Quantum mechanics applied to even the smallest polyatomic systems presents some problems. For instance, although molecules are presumably built up from individual atoms and molecular orbitals are built up (in some energy-minimizing way) from atomic orbitals, ideal MOs are typically delocalized over multiple atoms in a molecule, causing the atoms to strangely lose their identity in coming together to form the molecule in question. Faced with this situation, one could legitimately ask whether any aspect of the atom is transferable between different molecular systems or each molecular system is unique in itself. Does the atom retain its identity once part of a molecule, and if so, how?
Richard Bader’s “Quantum Theory of Atoms in Molecules,” or QTAIM, attempts to address this question and other quantum chemical questions that traditional QT has neither asked nor answered. Bader stresses that transferability of properties between the same functionality in two different molecules is essential for chemistry, and that quantum mechanics supports such transferability.
The cornerstone of QTAIM is its focus on the molecular electron density p (pretend that’s the letter rho for now). It can be derived using the total molecular wavefunction, and as such is 100% consistent with quantum mechanics. Apparently, Feynman came up with a theory that states that although p is derived quantum mechanically, it can be treated with classical methods. Take the ethylene molecule, for example. Its electron density in the molecular plane is plotted at right. Notice that the positions of the nuclei correspond to maxima in the electron density. Is there some non-arbitrary way to divide this electron density up between the individual atoms of the molecule?
The gradient of p(r) gives the vector in the direction of maximum change of p, and by tracing lines tangent to the gradient vectors, we get a plot like the one shown. The nuclei act as “attractors” for the electron density–all the gradient lines converge on one of the nuclei. The set of trajectories that terminate at a particular nucleus are called that nucleus’s “basin.” In QTAIM, an atom is simply the union of a nucleus (or more generally, an attractor) and its basin. The figure below shows the six basins of the ethylene molecule and the “basin-crossing” trajectories (in bold) that originate and terminate at two different attractors.
The trajectories between two attractors are, not surprisingly, chemical bonds according to QTAIM. The curved lines shown separate the different atomic basins of the molecule, and the points shown are called “bond critical points,” the location along a bond at which the two atomic basins involved meet. The trajectories are not always straight, particularly for very strained molecules. What you can count on is that the properties of these topological atoms are both transferable among and additive within molecular systems. Analyzing the electron density gets to the heart of why atomic or functional-group properties are transferable between molecules!
But wait there’s more! At the extrema of the electron density, the first derivative (gradient) is zero. The second derivative (the Laplacian) is negative for a maximum and positive for a minimum of charge density. At local charge concentrations, then, the negative Laplacian is more or less a maximum. Graphs of the negative Laplacian of the electron density match up very well with areas of charge concentration according to the VSEPR model. In fact, the electron density can be used to determine or calculate an amazingly large number of molecular properties (including reactivity) using only ab initio quantum methods.
For more on Bader’s fascinating theory, check out his website.