Excitation diagram

Because of the enormous amount of integration required, computation costs on conventional computers would have been prohibitive and thus, all of our computations were performed on the Cray 1 and Cray 2 supercomputers at the University of Minnesota. The results are shown in figure 2 where the forcing frequency w has been scaled by the system s natural frequency co0 

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Fig. 13.14. The excitation diagram for the Takoudis-Schmidt-Aris model showing resonance horns (Arnol d tongues) emerging from integer quotients of forcing and natural frequencies. For details of the behaviour in the closed broken curve see Fig. 13.16. (Reproduced with permission from McKarnin, M. A. et al. (1988). Proc. R. Soc., A417, 363-88.)...

Several codimension-two bifurcations have already been mentioned. Although they occur in restricted subspaces of parameter space and would therefore be difficult to locate experimentally, their usefulness lies in their role as centres for critical behaviour. Emanating from each local codimen-sion-two point will be two or more of the above codimension-one bifurcation curves. Their usefulness in studying dynamics is akin to that of the triple point in thermodynamic phase equilibria in which boundaries between three different phases come together at a point in a two-parameter diagram. Because some of these codimension-two points have been studied and classified analytically, finding one can provide clues about what other codimension-one bifurcation curves to expect near by and thus aids in the continuation of all of the bifurcation curves in the excitation diagram. 

The excitation diagram was found to contain saddle-node, Hopf, period doubling, and homoclinic bifurcations for the stroboscopic map. In addition, many of these co-dimension one bifurcation curves were found to meet at the following co-dimension two bifurcation points Bogdanov points (double +1 multipliers), points with double -1 multipliers, points with multipliers at li and H, metacritical period-doubling points, and saddle-node cusp points. 

Figure 9 Connected and disconnected wave function diagrams (a) second-order connected triple-excitation diagram, (b) second-order disconnected quadruple-excitation diagram, (c) third-order disconnected triple-excitation diagram, (d) third-order connected quadruple-excitation diagram...

The triple-excitation fourth-order energy, in contrast to the quadruple-excitation component, arises from connected wave function diagrams. The algorithm required to evaluate this energy component is considerably less tractable than that for the quadruple-excitation energy, depending on 7, where n is the number of basis functions. The triple-excitation diagrams can be written in terms of the intermediates. 

In the coupled pair approximation,160 all diagrams which can be formed by considering products of disconnected double-excitations are summed to infinite order. Thus in fourth-order, the coupled pair approximation includes the linked double-excitation and linked quadruple-excitation diagrams shown in Figure 6 and Figure 8, respectively. The coupled pair approximation may be represented diagrammatically as follows ... 

Another way of introducing cluster operators is to define the operator 7) to sum only connected /-fold excitation diagrams in P mbpt, and by virtue of defining Cl = exp(T) the disconnected but linked mbpt diagrams are summed as the quadratic and higher terms in the exp(T) expansion. This is the essential relationship of MBPT to coupled-cluster theory. 

The final rule pertains to the number of distinct permutations that are possible among the open lines. For any single excitation diagram there can only be one. However, for double excitations and higher, there is normally more than one distinct way for a double < > , triple etc.,... 

The total number of E diagrams produced by the connected T(f contribution amounts to 168. Adding the 84 fifth-order disconnected quadruple excitation diagrams, ds, results in a total of 252 antisymmetrized diagrams of quadruple excitation type. 

Finally, we mention that if one were to sum all double excitation diagrams (an imaginary horizontal line crosses only two hole and two particle lines for such diagrams) one would obtain the doubly excited MBPT (D-MBPT(oo)) of Bartlett and coworkers. This approximation is equivalent to the linear CCA discussed in Subsection 5.2.3. Thus, the appropriate correlation energy is (see Eq. (5.65))... 

Before stating the general rules for working with diagrams, we note that many apparently different diagrams will yield algebraically identical results. For instance, in the evaluation of (QH ) 0) we encounter double-excitation diagrams such as... 

The superscripts D and Q refer to the two components of fourth-order perturbation theory corresponding to double- and quadruple-excitation diagrams. This defines the perturbation-theory model DQ-MBPT(4). 

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