Noninteracting/noninteraction subsystems

Much of the recent literature on RDM reconstruction functionals is couched in terms of cumulant decompositions [13, 27-38]. Insofar as the p-RDM represents a quantum mechanical probability distribution for p-electron subsystems of an M-electron supersystem, the RDM cumulant formalism bears much similarity to the cumulant formalism of classical statistical mechanics, as formalized long ago by by Kubo [39]. (Quantum mechanics introduces important differences, however, as we shall discuss.) Within the cumulant formalism, the p-RDM is decomposed into connected and unconnected contributions, with the latter obtained in a known way from the lower-order -RDMs, q < p. The connected part defines the pth-order RDM cumulant (p-RDMC). In contrast to the p-RDM, the p-RDMC is an extensive quantity, meaning that it is additively separable in the case of a composite system composed of noninteracting subsystems. (The p-RDM is multiphcatively separable in such cases [28, 32]). The implication is that the RDMCs, and the connected equations that they satisfy, behave correctly in the limit of noninteracting subsystems by construction, whereas a 2-RDM obtained by approximate solution of the CSE may fail to preserve extensivity, or in other words may not be size-consistent [40, 42]. 

In this section we introduce the p-RDMC, Ap, which encapsulates the part of the p-RDM that is additively separable in the limit of noninteracting subsystems. Although the RDMCs have been discussed at length in the literature [27-38], this section provides an introduction and summary of the most important points. In this section we use the second-quantized formulation of the ROMs (see Eq. (8)), as separability properties are most easily introduced using this formalism. 

Although the ROMs provide a compact and appealing description of electronic structure, this description is unsatisfactory in at least one respect, namely, expectation values calculated from ROMs are not manifestly extensive, so do not necessarily become additively separable in the limit of noninteracting subsystems. This basic flaw ultimately arises because the RDMs are multiplicatively separable rather than additively separable [28-32]. 

To illustrate this point, consider a composite system composed of two noninteracting subsystems, one with p electrons (subsystem A) and the other with q = N — p electrons (subsystem B). This would be the case, for example, in the limit that a diatomic molecule A—B is stretched to infinite bond distance. Because subsystems A and B are noninteracting, there must exist disjoint sets Ba and Bb of orthonormal spin orbitals, one set associated with each subsystem, such that the composite system s Hamiltonian matrix can be written as a direct sum. 

In the present context, the way to ensure extensivity is to reformulate the CSE so that the RDMCs and not the RDMs are the basic variables. One can always recover the RDMs from the cumulants, but only the cumulants satisfy connected equations that do not admit the possibility of mixing noninteracting subsystems. Connected equations are derived in Section V. Before introducing that material, we first provide a general formulation of the p-RDMC for arbitrary p. 

From a theoretical point of view, experimental properties T can be distinguished with respect to their behavior in a system S. A molecular property T may be categorized in terms of its behavior under the hypothesis that a system S = Au B breaks up into two separate noninteracting subsystems A and 3 [Trinajstic, Randic et al, 1986]. 

Next, we consider the second-order energy and, in particular, we examine for a system consisting of two noninteracting subsystems a and h). For this case, the Hamiltonian // separates into... 

As we did above for the energy, it is instructive to analyze when it pertains to two noninteracting subsystems (a and b). For this special case, the sum in Eq. (3.46) separates into terms pertaining to each of the isolated systems ... 

Because, for two noninteracting subsystems, both of the disjoint sums occurring in Eq. (3.60) are size consistent (i.e., proportional to the size of the system), the product would not be size consistent. Hence, unlinked diagrams correspond directly to non-size-consistent factors, which should not be included. 

The Hamiltonian for the compound system of the noninteracting subsystems is given by... 

The use of a similarity-transformed Hamiltonian in linked coupled-cluster theory means that the energy and amplitude equations contain terms that consist either of the Hamiltonian itself or of nested commutators of the Hamiltonian with cluster operators. For a system containing two noninteracting subsystems A and B, these nested commutators separate additively into nested commutators, each involving a single subsystem, for example. 

Consider the compound system AB consisting of the two noninteracting subsystems A and B. In a basis where t, Tg, t b) and ta, tb, tab) constitute the row and column labels, respectively, the Jacobian may be written in the partitioned form... 

We consider a supersystem consisting of two noninteracting subsystems A and B. The Hamiltonian is separable... 

The operator may be the identity operator (in which case it does not affect the Hartree-Fock ket) or it may be a string of creation and annihilation operators for the occupied and virtual spin orbitals of system A, respectively (in which case it annihilates the Hartiee-Fock ket). The same considerations apply to Tg. Since operators referring to different noninteracting subsystems commute, we may write the matrix element in (14.2.56) as... 

The exact wave function is size-extensive in the sense that, for a system containing noninteracting subsystems, the total energy is equal to the sum of the energies of the individual systems. More specifically, if the total Hamiltonian can be written as a sum of M noninteracting Hamiltonians... 

For large systems, there is a noteworthy difference in the behaviour of the error in the Hartree-Fock energy and the error in the Hartree-Fock wave function. Consider m noninteracting subsystems. For monomer I, the exact wave function can be written as... 

In other words, we get the same result by considering (A+B) as a supersystem as when handling A and B subsystems separately. This is especially important for extended systems, involving n subsystems, in which case the limiting process n —> oo will only make sense if the energy is strictly linear in n in the noninteracting limit. 

The fact that the exact ground-state electron density can be obtained as a sum of two components a trial electron density belonging to the class of functions, such that Px = Po — p1 0,1 is noninteracting pure-state v-representable, and the electron density derived from Eq. 31 provides a formal basis for the electron-density-partitioning based strategy in multi-level modelling. If subsystem A represents a small part of the... 

The derivation of the TF-type models from the third construction shows the dramatic approximations in the TF-type theories. The link between subsystems in the third construction has been reduced to the Fermi energy only as L approaches 0. The other interactions have been turned off. The electrons subject only to the local potential where they are. This is consistent with the noninteracting electron gas picture of the... 

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