Localized Active Orbitals Research Articles

Assigning the Cerium Oxidation State for CH2CeF2 and OCeF2 Based on Multireference Wave Function Analysis.

The geometric and electronic structure of the recently experimentally studied molecules ZCeF2 (Z = CH2, O) was investigated by density functional theory (DFT) and wave function-based ab initio methods. Special attention was paid to the Ce-Z metal-ligand bonding, especially to the nature of the interaction between the Ce 4f and the Z 2p orbitals and the possible multiconfigurational character arising from it, as well as to the assignment of an oxidation state of Ce reflecting the electronic structure. Complete active space self-consistent field (CASSCF) calculations were performed, followed by orbital rotations in the active orbital space. The methylene compound CH2CeF2 has an open-shell singlet ground state, which is characterized by a two-configurational wave function in the basis of the strongly mixed natural CASSCF orbitals. The system can also be described in a very compact way by the dominant Ce 4f(1) C 2p(1) configuration, if nearly pure Ce 4f and C 2p orbitals are used. In the basis of these localized orbitals, the molecule is almost monoconfigurational and should be best described as a Ce(III) system. The singlet ground state of the oxygen OCeF2 complex is of closed-shell character when a monoconfigurational wave function with very strongly mixed Ce 4f and O 2p CASSCF natural orbitals is used for the description. The transformation to orbitals localized on the cerium and oxygen atoms leads to a multiconfigurational wave function and reveals characteristics of a mixed valent Ce(IV)/Ce(III) compound. Additionally, the interactions of the localized active orbitals were analyzed by evaluating the expectation values of the charge fluctuation operator and the local spin operator. The Ce 4f and C 2p orbital interaction of the CH2CeF2 compound is weakly covalent and resembles the interaction of the H 1s orbitals in a stretched hydrogen dimer. In contrast, the interaction of the localized active orbitals for OCeF2 shows ionic character. Calculated vibrational Ce-C and Ce-O stretching frequencies at the DFT, CASSCF, second-order Rayleigh-Schrödinger perturbation theory (RS2C), multireference configuration interaction (MRCI), as well as single, doubles, and perturbative triples coupled cluster (CCSD(T)) level are reported and compared to experimental infrared absorption data in a Ne and Ar matrix.

Aspects of Size-Consistency of Orbitally Noninvariant Size-Extensive Multireference Perturbation Theories: A Case Study Using UGA-SSMRPT2 as a Prototype.

Profiling a potential energy surface (PES), all the way to dissociate a molecular state into particular fragments and to display real or avoided crossings, requires a multireference description and the maintenance of size-consistency. The many body methods, which suit this purpose, should thus be size-extensive. Size-extensive theories, which are invariant with respect to transformation among active orbitals are, in principle, size-consistent. Relatively cheaper size-extensive theories, which do not possess this invariance, can still be size-consistent if the active orbitals are localized on the asymptotic fragments. Such methods, if perturbative in nature, require the use of an unperturbed Hamiltonian, which has orbital invariance with respect to the transformation within active, core, and virtual orbitals. The principal focus of this paper is to numerically realize size-consistency with localized active orbitals using our recently developed orbitally noninvariant Unitary Group Adapted State Specific Multireference second order Perturbation Theory (UGA-SSMRPT2) as a prototype method. Our findings expose certain generic potential pitfalls of size-extensive but orbitally noninvariant MRPT theories, which are mostly related to the inability of reaching proper localized active orbitals in the fragments due to the artifacts of the orbital generation procedure. Despite the invariance of the zeroth order CAS function, lack of invariance of the MRPT itself then leads to size-inconsistency. In particular, reaching symmetry broken fragment active orbitals is an issue of concern where suitable state-averaging might ameliorate the problem, but then one has to abandon full orbital optimization. Additionally, there can be situations where the orbitals of the fragment reached as an asymptote of the supermolecule are not the same as those obtained from the optimization of the fragments individually and will require additional transformation. Moreover, for a certain PES, one may either abandon the use of optimized orbitals for that state to preserve proper symmetry and degeneracy in the fragment orbitals or be satisfied with the use of optimized orbitals, which generate broken symmetric orbitals in the fragmentation limit. All these pathologies are illustrated using the PES of various electronic states of multiply bonded systems like N2, C2H2, HCN, C2, and O2. Subject to such proviso, the UGA-SSMRPT2 turns out to be an excellent theory for studying the PES leading to fragmentation of strongly correlated systems satisfying the requirements of size-consistency with localized active orbitals. An unexpected spin-off of our studies is the realization that the size-inextensive MRMP2, which bears a close structural similarity with our theory, might under certain situations display size-consistency. We analyze this feature concretely in our paper. Our studies may serve as a benchmark for monitoring numerically the size-consistency of any state specific multireference theory which is size-extensive but not invariant with respect to transformation of active orbitals.

Unitary group adapted state specific multireference perturbation theory: Formulation and pilot applications.

We present here a comprehensive account of the formulation and pilot applications of the second-order perturbative analogue of the recently proposed unitary group adapted state-specific multireference coupled cluster theory (UGA-SSMRCC), which we call as the UGA-SSMRPT2. We also discuss the essential similarities and differences between the UGA-SSMRPT2 and the allied SA-SSMRPT2. Our theory, like its parent UGA-SSMRCC formalism, is size-extensive. However, because of the noninvariance of the theory with respect to the transformation among the active orbitals, it requires the use of localized orbitals to ensure size-consistency. We have demonstrated the performance of the formalism with a set of pilot applications, exploring (a) the accuracy of the potential energy surface (PES) of a set of small prototypical difficult molecules in their various low-lying states, using natural, pseudocanonical and localized orbitals and compared the respective nonparallelity errors (NPE) and the mean average deviations (MAD) vis-a-vis the full CI results with the same basis; (b) the efficacy of localized active orbitals to ensure and demonstrate manifest size-consistency with respect to fragmentation. We found that natural orbitals lead to the best overall PES, as evidenced by the NPE and MAD values. The MRMP2 results for individual states and of the MCQDPT2 for multiple states displaying avoided curve crossings are uniformly poorer as compared with the UGA-SSMRPT2 results. The striking aspect of the size-consistency check is the complete insensitivity of the sum of fragment energies with given fragment spin-multiplicities, which are obtained as the asymptotic limit of super-molecules with different coupled spins.

Superior performance of Mukherjee’s state-specific multi-reference coupled-cluster theory at the singles and doubles truncation scheme with localized active orbitals

The state-specific multi-reference coupled-cluster (SS-MRCC) theory of Mukherjee et al., in its singles and doubles truncation scheme (SS-MRCCSD), misses important couplings between the virtual functions reached by single and double excitations from different model functions. Since the SS-MRCC theory is not invariant with respect to the transformations among the active orbitals, the results are dependent on the active orbitals chosen. We demonstrate in this paper with results for potential energy curves for several example molecules involving single and multiple bond dissociation that the performance of SS-MRCCSD is significantly improved if localized active orbitals are used. The improvement is remarkable both in terms of the non-parallelity error and the magnitude of correlation energy recovered vis-a-vis the full configuration interaction results with the same basis set. The results bolster our claim that SS-MRCCSD with localized orbitals is an accurate general theory for potential energy surfaces.

Quantum Monte Carlo with Jastrow-valence-bond wave functions

We consider the use in quantum Monte Carlo calculations of two types of valence bond wave functions based on strictly localized active orbitals, namely valence bond self-consistent-field and breathing-orbital valence bond wave functions. Complemented by a Jastrow factor, these Jastrow-valence-bond wave functions are tested by computing the equilibrium well depths of the four diatomic molecules C(2), N(2), O(2), and F(2) in both variational Monte Carlo and diffusion Monte Carlo. We show that it is possible to design compact wave functions based on chemical grounds that are capable of describing both static and dynamic electron correlations. These wave functions can be systematically improved by inclusion of valence bond structures corresponding to additional bonding patterns.

Full implementation and benchmark studies of Mukherjee’s state-specific multireference coupled-cluster ansatz

The state-specific multireference coupled-cluster (SS-MRCC) ansatz developed by Mukherjee and co-workers [J. Chem. Phys. 110, 6171 (1999)] has been implemented by means of string-based techniques. The implementation is general and allows for using arbitrary complete active spaces of any spin multiplicity and arbitrarily high excitations in the cluster operators. Several test calculations have been performed for single- and multiple-bond dissociations of molecular systems. Our experience shows that convergence problems are encountered when solving the working equations of the SS-MRCC in the case the weight of one or more reference functions tends to take on very small values. This is system specific and cannot yet be handled in a black-box fashion. The problem can be obviated by either dropping all the cluster amplitudes from the corresponding model functions with coefficients below a threshold or by a regularization procedure suggested by Tikhonov or a combination of both. In the current formulation the SS-MRCC is not invariant with respect to transformation of active orbitals among themselves. This feature has been extensively explored to test the degree of accuracy of the computed energies with both pseudocanonical and localized active orbitals. The performance of the method is assessed by comparing the results with the corresponding full configuration interaction (CI) values with the same set of orbitals (correlated and frozen). Relative efficacies of CI methods such as MRCI singles and doubles with the same active space and size-extensivity corrected ones such as MR averaged coupled pair functional and MR averaged quadratic CC have also been studied. Allied full-fledged CC methods have also been employed to see their relative performance vis-à-vis the SS-MRCC. These latter methods are the complete-active-space-inspired single-reference (SR) CC based SS theory and the single-root MR Brillouin-Wigner CC. Our benchmark results indicate that the performance of the SS-MRCC is generally quite good for localized active orbitals. The performance with the pseudocanonical orbitals, however, is sometimes not as satisfactory as for the localized orbitals.

Electronic Structure of Selected {FeNO}7 Complexes in Heme and Non-Heme Architectures: A Density Functional and Multireference ab Initio Study

The multiconfigurational CASSCF/CASPT2 approach, along with various functionals of density functional theory, is applied to selected iron(II)-nitrosyl ({FeNO}(7)) complexes, both with heme and nonheme groups. The energetics of the lowest doublet and quartet spin states at the correlated ab initio (CASPT2) level is presented for the first time. Comparison of the CASSCF and (unrestricted) DFT spin densities indicates that the nonhybrid functionals yield the spin densities most closely to the ab initio ones. The analysis of the multiconfigurational CASSCF wave function in terms of the localized active orbitals allows one to resolve the nature of Fe-NO bonding as a mixture of Fe(II)-NO(0) and Fe(III)-NO(-) resonance structures (in comparable contributions) for both spin states and various ligands.

Local orbitals for quasi-degenerate systems

A CAS-SCF algorithm based on molecular orbitals that conserve their physical character during the iterative process has been developed. The method is based on the iterative partial diagonalization of the one-body density matrix, obtained from a Configuration Interaction restricted to the space of single excitations from the CAS. When localized guess orbitals are used, the locality property is conserved for the final orbitals. This localization technique is particularly suitable for the treatment of quasi-degenerated systems, since it can be applied to those cases that cannot be correctly described at SCF level. For large systems, the use of localized active orbitals leads to a huge reduction of the computational effort, and permits MR-CI treatments that would be out of the possibilities of the standard delocalized approaches.

Correlated description of multiple bonds using localized active orbitals

It is shown that the use of atomic-like active orbitals instead of symmetry-adapted ones is computationally interesting for the treatment of strongly correlated problems, when the zero-order description involves a large complete active space (CAS). Using localized active orbitals makes it possible to reduce the number of references to the leading valence bond configurations, and hence to achieve the calculation of the dynamical correlation energy for a lower cost. This benefit would be especially dramatic for remote active centers but it remains significant in the most difficult problem of strong multiple bonds, as illustrated on N 2 and Mo 2Cl 8 4−. The valence bond analysis of the wavefunction makes clear the physical content of the non-dynamical correlation.

An intrinsic localization procedure for active CAS SCF orbitals

An intrinsic localization criterion for the active (valence) orbitals of a CAS SCF wavefunction is presented. The localization criterion is based on minimization of the energy of a “perfect pairing” configuration. Equations for carrying out the localization in terms of an exponential transformation are developed. The technique can easily be incorporated into any MC SCF program. The CAS SCF wavefunction obtained using these localized active orbitals corresponds to a full VB calculation where the VB structures are built from orthogonal “molecule adapted minimal basis set atomic orbitals” and thus offers an interpretational advantage over the use of canonical CAS SCF orbitals. The method is applied to the 1,3-dipole, nitrone.