Reduced density-matrix correlation densities

A description of the different terms contributing to the correlation effects in the third order reduced density matrix faking as reference the Hartree Fock results is given here. An analysis of the approximations of these terms as functions of the lower order reduced density matrices is carried out for the linear BeFl2 molecule. This study shows the importance of the role played by the homo s and lumo s of the symmetry-shells in the correlation effect. As a result, a new way for improving the third order reduced density matrix, correcting the error ofthe basic approximation, is also proposed here. 

Then, in the Old Ages (1940 or 1951-1967) some ingenious people became aware that, in the case of two-body interactions, it is the two-particle reduced density matrix (2-RDM) that carries in a compact way all the relevant information about the system (energy, correlations, etc.). Early insight by Husimi (1940) and challenges by Charles Coulson were completed by a clear realization and formulation of the A-representability problem by John Coleman in 1951 (for the history, see his book [1] and Chapters 1 and 17 of the present book). Then a series of theorems on A-representability followed, by John Coleman and many... 

G. Gidofalvi and D. A. Mazziotti, Boson correlation energies via variational minimization with the two-particle reduced density matrix exact iV-representabihty conditions for harmonic interactions. Phys. Rev. A 69, 042511 (2004). 

T. Yanai and G. K. L. Chan, Canonical transformation theory for dynamic correlations in multireference problems, in Reduced-Density-Matrix Mechanics With Application to Many-Electron Atoms and Molecules, A Special Volume of Advances in Chemical Physics, Volume 134 (D.A. Mazziotti, ed.), Wiley, Hoboken, NJ, 2007. 

So it is the probability that orbital i is empty and orbitals j and k are filled or that orbital i is filled and orbitals j and k are empty. It is interesting that these particular three-electron correlations can be evaluated using the two-electron reduced density matrix. 

The matrix elements of the reduced density matrix needed to calculate the entanglement can be written in terms of the spin-spin correlation functions and the average magnetization per spin. The spin-spin correlation functions for the ground state are dehned as [62]... 

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 ... 

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. 

Since the bath is made up of harmonic oscillators, it can be eliminated analytically (provided that one is only interested in the reduced density matrix of the reaction coordinate). After performing this elimination, the bath-plus-coupling part H of the Hamiltonian leads to an influence functional 0[cr] in terms of the spins cr, alone [38,39], and the correlation function takes the form... 

An important feature of the reduced-density-matrix approach is that it allows the bath to be treated at different levels of approximation. In the Redfield equation, the bath enters only through the correlation functions of the coupled bath variables in Eq. (18). This means that a substantial part of the complexity of a realistic condensed-phase environment is... 

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... 

Methods using the first-order reduced density matrix as variable can be chosen to strictly enforce the A-representability of yi, and employ the exact energy functional for all the terms except the correlation energy. The latter, however, inherently depends on 72, which in this approach must be approximated as a function of yi. One can argue that Hartree-Fock belongs to this class of methods, with imphcit neglect of the electron correlation. 

The exchange-correlation energy can thus be obtained by integrating the electron-electron interaction over the A variable and subtracting the Coulomb part. The right-hand side of eq. (B.18) can be written in terms of the second-order reduced density matrix eq. (6.14), and the definition of the exchange-correlation hole in eq. (6.21) allows the Coulomb energy to be separated out. 

In order to obtain a bond order formula for open-shell systems that can be applied for both the indep)endent-partide model and correlated wave functions and which simultaneously yields unique bond orders for all spin multiplet components (in the absence of a magnetic field), Alcoba et al. [151, 152] derived a general expression (in the Hilbert space partitioning scheme) from a second-order reduced density matrix. Furthermore, as the first- and second-order reduced density matrices are invariant with respect to the spin projection, they are only a function of the total spin or similarly of the maximum projection S = and the bond order can be evaluated for the highest spin-projected state = S. They arrived at the following expression for the bond order... 

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. 

Keyvrords Two-electron reduced density matrix N-representability conditions Strong electron correlation Hubbard models... 

The first-order reduced density matrix of the exact wave function also has a number of weakly occupied orbitals for which i > N. These orbitals never appear in a single Slater determinant method, but are important for a correct description of the correlation hole. 

The MRCI method is widely used to deal with molecules with substantial static correlation, but can only be applied to relatively small molecules. (The CASPT2 method of Section 16.3 can treat static correlation in somewhat larger molecules.) Another approach is to avoid using a wave function and instead deal with the two-electron reduced density matrix. 

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. 

We use correlation energy functionals which are sensible to the wave-function used, particularly the functionals proposed by Colle-Salvetti (CS) and Moscardd and P rez-Jimenez for the levels 1 (MPJl) and 5 (MPJ5) Both have a dependence in the reduced density matrix of second order (F). In the SCF procedure, and for simpUcity, we use the correlation energy potentials obtained from Moscardo and Perez-Jimdnez s functionals... 

The exchange-correlation energy can be subject to the same type of multipole expansion. For any wavefunction we can write the second order reduced density matrix as a function of molecular orbitals via... 

From the intuitive point of view, the concept of bond order is somewhat more ambiguous than that of atomic population. Whereas A a is usually construed as the number of electrons ascribed to an atom A, the bond order Bab niay be interpreted as a measure of bond strength, bond multiplicity, or the number of electron pairs involved in covalent interactions between A and B. Most of the currently known definitions of Bab stem from the last of these interpretations. Accordingly, they begin with the extraction of the exchange correlation term from the two-electron reduced density matrix y(ri, T2, r, rj). 

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