Long‐range interactions in embedded ionic cluster calculations

The effect of long‐range Coulomb interactions on bulk properties is studied for the ionic solids NaBr and NaCl. The embedded cluster approach in the framework of density functional theory is employed. The Madelung potential is calculated with the Evjen cube summation method. To explore the effects of the long‐range interactions on the electron densities and the Madelung constant, the Evjen cube size is varied from 310 to 19650 point charges for 33 atom clusters. To study the size effect of the quantum region, all‐electron clusters with 33 to 87 atoms, embedded in Evjen cubes of 6859 point charges, are investigated. The results show that for the 81 and 87 atom clusters the Madelung potential is constant from the center up to the second neighbor shell. For the same atoms, in all clusters, the electron density at the nuclei has nearly the same value. The largest difference found for the positive ions was 0.54%, and for the negative ions, 0.14%. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Scaling relations,virial theorem,and energy densities for long‐range and short‐range density functionals

Decomposition of the Coulomb electron–electron interaction into a long‐range and a short‐range part is described within the framework of density functional theory, deriving some scaling relations and the corresponding virial theorem. We study the behavior of the local density approximation in the high‐density limit for the long‐range and the short‐range functionals by carrying out a detailed analysis of the correlation energy of a uniform electron gas interacting via a long‐range‐only electron–electron repulsion. Possible definitions of exchange and correlation energy densities are discussed and clarified with some examples. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

High‐temperature superconductivity and long‐range order in strongly correlated electronic systems

A selective review of the question of how repulsive electron correlations might give rise to off‐diagonal long‐range order (ODLRO) in high‐temperature superconductors is presented. The article makes detailed explanations of the relevance to superconductivity of reduced electronic density matrices and how these can be used to understand whether ODLRO might arise from Coulombic repulsions in strongly correlated electronic systems. Time‐reversed electron pairs on alternant Cuprate and the iron‐based pnictide and chalcogenide lattices may have a weak long‐range attractive tail and much stronger short‐range repulsive Coulomb interaction. The long‐range attractive tail may find its origin in one of the many suggested proposals for high‐T_c superconductivity and thus has an uncertain origin. A phenomenological Hamiltonian is invoked whose model parameters are obtained by fitting to experimental data. A detailed summary is given of the arguments that such interacting electrons can cooperate to produce a superconducting state in which time‐reversed pairs of electrons effectively avoid the repulsive hard‐core of the Coulomb interaction but reside on average in the attractive well of the long‐range potential. Thus, the pairing of electrons itself provides an enhanced screening mechanism. The alternant lattice structure is the key to achieving robust high‐temperature superconductivity with dx²‐y² or sign alternating s‐wave or s± condensate symmetries in cuprates and iron‐based compounds. Some attention is also given to the question first raised by Leggett as to where the Coulombic energy is saved in the superconducting transition in cuprates. A mean‐field‐type model in which the condensate density serves as an order parameter is discussed. Many of the observed trends in the thermal properties of cuprate superconductors are reproduced giving strong support for the proposed model for high‐temperature superconductivity in such strongly correlated electronic systems. © 2015 Wiley Periodicals, Inc.     

Toward the complete range separation of non‐hybrid exchange–correlation functional

In this study, we use a very simple scheme to achieve range separation of a total exchange–correlation functional. We have utilized this methodology to combine a short‐range pure density functional theory (DFT) functional with a corresponding long‐range pure DFT, leading to a “Range‐separated eXchange–Correlation” (RXC) scheme. By examining the performance of a range of standard exchange–correlation functionals for prototypical short‐ and long‐range properties, we have chosen B‐LYP as the short‐range functional and PBE‐B95 as the long‐range counterpart. The results of our testing using a more diverse range of data sets show that, for properties that we deem to be short‐range in nature, the performance of this prescribed RXC‐DFT protocol does resemble that of B‐LYP in most cases, and vice versa. Thus, this RXC‐DFT protocol already provides meaningful numerical results. Furthermore, we envisage that the general RXC scheme can be easily implemented in computational chemistry software packages. This study paves a way for further refinement of such a range‐separation technique for the development of better performing DFT procedures. © 2015 Wiley Periodicals, Inc.     

Carbondioxide rare‐gas systems: Sensitivity of basis sets and double‐hybrid density functionals

This study emphasizes on the performance of six newly developed double‐hybrid density functionals (DHDF) in explaining the potential energy curves of different carbondioxide rare‐gas systems. The basis set sensitivity has also been explored with the use of three basis sets. Our results suggest that for lighter He/Ne‐CO₂ complexes, proper choice of DHDF and basis set lead to results those matches exactly with earlier calculations and also with the experiment. On the other hand, for heavier Ar/Kr‐CO₂ complexes although the equilibrium separation distance matches exactly with earlier observations, the interaction energy values lie far apart. The overall investigation emphasizes on the fact that one has to tune the methods and basis sets properly to achieve good and satisfactory results. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Low‐memory iterative density fitting

A new low‐memory modification of the density fitting approximation based on a combination of a continuous fast multipole method (CFMM) and a preconditioned conjugate gradient solver is presented. Iterative conjugate gradient solver uses preconditioners formed from blocks of the Coulomb metric matrix that decrease the number of iterations needed for convergence by up to one order of magnitude. The matrix‐vector products needed within the iterative algorithm are calculated using CFMM, which evaluates them with the linear scaling memory requirements only. Compared with the standard density fitting implementation, up to 15‐fold reduction of the memory requirements is achieved for the most efficient preconditioner at a cost of only 25% increase in computational time. The potential of the method is demonstrated by performing density functional theory calculations for zeolite fragment with 2592 atoms and 121,248 auxiliary basis functions on a single 12‐core CPU workstation. © 2015 Wiley Periodicals, Inc.     

Density functional theory for molecular and periodic systems using density fitting and continuous fast multipole method: Analytical gradients

A full implementation of analytical energy gradients for molecular and 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. Its key component is a combination of density fitting (DF) approximation and continuous fast multipole method (CFMM) that allows for an efficient calculation of the Coulomb energy gradient. For exchange‐correlation part the hierarchical numerical integration scheme (Burow and Sierka, Journal of Chemical Theory and Computation 2011, 7, 3097) is extended to energy gradients. Computational efficiency and asymptotic O(N) scaling behavior of the implementation is demonstrated for various molecular and periodic model systems, with the largest unit cell of hematite containing 640 atoms and 19,072 basis functions. The overall computational effort of energy gradient is comparable to that of the Kohn–Sham matrix formation. © 2016 Wiley Periodicals, Inc.     

Accelerating Kohn‐Sham response theory using density fitting and the auxiliary‐density‐matrix method

An extension of the formulation of the atomic‐orbital‐based response theory of Larsen et al., JCP 113, 8909 (2000) is presented. This new framework has been implemented in LSDalton and allows for the use of Kohn‐Sham density‐functional theory with approximate treatment of the Coulomb and Exchange contributions to the response equations via the popular resolution‐of‐the‐identity approximation as well as the auxiliary‐density matrix method (ADMM). We present benchmark calculations of ground‐state energies as well as the linear and quadratic response properties: vertical excitation energies, polarizabilities, and hyperpolarizabilities. The quality of these approximations in a range of basis sets is assessed against reference calculations in a large aug‐pcseg‐4 basis. Our results confirm that density fitting of the Coulomb contribution can be used without hesitation for all the studied properties. The ADMM treatment of exchange is shown to yield high accuracy for ground‐state and excitation energies, whereas for polarizabilities and hyperpolarizabilities the performance gain comes at a cost of accuracy. Excitation energies of a tetrameric model consisting of units of the P700 special pigment of photosystem I have been studied to demonstrate the applicability of the code for a large system.     

Long‐range heteronuclear coupling constants in 2,6‐disubstituted purine derivatives

Four‐ and five‐bond heteronuclear J‐couplings between the hydrogen H‐8 and carbons C‐6 and C‐2 in a series of 7‐ and 9‐benzyl substituted purine derivaties with variuous substituents in positions 2 and 6 were studied by coupled ¹³ C NMR and H,C‐HMBC experiments and by DFT calculations. We have found that for some of the derivatives, the five‐bond coupling H8‐C2 is higher than the four‐bond H8‐C6 coupling, which is also evidenced by a stronger crosspeak in the HMBC. This finding contradicts the generally accepted opinion that only strong three‐bond crosspeaks and one weak four‐bond H8‐C6 crosspeak can be observed in the HMBC spectra of purine derivatives. The misinterpretation of HMBC spectra may lead to an incorrect determination of the purine derivatives' structure. Copyright © 2012 John Wiley & Sons, Ltd.     

Long‐range 19F–15N heteronuclear shift correlation: examination of J‐modulations associated with broad range accordion excitation

The first demonstrated example of ¹⁹F–¹⁵N long‐range heteronuclear shift correlation spectroscopy at natural abundance is reported. Because of the very large variation in the size of ²J(N,F) vs ³J(N,F) long‐range heteronuclear couplings, the utilization of one of the new accordion‐optimized long‐range heteronuclear shift correlations experiments is essential if all possible correlations are to be observed in a single experiment. A modified IMPEACH‐MBC pulse sequence was used in conjunction with an optimization range from 4 to 50 Hz to demonstrate the technique using a mixture of 2‐ and 3‐fluoropyridine, which had ²J(N,F) and ³J(N,F) long‐range couplings of ?52 and 3.6 Hz, respectively. Because of the size of the ²J(N,F) long‐range coupling constant, a J‐modulation of the long‐range correlation response is observed in the spectrum resulting in a ‘doublet’ in F₁ due to amplitude modulation. The size of the ‘doublet’ is shown to be a function of the parameter selection (t_1max,T_max,T_min and spectral width in F₁). This behavior is similar to F₁ ‘skew’ associated with long‐range correlation responses in ACCORD‐HMBC spectra which has been analyzed in detail previously. Copyright © 2002 John Wiley & Sons, Ltd.     

Electron density distribution in potassium bis -(carbonato)cuprate(II)

The electron density distribution in potassiumbis-(carbonato)cuprate(II) has been analyzed using x-ray diffraction data from an earlier structure determination. While the copper-ligand geometry is close to square planar the deformation density near the metal is strongly asymmetric. There are local maxima near the copper atom along the line of the Cu-K vectors. These resemble features found in corresponding regions in normal length metal-metal bonds. The observation is consistent with the long range nature of the Coulomb potential associated with the potassium ion.     

Compatibility of correlation‐consistent basis sets with a hybrid Hartree–Fock/density functional method

The present study examines the feasibility of combining the correlation‐consistent basis sets developed by Dunning and coworkers with the hybrid Hartree–Fock/density functional method B3LYP. Furthermore, extrapolation to the complete basis set (CBS) limit minimizes errors due to the presence of an incomplete basis set and can act as a rigorous test of the limitations of the B3LYP method. Equilibrium geometries, energies, and harmonic vibrational frequencies were determined for a series of well‐studied, yet computationally challenging, small inorganics and their respective ions. The results were then extrapolated to the CBS limit, where applicable, and compared to experiment. It was found that a union between the hybrid Hartree–Fock/density functional B3LYP method and Dunning's augmented correlation‐consistent basis sets gave results that were comparable to molecular orbital methods that explicitly account for electron correlation. Furthermore, the minimum basis set necessary to attain reasonable results for the systems studied was aug‐cc‐pVTZ. Upgrading to the aug‐cc‐pVQZ level and subsequent extrapolation to the CBS limit further improved the overall agreement with the experiment. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 207–216, 1999     

High‐spin behavior of molecular crystals and extended π systems

The qualitative rules for the existence of high‐spin ground states in extended systems and molecular crystals are examined here on a firmer theoretical footing. Extended systems have been categorized into three groups, namely, type I, type II, and type III, depending on the type of bonding interactions. The general form of the spin Hamiltonian operators have been written down. The active spaces have been restricted to the minimum size for each of these three types of spin systems. The zeroth‐order state vectors and the Hartree–Fock ground‐state energies have been identified for unit species of each type. The extended system Hamiltonian operators are further truncated in such a way that only the nearest‐neighbor interactions are retained. Expressions have been derived for the energy gap from a molecular orbital approach. The relatively small effects of electron correlation on the energy gaps have been estimated for the type I systems, which belong to the systems of solid‐state physics. In particular, it has been shown that for the type I systems the singlet–triplet gap, and hence the ferromagnetic coupling constant, primarily depends upon the difference of one‐electron kinetic energies and not on the two‐electron exchange integrals. This result agrees with the concept of kinetic exchange that was introduced in the context of a resonating valence‐bond formalism. Type II systems are exemplified by extended systems that can be prepared from conjugated molecules while organic molecular crystals form examples of type III species. For these systems, however, the Coulomb exchange interaction has been shown to dominate the energy gap. A quick review of the Heisenberg spin Hamiltonian for the H₂ molecule is sufficient to point out that the sign of the calculated ferromagnetic coupling constant depends on the method of calculation, the nature of the basis set, and the bond length. This is amply supported by ab initio calculations on this species. Numerical data have also been obtained from computations on m‐phenylene‐coupled nitroxy radicals and stacks of α‐nitronyl nitroxide, but these calculations have been based on a semiempirical quantum chemical methodology (INDO) since some of the species involved are exceedingly large. Computed energy gaps are in good agreement with experimental and other theoretical (AM1, PM3) results. Nevertheless, for the dimer, trimer, tetramer, and pentamer of the type II specimen, the important π orbitals are far from being degenerate. The quantitative results clearly deviate from the criterion of degeneracy that was suggested from qualitative theories for the existence of a high‐spin ground state. Therefore, the criteria for the existence of high spins have been reformulated in terms of the monomer orbitals. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 308–324, 2000     

Grid‐based density functional calculations of many‐electron systems

Exploratory variational pseudopotential density functional calculations are performed for the electronic properties of many‐electron systems in the 3D cartesian coordinate grid (CCG). The atom‐centered localized gaussian basis set, electronic density, and the two‐body potentials are set up in the 3D cubic box. The classical Hartree potential is calculated accurately and efficiently through a Fourier convolution technique. As a first step, simple local density functionals of homogeneous electron gas are used for the exchange‐correlation potential, while Hay‐Wadt‐type effective core potentials are employed to eliminate the core electrons. No auxiliary basis set is invoked. Preliminary illustrative calculations on total energies, individual energy components, eigenvalues, potential energy curves, ionization energies, and atomization energies of a set of 12 molecules show excellent agreement with the corresponding reference values of atom‐centered grid as well as the grid‐free calculation. Results for three atoms are also given. Combination of CCG and the convolution procedure used for classical Coulomb potential can provide reasonably accurate and reliable results for many‐electron systems. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Auxiliary basis sets for density‐fitting second‐order Møller–Plesset perturbation theory: Weighted core‐valence correlation consistent basis sets for the 4d elements Y–Pd

Auxiliary basis sets (ABS) specifically matched to the cc‐pwCVnZ‐PP and aug‐cc‐pwCVnZ‐PP orbital basis sets (OBS) have been developed and optimized for the 4d elements Y‐Pd at the second‐order Møller‐Plesset perturbation theory level. Calculation of the core‐valence electron correlation energies for small to medium sized transition metal complexes demonstrates that the error due to the use of these new sets in density fitting is three to four orders of magnitude smaller than that due to the OBS incompleteness, and hence is considered negligible. Utilizing the ABSs in the resolution‐of‐the‐identity component of explicitly correlated calculations is also investigated, where it is shown that i‐type functions are important to produce well‐controlled errors in both integrals and correlation energy. Benchmarking at the explicitly correlated coupled cluster with single, double, and perturbative triple excitations level indicates impressive convergence with respect to basis set size for the spectroscopic constants of 4d monofluorides; explicitly correlated double‐ζ calculations produce results close to conventional quadruple‐ζ, and triple‐ζ is within chemical accuracy of the complete basis set limit. © 2013 Wiley Periodicals, Inc.     

Reaction energetics on long‐range corrected density functional theory: Diels–Alder reactions

The possibility of quantitative reaction analysis on the orbital energies of long‐range corrected density functional theory (LC‐DFT) is presented. First, we calculated the Diels–Alder reaction enthalpies that have been poorly given by conventional functionals including B3LYP functional. As a result, it is found that the long‐range correction drastically improves the reaction enthalpies. The barrier height energies were also computed for these reactions. Consequently, we found that dispersion correlation correction is also crucial to give accurate barrier height energies. It is, therefore, concluded that both long‐range exchange interactions and dispersion correlations are essentially required in conventional functionals to investigate Diels–Alder reactions quantitatively. After confirming that LC‐DFT accurately reproduces the orbital energies of the reactant and product molecules of the Diels–Alder reactions, the global hardness responses, the halves of highest occupied molecular orbital (HOMO)‐lowest unoccupied molecular orbital (LUMO) energy gaps, along the intrinsic reaction coordinates of two Diels–Alder reactions were computed. We noticed that LC‐DFT results satisfy the maximum hardness rule for overall reaction paths while conventional functionals violate this rule on the reaction pathways. Furthermore, our results also show that the HOMO‐LUMO gap variations are close to the reaction enthalpies for these Diels–Alder reactions. Based on these results, we foresee quantitative reaction analysis on the orbital energies. © 2012 Wiley Periodicals, Inc.     

On the basis‐set dependence of local and integrated electron density properties: Application of a new computer program for quantum‐chemical density analysis

A new computer program for post‐processing analysis of quantum‐chemical electron densities is described. The code can work with Slater‐ and Gaussian‐type basis functions of arbitrary angular momentum. It has been applied to explore the basis‐set dependence of the electron density and its Laplacian in terms of local and integrated topological properties. Our analysis, including Gaussian/Slater basis sets up to sextuple/quadruple‐zeta order, shows that these properties considerably depend on the choice of type and number of primitives utilized in the wavefunction expansion. Basis sets with high angular momentum (l = 5 or l = 6) are necessary to achieve convergence for local properties of the density and the Laplacian. In agreement with previous studies, atomic charges defined within Bader's Quantum Theory of Atoms in Molecules appear to be much more basis‐set dependent than the Hirshfeld's stockholder charges. The former ones converge only at the quadruple‐zeta/higher level with Gaussian/Slater functions. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2009     

Assessment of density functionals with long‐range and/or empirical dispersion corrections for conformational energy calculations of peptides

Density functionals with long‐range and/or empirical dispersion corrections, including LC‐ωPBE, B97‐D, ωB97X‐D, M06‐2X, B2PLYP‐D, and mPW2PLYP‐D functionals, are assessed for their ability to describe the conformational preferences of Ac‐Ala‐NHMe (the alanine dipeptide) and Ac‐Pro‐NHMe (the proline dipeptide) in the gas phase and in water, which have been used as prototypes for amino acid residues of peptides. For both dipeptides, the mean absolute deviation (MAD) is estimated to be 0.22–0.40 kcal/mol in conformational energy and 2.0–3.2° in torsion angles ? and ψ using these functionals with the 6‐311++G(d,p) basis set against the reference values calculated at the MP2/aug‐cc‐pVTZ//MP2/aug‐cc‐pVDZ level of theory in the gas phase. The overall performance is obtained in the order B2PLYP‐D ≈ mPW2PLYP‐D > ωB97X‐D ≈ M06‐2X > MP2 > LC‐ωPBE > B3LYP with the 6–311++G(d,p) basis set. The SMD model at the M06‐2X/6‐31+G(d) level of theory well reproduced experimental hydration free energies of the model compounds for backbone and side chains of peptides with MADs of 0.47 and 4.3 kcal/mol for 20 neutral and 5 charged molecules, respectively. The B2PLYP‐D/6‐311++G(d,p)//SMD M06‐2X/6‐31+G(d) level of theory provides the populations of backbone and/or prolyl peptide bond for the alanine and proline dipeptides in water that are consistent with the observed values. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

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.