We present an approach for the calculation of spin density distributions for molecules that require very large active spaces for a qualitatively correct description of their electronic structure. Our approach is based on the density-matrix renormalization group (DMRG) algorithm to calculate the spin density matrix elements as a basic quantity for the spatially resolved spin density distribution. The spin density matrix elements are directly determined from the second-quantized elementary operators optimized by the DMRG algorithm. As an analytic convergence criterion for the spin density distribution, we employ our recently developed sampling-reconstruction scheme [J. Chem. Phys.2011, 134, 224101] to build an accurate complete-active-space configuration-interaction (CASCI) wave function from the optimized matrix product states. The spin density matrix elements can then also be determined as an expectation value employing the reconstructed wave function expansion. Furthermore, the explicit reconstruction of a CASCI-type wave function provides insight into chemically interesting features of the molecule under study such as the distribution of α and β electrons in terms of Slater determinants, CI coefficients, and natural orbitals. The methodology is applied to an iron nitrosyl complex which we have identified as a challenging system for standard approaches [J. Chem. Theory Comput.2011, 7, 2740].
Entropy maximization has proven effective in treating certain aspects of the phase problem of X-ray diffraction. Much of its development has been expressed in probabilistic language, although image enhancement has been somewhat more physical or geometric in description. Here phasing and entropy maximization are embedded in the quantum mechanical problem of reconstructing an electronic one-matrix under experimental constraints. Entropy on an N-representable one-particle density matrix is well defined. The entropy is the expected form, and it is a simple function of the one-matrix eigenvalues which all must be non-negative. Certain other properties are pertinent to phasing which is implicit in one-matrix reconstruction governed by entropy maximization. Throughout this work reference is made to informational entropy, not the entropy of thermodynamics.
Quantum embedding based on the (one-electron-reduced) density matrix is revisited by means of the unitary Householder transformation. While being exact and equivalent to (but formally simpler than) density matrix embedding theory (DMET) in the noninteracting case, the resulting Householder-transformed density matrix functional embedding theory (Ht-DMFET) preserves, by construction, the single-particle character of the bath when electron correlation is introduced. In Ht-DMFET, the projected ``
impurity+bath'' cluster's Hamiltonian (from which approximate local properties of the interacting lattice can be extracted) becomes an explicit functional of the density matrix. In the spirit of single-impurity DMET, we consider in this work a closed (two-electron) cluster constructed from the full-size noninteracting density matrix. When the (Householder-transformed) interaction on the bath site is taken into account, per-site energies obtained for the half-filled one-dimensional Hubbard lattice match almost perfectly the exact Bethe ansatz results in all correlation regimes. In the strongly correlated regime, the results deteriorate away from half-filling. This can be related to the electron number fluctuations in the (two-site) cluster which are neither described in Ht-DMFET nor in regular DMET. As expected, the per-site energies dramatically improve when increasing the number of embedded impurities. Formal connections with density/density matrix functional theories have been briefly discussed and should be explored further. Work is currently in progress in this direction.