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Physical Review A

Approximatingq-order reduced density matrices in terms of the lower-order ones. II. Applications

Francisco Colmenero, Carmela Valdemoro

Published Feb 1, 1993• 210 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.47.979

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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.
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.
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.
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, H2{\mathrm{H}}_{2}H2​O, H3O+{H}_{3}{O}^{+}H3​O+, NH3{\mathrm{H}}_{3}H3​, CH4{\mathrm{H}}_{4}H4​, ${\mathrm{BH}}_{4}^{\phantom{\rule{0ex}{0ex}}\ensuremath{-}}$, NH4+{\mathrm{NH}}_{4}^{\phantom{\rule{0ex}{0ex}}+}NH4+​, and CH3{\mathrm{H}}_{3}H3​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.
For 50 years, progress toward the direct calculation of the ground-state two-electron reduced density matrix (2-RDM) was stymied from an inability to constrain the 2-RDM to represent an N-electron wave function. Recent advances in theory and optimization have realized the direct calculation of the 2-RDM. A variational 2-RDM procedure, using first-order semidefinite programming, has been shown to capture multireference correlation effects important at nonequilibrium geometries [Mazziotti, Phys. Rev. Lett. 2004, 93, No. 213001]. This method emerged from research on a nonvariational calculation of the 2-RDM by the contracted Schrödinger equation. Both approaches will be discussed and illustrated.
Quantum phase transitions are explored with reduced-density-matrix (RDM) mechanics. While in wave mechanics the quantum phase transition is identified by a crossing or avoided crossing between ground- and excited-state energies, in RDM mechanics the transition is characterized by movement of the ground-state two-electron RDM (2-RDM) along the boundary of the convex set of 2-RDMs between regions with dramatically different expectation values (order parameters) of one or more operators. With recent advances the ground-state 2-RDM can be directly computed without the many-particle wave function by variational optimization of the energy with the 2-RDM [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Because the variational calculation of the 2-RDM does not depend on a reference wave function, it can accurately predict the energies and properties of a system both near and far from the quantum phase transition.
An improved parametrization of the two-electron reduced density matrix (2-RDM) [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)] was recently shown to yield energies and properties that are markedly better than those calculated by traditional ab initio methods of similar computational scaling. In this paper a family of such energy functionals, generalizing the ones obtained previously, is derived through the use of (i) ppp-particle contraction relations based on the contraction of the cumulant expansions of ppp-particle RDMs and (ii) Cauchy-Schwarz relations that arise from an important set of NNN-representability constraints known as the two-positivity conditions. The 2-RDMs are explicitly parameterized in terms of the first-order part of the cumulant 2-RDM and, for the inclusion of single excitations, a second-order part of the 1-RDM. In contrast to earlier formulations based on the coefficients from configuration interaction with single and double excitations (CISD), the cumulant-based parametric 2-RDM methods, from the properties of cumulants, are rigorously size extensive. We also show that writing the energy functionals in terms of correlated 1-RDMs and cumulant 2-RDMs reduces the computational cost of the parametric 2-RDM methods to that of CISD. Applications are made to ground-state energies of several molecules, equilibrium bond distances, and frequencies of HF, F2{\mathrm{F}}_{2}F2​, and CO, the relative energy of the cis and trans isomers of HO3{\mathrm{HO}}_{3}HO3​${}^{\ensuremath{-}}$, and the HCN-HNC isomerization reaction. For bond breaking in hydrogen fluoride the improved and more efficient parametric 2-RDM methods yield energies with similar accuracies at both equilibrium and nonequilibrium geometries in 6-31G** and polarized valence quadruple-$\ensuremath{\zeta}$ basis sets. Computed 2-RDMs very nearly satisfy well-known NNN-representability conditions.