Ireneusz Grabowski

On the mutual exclusion of variationality and size consistency

Why do variational electron-correlation methods such as truncated configuration-interaction methods tend to be non-size-consistent (non-size-extensive)? Why are size-consistent (size-extensive) methods such as Møller–Plesset perturbation and coupled-cluster methods non-variational? We conjecture that the variational and size-consistent properties are mutually exclusive in an ab initio electron-correlation method (which thus excludes the Hartree–Fock and density-functional methods). We analyze some key examples that support the truth of this conjecture.

Applicability of valence‐universal multireference coupled‐cluster theories to quasidegenerate electronic states. I. Models involving at most two‐body amplitudes

The recently developed algebraic formulation of valence‐universal coupled‐cluster (VU‐CC) theories [Jeziorski and Paldus, J. Chem. Phys. 90, 2714 (1989)] for open‐shell systems has been employed in a systematic derivation of explicit equations defining cluster amplitudes assuming Lindgren’s normal ordered exponential ansatz for the wave operator. The latter is approximated by its one‐ and two‐electron components. Various aspects of the applicability of this version of the VU‐CC theory to quasidegenerate electronic states are studied for a model system consisting of two slightly stretched, interacting hydrogen molecules. A single parameter that determines the geometry of this system makes it possible to vary the extent of quasidegeneracy of the two lowest‐energy states over a wide range. Along with the complete theory, the linear version (VU‐LCC) is also examined. The results are compared with the full configuration interaction results as well as with those obtained using other approaches. It was found tha...

A coupled-cluster correction to the multi-reference configuration interaction method

Abstract A correction to the multi-reference configuration interaction method with singles and doubles (MR-CISD) is proposed. Assuming that a reasonable choice for the reference space permits good evaluation of the nondynamic correlation within the MR-CISD framework, the correction is designed to improve the description of the dynamic correlation effects. It is based on the single-reference cluster analysis of the wavefunction and intends to approximate the contribution from those triple and quadruple excitations which are lacking in the MR-CISD expansion. The correction is applied to several model systems, showing its ability to improve significantly the MR-CISD energies and, because of its simple noniterative character, can be seen as an attractive alternative to other methods designated for ground state energy calculations in the presence of quasi-degeneracy.

Ab initio density functional theory applied to quasidegenerate problems.

Ab initio density functional theory (DFT), previously applied primarily at the second-order many-body perturbation theory (MBPT) level, is generalized to selected infinite-order effects by using a new coupled-cluster perturbation theory (CCPT). This is accomplished by redefining the unperturbed Hamiltonian in ab initio DFT to correspond to the CCPT2 orbital dependent functional. These methods are applied to the Be-isoelectronic systems as an example of a quasidegenerate system. The CCPT2 variant shows better convergence to the exact quantum Monte Carlo correlation potential for Be than any prior attempt. When using MBPT2, the semicanonical choice of unperturbed Hamiltonian, plays a critical role in determining the quality of the obtained correlation potentials and obtaining convergence, while the usual Kohn-Sham choice invariably diverges. However, without the additional infinite-order effects, introduced by CCPT2, the final potentials and energies are not sufficiently accurate. The issue of the effects of the single excitations on the divergence in ordinary OEP2 is addressed, and it is shown that, whereas their individual values are small, their infinite-order summation is essential to the good convergence of ab initio DFT.

Comparing ab initio density-functional and wave function theories: The impact of correlation on the electronic density and the role of the correlation potential

The framework of ab initio density-functional theory (DFT) has been introduced as a way to provide a seamless connection between the Kohn-Sham (KS) formulation of DFT and wave-function based ab initio approaches [R. J. Bartlett, I. Grabowski, S. Hirata, and S. Ivanov, J. Chem. Phys. 122, 034104 (2005)]. Recently, an analysis of the impact of dynamical correlation effects on the density of the neon atom was presented [K. Jankowski, K. Nowakowski, I. Grabowski, and J. Wasilewski, J. Chem. Phys. 130, 164102 (2009)], contrasting the behaviour for a variety of standard density functionals with that of ab initio approaches based on second-order Møller-Plesset (MP2) and coupled cluster theories at the singles-doubles (CCSD) and singles-doubles perturbative triples [CCSD(T)] levels. In the present work, we consider ab initio density functionals based on second-order many-body perturbation theory and coupled cluster perturbation theory in a similar manner, for a range of small atomic and molecular systems. For comparison, we also consider results obtained from MP2, CCSD, and CCSD(T) calculations. In addition to this density based analysis, we determine the KS correlation potentials corresponding to these densities and compare them with those obtained for a range of ab initio density functionals via the optimized effective potential method. The correlation energies, densities, and potentials calculated using ab initio DFT display a similar systematic behaviour to those derived from electronic densities calculated using ab initio wave function theories. In contrast, typical explicit density functionals for the correlation energy, such as VWN5 and LYP, do not show behaviour consistent with this picture of dynamical correlation, although they may provide some degree of correction for already erroneous explicitly density-dependent exchange-only functionals. The results presented here using orbital dependent ab initio density functionals show that they provide a treatment of exchange and correlation contributions within the KS framework that is more consistent with traditional ab initio wave function based methods.

Connections between second-order Görling-Levy and many-body perturbation approaches in density functional theory

Formal connections between the high-density scaling limit of the correlation energy functional Ec[n] in density functional theory and second-order energy expressions from different perturbation theory formulations are presented. It is demonstrated that the second-order correlation potential considered by Grabowski et al. [J. Chem. Phys. 116, 4415 (2002)] is equivalent to the high-density limit of the exact correlation potential, and thus provides the first self-consistent finite-basis-set implementation of a Kohn–Sham (KS) potential correct through second-order. A different second-order correlation functional based on the exchange-only KS approach is introduced. It is shown that this second-order correlation functional leads to the same self-consistent KS realization as the one derived from the second-order component of Ec[n].

Approximate coupled-cluster methods employing split cluster amplitudes: Implementation of an almost-linear coupled-cluster formalism

A new approximation strategy, split-amplitude strategy, useful within the framework of the coupled-cluster (CC) methodology is proposed. It consists in representing the individual cluster amplitudes as a sum of two components, one of fixed value, which may be obtained from external sources, and the other determined from a set of modified CC equations. This approach provides new possibilities of absorbing information concerning the values of cluster amplitudes from independent calculations. By properly choosing the fixed amplitude components, one may substantially reduce the magnitudes of the most significant amplitudes to be determined for the state considered, which in turn causes that the known approximation procedures are more justifiable when applied to the modified CC equations than to the equations of the standard CC approaches. The split-amplitude strategy has been employed to setting up several almost-linear CC (AL-CC) approaches of a single reference type corresponding to the basic CC methods. Th...

Coverage of dynamic correlation effects by density functional theory functionals: Density-based analysis for neon

The problem of linking the dynamic electron correlation effects defined in traditional ab initio methods [or wave function theories (WFTs)] with the structure of the individual density functional theory (DFT) exchange and correlation functionals has been analyzed for the Ne atom, for which nondynamic correlation effects play a negligible role. A density-based approach directly hinged on difference radial-density (DRD) distributions defined with respect the Hartree-Fock radial density has been employed for analyzing the impact of dynamic correlation effects on the density. Attention has been paid to the elimination of basis-set incompleteness errors. The DRD distributions calculated by several ab initio methods have been compared to their DFT counterparts generated for representatives of several generations of broadly used exchange-correlation functionals and for the recently developed orbital-dependent OEP2 exchange-correlation functional [Bartlett et al., J. Chem. Phys. 122, 034104 (2005)]. For the local, generalized-gradient, and hybrid functionals it has been found that the dynamic correlation effects are to a large extend accounted for by densities resulting from exchange-only calculations. Additional calculations with self-interaction corrected exchange potentials indicate that this finding cannot be explained as an artifact caused by the self-interaction error. It has been demonstrated that the VWN5 and LYP correlation functionals do not represent any substantial dynamical correlation effects on the electron density, whereas these effects are well represented by the orbital-dependent OEP2 correlation functional. Critical comparison of the present results with their counterparts reported in literature has been made. Some attention has been paid to demonstrating the differences between the energy- and density-based perspectives. They indicate the usefulness of density-based criteria for developing new exchange-correlation functionals.

Approximate Coupled Cluster Methods: Combined Reduced Multireference and Almost–Linear Coupled Cluster Methods with Singles and Doubles 1

Abstract The recently introduced version of the externally corrected coupled cluster method with singles and doubles (CCSD), exploiting a small active space multireference (MR) configuration interaction with singles and doubles (CISD) wave function as a source of the three and four body connected cluster amplitudes, which is referred to as the reduced MR (RMR) CCSD method, and the so-called almost-linear (AL) CCSD method based on the split–amplitude strategy, are combined to achieve a considerable saving of the computational effort, while yielding almost exact RMR CCSD results. The performance of the suggested approach is exemplified on several models involving four and eight hydrogen atoms in various geometric configurations that enable the quasidegeneracy effects to be varied from the completely degenerate limit to the nondegenerate case, as well as on the symmetrically stretched double zeta model of the water molecule and of the fluorine molecule. In all cases the suggested approach provides an excellent approximation to the full RMR CCSD energies, which in turn faithfully account for the nondynamical correlation effects involved.

A density difference based analysis of orbital-dependent exchange-correlation functionals

We present a density difference based analysis for a range of orbital-dependent Kohn–Sham functionals. Results for atoms, some members of the neon isoelectronic series and small molecules are reported and compared with ab initio wave function calculations. Particular attention is paid to the quality of approximations to the exchange-only optimised effective potential (OEP) approach: we consider both the localised Hartree–Fock as well as the Krieger–Li–Iafrate methods. Analysis of density differences at the exchange-only level reveals the impact of the approximations on the resulting electronic densities. These differences are further quantified in terms of the ground state energies, frontier orbital energy differences and highest occupied orbital energies obtained. At the correlated level, an OEP approach based on a perturbative second-order correlation energy expression is shown to deliver results comparable with those from traditional wave function approaches, making it suitable for use as a benchmark against which to compare standard density functional approximations.