With the use of the density equation [H. Nakatsuji, Phys. Rev. A 14, 41 (1976)], the second-order density matrices are directly determined without any use of the wave functions. The third- and fourth-order reduced density matrices (RDM's) are decoupled into lower-order ones using the Green's function technique. This method is applied to Be, Ne, ${\mathrm{H}}_{2}$O, ${H}_{3}{O}^{+}$, N${\mathrm{H}}_{3}$, C${\mathrm{H}}_{4}$, ${\mathrm{BH}}_{4}^{\phantom{\rule{0ex}{0ex}}\ensuremath{-}}$, ${\mathrm{NH}}_{4}^{\phantom{\rule{0ex}{0ex}}+}$, and C${\mathrm{H}}_{3}$F, and the results are successfully compared with the full configuration interaction results. The convergence is fairly good, and the calculated second-order RDM's almost satisfy the necessary conditions of the N representability, the P, Q, and G conditions, and the first-order RDM's are exactly N representable. These results show that the present method is very promising.
With the use of the density equation [H. Nakatsuji, Phys. Rev. A 14, 41 (1976)], the second-order density matrices are directly determined without any use of the wave functions. The third- and fourth-order reduced density matrices (RDM's) are decoupled into lower-order ones using the Green's function technique. This method is applied to Be, Ne, H2O, H3O+, NH3, CH4, BH4\ensuremath−, NH4+, and CH3F, and the results are successfully compared with the full configuration interaction results. The convergence is fairly good, and the calculated second-order RDM's almost satisfy the necessary conditions of the N representability, the P, Q, and G conditions, and the first-order RDM's are exactly N representable. These results show that the present method is very promising.
The methods used to determine the reduced density matrix (RDM) of the ground and excited states, the finite-temperature systems, and the large systems without using the wave function by solving the density equation were discussed. We examined the foundations to reconstruct the higher-order RDMs of the ground and excited states and the finite-temperature systems in terms of the lower-order RDMs. We presented the equation to determine the RDMs of the finite-temperature systems directly and showed that only the exact RDMs satisfy the equation. Our previous approximation for third- and fourth-order RDMs of the ground state [H. Nakatsuji and K. Yasuda, Phys. Rev. Lett. 76, 1039 (1996)] was reformulated, and the accuracy of this approximation for the excited states was examined. The structure of the nth order energy density matrix (n-EDM) was analyzed, and the calculation method which sums up the Parquet diagram of the 2-EDM without explicitly constructing the third- and fourth-order RDMs was reported. This approximation is more accurate than the previous second-order approximation and also includes the infinite series of bubble and ladder Green's function diagrams. Such a method is necessary to apply the density-equation method to large systems, such as polymers, metals, and semiconductors. The new approximation together with the density equation was applied to the ground states of some molecules including CO, C2H2,C3H8, and C4H10, and the excited states of the Be atom and Li2 molecule. The calculated energies were as accurate as the exact or coupled-cluster single and double excitations with triples included noniteratively, and the energy errors of the second-order approximation were significantly reduced. The calculated 2-RDMs almost satisfied important representability conditions while the 1-RDMs were exactly ensemble representable. These results demonstrate that the density equation offers a new quantitative method for treating electron correlations. The relationship between the iterative procedure and the finite-temperature density-equation method was discussed.
The general equation linking a q-order reduced density matrix (q-RDM) with the corresponding q-order hole reduced density matrix (q-HRDM) has been reported in the preceding paper [F. Colmenero, C. Perez del Valle, and C. Valdemoro, preceding paper, Phys. Rev. A 47, 971 (1993)]. In this equation, neither Kronecker \ensuremath{\delta} functions nor mixed products of elements of RDM's and HRDM's appear, and the part involving hole terms has the same structure as that involving particle terms. Recently [C. Valdemoro, Phys. Rev. A 45, 4462 (1992)], a similar equation for q=2 has been used to approximate the 2-RDM from the knowledge of the 1-RDM. Here, the 3-RDM and the 4-RDM are approximated by using the 2-RDM as an initial datum. The ultimate aim of this research is to develop an iterative method for solving the contracted form of the Schr\"odinger equation [L. Cohen and C. Frishberg, Phys. Rev. A 13, 927 (1976); H. Nakatsuji, ibid. 14, 41 (1976)]. A matrix form in the orbital representation of this equation is reported here. Finally, the second-order hypervirial equation, also in its matrix form, has been derived so that the quality of the results can be judged.
Two-electron reduced density matrices (2-RDMs) of many-electron molecules are directly determined without calculation of their wave functions by solving the anti-Hermitian contracted Schrödinger equation. Approximation of the 3-RDM in the anti-Hermitian contracted Schrödinger equation by a corrected cumulant expansion [Mazziotti, Phys. Rev. A 60, 3618 (1999)] permits the direct calculation of the energy and 2-RDM with many high-order correlation effects included. The method is illustrated for the molecules BeH2, H2O, NH3, CH4, and CO as well as the dissociation of BH. Correlation energies are obtained within 95%-100% of full-configuration interaction, and 2-RDMs very nearly satisfy known N-representability conditions.
The contracted Schr\"odinger equation (CSE) technique through its direct determination of the two-particle reduced density matrix (2RDM) without the wave function may offer a fresh alternative to traditional many-body quantum calculations. Without additional information the CSE, also known as the density equation, cannot be solved for the 2RDM because it also requires a knowledge of the 4RDM. We provide theoretical foundations through a reconstruction theorem for recent attempts at generating higher RDMs from the 2RDM to remove the indeterminacy of the CSE. With Grassmann algebra a more concise representation for Valdemoro's reconstruction functionals [F. Colmenero, C. Perez del Valle, and C. Valdemoro, Phys. Rev. A 47, 971 (1993)] is presented. From the perspective of the particle-hole equivalence we obtain Nakatsuji and Yasuda's correction for the 4RDM formula [H. Nakatsuji and K. Yasuda, Phys. Rev. Lett. 76, 1039 (1996)] as well as a corrective approach for the 3RDM functional. A different reconstruction strategy, the ensemble representability method (ERM), is introduced to build the 3- and 4-RDMs by enforcing four-ensemble representability and contraction conditions. We derive the CSE in second quantization without Valdemoro's matrix contraction mapping and offer the first proof of Nakatsuji's theorem for the second-quantized CSE. Both the functional and ERM reconstruction strategies are employed with the CSE to solve for the energies and the 2RDMs of a quasispin model without wave functions. We elucidate the iterative solution of the CSE through an analogy with the power method for eigenvalue equations. Resulting energies of the CSE methods are comparable to single-double configuration-interaction (SDCI) energies, and the 2RDMs are more accurate by an order of magnitude than those from SDCI. While the CSE has been applied to systems with 14 electrons, we present results for as many as 40 particles. Results indicate that the 2RDM remains accurate as the number of particles increases. We also report a direct determination of excited-state 2RDMs through the CSE. By circumventing the wave function, the CSE presents new possibilities for treating electron correlation.
Preface. I: Properties of Reduced Density Matrices. 1. RDMs: How Did We Get Here? A.J. Coleman. 2. Some Theorems on Uniqueness and Reconstruction of Higher-Order Density Matrices M. Rosina. 3. Cumulant Expansions of Reduced Densities, Reduced Density Matrices, and Green's Functions P. Ziesche. 4. On Calculating Approximate and Exact Density Matrices R. Erdahl, B. Jin. II: The Contracted Schroedinger Equation. 5. Density Equation Theory in Chemical Physics H. Nakatsuji. 6. Critical Questions Concerning Iterative Solution of the Contracted Schroedinger Equation C. Valdemoro, L.M. Tel, E. Perez-Romero 7. Cumulants and the Contracted Schroedinger Equation D.A. Mazziotti.. III: Density Matrix Functional Theory. 8. Natural Orbital Functional Theory S. Goedecker, C.J. Umrigar. 9. The Pair Density in Approximate Density Functional Theory: The Hidden Agent N.T. Maitra, K. Burke. 10. Functional N-representability in Density Matrix and Density Functional Theory: An Illustration for Hooke's Atom E.V. Ludena, V. Karasiev, A. Artemiev, D. Gomez.. IV: Electron Intracule and Extracule Densities. 11. Intracule and Extracule Densities: Historical Perspectives and Future Prospects E. Valderrama, J.M. Ugalde, R.J. Boyd. 12. Topology of Electron Correlation J. Cioslowski. 13. Electron-Pair Densities of Atoms T. Koga. Index.
The commutation-anticommutation relations of q-electron operators imply a set of N representability conditions [A. J. Coleman, Rev. Mod. Phys. 31, 668 (1963)] for the corresponding q-order reduced density matrices (q-RDM) [C. Valdemoro, An. Fis. 79, 95 (1983); in Structure, Interaction and Reactivity, edited by S. Fraga (Elsevier, Amsterdam, 1992)]. From these conditions, a general and closed-form relation is obtained here. In this equation the part involving RDM's has the same structure as that involving hole reduced density matrices. This relation is the basis of a method for approximating a q-RDM in terms of the r-RDM's [C. Valdemoro, Phys. Rev. A 45, 4462 (1992)] with rq. The derivation of this relation can be simplified significantly by employing a graph method which is described here. These graphs are in a one-to-one correspondence with the elements of the symmetric group of permutations.