Configuration State Functions Research Articles

Second-order Rayleigh–Schrödinger perturbation theory for the GRASP2018 package: Core–valence correlations


 
 
 The General Relativistic Atomic Structure package [GRASP2018, C. Froese Fischer, G. Gaigalas, P. Jönsson, and J. Bieroń, Comput. Phys. Commun. (2019), DOI: 10.1016/j.cpc.2018.10.032] is based on multiconfiguration Dirac– Hartree–Fock and relativistic configuration interaction (RCI) methods for energy structure calculations. The atomic state function used in the program is built from the set of configuration state functions (CSFs). The valence–valence, core–valence and core–core correlations are explicitly included through expansions over CSFs in RCI. We present a combination of RCI and the stationary second-order Rayleigh–Schrödinger many-body perturbation theory in an irreducible tensorial form to account for electron core–valence correlations when an atom or ion has any number of valence electrons. This newly developed method, which offers two ways of use, allows a significant reduction of the CSF space for complex atoms and ions. We also demonstrate how the method and program works for the energy structure calculation of Cl III ion.
 
 

Spin-Adapted Externally Contracted Multireference Configuration Interaction Method Based on Selected Reference Configurations.

As one kind of approximation of the full configuration interaction solution, the selected configuration interaction (sCI) methods have been shown to be valuable for large active spaces. However, the inclusion of dynamic correlation beyond large active spaces is necessary for more quantitative results. Since the sCI wave function can provide a compact reference for multireference methods, previously, we proposed an externally contracted multireference configuration interaction method using the sCI reference reconstructed from the density matrix renormalization group wave function [J. Chem. Theory Comput. 2018, 14, 4747-4755]. The DMRG2sCI-EC-MRCI method is promising for dealing with more than 30 active orbitals and large basis sets. However, it suffers from two drawbacks: spin contamination and low efficiency when using Slater determinant bases. To solve these problems, in this work, we adopt configuration state function bases and introduce a new algorithm based on the hybrid of tree structure for convenient configuration space management and the graphical unitary group approach for efficient matrix element calculation. The test calculation of naphthalene shows that the spin-adapted version could achieve a speed-up of 6.0 compared with the previous version based on the Slater determinant. Examples of dinuclear copper(II) compound as well as Ln(III) and An(III) complexes show that the sCI-EC-MRCI can give quantitatively accurate results by including dynamic correlation over sCI for systems with large active spaces and basis sets.

Energies and radiative properties of 1s22l2l′ and 1s(2l)22l′ levels in tungsten ion, W70+

Multiconfiguration Dirac–Hartree–Fock (MCDHF) calculations of energy levels, oscillator strengths and transition rates of open K-shell transitions in beryllium-like tungsten ion, W[Formula: see text], are performed using the fully relativistic GRASP2018 code. For only principal quantum number [Formula: see text], energy levels of 1[Formula: see text] 2[Formula: see text], 1[Formula: see text]2[Formula: see text], 1[Formula: see text]2[Formula: see text], 1[Formula: see text]2p, 1[Formula: see text] and 1[Formula: see text] at different angular momenta and parity ([Formula: see text] are presented. Excitations of up to four electrons from the ground level, 1[Formula: see text] 2[Formula: see text], to excited levels are allowed. Correlations up to 5l orbitals are included in the configuration state function (CSF), [Formula: see text], p, d, f and g. The fine-structure calculations of E1, E2, M1 and M2 oscillator strengths and radiative rates between intercombination transitions in W[Formula: see text] ions are evaluated. The accuracy of these results is estimated using different ways, such as the comparison with previously published results and calculating the uncertainty of the transition data. The uncertainty of the transition probabilities showed low values for most of transitions.

Predicting the X-ray Absorption Spectrum of Ozone with Single Configuration State Functions.

X-ray absorption spectra (XAS) of biradicaloid species are often thought to represent a challenge to theoretical methods. This has led to the testing of recently developed multireference techniques on the XAS of ozone, but reproduction of the experimental spectral profile has proven difficult. We utilize a minimal model consisting of a single configuration state function (CSF) per excited state to model core-level excitations of ozone, with the orbitals of each CSF optimized using the restricted open-shell Kohn-Sham (ROKS) method. This protocol leads to semiquantitative agreement with experimental XAS. In fact, we find that low-lying core-hole excited states in biradicaloids can be approximated with individual CSFs, despite the presence of multireference character in the ground state. We also report that the 1s → π* and 1s → σ* transitions have quite distinct widths for O3. This reveals the importance of sampling over a representative range of geometries from the vibrational ground state for properly assessing the accuracy of electronic structure methods against experiments instead of the popular procedure of uniformly broadening stick spectra at the equilibrium geometry.

PyBEST: Improved functionality and enhanced performance

PyBEST: Improved functionality and enhanced performance

Modular Approach to Selected Configuration Interaction in an Arbitrary Spin Basis: Implementation and Comparison of Approaches.

A modular selected configuration interaction (SCI) code has been developed that is based on the existing Monte-Carlo configuration interaction code (MCCI). The modularity allows various selection protocols to be implemented with ease and allows for fair comparison between wave functions built via different criteria. We have initially implemented adaptations of existing SCI theories, which are based on either energy- or coefficient-driven selection schemes. These codes have been implemented not only in the basis of Slater determinants (SDs) but also in the basis of configuration state functions (CSFs) and extended to state-averaged regimes. This allows one to take advantage of the reduced dimensionality of the wave function in the CSF basis and also the guarantee of pure spin states. All SCI methods were found to be able to predict potential energy surfaces to high accuracy, producing compact wave functions, when compared to full configuration interaction (FCI) for a variety of bond-breaking potential energy surfaces. The compactness of the error-controlled adaptive configuration interaction approach, particularly in the CSF basis, was apparent with nonparallelity errors within chemical accuracy while containing as little as 0.02% of the FCI CSF space. The size-to-accuracy was also extended to FCI spaces approaching one billion configurations.

Local Oxidation States in {FeNO}6-8 Porphyrins: Insights from DMRG/CASSCF-CASPT2 Calculations.

A first DMRG/CASSCF-CASPT2 study of a series of paradigmatic {FeNO}6, {FeNO}7, and {FeNO}8 heme-nitrosyl complexes has led to substantial new insight as well as uncovered key shortcomings of the DFT approach. By virtue of its balanced treatment of static and dynamic correlation, the calculations have provided some of the most authoritative information available to date on the energetics of low- versus high-spin states of different classes of heme-nitrosyl complexes. Thus, the calculations indicate low doublet-quartet gaps of 1-4 kcal/mol for {FeNO}7 complexes and high singlet-triplet gaps of ≳20 kcal/mol for both {FeNO}6 and {FeNO}8 complexes. In contrast, DFT calculations yield widely divergent spin state gaps as a function of the exchange-correlation functional. DMRG-CASSCF calculations also help calibrate DFT spin densities for {FeNO}7 complexes, pointing to those obtained from classic pure functionals as the most accurate. The general picture appears to be that nearly all the spin density of Fe[P](NO) is localized on the Fe, while the axial ligand imidazole (ImH) in Fe[P](NO)(ImH) pushes a part of the spin density onto the NO moiety. An analysis of the DMRG-CASSCF wave function in terms of localized orbitals and of the resulting configuration state functions in terms of resonance forms with varying NO(π*) occupancies has allowed us to address the longstanding question of local oxidation states in heme-nitrosyl complexes. The analysis indicates NO(neutral) resonance forms [i.e., Fe(II)-NO0 and Fe(III)-NO0] as the major contributors to both {FeNO}6 and {FeNO}7 complexes. This finding is at variance with the common formulation of {FeNO}6 hemes as Fe(II)-NO+ species but is consonant with an Fe L-edge XAS analysis by Solomon and co-workers. For the {FeNO}8 complex {Fe[P](NO)}-, our analysis suggests a resonance hybrid description: Fe(I)-NO0 ↔ Fe(II)-NO-, in agreement with earlier DFT studies. Vibrational analyses of the compounds studied indicate an imperfect but fair correlation between the NO stretching frequency and NO(π*) occupancy, highlighting the usefulness of vibrational data as a preliminary indicator of the NO oxidation state.

The efficient conversion between linear wave function expansions and nonlinear graphically contracted function expansions

Several algorithms are examined for the transformation of a graphically contracted function (GCF) to a configuration state function (CSF) vector, denoted , and for the reverse transformation from CSF vector to GCF, denoted . The initial implementation of these algorithms is applied to a sequence of full-CI wave functions of similar molecules. The most efficient transformation is based on a depth-first search (DFS) and scales as where is an average facet count for the GCF wave function, is the dimension of the vector x, and is the dimension of the GCF basis. Two different transformations are proposed, one based on a pairwise recursive merge algorithm and the other based on a least-squares fitting algorithm. The recursive merge is a reliable algorithm, but our initial implementation does not display the expected scaling behaviour with respect to . The iterative least-squares fitting algorithm is based on an efficient DFS procedure to compute the error ε and its gradient g. However, the resulting iterative least-squares optimisation is slowly convergent, requiring excessive iterations to converge to a target error tolerance. For most of the calculations in this case study, the recursive merge procedure is found to be more efficient than the least-squares optimisation procedure.

Chemical Bonding in the C2 Molecule

Bonding in the C2 molecule is investigated with CAS(8,8) wave functions using canonical MOs. In a subsequent step, orthogonal atomic orbitals are constructed by localizing the CASSCF MOs on the two carbon atoms with an orthogonal transformation. This orbital transformation causes an orthogonal transformation of the configuration state functions (CSF) spanning the function space of the singlet ground state of C2. Instead of CSFs built from canonical MOs, one obtains CSFs of orthogonal deformed atomic orbitals (AO). This approach resembles the orthogonal valence bond (OVB) methods’ CSFs, which are very different from the conventional VB, based on non-orthogonal AOs. To become used to the different argumentation, the bonding situations in ethane (single bond), ethene (double bond), and the nitrogen molecule (triple bond) are also studied. The complex bonding situation in C2 is caused by the possibility to excite an electron with a spin flip from the doubly occupied 2s AO into the 2p subshell, and the resulting high-spin 5Su state of the carbon atom allows for a better reduction of the Pauli repulsion. However, the electron structure around the equilibrium distance does not allow one to say that C2 in its ground state has a double, or triple, or even a quadruple bond.

Fine-Tuning of Atomic Energies in Relativistic Multiconfiguration Calculations

Ab initio calculations sometimes do not reproduce the experimentally observed energy separations at a high enough accuracy. Fine-tuning of diagonal elements of the Hamiltonian matrix is a process which seeks to ensure that calculated energy separations of the states that mix are in agreement with experiment. The process gives more accurate measures of the mixing than can be obtained in ab initio calculations. Fine-tuning requires the Hamiltonian matrix to be diagonally dominant, which is generally not the case for calculations based on jj-coupled configuration state functions. We show that this problem can be circumvented by a method that transforms the Hamiltonian in jj-coupling to a Hamiltonian in LSJ-coupling for which fine-tuning applies. The fine-tuned matrix is then transformed back to a Hamiltonian in jj-coupling. The implementation of the method into the General Relativistic Atomic Structure Package is described and test runs to validate the program operations are reported. The new method is applied to the computation of the 2s21S0−2s2p1,3P1 transitions in C III and to the computation of Rydberg transitions in B I, for which the 2s2p22S1/2 perturber enters the 2s2ns2S1/2 series. Improved convergence patterns and results are found compared with ab initio calculations.

Measuring Electron Correlation: The Impact of Symmetry and Orbital Transformations.

In this perspective, the various measures of electron correlation used in wave function theory, density functional theory and quantum information theory are briefly reviewed. We then focus on a more traditional metric based on dominant weights in the full configuration solution and discuss its behavior with respect to the choice of the N-electron and the one-electron basis. The impact of symmetry is discussed, and we emphasize that the distinction among determinants, configuration state functions and configurations as reference functions is useful because the latter incorporate spin-coupling into the reference and should thus reduce the complexity of the wave function expansion. The corresponding notions of single determinant, single spin-coupling and single configuration wave functions are discussed and the effect of orbital rotations on the multireference character is reviewed by analyzing a simple model system. In molecular systems, the extent of correlation effects should be limited by finite system size and in most cases the appropriate choices of one-electron and N-electron bases should be able to incorporate these into a low-complexity reference function, often a single configurational one.

State-Specific Configuration Interaction for Excited States.

We introduce and benchmark a systematically improvable route for excited-state calculations, labeled state-specific configuration interaction (ΔCI), which is a particular realization of multiconfigurational self-consistent field and multireference configuration interaction. Starting with a reference built from optimized configuration state functions, separate CI calculations are performed for each targeted state (hence, state-specific orbitals and determinants). Accounting for single and double excitations produces the ΔCISD model, which can be improved with second-order Epstein-Nesbet perturbation theory (ΔCISD+EN2) or a posteriori Davidson corrections (ΔCISD+Q). These models were gauged against a vast and diverse set of 294 reference excitation energies. We have found that ΔCI is significantly more accurate than standard ground-state-based CI, whereas close performances were found between ΔCISD and EOM-CC2 and between ΔCISD+EN2 and EOM-CCSD. For larger systems, ΔCISD+Q delivers more accurate results than EOM-CC2 and EOM-CCSD. The ΔCI route can handle challenging multireference problems, singly and doubly excited states, from closed- and open-shell species, with overall comparable accuracy and thus represents a promising alternative to more established methodologies. In its current form, however, it is reliable only for relatively low-lying excited states.

Resolution of Electronic States in Heisenberg Cluster Models within the Unitary Group Approach.

In this work ground and excited electronic states of Heisenberg cluster models, in the form of configuration interaction many-body wave functions, are characterized within the spin-adapted Graphical Unitary Group Approach framework, and relying on a novel combined unitary and symmetric group approach. Finite-size cluster models of well-defined point-group symmetry and of general local-spin are presented, including J1-J2 triangular and tetrahedral clusters, which are often used to describe magnetic interactions in biological and biomimetic polynuclear transition metal clusters with unique catalytic activity, such as nitrogen fixation and photosynthesis. We show that a unique block-diagonal structure of the underlying Hamiltonian matrix in the spin-adapted basis emerges when an optimal lattice site ordering is chosen that reflects the internal symmetries of the model investigated. The block-diagonal structure is bound to the commutation relations between cumulative spin operators and the Hamiltonian operator, that in turn depend on the geometry of the cluster investigated. The many-body basis transformation, in the form of the orbital/site reordering, exposes such commutation relations. These commutation relations represent a rigorous and formal demonstration of the block-diagonal structure in Hamiltonian matrices and the compression of the corresponding spin-adapted many-body wave functions. As a direct consequence of the block-diagonal structure of the Hamiltonian matrix, it is possible to selectively optimize electronic excited states without the overhead of calculating the lower-energy states by simply relying on the initial ansatz for the targeted wave function. Additionally, more compact many-body wave functions are obtained. In extreme cases, electronic states are precisely described by a single configuration state function, despite the curse of dimensionality of the corresponding Hilbert space. These findings are crucial in the electronic structure theory framework, for they offer a conceptual route toward wave functions of reduced multireference character, that can be optimized more easily by approximated eigensolvers and are of more facile physical interpretation. They open the way to study larger ab initio and model Hamiltonians of increasingly larger number of correlated electrons, while keeping the computational costs at their lowest. In particular, these elements will expand the potential of electronic structure methods in understanding magnetic interactions in exchange-coupled polynuclear transition metal clusters.

Independently Optimized Orbital Sets in GRASP—The Case of Hyperfine Structure in Li I

In multiconfiguration Dirac–Hartree–Fock (MCDHF) calculations, there is a strong coupling between the localization of the orbital set and the configuration state function (CSF) expansion used to determine it. Furthermore, it is well known that an orbital set resulting from calculations, including CSFs describing core–core correlation and other effects, which aims to lower the weighted energies of a number of targeted states as much as possible, may be inadequate for building CSFs that account for correlation effects that are energetically unimportant but decisive for computed properties, e.g., hyperfine structures or transition rates. This inadequacy can be traced in irregular or oscillating convergence patterns of the computed properties as functions of the increasing orbital set. In order to alleviate the above problems, we propose a procedure in which the orbital set is obtained by merging several separately optimized, and mutually non-orthogonal, orbital sets. This computational strategy preserves the advantages of capturing electron correlation on the total energy through the variational MCDHF method and allows to target efficiently the correlation effects on the considered property. The orbital sets that are merged are successively orthogonalized against each other to retain orthonormality. The merged orbital set is used to build CSFs that efficiently lower the energy and also adequately account for the correlation effects that are important for the property. We apply the procedure to compute the hyperfine structure constants for the 1s22s2S1/2 and 1s22p2P1/2,3/2o states in 7Li and show that it leads to considerably improved convergence patterns with respect to the increasing orbital set compared to standard calculations based on a single orbital set, energy-optimized in the variational procedure. The perspectives of the new procedure are discussed in a broader context in the summary.

The rate constant for the automerization of vinyl radical: A theoretical approach using the Complete Active Space Self Consistent Field method

The Complete Active Space Self Consistent Field method (CASSCF) was utilized to investigate the rate constant for the automerization of vinyl radical. Using an enlarged active space including all valence electrons, thus producing a total number of 213,444 Configuration State Functions (CSFs) the rate constant for the automerization is of the order of 109 s−1 or 1010 s−1 in case of using the vibrational energy splitting tunneling method with the corresponding energy barrier height, involving the zero point vibrational energy corrections, varying between 4.69 and 4.71 kcal/mol (1639.69–1647.34 cm−1). Furthermore depending on the basis set, the ground state vibrational proton tunneling splitting, ΔEsplit, varies between 0.27 and 0.29 cm−1.

A recursive formulation of one‐electron coupling coefficients for spin‐adapted configuration interaction calculations featuring many unpaired electrons

AbstractThis work reports on a novel computational approach to the efficient evaluation of one‐electron coupling coefficients as they are required during spin‐adapted electronic structure calculations of the configuration interaction type. The presented approach relies on the equivalence of the representation matrix of excitation operators in the basis of configuration state functions and the representation matrix of permutation operators in the basis of genealogical spin eigenfunctions. After the details of this connection are established for every class of one‐electron excitation operator, a recursive scheme to evaluate permutation operator representations originally introduced by Yamanouchi and Kotani is recapitulated. On the basis of this scheme we have developed an efficient algorithm that allows the evaluation of all nonredundant coupling coefficients for systems with 20 unpaired electrons and a total spin of within only a few hours on a simple Desktop‐PC. Furthermore, a full‐CI implementation that utilizes the presented approach to one‐electron coupling coefficients is shown to perform well in terms of computational timings for CASCI calculations with comparably large active spaces. More importantly, however, this work paves the way to spin‐adapted and configuration driven selected configuration interaction calculations with many unpaired electrons.

Extended relativistic multiconfiguration calculations of energy levels and transition properties in singly ionized tin

Multiconfiguration Dirac-Hartree-Fock (MCDHF) and relativistic configuration interaction (RCI) calculations are performed for 22 states in singly ionized tin (Sn II) belonging to the 5s2ns(n=6,7), 5s2nd(n=5,6), 5s5p2 even parity configurations and the 5s2np(n=5,6,7), 5s24f odd parity configurations. Valence-valence and core-valence correlation effects are taken into account through configuration state function (CSF) expansions. Complete and consistent data sets of level energies, wavelengths, oscillator strengths, lifetimes and transition rates among all these states are given. The results are compared with existing theoretical and experimental results. There is an excellent agreement for calculated excitation energies with experimental data from the NIST database. Lifetimes and transition rates are also in agreement with the results from previous calculations and available measurements for most of the transitions.

Reducing the computational load – atomic multiconfiguration calculations based on configuration state function generators

In configuration interaction (CI) calculations the atomic wave functions are given as expansions over configuration state functions (CSFs) built on relativistic one-electron orbitals. The expansion coefficients of the configuration state functions are obtained by constructing and diagonalizing the Hamiltonian matrix. Here we show how a regrouping of the configuration state functions and the introduction of configuration state function generators (CSFGs) allow for a substantial reduction of the computational load in relativistic CI calculations. The computational methodology based on configuration state function generators, recently implemented in the General Relativistic Atomic Structure package (Grasp2018, Froese Fischer et al. (2019) [16]), is applied to a number of atomic systems and correlation models with increasing sets of one-electron orbitals. We demonstrate a reduction of the CPU time with factors between 10 and 14 for the largest CI calculations. The inclusion of the Breit interaction into the calculations is time consuming. By applying restrictions on the Breit integrals we show that it is possible to further reduce the CPU times with factors between 2 and 3, with negligible changes to the computed excitation energies. We also demonstrate that the introduction of configuration state function generators allows for efficient a priori condensation techniques, with reductions of the expansions sizes with factors between 1.5 and 2.5 and the CPU time with factors between 2.5 and 4.5, again with negligible changes to the excitation energies. In total we demonstrate reductions of the CPU time with factors up to 68 for CI calculations based on configuration state function generators, restrictions on the Breit integrals and with a priori condensed expansions compared to ordinary CI calculations without restrictions on the Breit integrals and with full expansions. Further perspectives of the new methodology based on configuration state function generators are given.

Large-scale Multiconfiguration Dirac–Hartree–Fock Calculations for Astrophysics: C-like Ions from O iii to Mg vii

Large-scale multiconfiguration Dirac–Hartree–Fock calculations are provided for the n ≤ 5 states in C-like ions from O iii to Mg vii. Electron correlation effects are accounted for by using large configuration state function expansions, built from sets of orbitals with principal quantum numbers n ≤ 10. An accurate and complete data set of excitation energies, wavelengths, radiative transition parameters, and lifetimes is offered for the 156 (196, 215, 272, 318) lowest states of the 2s 22p 2, 2s2p 3, 2p 4, 2s 22p3s, 2s 22p3p, 2s 22p3d, 2s2p 23s, 2s2p 23p, 2s2p 23d, 2p 33s, 2p 33p, 2p 33d, 2s 22p4s, 2s 22p4p, 2s 22p4d, 2s 22p4f, 2s2p 24s, 2s2p 24p, 2s2p 24d, 2s2p 24f, 2s 22p5s, 2s 22p5p, 2s 22p5d, 2s 22p5f, and 2s 22p5g configurations in O iii (F iv, Ne v, Na vi, Mg vii). By comparing available experimental wavelengths with the MCDHF results, the previous line identifications for the n = 5, 4, 3 → n = 2 transitions of Na vi in the X-ray and EUV wavelength range are revised. For several previous identifications discrepancies are found, and tentative new (or revised) identifications are proposed. A consistent atomic data set including both energy and transition data with spectroscopic accuracy is provided for the lowest hundreds of states for C-like ions from O iii to Mg vii.

Combined unitary and symmetric group approach applied to low-dimensional Heisenberg spin systems

A novel combined unitary and symmetric group approach is used to study the spin-$\frac{1}{2}$ Heisenberg model and related Fermionic systems in a total spin-adapted representation, using a linearly-parameterised Ansatz for the many-body wave function. We show that a more compact ground-state wave function representation---indicated by a larger leading ground-state coefficient---is obtained when combining the symmetric group ${\mathcal{S}}_{n}$, in the form of permutations of the underlying lattice site ordering, with the cumulative spin coupling based on the unitary group $\mathrm{U}(n)$. In one-dimensional systems the observed compression of the wave function is reminiscent of block-spin renormalization group approaches, and allows us to study larger lattices (here taken up to 80 sites) with the spin-adapted full configuration interaction quantum Monte Carlo method, which benefits from the sparsity of the Hamiltonian matrix and the corresponding sampled eigenstates that emerge from the reordering. We find that in an optimal lattice ordering the configuration state function with highest weight already captures with high accuracy the spin-spin correlation function of the exact ground-state wave function. This feature is found for more general lattice models, such as the Hubbard model, and ab initio quantum chemical models, exemplified by one-dimensional hydrogen chains. We also provide numerical evidence that the optimal lattice ordering for the unitary group approach is not generally equivalent to the optimal ordering obtained for methods based on matrix-product states, such as the density-matrix renormalization group approach.