Reduced density matrix methods

Before embarking on a more detailed analysis of how to design exchange-correlation energy functionals in Kohn-Sham theory, it may be instructive to take a slight detour 

We will ignore electron spin, except when required. The corresponding reduced spin density matrices are defined completely analogously to eq. (6.11), if the r variables are taken to represent both spatial and spin coordinates. 

The diagonal components of the first-order density matrix (setting r = ri) gives the electron density function pi, often written without the subscript 1 when higher order densities are not involved. 

The integral is the probability of finding an electron (it does not matter which, since they are indistinguishable) at position ri, and the A eiec prefactor ensures that the density integrates to the number of electrons. 

The corresponding second-order density matrix yields the electron pair-density upon setting x[ = ri and r2 = t2. 

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Z. Zhao, B. J. Braams, H. Fukuda, M. L. Overton, and J. K. Percus, The reduced density matrix method for electronic structure calculations and the role of three-index representabihty conditions. J. Chem. Phys. 120, 2095 (2004). 

G. Gidofalvi and D. A. Mazziotti, Spin and symmetry adaptation of the variational two-electron reduced-density-matrix method. Phys. Rev. A 72, 052505 (2005). 

Sand, A. M., Schwerdtfeger, C. A., Mazziotti, D. A. (2012). Strongly correlated barriers to rotation from parametric two-electron reduced-density-matrix methods in application to the isomerization of diazene. The Journal of Chemical Physics, 136, 034112. 

Husimi, K., Proc. Phys.-Math. Soc. Japan 22, 264, "Some formal properties of the density matrix." Introduction of the concept of reduced density matrix. Statistical-mechanical treatment of the Hartree-Fock approximation at an arbitrary temperature and an alternative method of obtaining the reduced density matrices are discussed. 

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

T. Juhasz and D. A. Mazziotti, Perturbation theory corrections to the two-particle reduced density matrix variational method. J. Chem. Phys. 121, 1201 (2004). 

R. M. Erdahl, Two algorithms for the lower bound method of reduced density matrix theory. Reports Math. Phys. 15, 147-162 (1979). 

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). 

The A-representability constraints presented in this chapter can also be applied to computational methods based on the variational optimization of the reduced density matrix subject to necessary conditions for A-representability. Because of their hierarchical structure, the (g, R) conditions are also directly applicable to computational approaches based on the contracted Schrodinger equation. For example, consider the (2, 4) contracted Schrodinger equation. Requiring that the reconstmcted 4-matrix in the (2, 4) contracted Schrodinger equation satisfies the (4, 4) conditions is sufficient to ensure that the 2-matrix satisfies the rather stringent (2, 4) conditions. Conversely, if the 2-matrix does not satisfy the (2, 4) conditions, then it is impossible to construct a 4-matrix that is consistent with this 2-matrix and also satisfies the (4, 4) conditions. It seems that the (g, R) conditions provide important constraints for maintaining consistency at different levels of the contracted Schrodinger equation hierarchy. 

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. 

As anticipated, the multipolar model is not the only technique available to refine electron density from a set of measured X-ray diffracted intensities. Alternative methods are possible, for example the direct refinement of reduced density matrix elements [73, 74] or even a wave function constrained to X-ray structure factor (XRCW) [75, 76]. Of course, in all these models an increasing amount of physical information is used from theoretical chemistry methods and of course one should carefully consider how experimental is the information obtained. 

Elements of second order reduced density matrix of a fermion system are written in geminal basis. Matrix elements are pointed out to be scalar product of special vectors. Based on elementary vector operations inequalities are formulated relating the density matrix elements. While the inequalities are based only on the features of scalar product, not the full information is exploited carried by the vectors D. Recently there are two object of research. The first is theoretical investigation of inequalities, reducibility of the large system of them. Further work may have the chance for reaching deeper insight of the so-called N-representability problem. The second object is a practical one examine the possibility of computational applications, associate conditions above with known methods and conditions for calculating density matrices. 

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]. 

Using the effective Hamiltonian method, the EOM for the reduced density matrix a and the structure of 1 have been derived. However, how to evaluate F is not provided. 

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