Rapid algorithm for computing the electron repulsion integral over higher order Gaussian-type orbitals: Accompanying coordinate expansion method

A general algorithm for rapidly computing the electron repulsion integral (ERI) is derived for the ACE-b3k3 formula, which has been derived previously. [K. Ishida, Int. J. Quantum Chem., 59, 209 (1996)]. A computer program code that is universal for all types of Gaussian-type orbitals (GTOs) up to h-type can be constructed by the use of this general algorithm. It is confirmed that the ACE-b3k3 algorithm is numerically very stable even for higher order GTOs. It is found that, in a floating-point-operation (FLOP) count assessment, the ACE-b3k3 algorithm is the fastest among all methods available in the literature for (dd|dd), (ff|ff), (gg|gg), and (hh|hh) ERIs when the degree of contraction of the GTO is high. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 923–934, 1998     

Accompanying coordinate expansion formulas derived with the solid harmonic gradient

A series of accompanying coordinate expansion (ACE) formulas for calculating the electron repulsion integral (ERI) over both generally and segmentally contracted solid harmonic (SH) Gaussian-type orbitals (GTOs) can be rederived by the use of the modified operator (called solid harmonic gradient here) of the spherical tensor gradient of Bayman and the reducing solid harmonic gradient defined in this article. The final general formulas contain the reducing mixed solid harmonics defined in a previous article [Ishida, K. J Chem Phys 1999, 111, 4913] and the reducing triply mixed solid harmonics defined previously [Ishida, K. J Chem Phys 2000, 113, 7818]. Each general formula in the series is named ACEb1k1, ACEb2k3, or ACEb3k3. New general algorithm can be obtained inductively from the general formula named ACEb2k3, in addition to the previously developed ACEb1k1 and ACEb3k3. For calculating ERI practically, we select one of these ACE algorithms, as it gives the minimum floating-point operation (FLOP) count. Theoretical assessment by the use of the FLOP count is performed for the (LL/LL) class of ERIs over both generally and segmentally contracted SH-GTOs (L = 1-3). It is found that the present ACE is theoretically the fastest among all rigorous methods in the literature.     

New recurrence relations for the rapid evaluation of electron repulsion integrals based on the accompanying coordinate expansion formula

We present an algorithm for the rapid computation of electron repulsion integrals (ERIs) over Gaussian basis functions based on the accompanying coordinate expansion (ACE) formula. The present algorithm uses equations termed angular momentum reduced expressions and introduces two types of recurrence relations to ACE formulas. Numerical efficiencies are assessed for (p pmid R:p p) and (sp spmid R:sp sp) ERIs by using the floating-point operation count. The algorithm is suitable for calculating ERIs for the same exponents but different angular momentum functions, such as L shells and derivatives of ERIs. The present algorithm is also capable of calculating ERIs with highly contracted Gaussian basis functions.     

A new algorithm of two-electron repulsion integral calculations: a combination of Pople–Hehre and McMurchie–Davidson methods

A new algorithm of two-electron repulsion integral (ERI) calculations has been developed. In this algorithm, Cartesian axes are rotated to make several coordinate components zero or constant using the Pople–Hehre algorithm, and ERIs are evaluated at the rotated coordinate by the McMurchie–Davidson algorithm. The new algorithm applicable to (ss|ss) to (dd|dd) ERIs is implemented in the quantum chemistry program GAMESS. Test calculations show that the new algorithm reduces the computational times by 10–40%, as compared with the Pople–Hehre and the Rys quadrature algorithms.     

Derivative of electron repulsion integral using accompanying coordinate expansion and transferred recurrence relation method for long contraction and high angular momentum

In this study, an early‐working algorithm is designed to evaluate derivatives of electron repulsion integrals (DERIs) for heavy‐element systems. The algorithm is constructed to extend the accompanying coordinate expansion and transferred recurrence relation (ACE‐TRR) method, which was developed for rapid evaluation of electron repulsion integrals (ERIs) in our previous article (M. Hayami, J. Seino, and H. Nakai, J. Chem. Phys. 2015, 142, 204110). The algorithm was formulated using the Gaussian derivative rule to decompose a DERI of two ERIs with the same sets of exponents, different sets of contraction coefficients, and different angular momenta. The algorithms designed for segmented and general contraction basis sets are presented as well. Numerical assessments of the central processing unit time of gradients for molecules were conducted to demonstrate the high efficiency of the ACE‐TRR method for systems containing heavy elements. These heavy elements may include a metal complex and metal clusters, whose basis sets contain functions with long contractions and high angular momenta.     

Evaluation of the Boys Function using Analytical Relations

The analytical relations for Boys function F _m (x) are presented. These relations are useful in the fast and more accurate calculations of multicenter molecular integrals over Gaussian type orbitals (GTOs). The formulas obtained are numerically stable for all values of m and x.     

Spectral representation of Hartree‐Fock exchange operator

We report a new mathematical result: it is possible to construct a spectral representation of the two particles Coulomb potential in the form of | r ? r ^′|^?1 = ∑_λλg( r ) g_λ( r ^′). We call this formula λ‐decomposition. Two special nontrivial cases of λ‐decomposition are reported together with the numerical analysis of the convergence for one of them. It is shown how λ‐decomposition allows to construct a new fast algorithm for Hartree‐Fock exchange operator calculation, in which the calculation of electron repulsion integrals (ERIs) is completely avoided. The connection between the new method and the resolution of identity and Cholesky decomposition based approaches has been established. Finally, the accuracy of ERIs evaluation within the new approach has been studied numerically. The results demonstrate that it is possible to achieve the accuracy of 10^?10 for the ERIs in wide range of their orbital exponents with relatively small number of terms in λ‐decomposition. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Extension of accompanying coordinate expansion and recurrence relation method for general‐contraction basis sets

An algorithm of the accompanying coordinate expansion and recurrence relation (ACE‐RR), which is used for the rapid evaluation of the electron repulsion integral (ERI), has been extended to the general‐contraction (GC) scheme. The present algorithm, denoted by GC‐ACE‐RR, is designed for molecular calculations including heavy elements, whose orbitals consist of many primitive functions with and without higher angular momentum such as d‐ and f‐orbitals. The performance of GC‐ACE‐RR was assessed for ‐, ‐, ‐, and ‐type ERIs in terms of contraction length and the number of GC orbitals. The present algorithm was found to reduce the central processing unit time compared with the ACE‐RR algorithm, especially for higher angular momentum and highly contracted orbitals. Compared with HONDOPLUS and GAMESS program packages, GC‐ACE‐RR computations for ERIs of three‐dimensional gold clusters Au_n (n = 1, 2, …, 10, 15, 20, and 25) are more than 10 times faster. © 2014 Wiley Periodicals, Inc.     

The PI index of phenylenes

The Padmakar–Ivan (PI) index is a graph invariant defined as the summation of the sums of n _eu (e|G) and n _ev (e|G) over all the edges e = uv of a connected graph G, i.e., PI(G) = ∑ _e∈E(G)[n _eu (e|G) + n _ev (e|G)], where n _eu (e|G) is the number of edges of G lying closer to u than to v and n _ev (e|G) is the number of edges of G lying closer to v than to u. An efficient formula for calculating the PI index of phenylenes is given, and a simple relation is established between the PI index of a phenylene and of the corresponding hexagonal squeeze.     

Reliability of atomic natural orbital basis sets in calculations involving pseudopotentials

The performance of Atomic Natural Orbital (ANO) basis sets for calculations involving nonempirical core pseudopotentials has been studied by comparing the results for atomic and molecular nitrogen obtained using contracted ANO basis sets with those obtained using both the primitive set and a segmented one. The primitive set has been optimized at the SCF level for atomic N treated as a five-electron pseudo-atom, and consists of 7s and 7p primitive GTOs supplemented by 2d and 1f GTOs optimized at the CI level. From this primitive set three contracted [3s 3p 2d 1f] sets have been obtained. The first one has been derived from the ANOs of the neutral atom, the second has been obtained from an averaged density matrix and the third one is a segmented set. For the atom, the segmented set gives a zero contraction error at the SCF level as it must be in valence-only calculations. The ANO basis sets show some small contraction error at the SCF level but perform better in CI calculations. However, for the diatomic N₂ molecule the ANO basis sets exhibit a rather large contraction error in the calculated SCF energy. A detailed analysis of the origin of this error is reported, which shows that the conventional strategy used to derive ANO basis sets does not work very well when pseudopotentials are involved.     

PI indices of pericondensed benzenoid graphs

The Padmakar–Ivan (PI) index is a graph invariant defined as the summation of the sums of n _eu (e|G) and n _ev (e|G) over all the edges e = uv of a connected graph G, i.e., , where n _eu (e|G) is the number of edges of G lying closer to u than to v and n _ev (e|G) is the number of edges of G lying closer to v than to u. An efficient formula for calculating the PI index of a class of pericondensed benzenoid graphs consisting of three rows of hexagonal of various lengths.     

Molecular weight-dependence of free volume in polystyrene studied by positron annihilation measurements

Positron annihilation lifetime measurements are reported for four monodisperse polystyrenes with molar mass M = 4,000, 9,200, 25,000, and 400,000. The temperature dependences of orthopositronium (o-Ps) lifetime (τ₃) and intensity (I₃) were measured from 5°C to T_g + 30°C for each sample. From these data, the free volume hole size, 〈v_f(τ₃)〉, and fractional free volume h_ps=CI₃〈v_f(τ₃)〉 were calculated. The temperature dependences of τ₃, 〈v_f(τ₃)〉 and h_ps show a discrete change in slope at an effective glass transition temperature, T_g,ps, which is measurably below the conventional bulk T_g. This suggests that τ₃ is sensitive to large holes which retain their liquid-like mobility in the glassy state. Good agreement was found for T > h_g,ps between h_ps and the theoretical free volume fraction h_th deduced from experimental P-V-T data for polystyrene using the statistical mechanical theory of Simha and Somcynsky. Below T_g,ps, deviations between h_ps and h_th are observed, h_ps falling increasingly below h_th as temperature decreases. Whereas h_ps and h_th depend strongly on M in the melt, each essentially independent of M in the glass. A free volume quantity, computed from the bulk volume, which is in good numerical agreement with the Simha-Somcynsky h-function in the melt, gives improved agreement with h_ps in the glassy state. © 1994 John Wiley & Sons, Inc.     

Synthesis and X-Ray Analysis of a Nickel Triphenyl-Cyclopropenyl Complex. A mo-interpretation of the geometry and bonding in the (C3H3)ML3-type of complexes

The reaction of [(C₃Ph₃)Ni(PPh₃)₂]ClO₄ with P(CH₂CH₂PPh₂)₃(pp₃) and NaBPh₄ yields the [(C₃Ph₃)Ni(pp₃)]BPh₄-complex. After long exposure of the solution of this compound in acetone/butanol to the air a new derivative [(C₃Ph₃)-Ni(pp₂po)]BPh₄· 0.5 C₄H₉OH, where pp₂po is (Ph₂PCH₂CH₂)₂P(CH₂ CH₂POPh₂), is obtained. Complete X-ray analysis has been carried out for the latter complex: a=18.303 (5); b=29.445 (6), c=13.305 (5) Å, β=112.70 (9)°; space group monoclinic, P2₁/a, Z=4. Disorder problems were encountered in the refinement of the structure. The best R is 0.093. One of the arms of the parent pp₃-molecule, not coordinated to the metal, undergoes oxidation. The Ni-atom, coordinated by the three remaining P-atoms of the ligand, is also linked in a roughly η³-mode to the cyclopropenium ligand. The geometry of the molecule is examined in detail. Extended HMO-calculations were performed to interpret how the variation of P? Ni? P angles affects the bonding between the NiP₃- and C₃H₃-fragments. The conclusion is that the overall energy of the complex may be lowered in spite of a weakening of the Ni-cyclopropenium linkage. Extensions are made to other systems containing a linkage between a metal and a X₃-ring (X=P,As).     

Gaussian fitting function basis sets for crystalline silicon: Bond-centered s-type vs. site-centered f-type

The lattice constant, cohesive energy, bulk modulus, and band gap for Si were calculated with the linear combinations of Gaussian-type orbitals-fitting function (LCGTO-FF) technique using three distinct types of charge density and exchange-correlation fitting function basis sets: (l) site-centered s-type GTOs only; (2) site- and bond-centered s-type GTOs; and (3) site-centered s-and f-type GTOs. All three basis sets produce good results for the lattice constant and bulk modulus of Si, but only the two larger basis sets determine the cohesive energy and LDA band gap accurately. The numerical results obtained with the two larger basis sets are in good quantitative agreement with each other and with results from other techniques. © 1996 John Wiley & Sons, Inc.     

Two-electron integral evaluation for uncontracted geometrical-type Gaussian functions

A new algorithm for efficient evaluation of two-electron repulsion integrals (ERIs) using uncontracted geometrical-type Gaussian basis functions is presented. Integrals are evaluated by the Habitz and Clementi method. The use of uncontracted geometrical basis sets allows grouping of basis functions into shells (s, sp, spd, or spdf) and processing of integrals in blocks (shell quartets). By utilizing information common to a block of integrals, this method achieves high efficiency. This technique has been incorporated into the KGNMOL molecular interaction program. Representative timings for a number of molecules with different basis sets are presented. The new code is found to be significantly faster than the previous program. For ERIs involving only s and p functions, the new algorithm is a factor of two faster than previously. The new program is also found to be competitive when compared with other standard molecular packages, such as HONDO-8 and Gaussian 86.     

Synthesis of thermally-induced phase separating polymer with well-defined polymer structure by living cationic polymerization. I. Synthesis of poly(vinyl ether)s with oxyethylene units in the pendant and its phase separation behavior in aqueous solution

Living cationic polymerization of alkoxyethyl vinyl ether [CH₂?CHOCH₂CH₂OR; R: CH₃ (MOVE), C₂H₅ (EOVE)] and related vinyl ethers with oxyethylene units in the pendant was achieved by 1-(isobutoxy)ethyl acetate ( 1 )/Et_1.5AlCl_1.5 initiating system in the presence of an added base (ethyl acetate or THF) in toluene at 0°C. The polymers had a very narrow molecular weight distribution (M?_w/M?_n = 1.1–1.2) and the M?_n proportionally increased with the progress of the polymerization reaction. On the other hand, the polymerization by 1 /EtAlCl₂ initiating system in the presence of ethyl acetate, which produces living polymer of isobutyl vinyl ether, yielded the nonliving polymer. When an aqueous solution of the polymers thus obtained was heated, the phase separation phenomenon was clearly observed in each polymer at a definite critical temperature (T_ps). For example, T_ps was 70°C for poly(MOVE), and 20°C for poly(EOVE) (1 wt % aqueous solution, M?_n ～ 2 × 10⁴). The phase separation for each case was quite sensitive (ΔT_ps = 0.3–0.5°C) and reversible on heating and cooling. The T_ps or ΔT_ps was clearly dependent not only on the structure of polymer side chains (oxyethylene chain length and ω-alkyl group), but also on the molecular weight (M?_n = 5 × 10³-7 × 10⁴) and its distribution. © 1992 John Wiley & Sons, Inc.     

Determination of binding constants of multiple charged cyclodextrin complexes by ACE using uncorrected and ionic strength corrected actual mobilities of the species involved

In this study, the apparent binding constants and limiting mobilities of the multiply charged complexes of the Δ− and Λ−enantiomers of Ru(II)- and Fe(II)-polypyridyl associates ([Ru(2,2′-bipyridine)₃]²⁺, [Ru(1,10-phenanthroline)₃]²⁺, and [Fe(1,10-phenanthroline)₃]²⁺) with single-isomer 2,3-diacetylated-6-sulfated-cyclodextrins (CDs) (12Ac-6S-α-CD, 14Ac-7S-β-CD, and 16Ac-8S-γ-CD) were determined by ACE using uncorrected and ionic strength corrected actual mobilities of the species involved. Two limiting models were tested for the ionic strength correction of the actual mobilities based on an empirical relation for the ionic strength correction of multivalent ionic species. In model 1, the nominal values of the charge numbers (z_S,nom) and analytical concentrations (c_S,nom) of the above CD selectors in the BGEs were applied for calculation of the BGE ionic strength, as usual. In model 2, the CD selectors were considered as singly charged species (z_S = −1) with |z_S,nom|-times higher concentrations in the BGE than their analytical concentrations (c_S = |z_S,nom| × c_S,nom) in the calculation of the BGE ionic strength. In all three cases–with uncorrected actual mobilities as well as with actual mobilities corrected according to the two limiting models–the measured effective mobilities of the above enantiomers fit well the theoretical curves of their mobility dependences on the CD selectors concentrations in the BGE, with high average coefficients of determination (R² = 0.9890–0.9995). Nevertheless, the best physico-chemically meaningful values of the apparent binding constants and the limiting mobilities of the enantiomer-CDs complexes with low RSDs were obtained using the actual mobilities of the species involved corrected according to model 2.     

Coordination modes of diphosphines in silver(I)–diphosphine complexes mediated by 2,4,6-tri(2-pyridyl)-1,3,5-triazine (tptz)

Silver (I) complexes [Ag₂(tptz)(dppm)₂(DMF)](BF₄)₂·2DMF (1), [Ag(tptz)(dppe)]_n(BF₄)_n·2nH₂O·nMeOH (2), [Ag₂(tptz)₂(dppp)₂](BF₄)₂ (3) and [Ag₂(tptz)₂(dppb)](BF₄)₂ (4) were obtained from the reactions of AgBF₄ and diphosphine Ph₂P(CH₂) _nPPh₂ (L_pp, n=1–4) in the presence of 2,4,6-tris(2-pyridyl)-1,3,5-triazine (tptz) in MeOH–DMF. Single crystal analyses showed that the closed metallocyclic unit [Ag₂(L_pp)₂]²⁺ with double L_pp bridges was obtained in (1) and (3) with an odd number of n, while an open metallochain (Ag₂L_pp)²⁺ with a single bridge formed in (2) and (4) with n being even. Coordination modes for the diphosphines are directly related to the rigid tptz ligand with a large -system.     

Planar Chiral Multiple Resonance Thermally Activated Delayed Fluorescence Materials for Efficient Circularly Polarized Electroluminescence

Chiral boron/nitrogen doped multiple resonance thermally activated delayed fluorescence (MR-TADF) emitters are promising for highly efficient and color-pure circularly polarized organic light-emitting diodes (CP-OLEDs). Herein, we report two pairs of MR-TADF materials (Czp-tBuCzB, Czp-POAB) based on planar chiral paracyclophane with photoluminescence quantum yields of up to 98 %. The enantiomers showed symmetric circularly polarized photoluminescence spectra with dissymmetry factors |g_PL| of up to 1.6×10⁻³ in doped films. Meanwhile, the sky-blue CP-OLEDs with (R/S)-Czp-tBuCzB showed an external quantum efficiency of 32.1 % with the narrowest full-width at half-maximum of 24 nm among the reported CP-OLEDs, while the devices with (R/S)-Czp-POAB displayed the first nearly pure green CP electroluminescence with |g_EL| factors at the 10⁻³ level. These results demonstrate the incorporation of planar chirality into MR-TADF emitter is a reliable strategy for constructing of efficient CP-OLEDs.     

Expression of theSU(2) rotation matrices D( j ) in terms of normalized irreducible tensorial matrices

TheSU(2) rotation matricesD ^(j), specified in terms of axis and angle of rotation, are expressed as linear combination of normalized irreducible tensorial matricces (NITM) of rankl = 0 to 2j rotated to the polar angles of the axis. The rotated NITM are constructed from spherical harmonics of the same rank. Since this formulation requires no matrix products, it may be computationally more efficient than Euler angle formulas, particularly for largej. Rotated NITM and formulas for theD ^(j) withj = 1/2 andj = 1 are written out explicitly. A formula for the structure constants of the products of conformable NITM is also given in terms of 3-j and 6-j symbols.