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The Journal of Chemical Physics

Linear scaling and the 1,2-contracted Schrödinger equation

David A. Mazziotti

Published Nov 8, 2001• 40 citations
Publisher

Abstract

A contracted Schrödinger equation (1,2-CSE) is derived for the class of Hamiltonians without explicit interactions including those from Hartree–Fock and density functional theories. With cumulant reconstruction of the two-particle reduced density matrix (2-RDM) from the one-particle-RDM (1-RDM), the 1,2-CSE may be expressed solely in terms of the 1-RDM. We prove that a 1-RDM satisfies the 1,2-CSE if and only if it is an eigenstate of the N-particle Schrödinger equation. The 1,2-CSE is solved through the development and implementation of a reduced, linear-scaling analog of the ordinary power method for finding matrix eigenvalues. The power formula for updating the 1-RDM requires fewer matrix operations than the gradient procedure derived by Li et al. [Phys. Rev. B 47, 10891 (1993)] and Daw [Phys. Rev. B 47, 10895 (1993)]. Convergence of the contracted power method with purification is illustrated with several molecules. While providing a new tool for semiempirical, Hartree–Fock, and density functional calculations, the 1,2-CSE also represents an initial step toward a linear-scaling algorithm for solving higher CSEs which explicitly treat electron correlation.

DOI
10.1063/1.1412002

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A contracted Schrödinger equation (1,2-CSE) is derived for the class of Hamiltonians without explicit interactions including those from Hartree–Fock and density functional theories. With cumulant reconstruction of the two-particle reduced density matrix (2-RDM) from the one-particle-RDM (1-RDM), the 1,2-CSE may be expressed solely in terms of the 1-RDM. We prove that a 1-RDM satisfies the 1,2-CSE if and only if it is an eigenstate of the N-particle Schrödinger equation. The 1,2-CSE is solved through the development and implementation of a reduced, linear-scaling analog of the ordinary power method for finding matrix eigenvalues. The power formula for updating the 1-RDM requires fewer matrix operations than the gradient procedure derived by Li et al. [Phys. Rev. B 47, 10891 (1993)] and Daw [Phys. Rev. B 47, 10895 (1993)]. Convergence of the contracted power method with purification is illustrated with several molecules. While providing a new tool for semiempirical, Hartree–Fock, and density functional calculations, the 1,2-CSE also represents an initial step toward a linear-scaling algorithm for solving higher CSEs which explicitly treat electron correlation.
The power method for solving N-particle eigenvalue equations is contracted onto the two-particle space to produce a reduced “variational” method for solving the contracted Schrödinger equation (CSE), also known as the density equation. In contrast to the methods which solve a system of approximate nonlinear equations to determine the two-particle reduced density matrix (2-RDM) nonvariationally, the contracted power method updates the 2-RDM iteratively through a “gradient” of the N-particle energy. After each power iteration we modify the 2-RDM to satisfy certain N-representability conditions through an extension of purification to correlated RDMs. The contracted power method is illustrated with a variety of molecules. Significant features of the present calculations include (i) accurate results for both first- and second-order functionals for building the 3- and the 4-RDM’s from the 2-RDM’s; (ii) the first molecular implementation of the Mazziotti correction within the CSE [Mazziotti, Phys. Rev. A 60, 3618 (1999)]; (iii) a spin–orbital formulation; (iv) the treatment of both core and valence orbitals as active; and; (v) a reduction of the CSE computational scaling through fast summation and the natural-orbital transformation.
Correlation energies and reduced density matrices (RDMs) of atoms and molecules are directly computed by solving the 1,3-contracted Schr\"odinger equation (1,3-CSE). The solution of the 1,3-CSE synthesizes two optimization strategies recently employed for the direct determination of the 2-RDM: (i) variational minimization of the energy with respect to a 2-RDM constrained by positivity conditions [D. A. Mazziotti, Phys. Rev. A 65,65,65, 062511 (2002)] and (ii) the contracted power method for solving the 2,4-CSE [D. A. Mazziotti, J. Chem. Phys. 116,116,116, 1239 (2002)]. While both the 3- and the 4-RDMs in the 2,4-CSE are reconstructed from the 2-RDM by cumulant expansions, similar techniques cannot be directly applied to the 1,3-CSE because constructing the 2-RDM from the 1-RDM with cumulant theory does not improve upon the mean-field approximation. We, however, establish a unique mapping from the 1-RDM to the 2-RDM by searching for the 2-RDM, constrained by contraction and N-representability conditions, which minimizes the energy. The 2-RDM constrained search is practically implemented through recent advances in positive semidefinite programming. With the variational reconstruction of the 2-RDM and a cumulant reconstruction of the 3-RDM, the 1,3-CSE may be solved via a contracted power method for the ground-state energy and RDMs. The initial RDMs, it is shown, need not be N representable for the contracted power method to converge; this allows us to choose the original RDMs from a variational calculation with approximate N-representability conditions on the 2-RDM. Application of the 1,3-CSE algorithm to atoms and molecules yields highly accurate correlation energies both near and far from equilibrium geometries.
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.
The theory of the contracted Schr\"odinger equation (CSE) [D. A. Mazziotti, Phys. Rev. A 57, 4219 (1998)] is connected with traditional methods of electronic structure including configuration-interaction (CI) and coupled-cluster (CC) theory. We derive a transition contracted Schr\"odinger equation (TCSE) which depends on the wave function $\ensuremath{\psi}$ as well as another N-particle function $\ensuremath{\chi}$ through the two-, three-, and four-particle reduced transition matrices (RTMs). By reconstructing the 3 and 4 RTMs approximately from the 2-RTM, the indeterminacy of the equation may be removed. The choice of the reconstruction and the function $\ensuremath{\chi}$ determines whether one obtains the CI, CC, or CSE theory. Through cumulant theory and Grassmann algebra we derive reconstruction formulas for the 3- and 4-RTMs which generalize both the reduced density matrix (RDM) cumulant expansions as well as the exponential ansatz for the CC wave function. This produces a fresh approach to CC theory through RTMs. Two theoretical differences between the CC and the CSE theories are established for energetically nondegenerate states: (i) while the CSE has a single exact solution when the 3- and 4-RDMs are N-representable, the CC equations with N-representable 3- and 4-RTMs have a family of solutions. Thus, N-representability conditions offer a medium for improving the CSE solution but not the CC solution, and (ii) while the 2-RDM for an electronic Hamiltonian reconstructs to unique N-representable 3- and 4-RDMs, the 2-RTM builds to a family of N-representable 3- and 4-RTMs. Hence, renormalized reconstructions beyond the cumulant expansion may be developed for the 2-RDM but not for the 2-RTM without explicit use of the Hamiltonian. In the applications we implement our recently developed reconstruction formula for the 3-RDM which extends beyond the cumulant approximation. Calculations compare the 3-RDM and 3-RTM reconstructions for the molecules LiH, BeH2,{\mathrm{BeH}}_{2},BeH2​, BH3,{\mathrm{BH}}_{3},BH3​, and H2O{\mathrm{H}}_{2}\mathrm{O}H2​O as well as for systems with more general two-particle interactions. The TCSE offers a unified approach to electronic structure.
A recent advance in the theory of the contracted Schr\"odinger equation (CSE), in which only the anti-Hermitian part of the equation is solved, permits the direct determination of ground-state two-electron reduced density matrices (2-RDM's) that yield 95%--100% of the correlation energy of atoms and molecules [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. Here we discuss in detail the anti-Hermitian contracted Schr\"odinger equation (ACSE) and its comparison to the CSE with regard to cumulant reconstruction of RDM's, the role of Nakatsuji's theorem, and the structure of the wave function. The ACSE is also formulated in the Heisenberg representation and related to canonical diagonalization. The solution of the ACSE is illustrated with a variety of molecules including H2O{\mathrm{H}}_{2}\mathrm{O}H2​O, CH2{\mathrm{CH}}_{2}CH2​, NH4+{{\mathrm{NH}}_{4}}^{+}NH4​+, HF\mathrm{HF}HF, and N2{\mathrm{N}}_{2}N2​, and potential energy and dipole-moment surfaces are computed for boron hydride in a polarized double-$\ensuremath{\zeta}$ basis set. The computed 2-RDM's very closely satisfy known NNN-representability conditions.
The solution of N-representability for the two-particle reduced density matrix (2-RDM) is intimately connected through the contracted Schr\"odinger equation with the problem of reconstructing the 3- and 4-RDM's from the 2-RDM [D. A. Mazziotti, Phys. Rev. A 57, 4219 (1998)]. We derive an explicit formula for building the 3-RDM from the 2-RDM which significantly improves the accuracy of the 3-RDM cumulant expansion. Calculations indicate that the improvement is better than that achieved by the correction of Nakatsuji and Yasuda (NY) [Phys. Rev. Lett. 76, 1039 (1996); Phys. Rev. A 56, 2648 (1997)]. We also present a more efficient formulation of the NY correction. Both the present and NY reconstructions are shown to satisfy the particle-hole duality. Comparison of the reconstruction methods is made through several molecules LiH, BeH2,{\mathrm{BeH}}_{2},BeH2​, BH3,{\mathrm{BH}}_{3},BH3​, H2O,{\mathrm{H}}_{2}\mathrm{O},H2​O, and CH4,{\mathrm{CH}}_{4},CH4​, as well as systems with more general pairwise correlation.