Quantum mechanics multielectron systems

Quantum mechanics describes molecules in terms of interactions between nuclei and electrons and molecular geometry in terms of minimum energy arrangements of nuclei. All quantum-mechanical methods ultimately trace back to Schrodinger s (time-independent) equation, which may be solved exactly for the hydrogen atom. For a multinuclear and multielectron system, the Schrodinger equation may be defined as ... 

The model that is outlined above is generated from a one-electron Hamiltonian and is only an approximation to the tme wavefimction for a multielectron system. As suggested earlier, other components may be added as a linear combination to the wavefimction that has just been derived. There are many techniques used to alter the original trial wavefimction. One of these is frequently used to improve wavefimctions for many types of quantum mechanical systems. Typically a small amount of an excited-state wavefimction is included with the minimal basis trial fimction. This process is called configuration interaction (Cl) because the new trial function is a combination of two molecular electron configurations. For example, in the H2+ system a new trial fimction can take the form... 

The basis of computational quantum mechanics is the equation posed by Erwin Schrbdinger in 1925 that bears his name. Solving this equation for multielectron systems remains as the central problem of computational quantum mechanics. The difficulty is that because of the interactions, the wave function of each electron in a molecule is affected by, and coupled to, the wave functions of all other electrons, requiring a computationally intense self-consistent iterative calculation. As computational equipment and methods have improved, quantum chemical calculations have become more accurate, and the molecules to which they have been applied more complex, now even including proteins and other biomolecules. 

In the present context, this example was intended to serve as a reminder of how one formulates a simple model for the quantum mechanics of electrons in metals and, also, how the Pauli principle leads to an explicit algorithm for the filling up of these energy levels in the case of multielectron systems. In addition, we have seen how this model allows for the explicit determination (in a model sense) of the cohesive energy and bulk modulus of metals. 

At the most fundamental level chemical phenomena are determined by the behaviors of valence electrons, which in turn are governed by the laws of quantum mechanics. Thus, a first principles or ab initio approach to chemistry would require solving Schrodinger s equation for the chemical system under study. Unfortunately, Schrodinger s equation cannot be solved exactly for molecules or multielectron atoms, so it became necessary to develop a variety of mathematical methods that made approximate computer solutions of the equation possible. 

An important thing to understand about both of these theories is that when properly applied, they can be used to understand any atomic or molecular system. By using more and more terms in a perturbation-theory treatment or more and better trial functions in variation-theory treatments, one can do approximation calculations that yield virtually exact results. So even though the Schrodinger equation cannot be solved analytically for multielectron systems, it can be solved numerically using these techniques. The lack of analytic solutions does not mean that quantum mechanics is wrong or incorrect or incomplete it just means that analytic solutions are not available. Quantum mechanics does provide tools for understanding any atomic or molecular system and so it replaces classical mechanics as a way to properly describe electron behavior. 

The above example and selection rules are also applicable to hydrogen-like ions, which have a single electron. However, such systems are in the vast minority of atomic species whose spectra need to be understood. Recall that one of the final failings of classical mechanics was the inability to explain spectra. Although quantum mechanics does not provide analytic solutions for wavefunctions of multielectron systems, it does provide tools for understanding it. 

The quantum mechanical description of a multielectron system is in principle the same than that of hydrogen atom. However the complexity of differential equations arising from building a Hamilton operator appropriate for multielectron systems prevents us from obtaining exact solutions. Nonetheless approximated methods, based fundamentally on reformulations of the problem... 

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