Diagrams hole line

In the Brueckner-Hartree-Fock (BHF) approximation, the Brueckner-Bethe-Goldstone (BBG) hole-line expansion is truncated at the two-hole-line level [5]. The short-range NN repulsion is treated by a resummation of the particle-particle ladder diagrams into a n effect vc i n tcract ion or G-matrix. Self-consistency is required at the level of the BHF single-particle spectrum eBHF(k),... 

The sign of the total expression corresponding to the pertinent Goldstone diagram is given by (—l),+ i where l is the number of closed loops and h is the number of hole lines. 

Figure 1 Diagram elements (a) one-electron operator, (b) two-electron operator, (c) particle line, (d) hole line...

Figure 1 Some basic components of coupled cluster diagrams (a) hole lines ...

Label all directed lines with appropriate indices. By the convention we have used so far, hole lines would be labeled with i, j, k, I,. .. and particle lines with a, b, c, d,. . . . Therefore, for the diagram abovewe label the... 

Therefore, the two energy diagrams are equivalent, because the two hole lines and the two particle lines from the T2 diagram both connect to the same diagram fragment ... 

Unlike the diagram in Eq. [165], this diagram contains a pair of equivalent vertices since both Tj fragments are connected to the same interaction line in exactly the same manner (each by a hole line and a particle line), a prefactor of Vi is multiplied into the final expression. Generally speaking, if there are n equivalent vertices in the diagram, they contribute a prefactor of Hn to the final expression. 

However, the first two of these fragments can connect to the T diagram in only one index—via either a single hole line or particle line—thus leaving an additional line extending below the Tj interaction line in the final diagram, for example,... 

If simultaneous interchange of the particle lines and the hole lines yields a topologically equivalent diagram and the hole and particle lines are not independently interchangeable, cPh = 2. 

The basic elements of the Brandow diagrams are shown in Figure 3. These are a one-electron matrix element, a two-electron matrix element, a particle line represented by a line with an upward directed arrow and a hole line represented by a line with a downward directed arrow. Particle lines represent the particle lines created above the Fermi level when an electron is excited whilst hole lines represent the hole which are simultaneously created below the Fermi level. A time-dependent physical interpretation of the diagram may be given. An example of such an interpretation is given in Figure 4. 

Figure 3 Basic elements of the diagrams introduced by Brandow (a) particle line (b) hole line (c) one-electron interaction (d) two-electron interaction...

There is a summation of the type for each part of the diagram lying between adjacent interaction lines. The first summation extends over all hole lines... 

Multiply each numerator by (—1), where h is the number of hole lines in the diagram and / the number of closed loops. A closed loop is formed when one can trace from one endpoint of an interaction along the direction of an arrow and end up back at the same point without ever having to cross an interaction (dashed) line (Fig. 3.3A contains three loops. Fig. 3.2C contains two loops, and Fig. 3.2B has one loop). 

Diagram A contains three hole lines and three closed loops. Diagram E may, in a similar way, be expressed as... 

Restricting f to only some simple fermion-like products of electron field operators will generate only certain types of diagrams that can then be summed to all orders. For instance, self-energy diagrams in third order of the ring and ladder types, which can easily be generalized in any order. It is notable that between consecutive interaction lines, there only occur one hole line and two particle lines or vice versa [see Fig. 9.1]. 

We resolve this by defining circular lines to be hole lines. It is easy to show that the diagrams we can generate in this way are... 

These expressions are identical to our previous results for the second- and third-order energies (Eqs. (6.12) and (6.15)) when i = 1. What if we want the perturbation expansion for some state i, which is not necessarily the lowest What do the diagrams look like One can easily verify that we get the same answers as before if we label our hole lines by the index i and the particle lines by the indices m, n, k,..., which can take on the values 1,2,..., i - 1, I + 1,..., N. Thus we now have a complete diagrammatic representation of RS perturbation theory, which is applicable to any perturbation and any zeroth-order state. 

---