Molecular systems in chemistry often have wave functions with substantial contributions from two-or-more electronic configurations. Because traditional complete-active-space self-consistent-field (CASSCF) methods scale exponentially with the number N of active electrons, their applicability is limited to small active spaces. In this paper we develop an active-space variational two-electron reduced-density-matrix (2-RDM) method in which the expensive diagonalization is replaced by a variational 2-RDM calculation where the 2-RDM is constrained by approximate N-representability conditions. Optimization of the constrained 2-RDM is accomplished by large-scale semidefinite programming [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Because the computational cost of the active-space 2-RDM method scales polynomially as r(a)(6) where r(a) is the number of active orbitals, the method can be applied to treat active spaces that are too large for conventional CASSCF. The active-space 2-RDM method performs two steps: (i) variational calculation of the 2-RDM in the active space and (ii) optimization of the active orbitals by Jacobi rotations. For large basis sets this two-step 2-RDM method is more efficient than the one-step, low-rank variational 2-RDM method [Gidofalvi and Mazziotti, J. Chem. Phys. 127, 244105 (2007)]. Applications are made to HF, H(2)O, and N(2) as well as n-acene chains for n=2-8. When n>4, the acenes cannot be treated by conventional CASSCF methods; for example, when n=8, CASSCF requires optimization over approximately 1.47x10(17) configuration state functions. The natural occupation numbers of the n-acenes show the emergence of bi- and polyradical character with increasing chain length.
Variationally calculating the ground state of a many-electron quantum system using only the two-electron reduced-density-matrix (2-RDM) requires
N-representability conditions that constrain the 2-RDM to correspond to an
N-electron wave function. A systematic hierarchy of
N-representability conditions, known as
p-positivity conditions, has been developed [D. A. Mazziotti and R. M. Erdahl, Phys. Rev. A 63, 042113 (2001)], and many-electron atoms and molecules in nonminimal basis sets have been solved with useful accuracy by a variational 2-RDM method with 2-positivity conditions [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. This paper considers two forms of partial 3-positivity conditions, the lifting conditions and the
T1∕T2 conditions, to further enhance the accuracy of the 2-RDM methods without the computational cost of full 3-positivity conditions. Variational 2-RDM methods with different
N-representability constraints including 2-positivity conditions, the two types of partial 3-positivity conditions, as well as the complete 3-positivity conditions are applied to compute the ground state of the Lipkin spin model. The energies and 2-RDMs are compared to the results from full and truncated configuration interaction, many-body perturbation theory, and couple cluster theory with single and double excitations. Implications of using partial 3-positivity for variational 2-RDM calculations of many-electron atoms and molecules will be discussed.
Full configuration interaction (FCI) restricted to a pairing space yields size-extensive correlation energies but its cost scales exponentially with molecular size. Restricting the variational two-electron reduced-density-matrix (2-RDM) method to represent the same pairing space yields an accurate lower bound to the pair FCI energy at a mean-field-like computational scaling of O(r3) where r is the number of orbitals. In this paper, we show that localized molecular orbitals can be employed to generate an efficient, approximately size-extensive pair 2-RDM method. The use of localized orbitals eliminates the substantial cost of optimizing iteratively the orbitals defining the pairing space without compromising accuracy. In contrast to the localized orbitals, the use of canonical Hartree-Fock molecular orbitals is shown to be both inaccurate and non-size-extensive. The pair 2-RDM has the flexibility to describe the spectra of one-electron RDM occupation numbers from all quantum states that are invariant to time-reversal symmetry. Applications are made to hydrogen chains and their dissociation, n-acene from naphthalene through octacene, and cadmium telluride 2-, 3-, and 4-unit polymers. For the hydrogen chains, the pair 2-RDM method recovers the majority of the energy obtained from similar calculations that iteratively optimize the orbitals. The localized-orbital pair 2-RDM method with its mean-field-like computational scaling and its ability to describe multi-reference correlation has important applications to a range of strongly correlated phenomena in chemistry and physics.
Molecular periodic systems have the challenge of not only representing an infinite, albeit periodically repeating, molecule but also treating the strong electron correlation that emerges in such systems. Many post-Hartree-Fock methods that are efficient in moderately correlated molecules often fail to describe strongly correlated electrons and their properties in periodic systems. Here we develop and apply an electronic structure approach to periodic systems based on the variational calculation of the two-electron reduced density matrix (2-RDM). The variational 2-RDM method calculates a lower bound to the ground-state energy of a molecular periodic system without computation or storage of its many-electron wave function. The 2-RDM is constrained by
N-representability conditions that are necessary for it to represent at least one
N-electron density matrix---conditions that allow it to treat strongly correlated systems. Two periodic systems are treated to demonstrate the methodology: (i) a metallic hydrogen chain and (ii) a strongly correlated acene chain. In the hydrogen chains the 2-RDM theory describes the strongly correlated Mott metal-to-insulator transition while in acene chains it describes the emergence of polyradical character with increasing chain length. The methodology is applicable to treating strong electron correlation in more general periodic molecules and materials.
An alternative approach to open-shell molecular calculations using the variational two-electron reduced-density-matrix (2-RDM) theory [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] is presented. The energy and 2-RDM of the open-shell molecule (or radical) are computed from the limit of dissociating one or more hydrogen atoms from a molecule in a singlet state. Because the ground-state energy of an ``infinitely'' separated hydrogen atom in a given finite basis is known, we can determine the energy of the radical by subtracting the energy of one or more hydrogen atoms from the energy of the total dissociated system. The 2-RDM is constrained to have singlet symmetry in all calculations. Two sets of
N-representability conditions are employed: (i) two-positivity conditions, and (ii) two-positivity conditions plus the
T2 condition, which is a subset of the three-positivity conditions. Optimization of the energy with respect to the 2-RDM is performed with a first-order algorithm for solving the semidefinite program within the variational 2-RDM method. We present calculations of several radicals near equilibrium as well as the dissociation curves of the diatomic radicals CH and OH.
In this thesis we develop the variational two-electron reduced-density-matrix method for extended systems. Extended systems are represented in two ways: i) lattice models describing the dominant valence electronic structure with periodic boundaries to account for their extended nature and ii) a crystalline-orbital basis built from atomic orbitals using the generalization of molecular orbital theory to polymers. The first part of this thesis (Ch. 3-4) examines the performance of the variational 2-RDM method on lattice systems with tunable electron correlation. The first of these systems is the classic Hubbard model with linear and ladder lattice topologies. Because electron correlation functions, such as charge- and spin-ordering, are linear functions of the 2-RDM, the difference in electronic structure between one- and quasi-one-dimensional systems is accurately characterized. The second model contains only two-body interactions and is unique among typical spin models in that it does not have a mean-field reference wave function. The ground state wave functions from all Hamiltonians in the model have the same 1-electron reduced density matrix; consequently, one-electron theories are largely inapplicable. The superconducting eta-pairing ground states make the model a unique tool for demonstrating the necessary N-representability in highly correlated environments. The second part of this thesis (Ch. 5-6) develops a formalism for modeling materials by solving the full Schrodinger equation. Crystalline-orbital Hartree-Fock provides a set of orbitals and integral tensors for the variational 2-RDM method. We demonstrate that time-reversal symmetry, which is implicitly included in position space electronic structure calculations, must be explicitly included as an N-representability constraint on the 2-RDM when using a momentum space basis. The necessity of these equality constraints is demonstrated by the accurate recovery of the binding energy of two polymers and the symmetry of their natural orbital occupations.