Reconstruction functionals

Approximate Reconstruction Functionals for the p-RDMs in Terms of Lower ROMs Where Corrections to Valdemoro s Functionals Are Underlined... 

Corrections for the 4-RDM and 5-RDM functionals may be obtained by searching for some terms involving the wedge products of lower ROMs, which cancel with the corresponding corrections for the HRDM functionals [20]. Consider the matrices and A describing the errors in Valdemoro s reconstruction functionals for the 2- and 3-RDMs as well as the matrices A and A describing the errors in Valdemoro s reconstruction functionals for the 2- and 3-HRDMs... 

The reconstruction functionals, derived in the previous section through the particle-hole duality, may also be produced through the theory of cumulants [21,22,24,26,39,55-57]. We begin by constructing a functional whose derivatives with respect to probe variables generate the reduced density matrices in second quantization. Because we require that additional derivatives increase the number of second quantization operators, we are led to the following exponential form ... 

The reconstruction functionals may be understood as substantially renormalized many-body perturbation expansions. When exact lower RDMs are employed in the functionals, contributions from all orders of perturbation theory are contained in the reconstructed RDMs. As mentioned previously, the reconstruction exactly accounts for configurations in which at least one particle is statistically isolated from the others. Since we know the unconnected p-RDM exactly, all of the error arises from our imprecise knowledge of the connected p-RDM. The connected nature of the connected p-RDM will allow us to estimate the size of its error. For a Hamiltonian with no more than two-particle interactions, the connected p-RDM will have its first nonvanishing term in the (p — 1) order of many-body perturbation theory (MBPT) with a Hartree-Fock reference. This assertion may be understood by noticing that the minimum number of pairwise potentials V required to connectp particles completely is (p — 1). It follows from this that as the number of particles p in the reconstmcted RDM increases, the accuracy of the functional approximation improves. The reconstmction formula in Table I for the 2-RDM is equivalent to the Hartree-Fock approximation since it assumes that the two particles are statistically independent. Correlation corrections first appear in the 3-RDM functional, which with A = 0 is correct through first order of MBPT, and the 4-RDM functional with A = 0 is correct through second order of MBPT. 

Rosina s theorem states that for an unspecified Hamiltonian with no more than two-particle interactions the ground-state 2-RDM alone has sufficient information to build the higher ROMs and the exact wavefunction [20, 51]. Cumulants allow us to divide the reconstruction functional into two parts (i) an unconnected part that may be written as antisymmetrized products of the lower RDMs, and (ii) a connected part that cannot be expressed as products of the lower RDMs. As shown in the previous section, cumulant theory alone generates all of the unconnected terms in RDM reconstruction, but cumulants do not directly indicate how to compute the connected portions of the 3- and 4-RDMs from the 2-RDM. In this section we discuss a systematic approximation of the connected (or cumulant) 3-RDM [24, 26]. 

The theory of cumulants allows us to partition an RDM into contributions that scale differently with the number N of particles. Because aU of the particles are connected by interactions, the cumulant RDMs scale linearly with the number N of particles. The unconnected terms in the p-RDM reconstruction formulas scale between N and W according to the number of connected RDMs in the wedge product. For example, the term scales as NP since all p particles are statistically independent of each other. By examining the scaling of terms with N in the contraction of higher reconstruction functionals, we may derive an important set of relations for the connected RDMs. 

Only the connected ROMs A and scale linearly with N in the reconstruction formulas for the 3- and 4-RDMs. However, the contraction of the 4-RDM reconstruction formula in Table I generates by transvection additional terms that scale linearly with N. Without approximation the terms that scale linearly with N on both sides of Eq. (47) may be set equal. These terms must be equal to preserve the validity of Eq. (47) for any integer value of N. In this manner we obtain a relation that reveals which terms of the 4-RDM reconstruction functional are mapped to the connected 3-RDM [26] ... 

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]. 

Using cumulant reconstruction functionals A3[Ai, A2] and A4[Ai, A2], one can certainly derive closed, nonlinear equations for the elements of Ai and A2, which could be solved using an iterative procedure that does not exploit the reconstruction functionals at each iteration. Of the RDM reconstruction functionals derived to date, several [7, 8, 11] utilize the cumulant decompositions in Eqs. (25c) and (25d) to obtain the unconnected portions of D3 and D4 exactly (in terms of the lower-order RDMs), then use many-body perturbation theory to estimate the connected parts A3 and A4 in terms of Aj and A2, the latter essentially serving as a renormalized pair interaction. Reconstruction functionals of this type are equally useful in solving ICSE(l) and ICSE(2), but the reconstruction functionals introduced by Valdemoro and co-workers [25, 26] cannot be used to solve the ICSEs because they contain no connected terms in D3 or D4 (and thus no contributions to A3 or A4). 

Next, we present some observations concerning the connection between the reconstruction process and the iterative solution of either CSE(p) or ICSE(p). The perturbative reconstruction functionals mentioned earlier each constitute a finite-order ladder-type approximation to the 3- and 4-RDMCs [46, 69] examples of the lowest-order corrections of this type are shown in Fig. 3. The hatched squares in these diagrams can be thought of as arising from the 2-RDM, which serves as an effective pair interaction for a form of many-body perturbation theory. Ordinarily, ladder-type perturbation expansions neglect three-electron (and higher) correlations, even when extended to infinite order in the effective pair interaction [46, 69], but iterative solution of the CSEs (or ICSEs) helps to... 

Piris and Otto (PO) achieved a reconstruction functional D[ D] satisfying the most general properties of the 2-RDM [58]. They kept the spin structure from Refs. [52, 53], but introduced a new spatial dependence in the correction term of the 2-RDM. Calculated values for polarizabilities [59], ionization energies, equilibrium geometries, and vibrational frequencies [60] in molecules were... 

The 2-RDM formulation, Eq. (38), allows us to generalize the constrained search to approximately V-representable sets of 2-RDMs. In order to approximate the unknown functional Eee[V, D], we use here a reconstructive functional D[ D] that is, we express the elements D h in terms of the We neglect any explicit dependence of on the NOs themselves because the energy functional already has a strong dependence on the NOs via the one- and two-electron integrals. 

Mammalian cells and tissues reside in mechanically dynamic microenvironments in the body. Abnormal or excessive physical loads on tissues result in tissue deformation. Numerous studies have confirmed that proper mechanical stimuli applied to cells or tissues contribute to maintaining cell/tissue morphology and inducing specialized functions. Mechanical stimuli have recently been applied to regenerate functional tissues, in particular tissues for the cardiovascular system. To reconstruct functionally active SM tissues that are comparable to native tissues, the re-creation in vitro of in vivo mechano-active microenvironments may be a necessary part of the tissueengineering process. 

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