Comparison of Variation and Perturbation Theories

Researchers in quantum mechanical calculations should understand the limitations and strengths of each method. Typically, the method used is the one that provides the information a particular researcher wants about a real system. If a well-defined Hamiltonian and wavefunction are desired, then perturbation theory provides that. If the absolute energy is important, variation theory provides a way to get better and better results. The calculational cost is also a factor. Those with access to supercomputers can work with a lot of equations in a relatively short time. Those without may find themselves limited to a small number of corrections to ideal wavefunctions. 

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. 

Unless otherwise noted, all art on this page is Cengage Learning 2014. 

Because most chemical systems are molecules, it is important to understand how quantum mechanics is applied to molecules. When we use the word molecule, we are usually speaking of some chemically bonded system that exists as discrete collections of atoms bonded to each other in some specific way. This contrasts with ionic compounds, which are atoms (or groups of covalently bonded atoms, the so-called polyatomic ions) held together by their opposing charge that is, they are composed of cations and anions. As one might expect from the previous discussions about wavefimctions in multi-electron atoms, wavefimctions of molecules get even more complicated. In reality there are some useful simplifications, which we will get to in the next chapter, but a general consideration of a simple diatomic molecule is useful at this point 

FIGURE 12.12 Max Bom (1882-1970). Not only did he develop the probabilistic interpretation of the wavefunction, but he also devised a quantum-mechanical description for molecules. 

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