Many-electron atoms wavefunctions

The procedure outlined in Fignre 2.3 is computationally intensive and requires sophisticated computer programs to accomplish. Fortunately for chemists, however, it is possible to understand a great deal abont many-electron atom wavefunctions with-ont having to explicitly solve for the SCF orbitals. Most importantly, the SCF orbitals can be described using the same set of quantum numbers (namely, n, I, and ttii) that index the atomic orbitals of the hydrogen atom. The restrictions on these three quan-tnm numbers are the same as those for hydrogen (namely = 1, 2, 3,... Z = 0,... 

This way of looking at the molecule appeals to us for the same reason that we use the one-electron atomic orbitals to write electron configurations that approximate the much more complicated many-electron atomic wavefunctions. Reducing the number of coordinates, the dimensionality of the problem, makes the problem more tractable and can lead to a useful intuition about how these systems behave. 

You are probably used to this idea from descriptive chemistry, where we build up the configurations for many-electron atoms in terms of atomic wavefunctions, and where we would write an electronic configuration for Ne as... 

Because the total Hamiltonian of a many-electron atom or molecule forms a mutually commutative set of operators with S2, Sz, and A = (V l/N )Ep sp P, the exact eigenfunctions of H must be eigenfunctions of these operators. Being an eigenfunction of A forces the eigenstates to be odd under all Pp. Any acceptable model or trial wavefunction should be constrained to also be an eigenfunction of these symmetry operators. 

Having decided to use AOs (or combinations of them) for yrA and pB> we will now look at the form these take. They are approximate solutions to the Schrodinger equation for the atom in question. The Schrodinger equation for many-electron atoms is usually solved approximately by writing the total electronic wavefunction as the product of one-electron functions (these are the AOs). Each AO 4>i is a function of the polar coordinates r, 0, and single electron and can be written as... 

Although equation (2) appears concise in the form above, it cannot be solved exactly for a molecular system. Electron-electron repulsions present a practical problem in the same way that they do in many-electron atoms the repulsive interaction between two electrons is a function of the position coordinates of each electron and cannot be separated into two functions dependent on each set of coordinates individually. However, even if a comphcated wavefunction that was expressed in the coordinates of aU the electrons in a molecule could be developed, it would have to be reported as an extensive grid of points in 3-D space, each associated with a different potential energy with unique solutions for... 

The second but often dominant effect is the so-called indirect relativistic effect. This occurs as a change in the radial distribution of the wavefunctions because in a many-electron atom the inner electrons contract and thus shield the outer ones more effectively. As a result, this effect often compensates the direct relativistic effect for the d-wavefunctions for the 5f-wavefimctions, however, this leads to an increased radius and the 4f-wavefunctions are hardly affected at all. As a consequence, the 5f-wavefunctions are chemically much more active in the Actinides than the 4f-wavefunctions in the Lanthanides. 

C. A. Nicolaides, D.R. Beck, Variational calculations of correlated wavefunctions and energies for ground, low-lying as well as highly excited discrete states in many-electron atoms, J. Phys. B 6 (1973) 535. 

Solutions of Schrodinger s wave equation give the allowed energy levels and the corresponding wavefunctions. By analogy with the orbits of electrons in the classical planetary model (see Topic AT), wavefunctions for atoms are known as atomic orbitals. Exact solutions of Schrodinger s equation can be obtained only for one-electron atoms and ions, but the atomic orbitals that result from these solutions provide pictures of the behavior of electrons that can be extended to many-electron atoms and molecules (see Topics A3 and C4-C7). 

Putting electrons into orbitals similar to those in the hydrogen atom gives a useful way of approximating the wavefunction of a many-electron atom. The electron configuration specifies the occupancy of orbitals, each of which has an associated energy. 

Of course it is still possible to use the complex mathematical expressions, corresponding to the different type of orbital solutions to the hydrogen atom problem, in order to build up a wavefunction that approximates that of a many-electron atom or molecule. In such cases, we are using orbitals in a purely instrumental fashion to model the wavefunction of the atom or molecule and there is no pretense, at least by experts in the field, that the constituent orbitals used in this modeling procedure possess any independent existence. Contrary to the recent claims which appeared in Natme magazine, as well as many other publications, orbitals have not been observed (Scerri 2000b, 2001). 

As will emerge, these properties are somewhat altered in heavy atoms there may be more than one radius for which exact balance occurs. Also, the centrifugal barrier may not rise high enough in a many-electron atom to exclude the radial wavefunction from within the core, although of course it will always exclude the wavefunction from the geometrical centre. 

The physical origin of the double well is also readily understood. In H (as opposed to many-electron atoms), if the centrifugal repulsive term is included within an effective potential then, since it grows as ( + l)h2/2mr2, there is a repulsive potential at small radius (r < ro, say) can only expel wavefunctions of high angular momentum into the outer Coulombic reaches. Note that, at all radii, the effective potential is the net... 

The early treatment of many-electron atoms was very crude. Hartree expressed the wavefunction for an atom as the product of one-electron orbitals. These orbitals were supposed to be the solutions to a set of coupled differential equations which seemed hopelessly difficult to solve. Eventually Hartree was able to obtain approximate solutions on a mechanical dilTeren-tial analyzer which he built out of Mechano parts. 

As can be seen, these orbitals are the product of a radial part R dependent on r, the distance from the electron to the nucleus, and Y, called a spherical harmonic, a function detailing the angular dependence of the atomic wavefunction. For all many-electron atoms, the radial term must be approximated because of the aforementioned problem of electron-electron repulsions. 

A wavefunction, ip, is a solution to the Schrodinger equation. For atoms, wavefunctions describe the energy and probabihty of location of the electrons in any region around the proton nucleus. The simplest wavefunctions are found for the hydrogen atom. Each of the solutions contains three integer terms called quantum numbers. They are n, the principal quantum number, I, the orbital angular momentum quantum number and mi, the magnetic quantum number. These simplest wavefunctions do not include the electron spin quantum number, m, which is introduced in more complete descriptions of atoms. Quantum numbers define the state of a system. More complex wavefunctions arise when many-electron atoms or molecules are considered. 

One approximate method for obtaining Eq. 7.182 in a central field form wa.s introduced by Hartree and named the self-consistent field approach. This method regards each electron in a many-electron atom as moving in the temporarily fixed field of the remaining electrons. The system can now be described in terms of one-electron wavefunctions (or orbitals) j(rj). The non-Coulomb potential energy for the jth electron is then V y(ry) atid this contains the other electronic coordinates only as parameters. Vjirj) can be chosen to be spherically symmetric. The computational procedure is to solve the Schrodinger equation for every electron in its own central field and then to make the wavefunctions, so found, self-consistent with their potential fields. The complete wavefunction for the system is a product of the one-electron functions. 

The wavefunctions that we shall discuss form the basis of our understanding of atomic structure in general, because the concepts introduced can be extended to many-electron atoms. They will also prove useful when we come to discuss chemical bonding. 

The wavefunction that we have just derived for the helium atom is incomplete because it does not include the spins of the two electrons. The occupation of atomic oritals in many-electron atoms is controlled by the Pauli exclusion principle, which states that ... 

In a many-electron atom an approximate wavefunction, 0., can be obtained for the ith electron by solving the following one-electron wave... 

The approach is rather different from that adopted in most books on quantum chemistry in that the Schrbdinger wave equation is introduced at a fairly late stage, after students have become familiar with the application of de Broglie-type wavefunctions to free particles and particles in a box. Likewise, the Hamiltonian operator and the concept of eigenfunctions and eigenvalues are not introduced until the last two chapters of the book, where approximate solutions to the wave equation for many-electron atoms and molecules are discussed. In this way, students receive a gradual introduction to the basic concepts of quantum mechanics. 

The Wavefunctions of Many-Electron Atoms Can Be Described to a Good Approximation Using Atomic Orbitals 127... 

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