Schrodinger equation justification

Some theoretical purists tend to view molecular mechanics calculations as merely a collection of empirical equations or as an interpolative recipe that has very little theoretical Justification. It should be understood, however, that molecular mechanics is not an ad hoc approach. As previously described, the Born-Oppenheimer approximation allows the division of the Schrodinger equation into electronic and nuclear parts, which allows one to study the motions of electrons and nuclei independently. From the molecular mechanics perspective, the positions of the nuclei are solved explicitly via Eq. (2). Whereas in quantum mechanics one solves, which describes the electronic behavior, in molecular mechanics one explicitly focuses on the various atomic interactions. The electronic system is implicitly taken into account through judicious parametrization of the carefully selected potential energy functions. 

The Schrodinger equation is a bit of a mystery the first few times it is encountered. And with good reason. It is more or less given as something worth studying without any real justification, except perhaps, some statement such as it is valuable because it works. The Schrodinger equation is... 

If zero-point vibration amplitudes of the dot are comparable with the Fermi length of the electrons, the shuttling takes place at small bias voltage. This is the case for cold dots. The constructive interference of electron waves in the tunnel gap center effectively charges the dot. In the quantum limit, this charging requires a justification of the tunnel-term concept based on the Schrodinger equation. In next section we address a more rigorous quantum mechanical picture based on the "ab-initio" SET model. 

The Born-Oppenheimer principle assumes separation of nuclear and electronic motions in a molecule. The justification in this approximation is that motion of the light electrons is much faster than that of the heavier nuclei, so that electronic and nuclear motions are separable. A formal definition of the Born-Oppenheimer principle can be made by considering the time-independent Schrodinger equation of a molecule, which is of the form... 

The expansion (10.42) finite (justification can be only diagrammatic, and is not given here) since in the Hamiltonian H we have only two-particle interactions. Substituting this into the Schrodinger equation we have ... 

The variation method has an appealingly intuitive justification, namely that the forces at work on the electrons can always push them into the lowest-energy distribution possible (see Fig. 4.11). All we have to do is try to find that minimum energy. A general approach to solving a Schrodinger equation variationally works like this ... 

The ability to trace a theory to its mathematical foundations can be used to distinguish an ab initio or first-principles theory from other types. In the context of alloy theory, for example, a theory remains first-principles as long as any approximations made as one proceeds from a formally exact expression of the Schrodinger equation are well understood in mathematical terms. Alternatively, the introduction of parameters (obtained through fits to experimental data or the consideration of certain limits in the mathematical expressions of the theory) in designing a model system break the smooth running of the mathematics. Theories so constructed may indeed provide a viable phenomenological description of natural phenomena but they cannot claim a priori justification of their content. 

Separation of Electronic and Nuclear Motion. Because, in general, electrons move with much greater velocities than nuclei, to a first approximation electron and nuclear motions can be separated (Born-Oppenheimer theorem [3]). The validity of this separation of electronic and nuclear motions provides the only real justification for the idea of a potential-energy curve of a molecule. The eigenfunction Y for the entire system of nuclei and electrons can be expressed as a product of two functions F< and T , where is an eigenfunction of the electronic coordinates found by solving Schrodinger s equation with the assumption that the nuclei are held fixed in space and Yn involves only the coordinates of the nuclei [4]. 

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