Two-body matrix

All three- and four-body contributions listed above are evaluated directly without storing the corresponding three- and four-body matrix elements. The final expressions are given in Appendix B. The only precomputed (and, possibly, stored) matrix elements are those which define one- and two-body components of Hu,open defined by Eq. (178). As it is demonstrated in Appendix A, we can express the corresponding one- and two-body matrix elements A and h psq in terms of A ( 1,2) and hpq( 1,2),... 

By using diagrammatic approach, we can show that one- and two-body matrix elements ft and ft , which define Hi and H2 [cf. Eqs. (143) and... 

The explicit formulas for hf and h , which represent one- and two-body matrix elements of H, can be found in Ref. 34 (for the list of misprints in expressions presented in Table 1 of Ref. 34, see Ref. 83). The only expressions that cannot be found in Ref. 34 are those for h a and h fb, since they vanish in the EOMCCSD case (they no longer vanish when the EOMXCCSD method using XCCSD amplitudes is employed). The explicit expressions for h and h%b (which are, of course, the left-hand sides of the SRCCSD equations, provided that we use XCCSD amplitudes) are... 

In this work we interpret the two-body matrix elements of in the representation of the LS-coupling scheme as done by Wildenthal, Brown and co-workers [48,63] and Hosaka and Toki [64]. This representation allows for a more direct comparison with the NN interaction, which is expressed in terms of partial waves. The LS-coupled matrix element of a given component (afi)LSJ Tj f J(yS)LSJ T), with a = Ha/ji, is related to the corresponding matrix elements of the total interaction in the jf/-scheme by... 

We conclude our discussion on the derived effective interactions by employing the two-body matrix elements in the calculation of the eigenvalues for nuclei with more than two valence nucleons. Here we limit our attention to two isotopes, i.e. and Sc. The effective interactions we use are those obtained by using the LS method with a third-order Q-box and by including excitations up to 6ho) in oscillator energy in the evaluation of the adhering diagrams. 

The reason we include in our discussion, is the fact that this nucleus with T =, the JT = 10 two-body matrix elements, discussed above in connection with the increased binding provided by a potential with a weak tensor force, come into play. For which has isospin T = 2, these matrix elements do not occur. It is therefore instructive to compare the spectra obtained with all three Bonn potentials for these two nuclei. 

It is easy to compute the energy shifts due to W in terms of these two-body matrix elements 8( , /) and, in turn, to express the results with the 8(0, /) only. Of course, one has to make use of the permutation properties of the wave functions. One gets... 

Ketone body synthesis occurs only in the mitochondrial matrix. The reactions responsible for the formation of ketone bodies are shown in Figure 24.28. The first reaction—the condensation of two molecules of acetyl-CoA to form acetoacetyl-CoA—is catalyzed by thiolase, which is also known as acetoacetyl-CoA thiolase or acetyl-CoA acetyltransferase. This is the same enzyme that carries out the thiolase reaction in /3-oxidation, but here it runs in reverse. The second reaction adds another molecule of acetyl-CoA to give (i-hydroxy-(i-methyl-glutaryl-CoA, commonly abbreviated HMG-CoA. These two mitochondrial matrix reactions are analogous to the first two steps in cholesterol biosynthesis, a cytosolic process, as we shall see in Chapter 25. HMG-CoA is converted to acetoacetate and acetyl-CoA by the action of HMG-CoA lyase in a mixed aldol-Claisen ester cleavage reaction. This reaction is mechanistically similar to the reverse of the citrate synthase reaction in the TCA cycle. A membrane-bound enzyme, /3-hydroxybutyrate dehydrogenase, then can reduce acetoacetate to /3-hydroxybutyrate. 

Comparatively little space will therefore be devoted to some rather recent approaches, such as the plasma model of Bohm and Pines, the two-body interaction method developed by Brueckner in connection with nuclear theory, Daudel s loge theory, and the method of variation of the second-order density matrix. This does not mean that these methods would be less powerful or less impor-... 

Then, in the Old Ages (1940 or 1951-1967) some ingenious people became aware that, in the case of two-body interactions, it is the two-particle reduced density matrix (2-RDM) that carries in a compact way all the relevant information about the system (energy, correlations, etc.). Early insight by Husimi (1940) and challenges by Charles Coulson were completed by a clear realization and formulation of the A-representability problem by John Coleman in 1951 (for the history, see his book [1] and Chapters 1 and 17 of the present book). Then a series of theorems on A-representability followed, by John Coleman and many... 

M. Rosina, (a) Direct variational calculation of the two-body density matrix (b) On the unique representation of the two-body density matrices corresponding to the AGP wave function (c) The characterization of the exposed points of a convex set bounded by matrix nonnegativity conditions (d) Hermitian operator method for calculations within the particle-hole space in Reduced Density Operators with Applications to Physical and Chemical Systems—II (R. M. Erdahl, ed.), Queen s Papers in Pure and Applied Mathematics No. 40, Queen s University, Kingston, Ontario, 1974, (a) p. 40, (b) p. 50, (c) p. 57, (d) p. 126. 

M. V. Mihailovic and M. Rosina, The particle-hole states in some light nuclei calculated with the two-body density matrix of the ground state. Nucl. Phys. A237, 229-234 (1975). 

M. Rosina, Direct variational calculation of the two-body density matrix, in The Nuclear Marty-Body Problem Proceedings the Symposium on Present Status and Novel Developments in the Nuclear Many-Body Problem, Rome 1972, (F. Calogero and C. Ciofi degli Atti, eds.), Editrice Compositori, Bologna, 1973. 

M. Rosina and M. V. Mihailovic, The determination of the particle—hole excited states by using the variational approach to the ground state two-body density matrix, in International Conference on Properties of Nuclear States, Montreal 1969, Les Presses de I Universite de Montreal, 1969. 

A. The Pure Two-Body Correlation Matrix within the 2-RDM Formalism... 

Equation (15) implies that the 2-RDM and 2-HRDM matrices contain the same information. Indeed, these matrices are two of the three different matrix representations of the 2-RDM on the two-body space, the third one being the second-order G-matrix (2-G) [16]. This matrix, which may be written [24, 25]... 

That is, all the information about the three important matrices 2-RDM, 2-HRDM, and 2-G is contained and available in the pure two-body correlation matrix. Moreover, the spin properties of both the pure two-body correlation matrix and the 2-G matrices play a central role in this purification procedure. 

A very important property of the 2-CM is that [15, 83] the decomposition of the 2-HRDM yields a two-body hole correlation matrix that coincides with the 2-CM. Thus... 

That is, both the 2-CM and the 2-G matrix have common elements, but a given element occupies different positions in each matrix. In other words, while the labels of the row/column of the 2-CM refer, as in the 2-RDM, to two particlesitwo holes, the labels of the row/column of the 2-G matrix refer to particle-hole/hole-particle. Thus, although both the 2-CM and the 2-G matrices describe similar types of correlation effects, only the 2-CM describes pure two-body correlation effects. This is because the 2-CM natural tensorial contractions vanish, and thus there is no contribution to the natural contraction of the 2-RDM into the one-body space whereas the 2-G natural tensorial contractions are functionals of the 1-RDM. 

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