The random Integration method (RIM)
The RIM is a numerical accurate method to treat the divergence of the Coulomb interaction in integrated quantities (like the exchange self-energy, the Bethe-Salpeter kernel, the Total energy expression).Motivation
Consider, for example, the exchange self-energy introduced in XX
integrated in the Brillouin Zone (BZ) of the reciprocal lattice (RL). It can be rewritten, in a more general form, as
The same general expression applies to the calculation of the correlation self-energy ( GW ) and of the Bethe Salpeter Kernel ( BSK ). The integral over the BZ is usually numerically evaluated as:
Here we have assumed the integrand to be a smooth function of the momenta and so it can be considered constant in each region of the BZ centered around each q point. While this is a reasonable assumption for the wave-function derived quantities (response functions, oscillators...) it may not be true for the Coulomb integral. In the Random-Integration-Method (RIM) the exchange self-energy is rewritten without approximating the Coulomb potential
in this way
and the assumption of smoothness is now restricted only to the oscillators and occupation numbers. The small Brillouin Zones (sBZ) relative to a given q-point are the Brillouin Zones of the momenta vectors lattice. They are chosen in such a way to cover the whole BZ. In the RIM run-level yambo calculates the integrals of the symmetrized Coulomb potential
using a simple Montecarlo technique via a set of RandQpts random points generated in the given sBZ. The integrals are evaluated up to RandGvec RL vectors. If the QpgFull flag is not activated only the diagonal elements (in G,G') are calculated.
The Numerical Integration
Given the sBZ region the Montecarlo technique used by yambo defines a box large enough to contain the sBZ region. Then an iterative procedure generates M random points in the box until N= RandQpts are found in the sBZ. At this point the integral is easily evaluated as
Example:
The RIM procedure become essential when dealing with reduced periodicity systems (e.g quasi 1D / 2D systems: polymers, atomic chains, sheets). For such systems, it is often not necessary to have 3d sampling of k-points in the BZ, but is enough to sample in the direction where the system is periodic. In these case caution has to be taken when considering the Coulomb integrals (e.g.in the exchange self-energy), because the integrals are converging in three dimensions but are not in lower dimensions. Here we report the value of the Exchange Self Energy in eV for the HOMO band at the X point for a chain of H2 molecules with an intermolecular distance of 2.5 a.u. and an intermolecular distance of 2 a.u. varying the number of the k points along the chain axis (taken along the Gamma-X direction):# k points without RIM with RIM 11 -15.33527 -12.82946 21 -18.88934 -13.93515 31 -22.77704 -13.99632 41 -26.74025 -14.01759 61 -34.77954 -14.03363 81 -42.88489 -14.03688The calculation with the RIM is performed using RandQpts = 1000000 and RandGvec = 1.
We can see that if the RIM procedure it is not adopted the exchange Self Energy value does not converge, and raising the number of k-points it tends to explode, because the q-points approach the singularity. We can see how in this case the RIM remove this problem, the coulomb integrals are evaluated correctly in three dimension with the Monte Carlo Method and the exchange Self Energy converges with respect with the k-point sampling in one dimension.
References
- Olivia Pulci, Giovanni Onida, Rodolfo Del Sole, and Lucia Reining, Phys. Rev. Lett. 81, 5374 (1998)
- Carlo A. Rozzi, Daniele Varsano, Andrea Marini, Eberhard K. U. Gross, Angel Rubio Phys. Rev. B 73, 205119 (2006)