Utilization of deformations in molecular quantum chemistry and application to density functional theory

The aim of this article is to present in a way accessible to most quantum chemists a general mathematical method which consists in deforming wave functions and density functions (in the spirit of the local scaling transformation). This deformation method allows us to obtain several new results, including a characterization of the set of wave functions that have the same given density function (which gives a new insight on a result of G. Zumbach and K. Maschke, Phys. Rev. A 28 , 544 (1983)) and an N-representability result where symmetry is taken into account. We also propose new theoretical ways to generate approximations of the exact density functional and give a numerical example. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 221–231, 1998     

The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals

We present two new hybrid meta exchange- correlation functionals, called M06 and M06-2X. The M06 functional is parametrized including both transition metals and nonmetals, whereas the M06-2X functional is a high-nonlocality functional with double the amount of nonlocal exchange (2X), and it is parametrized only for nonmetals.The functionals, along with the previously published M06-L local functional and the M06-HF full-Hartree–Fock functionals, constitute the M06 suite of complementary functionals. We assess these four functionals by comparing their performance to that of 12 other functionals and Hartree–Fock theory for 403 energetic data in 29 diverse databases, including ten databases for thermochemistry, four databases for kinetics, eight databases for noncovalent interactions, three databases for transition metal bonding, one database for metal atom excitation energies, and three databases for molecular excitation energies. We also illustrate the performance of these 17 methods for three databases containing 40 bond lengths and for databases containing 38 vibrational frequencies and 15 vibrational zero point energies. We recommend the M06-2X functional for applications involving main-group thermochemistry, kinetics, noncovalent interactions, and electronic excitation energies to valence and Rydberg states. We recommend the M06 functional for application in organometallic and inorganometallic chemistry and for noncovalent interactions. Electronic Supplementary Material The online version of this article (doi:) contains supplementary material, which is available to authorized users. Contribution to the Mark S. Gordon 65th Birthday Festschrift Issue.     

Comparison of one‐parameter and linearly scaled one‐parameter double‐hybrid density functionals for noncovalent interactions

We have compared the performances of the one‐parameter and linearly scaled one‐parameter double‐hybrid density functionals (1DH‐DFs and LS1DH‐DFs) for noncovalent interactions. The only one parameter related to the Hartree–Fock (HF) exchange for each of the tested 1DH‐DFs and LS1DH‐DFs has been fitted with the well‐designed S66 database. The obtained DHDFs are dubbed as 1DH‐PBE‐NC, LS1DH‐PBE‐NC, 1DH‐TPSS‐NC, LS1DH‐TPSS‐NC, 1DH‐PWB95‐NC, and LS1DH‐PWB95‐NC, where “NC” denotes noncovalent interactions. With a specific combination of exchange and correlation functionals, the dependent parameters related to the nonlocal second‐order perturbative energies are nearly identical for the 1DH and LS1DH models. According to our benchmark computations against the S66, S22B, NCCE31, ADIM6, and L7 databases, we suggest that the 1DH‐PWB95‐NC and LS1DH‐PWB95‐NC functionals are much more suitable for evaluating noncovalent interaction energies. Unlike the versatile DHDFs with dispersion corrections for general purpose, our optimized 1DH‐DFs and LS1DH‐DFs only aim at noncovalent interactions. © 2016 Wiley Periodicals, Inc.     

Towards an order-N DFT method

One of the most important steps in a Kohn-Sham (KS) type density functional theory calculation is the construction of the matrix of the KS operator (the “Fock” matrix). It is desirable to develop an algorithm for this step that scales linearly with system size. We discuss attempts to achieve linear scaling for the calculation of the matrix elements of the exchange-correlation and Coulomb potentials within a particular implementation (the Amsterdam density functional, ADF, code) of the KS method. In the ADF scheme the matrix elements are completely determined by 3D numerical integration, the value of the potentials in each grid point being determined with the help of an auxiliary function representation of the electronic density. Nearly linear scaling for building the total Fock matrix is demonstrated for systems of intermediate size (in the order of 1000 atoms). For larger systems further development is desirable for the treatment of the Coulomb potential. Received: 30 March 1998 / Accepted: 6 July 1998 / Published online: 15 September 1998     

A Benchmark of Density Functional Approximations For Thermochemistry and Kinetics of Hydride Reductions of Cyclohexanones

The performance of density functionals and wavefunction methods for describing the thermodynamics and kinetics of hydride reductions of 2-substituted cyclohexanones has been evaluated for the first time. A variety of exchange correlation functionals ranging from generalized gradient approximations to double hybrids have been tested and their performance to describe the facial selectivity of hydride reductions of cyclohexanones has been carefully assessed relative to the CCSD(T) method. Among the tested methods, an approach in which single-point energy calculations using the double hybrid B2PLYP−D3 functional on ωB97X−D optimized geometries provides the most accurate transition state energies for these kinetically-controlled reactions. Moreover, the role of torsional strain, temperature, solvation, noncovalent interactions on the stereoselectivity of these reductions was elucidated. Our results indicate a prominent role of the substituent on the cis/trans ratios driven by the delicate interplay between torsional strain and dispersion interactions.     

Structure,epr parameters,and reactivity of organic free radicals from a density functional approach

Summary A recently introduced density functional incorporating gradient corrections and some Hartree-Fock exchange has been used to study the structures, properties, and reactivity of representative organic free radicals. A general theoretical model has been introduced, in which standardized grid, functional, and orbital basis set are used to compute geometrical parameters, vibrational frequencies, and one-electron properties. The results are compared with available experimental data from diatomic to polyatomic radicals. All the geometric and electronic parameters compare favourably with available experimental data and with the results of refined post Hartree-Fock computations. Also the thermodynamics and kinetics of a representative unimolecular reaction (isomerization of formaldehyde radical cation) are well reproduced. These findings together with the very favourable scaling of the computations with the number of electrons suggest that the density functional approach is a promising theoretical tool for the study of relationships between structure and properties of large free radicals.     

Exchange-correlation functional with broad accuracy for metallic and nonmetallic compounds, kinetics, and noncovalent interactions

By incorporating kinetic-energy density in a balanced way in the exchange and correlational functionals and removing self-correlation effects, we have designed a density functional that is broadly applicable to organometallic, inorganometallic, and nonmetallic bonding, thermochemistry, thermochemical kinetics, and noncovalent interactions as well as satisfying the uniform electron gas limit. The average error is reduced by a factor of 1.3 compared with the best previously available functionals, but even more significantly, we find a functional that has a high accuracy for all four categories of interaction.     

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.     

Harmonic frequency scaling factors for Hartree-Fock, S-VWN, B-LYP, B3-LYP, B3-PW91 and MP2 with the Sadlej pVTZ electric property basis set

Scaling factors for obtaining fundamental vibrational frequencies from harmonic frequencies calculated at six of the most commonly used levels of theory have been determined from regression analysis for the polarized-valence triple-zeta (pVTZ) Sadlej electric property basis set. The Sadlej harmonic frequency scaling factors for first- and second-row molecules were derived from a comparison of a total of 900 individual vibrations for 111 molecules with available experimental frequencies. Overall, the best performers were the hybrid density functional theory (DFT) methods, Becke's three-parameter exchange functional with the Lee–Yang–Parr fit for the correlation functional (B3-LYP) and Becke's three-parameter exchange functional with Perdew and Wang's gradient-corrected correlation functional (B3-PW91). The uniform scaling factors for use with the Sadlej pVTZ basis set are 0.9066, 0.9946, 1.0047, 0.9726, 0.9674 and 0.9649 for Hartree–Fock, the Slater–Dirac exchange functional with the Vosko–Wilk–Nusair fit for the correlation functional (S-VWN), Becke's gradient-corrected exchange functional with the Lee–Yang–Parr fit for the correlation functional (B-LYP), B3-LYP, B3-PW91 and second-order M?ller–Plesset theory with frozen core (MP2(fc)), respectively. In addition to uniform frequency scaling factors, dual scaling factors were determined to improve the agreement between computed and observed frequencies. The scaling factors for the wavenumber regions below 1800 cm⁻¹ and above 1800 cm⁻¹ are 0.8981 and 0.9097, 1.0216 and 0.9857, 1.0352 and 0.9948, 0.9927 and 0.9659, 0.9873 and 0.9607, 0.9844 and 0.9584 for Hartree–Fock, S-VWN, B-LYP, B3-LYP, B3-PW91 and MP2(fc), respectively. Hybrid DFT methods along with the Sadlej pVTZ basis set provides reliable theoretical vibrational spectra in a cost-effective manner. Received: 22 May 2000 / Accepted: 30 August 2000 / Published online: 28 February 2001     

Symbolic algebra in functional derivative potential calculations

This article gives the details of the methodology used in constructing a symbolic algebra program designed for evaluating potentials as the functional derivatives of so-called functional generators in molecular density-functional theory. The derived formulae are used in illustrative examples involving partial functional integration, the comparison of the exchange potential arising from different mathematical representations of the electron density for a given functional generator, and the evaluation and comparison of the potential for different functional generators with a given density. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 300–307, 1998     

Reparameterization of a meta-generalized gradient approximation functional by combining TPSS exchange with τ1 correlation

We present a new local meta-GGA exchange-correlation density functional by combining the TPSS meta-GGA exchange and the τ1 meta-GGA correlation functionals. The TPSS meta-GGA exchange-correlation and the τ1 meta-GGA correlation functionals have been implemented in the deMon code. The parameters in the τ1 meta-GGA correlation model are reoptimized in a synchronized way to match the original TPSS meta-GGA exchange counterpart. This reparametrized meta-GGA functional is referred to as “TPSSτ3”. The TPSSτ3 and TPSSτ1 meta-GGAs are validated using a test set that consists of covalent molecules, hydrogen-bonded complexes, and van der Waals interactions. The calculated results from TPSSτ1 and TPSSτ3 are analyzed and compared with reliable experimental data and theoretical data, as well as with those from Bmτ1 and TPSS calculations. The τ1 correlation model describes the aromatic compounds better than TPSS. TPSSτ3 yields satisfactory results for the covalent molecules, the hydrogen-bonded complexes, and the van der Waals complexes in the test set compared with TPSS, Bmτ1 and TPSSτ1. Contribution to the Serafin Fraga Memorial Issue.     

Performance of parallel TURBOMOLE for density functional calculations

The parallelization of density functional treatments of molecular electronic energy and first-order gradients is described, and the performance is documented. The quadrature required for exchange correlation terms and the treatment of exact Coulomb interaction scales virtually linearly up to 100 nodes. The RI-J technique to approximate Coulomb interactions (by means of an auxiliary basis set approximation for the electron density) even shows superlinear speedup on distributed memory architectures. The bottleneck is then linear algebra. Demonstrative application examples include molecules with up to 300 atoms and 3000 basis functions that can now be treated in a few hours per geometry optimization cycle in C₁ symmetry. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1746–1757, 1998     

A new local density functional for main-group thermochemistry, transition metal bonding, thermochemical kinetics, and noncovalent interactions

We present a new local density functional, called M06-L, for main-group and transition element thermochemistry, thermochemical kinetics, and noncovalent interactions. The functional is designed to capture the main dependence of the exchange-correlation energy on local spin density, spin density gradient, and spin kinetic energy density, and it is parametrized to satisfy the uniform-electron-gas limit and to have good performance for both main-group chemistry and transition metal chemistry. The M06-L functional and 14 other functionals have been comparatively assessed against 22 energetic databases. Among the tested functionals, which include the popular B3LYP, BLYP, and BP86 functionals as well as our previous M05 functional, the M06-L functional gives the best overall performance for a combination of main-group thermochemistry, thermochemical kinetics, and organometallic, inorganometallic, biological, and noncovalent interactions. It also does very well for predicting geometries and vibrational frequencies. Because of the computational advantages of local functionals, the present functional should be very useful for many applications in chemistry, especially for simulations on moderate-sized and large systems and when long time scales must be addressed.     

Theoretical and Crystallographic Study of Lead(IV) Tetrel Bonding Interactions

The ability of tetrahedral lead(IV) to establish noncovalent σ-hole tetrel bonding interactions with electron-rich atoms (ElRs; anions and Lewis bases) has been studied at the PBE0-D3/def2-TZVPD level of theory. An analysis of the Cambridge Crystallographic Database (CSD), which is a convenient storehouse of geometric information, has been performed to investigate the existence of tetrel bonding interactions involving tetrahedral lead(IV) derivatives. Several examples of tetrel bonding interactions that are crucial in crystal packing, ranging from 0D to 2D assemblies, have been found. In addition to the energetic and theoretical study of several XPb(CH₃)₃⋅⋅⋅ElR complexes (X=F, CN, CF₃, and CH₃), Bader's theory of atoms in molecules has also been used to further analyze and characterize the noncovalent interactions described herein.     

Applications of density functional theory methods in millimeter‐wave spectroscopy

Density functional theory (DFT), using the most common functionals, and ab initio quantum chemistry methods are used to calculate the rotational constants and dipole moments of the astrophysically important molecules HCN, CH₃CN, CH₃CNH⁺, HCCCN, and HCCNC. As far as millimeter‐wave spectroscopy is of interest the DFT methods performed well with most functionals, giving results within ±1% of experiments for rotational constants and ±3% for dipole moments. Analyzing the results obtained with all theoretical models, it may be concluded that the Becke's three‐parameter exchange functional and the gradient‐corrected functional of Lee, Yang, and Paar (B3LYP) and Becke's three‐parameter functional with Perdew–Wang correlational functional [B3PW91/6‐31G(d, p)] give the best performances. A detailed analysis of the electron correlation effects shows that HCCCN is more stable than is HCCNC, by 1.16 eV, with important contribution arising from triple excitations. This result is also compared with those obtained with DFT methods. Despite occasional difficulties, DFT with the currently available functionals are of great utility in quickly assessing spectroscopic parameters of astrophysical interest. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Dual-hybrid direct random phase approximation and second-order screened exchange with nonlocal van der Waals correlations for noncovalent interactions

We have added the nonlocal van der Waals correlation energy functional of Vydrov and Van Voorhis in 2010 (VV10NL) to the dual-hybrid direct random phase approximation (dRPA75) and second-order screened exchange (SOSEX75) for noncovalent interactions. The obtained methods are denoted as dRPA75-NL and SOSEX75-NL, and the corresponding short-range attenuation parameters are fitted with the large aug-cc-pV5Z basis set against the S66 dataset. Therefore, the dRPA75-NL method overcomes the error cancellation problem of the dRPA75 method with the relatively small aug-cc-pVTZ basis set for noncovalent interactions. Based on our benchmark computations, the dRPA75-NL and SOSEX75-NL methods perform very well on evaluating noncovalent interaction energies. Compared with the double-hybrid density functionals (DHDFs) of DSD-PBEP86-NL and DOD-PBEP86-NL, the dRPA75-NL and SOSEX75-NL methods perform much better on charge transfer interactions. Furthermore, the SOSEX75-NL method also gives insight into the development of computational methods for both closed-shell and open-shell noncovalent interactions. In summary, our computations demonstrate that even the full dRPA and SOSEX correlations still need additional dispersion corrections for noncovalent interactions.     

Factors determining the enzyme catalytic power caused by noncovalent interactions: Charge alterations in enzyme active sites

Understanding the origin of the enormous catalytic power of enzymes is very important. Electrostatic interactions and desolvation are the phenomena that are most proposed to explain the catalysis of enzymes; however, they also decelerate enzymatic reactions. How enzymes catalyze reactions through noncovalent interactions is still not well-understood. In this study, we explored how enzyme-substrate noncovalent interactions affect the free energy barriers (ΔG³s) of reactions by using a theoretical derivation approach. We found that enzymes reduce ΔG³s of reactions by decreasing positive charges and/or increasing negative charges in the electron-donating centers and by decreasing negative charges and/or increasing positive charges in the electron-accepting centers of reactions. Enzyme-substrate noncovalent interactions are essential approaches through which the charge alterations lead to ΔG³ reductions. Validations with reported experimental data demonstrated that this charge alteration mechanism can explain the catalyses caused by diverse types of noncovalent interactions. Electrostatic interactions and desolvation are the most observed noncovalent interactions essential for ΔG³ reductions. This mechanism does not contradict any specific enzymatic catalysis and overcomes the shortages of the electrostatic interaction and desolvation mechanisms. This study can provide useful guidance in exploring enzymatic catalysis and designing catalyst.     

A semiempirical long-range corrected exchange correlation functional including a short-range Gaussian attenuation (LCgau-B97)

We applied an improved long‐range correction scheme including a short‐range Gaussian attenuation (LCgau) to the Becke97 (B97) exchange correlation functional. In the optimization of LCgau‐B97 functional, the linear parameters are determined by least squares fitting. Optimizing μ parameter (0.2) that controls long‐range portion of Hartree‐Fock (HF) exchange to excitation energies of large molecules (Chai and Head‐Gordon, J Chem Phys 2008, 128, 084106) and additional short‐range Gaussian parameters (a = 0.15 and k = 0.9) that controls HF exchange inclusion ranging from short‐range to mid‐range (0.5–3 Å) to ground state properties achieved high performances of LCgau‐B97 simultaneously on both ground state and excited state properties, which is better than other tested semiempirical density functional theory (DFT) functionals, such as ωB97, ωB97X, BMK, and M0x‐family. We also found that while a small μ value (～0.2) in LC‐DFT is appropriate to the local excitation and intramolecular charge‐transfer excitation energies, a larger μ value (0.42) is desirable in the Rydberg excitation‐energy calculations. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Engineering functional materials by halogen bonding

Engineering functional materials endowed with unprecedented properties require the exploitation of new intermolecular interactions, which can determine the characteristics of the bulk materials. The great potential of Halogen Bonding (XB), namely any noncovalent interaction involving halogens as electron acceptors, in the design of new and high‐value functional materials is now emerging clearly. This Highlight will give a detailed overview on the energetic and geometric features of XB, showing how some of them are quite constant in most of the formed supramolecular complexes (e.g., the angle formed by the covalent and the noncovalent bonds around the halogen atom), while some others depend strictly on the nature of the interacting partners. Then, several specific examples of halogen‐bonded supramolecular architectures, whose structural aspects as well as applications in fields as diverse as enantiomers' separation, crystal engineering, liquid crystals, natural, and synthetic receptors, will be fully described. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: PolymChem 45: 1–15, 2007     

Self-consistent implementation of a nonlocal van der Waals density functional with a Gaussian basis set

Nearly all common density functional approximations fail to properly describe dispersion interactions responsible for binding in van der Waals complexes. Empirical corrections can fix some of the failures but cannot fully grasp the complex physics and may not be reliable for systems dissimilar to the fitting set. In contrast, the recently proposed nonlocal van der Waals density functional (vdW-DF) was derived from first principles, describes dispersion interactions in a seamless fashion, and yields the correct asymptotics. Implementation of this functional is somewhat cumbersome: Nonlocal dependence on the electron density requires numerical double integration over the space variables and functional derivatives are nontrivial. This paper shows how vdW-DF can be implemented self-consistently with Gaussian basis functions. The gradients of the energy with respect to nuclear displacements have also been derived and coded, enabling efficient geometry optimizations. We test the vdW-DF correlation functional in combination with several exchange approximations. We also study the sensitivity of the method to the basis set size and to the quality of the numerical quadrature grid. For weakly interacting systems, acceptable accuracy in semilocal exchange is achieved only with fine grids, whereas for nonlocal vdW-DF correlation even rather coarse grids are sufficient. The current version of vdW-DF is not well suited for pairing with Hartree-Fock exchange, leading to considerable overbinding.