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Conditions to use different Schrodinger Equation forms


Every molecular system has an energy associated with it. This energy might show a shift depending on the surrounding environment or internal variations. Magnetic dipoles present in a molecular system in the form of electrons and the nuclei is one such reason for the energy shift. But the effect of these magnetic dipoles on the energy shifts will be much less which can be accounted by considering perturbation theory. Chemical binding energies on the other hand are responsible for the major shifts in energy of a molecular system. Thus the focus is shifted towards the valence electrons responsible for most of the chemical activity of a molecule. The valence electrons are separated from the core electrons. The core electrons are treated approximately. This reduces the efforts of considering the relativistic effects affecting the core electrons, and thus can now be ignored.

What are relativistic effect?
It has been nicely explained by Ermler and Pitzer in the introduction of their review on Relativisitic effects on chemical systems in Annual Review of Physical Chemistry (1985). As explained by them, velocity of light c is finite. But theoretical calculations often make the approximation of considering it to be ∞. This approximation works for light atoms like H,C,N,O. But for heavy elements, this non relativistic approximation of considering c as ∞ doesn't hold. Thus the relativistic effects are the differences between calculation for the correct value of the velocity of light and the results for c= ∞. This can be taken care of for c=∞ by working with Dirac equation yielding electron spin, or by supplementing the Schrodinger equation with an ad hoc assumption about spin.

The non relativistic Schrondinger equation without magnetic effects can then be used to obtain the structure and reactivity of molecular systems and systems in Solid state.

While the time independent Schrodinger equation is sufficient for the structure characterization of a molecular system, its interaction with electromagnetic radiation requires time dependent Schrodinger equation.

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