A variational reformulation of molecular properties in electronic-structure theory.

Conventional quantum-mechanical calculations of molecular properties, such as dipole moments and electronic excitation energies, give errors that depend linearly on the error in the wave function. An exception is the electronic energy, whose error depends quadratically on the error in wave function. We here describe how all properties may be calculated with a quadratic error, by setting up a variational Lagrangian for the property of interest. Because the construction of the Lagrangian is less expensive than the calculation of the wave function, this approach substantially improves the accuracy of quantum-chemical calculations without increasing cost. As illustrated for excitation energies, this approach enables the accurate calculation of molecular properties for larger systems, with a short time-to-solution and in a manner well suited for modern computer architectures.

Cluster perturbation theory IX: Perturbation series for the coupled cluster singles and doubles ground state energy.

In this paper, we develop and analyze a number of perturbation series that target the coupled cluster singles and doubles (CCSD) ground state energy. We show how classical Møller-Plesset perturbation theory series can be restructured to target the CCSD energy based on a reference CCS calculation and how the corresponding cluster perturbation series differs from the classical Møller-Plesset perturbation series. Subsequently, we reformulate these series using the coupled cluster Lagrangian framework to obtain series, where fourth and fifth order energies are determined only using parameters through second order. To test the methods, we perform a series of test calculations on molecular photoswitches of both total energies and reaction energies. We find that the fifth order reaction energies are of CCSD quality and that they are of comparable accuracy to state-of-the-art approximations to the CCSD energy based on local pair natural orbitals. The advantage of the present approach over local correlation methods is the absence of user defined threshold parameters for neglecting or approximating contributions to the correlation energy. Fixed threshold parameters lead to discontinuous energy surfaces, although this effect is often small enough to be ignored, but the present approach has a differentiable energy that will facilitate derivation and implementation of gradients and higher derivatives. A further advantage is that the calculation of the perturbation correction is non-iterative and can, therefore, be calculated in parallel, leading to a short time-to-solution.

Corrigendum: Coupled cluster theory on modern heterogeneous supercomputers.

[This corrects the article DOI: 10.3389/fchem.2023.1154526.].

Are European Blue Economy ambitions in conflict with European environmental visions?

We report the outcomes of a comprehensive study of the potential consequences of the implementation of the EU Maritime Spatial Planning Directive (MSPD) in Danish waters. The analyses are anchored in a framework developed in support of data-driven Ecosystem-Based Maritime Spatial Planning. The data for the models include not only human stressors but also information on the distribution of ecosystem components ranging from planktonic communities over benthic communities to fish, seabirds and marine mammals. We have established a baseline, based on state-of-the-art data sets, with respect to combined effects upon ecosystem components. Future scenarios for the developments in human stressors were estimated for 2030 and 2050 based on information on existing policies, strategies and plans and were compared to the baseline. In addition, we developed a scenario for implementation of the Marine Strategy Framework Directive (MSFD), i.e. working towards meeting the objectives of Good Environmental Status. Our results indicate that (1) combined human stressors will possibly increase in 2030 and 2050 compared to the baseline, (2) increased combined human stressors are likely to lead to a worsening of the environmental and ecological status sensu the Marine Strategy Framework Directive and the Water Framework Directive (WFD), and (3) the MSPD implementation process appears to conflict with the MSFD and WFD objectives. Accordingly, we are sceptical of claims of an untapped potential for Blue Growth in Danish marine waters.

A relativistic configuration interaction method with general expansions and initial applications to electronic g-factors.

A new implementation of the orbital-based two-component relativistic configuration interaction approach is reported and applied to calculations of the electronic g-shifts of three diatomic radicals: AlO, HgF, and PdH. The new implementation augments efficient routines for the calculation of nonrelativistic Hamiltonians with new vectorized routines for the calculation of the action of the one-electron spin-orbit operator and allows efficient calculations for the expansion of generalized active space type. The program makes full use of double group as well as time-reversal symmetry. Particle-hole reorganization of the operators is used to improve the efficiency for expansions with nearly fully occupied orbital spaces. The flexibility of the algorithm and program is used to investigate the convergence of electronic g-shifts for the three diatomic radicals as functions of the active space, states included in the orbital optimization, and excitation levels. It was possible to converge to the valence limits within a few percent using expansions containing up to quadruple excitations. However, when excitations from the core orbitals were added, it was not possible to demonstrate convergence to within a few percent with expansions containing at most 10 × 109 determinants.

Coupled cluster theory on modern heterogeneous supercomputers.

This study examines the computational challenges in elucidating intricate chemical systems, particularly through ab-initio methodologies. This work highlights the Divide-Expand-Consolidate (DEC) approach for coupled cluster (CC) theory-a linear-scaling, massively parallel framework-as a viable solution. Detailed scrutiny of the DEC framework reveals its extensive applicability for large chemical systems, yet it also acknowledges inherent limitations. To mitigate these constraints, the cluster perturbation theory is presented as an effective remedy. Attention is then directed towards the CPS (D-3) model, explicitly derived from a CC singles parent and a doubles auxiliary excitation space, for computing excitation energies. The reviewed new algorithms for the CPS (D-3) method efficiently capitalize on multiple nodes and graphical processing units, expediting heavy tensor contractions. As a result, CPS (D-3) emerges as a scalable, rapid, and precise solution for computing molecular properties in large molecular systems, marking it an efficient contender to conventional CC models.

Massively parallel GPU enabled third-order cluster perturbation excitation energies for cost-effective large scale excitation energy calculations.

We present here a massively parallel implementation of the recently developed CPS(D-3) excitation energy model that is based on cluster perturbation theory. The new algorithm extends the one developed in Baudin et al. [J. Chem. Phys., 150, 134110 (2019)] to leverage multiple nodes and utilize graphical processing units for the acceleration of heavy tensor contractions. Furthermore, we show that the extended algorithm scales efficiently with increasing amounts of computational resources and that the developed code enables CPS(D-3) excitation energy calculations on large molecular systems with a low time-to-solution. More specifically, calculations on systems with over 100 atoms and 1000 basis functions are possible in a few hours of wall clock time. This establishes CPS(D-3) excitation energies as a computationally efficient alternative to those obtained from the coupled-cluster singles and doubles model.

European coastal monitoring programmes may fail to identify impacts on benthic macrofauna caused by bottom trawling.

Bottom trawling (hereafter trawling) is the dominant human pressure impacting continental shelves globally. However, due to ongoing data deficiencies for smaller coastal vessels, the effects of trawling on nearshore seabed ecosystems are poorly understood. In Europe, the Water Framework Directive (WFD) provides a framework for the protection and improvement of coastal water bodies. It requires member states to track the status of 'biological quality elements' (including benthic macrofauna) using WFD-specific ecological indicators. While many of these metrics are sensitive to coastal pressures such as nutrient enrichment, little is known about their ability to detect trawling impacts. Here, we analysed a comprehensive data set of 5885 nearshore benthic samples - spatiotemporally matched to high-resolution trawling and environmental data - to examine how these pressures affect coastal benthos. In addition, we investigated the ability of 8 widely-used benthic monitoring metrics to detect impacts on benthic biological quality. We found that abundance (N) and species richness (S) were strongly impacted by bottom trawling. A clear response to trawling was also observed for the WFD-specific Benthic Quality Index (BQI). Relationships between N and S, and trawling were particularly consistent across the study area, indicating sensitivity across varying environmental conditions. In contrast, WFD indices such as AZTIs Marine Biotic Index (AMBI), multivariate AMBI (M-AMBI), and the Danish Quality Index (DKI), were unresponsive to trawling. In fact, some of the most heavily trawled areas examined were classified as being of 'high/good ecological status' by these indices. A likely explanation for this is that the indices are calculated using species sensitivity scores, based on expected species response to eutrophication and chemical pollution. While the BQI also uses species sensitivity scores, these are based on observed responses to disturbance gradients comprising a range of coastal pressures. Given the prominent use of AMBI and DKI throughout Europe, our results highlight the considerable risk that the metrics used to assess Good Ecological Status (GES) under the WFD may fail to identify trawling impacts. As trawling represents a widespread source of coastal disturbance, fishing impacts on benthic macrofauna may be underestimated, or go undetected, in many coastal monitoring programmes around Europe.

Capabilities and limits of the unitary coupled-cluster approach with generalized two-body cluster operators.

Unitary cluster expansions of the electronic wavefunction have recently gained much interest because of their use in conjunction with quantum algorithms. In this contribution, we investigate some aspects of an ansatz, using generalized two-body excitation operators, which have been considered in some recent studies on quantum algorithms for quantum chemistry. Our numerical results show that, in particular, two-body operators with effective particle-hole excitation level of one in connection with the usual particle-hole double excitation operators lead to a very accurate, yet compact representation of the wavefunction. Generalized two-body operators with effective excitation rank zero have a considerably less pronounced effect. We compare with standard and unitary coupled-cluster expansions and show that the above mentioned approach matches or even surpasses the accuracy of expansions with three-body particle-hole excitations, in particular at the onset of strong correlation. A downside of the approach is that it is rather difficult to rigorously converge it to its variational minimum.

Cluster perturbation theory. VIII. First order properties for a coupled cluster state.

We have extended cluster perturbation (CP) theory to comprehend the calculation of first order properties (FOPs). We have determined CP FOP series where FOPs are determined as a first energy derivative and also where the FOPs are determined as a generalized expectation value of the external perturbation operator over the coupled cluster state and its biorthonormal multiplier state. For S(D) orbital excitation spaces, we find that the CP series for FOPs that are determined as a first derivative, in general, in second order have errors of a few percent in the singles and doubles correlation contribution relative to the targeted coupled cluster (CC) results. For a SD(T) orbital excitation space, we find that the CP series for FOPs determined as a generalized expectation value in second order have errors of about ten percent in the triples correlation contribution relative to the targeted CC results. These second order models, therefore, constitute viable alternatives for determining high quality FOPs.

Cluster perturbation theory. VII. The convergence of cluster perturbation expansions.

The convergence of the recently developed cluster perturbation (CP) expansions [Pawlowski et al., J. Chem. Phys. 150, 134108 (2019)] is analyzed with the double purpose of developing the mathematical tools and concepts needed to describe these expansions at general order and to identify the factors that define the rate of convergence of CP series. To this end, the CP energy, amplitude, and Lagrangian multiplier equations as a function of the perturbation strength are developed. By determining the critical points, defined as the perturbation strengths for which the Jacobian becomes singular, the rate of convergence and the intruder and critical states are determined for five small molecules: BH, CO, H2O, NH3, and HF. To describe the patterns of convergence for these expansions at orders lower than the high-order asymptotic limit, a model is developed where the perturbation corrections arise from two critical points. It is shown that this model allows for rationalization of the behavior of the perturbation corrections at much lower order than required for the onset of the asymptotic convergence. For the H2O, CO, and HF molecules, the pattern and rate of convergence are defined by critical states where the Fock-operator underestimates the excitation energies, whereas the pattern and rate of convergence for BH are defined by critical states where the Fock-operator overestimates the excitation energy. For the NH3 molecule, both forms of critical points are required to describe the convergence behavior up to at least order 25.

Cluster perturbation theory. VI. Ground-state energy series using the Lagrangian.

We have extended cluster perturbation (CP) theory to comprehend the Lagrangian framework of coupled cluster (CC) theory and derived the CP Lagrangian energy series (LCP) where the 2n + 1/2n + 2 rules for the cluster amplitudes and multipliers are used to get the energy corrections. We have also developed the variational CP (LCP) series, where the total cluster amplitudes and multipliers are determined through the same orders as in the LCP series, but the energy is obtained by inserting the total cluster amplitudes and multipliers in the Lagrangian. The energies of the LCP series have errors that are bilinear in the errors of the total cluster amplitudes and multipliers. Test calculations have been performed for S(D) and SD(T) orbital excitation spaces. With the exception of molecular systems that have a low lying doubly excited state compared to the electronic ground state configuration, we find that the fourth order models LCPS (D-4), LCPSD (T-4), and LCPSD(T-4) give energies of CC target state quality. For the LCPS (D-4) model, CC target state quality is obtained as the LCPS (D-4) calculation determines more than 99.7% of the coupled cluster singles and doubles (CCSD) correlation energy as the numerical deviations of the LCPS (D-4) energy from the CCSD energy were more than an order of magnitude smaller than the triples correlation contribution. For the LCPSD (T-4) and LCPSD(T-4) models, CC target state quality was obtained, given that the LCPSD (T-4) and LCPSD(T-4) calculations recover more than 99% of the coupled cluster singles doubles and triples (CCSDT) correlation contribution and as the numerical deviations of the LCPSD (T-4) and LCPSD(T-4) energies from the CCSDT energy were nearly and order of magnitude smaller than the quadruples correlation contribution. We, thus, suggest that the fourth order models may replace the full target CC models with no or very limited loss of accuracy.

NBCn1 Increases NH4 + Reabsorption Across Thick Ascending Limbs, the Capacity for Urinary NH4 + Excretion, and Early Recovery from Metabolic Acidosis.

BACKGROUND: The electroneutral Na+/HCO3 - cotransporter NBCn1 (Slc4a7) is expressed in basolateral membranes of renal medullary thick ascending limbs (mTALs). However, direct evidence that NBCn1 contributes to acid-base handling in mTALs, urinary net acid excretion, and systemic acid-base homeostasis has been lacking. METHODS: Metabolic acidosis was induced in wild-type and NBCn1 knockout mice. Fluorescence-based intracellular pH recordings were performed and NH4 + transport measured in isolated perfused mTALs. Quantitative RT-PCR and immunoblotting were used to evaluate NBCn1 expression. Tissue [NH4 +] was measured in renal biopsies, NH4 + excretion and titratable acid quantified in spot urine, and arterial blood gasses evaluated in normoventilated mice. RESULTS: Basolateral Na+/HCO3 - cotransport activity was similar in isolated perfused mTALs from wild-type and NBCn1 knockout mice under control conditions. During metabolic acidosis, basolateral Na+/HCO3 - cotransport activity increased four-fold in mTALs from wild-type mice, but remained unchanged in mTALs from NBCn1 knockout mice. Correspondingly, NBCn1 protein expression in wild-type mice increased ten-fold in the inner stripe of renal outer medulla during metabolic acidosis. During systemic acid loading, knockout of NBCn1 inhibited the net NH4 + reabsorption across mTALs by approximately 60%, abolished the renal corticomedullary NH4 + gradient, reduced the capacity for urinary NH4 + excretion by approximately 50%, and delayed recovery of arterial blood pH and standard [HCO3 -] from their initial decline. CONCLUSIONS: During metabolic acidosis, NBCn1 is required for the upregulated basolateral HCO3 - uptake and transepithelial NH4 + reabsorption in mTALs, renal medullary NH4 + accumulation, urinary NH4 + excretion, and early recovery of arterial blood pH and standard [HCO3 -]. These findings support that NBCn1 facilitates urinary net acid excretion by neutralizing intracellular H+ released during NH4 + reabsorption across mTALs.

Structure of the human lipid exporter ABCB4 in a lipid environment.

ABCB4 is an ATP-binding cassette transporter that extrudes phosphatidylcholine into the bile canaliculi of the liver. Its dysfunction or inhibition by drugs can cause severe, chronic liver disease or drug-induced liver injury. We determined the cryo-EM structure of nanodisc-reconstituted human ABCB4 trapped in an ATP-bound state at a resolution of 3.2 Å. The nucleotide binding domains form a closed conformation containing two bound ATP molecules, but only one of the ATPase sites contains bound Mg2+. The transmembrane domains adopt a collapsed conformation at the level of the lipid bilayer, but we observed a large, hydrophilic and fully occluded cavity at the level of the cytoplasmic membrane boundary, with no ligand bound. This indicates a state following substrate release but prior to ATP hydrolysis. Our results rationalize disease-causing mutations in human ABCB4 and suggest an 'alternating access' mechanism of lipid extrusion, distinct from the 'credit card swipe' model of other lipid transporters.

Convergence patterns and rates in two-state perturbation expansions.

A simple two-state model has previously been shown to be able to describe and rationalize the convergence of the most common perturbation method for including electron correlation, the Møller-Plesset expansion. In particular, this simple model has been able to predict the convergence rate and the form of the higher-order corrections for typical Møller-Plesset expansions of the correlation energy. In this paper, the convergence of nondegenerate perturbation expansions in the two-state model is analyzed in detail for a general form of two-state perturbation expansion by examining the analytic expressions of the corrections and series of the values of the corrections for various choices of the perturbation. The previous analysis that covered only a single form of the perturbation is thereby generalized to arbitrary forms of the perturbation. It is shown that the convergence may be described in terms of four characteristics: archetype, rate of convergence, length of recurring period, and sign pattern. The archetype defines the overall form of a plot of the energy-corrections, and the remaining characteristics specify details of the archetype. For symmetric (Hermitian) perturbations, five archetypes are observed: zigzag, interspersed zigzag, triadic, ripples, and geometric. Two additional archetypes are obtained for an asymmetric perturbation: zigzag-geometric and convex-geometric. For symmetric perturbations, each archetype has a distinctive pattern that recurs with a period which depends on the perturbation parameters, whereas no such recurrence exists for asymmetric perturbations from a series of numerical corrections. The obtained relations between the form of a two-state perturbation and the energy corrections allow us to obtain additional insights into the convergence behavior of the Møller-Plesset and other forms of perturbation expansions. This is demonstrated by analyzing several diverging or slowly converging perturbation expansions of ground state and excitation energies. It is demonstrated that the higher-order corrections of these expansions can be described using the two-state model and each expansion can therefore be described in terms of an archetype and the other three characteristics. Examples of all archetypes except the zigzag and convex-geometric archetypes are given. For each example, it is shown how the characteristics may be extracted from the higher-order corrections and used to identify the term in the perturbation that is the cause of the observed slow convergence or divergence.

Cluster perturbation theory. II. Excitation energies for a coupled cluster target state.

In cluster perturbation (CP) theory, we consider a target excitation space relative to a Hartree-Fock state and partition the target excitation space into a parent excitation space and an auxiliary excitation space. The zeroth-order state is in CP theory a coupled cluster (CC) state in the parent excitation space, and the target state is a CC state in the target excitation space. In this paper, we derive CP series for excitation energies in orders of the CC parent-state similarity-transformed fluctuation potential where the zeroth-order term in the series is an excitation energy for the CC parent state response eigenvalue equation and where the series formally converge to an excitation energy for the CC target state response eigenvalue equation. We give explicit expressions for the lowest-order excitation energy corrections. We also report calculations for CP excitation energy series for various parent and target excitation spaces and examine how well the lower-order corrections can reproduce the total excitation energies. Considering the fast local convergence we have observed for the CP excitation energy series, it becomes computationally attractive to use low-order corrections in CP series to obtain excitation energies of CC target state quality. For the CPS(D-n) series, the first-order correction vanishes, the second-order correction becomes the CIS(D) model, and for the CPS(D-3) model, our calculations suggest that excitation energies of CCSD quality are obtained. The numerical results also suggest that a similar behavior can be seen for the low-order excitation energy corrections for CP series where the parent state contains more than a singles excitation space, e.g., for the CPSD(T) model. We therefore expect the low-order excitation energy corrections in CP series soon to become state-of-the-art models for determining excitation energies of CC target state quality.

Cluster perturbation theory. I. Theoretical foundation for a coupled cluster target state and ground-state energies.

We introduce a new class of perturbation models-the cluster perturbation (CP) models-where the major drawbacks of Møller-Plesset perturbation theory and coupled cluster perturbation theory have been eliminated. In CP theory, we consider a target excitation space relative to the Hartree-Fock state and partition the target excitation space into a parent and an auxiliary excitation space. The zeroth-order state is a coupled cluster (CC) state in the parent excitation space, and the target state is either a cluster linear or a CC state in the target excitation space. In CP theory, perturbation series are determined in orders of the CC parent state similarity-transformed fluctuation potential for the energy and for a molecular property, where the zeroth-order term in the series is the energy or a molecular property for the CC parent state and where the series formally converge to the energy or a molecular property for the target state. In CP theory, we use a generalized order concept, where the zeroth-order component of the extended parent-state Jacobian contains a fluctuation potential contribution, and use this new generalized order to treat internal relaxation in the parent excitation space at zeroth order and hence remove it from the perturbation calculation. Even more importantly, using this new generalized order concept, CP series can be determined for molecular properties of ground and excited states and for transition properties between these states, including excitation energies and energies of the excited states. The applicability of CP theory to both the energy and molecular properties and numerical results for the CP energy and molecular property series demonstrate the superiority of CP theory compared to previous perturbation models. Low-order corrections in the CP perturbation series can be expected soon to become state-of-the-art electronic structure models for the determination of energies and molecular properties of target-state quality for single-configuration dominated molecular systems.

Cluster perturbation theory. IV. Convergence of cluster perturbation series for energies and molecular properties.

The theoretical foundation has been developed for establishing whether cluster perturbation (CP) series for the energy, molecular properties, and excitation energies are convergent or divergent and for using a two-state model to describe the convergence rate and convergence patterns of the higher-order terms in the CP series. To establish whether the perturbation series are convergent or divergent, a fictitious system is introduced, for which the perturbation is multiplied by a complex scaling parameter z. The requirement for convergent perturbation series becomes that the energy or molecular property, including an excitation energy, for the fictitious system is an analytic, algebraic function of z that has no singularities when the norm |z| is smaller than one. Examples of CP series for the energy and molecular properties, including excitation energies, are also presented, and the two-state model is used for the interpretation of the convergence rate and the convergence patterns of the higher-order terms in these series. The calculations show that the perturbation series effectively become a two-state model at higher orders.

Cluster perturbation theory. III. Perturbation series for coupled cluster singles and doubles excitation energies.

The cluster perturbation series, CPS(D), for coupled cluster singles and doubles excitation energies is considered. It is demonstrated that the second-order model CPS(D-2) is identical to the configuration interaction singles with perturbative doubles, CIS(D) model. The third-order model, CPS(D-3), provides excitation energies of coupled cluster singles and doubles (CCSD) quality in the sense that the difference between CPS(D-3) and CCSD excitation energies is of the same size or smaller than the effect of adding triples corrections to CCSD excitation energies. We further show that the third-order corrections can be efficiently implemented, in particular, when the resolution of the identity approximation is used for integrals. We also show that the CPS(D-3) excitation energies can be determined for system sizes that are far beyond what can be considered in conventional CCSD excitation energy calculations.

Cluster perturbation theory. V. Theoretical foundation for cluster linear target states.

Cluster perturbation (CP) theory was developed in Paper I [F. Pawlowski et al., J. Chem. Phys. 150, 134108 (2019)] for a coupled cluster (CC) target state and is extended in this paper to comprehend a cluster linear (CL) target state, for which the embedding of a CC parent state in the target excitation space is described using a linear parametrization. The theory is developed for determining the energy and molecular properties for a CL state. When CP theory is applied to a CL target state, a series of corrections is determined in orders of the CC parent-state similarity-transformed fluctuation potential, where the zeroth-order term is the energy or molecular property of the CC parent state and where the series formally converges to the energy or molecular property of the CL target state. The determination of energies and molecular properties is simpler for a CL state than for a CC state because the CL state is linearly parametrized. The amplitude equations are quadratic for a CL target state, while quartic for a CC target state, and molecular property expressions for a CL target state have the same simple structure as for a configuration interaction state. The linear parametrization introduces non-size-extensive contributions in the energy and molecular property expressions. However, since the linear parametrization describes the embedding of the CC parent state in the target excitation space, the energy and molecular properties for a CL state are weakly size-extensive. For the energy, weak size-extensivity means that non-size-extensive contributions enter in sixth and higher orders in the CP energy series, whereas for molecular properties, weak size-extensivity means that non-size-extensive contributions enter in second and higher orders. Weak size-extensivity therefore has a little or vanishing effect on calculated energies or molecular properties. The determination of the CP energy and molecular property corrections does not require that amplitude or response equations are solved explicitly for the target state and it becomes computationally tractable to use low-order corrections from these series to obtain energies and molecular properties of CL target state quality. For three simple molecules, HF, N2, and CH2, the accuracy of the CL approach for ground-state energies is tested using a parent state including single and double excitations (i.e., the CC singles-and-doubles state, CCSD) and a target state that includes triple excitations. It is found that the size-extensive fifth-order CL energies deviate by less than 0.0001 hartree from the energies of a target CC that includes triple excitations (i.e., the CC singles-doubles-and-triples state, CCSDT). CP theory with a CL target state therefore becomes a very attractive replacement of standard CC theory for high-accuracy energy and molecular property calculations, in which triple and higher excitation levels are considered.