Reference Wave Function Research Articles

Metal-to-metal charge-transfer transitions: reliable excitation energies from ab initio calculations

A computational strategy is presented to describe excited states, involving the transfer of an electron from one metallic site to a neighboring metal center, the so-called metal-to-metal charge-transfer (MMCT) states. An accurate ab initio treatment of these states in transition metal compounds is intrinsically difficult for both time-dependent density functional and wave function-based methods. The rather large dependence of the MMCT energies on the applied functional makes difficult to extract reliable estimates from density functional theory, while the standard multiconfigurational approach (complete active space SCF + second-order perturbation theory) leads to severe intruder state problems and unrealistic, negative energies. The analysis of the failure of the multiconfigurational approach shows that the state-average orbitals are biased toward the ground state and strongly deficient to describe the MMCT state. We propose a method to improve the orbitals by gradually approaching as much as possible the state-specific description of the MMCT state in the reference wave function for the second-order perturbation treatment of the dynamic electron correlation.

Orbitally invariant internally contracted multireference unitary coupled cluster theory and its perturbative approximation: Theory and test calculations of second order approximation

A unitary wave operator, exp (G), G(+) = -G, is considered to transform a multiconfigurational reference wave function Φ to the potentially exact, within basis set limit, wave function Ψ = exp (G)Φ. To obtain a useful approximation, the Hausdorff expansion of the similarity transformed effective Hamiltonian, exp (-G)Hexp (G), is truncated at second order and the excitation manifold is limited; an additional separate perturbation approximation can also be made. In the perturbation approximation, which we refer to as multireference unitary second-order perturbation theory (MRUPT2), the Hamiltonian operator in the highest order commutator is approximated by a Mo̸ller-Plesset-type one-body zero-order Hamiltonian. If a complete active space self-consistent field wave function is used as reference, then the energy is invariant under orbital rotations within the inactive, active, and virtual orbital subspaces for both the second-order unitary coupled cluster method and its perturbative approximation. Furthermore, the redundancies of the excitation operators are addressed in a novel way, which is potentially more efficient compared to the usual full diagonalization of the metric of the excited configurations. Despite the loss of rigorous size-extensivity possibly due to the use of a variational approach rather than a projective one in the solution of the amplitudes, test calculations show that the size-extensivity errors are very small. Compared to other internally contracted multireference perturbation theories, MRUPT2 only needs reduced density matrices up to three-body even with a non-complete active space reference wave function when two-body excitations within the active orbital subspace are involved in the wave operator, exp (G). Both the coupled cluster and perturbation theory variants are amenable to large, incomplete model spaces. Applications to some widely studied model systems that can be problematic because of geometry dependent quasidegeneracy, H4, P4, and BeH(2), are performed in order to test the new methods on problems where full configuration interaction results are available.

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.

Investigation of electron–hole correlation using explicitly correlated configuration interaction method

The explicitly correlated configuration interaction (XCCI) method is presented as a general technique for accurate description of electron–hole interaction. The XCCI method is a variational method where an R12-explicitly correlated wavefunction that depends on the electron–hole separation is used as the zeroth order reference wavefunction for performing the CI expansion. This is the main feature that differentiates the XCCI from the conventional CI method. Benchmark calculations on parabolic quantum dot using XCCI, FCI, and R12-FCI methods are presented and the results demonstrate that XCCI wavefunction is variationally superior to the FCI wavefunction for the ground and the excited states.

Thermodynamics of Titanium and Vanadium Reduction in Non-Aqueous Environment Calculated at Various Levels of Theory

Reduction of titanium and vanadium compounds is a process accompanying the activation of coordinative olefin polymerization catalysts. Four density functional theory (DFT) functionals, coupled cluster with single, double, and perturbative triple excitations method CCSD(T) as well as complete active-space second-order perturbation theory method CASPT2 with a complete active-space self-consistent field CASSCF reference wave function were applied to investigate the thermodynamics of titanium and vanadium reduction. The performance of these theoretical methods was assessed and compared with experimental values. The calculations indicate that vanadium(IV) chloride is more easily reduced by trimethylaluminum than the corresponding titanium compound; the energies of reaction calculated at the CCSD(T) level are equal -57.21 and -33.10 kcal/mol, respectively. The calculations deal with the redox reactions of metal chlorides in the gas phase, rather than solvated ions in the aqueous solution. This approach may be more appropriate for olefin polymerization, usually carried out in nonpolar solvents.

Strongly correlated barriers to rotation from parametric two-electron reduced-density-matrix methods in application to the isomerization of diazene

Recently, parameterization of the two-electron reduced density matrix (2-RDM) has made possible the determination of electronic energies with greater accuracy and lower cost than traditional electron-pair theories including coupled cluster with single and double excitations [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)]. We examine the method's performance for strongly correlated barriers to rotation; in particular, we study two distinct pathways in the isomerization of diazene (N(2)H(2)) from cis to trans: (i) a strongly correlated rotational pathway and (ii) a moderately correlated inversion pathway. While single reference wavefunction methods predict that the rotational barrier is higher than the inversional barrier, the parametric 2-RDM method predicts that the rotational barrier is lower than the inversional barrier by 3.1 kcal/mol in the extrapolated basis set limit. The parametric 2-RDM results are in agreement with those from multireference methods including multireference perturbation theory and the solution to the anti-Hermitian contracted Schrödinger equation. We report energies, optimized structures, and natural orbital occupation numbers for three diazene minima and two transition states.

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

On the Importance of the Orbital Relaxation in Ground-State Coupled Cluster Calculations in Solution with the Polarizable Continuum Model of Solvation.

In the description of the electrostatic interaction between a solute treated at coupled cluster (CC) level of theory and a solvent modeled as a continuum dielectric, the solvent response depends on various contributions: the choice of the reference wave function, the correlation density, and the orbital relaxation. In previous work, we examined the first two factors with the coupled cluster singles and doubles (CCSD) method and its variant Brueckner doubles (BD) method. The CC wave function was combined with the polarizable continuum model (PCM) of solvation in an integrated and efficient method able to describe energy and molecular properties through analytic energy gradients. Additionally, we investigated some approximations, and proposed new ones, that reduce the computational cost to nearly that of gas phase CC while keeping most of the complete model description. In this work, we study the contribution of the orbital relaxation and compare it to the other effects. Such contribution is introduced with a self-consistent macroiteration procedure, where the reaction field is updated with a refined density. The results presented here show that the effect of the orbital relaxation is small for CCSD, while for BD the integrated and self-consistent approaches are equivalent. Thus, these results further confirm that the integrated CCSD-PCM and BD-PCM methods, especially with their respective approximations, are an efficient approach to perform high-level electronic structure calculations in solution.

The optimized orbital coupled cluster doubles method and optical rotation

The impact of variational optimization of the molecular orbitals used in coupled cluster response theory is compared to the use of conventional Hartree–Fock orbitals for the computation of optical rotations in a number of chiral compounds. For molecular structures near equilibrium, where the coupled cluster method is relatively insensitive to the choice of reference wave function, the two approaches yield similar results for test cases such as (S)-methyloxirane, (S)-methylthiirane, (S)-2-chloropropionitrile, and (1S,4S)-norbornenone. However, small distortions away from equilibrium affect the optimized-orbital coupled cluster approach to a much greater extent than its Hartree–Fock counterpart. This effect is tested for methyloxirane in which the C–O bond associated with the stereogenic carbon is stretched and for a pyramidalized fluoroformaldehyde where the corresponding C=O bond is similarly extended. The potential repercussions of this strong geometry dependence on the magnitude of vibrational corrections are discussed.

Multiconfigurational Second-Order Perturbation Theory Restricted Active Space (RASPT2) Studies on Mononuclear First-Row Transition-Metal Systems.

A series of model transition-metal complexes, CrF6, ferrocene, Cr(CO)6, ferrous porphin, cobalt corrole, and FeO/FeO(-), have been studied using second-order perturbation theory based on a restricted active space self-consistent field reference wave function (RASPT2). Several important properties (structures, relative energies of different structural minima, binding energies, spin state energetics, and electronic excitation energies) were investigated. A systematic investigation was performed on the effect of: (a) the size and composition of the global RAS space, (b) different (RAS1/RAS2/RAS3) subpartitions of the global RAS space, and (c) different excitation levels (out of RAS1/into RAS3) within the RAS space. Calculations with active spaces, including up to 35 orbitals, are presented. The results obtained with smaller active spaces (up to 16 orbitals) were compared to previous and current results obtained with a complete active space self-consistent field reference wave function (CASPT2). Higly accurate RASPT2 results were obtained for the heterolytic binding energy of ferrocene and for the electronic spectrum of Cr(CO)6, with errors within chemical accuracy. For ferrous porphyrin the intermediate spin (3)A2g ground state is (for the first time with a wave function-based method) correctly predicted, while its high magnetic moment (4.4 μB) is attributed to spin-orbit coupling with very close-lying (5)A1g and (3)Eg states. The toughest case met in this work is cobalt corrole, for which we studied the relative energy of several low-lying Co(II)-corrole π radical states with respect to the Co(III) ground state. Very large RAS spaces (25-33 orbitals) are required for this system, making compromises on the size of RAS2 and/or the excitation level unavoidable, thus increasing the uncertainty of the RASPT2 results by 0.1-0.2 eV. Still, also for this system, the RASPT2 method is shown to provide distinct improvements over CASPT2, by overcoming the strict limitations in the size of the active space inherent to the latter method.

Meaning and magnitude of the reduced density matrix cumulants

Within the framework of a generalized normal ordering (GNO), invented by Mukherjee [1], the reduced density matrix cumulants of the (multiconfigurational) reference wave function play a central role, as they arise directly from the contraction rules. The extended Wick theorem allows contractions of an arbitrary number of active annihilators and creators through a cumulant of corresponding rank. Because the cumulant rank truncates naturally only at the number of active spin orbitals, practical applications of the GNO concept seem to rely on a fast convergence of the cumulant series, allowing one to neglect cumulants with high rank. By computing cumulant norms for selected systems (and up to rank 16), we demonstrate that the cumulants decay approximately exponentially with increasing rank for single reference cases, while the convergence is generally slower for multireference cases. When strong left–right correlation is present as in the singlet state of a dissociated N2 molecule, even the cumulant with maximum rank, λ12≈−1.5 for a CAS(6,6), is not negligible per se. Besides reporting numerical results, the authors reformulate the theory of reduced density matrices and their cumulants using a notation that is particularly easy to handle, highlighting the close connection to conventional statistics. From this statistical approach, a simple interpretation of reduced density matrices and cumulants follows, according to which an n-body cumulant is a measure for the correlation between the occupation numbers of n spin orbitals. This interpretation is also valid for cumulants with ranks exceeding the number of electrons in the system.

Approximate variational coupled cluster theory

We show that it is possible to construct an accurate approximation to the variational coupled cluster method, limited to double substitutions, from the minimization of a functional that is rigorously extensive, exact for isolated two-electron subsystems and invariant to transformations of the underlying orbital basis. This approximate variational coupled cluster theory is a modification and enhancement of our earlier linked pair functional theory. It is first motivated by the constraint that the inverse square root of the matrix that transforms the cluster amplitudes must exist. Low-order corrections are then included to enhance the accuracy of the approximation of variational coupled cluster, while ensuring that the computational complexity of the method never exceeds that of the standard traditional coupled cluster method. The effects of single excitations are included by energy minimization with respect to the orbitals defining the reference wavefunction. The resulting quantum chemical method is demonstrated to be a robust approach to the calculation of molecular electronic structure and performs well when static correlation effects are strong.

Quasi-degenerate second-order perturbation theory for occupation restricted multiple active space self-consistent field reference functions

A multi-configuration quasi-degenerate second-order perturbation method based on the occupation restricted multiple active space (ORMAS-PT/ORMAS) reference wavefunction is presented. ORMAS gives one the ability to approximate a complete active space self-consistent field (CASSCF) wavefunction using only a subset of the configurations from the CASSCF space. The essential idea behind ORMAS-PT is to use the multi-reference Møller-Plesset formalism to correct the ORMAS reference energy. A computational scheme employing direct CI methodology is presented. Several tests are presented to demonstrate the performance of the ORMAS-PT method.

Brueckner doubles coupled cluster method with the polarizable continuum model of solvation

We present the theory and implementation for computing the (free) energy and its analytical gradients with the Brueckner doubles (BD) coupled cluster method in solution, in combination with the polarizable continuum model of solvation (PCM). The complete model, called PTED, and an efficient approximation, called PTE, are introduced and tested with numerical examples. Implementation details are also discussed. A comparison with the coupled-cluster singles and doubles CCSD-PCM-PTED and CCSD-PCM-PTE schemes, which use Hartree-Fock (HF) orbitals, is presented. The results show that the two PTED approaches are mostly equivalent, while BD-PCM-PTE is shown to be superior to the corresponding CCSD scheme when the HF reference wave function is unstable. The BD-PCM-PTE scheme, whose computational cost is equivalent to gas phase BD, is therefore a promising approach to study molecular systems with complicated electronic structure in solution.

Correlated wavefunction methods in bioinorganic chemistry

In this commentary the challenges faced in the application of wavefunction-based ab initio methods to (open-shell) transition metal complexes of (bio)inorganic interest are briefly touched on. Both single-reference and multireference methods are covered. It is stressed that the generation and nature of the reference wavefunction is a subject of major importance. How erroneous results can be easily obtained even with coupled-cluster theory is illustrated through the example of the septet-quintet separation in iron(IV)-oxo complexes. Second, the interplay between relativistic and correlation effects is important. This is demonstrated with coupled-cluster calculations on models for dinuclear copper active sites, where relativity has a major influence on the relative stabilities of the bis(μ-oxo) and side-on peroxo species.

Ab initio multireference singles and doubles configuration interaction study of the low‐energy states of iron mononitride

AbstractLow‐lying states of iron mononitride are studied using ab initio multireference singles and doubles configuration interaction (MR‐SDCI) calculations. For each one of the 2Δ, 4Π, 6Σ+, 2Π, 4Δ, 6Π, and 6Δ states the reference wavefunction has been obtained at the complete active space self consistent field (CASSCF) level and relativistic corrections were included. Potential energy curves are presented for all states. Spectroscopic constants have been determined for the 2Δ, 4Π, and 6Σ+ states assuming a Morse‐type potential function using a code written by one of us. For the 2Δ state, the calculated spectroscopic constants are Re = 1.567 Å, ωe = 827 cm−1, and μ = 1.68 D. Rotovibrational analysis suggests that the third vibrational level of the 2Δ state already surpasses the first vibrational levels of the 4Π and 6Σ+ states, what reassures the difficulty in correctly attributing the lines of the experimental spectrum. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

Determination of molecular vibrational state energies using the ab initio semiclassical initial value representation: Application to formaldehyde

We have demonstrated the use of ab initio molecular dynamics (AIMD) trajectories to compute the vibrational energy levels of molecular systems in the context of the semiclassical initial value representation (SC-IVR). A relatively low level of electronic structure theory (HF/3-21G) was used in this proof-of-principle study. Formaldehyde was used as a test case for the determination of accurate excited vibrational states. The AIMD-SC-IVR vibrational energies have been compared to those from curvilinear and rectilinear vibrational self-consistent field/vibrational configuration interaction with perturbation selected interactions-second-order perturbation theory (VSCF/VCIPSI-PT2) and correlation-corrected vibrational self-consistent field (cc-VSCF) methods. The survival amplitudes were obtained from selecting different reference wavefunctions using only a single set of molecular dynamics trajectories. We conclude that our approach is a further step in making the SC-IVR method a practical tool for first-principles quantum dynamics simulations.

Strong correlation treated via effective hamiltonians and perturbation theory

We propose a new approach to determine a suitable zeroth-order wavefunction for multiconfigurational perturbation theory. The same ansatz as in complete active space (CAS) wavefunction optimization is used but it is split in two parts, a principal space (A) and a much larger extended space (B). Löwdin's partitioning technique is employed to map the initial eigenvalue problem to a dimensionality equal to that of (A) only. Combined with a simplified expression for the (B) portion of the wavefunction, we are able to drastically reduce the storage and computational demands of the wavefunction optimization. This scheme is used to produce reference wavefunctions and energies for subsequent second-order perturbation theory (PT2) corrections. Releasing the constraint of computing the exact CAS energy and wavefunction prior to the PT2 treatment introduces a nonstandard paradigm for multiconfigurational methods. Based on the results of test calculations, we argue that principal parts with only few percents of the total number of CAS configurations could provide final multiconfigurational PT2 energies of the same accuracy as in the standard paradigm. In the future, algorithmic improvements for this scheme will bring into reach active spaces much beyond the present limit of CAS-based methods, therefore allowing for accurate studies of systems featuring strong correlation.

Electronic spectrum of F2CO: theoretical calculations of vertical excitation energies and intensities

In this work, the linear response formalism with a triples-corrected CCSD reference wave function, LR-CCSDR(3), is applied to the calculation of vertical excitation energies of singlet states of the F2CO molecule. A basis set of atomic natural orbitals augmented with a series of Rydberg functions has been used in the calculations. A large number of electronically excited states were calculated, and the valence, Rydberg, or mixed character of the states were investigated. In addition, the molecular quantum defect orbital (MQDO) method has been used to determine transition intensities involving Rydberg states. Excitation energies and transition intensities for Rydberg states with n > 3 are reported for the first time.

The correlation Consistent composite Approach: The spin contamination effect on an MP2-based composite methodology

The correlation consistent composite approach (ccCA) is reformulated utilizing a restricted open-shell reference wavefunction to remove spin-contamination from open-shell systems. The RO-ccCA-PS3 variant has an MAD of 0.65kcalmol−1 for the 32 open-shell systems in the G3/99 enthalpy of formation test set, as compared to 0.77kcalmol−1 using the unrestricted formulation. For the 22 most highly spin-contaminated species in the G3/99 test set, RO-ccCA produces a lower MAD; 1.20 versus 2.17kcalmol−1. Additionally, the performance of RO-ccCA is examined for 30 bond dissociation energies and found to show improved accuracy over unrestricted ccCA.