Density matrices from position and momentum densities

Summary One-electron density matrices, which are representable in single-centers-orbital basis sets, have been investigated with respect to their reconstruction from densities. The maximum allowed dimension for reconstruction from a combination of position & momentum density dependent properties is only slightly bigger than the dimension in the case of position (or momentum) densities only. Since for a given one-particle basis of dimensionM, the number of one-matrix elements which can be determined is also of orderM only, while the total number of one-matrix elements is of orderM ², it is in general necessary to introduce severe constraints and restrictions. The accuracy demands on the data and algorithms increase exponentially for linearly increasing size of basis set.     

Application of Löwdin's canonical orthogonalization method to the Slater‐type orbital configuration‐interaction basis set

We apply Löwdin's canonical orthogonalization method to investigate the linearly dependent problem arising from the variational calculation of atomic systems using Slater‐type orbital configuration‐interaction (STO‐CI) basis functions. With a specific arithmetic precision used in numerical computations, the nonorthogonal STO‐CI basis is easily linearly dependent when the number of basis functions is sufficiently large. We show that Löwdin's canonical orthogonalization method can successfully overcome such problem and simultaneously reduce the dimension of basis set. This is illustrated first through an S‐wave model He atom, and then the real two‐electron atoms in both the ground and excited states. In all of these calculations, the variational bound state energies of the two‐electron systems are obtained in reasonably high accuracy using over‐redundant STO‐CI bases, however, without using extended high‐precision technique. © 2015 Wiley Periodicals, Inc.     

Removal of dependencies from nearly complete basis sets. Calculations on the helium dimer

A method is devised for dealing with almost linearly dependent basis sets that contain large sets of bond functions. Using the largest of such basis sets, LARSAT, the second-order Møller-Plesset polarization dispersion energy of the helium dimer is calculated to be - 17.08 K at R = 5.6 bohrs. MR-SDCI calculations, employing a set of 37 reference configurations, were performed for the helium dimer with several basis sets at 4.0 and 5.6 bohrs. Size-extensivity corrections were included to take into account the R dependency of the size-extensivity error in MR-SDCI calculations. The He₂ interaction energies computed with basis LARSAT are - 10.92 K at 5.6 bohrs and 295.1 K at 4.0 bohrs. The 37-MR-SDCI calculations with basis LARSAT almost reproduce the He₂ full configuration interaction (CI) interaction energies computed with the same basis, at notably smaller cost. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 805–815, 1997     

Orthogonal Waller–Hartree spin eigenfunctions

It was shown recently by the present author that the double symmetrization and the double antisymmetrization are essential in the spin-dependent and the spin-free formalisms, respectively, to perform the exclusion of all the unnecessary spin eigenfunctions and the selection of a unique set of linearly independent spin eigenfunctions. The double antisymmetrized Wigner matric basis and the Wigner double symmetrized matric basis are presented in this article for N up to 6. The double symmetrization or the double antisymmetrization also results in a direct expansion method for the calculation of the orthogonal spin coefficients; this direct method does not require the knowledge of the wave functions of the N ? 1 electron system. The modified method of Graebenstetter can also be used to calculate these orthogonal spin coefficients.     

Highly efficient implementation of pseudospectral time‐dependent density‐functional theory for the calculation of excitation energies of large molecules

We have developed and implemented pseudospectral time‐dependent density‐functional theory (TDDFT) in the quantum mechanics package Jaguar to calculate restricted singlet and restricted triplet, as well as unrestricted excitation energies with either full linear response (FLR) or the Tamm–Dancoff approximation (TDA) with the pseudospectral length scales, pseudospectral atomic corrections, and pseudospectral multigrid strategy included in the implementations to improve the chemical accuracy and to speed the pseudospectral calculations. The calculations based on pseudospectral time‐dependent density‐functional theory with full linear response (PS‐FLR‐TDDFT) and within the Tamm–Dancoff approximation (PS‐TDA‐TDDFT) for G2 set molecules using B3LYP/6‐31G*^* show mean and maximum absolute deviations of 0.0015 eV and 0.0081 eV, 0.0007 eV and 0.0064 eV, 0.0004 eV and 0.0022 eV for restricted singlet excitation energies, restricted triplet excitation energies, and unrestricted excitation energies, respectively; compared with the results calculated from the conventional spectral method. The application of PS‐FLR‐TDDFT to OLED molecules and organic dyes, as well as the comparisons for results calculated from PS‐FLR‐TDDFT and best estimations demonstrate that the accuracy of both PS‐FLR‐TDDFT and PS‐TDA‐TDDFT. Calculations for a set of medium‐sized molecules, including C_n fullerenes and nanotubes, using the B3LYP functional and 6‐31G^** basis set show PS‐TDA‐TDDFT provides 19‐ to 34‐fold speedups for C_n fullerenes with 450–1470 basis functions, 11‐ to 32‐fold speedups for nanotubes with 660–3180 basis functions, and 9‐ to 16‐fold speedups for organic molecules with 540–1340 basis functions compared to fully analytic calculations without sacrificing chemical accuracy. The calculations on a set of larger molecules, including the antibiotic drug Ramoplanin, the 46‐residue crambin protein, fullerenes up to C₅₄₀ and nanotubes up to 14×(6,6), using the B3LYP functional and 6‐31G^** basis set with up to 8100 basis functions show that PS‐FLR‐TDDFT CPU time scales as N^2.05 with the number of basis functions. © 2016 Wiley Periodicals, Inc.     

New global potential energy surface of the MgH2 system and dynamics studies of the reaction H + MgH → Mg + H2

A new global potential energy surface for the ground state of MgH₂ was constructed using the permutation invariant polynomial neural network method. About 70 000 ab initio energy points were calculated via the multi‐reference configuration interaction method method with aug‐cc‐pVTZ and aug‐cc‐pVQZ basis sets, and these points were used to construct the potential energy surface (PES). To avoid basis set superposition error, the basis set was extrapolated to the complete basis set limit using the two point energy extrapolation formula. The root mean square error of the present PES is only 8.85 meV. Initial state (v = 0, j = 0) dynamics studies were performed using the time‐dependent wave packet method with a second‐order split operator for the total angular momentum J up to a value of 50. Furthermore, the reaction probability, integral cross section, and thermal rate constant are reported and compared with available theoretical studies.     

Combined bond polarization function basis sets for accurate ab initio calculation of the dissociation energies of AHn molecules (A=LI to F)

An alternative route toward developing basis sets for post-Hartree-Fock calculations, the hybrid bond polarization function method, is investigated. Two new basis sets, denoted 6-31G(d, p)+ B and 6-31 + G(d,p)+B, are defined for the first-row hydrides. The dissociation energies of the first-row hydride species in their respective ground states are computed using full fourth-order Møller-Plesset theory, and compared with results obtained with large polarized basis sets containing no bond functions. It is shown that results are competitive even with basis sets as large as 6-311++G(3df,3pd), while computation times are reduced by a factor of 4 to 20. On empirical grounds, the basis set superposition error should be neglected entirely.     

Combined bond-polarization basis sets for accurate determination of dissociation energies. II. Basis set superposition error as a function of the parent basis set

The effect of the parent basis set on the basis set superposition error caused by bond functions is investigated systematically. An important difference between BSSE at the SCF and correlated levels is pointed out. Three new basis sets are defined, denoted 6-311 + G(d,p)B, 6-311 + G(2d,p)B, and 6-311 + G(2df,p)B. BSSE for the first-row hydrides seems to increase uniformly with increasing atomic number of the central atom. Expansion of the valence part of the basis set from 6-31G to 6-311G, as well as adding f functions, has a significant effect on the BSSE. Additional BSSEs incurred by bond functions are less than or equal to 1 kcal/mol for the 6-311 + G(2df,p)B basis set. For the dissociation energies of the first-row hydride species, agreement with experiment within only a few kcal/mol can be obtained even without resorting to isogyric reaction cycles. For high-quality calculations, adding bond functions seems to have definite advantages over expanding the polarization space beyond the [2d1f] level.     

Discrete sets of many‐body Sturmians

In this work, we present a method to obtain two‐particle Coulomb Sturmians Functions (CSF) with an expansion in a set of L² basis functions. In the two‐body case, we recover the exact (discrete) spectrum of the CSFs for negative energies and a discretized approximation for positive ones. Besides, we make use of this method to analyze the two‐independent electron problem as a Generalized Sturmian problem. We propose a discretized version of the wave function in terms of the CSF states, and show that the problem reduces to find numerical coincidences between energy‐dependent eigencharges of the mutually independent one‐electron systems. This expansion methodology includes the continuum information which is lost in the sets used previously in the literature, and is complete when the size of the basis goes to infinity. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Solution of the Hartree–Fock problem by expansion onto nested bases

A method for solving the Hartree–Fock problem in a finite basis set is derived, which permits each orbital to be expanded in a different basis. If the basis set for each orbital ?_i contains the basis functions for the preceding orbitals, ?_i?1, ?_i?2,… ?₁, then the ?_i form an orthonormal set. One advantage over the standard Hartree–Fock method is that a different long range behavior for each orbital, as for example is required in the Hartree–Fock-Slater method, can be forced. A calculation on the ground state of beryllium is performed using the nested procedure. Very little energy is lost because of nesting, and the node in the 1s orbital disappears.     

The role of material behavior in the performances of electroactive polymer energy harvesters

Electroactive polymer energy harvesters are promising devices for the conversion of mechanical work to electrical energy. The performances of these devices are strongly dependent on the mechanical response of the polymeric material and on the type of electromechanical cycle, and these are limited by the occurrence of dielectric breakdown, compression induced wrinkling and electromechanical instability (pull‐in). To identify the optimal electromechanical cycle that complies with all of these limitations, we set‐up and solve a constraint optimization problem and we critically discuss the influence of material behavior of the polymer in the optimal performances of the energy harvesting device. Finally, we show that if the rate‐independent dissipative behavior of the polymer (Mullins effect) is neglected, the optimization procedure may lead to quite unsatisfactory predictions: by making reference to explicit experimental data from literature we show that an optimal harvesting cycle deduced by neglecting the Mullins effect is far from being optimal when this is taken in consideration. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 1303–1314     

Relation between exchange-only optimized potential and Kohn-Sham methods with finite basis sets, and effect of linearly dependent products of orbital basis functions

Recently, Staroverov, Scuseria, and Davidson [J. Chem. Phys. 124, 141103 (2006)] presented examples of exchange-only optimized effective potential (xOEP) calculations that yield exactly the Hartree-Fock (HF) total energy. Here, building on their work, arguments showing under which conditions xOEP methods, with finite basis sets, do or do not yield the HF ground state energy but a higher one, are given. While the orbital products of a complete basis are linearly dependent, the HF ground state energy can only be obtained via a finite basis set xOEP scheme in the case that all products of occupied and unoccupied orbitals emerging from the employed orbital basis set are linearly independent of each other. Further, exchange potentials leading to the HF ground state energy likely exhibit unphysical oscillations and do not represent a Kohn-Sham (KS) exchange potential as a functional derivative of the exchange energy. These findings appear to explain the seemingly paradoxical results of Staroverov et al. that certain finite basis set xOEP calculations lead to the HF ground state energy despite the fact that within a real space (or complete basis) representation, the xOEP ground state energy is always higher than the HF energy. Moreover, independent of whether or not the occupied and unoccupied orbital products are linearly dependent, it is shown that finite basis set xOEP methods only represent exact exchange-only (EXX) KS methods, i.e., proper density-functional methods, if the orbital basis set and the auxiliary basis set representing the exchange potential are balanced to each other, i.e., if the orbital basis is comprehensive enough for a given auxiliary basis. Otherwise xOEP methods do not represent EXX KS methods and yield unphysical exchange potentials. The question whether a xOEP method properly represents a KS method with an exchange potential that is a functional derivative of the exchange energy is related to the problem of the definition of local multiplicative operators in finite basis representations. Plane wave calculations for bulk silicon illustrate the findings of this work.     

A new algebraic approach to the eigenvalue problems of linear differential operators without integrations

In this work, a new approximation scheme based on the evaluation of the pointwise expectation of the Hamiltonian (H) via a conveniently chosen basis set is proposed. This scheme does not necessitate integration; however, physical and mathematical considerations in choosing the basis set are considerably important when very precise and rapidly convergent results are desired. In this method, the best linear combination of “well-selected” basis functions are sought in a way such that H ψ / ψ is flat in the neighborhood of a conveniently chosen point in the domain of H. This yields an algebraic eigenvalue problem. Some concrete applications that have already been realized confirm the efficiency of this approach.     

H2O,H2, HF,F2 and F2O nuclear magnetic shielding constants and indirect nuclear spin–spin coupling constants (SSCCs) in the BHandH/pcJ‐n and BHandH/XZP Kohn–Sham limits

Good performance of segmented contracted basis sets XZP, where X = D, T, Q and 5, for obtaining H₂O, H₂, HF, F₂ and F₂O nuclear isotropic shielding constants in the BHandH Kohn–Sham basis set limit was shown. The results of two‐ and three‐parameter complete basis set limit extrapolation schemes were compared with experimental results, earlier literature data and benchmark ab initio results. Similar convergence patterns of shieldings obtained from calculations using general purpose XZP basis sets and from polarization‐consistent basis sets pcS‐n and pcJ‐n, where n = 0, 1, 2, 3 and 4, designed to accurately predict magnetic properties were observed. On the contrary, the SSCCs were more sensitive to the XZP basis set size and generally less accurate than those estimated using pcJ‐n basis set family. The BHandH density functional markedly outperforms B3LYP method in predicting heavy atom shieldings and SSCCs values in the studied systems. Copyright © 2009 John Wiley & Sons, Ltd.     

Accurate prediction of vertical electronic transitions of Ni(II) coordination compounds via time dependent density functional theory

Time dependent density functional theory calculations are completed for five Ni(II) complexes formed by polydentate peptides to predict the electronic absorption spectrum. The ligands examined were glycyl‐glycyl‐glycine (GGG), glycyl‐glycyl‐glycyl‐glycine (GGGG), glycyl‐glycyl‐histidine (GGH), glycyl‐glycyl‐cysteine (GGC), and triethylenetetramine (trien). Fifteen functionals and two basis sets were tested. On the basis of the mean absolute percent deviation (MAPD), the ranking among the functionals is: HSE06 ∼ MPW1PW91 ∼ PBE0 > ω‐B97x‐D ∼ B3P86 ∼ B3LYP ∼ CAM‐B3LYP > PBE ∼ BLYP ∼ BP86 > TPSS > TPSSh > BHandHLYP > M06 ≫ M06‐2X. Concerning the basis sets, the triple‐ζ def2‐TZVP performs better than the double‐ζ LANL2DZ. With the functional HSE06 and basis set def2‐TZVP the MAPD with respect to the experimental λ_max is 1.65% with a standard deviation of 1.26%. The absorption electronic spectra were interpreted in terms of vertical excitations between occupied and virtual MOs based on Ni‐d atomic orbitals. The electronic structure of the Ni(II) species is also discussed.     

Theoretical prediction of the contact distance dependence of the electron transfer reactivity of the ClO/ClO− coupling system

On the basis of the structures and properties of the ClO/ClO^? system obtained at the density functional theory (DFT) (UB3LYP) level, employing the 6‐311+G(3df) standard basis set, the electron transfer reactivity of this system is investigated. The results indicate that there are five possible stable coupling complexes that correspond to the generous minima on the global potential energy surfaces (PES). The most stable coupling complex is planar EC4, in which there is a O? O linkage with two trans‐Cl atoms. Their stabilization energies are calculated to be 20.57 (EC1: C₁), 20.54 (EC2: C₂, ²B), 20.69 (EC3: C₁), 20.70 (EC4: C_s, ²A′), and 20.69 (EC.5: C_2h, ²B_u) kcal/mol at the B3LYP/6‐311+G(3df) level; with the correction of the basis set superposition error (BSSE), the stability order of these encounter complexes is EC4 > EC.5 > EC3 > EC1 > EC2. Based on the five encounter complexes, five coupling modes are designed for the study of the electron transfer reactivity of this system. The dissociation energy curves at the activated states and the corresponding activation energies of these five coupling modes are obtained and are compared at the B3LYP/6‐311+G(3df) and MP2/6‐311+G* levels. The inapplicability of DFT methods has also been discussed in this article in predicting the energy curves, especially with a long contact distance, in which DFT methods give the abnormal behavior for the dissociations of the complexes caused by the “inverse symmetry breaking” problem. On the basis of the golden rule of the time‐dependent perturbation theory, the electron transfer reactivity and the contact distance dependence of the various electron transfer kinetics parameters (e.g., activation energy, coupling matrix element) have been analyzed at the UMP2(full)/6‐311+G* level. The electron transfer can take place over a range of contact distances, but the most effective coupling distance corresponds to only a small range. The coupling orientation analyses also indicate that the most favorable coupling mode to the electron transfer does not always correspond to the most stable encounter complex mechanism. Some highly energetic coupling modes are more favorable for the electron transfer. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Combined experimental and computational modeling studies on 3,5‐dimethyl‐pyrazole‐1‐carbodithioic acid benzyl ester

The title compound, 3,5‐Dimethyl‐pyrazole‐1‐carbodithioic acid benzyl ester, has been synthesized and structurally characterized by X‐ray single crystal diffraction, elemental analysis, IR spectra, and UV‐Vis spectrum. The crystal belongs to orthorhombic, space group P212121, with a = 5.3829(15), b = 11.193(3), c = 21.824(6) Å, V = 1315.0(6) ų, and Z = 4. The molecules are connected via intermolecular C–H···N hydrogen bonds into 1D infinite chains. The crystal structure is consolidated by the intramolecular C–H···S hydrogen bonds. Furthermore, Density functional theory (DFT) calculations of the structure, stabilities, orbital energies, composition characteristics of some frontier molecular orbitals and Mulliken charge distributions of the title compound were performed by means of Gaussian 03W package and taking B3LYP/6‐31G(d) basis set. The time‐dependent DFT (TD‐DFT) calculations have been employed to calculate the electronic spectrum of the title compound, and the UV‐Vis spectra has been discussed on this basis. The results show that DFT method at B3LYP/6‐31G(d) level can well reproduce the structure of the title compound. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Investigation of correlation between impact sensitivities and bond dissociation energies in some triazole energetic compounds

In this article, density functional theory has been utilized to study on the correlation between impact sensitivities h _50% and the bond dissociation energies (BDEs) of nine triazole energetic explosives. By employing B3LYP and B3P86 method with the 6-311G** basis set, all the molecules have been fully optimized. The BDEs for removal of the NO₂ group in these compounds have also been calculated at the same level. Computed results show that BDEs calculated by B3LYP method are all less than those by B3P86 method. The relationship between the impact sensitivities and the weakest C–NO₂ bond dissociation energy (BDE) values have been investigated. The results indicate a good linear correlation between the impact sensitivity h _50% and the ratio (BDE/E) of the weakest BDE to the total energy E.     

Prediction of the reduction potential in transition‐metal containing complexes: How expensive? For what accuracy?

Accurate computationally derived reduction potentials are important for catalyst design. In this contribution, relatively inexpensive density functional theory methods are evaluated for computing reduction potentials of a wide variety of organic, inorganic, and organometallic complexes. Astonishingly, SCRF single points on B3LYP optimized geometries with a reasonably small basis set/ECP combination works quite well‐‐B3LYP with the BS1 [modified‐LANL2DZ basis set/ECP (effective core potential) for metals, LANL2DZ(d,p) basis set/LANL2DZ ECP for heavy nonmetals (Si, P, S, Cl, and Br), and 6‐31G(d') for other elements (H, C, N, O, and F)] and implicit PCM solvation models, SMD (solvation model based on density) or IEFPCM (integral equation formalism polarizable continuum model with Bondi atomic radii and α = 1.1 reaction field correction factor). The IEFPCM‐Bondi‐B3LYP/BS1 methodology was found to be one of the least expensive and most accurate protocols, among six different density functionals tested (BP86, PBEPBE, B3LYP, B3P86, PBE0, and M06) with thirteen different basis sets (Pople split‐valence basis sets, correlation consistent basis sets, or Los Alamos National Laboratory ECP/basis sets) and four solvation models (SMD, IEFPCM, IPCM, and CPCM). The MAD (mean absolute deviation) values of SCRF‐B3LYP/BS1 of 49 studied species were 0.263 V for SMD and 0.233 V for IEFPCM‐Bondi; and the linear correlations had respectable R ² values (R ² = 0.94 for SMD and R ² = 0.93 for IEFPCM‐Bondi). These methodologies demonstrate relatively reliable, convenient, and time‐saving functional/basis set/solvation model combinations in computing the reduction potentials of transition metal complexes with moderate accuracy. © 2017 Wiley Periodicals, Inc.