Particle-Breaking Unrestricted Hartree-Fock Theory for Open Molecular Systems.

We recently introduced the particle-breaking restricted Hartree-Fock (PBRHF) model, a mean-field approach to address the fractional charging of molecules when they interact with an electronic environment. In this paper, we present an extension of the model referred to as particle-breaking unrestricted Hartree-Fock (PBUHF). The unrestricted formulation contains odd-electron states necessary for a realistic description of fractional charging. Within the PBUHF parametrization, we use two-body operators as they yield convenient operator transformations. However, two-body operators can change only the particle number by two. Therefore, we include noninteracting zero-energy bath orbitals to generate a linear combination of even and odd electron states. Depending on whether the occupied or virtual orbitals of a molecule interact with the environment, the average number of electrons is either decreased or increased. Without interaction, PBUHF reduces to the unrestricted Hartree-Fock wave function.

Particle-Breaking Hartree-Fock Theory for Open Molecular Systems.

In this work we present the particle-breaking Hartree-Fock (PBHF) model which is a mean-field approach to open molecular systems. The interaction of a system with the environment is parametrized through a particle-breaking term in the molecular Hamiltonian. The PBHF wave function is constructed through an exponential unitary transformation of a Slater determinant with a given number of electrons. We consider only the closed-shell formalism. The parametrization results in a linear combination of Slater determinants with different numbers of electrons, i.e., the PBHF wave function is not an eigenfunction of the number operator. As a result, the density matrix may have fractional occupations which are, because of the unitary parametrization, always between 0.0 and 2.0. The occupations are optimized simultaneously with the orbitals, using the trust-region optimization procedure. In the limit of a particle-conserving Hamiltonian, the PBHF optimization will converge to a standard Hartree-Fock wave function. We show that the average number of electrons may be decreased or increased depending on whether the particle-breaking term affects occupied or virtual orbitals.

The effect of midbond functions on interaction energies computed using MP2 and CCSD(T).

In this article we use MP2 and CCSD(T) calculations for the A24 and S66 data sets to explore how midbond functions can be used to generate cost effective counterpoise corrected supramolecular interaction energies of noncovalent complexes. We use the A24 data set to show that the primary role of midbond functions is not to approach the complete basis set limit, but rather to ensure a balanced description of the molecules and the interaction region (unrelated to the basis set superposition error). The need for balance is a consequence of using atom centered basis sets. In the complete basis set limit, the error will disappear, but reaching the complete basis set limit is not feasible beyond small systems. For S66 we investigate the need for increasing the number of midbond centers. Results show that adding a second midbond center increases the accuracy, but the effect is secondary to changing the atom centered basis set. Further, by comparing calculations using the 3s3p2d1f1g midbond set with using aug-cc-pVDZ and aug-cc-pVTZ as midbond sets, we see that the requirements for the midbond set to be effective, is not just that it contains diffuse functions, but also that high angular momentum functions are included. By comparing two approaches for placing midbond centers we show that results are not particularly sensitive to placement as long as the placement is reasonable.

Unimolecular Decomposition Reactions of Picric Acid and Its Methylated Derivatives─A DFT Study.

To handle energetic materials safely, it is important to have knowledge about their sensitivity. Density functional theory (DFT) has proven a valuable tool in the study of energetic materials, and in the current work, DFT is employed to study the thermal unimolecular decomposition of 2,4,6-trinitrophenol (picric acid, PA), 3-methyl-2,4,6-trinitrophenol (methyl picric acid, mPA), and 3,5-dimethyl-2,4,6-trinitrophenol (dimethyl picric acid, dmPA). These compounds have similar molecular structures, but according to the literature, mPA is far less sensitive to impact than the other two compounds. Three pathways believed important for the initiation reactions are investigated at 0 and 298.15 K. We compare the computed energetics of the reaction pathways with the objective of rationalizing the unexpected sensitivity behavior. Our results reveal a few if any significant differences in the energetics of the three molecules, and thus do not reflect the sensitivity deviations observed in experiments. These findings point toward the potential importance of crystal structure, crystal morphology, bimolecular reactions, or combinations thereof on the impact sensitivity of nitroaromatics.

Describing ground and excited state potential energy surfaces for molecular photoswitches using coupled cluster models.

In this article, we use two extensively studied systems, a retinal model system and azobenzene, to explore the use of coupled cluster models for describing ground and singlet excited state potential energy surfaces of photoswitchable systems. While not being suitable for describing nuclear dynamics of photoisomerization, coupled cluster models have useful attributes, such as the inclusion of dynamical correlation, their black box nature, and the systematic improvement offered by truncation level. Results for the studied systems show that when triple excitations (here through the CC3 model) are included, ground and excited state potential energy surfaces for isomerization paths may reliably be generated, also for states of doubly excited character. For ground state equilibrium cis- and trans-azobenzene, the molecular geometry and basis set is seen to significantly impact the vertical excitation energies for the two lowest excited states. Efficient implementations of coupled cluster models can therefore constitute valuable tools for investigating photoswitchable systems and can be used for preliminary black box studies to gather information before more complicated excited state dynamics approaches are pursued.

eT 1.0: An open source electronic structure program with emphasis on coupled cluster and multilevel methods.

The eT program is an open source electronic structure package with emphasis on coupled cluster and multilevel methods. It includes efficient spin adapted implementations of ground and excited singlet states, as well as equation of motion oscillator strengths, for CCS, CC2, CCSD, and CC3. Furthermore, eT provides unique capabilities such as multilevel Hartree-Fock and multilevel CC2, real-time propagation for CCS and CCSD, and efficient CC3 oscillator strengths. With a coupled cluster code based on an efficient Cholesky decomposition algorithm for the electronic repulsion integrals, eT has similar advantages as codes using density fitting, but with strict error control. Here, we present the main features of the program and demonstrate its performance through example calculations. Because of its availability, performance, and unique capabilities, we expect eT to become a valuable resource to the electronic structure community.

Characterization and Generation of Local Occupied and Virtual Hartree-Fock Orbitals.

The scope of this review article is to discuss the locality of occupied and virtual orthogonal Hartree-Fock orbitals generated by localization function optimization. Locality is discussed from the stand that an orbital is local if it is confined to a small region in space. Focusing on locality measures that reflects the spatial extent of the bulk of an orbital and the thickness of orbital tails, we discuss, with numerical illustrations, how the locality may be reported for individual orbitals as well as for sets of orbitals. Traditional and more recent orbital localization functions are reviewed, and the locality measures are used to compare the locality of the orbitals generated by the different localization functions, both for occupied and virtual orbitals. Numerical illustrations are given also for large molecular systems and for cases where diffuse functions are included in the atomic orbital basis. In addition, we have included a discussion on the physical and mathematical limitations on orbital locality.

Tautomerization of Thymine Using Ultraviolet Light.

Ultraviolet-light-induced changes to the nucleobase thymine deposited onto a MoS2 surface were studied using photoelectron spectroscopy and first-principles calculations. These measurements suggest changes in the molecular structure indicated by changes in core electron binding energies. The experimental work has been interpreted by means of ab initio calculations using coupled cluster singles and doubles (CCSD) linear response theory. Contrary to the expected behavior, i.e., the dimerization of two thymine molecules into a pyrimidine dimer, a shift between two tautomeric forms was observed upon UV-exposure. Exposure to ionizing radiation is known to induce damage in many biological molecules, and the present work gives additional insight into its effects on thymine, the interactions of the molecules, and finally how certain UV photoproducts may be avoided.

Correlated natural transition orbitals for core excitation energies in multilevel coupled cluster models.

In this article, we present a black-box approach for the selection of orbital spaces when computing core excitation energies in the multilevel coupled cluster (MLCC) framework. Information available from the lower level of theory is used to generate correlated natural transition orbitals (CNTOs) for the high-level calculation by including both singles and doubles information in the construction of the transition orbitals. The inclusion of the doubles excitation information is essential to obtain a set of orbitals that all contain physical information, in contrast to the natural transition orbitals where only a small subset of the virtual orbitals contains physical information. The CNTOs may be included in an active space based on a cutoff threshold for the eigenvalues corresponding to the orbitals. We present MLCC results for core excitation energies calculated using coupled cluster singles and doubles (CCSD) in the inactive space and CCSD with perturbative triples (CC3) in the active space. The use of CNTOs results in small errors compared to full CC3.

Local Hartree-Fock orbitals using a three-level optimization strategy for the energy.

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.

Pipek-Mezey localization of occupied and virtual orbitals.

Recent advances in orbital localization algorithms are used to minimize the Pipek-Mezey localization function for both occupied and virtual Hartree-Fock orbitals. Virtual Pipek-Mezey orbitals for large molecular systems have previously not been considered in the literature. For this work, the Pipek-Mezey (PM) localization function is implemented for both the Mulliken and a Löwdin population analysis. The results show that the standard PM localization function (using either Mulliken or Löwdin population analyses) may yield local occupied orbitals, although for some systems the occupied orbitals are only semilocal as compared to state-of-the-art localized occupied orbitals. For the virtual orbitals, a Löwdin population analysis shows improvement in locality compared to a Mulliken population analysis, but for both Mulliken and Löwdin population analyses, the virtual orbitals are seen to be considerably less local compared to state-of-the-art localized orbitals.

Localized orbitals from basis sets augmented with diffuse functions.

Orbital localization of occupied and virtual Hartree-Fock orbitals generated from basis sets augmented with diffuse functions is performed using the Pipek-Mezey, Boys, powers of the second central moment, and powers of the fourth central moment localizations. The locality of the obtained orbital sets are presented in terms of second and fourth moment orbital spreads. The results show that both local occupied and virtual orbitals may be obtained when using powers of the second central moment and powers of the fourth central moment localizations, while the Pipek-Mezey and Boys localizations fail to produce sets of local virtual orbitals. The locality of the fourth central moment virtual orbitals exhibits a locality similar to the locality of a Boys localization for non-augmented basis sets.

MP2 energy and density for large molecular systems with internal error control using the Divide-Expand-Consolidate scheme.

Divide-Expand-Consolidate (DEC) is a local correlation method where the inherent locality of the electron correlation problem is used to express the correlated calculation on a large molecular system in terms of small independent fragment calculations employing small subsets of local HF orbitals. A crucial feature of the DEC scheme is that the sizes of the local orbital spaces are determined in a black box manner during the calculation. In this way it is ensured that the correlation energy has been determined to a predefined precision compared to a conventional calculation. In the present work we apply the DEC scheme to calculate the correlation energy as well as the electron density matrix for the insulin molecule using second order Møller-Plesset (MP2) theory. This is the first DEC calculation on a molecular system which is too large to be treated using a conventional MP2 implementation. The fragmentation errors for the insulin DEC calculation are carefully analyzed using internal consistency checks. It is demonstrated that size-intensive properties are determined to the same precision for small and large molecules. For example, the percentage of correlation energy recovered and the error per electron in the correlated density matrix depend only on the predefined precision and not on the molecular size.

Orbital localization using fourth central moment minimization.

We present a new orbital localization function based on the sum of the fourth central moments of the orbitals. To improve the locality, we impose a power on the fourth central moment to act as a penalty on the least local orbitals. With power two, the occupied and virtual Hartree-Fock orbitals exhibit a more rapid tail decay than orbitals from other localization schemes, making them suitable for use in local correlation methods. We propose that the standard orbital spread (the square root of the second central moment) and fourth moment orbital spread (the fourth root of the fourth central moment) are used as complementary measures to characterize the locality of an orbital, irrespective of localization scheme.

The divide-expand-consolidate family of coupled cluster methods: numerical illustrations using second order Møller-Plesset perturbation theory.

Previously, we have introduced the linear scaling coupled cluster (CC) divide-expand-consolidate (DEC) method, using an occupied space partitioning of the standard correlation energy. In this article, we show that the correlation energy may alternatively be expressed using a virtual space partitioning, and that the Lagrangian correlation energy may be partitioned using elements from both the occupied and virtual partitioning schemes. The partitionings of the correlation energy leads to atomic site and pair interaction energies which are term-wise invariant with respect to an orthogonal transformation among the occupied or the virtual orbitals. Evaluating the atomic site and pair interaction energies using local orbitals leads to a linear scaling algorithm and a distinction between Coulomb hole and dispersion energy contributions to the correlation energy. Further, a detailed error analysis is performed illustrating the error control imposed on all components of the energy by the chosen energy threshold. This error control is ultimately used to show how to reduce the computational cost for evaluating dispersion energy contributions in DEC.

Implementation of Occupied and Virtual Edmiston-Ruedenberg Orbitals Using Cholesky Decomposed Integrals.

We present a trust-region optimization of the Edmiston-Ruedenberg orbital localization function. The approach is used to localize both the occupied and the virtual orbitals and is the first demonstration of general virtual orbital localization using the Edmiston-Ruedenberg localization function. In the Edmiston-Ruedenberg approach, the sum of the orbital self-repulsion energies is maximized to obtain the localized orbitals. The Cholesky decomposition reduces the cost of transforming the electron repulsion integrals, and the overall scaling of our implementation is O(N4). The optimization is performed with all quantities in the molecular orbital basis, and the localization of the occupied orbitals is often less expensive than the corresponding self-consistent field (SCF) optimization. Furthermore, the occupied orbital localization scales linearly with the basis set. For the virtual space, the cost is significantly higher than the SCF optimization. The orbital spreads of the resulting virtual Edmiston-Ruedenberg orbitals are larger than for other, less expensive, orbital localization functions. This indicates that other localization procedures are more suitable for applications such as local post-Hartree-Fock calculations.

Linear-Scaling Implementation of Multilevel Hartree-Fock Theory.

We introduce a new algorithm for the construction of the two-electron contributions to the Fock matrix in multilevel Hartree-Fock (MLHF) theory. In MLHF, the density of an active molecular region is optimized, while the density of an inactive region is fixed. The MLHF equations are solved in a reduced molecular orbital (MO) basis localized to the active region. The locality of the MOs can be exploited to reduce the computational cost of the Fock matrix: the cost related to the inactive density becomes linear scaling, while the iterative cost related to the active density is independent of the system size. We demonstrate the performance of this new algorithm on a variety of systems, including amino acid chains, water clusters, and solvated systems.

Potential energy surfaces for vibrational structure calculations from a multiresolution adaptive density-guided approach: implementation and test calculations.

A multiresolution procedure to construct potential energy surfaces (PESs) for use in vibrational structure calculations is developed in the framework of the adaptive density-guided approach. The implementation of the method allows the construction of hybrid PESs with different mode-coupling terms calculated with a variety of combinations of electronic structure methods and basis sets. Furthermore, the procedure allows the construction of hybrid PESs that incorporate a variety of contributions and corrections to the electronic energy, such as infinite basis set extrapolation and core correlation effects. A full account of the procedure is given together with a rather large set of benchmark calculations on a set of 20 small molecules, from diatomics to tetratomics.

Density-Based Multilevel Hartree-Fock Model.

We introduce a density-based multilevel Hartree-Fock (HF) method where the electronic density is optimized in a given region of the molecule (the active region). Active molecular orbitals (MOs) are generated by a decomposition of a starting guess atomic orbital (AO) density, whereas the inactive MOs (which constitute the remainder of the density) are never generated or referenced. The MO formulation allows for a significant dimension reduction by transforming from the AO basis to the active MO basis. All interactions between the inactive and active regions of the molecule are retained, and an exponential parametrization of orbital rotations ensures that the active and inactive density matrices separately, and in sum, satisfy the symmetry, trace, and idempotency requirements. Thus, the orbital spaces stay orthogonal, and furthermore, the total density matrix represents a single Slater determinant. In each iteration, the (level-shifted) Newton equations in the active MO basis are solved to obtain the orbital transformation matrix. The approach is equivalent to variationally optimizing only a subset of the MOs of the total system. In this orbital space partitioning, no bonds are broken and no a priori orbital assignments are carried out. In the limit of including all orbitals in the active space, we obtain an MO density-based formulation of full HF.

The Dalton quantum chemistry program system.

Dalton is a powerful general-purpose program system for the study of molecular electronic structure at the Hartree-Fock, Kohn-Sham, multiconfigurational self-consistent-field, Møller-Plesset, configuration-interaction, and coupled-cluster levels of theory. Apart from the total energy, a wide variety of molecular properties may be calculated using these electronic-structure models. Molecular gradients and Hessians are available for geometry optimizations, molecular dynamics, and vibrational studies, whereas magnetic resonance and optical activity can be studied in a gauge-origin-invariant manner. Frequency-dependent molecular properties can be calculated using linear, quadratic, and cubic response theory. A large number of singlet and triplet perturbation operators are available for the study of one-, two-, and three-photon processes. Environmental effects may be included using various dielectric-medium and quantum-mechanics/molecular-mechanics models. Large molecules may be studied using linear-scaling and massively parallel algorithms. Dalton is distributed at no cost from http://www.daltonprogram.org for a number of UNIX platforms.