Reduced density matrices development

Reduced Density Matrix versus Wave Function Recent Developments... 

On the other hand, there exist well-developed methods for calculating states of subsystems using the Markov approximation for the reduced density matrix... 

In Bohmian mechanics, the way the full problem is tackled in order to obtain operational formulas can determine dramatically the final solution due to the context-dependence of this theory. More specifically, developing a Bohmian description within the many-body framework and then focusing on a particle is not equivalent to directly starting from the reduced density matrix or from the one-particle TD-DFT equation. Being well aware of the severe computational problems coming from the first and second approaches, we are still tempted to claim that those are the most natural ways to deal with a many-body problem in a Bohmian context. 

C. Valdemoro, D. R. Alcoba, and L. M.Tel, Recent developments in the contracted Schrddinger equation method controlling the iV-representabiUty of the second-order reduced density matrix. Int. J. Quantum Chem 93, 212 (2003). 

C. Valdemoro, Reduced density matrix versus wave function recent developments, in Strategies and Applications in Quantum Chemistry, (Y. Ellinger and M. Defranceschi, eds.), Kluwer, Dordrecht, 1996, p. 55. 

A direction for improving DPT lies in the development of a functional theory based on the one-particle reduced density matrix (1-RDM) D rather than on the one-electron density p. Like 2-RDM, the 1-RDM is a much simpler object than the A-particle wavefunction, but the ensemble A-representability conditions that have to be imposed on variations of are well known [1]. The existence [10] and properties [11] of the total energy functional of the 1-RDM are well established. Its development may be greatly aided by imposition of multiple constraints that are more strict and abundant than their DPT counterparts [12, 13]. 

Almost all of the available A-representability constraints are based on the orbital representation of the reduced density matrix, F, jg, instead of the spatial representation, rg(xi,..., xg x i,..., x g). This is not problematic when the reduced density matrix is available, because it is easy to convert 2-matrices to and from the spatial representation (cf. Eqs. (18) and (19)). There has been a lot of recent interest in developing computational algorithms based on... 

This volume in Advances in Chemical Physics provides a broad yet detailed survey of the recent advances and applications of reduced-density-matrix mechanics in chemistry and physics. With advances in theory and optimization, Coulson s challenge for the direct calculation of the 2-RDM has been answered. While significant progress has been made, as evident from the many contributions to this book, there remain many open questions and exciting opportunities for further development of 2-RDM methods and applications. It is the hope of the editor and the contributors that this book will serve as a guide for many further advenmres and advancements in RDM mechanics. 

To describe consistently cotunneling, level broadening and higher-order (in tunneling) processes, more sophisticated methods to calculate the reduced density matrix were developed, based on the Liouville - von Neumann equation [186-193] or real-time diagrammatic technique [194-201]. Different approaches were reviewed recently in Ref. [202]. 

As for the case of a bosonic bath, the starting point here is the TL approach and a TL QME based on a second-order perturbation theory in the molecule-lead coupling was developed for the reduced density matrix p(t) of the molecule [38,65]... 

Various methods have been developed that interpolate between the coherent and incoherent regimes (for reviews see, e.g. (3)-(5)). Well-known approaches use the stochastic Liouville equation, of which the Haken-Strobl-Reineker (3) model is an example, and the generalized master equation (4). A powerful technique, which in principle deals with all aspects of the problem, uses the reduced density matrix of the exciton subsystem, which is obtained by projecting out all degrees of freedom (the bath) from the total statistical operator (6). This reduced density operator obeys a closed non-Markovian (integrodifferential) equation with a memory kernel that includes the effects of (multiple) interactions between the excitons and the bath. In practice, one is often forced to truncate this kernel at the level of two interactions. In the Markov approximation, the resulting description is known as Redfield theory (7). 

In Appendix A we have derived a reduced density matrix which exhibits the so-called extreme configuration that may develop ofif-diagonal long-range order (ODLRO), see equations (A6) and (A7). 

In passing we note that the functions in the set g are completely delocalized over the region of sites defined by the localized particle-antiparticle basis h, while the f-basis contains all possible phase-shifted contributions from each site in accordance with Eqs. (56) and (57) above. Some interrelationships can be recognized here. The first connection concerns Coleman s so-called extreme state [18], cf. the theories of superconductivity and superfluidity based on ODLRO. The second observation relates to the identification of the present finite dimensional representation as a precursor for possible condensations, developing correlations and coherences that may extend over macroscopic dimensions. If h is a set of two-particle determinants and the iV-particle fermionic wave function is constructed from an AGP, antisymmetrized geminal power, based on i, see Eq. (57), then the reduced density matrix can be represented as... 

Analogically to the representation of the wave-function in structural terms, there is a way to separate (hyper)polarizabilities into the individual contributions from individual atoms. A method for such separation was developed by Bredas [15, 16] and is called the real-space finite-field method. The approach can be easily implemented for a post-Hartree-Fock method in the r-electron approximation due to the simplicity of e calculation of the one-electron reduced density matrix (RDMl) elements. In our calculations we are using a simple munerical-derivative two-points formula for RDMl matrix elements (Z ) [88] (see also [48]) ... 

In principle, the theory of nonlinear spectroscopy with femtosecond laser pulses is well developed. A comprehensive and up-to-date exposition of nonlinear optical spectroscopy in the femtosecond time domain is provided by the monograph of Mukamel. ° For additional reviews, see Refs. 7 and 11-14. While many theoretical papers have dealt with the analysis or prediction of femtosecond time-resolved spectra, very few of these studies have explicitly addressed the dynamics associated with conical intersections. In the majority of theoretical studies, the description of the chemical dynamics is based on rather simple models of the system that couples to the laser fields, usually a few-level system or a set of harmonic oscillators. In the case of condensed-phase spectroscopy, dissipation is additionally introduced by coupling the system to a thermal bath, either at a phenomenological level or in a more microscopic maimer via reduced density-matrix theory. 

There are correlation functionals which a dependence in the natural orbitals or in the reduced density matrix (F), which are applicable to MD-wavc-functions 18.19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39 including a family of c[F] recently developed in onr laboratory 36,37,38 jj g been fonnd that for the H2 and the F2 molecnles, when Ec,md is approximated by some of these functionals, a considerable improvement of the potential energy snrface and of the spectroscopic constants is obteiined over the results of the MD wave function without these functionals. 

In the last decade, a series of functionals has been developed [1, 2] using a reconstruction proposed by Piris [3] of the two-particle reduced density matrix (2-RDM) in terms of the one-particle RDM (1-RDM). In particular, the Piris natural orbital functional 5 (PNOF5) [4, 5] has proved to... 

By its size, this chapter fails to address the entire background of MQS and for more information, the reader is referred to several reviews that have been published on the topic. Also it could not address many related approaches, such as the density matrix similarity ideas of Ciosloswki and Fleischmann [79,80], the work of Leherte et al. [81-83] describing simplified alignment algorithms based on quantum similarity or the empirical procedure of Popelier et al. on using only a reduced number of points of the density function to express similarity [84-88]. It is worth noting that MQS is not restricted to the most commonly used electron density in position space. Many concepts and theoretical developments in the theory can be extended to momentum space where one deals with the three components of linear momentum... 

A quantum system of N particles may also be interpreted as a system of (r — N) holes, where r is the rank of the one-particle basis set. The complementary nature of these two perspectives is known as the particle-hole duality [13, 44, 45]. Even though we treated only the iV-representability for the particles in the formal solution, any p-hole RDM must also be derivable from an (r — A)-hole density matrix. While the development of the formal solution in the literature only considers the particle reduced Hamiltonian, both the particle and the hole representations for the reduced Hamiltonian are critical in the practical solution of N-representability problem for the 1-RDM [6, 7]. The hole definitions for the sets and are analogous to the definitions for particles except that the number (r — N) of holes is substituted for the number of particles. In defining the hole RDMs, we assume that the rank r of the one-particle basis set is finite, which is reasonable for practical calculations, but the case of infinite r may be considered through the limiting process as r —> oo. 

We have focused on the lower bound method, but density matrix research has moved forward on a much broader front than that. In particular, work on the contracted Schrodinger equation played an important role in developments. A more complete picture can be found in Coleman and Yukalov s book [23]. It has taken 55 years and work by many scientists to fulfill Coleman s 1951 claim at Chalk River that except for a few details which would be easily overcome in a couple of weeks—the A-body problem has been reduced to a 2.5-body problem ... 

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