The variational two-electron reduced density matrix (v2RDM) method is generalized for the description of total angular momentum ( J ) and projection of total angular momentum ( M J ) states in atomic systems described by nonrelativistic Hamiltonians, and it is shown that the approach exhibits serious deficiencies. Under ensemble N -representability constraints, v2RDM theory fails to retain the appropriate degeneracies among various J states for fixed spin ( S ) and orbital angular momentum ( L ), and for fixed L, S, and J, the manifold of M J states is not necessarily degenerate. Moreover, a substantial energy error is observed for a system for which the two-electron reduced density matrix is exactly ensemble N -representable; in this case, the error stems from violations in pure-state N -representability conditions. Unfortunately, such violations do not appear to be good indicators of the reliability of energies from v2RDM theory in general. Several states are identified for which energy errors are near zero and yet pure-state conditions are clearly violated.
We establish necessary and sufficient conditions for the N-representability of the universal one-electron reduced density matrix functional. Functionals satisfying these conditions are guaranteed to yield variational upper bounds on the true energy in one-electron reduced density matrix functional theory, regardless of the strength of the interparticle repulsion. Conversely, any functional violating these conditions will necessarily underestimate the true energy for certain systems. These exact constraints impose a stringent restriction on density matrix functional approximations, as many existing functionals-including the Hartree-Fock functional-appear to violate them. This mathematical formalism, therefore, can guide the development of new approximate functionals and numerical algorithms.
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
T2 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
T2 condition that significantly decreases the size of the largest block in the
T2 matrix. The
T2 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
T2 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
N2. Results are compared with the 2RDM method without the
T2 condition as well as several wave-function methods.
Three strict constraints upon the electron-electron repulsion energy functional of the one-electron reduced density matrix (the 1-matrix) are obtained by combining its invariance and stationary properties with the extended Koopmans' theorem. The constraints relate the quantities derived from the functional pertaining to an N-electron system with those of its (N-1)-electron congener. Together with the N-representability requirement for the 1-matrix of the congener, identities involving the electron-electron repulsion energies of the two systems and their derivatives with respect to the 1-matrices seriously narrow down the choices for potential approximate density-matrix functionals. This fact is well illustrated in the case of two-electron systems, where the validity of the new constraints is confirmed and found to originate from a nontrivial cancellation among different terms. Thus, the constraints provide a new tool for the construction and testing of new functionals that complements the previously known conditions such as the reproduction of the homogeneous gas energies and momentum distributions, convexity, and the N-representability of the associated 2-matrices.
Molecular two-electron reduced density matrices (2-RDMs) are computed variationally without the many-electron wave function by constraining the 2-RDM with a set of three-particle
N-representability conditions known as three-positivity conditions. These constraints restrict four distinct three-particle probability distributions, which can be defined for any
N-particle system, to be nonnegative. The variational calculation of the 2-RDM with full three-positivity conditions is implemented with the first-order semidefinite programming algorithm [D.A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. We also derive and implement a generalization of the
T2 representability condition, which is a subset of the three-positivity conditions. Energies and 2-RDMs are computed for several molecules as well as the nitrogen molecule at both equilibrium and nonequilibrium geometries. The ground-state energies for nitrogen are within
0.0015a.u. of the values from full configuration interaction at all internuclear distances.
Usually, variational calculations for the second-order reduced density matrix are performed subject to the constraint that the ``
P'' and ``
Q'' matrices are positive semidefinite, which only constrains the lowest eigenvalue of these matrices. We characterize the highest eigenvalue of these matrices and discuss how the associated constraint (which is related to the ground-state energy of the Hamiltonians $H=\ensuremath{-}P$ and $H=\ensuremath{-}Q$) can be implemented in practical calculations. This necessary condition for
N-representability should help ensure that the second-order reduced density matrix is not ``overcorrelated.''