Reduced density matrix structure

The structure of the left-hand side of Eq. (4.2.14) is the same as the homogeneous equation of Ref. 148 for the matrix elements Pon, n, °f the reduced density matrix... 

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). 

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. 

The structure of the reduced density matrix follows from the symmetry properties of the Hamiltonian. However, for this case the concurrence C iJ) depends on ij and the location of the impurity and not only on the difference i—j as for the pure case. Using the operator expansion for the density matrix and the symmetries of the Hamiltonian leads to the general form... 

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. 

The concept of the molecular orbital is, however, not restricted to the Hartree-Fock model. Sets of orbitals can also be constructed for more complex wave functions, which include correlation effects. They can be used to obtain insight into the detailed features of the electron structure. One choice of orbitals are the natural orbitals, which are obtained by diagonalizing the spinless first-order reduced density matrix. The occupation numbers (T ) of the natural orbitals are not restricted to 2, 1, or 0. Instead they fulfill the condition ... 

E. Brandas, C.A. Chatzidimitriou-Dreismann, On the Connection Between Certain Properties of the Second-Order Reduced Density Matrix and the Occurrence of Coherent-Dissipative Structures in Disordered Condensed Matter, Int. J. Quant. Chem. 40 (1991) 649. 

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. 

Back in the 1960s, hopes for future progress in electronic structure theory were associated with correlated wave function techniques and the tantalizing possibility of variational calculations based on the two-electron reduced density matrix [294]. DFT was not on the quantum chemistry agenda at that time. The progress of wave function techniques has been remarkable, as documented elsewhere in this volume. In contrast, the density matrix approach has not yet materialized into a competitive computational method, despite many persistent efforts [295]. Meanwhile, approximate DFT has become the most widely used method of quantum chemistry, offering an unprecedented accuracy-to-cost ratio. 

Density matrices, in particular, the so-called first- and second-order reduced density matrices, are important quantities in the theoretical description of electronic structures because they contain all the essential information of the system under study. Given a set of orthonormal MOs, we define the first-order reduced density matrix D with matrix elements as the expectation value of the excitation operator E = aLa, -I- with respect to some electronic wave function Fgi,... 

The authors are honored to dedicate this article in memory of Isaiah Shavitt whose remarkable contributions transformed electronic structure theory. Observables that depend on pairwise interactions can be directly computed with the two-electron reduced density matrix (2-RDM) without the A-electron wave function [10, 36]. Integration of the A-electron density matrix over all electrons save two yields the 2-RDM... 

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 the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

By replacing the wavefunction with a density matrix, the electronic structure problem is reduced in size to that for a two- or three-electron system. Rather than solve the Schrodinger equation to determine the wavefunction, the lower bound method is invoked to determine the density matrix this requires adjusting parameters so that the energy content of the density matrix is minimized. More precisely, the lower bound method requires finding a solution to the energy problem,... 

The expressions eqs. (1.197), (1.199), (1.200), (1.201) are completely general. From them it is clear that the reduced density matrices are much more economical tools for representing the electronic structure than the wave functions. The two-electron density (more demanding quantity of the two) depends only on two pairs of electronic variables (either continuous or discrete) instead of N electronic variables required by the wave function representation. The one-electron density is even simpler since it depends only on one pair of such coordinates. That means that in the density matrix representation only about (2M)4 numbers are necessary to describe the system (in fact - less due to antisymmetry), whereas the description in terms of the wave function requires, as we know n 2m-n) numbers (FCI expansion amplitudes). However, the density matrices are rarely used directly in quantum chemistry procedures. The reason is the serious problem which appears when one is trying to construct the adequate representation for the left hand sides of the above definitions without addressing any wave functions in the right hand sides. This is known as the (V-representability problem, unsolved until now [51] for the two-electron density matrices. The second is that the symmetry conditions for the electronic states are much easier formulated and controlled in terms of the wave functions (Density matrices are the entities of the second power with respect to the wave functions so their symmetries are described by the second tensor powers of those of the wave functions). 

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