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We present a local framework for the calculation of coupled cluster excitation energies of large molecules (LoFEx). The method utilizes time-dependent Hartree-Fock information about the transitions of interest through the concept of natural transition orbitals (NTOs). The NTOs are used in combination with localized occupied and virtual Hartree-Fock orbitals to generate a reduced excitation orbital space (XOS) specific to each transition where a standard coupled cluster calculation is carried out. Each XOS is optimized to ensure that the excitation energies are determined to a predefined precision. We apply LoFEx in combination with the RI-CC2 model to calculate the lowest excitation energies of a set of medium-sized organic molecules. The results demonstrate the black-box nature of the LoFEx approach and show that significant computational savings can be gained without affecting the accuracy of CC2 excitation energies. [ABSTRACT FROM AUTHOR]

The theoretical foundation for solving coupled cluster singles and doubles (CCSD) amplitude equations to a desired precision in terms of independent fragment calculations using restricted local orbital spaces is reinvestigated with focus on the individual error sources. Four different error sources are identified theoretically and numerically and it is demonstrated that, for practical purposes, local orbital spaces for CCSD calculations can be identified from calculations at the MP2 level. The development establishes a solid theoretical foundation for local CCSD calculations for the independent fragments, and thus for divide-expand-consolidate coupled cluster calculations for large molecular systems with rigorous error control. Based on this theoretical foundation, we have developed an algorithm for determining the orbital spaces needed for obtaining the single fragment energies to a requested precision and numerically demonstrated the robustness and precision of this algorithm. [ABSTRACT FROM AUTHOR]

It is demonstrated that a set of local orthonormal Hartree-Fock (HF) molecular orbitals can be obtained for both the occupied and virtual orbital spaces by minimizing powers of the orbital variance using the trust-region algorithm. For a power exponent equal to one, the Boys localization function is obtained. For increasing power exponents, the penalty for delocalized orbitals is increased and smaller maximum orbital spreads are encountered. Calculations on superbenzene, C60, and a fragment of the titin protein show that for a power exponent equal to one, delocalized outlier orbitals may be encountered. These disappear when the exponent is larger than one. For a small penalty, the occupied orbitals are more local than the virtual ones. When the penalty is increased, the locality of the occupied and virtual orbitals becomes similar. In fact, when increasing the cardinal number for Dunning's correlation consistent basis sets, it is seen that for larger penalties, the virtual orbitals become more local than the occupied ones. We also show that the local virtual HF orbitals are significantly more local than the redundant projected atomic orbitals, which often have been used to span the virtual orbital space in local correlated wave function calculations. Our local molecular orbitals thus appear to be a good candidate for local correlation methods. [ABSTRACT FROM AUTHOR]

The recently developed Local Framework for calculating Excitation energies (LoFEx) is extended to the coupled cluster singles and doubles (CCSD) model. In the new scheme, a standard CCSD excitation energy calculation is carried out within a reduced excitation orbital space (XOS), which is composed of localised molecular orbitals and natural transition orbitals determined from time-dependent Hartree–Fock theory. The presented algorithm uses a series of reduced second-order approximate coupled cluster singles and doubles (CC2) calculations to optimise the XOS in a black-box manner. This ensures that the requested CCSD excitation energies have been determined to a predefined accuracy compared to a conventional CCSD calculation. We present numerical LoFEx-CCSD results for a set of medium-sized organic molecules, which illustrate the black-box nature of the approach and the computational savings obtained for transitions that are local compared to the size of the molecule. In fact, for such local transitions, the LoFEx-CCSD scheme can be applied to molecular systems where a conventional CCSD implementation is intractable. [ABSTRACT FROM AUTHOR]

A common perception about molecular systems with a nonlocal electronic structure (as manifested by a nonlocal Hartree-Fock (HF) density matrix), such as conjugated π-systems, is that they can only be described in terms of nonlocal molecular orbitals. This view is mostly founded on chemical intuition, and further, this view is strengthened by traditional approaches for obtaining local occupied and virtual orbital spaces, such as the occupied Pipek-Mezey orbitals, and projected atomic orbitals. In this article, we discuss the limitations for localizability of HF orbitals in terms of restrictions posed by the delocalized character of the underlying density matrix for the molecular system and by the orthogonality constraint on the molecular orbitals. We show that the locality of the orbitals, in terms of nonvanishing charge distributions of orbitals centered far apart, is much more strongly affected by the orthogonality constraint than by the physical requirement that the occupied orbitals must represent the electron density. Thus, the freedom of carrying out unitary transformations among the orbitals provides the flexibility to obtain highly local occupied and virtual molecular orbitals, even for molecular systems with a nonlocal density matrix, provided that a proper localization function is used. As an additional consideration, we clear up the common misconception that projected atomic orbitals in general are more local than localized orthogonal virtual orbitals. [ABSTRACT FROM AUTHOR]

Using the three-level energy optimization procedure combined with a refined version of the least-change strategy for the orbitals-where an explicit localization is performed at the valence basis level-it is shown how to more efficiently determine a set of local Hartree-Fock orbitals. Further, a core-valence separation of the least-change occupied orbital space is introduced. Numerical results comparing valence basis localized orbitals and canonical molecular orbitals as starting guesses for the full basis localization are presented. The results show that the localization of the occupied orbitals may be performed at a small computational cost if valence basis localized orbitals are used as a starting guess. For the unoccupied space, about half the number of iterations are required if valence localized orbitals are used as a starting guess compared to a canonical set of unoccupied Hartree-Fock orbitals. Different local minima may be obtained when different starting guesses are used. However, the different minima all correspond to orbitals with approximately the same locality. © 2013 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

We demonstrate that the divide-expand-consolidate (DEC) scheme - which has previously been used to determine the second-order Mo\ller-Plesset (MP2) correlation energy - can be applied to evaluate the MP2 molecular gradient in a linear-scaling and embarrassingly parallel manner using a set of local Hartree-Fock orbitals. All manipulations of four-index quantities (describing electron correlation effects) are carried out using small local orbital fragment spaces, whereas two-index quantities are treated for the full molecular system. The sizes of the orbital fragment spaces are determined in a black-box manner to ensure that the error in the DEC-MP2 correlation energy compared to a standard MP2 calculation is proportional to a single input threshold denoted the fragment optimization threshold (FOT). The FOT also implicitly controls the error in the DEC-MP2 molecular gradient as substantiated by a theoretical analysis and numerical results. The development of the DEC-MP2 molecular gradient is the initial step towards calculating higher order energy derivatives for large molecular systems using the DEC framework, both at the MP2 level of theory and for more accurate coupled-cluster methods. [ABSTRACT FROM AUTHOR]