Complete Active Space Reference Research Articles

Dynamic correlation for non-orthogonal reference states: Improved perturbational and variational methods.

The use of non-orthogonal orbitals allows the construction and use of more compact wave functions than offered by standard methods using orthogonal molecular orbitals; in particular, for molecules containing partly occupied atomic orbitals as present, for example, in transition metal complexes. With the purpose of developing efficient dynamic correlation methods, we discuss several new internal correlation methods employing a reference state containing non-orthogonal active orbitals. The non-orthogonal internally contracted perturbation theory approach is improved in several directions. The major improvements are the use of the Dyall Hamiltonian including two-electron interactions within the active space as the zero-order operator, the calculation of third-order energy-corrections, and the inclusion of excitations in the space of active orbitals. The latter improvement corrects for the use of an incomplete reference state. The improvements are tested for the nitrogen molecule and the challenging chromium dimer. The combined use of the improved zero-order Hamiltonian and the inclusion of active space excitations allow us to obtain potential curves for the chromium dimer that are close to those obtained using the larger complete active space reference wave function.

A second-order multi-reference quasiparticle-based perturbation theory

The purpose of this paper is to introduce a second-order perturbation theory derived from the mathematical framework of the quasiparticle-based multi-reference coupled-cluster approach (Rolik and Kallay in J Chem Phys 141:134112, 2014). The quasiparticles are introduced via a unitary transformation which allows us to represent a complete active space reference function and other elements of an orthonormal multi-reference basis in a determinant-like form. The quasiparticle creation and annihilation operators satisfy the fermion anti-commutation relations. As the consequence of the many-particle nature of the applied unitary transformation these quasiparticles are also many-particle objects, and the Hamilton operator in the quasiparticle basis contains higher than two-body terms. The definition of the new theory strictly follows the form of the single-reference many-body perturbation theory and retains several of its beneficial properties like the extensivity. The efficient implementation of the method is briefly discussed, and test results are also presented.

Discontinuities-free complete-active-space state-specific multi-reference coupled cluster theory for describing bond stretching and dissociation.

The earlier proposed multi-reference state-specific coupled-cluster theory with the complete active space reference [CASCC; Lyakh et al., J. Chem. Phys. 122, 024108 (2005)] suffered from a problem of energy discontinuities when the formal reference state was changing in the calculation of the potential energy curve (PEC). A simple remedy to the discontinuity problem is found and is presented in this work. It involves using natural complete active space self-consistent field active orbitals in the complete active space coupled-cluster calculations. The approach gives smooth PECs for different types of dissociation problems, as illustrated in the calculations of the dissociation of the single bond in the hydrogen fluorine molecule and of the symmetric double-bond dissociation in the water molecule.

A quasiparticle-based multi-reference coupled-cluster method.

The purpose of this paper is to introduce a quasiparticle-based multi-reference coupled-cluster (MRCC) approach. The quasiparticles are introduced via a unitary transformation which allows us to represent a complete active space reference function and other elements of an orthonormal multi-reference (MR) basis in a determinant-like form. The quasiparticle creation and annihilation operators satisfy the fermion anti-commutation relations. On the basis of these quasiparticles, a generalization of the normal-ordered operator products for the MR case can be introduced as an alternative to the approach of Mukherjee and Kutzelnigg [Recent Prog. Many-Body Theor. 4, 127 (1995); Mukherjee and Kutzelnigg, J. Chem. Phys. 107, 432 (1997)]. Based on the new normal ordering any quasiparticle-based theory can be formulated using the well-known diagram techniques. Beyond the general quasiparticle framework we also present a possible realization of the unitary transformation. The suggested transformation has an exponential form where the parameters, holding exclusively active indices, are defined in a form similar to the wave operator of the unitary coupled-cluster approach. The definition of our quasiparticle-based MRCC approach strictly follows the form of the single-reference coupled-cluster method and retains several of its beneficial properties. Test results for small systems are presented using a pilot implementation of the new approach and compared to those obtained by other MR methods.

On the Deactivation Mechanisms of Adenine–Thymine Base Pair

In this contribution, the multiconfigurational second-order perturbation theory method based on a complete active space reference wave function (CASSCF/CASPT2) is applied to study all possible single and double proton/hydrogen transfers between the nucleobases in the adenine-thymine (AT) base pair, analyzing the role of excited states with different nature [localized (LE) and charge transfer (CT)], and considering concerted as well as step-wise mechanisms. According to the findings, once the lowest excited states, localized in adenine, are populated during UV irradiation of the Watson-Crick base pair, the proton transfer in the N-O bridge does not require high energy in order to populate a CT state. The latter state will immediately relax toward a crossing with the ground state, which will funnel the system to either the canonical structure or the imino-enol tautomer. The base pair is also capable of repairing itself easily since the imino-enol species is unstable to thermal conversion.

Multiconfiguration second‐order perturbation theory approach to strong electron correlation in chemistry and photochemistry

AbstractRooted in the very fundamental aspects of many chemical phenomena, and for the majority of photochemistry, is the onset of strongly interacting electronic configurations. To describe this challenging regime of strong electron correlation, an extraordinary effort has been put forward by a young generation of scientists in the development of theories ‘beyond’ standard wave function and density functional models. Despite their encouraging results, a twenty‐and‐more‐year old approach still stands as the gold standard for these problems: multiconfiguration second‐order perturbation theory based on complete active space reference wave function (CASSCF/CASPT2). We will present here a brief overview of the CASSCF/CASPT2 computational protocol, and of its role in our understanding of chemical and photochemical processes. © 2011 John Wiley & Sons, Ltd.This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods

State-specific multireference coupled-cluster theory of molecular electronic excited states

A state-specific coupled cluster (CC) theory with the CAS (complete active space) reference (CASCC) and based on an approach that employs single reference determinant (so-called “formal reference” determinant) in the expression for the wave function has been developed to study electronic excited states with different spatial and spin symmetries. The formal reference determinant is used to generate the reference wave function in the form of a linear combination of the CAS determinants contracted to configurations with the spin and spatial symmetries of the target state. Such properly symmetrized multideterminantal reference provides the zero-order description of the state under consideration. To that reference an exponential CC operator is applied to describe the dynamic electron correlation effects in the CASCC wave function of the considered state. All necessary equations for the CASCC energy and configurational amplitudes are generated using an automated computer-based method which constructs all necessary coupled cluster diagrams for any arbitrary level of the electronic excitation.

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.

A calculation of spectroscopic parameters for hydrogen fluoride with multi-reference state-specific coupled-cluster method

The multi-reference state-specific coupled-cluster method with a complete-active-space reference (CASCC) developed by our group has been used to calculate the potential energy curves, spectroscopic parameters, and vibrational levels for the ground ( X 1 Σ + ) and excited ( B 1 Σ + ) states of 19FH. The working algorithm for the CASCC method was derived with a computer-based automated approach that generates the CC energy and amplitude equations and the corresponding coupled-cluster diagrams. The spin–orbital sets for the CASCC calculations have been obtained using the complete-active-space self-consistent-field method (CASSCF). As determined by comparing the theoretical results with experimental ones reconstructed using the Rydberg–Klein–Reese method, the CASCC energies are better than those obtained with the CCSD and EOM-CCSD methods.

What Active Space Adequately Describes Oxygen Activation by a Late Transition Metal? CASPT2 and RASPT2 Applied to Intermediates from the Reaction of O2 with a Cu(I)-α-Ketocarboxylate.

Multiconfigurational second-order perturbation theory calculations based on a complete active space reference wave function (CASPT2), employing active spaces of increasing size, are well converged at the level of 12 electrons in 12 orbitals for the singlet-triplet state-energy splittings of three supported copper-dioxygen and two supported copper-oxo complexes. Corresponding calculations using the restricted active space approach (RASPT2) offer similar accuracy with a significantly reduced computational overhead provided an inner (2,2) complete active space is included in the overall RAS space in order to account for strong biradical character in most of the compounds. The effects of the different active space choices and the outer RAS space excitations are examined, and conclusions are drawn with respect to the general applicability of the RASPT2 protocol.

Second-order perturbation theory with complete and restricted active space reference functions applied to oligomeric unsaturated hydrocarbons

Multiconfigurational second-order perturbation theory based on either a complete active space reference wave function (CASSCF/CASPT2) or a restricted active space reference wave function (RASSCF/RASPT2) has been applied to compute one-electron ionization potentials and vertical electronic energy differences of oligomers of length n formed from ethylene (n = 1-10), acetylene (n = 1-5), and phenylene (n = 1-3) subunits. The RASSCF/RASPT2 approach offers an accuracy similar to CASSCF/CASPT2 at significantly reduced computational expense (both methods show good agreement with experimental data where available). It is shown that RASPT2 extends the range of CASPT2-like approaches by permitting the use of larger active spaces.

The multi-reference retaining the excitation degree perturbation theory: A size-consistent, unitary invariant, and rapidly convergent wavefunction based ab initio approach

The retaining the excitation degree (RE) partitioning [R.F. Fink, Chem. Phys. Lett. 428 (2006) 461(20 September)] is reformulated and applied to multi-reference cases with complete active space (CAS) reference wave functions. The generalised van Vleck perturbation theory is employed to set up the perturbation equations. It is demonstrated that this leads to a consistent and well defined theory which fulfils all important criteria of a generally applicable ab initio method: The theory is proven numerically and analytically to be size-consistent and invariant with respect to unitary orbital transformations within the inactive, active and virtual orbital spaces. In contrast to most previously proposed multi-reference perturbation theories the necessary condition for a proper perturbation theory to fulfil the zeroth order perturbation equation is exactly satisfied with the RE partitioning itself without additional projectors on configurational spaces. The theory is applied to several excited states of the benchmark systems CH2, SiH2, and NH2, as well as to the lowest states of the carbon, nitrogen and oxygen atoms. In all cases comparisons are made with full configuration interaction results. The multi-reference (MR)-RE method is shown to provide very rapidly converging perturbation series. Energy differences between states of similar configurations converge even faster.

Restoring the size consistency of multireference configuration interactions through class dressings: Applications to ground and excited states

The present paper presents a revised version of a size-consistency correction to the multireference configuration interaction techniques previously proposed by Szalay et al. [J. Phys. Chem. 100, 6288 (1996)]. The method assumes a complete active space reference space and separates the nonreference determinants in several classes according to their number of inactive holes and particles. The correction is formulated as a dressing of the diagonal energies of these determinants, which depends on their class, as originally proposed by Ruttink et al. [J. Chem. Phys. 94, 7212 (1991)]. The exclusion principle violating corrections are evaluated through a simple counting of the various excitation processes which remain possible on each class. The efficiency of the method has been tested on a series of multireference problems for which full configuration interaction results are available (OH(2) bond breaking, Be insertion in H(2), excited states of CH(2)). The dressing of a given state not only provides excellent results for this state but also provides accurate excited roots. The efficiency of state-specific dressings is dramatic. The adaptation of this proposal to difference-dedicated configuration interactions can be extremely fruitful, as illustrated in the calculation of the 1 (1)A(g)-1(1)B(u)(pi->pi(*)) transition energy of the trans-butadiene molecule.

Potential energy surface of the electron excited states in the state-specific multi-reference coupled cluster theory. Hydrogen fluoride dissociation

The recently proposed state-specific multi-reference coupled cluster (SSMRCC) theory with the complete active space (CAS) reference has been tested in calculations of the potential energy surfaces of electronically-excited states. The algorithm for the method is derived using the computer-based automated approach for generating the coupled cluster diagrams and the energy and amplitude equations. The active space for the calculation of the target state is constructed using natural orbital expansion obtained from the configuration interaction method with single and double excitations (CISD) one-particle density matrix. The spin–orbitals for the CASCC calculation are obtained using the complete active space self consistent field (CASSCF) method. The numerical example shown concerns several electronically excited states with different spatial and spin symmetries of the hydrogen fluoride molecule. The calculations are performed at several internuclear distances and compared with the results obtained using other methods such as the full configuration interaction (FCI) method, the equation-of-motion (EOM) method, and the multi-reference perturbation theory (MRPT). The comparison shows that the CASCC results are better than the EOM and MRPT results.

Excited states in the multireference state-specific coupled-cluster theory with the complete active space reference

The recently proposed multireference state-specific coupled-cluster theory with the complete active space reference has been used to study electronically excited states with different spatial and spin symmetries. The algorithm for the method has been obtained using the computerized approach for automatic generation of coupled-cluster diagrams with an arbitrary level of the electronic excitation from a formal reference determinant. The formal reference is also used to generate the genuine reference state in the form of a linear combination of determinants contracted to a configuration with the spin and spatial symmetries of the target state. The natural-orbital expansions of the one-electron configuration inferaction density matrix allowed us to obtain the most compact orbital space for the expansion of the reference function. We applied our approach in the calculations of singlet and triplet states of different spatial symmetries of the water molecule. The comparisons of the results with values obtained using other many-particle methods and with the full configuration interaction results demonstrate good ability of the approach to deal with electronic excited states.

Numerically oriented static response approach based on state-specific multi-reference coupled electron-pair approximation (SS-MRCEPA) like methods

In this paper, we have exploited the recently developed state-specific multi-reference coupled electron-pair approximation (SS-MRCEPA) like methods for computing the electrostatic response properties. The SS-MRCEPA methods are formulated on complete active space reference functions and the required energies are computed via the diagonalization of an effective operator within this space. The SS-MRCEPA theories are size-consistent and size-extensive. They are very efficient for treating quasidegeneracy of varying extent and for bypassing the intruder problem. The efficacy of the methods is illustrated via the computation of the static dipole moment and polarizability of the ground state and the corresponding first excited state of the trapezoidal H4 model (H4) as well as the lowest two singlet states of the CH2 system using perturbed orbitals generated via the finite-field strategy, a numerically oriented static response approach.

Analytical energy gradients for internally contracted second-order multireference perturbation theory

The theory for computing analytical energy gradients for second-order multireference perturbation theory (MRPT2) with arbitrary multiconfiguration self-consistent field (MCSCF) reference functions is derived and implemented. MRPT2 with complete active space reference functions (CASPT2) is a special case. In our method the configurations with two electrons in the external orbital space are internally contracted. This ansatz strongly reduces the length of the configuration expansion as compared to uncontracted wave functions, but avoids bottlenecks occurring when fully contracted first-order wave functions are used. The method, which also allows the use of state-averaged MCSCF reference functions, is applied to optimize the structures of low-lying valence and Rydberg states of Pyrrole. Nonplanar equilibrium structures and large geometry relaxation effects on the excitation energies are found for the valence states. From the results it can be concluded that CASPT2 underestimates the excitation energies of the valence states, but leads to accurate results for the Rydberg states.

Basis set limit extrapolation of ACPF and CCSD(T) results for the third and fourth lanthanide ionization potentials

Theoretical values for the third and fourth ionization potentials of the lanthanide elements from La to Lu are presented. The results are based on large-scale multi-reference averaged coupled-pair functional calculations starting from a complete active space reference wavefunction and single-reference coupled-cluster calculations. Relativistic effects were accounted for by small-core energy-consistent ab initio pseudopotentials. The results obtained for optimized uncontracted valence basis sets including up to i functions were extrapolated to the basis set limit. The importance of higher angular momentum basis functions in ab initio calculations for energy differences between configurations with different 4f occupation numbers is discussed.

New scheme for solving the amplitude equations in the state-specific coupled cluster theory with complete active space reference for ground and excited states

A new scheme for solving the coupled cluster (CC) amplitude equations for ground and excited electronic states in the state-selective multireference (MR) CC method based on the complete active space (CAS) reference wave function (CASCC) is proposed. The CASCC wave function is generated using a single formal reference determinant, which is one of the CAS determinants, as the origin of the configuration expansion. Some single and double excitations from other CAS determinants may be triple, quadruple, and higher excitations from the reference determinant. We show that one may include the contribution from these higher excitations indirectly by modifying the coupled cluster amplitude equations corresponding to the single and double excitations. The modification involves including projections against the higher excitations in the equations for the singles and doubles. Test calculations for the ground and the first excited state of the H8-model system and for the singlet–triplet splitting of the CH2-biradical shows that the results improve with the use of the new scheme and become to the FCI benchmarks.

CASCCD: Coupled-cluster method with double excitations and the CAS reference

A new multireference coupled-cluster method which includes double excitations and is based on the complete active space (CAS) multiconfigurational reference wave function is proposed. By partitioning the CAS orbitals into active and nonactive sets a two-component, coupled-cluster wave function involving excitations into orbitals of the different sets was constructed. The first component includes all the CAS excitations and the second component, which has the exponential form, consists of double external and semi-external excitations. The coupled-cluster equations for the energy and for the amplitudes involved in the two components of the wave function were derived and illustrated using a diagrammatic formalism. Several numerical tests were performed, and the results demonstrate a very good performance of the method as compared to the full configuration interaction results.