Electron density atomic subspace

Perhaps the most rigorous way of dividing a molecular volume into atomic subspaces is the Atoms In Molecules (AIM) method of Bader.The electron density is the square of the wave function integrated over N — coordinates (it does not matter which coordinates since all electrons are identical). 

The initial geometry for formamide was set to the enol form as in Fig. 7.10. The atoms O, C, and N make a molecular plane, and the bridging water molecule is also placed initially so as to lie in that plane. In what follows, we refer to the molecular orbitals approximately lying on the plane and to those approximately perpendicular to the plane as a and tt orbitals, respectively. Likewise, using only tt orbitals in 7(r,f) and Bab (r.f)) we estimate the tt electron density and tt bond-order, respectively. Similarly, the a electron density and a bond-order are made available. This distinction between the a and tt subspaces is just a matter of convenience, and of course they are not physical observables individually, since Cs symmetry is not imposed on the molecular system. On the contrary, all the vibrational modes are active in the present SET calculations. Since the aim of this study is not to estimate the reaction probability but the mechanism of the electron dynamics associated with proton transfer, we chose somewhat artificial initial conditions of nuclear motion to sample as many paths achieving proton transfer as possible. 

This leads to a very ambitious idea if you can extend the Hohenberg-Kohn theorem to an atom located in two different molecular systems but with equal electronic densities on two arbitrary quantum subdomains, Pjjj(x) = P2n( ), one can conclude that the electron densities of the atom on the quantum space of the two molecules are also equal, Pj(x) = P2(x), and the density functionals E(Q) can be properly extended to the kinetie energy average for an atom, from an arbitrary molecular subspace to another. 

Fig. 1 A collection of gradient paths forming web-like patterns, which each span the subspace of each of the three topological atoms in HC=N. Each atomic basin consists of an infinite multitude of gradient paths, here represented by a few dozen paths originating at infinity and terminating at the respective nuclei. Electron density contour plots are superimposed onto the field of gradient paths. The values of the contour lines start with 1 x 10 and continue with the sequence 2 x 10", 4 X 10"and 8 X 10"a.u., where n starts at —3 and increases with unity increments...

A < 1. Some of these coefficients can be close to 1 (strongly occupied NSO) and some close to 0 (weakly occupied NSO), but in any case the post-HF density matrix loses its property = p. Turning to the basis of natural orbitals (NOs) results in the appearance of two spin components of the density matrix, p° and p. For RHF, ROHF, and UHF cases, these spin components are idempotent, being projectors on the subspace of the full MO space spanned by the occupied density operator is the Hermitian positive-semidefinite operator with a spur equal to the number of electrons. At the same time, in general, this operator does not possess any other specific properties such as, for example, idempontency. After the convolution over the spin variables, the density operator breaks down into two components whose matrix representation in the basis set of atomic orbitals (AOs) has the form... 

---