k -particle cumulants λk (for 2⩽k⩽n) corresponding to the k-particle reduced density matrices γk for an n-fermion system are defined via a generating function. The two-particle cumulant λ2 describes two-particle correlations (excluding exchange), λ3 genuine three-particle correlations etc. The properties of these cumulants are analyzed. Conditions for vanishing of certain λk are formulated. Necessary and sufficient for λ2=0 is the well-known idempotency condition γ2=γ for γ≡γ1. For λ3=0 to hold, a general necessary condition is Tr{2γ3−3γ2+γ}=0, for three special forms of the wave function (arbitrary two-electron state, antisymmetrized product of strongly orthogonal geminals on antisymmetrized geminal power wave function of extreme type) 2γ3−3γ2+γ=0 turns out to be necessary and sufficient. For a multiconfiguration self-consistent field wave function the only nonvanishing matrix elements of the cumulants are those where all labels refer to active (partially occupied) spin orbitals. Spin-free cumulants Λk corresponding to the spin-free reduced density matrices Γk are also defined and analyzed. The main interest in the density cumulants is in connection with the recently formulated normal ordering and the corresponding Wick theorem for arbitrary reference functions, but they are also useful for an analysis of electron correlation.
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)
p-particle contraction relations based on the contraction of the cumulant expansions of
p-particle RDMs and (ii) Cauchy-Schwarz relations that arise from an important set of
N-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, and CO, the relative energy of the cis and trans isomers of
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
N-representability conditions.
The fundamentality of the exponential representation of a second-quantised correlated wave function is emphasised with an accent on the physical sense of cluster amplitudes as cumulants of the correlated ansatz. Three main wave function formalisms, namely, the configuration-interaction theory, the coupled-cluster approach, and the many-body perturbation theory (as well as their extensions, e.g. the equation-of-motion coupled-cluster method, multireference schemes, etc.), are represented in an exponential form, leading to a formulation of the working equations in terms of cluster amplitudes. By expressing the corresponding many-body tensor equations in terms of cluster amplitudes, we could unambiguously check connectivity types and the asymptotic behaviour of all tensors/scalars involved (in the formal limit of an infinite number of correlated particles). In particular, the appearance of disconnected cluster amplitudes corresponds to unphysical correlations. Besides, we demonstrate that the equation-of-motion coupled-cluster approach, as well as certain excited-state configuration-interaction methods, can be recast in a fully connected (exponential) form, thus breaking the common belief that all truncated configuration-interaction methods violate connectivity. Our work is based on the recently developed algebraic framework which can be viewed as a complement to the classical diagrammatic analysis.
The one-particle reduced density-matrix (1-RDM) functional theory is a promising alternative to density-functional theory (DFT) that uses the 1-RDM rather than the electronic density as a basic variable. However, long-standing challenges such as the lack of the Kohn-Sham scheme and the complexity of the pure N-representability conditions are still impeding its wild utilization. Fortunately, ensemble N-representability conditions derived in the natural orbital basis are known and trivial such that almost every functional of the 1-RDM is actually natural orbital functional, which does not perform well for all the correlation regimes. In this work, we propose a variational minimization scheme in the ensemble N-representable domain that is not restricted to the natural orbital representation of the 1-RDM. We show that splitting the minimization into the diagonal and off-diagonal parts of the 1-RDM can open the way toward the development of functionals of the orbital occupations, which remains a challenge for the generalization of site-occupation functional theory in chemistry. Our approach is tested on the uniform Hubbard model using the Müller and the Töws-Pastor functionals, as well as on the dihydrogen molecule using the Müller functional.
Electronic excited states are computed through a technique for extracting them from a knowledge of the ground-state two-particle reduced density matrix (2-RDM). Diagonalization of the Hamiltonian in the basis set of one-particle excitations, deexcitations, and projections from the ground-state wave function yields those excited states dominated by one-particle excitations; however, explicit evaluation of the Hamiltonian requires a knowledge of the ground-state 4-RDM. Two techniques are developed for evaluating the Hamiltonian elements with only a knowledge of the 2-RDM: (i) the cumulant method in which the higher RDMs are reconstructed from the 2-RDM via cumulant theory and (ii) the Hermitian or anti-Hermitian operator method in which the higher RDMs are eliminated by selecting the basis functions to be Hermitian or anti-Hermitian. These techniques are extendable to the diagonalization of the Hamiltonian in basis sets of m-excitations where
m>1. Excited-state energies with an accuracy between 0.1% and 0.001% of their absolute values are determined in illustrative calculations on Be, LiH,
H2O, and BH.
Strong electron correlation in materials is explored within a class of model Hamiltonians that treat only pair interactions between electrons. The model is unique among typical spin Hamiltonians in that it does not have an effective mean-field reference wave function. The ground-state wave functions from all Hamiltonians in the model have the same one-electron reduced density matrix (1-RDM); consequently, one-electron theories such as the Hartree–Fock and density functional theories are inapplicable. In contrast, the ground-state two-electron reduced density matrix (2-RDM) has a one-to-one mapping to the ground-state wave function. For a range of lattices including linear, ladder, and square topologies, we variationally compute the 2-RDM subject to constraints, known as N -representability conditions, that are necessary for the 2-RDM to represent an N -electron ensemble density matrix. We find that for all model Hamiltonians the 2-RDM is accurately computed as long as the D, Q, and G N -representability conditions are supplemented with the T 2 condition. Energies, orbital and geminal occupation numbers, and correlation functions are computed. The model has applications to superconductivity as well as more general pairing phenomena in electronic systems. Effective methods are needed to treat strong electron correlation arising in the study of materials including the study of high-temperature superconductivity and phase transitions.