Two-body operators

The direct terms of both two-body operators are zero with a single integration in one of the moments. The exchange contribution of the first one is given by (using conventions from the previous section) ... 

With the two-body operator t ifi = a ajaiaic the change in 2-RDM with X can be written... 

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... 

The rest of this paper will deal exclusively with algorithms for construction of electronic wavefunctions because these are central to the overall problem. In order to appreciate the methods used, one must recall that we are interested in solving a partial differential equation eigenvalue problem for several wavefunctions at several different arrangements of the nuclei. This differential equation involves one- and two-body operators in the potential energy operator and partial derivatives with respect to 3N coordinates (where N is the number of electrons). 

The irreducible tensor product between two (spherical) vectors is defined in Eq. (37). An important feature of this Hamiltonian is that it explicitly describes the dependence of the coupling constants J, Am, and T, on the distance vectors rPP between the molecules and on the orientations phenomenological Hamiltonian (139). Another important difference with the latter is that the ad hoc single-particle spin anisotropy term BS2y, which probably stands implicitly for the magnetic dipole-dipole interactions, has been replaced by a two-body operator that correctly represents these interactions. The distance and orientational dependence of the coupling parameters J, A, , and Tm has been obtained as follows. 

In this basis set only the two-body operator Fq is diagonal with respect to J and M. A fully coupled basis set is then obtained by coupling together j and J to give Jy... 

We can now consider each term separately. The two-body operator Fq has the same matrix representation in the two-body coupled basis set and in the present three-body coupled basis set. The next term in S S" is diagonal in the chosen basis set [4]. Then the matrix representation of these diagonal terms is... 

The effective interaction discussed in the previous subsections is a scalar two-body operator. A general scalar two-body operator f can be written as... 

For the two-body operators we consider two models first only pion-exchange processes are retained and evaluated in the soft-pion limit, and second pion-exchange processes are evaluated at finite pion energy and momentum, the hard-pion limit, and augmented by heavy-meson-exchange processes (HPPHM). In the latter case, all the heavy-meson coupling constants and form factors are taken from the Bonn potential, OBEPR [13]. The explicit form of the two-body operators are given in Eq. (43) of Ref. [9]. 

We must emphasize, however, that this correspondence between the vector operators C, C and the second-quantization operators Xt,X is only a formal similarity indeed, the operators and C act on the orbital group U(n) modules while the second-quantization operators and X act on the spin-orbital group U(2n). The real meaning of the correspondence (26) is that we can either couple the U(n) tensors O and C within the U(n) framework to obtain higher rank U(n) tensors, like two-body operators (26) or we can express these U(n) operators as spin contractions of U(2n) operators. 

The segmentation of MEs of spin-dependent one-body operators is very similar to that for spin-independent two-body operators. This is a very useful fact in view of an actual implementation of this formalism, since it enables an evaluation of MEs of spin-dependent one-body operators by exploiting presently available UGA or GUGA codes. We now compare both formalisms in greater detail. 

Table 2 Coefficients relating MBs of spin-dependent, one-body operators and spin-independent, two-body operators, Eqs. (70), (73), and (76), for three possible shifts, as functions of the SU(2) labels. See footnote to Table 1 for the notation convention used...

We have shown that one-electron spin-dependent terms, Eq. (36), in the electronic Hamiltonian, Eq. (33), may be efficiently handled in much the same way as the standard spin-independent two-electron (i.e., Coulomb) terms. Indeed, as clearly implied by Eqs. (70), (73), and (76), the MEs of spin-dependent one-body operators in the U(2 ) D U(n)(8> SU(2) basis may be evaluated as MEs of spin-independent two-body operators in a standard U( -1-1) electronic G-T basis (see also [37, 39]). Since the MEs of generator products within the spin-independent UGA approach are well known, the above presented development should facilitate the implementation of the spin-dependent UGA formalism. This opens a possible avenue enabling us to handle spin-dependent MBPT terms via a simple modification of the existing UGA and GUGA codes, similarly as done by Yabushita et al. [37]. We emphasize that all the required segment values may be found in [36], and a few additional ones that are specific to a spin-dependent case are given in Tables 1 and 2. 

When expressed in the bonding and antibonding basis, the interaction term decomposes into four two-body operators representing inter- and intra-subspace pair scattering and inter- and intra-subspace pair exchange. 

The exactness of exponential cluster expansions employing two-body operators... 

It has recently been suggested that it may be possible to represent the exact ground-state wave function of an arbitrary many-fermion pairwise interacting system, defined by the Hamiltonian iJ, Eq. (22), by an exponential cluster expansion involving a general two-body operator [92-98]. If these statements were true, completely new ways of performing ab initio quantum calculations for many-fermion (e.g., many-electron) systems might... 

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