Analytical Gradient Research Articles

Density functional theory for molecular and periodic systems using density fitting and continuous fast multipole method: Stress tensor.

A full implementation of the analytical stress tensor for periodic systems is reported in the TURBOMOLE program package within the framework of Kohn-Sham density functional theory using Gaussian-type orbitals as basis functions. It is the extension of the implementation of analytical energy gradients (Lazarski et al., Journal of Computational Chemistry 2016, 37, 2518-2526) to the stress tensor for the purpose of optimization of lattice vectors. Its key component is the efficient calculation of the Coulomb contribution by combining density fitting approximation and continuous fast multipole method. For the exchange-correlation (XC) part the hierarchical numerical integration scheme (Burow and Sierka, Journal of Chemical Theory and Computation 2011, 7, 3097-3104) is extended to XC weight derivatives and stress tensor. The computational efficiency and favorable scaling behavior of the stress tensor implementation are demonstrated for various model systems. The overall computational effort for energy gradient and stress tensor for the largest systems investigated is shown to be at most two and a half times the computational effort for the Kohn-Sham matrix formation. © 2019 Wiley Periodicals, Inc.

Conical Intersections in Organic Molecules: Benchmarking Mixed-Reference Spin-Flip Time-Dependent DFT (MRSF-TD-DFT) vs Spin-Flip TD-DFT.

The mixed-reference spin-flip time-dependent density functional theory (MRSF-TD-DFT) method eliminates the erroneous spin contamination of the SF-TD-DFT methodology, while retaining the conceptual and practical simplicity of the latter. The availability of the analytic gradient of the energy of the MRSF-TD-DFT response states enables automatic geometry optimization of the targeted states. Here, we apply the new method to optimize the geometry of several S1/S0 conical intersections occurring in typical organic molecules. We demonstrate that MRSF-TD-DFT is capable of producing the correct double-cone topology of the intersections and describing the geometry of the lowest-energy conical intersections and their relative energies with accuracy matching that of the best multireference wavefunction ab initio methods. In this regard, MRSF-TD-DFT differs from many popular single-reference methods, such as, e.g., the linear response TD-DFT method, which fail to produce the correct topology of the intersections. As the new methodology completely eliminates the ambiguity with the identification of the response states as proper singlets or triplets, which is plaguing the SF-TD-DFT calculations, it can be used for automatic geometry optimization and molecular dynamic simulations not requiring constant human intervention.

Implementation of analytic gradients for CCSD and EOM-CCSD using Cholesky decomposition of the electron-repulsion integrals and their derivatives: Theory and benchmarks.

We present a general formulation of analytic nuclear gradients for the coupled-cluster with single and double substitution (CCSD) and equation-of-motion (EOM) CCSD energies computed using Cholesky decomposition (CD) representations of the electron repulsion integrals. By rewriting the correlated energy and response equations such that the storage of the largest four-index intermediates is eliminated, CD leads to a significant reduction in disk storage requirements, reduced I/O penalties, and an improved parallel performance. CD thus extends the scope of the systems that can be treated by (EOM-)CCSD methods, although analytic gradients in the framework of CD are needed to extend the applicability of (EOM-)CCSD methods in the context of geometry optimizations. This paper presents a formulation of analytic (EOM-)CCSD gradient within the CD framework and reports on the salient details of the corresponding implementation. The accuracy and the capabilities of analytic CD-based (EOM-)CCSD gradients are illustrated by benchmark calculations and several illustrative examples.

Wind farm power optimization through wake steering

Global power production increasingly relies on wind farms to supply low-carbon energy. The recent Intergovernmental Panel on Climate Change (IPCC) Special Report predicted that renewable energy production must leap from [Formula: see text] of the global energy mix in 2018 to [Formula: see text] by 2050 to keep global temperatures from rising 1.5°C above preindustrial levels. This increase requires reliable, low-cost energy production. However, wind turbines are often placed in close proximity within wind farms due to land and transmission line constraints, which results in wind farm efficiency degradation of up to [Formula: see text] for wind directions aligned with columns of turbines. To increase wind farm power production, we developed a wake steering control scheme. This approach maximizes the power of a wind farm through yaw misalignment that deflects wakes away from downstream turbines. Optimization was performed with site-specific analytic gradient ascent relying on historical operational data. The protocol was tested in an operational wind farm in Alberta, Canada, resulting in statistically significant ([Formula: see text]) power increases of 7-[Formula: see text] for wind speeds near the site average and wind directions which occur during less than [Formula: see text] of nocturnal operation and 28-[Formula: see text] for low wind speeds in the same wind directions. Wake steering also decreased the variability in the power production of the wind farm by up to [Formula: see text] Although the resulting gains in annual energy production were insignificant at this farm, these statistically significant wake steering results demonstrate the potential to increase the efficiency and predictability of power production through the reduction of wake losses.

The all configuration mean energy multiconfiguration self-consistent-field method. I. Equal configuration weights

ABSTRACTThe All Configuration Mean Energy (ACME) conditions are a special case of state averaging for Multiconfigurational Self-Consistent-Field (MCSCF) orbital optimisation. The method is formulated using the Graphical Unitary Group Approach (GUGA) in which the Configuration State Function (CSF) basis is represented as walks within a Shavitt graph. This graphical formulation leads to efficient recursive algorithms for the energy and reduced density matrices (RDM) that are independent of the CSF dimension and that scale only as O(n2) where n is the number of occupied orbitals. The Hamiltonian matrix diagonalization step is obviated and the CSF expansion coefficients are neither referenced nor required during the orbital optimisation. This allows MCSCF orbital optimisation to be performed for essentially unlimited numbers of active orbitals and arbitrarily large CSF expansions. The discussion includes various types of CSF expansion spaces, the partitioning of the essential and redundant orbital optimisation parameters, the computation of the spin-density, and the formulation of state-specific analytic gradients and nonadiabatic coupling for high-level electronic structure methods that use the ACME MCSCF orbitals.

Variational Formulation of the Generalized Many-Body Expansion with Self-Consistent Charge Embedding: Simple and Correct Analytic Energy Gradient for Fragment-Based ab Initio Molecular Dynamics.

The many-body expansion (MBE) and its extension to overlapping fragments, the generalized (G)MBE, constitute the theoretical basis for most fragment-based approaches for large-scale quantum chemistry. We reformulate the GMBE for use with embedding charges determined self-consistently from the fragment wave functions, in a manner that preserves the variational nature of the underlying self-consistent field method. As a result, the analytic gradient retains the simple "sum of fragment gradients" form that is often assumed in practice, sometimes incorrectly. This obviates (without approximation) the need to solve coupled-perturbed equations, and we demonstrate stable, fragment-based ab initio molecular dynamics simulations using this technique. Energy conservation fails when charge-response contributions to the Fock matrix are neglected, even while geometry optimizations and vibrational frequency calculations may yet be accurate. Stable simulations can be recovered by means of straightforward modifications introduced here, providing a general paradigm for fragment-based ab initio molecular dynamics.

Imaginary Shift in CASPT2 Nuclear Gradient and Derivative Coupling Theory.

We report the analytical nuclear gradient theory for complete active space second-order perturbation theory (CASPT2) with imaginary shift, which is commonly used to avoid divergence of the perturbation expression. Our formulation is based on the Lagrangian approach and is an extension of the algorithm for CASPT2 nuclear gradients with real shift. The working equations are derived and implemented into an efficient parallel program. Numerical examples are presented for the ground- and excited-state geometries and conical intersections of a green fluorescent protein model chromophore, p-HBDI-. We also report timing benchmarks with adenine, p-HBDI-, and iron porphyrin. It is demonstrated that the energies and geometries obtained with the imaginary shift improve accuracy at a minor additional cost which is mainly associated with evaluating the effective density matrix elements for the imaginary shift term.

Ab initio aided strain gradient elasticity theory in prediction of nanocomponent fracture

The aim of the paper is to address fracture problems in nanoscale-sized cracked components using a simplified form of the strain gradient elasticity theory aided by ab initio calculations. Quantification of the material length scale parameter l1 of the simplified form of the strain gradient elasticity theory plays a key role in the analysis. The parameter l1 is identified for silicon and tungsten single crystals using first principles calculations. Specifically, the parameter l1 is extracted from phonon-dispersions generated by ab-initio calculations and, for comparison, by adjusting the analytical strain gradient elasticity theory solution for the displacement field near the screw dislocation with the ab-initio calculations of this field. The obtained results are further used in the strain gradient elasticity modeling of crack stability in nano-panels made of silicon and tungsten single crystals, where due to size effects and nonlocal material point interactions the classical linear fracture mechanics breaks down. The cusp-like crack tip opening profiles determined by the gradient elasticity theory and a hybrid atomistic approach at the moment of nano-panels fracture revealed a very good mutual agreement.

Analytic evaluation of first-order properties within the mean-field variant of spin-free exact two-component theory.

We present a scheme for the calculation of energies and analytic energy gradients within spin-free exact two-component (SFX2C) theory in its mean-field variant, which we refer to as SFX2C-mf. In the presented scheme, the Foldy-Wouthuysen transformation is carried out after the spin-free four-component Hartree-Fock treatment such that in electron-correlated calculations only the non-mean-field part of the two-electron interactions is handled in an untransformed manner. The formulation of analytic gradients requires some adjustments in comparison with the nonrelativistic case, i.e., the additional solution of the spin-free Dirac Coulomb coupled-perturbed Hartee-Fock equations together with a simplified treatment of orbital relaxation at the SFX2C-mf level. The improved accuracy of SFX2C-mf in comparison with SFX2C-1e is demonstrated in the calculation of energies, dipole moments, and electric-field gradients for the hydrogen halides HX, X = F-At. It is shown that the main contribution to the improvement stems from the elimination of the error at the Hartree-Fock (HF) level; however, the corresponding correlation contribution is also improved such that SFX2C-mf can be considered a suitable scheme for the treatment of heavy-element compounds for which the error of SFX2C-1e is rather substantial.

Single-State Single-Reference and Multistate Multireference Zeroth-Order Hamiltonians in MS-CASPT2 and Conical Intersections.

Multistate complete active space second-order perturbation theory (MS-CASPT2) is one of the most successful quantum chemical methods for both static and dynamical correlations in photochemistry. In the literature, there are two definitions of the zeroth-order Hamiltonian depending on the form of the one-electron operator: multistate multireference MS-CASPT2 (MS-MR-MS-CASPT2) and single-state single-reference MS-CASPT2 (SS-SR-MS-CASPT2). Here, we implement an analytical gradient and derivative coupling for SS-SR-MS-CASPT2 and test SS-SR-MS-CASPT2 against MS-MR-(X)MS-CASPT2 in optimizing the molecular geometry critical points [the most stable geometries and minimum energy conical intersections (MECIs)] of a rhodopsin protein chromophore model (PSB3) and a green fluorescent protein chromophore model. In both cases, the MECIs in MS-MR-XMS-CASPT2 tend to have bridge hydrogen bonds that are more out-of-plane than those in SS-SR-MS-CASPT2, and for PSB3, the topology of potential energy surfaces (PESs) near the conical intersections is different. This result implies that caution is needed in analyzing the simulation results with different zeroth-order Hamiltonians and state-averaging schemes. The MS-MR-XMS-CASPT2 theory yields smooth PESs near the MECIs in all the tested cases, while the SS-SR-MS-CASPT2 theory does not. Therefore, we recommend using MS-MR-XMS-CASPT2 in conical intersection simulations, which require smooth PESs.

Analytical Energy Gradients for the Cluster-in-Molecule MP2 Method and Its Application to Geometry Optimizations of Large Systems.

An efficient analytical energy gradient algorithm for the cluster-in-molecule (CIM) second order Møller-Plesset perturbation theory (MP2) method is presented. In our algorithm, the gradient contributions from the nonseparable term of the two-body density matrix on a given atom is extracted from calculations on a cluster constructed for this atom. The other terms in the CIM-MP2 energy gradient expression are evaluated by constructing the density matrices of the whole system with the contributions from all clusters constructed. For basis sets with diffuse functions, tight CIM parameters are necessary to obtain accurate gradients. Benchmark calculations show that the CIM-MP2 method can accurately reproduce the conventional MP2 gradients and geometries for larger systems. The optimized structure of a 174-atom oligopeptide using the CIM-MP2 method with the cc-pVDZ basis set is in good agreement with the corresponding crystal structure. The present CIM-MP2 gradient program can be used for optimizing the geometries of large systems with hundreds of atoms on ordinary workstations.

Efficient implementations of analytic energy gradient for mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT).

Analytic energy gradients of individual singlet and triplet states with respect to nuclear coordinates are derived and implemented for the collinear mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT), which eliminates the problematic spin-contamination of SF-TDDFT. Dimensional-transformation matrices for the singlet and triplet response spaces are introduced, simplifying the subsequent derivations. These matrices enable the general forms of MRSF-TDDFT equations to be similar to those of SF-TDDFT, suggesting that the computational overhead of singlet or triplet states for MRSF-TDDFT is nearly identical to that of SF-TDDFT. In test calculations, the new MRSF-TDDFT yields quite different optimized structures and energies as compared to SF-TDDFT. These differences turned out to mainly come from the spin-contamination of SF-TDDFT, which are largely cured by MRSF-TDDFT. In addition, it was demonstrated that the clear separation of singlet states from triplets dramatically simplifies the location of minimum energy conical intersection. As a result, it is clear that the MRSF-TDDFT has advantages over SF-TDDFT in terms of both accuracy and practicality. Therefore, it can be a preferred method, which is readily applied to other "black-box" type applications, such as the minimum-energy optimization, reaction path following, and molecular dynamics simulations.

High-precision determination of Lorentz-symmetry-violating parameters in Ca+

We present high-precision calculations of local Lorentz invariance (LLI) violating parameters of the $3d{\phantom{\rule{0.16em}{0ex}}}^{2}{D}_{3/2;5/2}$ states in ${\mathrm{Ca}}^{+}$. We have employed three variants of the relativistic coupled-cluster (RCC) theory to determine these coefficients by gradually including electron correlation effects through the single, double, and triple excitation determinants from the Dirac-Hartree-Fock wave function. A precise estimate of the energy shift due to LLI violation depends on accurate evaluations of the expectation values of the square of the momentum operator ($\ensuremath{\langle}{p}^{2}\ensuremath{\rangle}$) and a second rank tensor ($\ensuremath{\langle}{T}_{0}^{(2)}\ensuremath{\rangle}$). It is found that the $\ensuremath{\langle}{T}_{0}^{(2)}\ensuremath{\rangle}$ values converge smoothly with the systematic inclusion of higher-order correlation effects in an expectation value evaluation approach, however, that is not the case for $\ensuremath{\langle}{p}^{2}\ensuremath{\rangle}$. Similar trends were also observed in the finite-field approach. To circumvent these problems, we determine $\ensuremath{\langle}{p}^{2}\ensuremath{\rangle}$ values very precisely by developing and applying an analytic gradient approach in the RCC framework. Corrections due to the Breit and quantum electrodynamics interactions are also estimated. Further, these calculations are validated by evaluating the energies and the quadrupole moments of the above states at different levels of approximations.

Analytic nuclear gradients of the algebraic-diagrammatic construction scheme for the polarization propagator up to third order of perturbation theory.

Analytic gradient expressions for the algebraic diagrammatic construction (ADC) scheme for the polarization propagator up to third order are derived using a Lagrangian approach. An implementation within the Q-CHEM electronic structure package for excited-state nuclear gradients of the ADC(2), ADC(2)-x, and ADC(3) models based on restricted and unrestricted Hartree-Fock references is presented. Details of the implementation and the applicability of the newly derived gradients for geometry optimizations and the quality of the resulting structures are discussed.

Excited-State Potential Energy Surfaces, Conical Intersections, and Analytical Gradients from Ground-State Density Functional Theory.

Kohn-Sham density functional theory (KS-DFT) has been a well-established theoretical foundation for ground-state electronic structure and has achieved great success in practical calculations. Recently, utilizing the eigenvalues from KS or generalized KS (GKS) calculations as an approximation to the quasiparticle energies, our group demonstrated a method to calculate the excitation energies from (G)KS calculation on the ground-state ( N - 1)-electron system. This method is now called QE-DFT (quasiparticle energies from DFT). In this work, we extend this QE-DFT method to describe excited-state potential energy surfaces (PESs), conical intersections, and the analytical gradients of excited-state PESs. The analytical gradients were applied to perform geometry optimization for excited states. In conjunction with several commonly used density functional approximations, QE-DFT can yield PESs in the vicinity of the equilibrium structure with accuracy similar to that from time-dependent DFT (TD-DFT). Furthermore, it describes conical intersection well, in contrast to TD-DFT. Good results for geometry optimization, especially bond length, of low-lying excitations for 14 small molecules are presented. The capability of describing excited-state PESs, conical intersections, and analytical gradients from QE-DFT and its efficiency based on just ground-state DFT calculations should be of great interest for describing photochemical and photophysical processes in complex systems.

Linear chains of hydrogen molecules under pressure: An extreme-pressure continuum model study

New analytical gradients of the electronic energy of a confined molecular system within the extreme-pressure continuum model are presented and applied to the study of the equilibrium geometries of linear chains of hydrogen molecules nH2 under pressures. The decrease in inter- and intramolecular H-H distances with the increase in the pressure has been studied up to 80 GPa. We have also shown that the compression of the bond-lengths can be interpreted in terms of the effect of the confining potential of the electron density of the molecular systems.

Multithreaded parallelization of the energy and analytic gradient in the fragment molecular orbital method

AbstractThe fragment molecular orbital method in GAMESS is parallelized in a multithreaded OpenMP implementation combined with the MPI version of the two‐level generalized distributed data interface. The energy and analytic gradient in gas phase and the polarizable continuum model of solvation are parallelized in this hybrid three‐level scheme, achieving a large memory footprint reduction and a high parallel efficiency on Intel Xeon Phi processors. The parallel efficiency is demonstrated on the Stampede2 and Theta supercomputers using up to 2048 nodes (262 144 threads).

Analytical gradient for the domain-based local pair natural orbital second order Møller-Plesset perturbation theory method (DLPNO-MP2).

Building upon our previously published work [P. Pinski and F. Neese, J. Chem. Phys. 148, 031101 (2018)], we derive the formally complete analytical gradient for the domain-based local pair natural orbital second order Møller-Plesset (MP2) perturbation theory method. Extensive testing of geometry optimizations shows that the deviations from resolution of the identity-based MP2 structures are small. Covalent bond lengths are reproduced to within 0.1 pm, whereas errors in interatomic distances between noncovalently interacting system parts do not exceed 1% with default truncation thresholds and 0.3% with tight thresholds. Moreover, we introduce a procedure to circumvent instabilities of the gradient caused by singular coupled-perturbed localization equations, as they occur for some symmetric systems with continuously degenerate localized orbitals. The largest system for which a geometry optimization was completed is a host-guest complex with over 200 atoms and more than 4000 basis functions (triple-zeta basis). The most demanding single-point gradient calculation was performed for the small protein crambin containing 644 atoms and over 12 000 basis functions.

Analytical energy gradients for ionized states using equation-of-motion coupled-cluster theory with spin-orbit coupling.

Spin-orbit coupling (SOC) may have a significant effect on the structure and harmonic frequencies of particularly heavy p-block element compounds. However, reports on analytical energy gradients with SOC are scarce, especially for excited states. In this work, we implemented analytical energy gradients for ionized states using the equation-of-motion coupled-cluster (CC) theory at the CC singles and doubles level (EOM-IP-CCSD) with SOC. Effects of SOC on structure and harmonic frequencies as well as properties for both the ground and some excited states of open-shell compounds with one unpaired electron can be investigated efficiently with the present implementation. A closed-shell reference is required in the calculations, and SOC is included in post-Hartree-Fock treatment. Relativistic effective core potentials are employed in dealing with both scalar relativistic effects and SOC, and we treat perturbations that are even under time reversal in this work. Both time-reversal symmetry and double point group symmetry for D2h * and its subgroups are exploited in the implementation. The method is applicable to states which can be reached by removing one electron from a closed-shell reference state. The results of some open-shell cations indicate the importance of SOC on structures and harmonic frequencies of heavy element compounds.

The Fragment Molecular Orbital Method Based on Long-Range Corrected Density-Functional Tight-Binding.

The presently available linear scaling approaches to density-functional tight-binding (DFTB) based on the fragment molecular orbital (FMO) method are severely impacted by the problem of artificial charge transfer due to the self-interaction error (SIE), which hampers the simulation of zwitterionic systems such as biopolymers or ionic liquids. Here we report an extension of FMO-DFTB where we included a long-range corrected (LC) functional designed to mitigate the DFTB SIE, called the FMO-LC-DFTB method, resulting in a robust method which succeeds in simulating zwitterionic systems. Both energy and analytic gradient are developed for the gas phase and the polarizable continuum model of solvation. The scaling of FMO-LC-DFTB with system size N is shown to be almost linear, O( N1.13-1.28), and its numerical accuracy is established for a variety of representative systems including neutral and charged polypeptides. It is shown that pair interaction energies between fragments for two mini-proteins are in excellent agreement with results from long-range corrected density functional theory. The new method was employed in long time scale (1 ns) molecular dynamics simulations of the tryptophan cage protein (PDB: 1L2Y ) in the gas phase for four different protonation states and in stochastic global minimum structure searches for 1-ethyl-3-methylimidazolium nitrate ionic liquid clusters containing up to 2300 atoms.