Two-Body Matrix Elements

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... 

The principal assumption needed to relate the elastic channel matrix element of tj to that of an effective two-body operator, is that the nuclear wave function can be adequately approximated by the independent particle model [To 77]. With this assumption, the relevant many-body matrix element of the two-body interaction, vSL, reduces to where ) is the intrinsic /th excited... 

Due to the size of the variational problem, a large Cl is usually not a practicable method for recovering dynamic correlation. Instead, one usually resorts to some form of treatment based on many-body perturbation theory where an explicit calculation of all off-diagonal Cl matrix elements (and the diagonalization of the matrix) are avoided. For a detailed description of such methods, which is beyond the scope of this review, the reader is referred to appropriate textbooks295. For the present purpose, it suffices to mention two important aspects. 

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. 

There are useful two- and many-electron analogues of the functions discussed above, but when the Hamiltonian contains only one- and two-body operators it is sufficient to consider the pair functions thus the analogue of p(x x ) is the pair density matrix 7t(xi,X2 x i,x ) while that of which follows on identifying and integrating over spin variables as in (4), is H(ri,r2 r i,r2)- When the electron-electron interaction is purely coulombic, only the diagonal element H(ri,r2) is required and the expression for the total interaction energy becomes... 

Strutinsky procedure. Because asymmetric shapes is so shallow, it is worthwhile to deal with octupole correlation effects using a microscopic, two-body interaction treatment of the octupole-octupole residual interaction [CHA80]. The pairing force matrix elements, G. come from a density dependent delta interaction. This set of matrix elements [CHA77] explains many features of the actinides at low and... 

The connection between these S -matrix elements and the measurable cross-sections is worked out by arguments that are similar to the two-body case (see Eqs (4.148) and (4.151)). To that end, the scattering probability must be averaged over the relevant initial states (corresponding to different impact points) and these states will be assumed to be sharply centered around the momentum p = p0. 

The first term in eq. (1) Ho represents the spherical part of a free ion Hamiltonian and can be omitted without lack of generality. F s are the Slater parameters and ff is the spin-orbit interaction constant /<- and A so are the angular parts of electrostatic and spin-orbit interactions, respectively. Two-body correction terms (including Trees correction) are described by the fourth, fifth and sixth terms, correspondingly, whereas three-particle interactions (for ions with three or more equivalent f electrons) are represented by the seventh term. Finally, magnetic interactions (spin-spin and spin-other orbit corrections) are described by the terms with operators m and p/. Matrix elements of all operators entering eq. (1) can be taken from the book by Nielsen and Koster (1963) or from the Argonne National Laboratory s web site (Hannah Crosswhite s datafiles) http //chemistry.anl.gov/downloads/index.html. In what follows, the Hamiltonian (1) without Hcf will be referred to as the free ion Hamiltonian. 

In principle, the most general representation of the matrix elements of equation (68) includes a many-body expansion with one-, two-, and three-body terms. However, only two conditions [(a) and (b) in equations (66), (67)] may be used in the determination of the one- and two-body energy terms. If more conditions were to exist, the dimension of the matrix necessary to represent the adiabatic potential would be larger.16 In what follows we shall argue that the definition of V12 must depend on the type of conical intersection, namely on whether its locus is finite or infinite in extent (Section II.C). For the cases of HzO and 03 [equations (66), (67)] the conical intersection occurs along the C projection line, but only for finite values of the molecular perimeter (for H20, there is also a simple intersection at the O + H2 asymptotic channel, which is avoided for finite values of the OH distances117). For example, one gets for the linear dissociation of 03... 

This completes the specification of the pseudopotential as a perturbation in a perfect crystal. We have obtained all of the matrix elements between the plane-wave states, which arc the electronic states of zero order in the pseudopotcntial. We have found that they vanish unless the difference in wave number between the two coupled states is a lattice wave number, and in that case they are given by the pseudopotential form factor for that wave number difference by Eq. (16-7), assuming that there is only one ion per primitive cell, as in the face-centered and body-centered cubic structures. We discuss only cases with more than one ion per primitive cell when we apply pseudopotential theory to semiconductors in Chapter 18. Tlicn the matrix element will be given by a structure factor, Eq. (16-17),... 

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