Complete Active Space Wave Function Research Articles

Automatic State Interaction with Large Localized Active Spaces for Multimetallic Systems.

The localized active space self-consistent field method factorizes a complete active space wave function into an antisymmetrized product of localized active space wave function fragments. Correlation between fragments is then reintroduced through localized active space state interaction (LASSI), in which the Hamiltonian is diagonalized in a model space of LAS states. However, the optimal procedure for defining the LAS fragments and LASSI model space is unknown. We here present an automated framework to explore systematically convergent sets of model spaces, which we call LASSI[r, q]. This method requires the user to select only r, the number of electron hops from one fragment to another, and q, the number of fragment basis functions per Hilbert space, which converges to CASCI in the limit of r, q → ∞. Numerical tests of this method on the trimetal oxo-centered complexes [Fe(III)Al(III)Fe(II)(μ3-O)(HCOO)6] and [Fe(III)2Fe(II)(μ3-O)(HCOO)6] show efficient convergence to the CASCI limit with 4-10 orders of magnitude fewer states than CASCI.

Understanding the impact of correlation on bond length alternation in polyenes

This article analyzes the electronic factors governing bond length alternation (BLA) in linear polyenes. The impact of the various effects is illustrated on small all-trans polyenes, namely butadiene, hexatriene and octatetraene prototype molecules. It is well known that self-consistent-field single determinant treatments overestimate the bond length alternation and the paper aims to identify physical effects of correlation which correct this defect. The question is addressed using an orthogonal valence bond-type formalism in which the wave function is expressed in terms of strongly localized bonding and antibonding molecular orbitals. This paper shows that dynamic polarization effects of π orbitals accounted for in the full-π complete active space wave function significantly reduce bond alternation. These effects are brought by single excitations applied on the inter-bond charge transfer determinants. The dynamic polarization of σ bonds, of either CC or CH character, is analyzed afterward by either enlarging the active space or by adding the 1hole-1particle excitations. It is shown that these effects also decrease the BLA and increase the coefficients of the charge transfer determinants. Moreover, the relation with dynamic polarization of ligand-to-metal and metal-to-ligand charge transfer (LMCT and MLCT, respectively) components in magnetic transition-metal compounds is discussed.

Multiconfiguration Pair-Density Functional Theory for Transition Metal Silicide Bond Dissociation Energies, Bond Lengths, and State Orderings.

Transition metal silicides are promising materials for improved electronic devices, and this motivates achieving a better understanding of transition metal bonds to silicon. Here we model the ground and excited state bond dissociations of VSi, NbSi, and TaSi using a complete active space (CAS) wave function and a separated-pair (SP) wave function combined with two post-self-consistent field techniques: complete active space with perturbation theory at second order and multiconfiguration pair-density functional theory. The SP approximation is a multiconfiguration self-consistent field method with a selection of configurations based on generalized valence bond theory without the perfect pairing approximation. For both CAS and SP, the active-space composition corresponds to the nominal correlated-participating-orbital scheme. The ground state and low-lying excited states are explored to predict the state ordering for each molecule, and potential energy curves are calculated for the ground state to compare to experiment. The experimental bond dissociation energies of the three diatomic molecules are predicted with eight on-top pair-density functionals with a typical error of 0.2 eV for a CAS wave function and a typical error of 0.3 eV for the SP approximation. We also provide a survey of the accuracy achieved by the SP and extended separated-pair approximations for a broader set of 25 transition metal–ligand bond dissociation energies.

Local Enhancement of Dynamic Correlation in Excited States: Fresh Perspective on Ionicity and Development of Correlation Density Functional Approximation Based on the On-Top Pair Density.

We discuss the interplay between the nondynamic and dynamic electron correlation in excited states from the perspective of the suppression of dynamic correlation (SDC) and enhancement of dynamic correlation (EDC) effects. We reveal that there exists a connection between the ionic character of a wave function and EDC. Following this finding we introduce a quantitative measure of ionicity based solely on local functions without referring to valence bond models. The ability to recognize both the SDC and EDC regions underlies the presented method, named CASΠDFT, combining complete active space (CAS) wave function and density functional theory (DFT) via the on-top pair density (Π) function. We extend this approach to excited states by devising an improved representation of the EDC effect in the correlation functional. The generalized CASΠDFT uses different DFT functionals for ground and excited states. Numerical demonstration for singlet π → π* excitations shows that CASΠDFT offers satisfactory accuracy at a fraction of the cost of the ab initio approaches.

Full Wave Function Optimization with Quantum Monte Carlo-A Study of the Dissociation Energies of ZnO, FeO, FeH, and CrS.

The dissociation energies of four transition-metal dimers are determined using diffusion Monte Carlo. The Jastrow, configuration interaction, and molecular orbital parameters of the wave function are both partially and fully optimized with respect to the variational energy. The pivotal role is thereby ascribable to the optimization of the molecular orbital parameters of a complete active space wave function in the presence of a Jastrow correlation function. Excellent results are obtained for ZnO, FeO, FeH, and CrS. In addition, potential energy curves are computed for the first three compounds at the multireference diffusion Monte Carlo (MR-DMC) level, from which spectroscopic constants such as the equilibrium bond distance, the harmonic frequency, and the anharmonicity are extracted. All of these quantities agree well with the experiment. Furthermore, it is shown for CrS that a restricted active space calculation can yield improved initial orbitals by including single and double excitations from the original active space into a set of virtual orbitals. We demonstrated in this study that the fixed-node error in DMC can be systematically reduced for multireference systems by orbital optimization in compact active spaces. Although DMC calculations with a large number of determinants are possible and very accurate, our results demonstrate that compact wave functions may be sufficient in order to obtain accurate nodal surfaces, which determine the accuracy of DMC, even in the case of transition-metal compounds.

Capturing the Dynamic Correlation for Arbitrary Spin-Symmetry CASSCF Reference with Adiabatic Connection Approaches: Insights into the Electronic Structure of the Tetramethyleneethane Diradical.

The recently proposed approach to multireference dynamic correlation energy based on the adiabatic connection (AC) is extended to an arbitrary spin symmetry of the reference state. We show that both the spin-free AC approach and its computationally inexpensive approximation, AC0, when combined with a complete active space wave function, constitute viable alternatives to the perturbation-based and density-functional-based multiconfiguration methods. In particular, the AC0 approach, thanks to its favorable scaling with the system size and the size of the active space, allows for treating larger systems than its perturbation-based counterparts while maintaining comparable accuracy. We show the method's robustness on illustrative chemical systems, including the elusive tetramethyleneethane (TME) diradical, potential energy surfaces of which present a challenge to most computational approaches. For the latter system, AC0 outperforms other methods, staying in close agreement with the full configuration interaction quantum Monte Carlo benchmark. A careful analysis of the contributions to the correlation energy of TME's lowest singlet and triplet states reveals the subtle interplay of the dynamic and static correlation as the key to understanding the shape of the diradical's potential energy surfaces.

Complete active space and corrected density functional theories helping each other to describe vertical electronic π → π* excitations in prototype multiple-bonded molecules.

The CASΠDFT method, which combines the complete active space (CAS) wave function approach and density functional theory (DFT), offers an efficient description of important excitations to the lowest excited states. CASΠDFT employs a correlation DFT functional corrected with a function P[x] of the ratio xr of the conditional and conventional electron densities obtained with the CAS on-top pair density Π(r). The sectors of P[x] for x(r) ≤ 1 and x(r) > 1 represent the opposite effects of the suppression of dynamic correlation with nondynamic correlation and its enhancement due to the ionic-type excitation. The present combination of the self-consistent-field CAS and the corrected Lee-Yang-Parr correlation functional closely reproduces in the relatively small double-zeta basis the benchmark experimental lowest singlet vertical π → π* excitations in the prototype multiple-bonded molecules N2, CO, C2H2, and C2H4.

Electronic Excited States from the Adiabatic-Connection Formalism with Complete Active Space Wave Functions.

It is demonstrated how the recently proposed multireference adiabatic-connection (AC) approximation for electron correlation energy ( Pernal , K. Electron Correlation from the Adiabatic Connection for Multireference Wave Functions . Phys. Rev. Lett. 2018 , 120 , 013001 ) can be extended to predicting correlation energy in excited states of molecules. It is the first successful application of the AC approach to computing excited-states energies of molecules using a complete active space (CAS) wave function as a reference. The unique feature of the AC-CAS approach with respect to popular methods such as CASPT2 and NEVPT2 is that it requires only one- and two-particle reduced density matrices, making it possible to efficiently treat large spaces of active electrons. Application of the simpler variant of AC, the AC0, which is based on the first-order expansion of the AC integrand at the uncorrelated system limit, to excited states yields excitation energies with accuracy rivaling that of the NEVPT2 method but at greatly reduced computational cost.

Correlation Energy from the Adiabatic Connection Formalism for Complete Active Space Wave Functions.

Recently, the adiabatic connection (AC) formula for the electron correlation energy has been proposed for a broad class of multireference wave functions (Pernal, K. Electron Correlation from the Adiabatic Connection for Multireference Wave Functions. Phys. Rev. Lett. 2018, 120, 013001). We show that the AC formula used together with the extended random phase approximation (ERPA) can be efficiently applied to complete active space (CAS) wave functions to recover the remaining electron correlation. Unlike most of the perturbation theory approaches, the proposed AC-CAS method does not require construction of higher than two-electron reduced density matrices, which offers an immediate computational saving. In addition, we show that typically the AC-CAS systematically reduces the errors of both the absolute value of energy and of the energy differences (energy barrier) upon enlarging active spaces for electrons and orbitals. AC-CAS consistently gains in accuracy from including more active electrons. We also propose and study that the performance of the AC0 approach resulting from the first-order expansion of the AC integrand at the coupling constant equal to 0. AC0 avoids solving the full ERPA eigenequation, replacing it with small-dimension eigenproblems, while retaining good accuracy of the AC-CAS method. Low computational cost, compared to AC-CAS or perturbational approaches, makes AC0 the most efficient ab initio approach to capturing electron correlation for the CAS wave functions.

Full Wave Function Optimization with Quantum Monte Carlo and Its Effect on the Dissociation Energy of FeS.

Diffusion quantum Monte Carlo calculations with partial and full optimization of the guide function are carried out for the dissociation of the FeS molecule. For the first time, quantum Monte Carlo orbital optimization for transition metal compounds is performed. It is demonstrated that energy optimization of the orbitals of a complete active space wave function in the presence of a Jastrow correlation function is required to obtain agreement with the experimental dissociation energy. Furthermore, it is shown that orbital optimization leads to a 5Δ ground state, in agreement with experiments but in disagreement with other high-level ab initio wave function calculations which all predict a 5Σ+ ground state. The role of the Jastrow factor in DMC calculations with pseudopotentials is investigated. The results suggest that a large Jastrow factor may improve the DMC accuracy substantially at small additional cost.

On-Top Pair Density as a Measure of Left-Right Correlation in Bond Breaking.

To better understand left-right electron correlation and the effect of bond breaking on the on-top pair density, analytic expressions for the total density, the on-top pair density, and the ratio of the on-top pair density to the square of the total density were derived for H2 for both a restricted Hartree-Fock wave function and a complete active space self-consistent-field wave function with two electrons in two active orbitals. These quantities are compared for the two wave functions for various points in space around the molecule as functions of internuclear distance. At some points in space, in the CASSCF(2,2) wave function, electron correlation, perhaps counterintuitively, increases the probability that two electrons are at the same point in space. At the Coulson-Fischer point, the on-top pair density for the complete active space wave function starts to rapidly approach zero, and this can be taken as an indicator of bond breaking.

A globally accurate potential energy surface of $$\mathrm{HS_2}{(A\,^2A^\prime )}$$ HS 2 ( A 2 A ′ ) and studies on the reaction dynamic of $$\mathrm{H}(^2\mathrm{S})+\mathrm{S_2}(a\,^1{\varDelta }_g)$$ H ( 2 S ) + S 2 ( a 1 Δ g )

A global potential energy surface (PES) for the first excited electronic state of $$\mathrm{HS_2}$$ is reported. Analytic representation of the PES is accomplished in the frame of a many-body expansion. A grid of 2069 ab initio points is used in the least-square fitting procedure. Calculations are carried out at the multireference configuration interaction level, using the complete active space wave function as reference. The aug-cc-pVQZ Dunning’s basis set is used. Topographical features of main stationary points of the PES are presented and compared with the literature. The newly obtained function is then used in quasi-classical trajectory calculations for $$\mathrm{H}(^2\mathrm{S})+\mathrm{S_2}(a\,^1{\varDelta }_g)$$ $$\rightarrow$$ $$\mathrm{SH}(X\,^2{\varPi }) + \mathrm{S} (^3P)$$ reaction. Differential and integral cross section for such a reaction is here calculated. State-specific reaction rate constant is presented as well.

Second-Order Perturbation Theory for Generalized Active Space Self-Consistent-Field Wave Functions.

A multireference second-order perturbation theory approach based on the generalized active space self-consistent-field (GASSCF) wave function is presented. Compared with the complete active space (CAS) and restricted active space (RAS) wave functions, GAS wave functions are more flexible and can employ larger active spaces and/or different truncations of the configuration interaction expansion. With GASSCF, one can explore chemical systems that are not affordable with either CASSCF or RASSCF. Perturbation theory to second order on top of GAS wave functions (GASPT2) has been implemented to recover the remaining electron correlation. The method has been benchmarked by computing the chromium dimer ground-state potential energy curve. These calculations show that GASPT2 gives results similar to CASPT2 even with a configuration interaction expansion much smaller than the corresponding CAS expansion.

A Density Functional Theory and ab Initio Investigation of the Oxidation Reaction of CO by IO Radicals.

To get an insight into the possible reactivity between iodine oxides and CO, a first step was to study the thermochemical properties and kinetic parameters of the reaction between IO and CO using theoretical chemistry tools. All stationary points involved were optimized using the Becke's three-parameter hybrid exchange functional coupled with the Lee-Yang-Parr nonlocal correlation functional (B3LYP) and the Møller-Plesset second-order perturbation theory (MP2). Single-point energy calculations were performed using the coupled cluster theory with the iterative inclusion of singles and doubles and the perturbative estimation for triple excitations (CCSD(T)) and the aug-cc-pVnZ (n = T, Q, and 5) basis sets on geometries previously optimized at the aug-cc-pVTZ level. The energetics was then recalculated using the one-component DK-CCSD(T) approach with the relativistic ANO basis sets. The spin-orbit coupling for the iodine containing species was calculated a posteriori using the restricted active space state interaction method in conjunction with the multiconfigurational perturbation theory (CASPT2/RASSI) employing the complete active space (CASSCF) wave function as the reference. The CCSD(T) energies were also corrected for BSSE for molecular complexes and refined with the extrapolation to CBS limit while the DK-CCSD(T) values were refined with the extrapolation to FCI. The exploration of the potential energy surface revealed a two-steps mechanism with a trans and a cis pathway. The rate constants for the direct and complex mechanism were computed as a function of temperature (250-2500 K) using the canonical transition state theory. The three-parameter Arrhenius expressions obtained for the direct and indirect mechanism at the DK-CCSD(T)-cf level of theory is 1.49 × 10(-17) × T(1.77) exp(-47.4 (kJ mol(-1))/RT).

Globally accurate ab initio based potential energy surface of H2O+(X 4A″)**Project supported by the National Natural Science Foundation of China (Grant Nos. 11304185 and 11074151), China Postdoctoral Science Foundation (Grant No. 2014M561957), the Postdoctoral Innovation Project of Shandong Province, China (Grant No. 201402013), and the Shandong Provincial Natural Science Foundation, China (Grant No. ZR2014AM022).

A globally accurate potential energy surface is reported for the electronic ground-state H2O+. The ab initio energies utilized to map the potential energy surface are calculated at the multireference configuration interaction method employing the aug-cc-pVQZ basis set and the full valence complete active space wave function as reference. In order to improve accuracy of the resulting raw ab initio energies, they are then extrapolated to the complete basis set limit and most importantly to the full configuration-interaction limit by semiempirically correcting the dynamical correlation using the double many-body expansion-scaled external correlation method. The topographical features of the current potential energy surface were examined in detail, which agree nicely with those of other theoretical work.

Multireference Character for 3d Transition-Metal-Containing Molecules.

Coupled cluster and configuration interaction diagnostics have been examined in order to assess the reliability of single reference quantum methods for a series of 3d transition metal species including hydrides, nitrides, chalcogenides, halides, small clusters, coordination complexes, and metal dimers. Several means of diagnostics have been considered including T1 and D1 diagnostics (the Frobenius norm and matrix 2-norm of coupled cluster amplitudes for single excitations, respectively), C0(2) (the weight of leading configuration of a complete active space wave function), and %TAE (percent total atomization energy). T1 and D1 diagnostics are strongly correlated for certain metal-ligand bonding types. The use of T1 and D1 together with %TAE can provide more reliable assessment of the severity of nondynamical correlation than a single indicator can provide. New criteria, namely T1 > 0.05, D1 > 0.15, and |%TAE| > 10, are suggested to identify inorganic species with substantial nondynamical correlation. For these systems, energies and spectroscopic properties computed using single reference electronic correlation methods may suffer from large errors and unpredictable behavior. Conversely, a computation where a molecule is below one or more of these thresholds does not always imply domination by a single reference. Some historically pathological molecules such as Mn2 and Cr2 show T1 < 0.05 and D1 < 0.15. Current implementations of coupled cluster diagnostics may still be insufficient for categorization of molecules that have pronounced nondynamical correlation.

The generalized active space concept in multiconfigurational self-consistent field methods

A multiconfigurational self-consistent field method based on the concept of generalized active space (GAS) is presented. GAS wave functions are obtained by defining an arbitrary number of active spaces with arbitrary occupation constraints. By a suitable choice of the GAS spaces, numerous ineffective configurations present in a large complete active space (CAS) can be removed, while keeping the important ones in the CI space. As a consequence, the GAS self-consistent field approach retains the accuracy of the CAS self-consistent field (CASSCF) ansatz and, at the same time, can deal with larger active spaces, which would be unaffordable at the CASSCF level. Test calculations on the Gd atom, Gd(2) molecule, and oxoMn(salen) complex are presented. They show that GAS wave functions achieve the same accuracy as CAS wave functions on systems that would be prohibitive at the CAS level.

Copper Corroles: the Question of Noninnocence

In this paper, the results are presented from a comparative study of the electronic and geometric structure of copper corroles by means of either density functional theory (DFT, using both pure and hybrid functionals) and multiconfigurational ab initio methods, starting from either a complete active space (CASSCF) or restricted active space (RASSCF) reference wave function and including dynamic correlation by means of second-order perturbation theory (CASPT2/RASPT2). DFT geometry optimizations were performed for the lowest singlet and triplet states of copper corrole, both unsubstituted and meso-substituted with three phenyl groups. The effect of saddling on the electronic structure was investigated by comparing the results obtained for planar (C(2v)) and saddled (C(2)) structures. With DFT, the origin of the saddling distortion is found to be dependent on the applied functional: covalent Cu 3d-corrole π interactions with pure functionals (BP86, OLYP), antiferromagnetic exchange coupling between an electron in the corrolate (C(2)) b type π orbital, and an unpaired Cu(II) 3d electron with hybrid functionals (B3LYP, PBE0). The CASPT2 results essentially confirm the suggestion from the hybrid functionals that copper corroles are noninnocent, although the contribution of diradical character to the copper-corrole bond is found to be limited to 50% or less. The lowest triplet state is calculated at 0-10 kcal/mol and conform with the experimental observation (variable temperature NMR) that this state should be thermally accessible.

Ab-Initio-Based Global Double Many-Body Expansion Potential Energy Surface for the Electronic Ground State of the Ammonia Molecule

We report a single-sheeted global double many-body expansion potential energy surface for the ground electronic state of NH(3) that has been calibrated from newly calculated ab initio energies. It employs realistic double many-body expansion functions previously reported from accurate ab initio data for the triatomic fragments plus four-body energy terms that have been calibrated from multireference configuration calculations carried out in the present work for the title system using the full valence complete active space wave function as reference and a triple-zeta basis set of the correlation consistent type. It provides therefore a reliable description of both short-range and long-range forces while dissociating correctly at all asymptotic limits. The major attributes of the NH(3) double many-body expansion potential energy surface have been characterized and found to be in good agreement both with the calculated ones from the raw ab initio energies and theoretical results available in the literature. It can then be recommended both as a reliable functional form on which dynamics calculations can be performed and as a model for improvement that is open to refinement through further accurate ab initio calculations, vibrational calculations, or both.

Excited states of methylene from quantum Monte Carlo

The ground and lowest three adiabatic excited states of methylene are computed using the variational Monte Carlo and diffusion Monte Carlo (DMC) methods using progressively larger Jastrow-Slater multideterminant complete active space (CAS) wave functions. The highest of these states has the same symmetry, (1)A(1), as the first excited state. The DMC excitation energies obtained using any of the CAS wave functions are in excellent agreement with experiment, but single-determinant wave functions do not yield accurate DMC energies of the states of (1)A(1) symmetry, indicating that it is important to include in the wave function Slater determinants that describe static (strong) correlation. Excitation energies obtained using recently proposed pseudopotentials [Burkatzki et al., J. Chem. Phys. 126, 234105 (2007)] differ from the all-electron excitation energies by at most 0.04 eV.