A study investigated the multiconfiguration self-consistent field and multireference configuration interaction methods and applications to the field of chemistry. These methods are applicable to ...arbitrary electronics states and molecular configurations.
A large-scale implementation of the complete active space self-consistent field (CASSCF) method is presented. The active space is described using the variational two-electron reduced-density-matrix ...(v2RDM) approach, and the algorithm is applicable to much larger active spaces than can be treated using configuration-interaction-driven methods. Density fitting or Cholesky decomposition approximations to the electron repulsion integral tensor allow for the simultaneous optimization of large numbers of external orbitals. We have tested the implementation by evaluating singlet–triplet energy gaps in the linear polyacene series and two dinitrene biradical compounds. For the acene series, we report computations that involve active spaces consisting of as many as 50 electrons in 50 orbitals and the simultaneous optimization of 1892 orbitals. For the dinitrene compounds, we find that the singlet–triplet gaps obtained from v2RDM-driven CASSCF with partial three-electron N-representability conditions agree with those obtained from configuration-interaction-driven approaches to within one-third of 1 kcal mol–1. When enforcing only the two-electron N-representability conditions, v2RDM-driven CASSCF yields less accurate singlet–triplet energy gaps in these systems, but the quality of the results is still far superior to those obtained from standard single-reference approaches.
The basic formulation for the multifacet generalization of the graphically contracted function (MFGCF) electronic structure method is presented. The analysis includes the discussion of linear ...dependency and redundancy of the arc factor parameters, the computation of reduced density matrices, Hamiltonian matrix construction, spin-density matrix construction, the computation of optimization gradients for single-state and state-averaged calculations, graphical wave function analysis, and the efficient computation of configuration state function and Slater determinant expansion coefficients. Timings are given for Hamiltonian matrix element and analytic optimization gradient computations for a range of model problems for full-CI Shavitt graphs, and it is observed that both the energy and the gradient computation scale as O(N(2)n(4)) for N electrons and n orbitals. The important arithmetic operations are within dense matrix-matrix product computational kernels, resulting in a computationally efficient procedure. An initial implementation of the method is used to present applications to several challenging chemical systems, including N2 dissociation, cubic H8 dissociation, the symmetric dissociation of H2O, and the insertion of Be into H2. The results are compared to the exact full-CI values and also to those of the previous single-facet GCF expansion form.
Polyaromatic hydrocarbons (PAHs) are a class of organic molecules with importance in several branches of science, including medicine, combustion chemistry, and materials science. The delocalized ...π-orbital systems in PAHs require highly accurate electronic structure methods to capture strong electron correlation. Treating correlation in PAHs has been challenging because (i) traditional wave function methods for strong correlation have not been applicable since they scale exponentially in the number of strongly correlated orbitals, and (ii) alternative methods such as the density-matrix renormalization group and variational two-electron reduced density matrix (2-RDM) methods have not been applied beyond linear acene chains. In this paper we extend the earlier results from active-space variational 2-RDM theory Gidofalvi, G.; Mazziotti, D. A. J. Chem. Phys. 2008, 129, 134108 to the more general two-dimensional arrangement of ringsacene sheetsto study the relationship between geometry and electron correlation in PAHs. The acene-sheet calculations, if performed with conventional wave function methods, would require wave function expansions with as many as 1.5 × 1017 configuration state functions. To measure electron correlation, we employ several RDM-based metrics: (i) natural-orbital occupation numbers, (ii) the 1-RDM von Neumann entropy, (iii) the correlation energy per carbon atom, and (iv) the squared Frobenius norm of the cumulant 2-RDM. The results confirm a trend of increasing polyradical character with increasing molecular size previously observed in linear PAHs and reveal a corresponding trend in two-dimensional (arch-shaped) PAHs. Furthermore, in PAHs of similar size they show significant variations in correlation with geometry. PAHs with the strictly linear geometry (chains) exhibit more electron correlation than PAHs with nonlinear geometries (sheets).
Practical algorithms are presented for the parameterization of orthogonal matrices Q ∈ R(m×n) in terms of the minimal number of essential parameters {φ}. Both square n = m and rectangular n < m ...situations are examined. Two separate kinds of parameterizations are considered, one in which the individual columns of Q are distinct, and the other in which only Span(Q) is significant. The latter is relevant to chemical applications such as the representation of the arc factors in the multifacet graphically contracted function method and the representation of orbital coefficients in SCF and DFT methods. The parameterizations are represented formally using products of elementary Householder reflector matrices. Standard mathematical libraries, such as LAPACK, may be used to perform the basic low-level factorization, reduction, and other algebraic operations. Some care must be taken with the choice of phase factors in order to ensure stability and continuity. The transformation of gradient arrays between the Q and {φ} parameterizations is also considered. Operation counts for all factorizations and transformations are determined. Numerical results are presented which demonstrate the robustness, stability, and accuracy of these algorithms.
Most electronic structure methods express the wavefunction as an expansion of N-electron basis functions that are chosen to be either Slater determinants or configuration state functions. Although ...the expansion coefficient of a single determinant may be readily computed from configuration state function coefficients for small wavefunction expansions, traditional algorithms are impractical for systems with a large number of electrons and spatial orbitals. In this work, we describe an efficient algorithm for the evaluation of a single determinant expansion coefficient for wavefunctions expanded as a linear combination of graphically contracted functions. Each graphically contracted function has significant multiconfigurational character and depends on a relatively small number of variational parameters called arc factors. Because the graphically contracted function approach expresses the configuration state function coefficients as products of arc factors, a determinant expansion coefficient may be computed recursively more efficiently than with traditional configuration interaction methods. Although the cost of computing determinant coefficients scales exponentially with the number of spatial orbitals for traditional methods, the algorithm presented here exploits two levels of recursion and scales polynomially with system size. Hence, as demonstrated through applications to systems with hundreds of electrons and orbitals, it may readily be applied to very large systems.
The Shavitt graph is a visual representation of a distinct row table (DRT) within the graphical unitary group approach. The DRT is a compact representation of the entire configuration state function ...expansion space within a molecular electronic structure calculation. Each node of the graph is associated with an integer triple (ak, bk, ck). These integers may be mapped to other quantum numbers, including the number of orbitals, number of electrons, and spin quantum number, and used to display Shavitt graphs in various ways that emphasize different aspects of the expansion space or that reveal different aspects of computed wave functions. In conclusion, the features of several graph density plots are discussed, including electron-hole symmetries and the bonding-antibonding wave function character.
PDF
