The Brückner G-matrix

The $G$ -matrix is our secret weapon for the nuclear many-body problem. In essence, it provides an in-medium strength of interaction given a bare nucleon-nucleon potential.
The bare nucleon-nucleon force has certain features that are difficult to handle in practice, like a hard core that makes mean field theory methods inapplicable. This bare force can be replaced by an effective interaction, which is an infinite resummation of scattering processes of two nucleons in the nuclear medium. This gets rid of the hard core problem and sums up many-body effects at the same time!

These microscopic effective interactions can be applied to the interactions between nucleons within a medium, but also to the interactions between valence nucleons and ph interactions.
(And there are also effective three body forces?)

We start with the Lippmann-Schwinger equation for the scattering of two particles in free space:

T_{k_{1} k_{2}, {k^{'}}_{1} {k^{'}}_{2}}^{E} = {\bar{v}}_{k_{1} k_{2}, {k^{'}}_{1} {k^{'}}_{2}} + \frac{1}{2} \sum_{p_{1} p_{2}} {\bar{v}}_{k_{1} k_{2}, p_{1} p_{2}} \frac{1}{E - (p_{1}^{2} / 2 m) - (p_{2}^{2} / 2 m) + i η} T_{p_{1} p_{2}, {k^{'}}_{1} {k^{'}}_{2}}^{E},

where $k_{1}, k_{2}, {k^{'}}_{1}, {k^{'}}_{2}$ are the momenta of the incoming and outgoing particles. It makes perfect sense (😂) to define an analogous scattering matrix within the nuclear medium. The plane wave indices for the initial and final states will be replaced with shell model indices, and the kinetic single particle energies by the corresponding shell model energies. Finally, the sum over intermediate states is restricted to levels above the Fermi surface (because two nucleons below the Fermi surface can only scatter into states above it!).

Question

What happens when we use quasiparticles and the Fermi surface is "blurred"?

So here it goes:

G_{a b, c d}^{E} = {\bar{v}}_{a b, c d} + \frac{1}{2} \sum_{m n > ϵ_{F}} {\bar{v}}_{a b, m n} \frac{1}{E - ϵ_{m} - ϵ_{n} + i η} G_{m n, c d}^{E} .

This is the Bethe-Goldstone equation. Graphical representation (will remind you of the good old days of QFT):
Pasted image 20221221102332.png

We can also write it in operator form

G = \bar{v} + \bar{v} \frac{Q_{F}}{E - H_{0}} G,

where $H_{0}$ is the shell model hamiltonian and $Q_{F} = \sum_{(m < n) > ϵ_{F}} | m n ⟩ ⟨ m n |$ a projector excluding occupied states.

Again from analogy with free particle scattering, we can write for the scattered WF:

| ψ_{a b} ⟩ = {\bar{v}}^{- 1} G | a b ⟩ = (1 + \frac{Q_{F}}{E - H_{0}}) | a b ⟩ = | a b ⟩ + \frac{Q_{F}}{E - H_{0}} \bar{v} | ψ_{a b} ⟩

Application to Hartree-Fock

Using microscopic effective forces for Hartree-Fock is straightforward in principle: you just have to swap the antisymmetrized potential ${\bar{v}}_{a b, c d}$ by the $G$ -matrix elements $G_{a b, c d}^{E}$ . It's not so simple in practice, however, as you have to enter a doubly self-consistent procedure:

Calculate G-matrix in a basis of first choice.
Diagonalize once (i.e., solve the HF equations), given a new basis.
Calculate a new G-matrix in this basis, and so on.

Furthermore, there is a certain ambiguity concerning the energy dependence of the $G$ -matrix within the HF equations.

Expand this!

This procedure can be simplified by using the Local density approximation, where $G$ ends up depending only on two variables $r$ and $r^{'}$ .