Graphically Contracted Function Construction with the Recursive Pairwise Merge Algorithm.

A new implementation of a recursive pairwise merge algorithm to construct a GCF from a list of CSF expansion coefficients is presented. The essential new feature is the preallocation of some work arrays used within the intermediate steps of the merge procedure. This results in roughly an order of magnitude improvement in overall efficiency and also approximately eliminates a factor of n, the molecular orbital dimension, from the original implementation. Initial application of this merge procedure to a series of Hm molecules shows that the GCF wave functions can be represented well both with delocalized canonical Hartree-Fock orbitals and with localized molecular orbitals. For a given wave function complexity, as measured by the average facet count, f̅, the delocalized Hartree-Fock orbitals show smaller errors for small f̅ values, while the localized orbitals show smaller errors for larger f̅ values.

Quantification of the Ionic Character of Multiconfigurational Wave Functions: The Qat Diagnostic.

The complete active space self-consistent field (CASSCF) method is a cornerstone in modern excited-state quantum chemistry providing the starting point for most common multireference computations. However, CASSCF, when used with a minimal active space, can produce significant errors (>2 eV) even for the excitation energies of simple hydrocarbons if the states of interest possess ionic character. After illustrating this problem in some detail, we present a diagnostic for ionic character, denoted as Q at, that is readily computed from the transition density. A set of 11 molecules is considered to study errors in vertical excitation energies. State-averaged CASSCF obtains a mean absolute error (MAE) of 0.87 eV for the 34 singlet states considered. We highlight a strong correlation between the obtained errors and the Q at diagnostic, illustrating its power to predict problematic cases. Conversely, using multireference configuration interaction with single and double excitations and Pople's size extensivity correction (MR-CISD+P), excellent results are obtained with an MAE of 0.11 eV. Furthermore, correlations with the Q at diagnostic disappear. In summary, we hope that the presented diagnostic will facilitate reliable and user-friendly multireference computations on conjugated organic molecules.

A Perspective on Sustainable Computational Chemistry Software Development and Integration.

The power of quantum chemistry to predict the ground and excited state properties of complex chemical systems has driven the development of computational quantum chemistry software, integrating advances in theory, applied mathematics, and computer science. The emergence of new computational paradigms associated with exascale technologies also poses significant challenges that require a flexible forward strategy to take full advantage of existing and forthcoming computational resources. In this context, the sustainability and interoperability of computational chemistry software development are among the most pressing issues. In this perspective, we discuss software infrastructure needs and investments with an eye to fully utilize exascale resources and provide unique computational tools for next-generation science problems and scientific discoveries.

Observation of site-selective chemical bond changes via ultrafast chemical shifts.

The concomitant motion of electrons and nuclei on the femtosecond time scale marks the fate of chemical and biological processes. Here we demonstrate the ability to initiate and track the ultrafast electron rearrangement and chemical bond breaking site-specifically in real time for the carbon monoxide diatomic molecule. We employ a local resonant x-ray pump at the oxygen atom and probe the chemical shifts of the carbon core-electron binding energy. We observe charge redistribution accompanying core-excitation followed by Auger decay, eventually leading to dissociation and hole trapping at one site of the molecule. The presented technique is general in nature with sensitivity to chemical environment changes including transient electronic excited state dynamics. This work provides a route to investigate energy and charge transport processes in more complex systems by tracking selective chemical bond changes on their natural timescale.

The generality of the GUGA MRCI approach in COLUMBUS for treating complex quantum chemistry.

The core part of the program system COLUMBUS allows highly efficient calculations using variational multireference (MR) methods in the framework of configuration interaction with single and double excitations (MR-CISD) and averaged quadratic coupled-cluster calculations (MR-AQCC), based on uncontracted sets of configurations and the graphical unitary group approach (GUGA). The availability of analytic MR-CISD and MR-AQCC energy gradients and analytic nonadiabatic couplings for MR-CISD enables exciting applications including, e.g., investigations of π-conjugated biradicaloid compounds, calculations of multitudes of excited states, development of diabatization procedures, and furnishing the electronic structure information for on-the-fly surface nonadiabatic dynamics. With fully variational uncontracted spin-orbit MRCI, COLUMBUS provides a unique possibility of performing high-level calculations on compounds containing heavy atoms up to lanthanides and actinides. Crucial for carrying out all of these calculations effectively is the availability of an efficient parallel code for the CI step. Configuration spaces of several billion in size now can be treated quite routinely on standard parallel computer clusters. Emerging developments in COLUMBUS, including the all configuration mean energy multiconfiguration self-consistent field method and the graphically contracted function method, promise to allow practically unlimited configuration space dimensions. Spin density based on the GUGA approach, analytic spin-orbit energy gradients, possibilities for local electron correlation MR calculations, development of general interfaces for nonadiabatic dynamics, and MRCI linear vibronic coupling models conclude this overview.

Representations of Shavitt Graphs Within the Graphical Unitary Group Approach.

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 (a k ,b k ,c k ). 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. The features of several graph density plots are discussed, including electron-hole symmetries and the bonding-antibonding wave function character. © 2019 Wiley Periodicals, Inc.

The Representation and Parametrization of Orthogonal Matrices.

Four representations and parametrizations of orthogonal matrices Q ∈ R(m×n) in terms of the minimal number of essential parameters {φ} are discussed: the exponential representation, the Householder reflector representation, the Givens rotation representation, and the rational Cayley transform representation. Both square n = m and rectangular n < m situations are considered. Two separate kinds of parametrizations are considered: one in which the individual columns of Q are distinct, the Stiefel manifold, and the other in which only span(Q) is significant, the Grassmann manifold. The practical issues of numerical stability, continuity, and uniqueness are discussed. The computation of Q in terms of the essential parameters {φ}, and also the extraction of {φ} for a given Q are considered for all of the parametrizations. The transformation of gradient arrays between the Q and {φ} variables is discussed for all representations. It is our hope that developers of new methods will benefit from this comparative presentation of an important but rarely analyzed subject.

The multifacet graphically contracted function method. I. Formulation and implementation.

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.

The multifacet graphically contracted function method. II. A general procedure for the parameterization of orthogonal matrices and its application to arc factors.

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.

The multiradical character of one- and two-dimensional graphene nanoribbons.

When is an acene stable? The pronounced multiradical character of graphene nanoribbons of different size and shape was investigated with high-level multireference methods. Quantitative information based on the number of effectively unpaired electrons leads to specific estimates of the chemical stability of graphene nanostructures.

Evaluation of the spin-orbit interaction within the graphically contracted function method.

The graphically contracted function (GCF) method is extended to include an effective one-electron spin-orbit (SO) operator in the Hamiltonian matrix construction. Our initial implementation is based on a multiheaded Shavitt graph approach that allows for the efficient simultaneous computation of entire blocks of Hamiltonian matrix elements. Two algorithms are implemented. The SO-GCF method expands the spin-orbit wave function in the basis of GCFs and results in a Hamiltonian matrix of dimension N(dim)=N(alpha)((S(max) + 1)(2) - S(min)(2)). N(alpha) is the number of sets of nonlinear arc factor parameters, and S(min) and S(max) are respectively the minimum and maximum values of an allowed spin range in the wave function expansion. The SO-SCGCF (SO spin contracted GCF) method expands the wave function in a basis of spin contracted functions and results in a Hamiltonian matrix of dimension N(dim) = N(alpha). For a given N(alpha) and spin range, the number of parameters defining the wave function is the same in the two methods after accounting for normalization. The full Hamiltonian matrix construction with both approaches scales formally as O(N(alpha)(2)omegan(4)) for n molecular orbitals. The omega factor depends on the complexity of the Shavitt graph and includes factors such as the number of electrons, N, and the number of interacting spin states. Timings are given for Hamiltonian matrix construction for both algorithms for a range of wave functions with up to N = n = 128 and that correspond to an underlying linear full-CI CSF expansion dimension of over 10(75) CSFs, many orders of magnitude larger than can be considered using traditional CSF-based spin-orbit CI approaches. For Hamiltonian matrix construction, the SO-SCGCF method is slightly faster than the SO-GCF method for a given N(alpha) and spin range. The SO-GCF method may be more suitable for describing multiple states, whereas the SO-SCGCF method may be more suitable for describing single states.

Computation of determinant expansion coefficients within the graphically contracted function method.

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.

Hamiltonian matrix and reduced density matrix construction with nonlinear wave functions.

An efficient procedure to compute Hamiltonian matrix elements and reduced one- and two-particle density matrices for electronic wave functions using a new graphical-based nonlinear expansion form is presented. This method is based on spin eigenfunctions using the graphical unitary group approach (GUGA), and the wave function is expanded in a basis of product functions (each of which is equivalent to some linear combination of all of the configuration state functions), allowing application to closed- and open-shell systems and to ground and excited electronic states. In general, the effort required to construct an individual Hamiltonian matrix element between two product basis functions H(MN) = M|H|N scales as theta (beta n4) for a wave function expanded in n molecular orbitals. The prefactor beta itself scales between N0 and N2, for N electrons, depending on the complexity of the underlying Shavitt graph. Timings with our initial implementation of this method are very promising. Wave function expansions that are orders of magnitude larger than can be treated with traditional CI methods require only modest effort with our new method.

A general nonlinear expansion form for electronic wave functions.

A new expansion form is presented for electronic wave functions. The wave function is a linear combination of product basis functions, and each product basis function in turn is formally equivalent to a linear combination of configuration state functions that comprise an underlying linear expansion space. The expansion coefficients that define the basis functions are nonlinear functions of a smaller number of variables. The expansion form is appropriate for both ground and excited states and to both closed and open shell molecules. The method is formulated in terms of spin-eigenfunctions using the graphical unitary group approach (GUGA), and consequently it does not suffer from spin contamination.

Ab initio study of AmCl(+): f-f spectroscopy and chemical binding.

A valence full configuration interaction study with a polarized double-zeta quality basis set has been carried out for the lowest 49 electronic states of AmCl(+). The calculations use a pseudopotential treatment for the core electrons and incorporate a one-electron spin-orbit interaction operator. Electrons in the valence s, p, d, and f subshells were included in the active space. The resulting electronic potential energy curves are largely repulsive. The chemical bonding is ionic in character with negligible participation of 5f electrons. The molecular f-f spectroscopy of AmCl(+) arises essentially from an in situ Am(2+) core with states slightly redshifted by the presence of chloride ion. Am(+)+Cl asymptotes which give rise to the few attractive potential energy curves can be predicted by analysis of the f-f spectroscopy of isolated Am(+) and Am(2+). The attractive curves have substantial binding energies, on the order of 75-80 kcal/mol, and are noticeably lower than recent indirect measurements on the isovalent EuCl(+). An independent empirical correlation supports the predicted reduction in AmCl(+) binding energy. The energies of the repulsive curves are strongly dependent on the selection of the underlying atomic orbitals while the energies of the attractive curves do not display this sensitivity. The calculations were carried out using our recently developed parallel spin-orbit configuration interaction software.

Analytic evaluation of nonadiabatic coupling terms at the MR-CI level. I. Formalism.

An efficient and general method for the analytic computation of the nonandiabatic coupling vector at the multireference configuration interaction (MR-CI) level is presented. This method is based on a previously developed formalism for analytic MR-CI gradients adapted to the use for the computation of nonadiabatic coupling terms. As was the case for the analytic energy gradients, very general, separate choices of invariant orbital subspaces at the multiconfiguration self-consistent field and MR-CI levels are possible, allowing flexible selections of MR-CI wave functions. The computational cost for the calculation of the nonadiabatic coupling vector at the MR-CI level is far below the cost for the energy calculation. In this paper the formalism of the method is presented and in the following paper [Dallos et al., J. Chem. Phys. 120, 7330 (2004)] applications concerning the optimization of minima on the crossing seam are described.

Analytic evaluation of nonadiabatic coupling terms at the MR-CI level. II. Minima on the crossing seam: formaldehyde and the photodimerization of ethylene.

The method for the analytic calculation of the nonadiabatic coupling vector at the multireference configuration-interaction (MR-CI) level and its program implementation into the COLUMBUS program system described in the preceding paper [Lischka et al., J. Chem. Phys. 120, 7322 (2004)] has been combined with automatic searches for minima on the crossing seam (MXS). Based on a perturbative description of the vicinity of a conical intersection, a Lagrange formalism for the determination of MXS has been derived. Geometry optimization by direct inversion in the iterative subspace extrapolation is used to improve the convergence properties of the corresponding Newton-Raphson procedure. Three examples have been investigated: the crossing between the 1(1)B1/2(1)A1 valence states in formaldehyde, the crossing between the 2(1)A1/3(1)A1 pi-pi* valence and ny-3py Rydberg states in formaldehyde, and three crossings in the case of the photodimerization of ethylene. The methods developed allow MXS searches of significantly larger systems at the MR-CI level than have been possible before and significantly more accurate calculations as compared to previous complete-active space self-consistent field approaches.

Reducing I/O costs for the eigenvalue procedure in large-scale configuration interaction calculations.

Several aspects of the matrix diagonalization method used for CI calculations in the COLUMBUS Program System are discussed, including a linear basis-contraction algorithm and the use of a nonorthogonal expansion basis. Both of these features significantly reduce the I/O requirements during the iterations.