Correlation cusp condition

The explicitly correlated wave function (we will get to it in a moment) has the interelectronic distance built in its mathematical form. We may compare this to making the electrons wear spectacles. Now th avoid each other. One of my students said that it would be the best if the electrons moved apart to infinity. Well, they cannot. They are attracted by the nucleus (energy gain), and being close to it, are necessarily close to each other too (energy loss). There is a compromise to achieve. 

Let us consider two particles with charges qt and qj and masses mi and rrij separated from other particles. This, of course, makes sense since simultaneous collisions of three or more particles occur very rarely in comparison with two-particle collisions. Let us introduce a Cartesian system of coordinates (say, in the middle of the beautiful market square in Brussels), so that the system of two particles is described with six coordinates. Then (atomic units are used) the sum of the kinetic energy operators of the particles is 

Now we separate the motion of the centre of mass of the two particles with position vectors ri and rj. The centre of mass in our coordinate system is indicated by the vector Rcu = Xcm, cm -Zcm) 

Let us also introduce the total mass of the system M = mimj, the reduced mass of the two particles p, = and the vector of their relative positions r = Ti— rj. Introducing the three coordinates of the centre of mass measured with respect to the market square in Brussels and the three coordinates x, y, z which are components of the vector r, we get (Appendix I, Example 1) 

After this operation, the Schrddinger equation for the stem is separated (as always in the case of two particles, see Appendix I, p. 971) into two equations the first describing the motion of the centre of mass (seen from Brussels) and the second describing the relative motion of the two particles (with Laplacian of x, y, z and reduced mass p). We are not interested in the first equation, but the second one is what we are after. Let us write down the Hamiltonian corresponding to the second equation 

---

For He 12-figure accuracy was reported [36, 16]. This is surprising since this ansatz neither fulfills the nuclear cusp nor the correlation cusp conditions. Although it is not yet fully understood why this works, some preliminary comments can be made. 

It has been shown [29,21] that, as a consequence of the correlation cusp condition [30] in the coalescence region (ri = r ), is approximately known and takes the form... 

The Gaussian geminals do not satisfy the correlation cusp condition (p. 587). because of factor exp —brfj). It is required (for simplicity, we write = r) that... 

The Coulomb hole that wavefunctions are predicted to have for close anti-parallel-spin electrons is also called a correlation hole. As a condition for wavefunctions containing correlation holes, Kato proposed a correlation cusp condition (Kato 1957),... 

The Hartree-Fock wavefunction violates this condition, because it gives zero for the left-hand side of this equation. As shown in Fig. 3.1, a wavefunction satisfying this condition contains a correlation hole, which contains a sharp dip, called a cusp, near ri2 = 0. This correlation hole causes anti-parallel-spin electrons to be further apart, and therefore reduces Coulomb interactions, thus lowering the total electronic energies. Sinanoglu named this electron correlation in the correlation cusp condition as dynamical correlation (Sinanoglu 1964). 

The correlation cusp condition shows that the wave function is not differentiable at r = 0. 

An exact wave function satisfies the correlation cusp condition, ( )r=o = =... 

---