ACE algorithm for the rapid evaluation of the electron-repulsion integral over Gaussian-type orbitals

A new series of general formulas to evaluate the electron-repulsion integral (ERI) can be derived from modifying the Gauss-Rys quadrature formula. These named as “accompanying coordinate expansion (ACE) formulas” are capable of evaluating very fast ERIs, especially for contracted Gaussian-type orbitals (GTOs). According to the degree of the contraction of GTOs, the optimum formula can be selected among these ACEs. Numerical examples are shown for (ps|ps) and (pp|pp) ERIs as typical examples. It is found that the present ACE algorithm is numerically stable and is most efficient among all algorithms in the literature in the floating-point-operation (FLOP) count for all varieties of the degree of contraction. © 1996 John Wiley & Sons, Inc.     

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     

Molecular integrals over the gauge-including atomic orbitals. II. The Breit-Pauli interaction

Each accompanying coordinate expansion (ACE) formula is derived for each of the orbit-orbit interaction, the spin-orbit coupling, the spin-spin coupling, and the contact interaction integrals over the gauge-including atomic orbitals (GIAOs) by the use of the solid harmonic gradient (SHG) operator. Each ACE formula is the general formula derived at the first time for each of the above molecular integrals over GIAOs. These molecular integrals are arising in the Breit-Pauli two-electron interaction for a relativistic calculation. We may conclude that we can derive a certain ACE formula for any kind of molecular integral over solid harmonic Gaussian-type orbitals by using the SHG operator. The present ACE formulas will be useful, for example, for a calculation of a molecule in a uniform magnetic field, for a relativistic calculation, and so on, with the GIAO as a basis function.     

Exchange energy gradients with respect to atomic positions and cell parameters within the Hartree-Fock Gamma-point approximation

Recently, linear scaling construction of the periodic exact Hartree-Fock exchange matrix within the Gamma-point approximation has been introduced [J. Chem. Phys. 122, 124105 (2005)]. In this article, a formalism for evaluation of analytical Hartree-Fock exchange energy gradients with respect to atomic positions and cell parameters at the Gamma-point approximation is presented. While the evaluation of exchange gradients with respect to atomic positions is similar to those in the gas phase limit, the gradients with respect to cell parameters involve the accumulation of atomic gradients multiplied by appropriate factors and a modified electron repulsion integral (ERI). This latter integral arises from use of the minimum image convention in the definition of the Gamma-point Hartree-Fock approximation. We demonstrate how this new ERI can be computed with the help of a modified vertical recurrence relation in the frame of the Obara-Saika and Head-Gordon-Pople algorithm. As an illustration, the analytical gradients have been used in conjunction with the QUICCA algorithm [K. Nemeth and M. Challacombe, J. Chem. Phys. 121, 2877 (2004)] to optimize periodic systems at the Hartree-Fock level of theory.     

Projected gradient algorithms for Hartree-Fock and density matrix functional theory calculations

We present projected gradient algorithms designed for optimizing various functionals defined on the set of N-representable one-electron reduced density matrices. We show that projected gradient algorithms are efficient in minimizing the Hartree-Fock or the Muller-Buijse-Baerends functional. On the other hand, they converge very slowly when applied to the recently proposed BBk (k=1,2,3) functionals [O. Gritsenko et al., J. Chem. Phys. 122, 204102 (2005)]. This is due to the fact that the BBk functionals are not proper functionals of the density matrix.     

Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

We present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

     

Polymorphisms in two genes, IL-1B and ACE, are associated with erythropoietin resistance in Korean patients on maintenance hemodialysis

Genetic polymorphisms may be linked to inter-individual differences in erythropoietin (EPO) resistance. We investigated the -511C/T polymorphism of the IL-1B gene and the I/D polymorphism of the ACE gene for any association with EPO resistance index (ERI) in maintenance hemodialysis patients (n=167). Because EPO responsiveness is multi-factorial, we also included other possible influences (age, sex, time on dialysis, ACE inhibitor or angiotensin receptor blocker use, ferritin, transferrin saturation, intact PTH, high sensitivity C-reactive protein, albumin, Kt/V, and presence of diabetes mellitus) on ERI in our analyses. Multiple regression analysis showed significant association of the IL-1B-511CC and ACE DD polymorphisms with ERI (P=0.038 and P=0.004 in the recessive model, respectively). The combination (C) of alleles of two loci showed that C1 (I-T) was significantly associated with ERI in the co-dominant and recessive models (P=0.005 and P=0.0001, respectively). Subjects who did not carry C1 showed significantly decreased ERI (10.10+/-5.15 IU/kg weight/g hemoglobin) compared to other study subjects (C1/C1 and C1/-; 12.97+/-4.90 and 15.12+/-7.43 IU/kg weight/g hemoglobin, respectively). Our study indicates that the IL-1B-511C/T and ACE I/D polymorphisms may be useful genetic markers of EPO requirement in hemodialysis patients. These findings might also provide a new perspective on therapeutic approaches to the treatment of end stage renal disease patients with anemia.     

Communication: extended multi-state complete active space second-order perturbation theory: energy and nuclear gradients

The extended multireference quasi-degenerate perturbation theory, proposed by Granovsky [J. Chem. Phys. 134, 214113 (2011)], is combined with internally contracted multi-state complete active space second-order perturbation theory (XMS-CASPT2). The first-order wavefunction is expanded in terms of the union of internally contracted basis functions generated from all the reference functions, which guarantees invariance of the theory with respect to unitary rotations of the reference functions. The method yields improved potentials in the vicinity of avoided crossings and conical intersections. The theory for computing nuclear energy gradients for MS-CASPT2 and XMS-CASPT2 is also presented and the first implementation of these gradient methods is reported. A number of illustrative applications of the new methods are presented.     

Alternative single-reference coupled cluster approaches for multireference problems: the simpler, the better

We report a general implementation of alternative formulations of single-reference coupled cluster theory (extended, unitary, and variational) with arbitrary-order truncation of the cluster operator. These methods are applied to compute the energy of Ne and the equilibrium properties of HF and C(2). Potential energy curves for the dissociation of HF and the BeH(2) model computed with the extended, variational, and unitary coupled cluster approaches are compared to those obtained from the multireference coupled cluster approach of Mukherjee et al. [J. Chem. Phys. 110, 6171 (1999)] and the internally contracted multireference coupled cluster approach [F. A. Evangelista and J. Gauss, J. Chem. Phys. 134, 114102 (2011)]. In the case of Ne, HF, and C(2), the alternative coupled cluster approaches yield almost identical bond length, harmonic vibrational frequency, and anharmonic constant, which are more accurate than those from traditional coupled cluster theory. For potential energy curves, the alternative coupled cluster methods are found to be more accurate than traditional coupled cluster theory, but are three to ten times less accurate than multireference coupled cluster approaches. The most challenging benchmark, the BeH(2) model, highlights the strong dependence of the alternative coupled cluster theories on the choice of the Fermi vacuum. When evaluated by the accuracy to cost ratio, the alternative coupled cluster methods are not competitive with respect to traditional CC theory, in other words, the simplest theory is found to be the most effective one.     

Accurate and efficient method for the treatment of exchange in a plane-wave basis

We describe an accurate and efficient extension of Chawla and Voth's [J. Chem. Phys. 108, 4697 (1998)] plane-wave based algorithm for calculating exchange energies, exchange energy densities, and exchange energy gradients with respect to wave-function parameters in systems of electrons subject to periodic boundary conditions. The theory and numerical results show that the computational effort scales almost linearly with the number of plane waves and quadratically with the number of k vectors. To obtain high accuracy with relatively few k vectors, we use an adaptation of Gygi and Baldereschi's [Phys. Rev. B 34, 4405 (1986)] method for reducing Brillouin-zone integration errors.     

Steric guided change of electron transfer mechanism in benzene

In fluorescence quenching study via electron transfer (ET), the quenching rate constant (k(q)) values generally decrease with lowering of quencher concentration, since smaller concentration of quencher always leads to a red shift in the donor-acceptor (D-A) distance in ET [M. Tachiya, S. Murata, J. Phys. Chem. 96 (1992) 8441; S. Murata, M. Tachiya, J. Phys. Chem. 100 (1996) 4064; L. Burel, M. Mastafavi, S. Murata, M. Tachiya, J. Phys. Chem. A 103 (1999) 5882]. However, while doing a comparative study with different carbazole (CZ) derivatives-1,4-dicyanobenzene (DCB) systems in benzene (BZ), we observed a deviation from that normal behaviour. It was found that for all of them with lower quencher (DCB) concentration, k(q) values actually increase instead of the expected reduction. Exceptionally, for simple CZ (C12H9N) with decrease in concentration of DCB, k(q) values can even reach the order of energy transfer (10(11) s(-1)). Interestingly, it is not observed when toluene (TL) or xylene (XY) is used as solvent. To explain this unique observation, a sandwich type of molecular structure is predicted, where BZ sliding in between CZ and DCB brings them closer enough, imparting more through bond character to CZ-DCB interaction and hence a higher rate of ET (k(q)) is observed [L. Burel, M. Mastafavi, S. Murata, M. Tachiya, J. Phys. Chem. A. 103 (1999) 5882].     

Low-temperature inelastic collisions between hydrogen molecules and helium atoms

Inelastic H(2):He collisions are studied from the experimental and theoretical points of view between 22 and 180 K. State-to-state cross sections and rates are calculated at the converged close-coupling level employing recent potential energy surfaces (PES): The MR-PES [J. Chem. Phys. 100, 4336 (1994)], and the MMR-PES and BMP-PESs [J. Chem. Phys. 119, 3187 (2003)]. The fundamental rates k(2-->0) and k(3-->1) for H(2):He collisions are assessed experimentally on the basis of a master equation describing the time evolution of rotational populations of H(2) in the vibrational ground state. These populations are measured in the paraxial region of supersonic jets of H(2)+He mixtures by means of high-sensitivity and high spatial resolution Raman spectroscopy. Good agreement between theory and experiment is found for the k(2-->0) rate derived from the MR-PES, but not for the BMP-PES. For the k(3-->1) rate, which is about one-third to one-half of k(2-->0), the result is less conclusive. The experimental k(3-->1) rate is compatible within experimental error with the values calculated from both PESs. In spite of this uncertainty, the global consistence of experiment and theory in the framework of Boltzmann equation supports the MR-PES and MMR-PESs, and the set of gas-dynamic equations employed to describe the paraxial region of the jet at a molecular level.     

SACM/CT Study of the dissociation/recombination dynamics of hydrogen peroxide on an ab initio potential energy surface. Part II. Specific rate constants k(E,J), thermal rate constants k infinity(T), and lifetime distributions

Statistical adiabatic channel model/classical trajectory (SACM/CT) calculations of the dissociation/recombination dynamics of hydrogen peroxide, H(2)O(2) <--> 2HO, have been performed on an ab initio potential energy surface by Kuhn, Rizzo, Luckhaus, Quack, and Suhm (J. Chem. Phys. 1999, 111, 2565). Specific rate constants k(E,J), thermal rate constants k(infinity)(T), and lifetime distributions are determined. After averaging over J, the derived k(E,J) are in quantitative agreement with non-exponential time-profiles of HO formation recorded after overtone excitation of H(2)O(2) near the dissociation threshold by Scherer and Zewail (J. Chem. Phys. 1987, 87, 97). The thermal high pressure rate constants for HO recombination agree with experimental data as well and can be represented by k(rec,infinity)/10(-10) cm(3) molecule(-1) s(-1) approximately [0.376 (298 K/T)(0.47) + 0.013 (T/298 K)(0.74)] over the range 60-1500 K. Non-statistical lifetime distributions are suggested not to have been of major relevance for the available experiment.     

Parent-molecule rotational depolarization of photofragment angular momentum distributions: diatomic and polyatomic molecules

We extend the A(q)(k) polarization-parameter model, which describes product angular momentum polarization from one photon photodissociation of polyatomic molecules in the molecular frame [J. Chem. Phys., 2010, 132, 224310], to the case of rotating parent molecules. The depolarization of the A(q)(k) is described by a set of rotational depolarization factors that depend on the angle of rotation of the molecular axis γ. We evaluate these rotational depolarization factors for the case of dissociating diatomic molecules and demonstrate that they are in complete agreement with the results of Kuznetsov and Vasyutinskii [J. Chem. Phys., 2005, 123, 034307] obtained from a fully quantum mechanical approach of the same problem, showing the effective equivalence of the two approaches. We further evaluate the set of rotational depolarization factors for the case of dissociating polyatomic molecules that have three (near) equal moments of inertia, thus extending these calculations to polyatomic systems. This ideal case yields insights for the dissociation of polyatomic molecules of various symmetries when we compare the long lifetime limit with the results obtained for the diatomic case. In particular, in the long lifetime limit the depolarization factors of the A(0)(k) (odd k), Re(A(1)(k)) (even k) and Im(A(1)(k)) (odd k) for diatomic molecules vanish; in contrast, for polyatomic molecules the depolarization factors for the A(0)(k) (odd k) reduce to a value of 1/3, whereas for the Re(A(1)(k)) (even k) and Im(A(1)(k)) (odd k) they reduce to 1/5.     

Generalized gradient approximation model exchange holes for range-separated hybrids

We propose a general model for the spherically averaged exchange hole corresponding to a generalized gradient approximation (GGA) exchange functional. Parameters are reported for several common GGAs. Our model is based upon that of Ernzerhof and Perdew [J. Chem. Phys. 109, 3313 (1998)]. It improves upon the former by precisely reproducing the energy of the parent GGA, and by enabling fully analytic evaluation of range-separated hybrid density functionals. Analytic results and preliminary thermochemical tests indicate that our model also improves upon the simple, local-density-based exchange hole model of Iikura et al. [J. Chem. Phys. 115, 3540 (2001)].     

Characterization of silicon-carbon clusters by infrared laser spectroscopy: the nu 3(sigma u) band of linear Si2C3

The nu 3(sigma u) fundamental vibration of 1 sigma g+ Si2C3 has been observed using a laser vaporization-supersonic cluster beam-diode laser spectrometer. Forty rovibrational transitions were measured in the range of 1965.8 to 1970.9 cm-1 with a rotational temperature of 10-15 K. A least-squares fit of these transitions yielded the following molecular constants: nu 3(sigma u)=1968.188 31(18) cm-1, B"=0.031 575 1(60) cm-1, and B'=0.031 437 4(57) cm-1. These results are in excellent agreement with recent Fourier transform infrared (FTIR) measurements of Si2C3 trapped in a solid Ar matrix [J. Chem. Phys. 100, 181(1994)] and with ab initio calculations [J. Chem. Phys. 100, 175 (1994)] which suggest cumulenic-like bonding for Si2C3, analogous to the isovalent C5 carbon cluster.     

Semiclassical treatment of thermally activated electron transfer in the intermediate to strong electronic coupling regime under the fast dielectric relaxation

The generalized nonadiabatic transition-state theory (NA-TST) (Zhao, Y.; et al. J. Chem. Phys. 2004, 121, 8854) is used to study electron transfer with use of the Zhu-Nakamura (ZN) formulas of nonadiabatic transition in the case of fast dielectric relaxation. The rate constant is expressed as a product of the well-known Marcus formula and a coefficient which represents the correction due to the strong electronic coupling. In the case of general multidimensional systems, the Monte Carlo approach is utilized to evaluate the rate by taking into account the multidimensionality of the crossing seam surface. Numerical demonstration is made by using a model system of a collection of harmonic oscillators in the Marcus normal region. The results are naturally coincident with the perturbation theory in the weak electronic coupling limit; while in the intermediate to strong electronic coupling regime where the perturbation theory breaks down the present results are in good agreement with those from the quantum mechanical flux-flux correlation function within the model of effective one-dimensional mode.     

Spherical hybrids

A general solution is presented to the problem of finding hybrid functions which point from the center toward the vertices of a regular solid. Results are given for the examples of 6, 8, 12, and 24 vertices. By taking linear combinations with suitable alias functions, the hybrids have been given nodes passing through all vertices except their own. This allows them to be used for interpolation and quadrature. The various functions required are defined as solid harmonics and group theoretical methods of generating these are discussed fully. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 197–214, 1997     

Porous agglomerates in the general linear flow field

We calculate the flow within and around a porous spherical agglomerate suspended in the general linear flow field, and also the flow induced by its rotation. We use the Stokes equations exterior to the particle and the Brinkman equations inside it. The effect of particle permeability on the flow is expressed via the Brinkman parameter beta = r(0)/square root of k, where r0 is particle radius and k is its permeability. With translational creeping motion of porous spheres in a quiet fluid investigated by Debye and Bueche [P. Debye, A.M. Bueche, J. Chem. Phys. 16 (6) (1943) 573-579], this study provides information necessary for investigating dynamics of porous particles moving in creeping shear flows under the action of external forces and torques. The agglomerate flow field solutions are used to calculate the effective viscosity of a dilute suspension of porous solid aggregates, which generalizes the well-known Einstein's equation for solid suspensions. The agglomerate effective viscosity diameter is proposed which allows using the Einstein's formula evaluation of the agglomerates suspension viscosity.