Many-electron atoms and the periodic table

Chemists in the nineteenth century recognized periodic trends in the physical and chemical properties of elements, long before quantum theory came onto the scene. Although these chemists were not aware of the existence of electrons and protons, their efforts to systematize the chemistry of the elements were remarkably successful. Their main sources of information were the atomic masses of the elements and other known physical and chemical properties. Modem quantum theory allows us to understand these periodic trends in terms of the ways in which the electrons are distributed among the atomic orbitals of an atom. 

1 The Wavefunctions of Many-Electron Atoms Can Be Described to a Good Approximation Using Atomic Orbitals 

The Schixidinger equadon works nicely for the simple hydrogen atom (Eqnadon 1.36), but it cannot be solved exactly for atoms containing more than one electron Fortunately, the atomic orbital concepts finom the hydrogen atom can be used to construct approximate (but reliable) solutions to the Schibdinger equation for many-electron atoms (that is, 

For the hydrogen atom with its single electron, the electronic wavefunction is a function of three variables, namely the x, y, and z coordinates of the electron. For a many-electron atom, the wavefunction is a function of the positions of all electrons. The wavefunction for helium (He), for example, is a function of six position variables JCi, yi, and zi for electron 1, and JCa, ya, and Z2 for electron 2  

Like that for the hydrogen atom, the SchrOdinger equation for the helium atom (Equation 2.2) contains a kinetic energy term as well as potential energy terms arising 

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Chapter 2 Many-Electron Atoms and the Periodic Table... 

Rutherford s discovery of the atomic nucleus was his greatest contribution to physics and it established him as the leading experimental physicist of his day. However, it was only a beginning, and many questions about the atom remained unanswered. As yet nothing was known about electron orbits or about the relationship between the structure of the atom and the periodic table. Before Rutherford performed his experiments, it was thought that the atom was understood. Now it was apparent that much remained to be learned. But then great discoveries in physics seem always to suggest new questions and open up new lines of research. The more that is known, the better the picture scientists have of what remains unknown. 

As readers of this volume are also aware, the best of both approaches have been blended together with the result that many computations are now performed by a careful mixture of wavefunction and density approaches within the same computations (Hehre et al., 1986). But the unfortunate fact is that, as yet, there is really no such thing as a pure density functional method for performing calculations. The philosophical appeal of a universal solution for all the atoms in the periodic table based on observable electron density, rather than fictional orbitals, has not yet borne fruit.21,22... 

The next step in our journey takes us from hydrogen and its single electron to the atoms of all the other elements in the periodic table. A neutral atom other than a hydrogen atom has more than one electron and is known as a many-electron atom. In the next three sections, we build on what we have learned about the hydrogen atom to see how the presence of more than one electron affects the energies of... 

Hydrogen (H) is the simplest and lightest atom in the periodic table. We drink it every day it is an essential component of water in fact, hydro-gen means water-generating, It has played a crucial role in many developments of modern physics. In this book we will model the hydrogen atom by a single quantum particle (the electron) moving in a spherically symmetric force field (created by the proton in the nucleus). There are certainly more sophisticated models available — for example, it is more precise to model the hydrogen atom as the mutual interaction of two particles, a proton and an electron" — but our model is simple and quite accurate. 

In fact, it is the only book in which you can find successive general non-relativistic and relativistic descriptions of the theory of energy spectra and transition probabilities in complex many-electron atoms and ions. The formulas and tables presented give the possibility, at least in principle, of calculating the energy spectra and electronic transitions of any multipolarity for any atom or ion of the Periodical Table. This book contains the bulk of new achievements in the non-relativistic and relativistic theory of an atom, especially as concerns the many-particle aspects of the non-relativistic and relativistic problem. It therefore complements books already available. 

Finally, although it is not precisely correct to assume that the N electrons in an atom occupy N independent one-electron orbitals, this remains a very useful idea for understanding many atomic properties, including the organization of the periodic table. Recall that for us to account for the arrangement of the atoms on the periodic table, the orbitals that correspond to a given value of n must fill in the order ns, then np, then nd, and, finally, nf. From this observation we would expect the energies of the one-electron SCF orbitals to vary in the order... 

The formal analysis of the mathematics required incorporating the linear combination of atomic orbitals molecular orbital approximation into the self-consistent field method was a major step in the development of modem Hartree-Fock-Slater theory. Independently, Hall (57) and Roothaan (58) worked out the appropriate equations in 1951. Then, Clement (8,9,63) applied the procedure to calculate the electronic structures of many of the atoms in the Periodic Table using linear combinations of Slater orbitals. Nowadays linear combinations of Gaussian functions are the standard approximations in modem LCAO-MO theory, but the Clement atomic calculations for helium are recognized to be very instructive examples to illustrate the fundamentals of this theory (67-69). 

The orbital concept and the Pauli exclusion principle allow us to understand the periodic table of the elements. An orbital is a one-electron spatial wave function. We have used orbiteils to obteiin approximate wave functions for many-electron atoms, writing the wave function as a Slater determinant of one-electron spin-orbitals. In the crudest approximation, we neglect all interelectronic repulsions and obtain hydrogenlike orbitals. The best possible orbitals are the Heu tree-Fock SCF functions. We build up the periodic table by feeding electrons into these orbitals, each of which can hold a pair of electrons with opposite spin. 

There is perhaps a simple solution, or perhaps a dissolution, of the problem. In posing the paradox regarding the 4s and 3d orbitals, many authors appear to have overlooked one very important feature, which makes the comparison problematic. In considering the buildup of atoms across the periodic table, one is concerned with the successive addition of one proton and one electron to each previous atom. However, in considering the ionization of any particular atom, one is concerned only with the successive removal of electrons and not the removal... 

In 1924, Pauli suggested that a fourth quantum number is necessary to explain the properties of the atoms in the periodic table. In the original version of his exclusion principle, no two electrons can have the same set of quantum numbers. At this time, Pauli did not want to connect the fourth quantum number to a spin of the particle since a spinning particle should have extension in space and this would lead to further inconsistencies. On the other hand, the mere existence of the spin seemed to explain many features of atomic spectra, as was discovered but not... 

First, we examine each atom and, using the periodic table, we determine how many valence electrons it would have if it were an atom not bonded to any other atoms. This is equal to the group number of the atom in the periodic table. For hydrogen this number equals 1, for carbon it equals 4, for nitrogen it equals 5, and for oxygen it equals 6. 

Our present views on the electronic structure of atoms are based on a variety of experimental results and theoretical models which are fully discussed in many elementary texts. In summary, an atom comprises a central, massive, positively charged nucleus surrounded by a more tenuous envelope of negative electrons. The nucleus is composed of neutrons ( n) and protons ([p, i.e. H ) of approximately equal mass tightly bound by the force field of mesons. The number of protons (2) is called the atomic number and this, together with the number of neutrons (A ), gives the atomic mass number of the nuclide (A = N + Z). An element consists of atoms all of which have the same number of protons (2) and this number determines the position of the element in the periodic table (H. G. J. Moseley, 191.3). Isotopes of an element all have the same value of 2 but differ in the number of neutrons in their nuclei. The charge on the electron (e ) is equal in size but opposite in sign to that of the proton and the ratio of their masses is 1/1836.1527. 

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