Coupled-perturbed Self-consistent Field Research Articles

Large-scale excited-state calculation using dynamical polarizability evaluated by divide-and-conquer based coupled cluster linear response method.

This study attempted to propose an efficient scheme at the coupled cluster linear response (CCLR) level to perform large-scale excited-state calculations of not only local excitations but also nonlocal ones such as charge transfers and transitions between delocalized orbitals. Although standard applications of fragmentation techniques to the excited-state calculations brought about the limitations that could only deal with local excitations, this study solved the problem by evaluating the excited states as the poles of dynamical polarizability. Because such an approach previously succeeded at the time-dependent density functional theory level [H. Nakai and T. Yoshikawa, J. Chem. Phys. 146, 124123 (2017)], this study was considered as an extension to the CCLR level. To evaluate the dynamical polarizability at the CCLR level, we revisited three equivalent formulas, namely, coupled-perturbed self-consistent field (CPSCF), random phase approximation (RPA), and Green's function (GF). We further extended these formulas to the linear-scaling methods based on the divide-and-conquer (DC) technique. We implemented the CCLR with singles and doubles (CCSDLR) program for the six schemes, i.e., the standard and DC-type CPSCF, RPA, and GF. Illustrative applications of the present methods demonstrated the accuracy and efficiency. Although the standard three treatments could exactly reproduced the conventional frequency-domain CCSDLR results, their computational costs were commonly higher than that of the conventional ones due to large amount of computations for individual frequencies of the external electric field. The DC-type treatments, which approximately reproduced the conventional results, could achieve quasilinear scaling computational costs. Among them, DC-GF was found to exhibit the best performance.

Numerical Nuclear Second Derivatives on a Computing Grid: Enabling and Accelerating Frequency Calculations on Complex Molecular Systems.

The computation of nuclear second derivatives of energy, or the nuclear Hessian, is an essential routine in quantum chemical investigations of ground and transition states, thermodynamic calculations, and molecular vibrations. Analytic nuclear Hessian computations require the resolution of costly coupled-perturbed self-consistent field (CP-SCF) equations, while numerical differentiation of analytic first derivatives has an unfavorable 6 N ( N = number of atoms) prefactor. Herein, we present a new method in which grid computing is used to accelerate and/or enable the evaluation of the nuclear Hessian via numerical differentiation: NUMFREQ@Grid. Nuclear Hessians were successfully evaluated by NUMFREQ@Grid at the DFT level as well as using RIJCOSX-ZORA-MP2 or RIJCOSX-ZORA-B2PLYP for a set of linear polyacenes with systematically increasing size. For the larger members of this group, NUMFREQ@Grid was found to outperform the wall clock time of analytic Hessian evaluation; at the MP2 or B2LYP levels, these Hessians cannot even be evaluated analytically. We also evaluated a 156-atom catalytically relevant open-shell transition metal complex and found that NUMFREQ@Grid is faster (7.7 times shorter wall clock time) and less demanding (4.4 times less memory requirement) than an analytic Hessian. Capitalizing on the capabilities of parallel grid computing, NUMFREQ@Grid can outperform analytic methods in terms of wall time, memory requirements, and treatable system size. The NUMFREQ@Grid method presented herein demonstrates how grid computing can be used to facilitate embarrassingly parallel computational procedures and is a pioneer for future implementations.

Development of an excited-state calculation method for large systems using dynamical polarizability: A divide-and-conquer approach at the time-dependent density functional level.

In this study, we developed an excited-state calculation method for large systems using dynamical polarizabilities at the time-dependent density functional theory level. Three equivalent theories, namely, coupled-perturbed self-consistent field (CPSCF), random phase approximation (RPA), and Green function (GF), were extended to linear-scaling methods using the divide-and-conquer (DC) technique. The implementations of the standard and DC-based CPSCF, RPA, and GF methods are described. Numerical applications of these methods to polyene chains, single-wall carbon nanotubes, and water clusters confirmed the accuracy and efficiency of the DC-based methods, especially DC-GF.

Approximate Force Constants from Uncoupled Self-Consistent Field Perturbation Theory Using Nonhybrid Density Functional Theory.

Second derivatives of the molecular energy with respect to the nuclear coordinates (the nuclear Hessian or force constant matrix) are important for predicting infrared and Raman spectra, for calculating thermodynamic properties, for characterizing stationary states, and for guiding geometry optimization. However, their calculation for larger systems scales with molecular size one power higher than the calculation of the energy and the forces. The step responsible for the steep scaling of the nuclear Hessian is the coupled-perturbed self-consistent field (CP-SCF) iteration. This is omitted in the uncoupled SCF (UC-SCF) approximation. We have found that, though UC-SCF performs rather poorly at the Hartree-Fock and hybrid DFT levels, its performance for "pure" (non-hybrid) DFT is remarkably good. This is valid also for imaginary frequencies that characterize transition states. UC-SCF vibrational frequencies and normal modes are compared with coupled calculations for various exchange-correlation functionals including Hartree-Fock, and with basis sets ranging from simple to large for a variety of organic and some organometallic molecules. Their unexpectedly good performance makes them good candidates for calculating thermodynamic properties and for guiding difficult geometry optimizations, including the determination of transition states.

Efficient implementation of the analytic second derivatives of Hartree–Fock and hybrid DFT energies: a detailed analysis of different approximations

In this paper, various implementations of the analytic Hartree–Fock and hybrid density functional energy second derivatives are studied. An approximation-free four-centre implementation is presented, and its accuracy is rigorously analysed in terms of self-consistent field (SCF), coupled-perturbed SCF (CP-SCF) convergence and prescreening criteria. The CP-SCF residual norm convergence threshold turns out to be the most important of these. Final choices of convergence thresholds are made such that an accuracy of the vibrational frequencies of better than 5 cm−1 compared to the numerical noise-free results is obtained, even for the highly sensitive low frequencies (<100–200 cm−1). The effects of the choice of numerical grid for density functional exchange–correlation integrations are studied and various weight derivative schemes are analysed in detail. In the second step of the work, approximations are introduced in order to speed up the computation without compromising its accuracy. To this end, the accuracy and efficiency of the resolution of identity approximation for the Coulomb terms and the semi-numerical chain of spheres approximation to the exchange terms are carefully analysed. It is shown that the largest performance improvements are realised if either Hartree–Fock exchange is absent (pure density functionals) and otherwise, if the exchange terms in the CP-SCF step of the calculation are approximated by the COSX method in conjunction with a small integration grid. Default values for all the involved truncation parameters are suggested. For vancomycine (176 atoms and 3593 basis functions), the RIJCOSX Hessian calculation with the B3LYP functional and the def2-TZVP basis set takes ∼3 days using 16 Intel® Xeon® 2.60GHz processors with the COSX algorithm having a net parallelisation scaling of 11.9 which is at least ∼20 times faster than the calculation without the RIJCOSX approximation.

Coupled-Perturbed SCF Approach for Calculating Static Polarizabilities and Hyperpolarizabilities with Nonorthogonal Localized Molecular Orbitals.

Coupled-perturbed self-consistent-field (CPSCF) approach has been broadly used for polarizabilities and hyperpolarizabilities computation. To extend this application to large systems, we have reformulated the CPSCF equations with nonorthogonal localized molecular orbitals (NOLMOs). NOLMOs are the most localized representation of electronic degrees of freedom. Methods based on NOLMOs are potentially ideal for investigating large systems. In atomic orbital representation, with a static external electric field added, the wave function and SCF operator of unperturbed NOLMO-SCF wave function/orbitals are expanded to different orders of perturbations. We have derived the corresponding equations up to the third order, which are significantly different from those of a conventional CPSCF method because of the release of the orthogonal restrictions on MOs. The solution to these equations has been implemented. Several chemical systems are used to verify our method. This work represents the first step toward efficient calculations of molecular response and excitation properties with NOLMOs.

Linear-scaling symmetry-adapted perturbation theory with scaled dispersion

We present a linear-scaling symmetry-adapted perturbation theory (SAPT) method that is based on an atomic orbital (AO) formulation of zeroth-order SAPT (SAPT0). The non-dispersive terms are realized with linear-scaling cost using both the continuous fast multipole method (CFMM) and the linear exchange (LinK) approach for integral contractions as well as our efficient Laplace-based coupled-perturbed self-consistent field method (DL-CPSCF) for evaluating response densities. The reformulation of the dispersion term is based on our linear-scaling AO Møller-Plesset second-order perturbation theory (AO-MP2) method, that uses our recently introduced QQR-type screening [S. A. Maurer, D. S. Lambrecht, J. Kussmann, and C. Ochsenfeld, J. Chem. Phys. 138, 014101 (2013)] for preselecting numerically significant energy contributions. Similar to scaled opposite-spin MP2, we neglect the exchange-dispersion term in SAPT and introduce a scaling factor for the dispersion term, which compensates for the error and at the same time accounts for basis set incompleteness effects and intramonomer correlation. We show in extensive benchmark calculations that the new scaled-dispersion (sd-)SAPT0 approach provides reliable results for small and large interacting systems where the results with a small 6-31G** basis are roughly comparable to supermolecular MP2 calculations in a triple-zeta basis. The performance of our method is demonstrated with timings on cellulose fragments, DNA systems, and cutouts of a protein-ligand complex with up to 1100 atoms on a single computer core.

Efficient Calculation of QM/MM Frequencies with the Mobile Block Hessian.

The calculation of the analytical second derivative matrix (Hessian) is the bottleneck for vibrational analysis in QM/MM systems when an electrostatic embedding scheme is employed. Even with a small number of QM atoms in the system, the presence of MM atoms increases the computational cost dramatically: the long-range Coulomb interactions require that additional coupled perturbed self-consistent field (CPSCF) equations need to be solved for each MM atom displacement. This paper presents an extension to the Mobile Block Hessian (MBH) formalism for QM/MM calculations with blocks in the MM region and its implementation in a parallel version of the Q-Chem/CHARMM interface. MBH reduces both the CPU time and the memory requirements compared to the standard full Hessian QM/MM analysis, without the need to use a cutoff distance for the electrostatic interactions. Special attention is given to the treatment of link atoms which are usually present when the QM/MM border cuts through a covalent bond. Computational efficiency improvements are highlighted using a reduced chorismate mutase enzyme system, consisting of 24 QM atoms and 306 MM atoms, as a test example. In addition, the drug bortezomib, used for cancer treatment of myeloma, has been studied as a test case with multiple MBH block choices and both a QM and QM/MM description. The accuracy of the calculated Hessians is quantified by imposing Eckart constraints, which allows for the assessment of numerical errors in second derivative procedures. The results show that MBH within the QM/MM description not only is a computationally attractive method but also produces accurate results.

Efficient linear-scaling calculation of response properties: Density matrix-based Laplace-transformed coupled-perturbed self-consistent field theory

A density matrix-based Laplace reformulation of coupled-perturbed self-consistent field (CPSCF) theory is presented. It allows a direct, instead of iterative, solution for the integral-independent part of the density matrix-based CPSCF (D-CPSCF) equations [J. Kussmann and C. Ochsenfeld, J. Chem. Phys. 127, 054103 (2007)]. In this way, the matrix-multiplication overhead compared to molecular orbital-based solutions is reduced to a minimum, while at the same time, the linear-scaling behavior of D-CPSCF theory is preserved. The present Laplace-based equation solver is expected to be of general applicability.

Calculation of the zero-field splitting tensor on the basis of hybrid density functional and Hartree-Fock theory

The zero-field splitting (ZFS) (expressed in terms of the D tensor) is the leading spin-Hamiltonian parameter for systems with a ground state spin S>12. To first order in perturbation theory, the ZFS arises from the direct spin-spin dipole-dipole interaction. To second order, contributions arise from spin-orbit coupling (SOC). The latter contributions are difficult to treat since the SOC mixes states of different multiplicities. This is an aspect of dominant importance for the correct prediction of the D tensor. In this work, the theory of the D tensor is discussed from the point of view of analytic derivative theory. Starting from a general earlier perturbation treatment [F. Neese and E. I. Soloman, Inorg. Chem. 37, 6568 (1998)], straightforward response equations are derived that are readily transferred to the self-consistent field (SCF) Hartree-Fock (HF) or density functional theory (DFT) framework. The main additional effort in such calculations arises from the solution of nine sets of nonstandard coupled-perturbed SCF equations. These equations have been implemented together with the spin-orbit mean-field representation of the SOC operator and a mean-field treatment of the direct spin-spin interaction into the ORCA electronic structure program. A series of test calculations on diatomic molecules with accurately known zero-field splittings shows that the new approach corrects most of the shortcomings of previous DFT based methods and, on average, leads to predictions within 10% of the experimental values. The slope of the correlation line is essentially unity for the B3LYP and BLYP functionals compared to approximately 0.5 in previous treatments.

The analytical gradient of dual-basis resolution-of-the-identity second-order Møller–Plesset perturbation theory

In this work, we present the analytical gradient of dual-basis second-order Møller–Plesset perturbation theory within the resolution-of-the-identity approximation (DB-RI-MP2). Interestingly, analytical DB-RI-MP2 gradient theory involves significant changes to both the theory and computation of the coupled-perturbed self-consistent field equations (CPSCF). From a theoretical point of view, the number of orbital responses required in DB-RI-MP2 analytical gradient theory has been reduced to the product of the number of occupied and virtual orbitals determined by the rank of the small atomic orbital (AO) basis. From a computational point of view, the DB-CPSCF equations can be solved within this smaller space at a fraction of the computational cost. Additional computational savings can be obtained during the digestion of the four-centered AO integral derivatives and the efficient underlying DB-SCF procedure, which lead to a significant overall reduction in the computational cost necessary for treating molecular systems containing less than 100 atoms. Based on stringent chemical tests and a detailed computational timings analysis, it was found that DB-RI-MP2 reproduces molecular equilibrium structures with an accuracy that approximates RI-MP2 using the target AO basis but at a dramatically reduced cost, thereby enabling more routine use of large AO basis sets during geometry optimizations at the MP2 level of theory.

Linear-scaling method for calculating nuclear magnetic resonance chemical shifts using gauge-including atomic orbitals within Hartree-Fock and density-functional theory

Details of a new density matrix-based formulation for calculating nuclear magnetic resonance chemical shifts at both Hartree-Fock and density functional theory levels are presented. For systems with a nonvanishing highest occupied molecular orbital-lowest unoccupied molecular orbital gap, the method allows us to reduce the asymptotic scaling order of the computational effort from cubic to linear, so that molecular systems with 1000 and more atoms can be tackled with today's computers. The key feature is a reformulation of the coupled-perturbed self-consistent field (CPSCF) theory in terms of the one-particle density matrix (D-CPSCF), which avoids entirely the use of canonical MOs. By means of a direct solution for the required perturbed density matrices and the adaptation of linear-scaling integral contraction schemes, the overall scaling of the computational effort is reduced to linear. A particular focus of our formulation is to ensure numerical stability when sparse-algebra routines are used to obtain an overall linear-scaling behavior.

An efficient approach for self-consistent-field energy and energy second derivatives in the atomic-orbital basis

Based on self-consistent-field (SCF) perturbation theory, we recast the SCF and the coupled-perturbed SCF (CPSCF) equations for time-independent molecular properties into the atomic-orbital basis. The density matrix and the perturbed density matrix are obtained iteratively by solving linear equations. Only matrix multiplications and additions are required, and this approach can exploit sparse matrix multiplications and thereby offer the possibility of evaluating second-order properties in computational effort that scales linearly with system size. Convergence properties are similar to conventional molecular-orbital-based CPSCF procedures, in terms of the number of derivative Fock matrices that must be constructed. We also carefully address the issue of the numerical accuracy of the calculated second derivatives of the energy, in order to specify the minimum precision necessary in the CPSCF procedure. It is found that much looser tolerances for the perturbed density matrices are adequate when using an expression for the second derivatives that is correct through second order in the CPSCF error.

A reformulation of the coupled perturbed self-consistent field equations entirely within a local atomic orbital density matrix-based scheme

To exploit the exponential decay found in numerical studies for the density matrix and its derivative with respect to nuclear displacements, we reformulate the coupled perturbed self-consistent field (CPSCF) equations and a quadratically convergent SCF (QCSCF) method for Hartree-Fock and density functional theory within a local density matrix-based scheme. Our D-CPSCF (density matrix-based CPSCF) and D-QCSCF schemes open the way for exploiting sparsity and to achieve asymptotically linear scaling of computational complexity with molecular size ( M), in case of D-CPSCF for all O ( M) derivative densities. Furthermore, these methods are even for small molecules strongly competitive to conventional algorithms.