the variation method, in which a variation trial function is chosen to minimize the approximate ground-state energy calculated with it. A simple orbital variation trial function was found to correspond to a reduced nuclear charge in the helium atom. This result was interpreted to mean that each electron in a helium atom shields the other electron from the full charge of the nucleus. A better variation trial function includes electron correlation, a dependence of the wave function on the electron lectrcm distance. ... 

To illustrate how this works with a multielectron atom, consider the N2 molecule. Here we must accommodate 14 electrons, 7 from each atom. As the two N atoms are brought together, each of the atomic levels split into a bonding and an antibonding state as shown in Figure 3.7. 

The quantum mechanical wave function for a multielectron atom can be approximated as a superposition of orbitals, each bearing some resemblance to those describing the quantum states of the hydrogen atom. Each orbital in a multielectron atom describes how a single electron behaves in the field of a nucleus under the average influence of all the other electrons. 

The observed ground-state electron configuration is always the one that gives the lowest total energy for the atom. As discussed in the text, electron motions in a multielectron atom are highly correlated consequently, the total energy of an atom is, in some cases, a very delicate balance between electron-nuclear attractions and electron-electron repulsions. 

The Pauli Exclusion Principle states that no two electrons of any single atom may simultaneously occupy a slate described by only a single set of quantum numbers. Five such numbers arc needed to describe fully the quantum-mechanical conditions of an electron. For j-j coupling this set is generally ti. I., v. j. iij. and for l.-S it is /t. /. j. u(. nr,. From die coupling of the angular momentum associated with the latter sets a full description of the multielectron stale, described by it, L. S, J. Mis determined. 

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