Many-electron atoms approximate solution

Exact solutions to the electronic Schrodinger equation are not possible for many-electron atoms, but atomic HF calculations have been done both numerically and within the LCAO model. In approximate work, and for molecular applications, it is desirable to use basis functions that are simple in form. A polyelectron atom is quite different from a one-electron atom because of the phenomenon of shielding", for a particular electron, the other electrons partially screen the effect of the positively charged nucleus. Both Zener (1930) and Slater (1930) used very simple hydrogen-like orbitals of the form... 

In many-electron atoms, the Schrodinger equation cannot be solved exactly, so approximations must be made. The simplest and crudest approximation is to neglect entirely electron-electron interactions (repulsions) and electron spin. In this way, hydrogenic orbitals are found as solutions. Into these orbitals we then place the electrons, according to the aufbau principle, and thus derive electron... 

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... 

Exact solutions such as those given above have not yet been obtained for the usual many-electron molecules encountered by chemists. The approximate method which retains tile idea of orbitals for individual electrons is called molecular-orbital theory (M. O. theory). Its approach to the problem is similar to that used to describe atomic orbitals in the many-electron atom. Electrons are assumed to occupy the lowest energy orbitals with a maximum population of two electrons per orbital (to satisfy the Pauli exclusion principle). Furthermore, just as in the case of atoms, electron-electron repulsion is considered to cause degenerate (of equal energy) orbitals to be singly occupied before pairing occurs. 

Given these difficulties of classical physics, it is not surprising that the quantum theory of many-electron atoms has no exact solutions either. Yet in some ways the quantum theory puts us in a more favourable situation. We do not try to compute detailed trajectories, because we accept that they do not exist in the classical sense. What the wavefiinction should tell us is a probability distribution. The fact that we require less specific information makes it easier to develop an approximation which is accurate enough to be... 

Better approximations can be made, and numerical calculations leave no doubt that Schrodinger s equation works very accurately for many-electron atoms, as it does for hydrogen. However, the orbital approximation is good enough for most purposes, and it leads to the very appealing picture of a many-electron atom in which each electron occupies an orbital which is similar to, although not identical with, the orbitals which form the exact solutions of the hydrogen atom. 

One of the benefits that quantum theory has for chemistry is an improved understanding of elemental periodicity, spectroscopy and statistical thermodynamics topics which can be developed without reference to the nature of electrons, atoms or molecules. The success of these applications depend on approximations to model many-electron atoms on the hydrogen solution and the recognition of spin as a further component of electronic angular momentum, subject to the secondary condition known as (Pauli s) exclusion principle. 

Although wave equations are readily composed for more-electron atoms, they are impossible to solve in closed form. Approximate solutions for many-electron atoms are all based on the assumption that the same set of hydrogen-atom quantum numbers regulates their electronic configurations, subject to the effects of interelectronic repulsions. The wave functions are likewise assumed to be hydrogen-like, but modified by the increased nuclear charge. The method of solution is known as the self-consistent-field procedure. 

There is interaction among all the electrons in a many-electron atom. Thus, the wave function for even one electron in a many-electron system will, in principle, be different from the wave function for the one electron in the hydrogen atom. Since the electrons are mutually indistinguishable, it is not possible to describe rigorously the properties of a single electron in such a system. There is no exact solution to this problem, and approximate methods must be adopted. 

Until this point, the consideration of electron-electron repulsion terms has been neglected in the molecular Hamiltonian. Of course, an accurate molecular Hamiltonian must account for these forces, even though an explicit term of this type renders exact solution of the Schrddinger equation impossible. The way around this obstacle is the same Hartree-Fock technique that is used for the solution of the Schrddinger equation in many-electron atoms. A Hamiltonian is constructed in which an effective potential of the other electrons substitutes for a true electron-electron reg sion term. The new operator is called the Lock operator, F. The orbital approximation is still used so that F can be separated into i (the total number of electrons) one-electron operators, Fi (19). 

While a great deal of progress has proved possible for the case of the hydrogen atom by direct solution of the Schrodinger wave equation, some of which will be summarized below, at the time of writing the treatment of many-electron atoms necessitates a simpler approach. This is afforded by the semi-classical Thomas-Fermi theory [4-6], the first explicit form of what today is termed density functional theory [7,8]. We shall summarize below the work of Hill et al. [9], who solved the Thomas-Fermi (TF) equation for heavy positive ions in the limit of extremely strong magnetic fields. This will lead naturally into the formulation of relativistic Thomas-Fermi (TF) theory [10] and to a discussion of the role of the virial in this approximate theory [11]. 

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). 

The system of equations (1.8) is based on the central field approximation, and therefore its application to real atoms is entirely dependent on the existence of closed shells, which restore spherical symmetry in each successive row of the periodic table. For spherically symmetric atoms with closed shells, the Hartree-Fock equations do not involve neglecting noncentral electrostatic interactions and are therefore said to apply exactly. This does not mean that they are expected to yield exact values for the experimental energies, but merely that they will apply better than for atoms which are not centrally symmetric. One should bear in mind that, in any real atom, there are many excited configurations, which mix in even with the ground state and which are not spherically symmetric. Even if one could include all of them in a Hartree-Fock multiconfigura-tional calculation, they would not be exactly represented. Consequently, there is no such thing as an exact solution for any many-electron atom, even under the most favourable assumptions of spherical symmetry. 

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. 

Like the Bohr model, the Schrodinger equation does not give exact solutions for many-electron atoms. However, unlike the Bohr model, the Schrodinger equation gives very good approximate solutions. These solutions show that the atomic orbitals of many-electron atoms resemble those of the H atom, which means we can use the same quantum numbers that we used for the H atom to describe the orbitals of other atoms. 

For many-electron atoms, no exact solutions to the corresponding Schrodinger equation exist because of the electron-electron repulsions. However, various approximations can be used to locate the electrons in these atoms. The common procedure for predicting where electrons are located in larger atoms is the Aufbau (building up) principle. 

This chapter revises basic atomic orbital theory. The chapter begins with the exact results for the case of the hydrogen atom and the orbital concept for many-electron atoms. It is very important to understand these details about atomic orbitals, since the orbital concept is essential in the approximation to chemical bonding known as the Linear Combination of Atomic Orbitals — Molecular Orbital [LCAO-MO] theory. Only in the case of the hydrogen atom are these atomic orbitals, as exact solutions to Schrddinger s equation, available as functions. 

The Schrddinger equation for the one-electron atom (Chapter 6) is exactly solvable. However, because of the interelectronic repulsion terms in the Hamiltonian, the Schrbdinger equation for many-electron atoms and molecules is not separable in any coordinate system and cannot be solved exactly. Hence we must seek approximate methods of solution. The two main approximation methods, the variation method and perturbation theory, will be presented in Chapters 8 and 9. To derive these methods, we must develop further the theory of quantum mechanics, which is what is done in this chapter. 

In this chapter we apply dimensional scaling techniques to the problem of electronic structure in many-electron atoms. As usual in the dimensional scaling approach, the motivating idea is to generalize the problem to spaces of arbitrary dimensionality Z , treat it at one or more values of D where it s particxilarly easy to do so, and finally relate the results obtained back to jD = 3. The D - oo limit turns out to be the easiest place to treat the many-electron atom. In fact, one can obtain [1] analytic solutions for this limit, as well as for the first-order corrections at finite D. With some work these results can be used to calculate approximate solutions at D = Z. However, the raw jD— oo solutions do not correspond very well with our common notions of what an atom looks like. 

In Chapter 7 we saw that it is not possible to obtain exact solutions of the Schrodinger equation for many-electron atoms, even within the one-electron approximation, and the same applies to molecules. For these systems it is necessary to use approximate solutions, which are based on chemical insight and chosen for mathematical convenience. 

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 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,... 

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