Hypersurface of the potential energy for nuclear motion

Theoretical chemistry is still in a stage which experts in the field characterized as the primitive beginnings of chemical ab initio dynamics . The majority of the systems studied so far are three-atomic stems.  

The Born-Oppenheimer approximation works wonders, as it is possible to consider the (classical or quantum) dynamics of the nuclei, whfle the electrons disappear from the scene (their role became, after determining the potential energy for the motion of the nuclei, described in the electronic energy, the quantiqr corresponding to B (R) from eq. (6.8) on p. 225). 

It is quite easy to see where the fundamental difficult is. Each of the hypersurfaces R) for the motion of 2 nuclei depends on - 6 atomic coordinates 

The number of points on the hypersurface which have to be calculated is extremely large and increases exponentially with the system size.  

There is no general methodology telling us what to do with the calculated points. There is a consensus that we should approximate the hypersurface by a smooth analytical function, but no general solution has yet been offered.  

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In the last section, we introduced methods which can be employed to calculate the total electronic energy Ea for a given molecule with fixed nuclei. The stepwise calculation of electronic energies for various nuclear configurations allows us to scan essential parts of the potential energy hypersurface (the so-called Born-Oppenheimer surface) on which nuclear motions and chemical reactions... 

In the first part (after using the Born-Oppenheimer approximation, fundamental to this chapter), we assume that we have calculated the ground-state electronic energy (i.e., the potential energy for the nuclear motion). It will turn out that the hypersurface has a characteristic drainpipe shape, and the bottom in the central section, in many cases, exhibits a barrier. Taking a three-atom example, we will show how the problem could be solved, if we were capable of calculating the quantum dynamics of the system accurately. 

This expression constitutes the basis of current interpretations of electron transfer processes in biological systems. From Eq. (9), the functions Hg, (Q) and Hbb (Q) represent potential energy surfaces for the nuclear motion described by Xav and Xbw respectively, if the weak diagonal corrections Taa and T b are neglected. Then, the region Q Q where Xav and Xbw overlap significantly corresponds to the minimum of the intersection hypersurface between Hga (Q) and Hbb (Q)- Referring to definition (5), this implies ... 

The problem of the shape of V (R), as well as of the nuclear motion on the V (R) potential energy hypersurface, will be the subject of this chapter. It will be seen that electronic energy can be computed with high accuracy as a function of R only for very simple systems (such as an atom plus a diatomic molecule system), for which quite a lot of detailed information can be obtained. 

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