Intracule functional models: Part III. The dot intracule and its Fourier transform

The dot intracule D(x) of a system gives the Wigner quasi-probability of finding two of its electrons with u·v = x, where u and v are the interelectronic distance vectors in position and momentum space, respectively. In this paper, we discuss D(x) and show that its Fourier transform d(k) can be obtained in closed form for any system whose wavefunction is expanded in a Gaussian basis set. We then invoke Parseval's theorem to transform our intracule-based correlation energy method into a d(k)-based model that requires, at most, a one-dimensional quadrature.