The energy of any atom or molecule with an even number N of electrons is shown to be an exact functional of a single geminal where the functionals for both the kinetic energy and the external potential are explicitly known. We derive the foundations for geminal functional theory (GFT) through a generalized constrained search and the use of two theorems which demonstrate that all one-particle properties of atoms and molecules with even N may be parametrized by a single geminal [A. J. Coleman, Int. J. Quantum Chem. 63, 23 (1997); D. W. Smith, Phys. Rev. 147, 896 (1966)]. By generalizing constrained search to optimize the universal functionals with respect to the 2-RDM (two particle reduced density matrix) rather than the wave function, we closely connect the one-density, the 1-RDM (one-particle reduced density matrix), and the geminal functional theories with 2-RDM minimization of the energy. Constrained search with the 2-RDM emphasizes that all energy functional methods must implicitly account for the N-representability of the 2-RDM within their universal functionals. An approximate universal functional for GFT, equivalent to a variational ansatz using the antisymmetrized geminal power wave function, yields energies that are significantly better than those from Hartree–Fock and yet rigorously above the exact energy.
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.
It is virtually impossible to directly solve the Schrödinger equation for a many-electron wave function due to the exponential growth in degrees of freedom with increasing particle number. The two-body reduced density matrix (2-RDM) formalism reduces this coordinate dependence to that of four particles irrespective of the wave function's dimensionality, providing a promising path to solve the many-body problem. Unfortunately, errors arise in this approach because the 2-RDM cannot practically be constrained to guarantee that it corresponds to a valid wave function. Here we approach this so-called
N-representability problem by expanding the 2-RDM in a complete basis of two-electron wave functions and studying the matrix formed by the expansion coefficients. This quantity, which we call the geminal density matrix (GDM), is found to evolve in time by a unitary transformation that preserves
N-representability. This evolution law enables us to calculate eigenstates of strongly correlated systems by a fictitious adiabatic evolution in which the electron-electron interaction is slowly switched on. We show how this technique is used to diagonalize atomic Hamiltonians, finding that the problem reduces to the solution of
∼N(N−1)/2 two-electron eigenstates of the Helium atom on a grid of electron-electron interaction scaling factors.
Variational reduced-density-matrix theory is applied to calculating the ground-state energy and two-electron reduced density matrices (2-RDMs) of the one-dimensional Hubbard model for a range of interaction strengths. The 2-RDM is constrained to represent an
N-particle wave function by two sets of
N-representability conditions, known as the 2- and partial 3-positivity conditions. Variational optimization of the energy with the 2-RDM constrained by
N-representability conditions is performed using a first-order semidefinite-programming algorithm that was developed for treating atoms and molecules [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Accurate energies for a broad range of interaction strengths indicate that the variational 2-RDM method is a valuable tool for studying strongly correlated electrons.
In methods like geminal-based approaches or coupled cluster that are solved using the projected Schrödinger equation, direct computation of the 2-electron reduced density matrix (2-RDM) is impractical and one falls back to a 2-RDM based on response theory. However, the 2-RDMs from response theory are not N-representable. That is, the response 2-RDM does not correspond to an actual physical N-electron wave function. We present a new algorithm for making these non-N-representable 2-RDMs approximately N-representable, i.e., it has the right symmetry and normalization and it fulfills the P-, Q-, and G-conditions. Next to an algorithm which can be applied to any 2-RDM, we have also developed a 2-RDM optimization procedure specifically for seniority-zero 2-RDMs. We aim to find the 2-RDM with the right properties which is the closest (in the sense of the Frobenius norm) to the non-N-representable 2-RDM by minimizing the square norm of the difference between this initial response 2-RDM and the targeted 2-RDM under the constraint that the trace is normalized and the 2-RDM, Q-matrix, and G-matrix are positive semidefinite, i.e., their eigenvalues are non-negative. Our method is suitable for fixing non-N-representable 2-RDMs which are close to being N-representable. Through the N-representability optimization algorithm we add a small correction to the initial 2-RDM such that it fulfills the most important N-representability conditions.
The minimum parameterization of the wave function is derived for the time-independent many-body problem. In this parameterization all electrons and electron pairs are indistinguishable which eliminates the exponential scaling with the number of particles plaguing all other correlation methods. An exponential scaling shown to stem from the expansion of the wave function in one-particle basis sets. It is demonstrated that using a geminal basis with a skew-symmetric orthogonal coefficient matrix, which fulfill a Lie algebra, the parameterization of the exact wave function becomes independent of the number of particles and only scale quadratic with the number of basis functions in the optimized basis and quartic in an arbitrary basis. The resulting antisymmetrized geminal power wave function is shown to fulfill the necessary and sufficient conditions for the exact wave function, treat all electrons and electron pairs as indistinguishable and be the most compact representation of the exact wave function possible. The number of parameters in the antisymmetrized geminal power wave function is exactly the same number of parameters as in the two-particle reduced density matrix thereby showing that these representations are equivalent just like the representations for a one-particle Hamiltonian. A detailed discussion of how the most compact wave function can be derived in general is also presented along with different geminal ansatze and sufficient closure relations. Due to the breaking of the scaling wall for the exact wave function it is expected that, if a general set of constraints can be found, even systems of biological relevance can be treated exactly in the future.
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.