Two-electron Reduced Density Matrix Research Articles

Reduced-density-matrix cumulants and correlation

Spin components for the Grassmann product of the one-electron reduced density matrix with itself are compared with those of the two-electron reduced density matrix for the same state. It is found that the Grassmann product introduces some inappropriate behaviors which must be removed by the cumulant but are not what is usually thought of as correlation.

First-order semidefinite programming for the two-electron treatment of many-electron atoms and molecules

The ground-state energy and properties of any many-electron atom or molecule may be rigorously computed by variationally computing the two-electron reduced density matrix rather than the many-electron wavefunction. While early attempts fifty years ago to compute the ground-state 2-RDM directly were stymied because the 2-RDM must be constrained to represent an N -electron wavefunction, recent advances in theory and optimization have made direct computation of the 2-RDM possible. The constraints in the variational calculation of the 2-RDM require a special optimization known as a semidefinite programming. Development of first-order semidefinite programming for the 2-RDM method has reduced the computational costs of the calculation by orders of magnitude [Mazziotti, Phys. Rev. Lett. 93 (2004) 213001]. The variational 2-RDM approach is effective at capturing multi-reference correlation effects that are especially important at non-equilibrium molecular geometries. Recent work on 2-RDM methods will be reviewed and illustrated with particular emphasis on the importance of advances in large-scale semidefinite programming.

Computation of dipole, quadrupole, and octupole surfaces from the variational two-electron reduced density matrix method

Recent advances in the direct determination of the two-electron reduced density matrix (2-RDM) by imposing known N-representability conditions have mostly focused on the accuracy of molecular potential energy surfaces where multireference effects are significant. While the norm of the 2-RDM's deviation from full configuration interaction has been computed, few properties have been carefully investigated as a function of molecular geometry. Here the dipole, quadrupole, and octupole moments are computed for a range of molecular geometries. The addition of Erdahl's T2 condition [Int. J. Quantum Chem. 13, 697 (1978)] to the D, Q, and G conditions produces dipole and multipole moments that agree with full configuration interaction in a double-zeta basis set at all internuclear distances.

Variational Reduced-Density-Matrix Theory Applied to the Potential Energy Surfaces of Carbon Monoxide in the Presence of Electric Fields

The variational optimization of the energy with respect to the two-electron reduced-density matrix (2-RDM), constrained by N-representability conditions, can determine the shape of molecular potential energy surfaces with useful accuracy. In this paper, we apply the 2-RDM method with a first-order optimization algorithm [Mazziotti, D. A. Phys. Rev. Lett. 2004, 93, 213001] to investigating the potential energy surfaces of carbon monoxide in the presence and absence of an electric field. Two beneficial characteristics of the 2-RDM method for computing potential energy surfaces include the following: (i) its ability to capture multireference effects without specifying any reference wave function or density matrix and (ii) its guarantee of a global energy minimum in the variational optimization. The 2-RDM method produces electronic ground-state energies with similar accuracy at equilibrium and nonequilibrium geometries in both the presence and the absence of the electric field. Computed dipole moments are similar in accuracy to the values from the computationally expensive configuration interaction with single, double, triple, and quadruple excitations. These surfaces have important applications in quantum molecular control theory.

Quantum Chemistry without Wave Functions: Two-Electron Reduced Density Matrices

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.

Variational two-electron reduced density matrix theory for many-electron atoms and molecules: Implementation of the spin- and symmetry-adaptedT2condition through first-order semidefinite programming

The energy and properties of a many-electron atom or molecule may be directly computed from a variational optimization of a two-electron reduced density matrix (2RDM) that is constrained to represent many-electron quantum systems. In this paper we implement a variational 2RDM method with a representability constraint, known as the ${T}_{2}$ condition. The optimization of the 2RDM is performed with a first-order algorithm for semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] which, because of its lower computational cost in comparison to second-order methods, allows the treatment of larger basis sets. We also derive and implement a spin- and symmetry-adapted formulation of the ${T}_{2}$ condition that significantly decreases the size of the largest block in the ${T}_{2}$ matrix. The ${T}_{2}$ condition, originally derived by Erdahl [Int. J. Quantum Chem. 13, 697 (1978)], was recently applied via a second-order algorithm to atoms and molecules [Z. Zhao et al., J. Chem. Phys. 120, 2095 (2004)]. While these calculations were restricted to molecules at equilibrium geometries in minimal basis sets, we apply the 2RDM method with the ${T}_{2}$ condition to compute the electronic energies of molecules in both minimal and nonminimal basis sets at equilibrium as well as nonequilibrium geometries. Accurate potential energies curves are produced for BH, HF, and ${\mathrm{N}}_{2}$. Results are compared with the 2RDM method without the ${T}_{2}$ condition as well as several wave-function methods.

Application of variational reduced-density-matrix theory to organic molecules

Variational calculation of the two-electron reduced-density matrix (2-RDM), using a new first-order algorithm [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)], is applied to medium-sized organic molecules. The calculations reveal systematic trends in the accuracy of the energy with well-known chemical concepts such as hybridization, electronegativity, and atomic size. Furthermore, correlation energies from hydrocarbon chains indicate that the calculation of the 2-RDM subject to two-positivity conditions is size extensive, that is, the energy grows linearly with the number of electrons. Because organic molecules have a well-defined set of functional groups, we employ the trends in energy accuracy of the functional groups to design a correction to the 2-RDM energy for an arbitrary organic molecule. We apply the 2-RDM calculations with the functional-group correction to a large set of organic molecules with different functional groups. Energies with millihartree accuracy are obtained both at equilibrium and nonequilibrium geometries.

Extensivity and the contracted Schrödinger equation

We provide an extensive formulation of the contracted Schrödinger equation and other reduced eigenvalue equations. Nonextensive (unconnected) terms in these equations cancel exactly, leading to completely connected one- and two-electron equations that together are equivalent to the Schrödinger equation. We discuss how these equations can be solved for the one- and two-electron cumulants. These cumulants yield a two-electron reduced density matrix that is necessarily size consistent, even for an approximate solution. A diagram technique, introduced to aid the formal manipulations, clarifies the connection between density matrix reconstruction and solution of the CSE.

Application of two-electron reduced density matrices for determining correlation energy

The simplest mathematical object containing electron correlation is the two-electron reduced density matrix (2-matrix). Recently an efficient algorithm has been proposed for the construction of a perfectly N-representable 2-matrix (J. Mol. Struct. (Theochem), 501–502 (2000) 125). The method for energy optimization based on this algorithm is equivalent to an unrestricted full-CI scheme. The economic behavior of the algorithm allows us to extend the size range of computations. In this paper, a brief description of the method and illustrative calculations are presented.

Energy partitioning using two-electron average matrices

A partition of the molecular energy using the two-electron Hamiltonian matrix as well as the two-electron reduced density matrix for any type of wavefunction is proposed. Consequently, the molecular structure is analyzed using the pair electron distribution only. The dissociations of the Li 2 and N 2 molecules using different types of wavefunctions are studied according to this energy partition.