Active-space two-electron reduced-density-matrix method: Complete active-space calculations without diagonalization of the N-electron Hamiltonian

Abstract

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.

Effect of strong electron correlation on the efficiency of photosynthetic light harvesting

“…In reality, however, the chromophores are assembled from chlorophyll molecules that contain an extensive network of conjugated carbon-carbon bonds surrounding magnesium ions, from which significant strong electron correlation, including polyradical character, has been shown to emerge [17,20]. Recent experimental efforts, not yet published, provide further insights into the model developed here.…”

“…Orbital occupations from recent 2-RDM calculations of correlation in polyaromatic hydrocarbons [20] suggest V = 0.8 to be a conservative estimate of the correlation within the LMG models of the 7 chromophores. Using this estimate with coupling energies screened to match experimental and computational data, we project a greater than 100% enhancement from the strong electron correlation relative to the theoretical limit in which electron correlation within the chromophore is absent.…”

“…[34]; when V > 0, the parameters {ǫ s } are adjusted, as shown in Table I, to ensure that the excitations of the correlated LMG models agree with the excitation energies of the seven chromophores. Based on acene-chain [20] and metalloporphyrin-ring data [35], which imply that the population of the lowest unoccupied orbital (unoccupied in a mean-field treatment) is at least 20%, we estimate the interaction strength V with N = 4 to be 0.8. The estimate of V is made by finding the value of V that gives an 80% probability of finding an electron in the highest occupied orbital (occupied in a mean-field treatment) and a 20% probability of finding an electron in the lowest unoccupied orbital.…”

“…While the chromophores are chlorophyll molecules containing large networks of conjugated carbon bonds that surround a charged magnesium ion, they have largely been represented in theoretical studies [4][5][6][7][8][9][10][11][12][13] by one-electron models that neglect the effects of electron correlation and entanglement within chromophores. Two advanced methods in electronic structure, density-matrix renormalization group [17] and two-electron reduceddensity-matrix theory [18,19], have recently shown that networks of conjugated bonds as in acene chains [17,20], acene sheets [20], and chlorophyll are associated with polyradical character that cannot be adequately described without a strongly correlated many-electron quantum model. In this paper we examine the efficiency of light harvesting where we represent each chromophore by a correlated N-electron model to treat strong electron correlation.…”

Active-SpaceN-Representability Constraints for Variational Two-Particle Reduced Density Matrix Calculations

“…Thus, we can effectively combine information which would be obtained from different CASSCF calculations, an approach that would not be possible apart from RDM methodology. The current approach should also be contrasted with other active-space RDM approaches which implement approximations to the 2-positivity constraints, but which are unrelated to the active-space constraints derived in this Letter [20,21].…”