Distance-dependent Schwarz-based integral estimates for two-electron integrals: reliable tightness vs. rigorous upper bounds

A new integral estimate for four-center two-electron integrals is introduced that accounts for distance information between the bra- and ket-charge distributions describing the two electrons. The screening is denoted as QQR and combines the most important features of the conventional Schwarz screening by Ha?ser and Ahlrichs published in 1989 [J. Comput. Chem. 10, 104 (1989)] and our multipole-based integral estimates (MBIE) introduced in 2005 [D. S. Lambrecht and C. Ochsenfeld, J. Chem. Phys. 123, 184101 (2005)]. At the same time the estimates are not only tighter but also much easier to implement, so that we recommend them instead of our MBIE bounds introduced first for accounting for charge-distance information. The inclusion of distance dependence between charge distributions is not only useful at the SCF level but is particularly important for describing electron-correlation effects, e.g., within AO-MP2 theory, where the decay behavior is at least 1/R(4) or even 1/R(6). In our present work, we focus on studying the efficiency of our QQR estimates within SCF theory and demonstrate the performance for a benchmark set of 44 medium to large molecules, where savings of up to a factor of 2 for exchange integrals are observed for larger systems. Based on the results of the benchmark set we show that reliable tightness of integral estimates is more important for the screening performance than rigorous upper bound properties.     

Rigorous integral screening for electron correlation methods

We derive rigorous multipole-based integral estimates (MBIE) in order to account for the distance dependence occurring in atomic-orbital (AO) formulations of electron correlation theory, where our focus is on AO-MP2 theory within a Laplace scheme. We find for the exact transformed integral products an extremely early onset of a linear-scaling behavior and a very small number of significant products. To preselect the significant integral products we adapt our MBIE method as rigorous upper bound. In this way it is possible to exploit the favorable scaling behavior observed and to reduce the scaling of estimated products asymptotically to linear, without sacrificing accuracy or reliability. By separating Coulomb- and exchange-type contractions only half-transformed integrals need to be computed. Furthermore, our scheme of rigorously preselecting transformed integral products via MBIE seems to offer particularly interesting perspectives for a direct formation of half- or fully transformed integrals by using multipole expansions and auxiliary basis sets.     

Tighter multipole-based integral estimates and parallel implementation of linear-scaling AO-MP2 theory

Within an atomic-orbital-based (AO-based) formulation of second-order M?ller-Plesset perturbation theory (MP2), we present a novel screening procedure which allows us to preselect numerically significant two-electron integrals more efficiently, especially for large basis sets. The screening is based on our recently introduced multipole-based integral estimates (MBIE) method [J. Chem. Phys., 2005, 123, 184102], that allows to exploit the 1/R(4) or 1/R(6) coupling between electronic charge distributions in transformed integral products within AO-MP2. In this way, linear scaling is attained with fully-controlled numerical accuracy. Furthermore, a parallel implementation of our linear-scaling AO-MP2 method is described, which also allows us to perform calculations with larger basis sets. First calculations reveal that for e.g. linear alkanes the scaling of the number of required transformed integral products is almost equal for 6-31G* and cc-pVTZ basis sets. Using the improved MBIE screening, the largest parallel calculation was performed for a ribozyme fragment consisting of 497 atoms and 5697 basis functions, while our largest AO-MP2 calculation was performed for a stacked DNA system (16 base pairs) comprising 1052 atoms and 10 674 basis functions on a single processor.     

Efficient evaluation of short-range Hartree-Fock exchange in large molecules and periodic systems

We present an efficient algorithm for the evaluation of short-range Hartree-Fock exchange energies and geometry gradients in Gaussian basis sets. Our method uses a hierarchy of screening levels to eliminate negligible two-electron integrals whose evaluation is the fundamental computational bottleneck of the procedure. By applying our screening technique to the Heyd-Scuseria-Ernzerhof [J. Chem. Phys. 118, 8207 (2003)] short-range Coulomb hybrid density functional, we achieve a computational efficiency comparable with that of standard nonhybrid density functional calculations.     

Explicitly correlated coupled cluster F12 theory with single and double excitations

Full explicitly correlated F12 coupled cluster theory with single and double excitations and with Slater-type geminal as a correlation factor is introduced and implemented within the standard approximation. The variant "C" that does not require integrals over the commutator between the kinetic operator and the correlation factor has been used. All the necessary integrals are analytically calculated. With variant C also, first results are reported for the correlation factor being the interelectronic distance coordinate, i.e., for original R12 method. Calculations have been performed for a set of eight molecules including CH(2)((1)A(1)), CH(4), NH(3), H(2)O, HF, CO, N(2), and F(2), as well as for the constituting atoms. Atomization energies are reported too.     

Method for performing LCAO band structure calculations in crystalline solids: Application to rubidium

In this work we present a method for performing LCAO (linear combination of atomic orbitals) band structure calculations (tight binding) in crystalline solids. In the first part of the article we apply group theoretical methods to the establishment of a least‐squares scheme for the calculation of the matrix of the crystal potential: This scheme is based on a well‐defined choice of independent parameters for the Bloch vector‐dependent matrix elements and on the considerations of the symmetries between these independent parameters and their Fourier coefficients. In the second part of this work we deal with the representation of the matrices of the identity operator and of the operator of the kinetic energy by linear combinations in terms of two center integrals: We express these linear combinations by a closed formula, which can be easily programmed on a computer, and we mention a method by which the two center integrals can be evaluated numerically fast and accurately. Finally we apply our theory to the derivation of numerical results: We determine the electronic states and the high‐momentum components of Compton rates in the alkali metal rubidium and we compare the results obtained with those of augmented plane‐wave (APW) calculations. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 212–225, 2000     

New implementation of second-order Møller-Plesset perturbation theory with an analytic Slater-type geminal

The author introduces a new method for the exchange commutator integrals in explicitly correlated M?ller-Plesset second order perturbation theory. The method is well suited with an analytic Slater-type geminal correlation factor. He also explains the scheme for auxiliary integrals needed for the correlation factor. Based on different Ans?tze, he analyzes the performance of the method on correlation energies and reaction enthalpies in detail.     

On the Rotational Invariance of the INDO Method Including f Orbitals into the Basis Set

In order to ensure the rotational invariance of the one-center-two-electron integrals in the INDO method including f orbitals into the basis set in our previous paper two methods are proposed in the present investigation to meet this requirement.The first one is to use the weighted average values for replacing the integrals respectively.The second is a more stringent alternative,in which we have derived 690 nonzero hybrid one-center-two-electron integrals in addition to Coulomb and exchange types as expressions of Slater-Condon parameters.     

Density fitting of two-electron integrals in extended systems with translational periodicity: the Coulomb problem

Density fitting approach to Coulomb integrals for infinite systems with translational periodicity is reformulated in direct space. Despite of the Coulomb infinite decay of some integrals, direct-space calculation is shown to be feasible. Moreover, we show that the direct-space ansatz is completely equivalent to our previous formulation in reciprocal space. Computational demands scale linearly with the number of unit cells. In addition, direct-space treatment has some practical advantages over the reciprocal-space formulation. The efficiency of our scheme is demonstrated on systems with translational periodicity in one dimension. Computation time takes only a small fraction of the conventional calculation with exact integrals. We show that for infinite systems auxiliary basis sets of equally good quality as for molecules can be constructed in a systematic way.     

Evaluation of protein 15N relaxation times by inverse Laplace transformation

Relaxation times (T1, T2, T1rho) are usually evaluated from exponential decay data by least-squares fitting methods. For this procedure, the integrals or amplitudes of signals must be determined, which can be laborious with large data sets. Moreover, the fitting requires a priori knowledge of the number of exponential components responsible for the decay. We have adapted inverse Laplace transformation (ILT) for the analysis of relaxation data. Exponential components are resolved with ILT to reciprocal space on their corresponding relaxation rate values. The ILT approach was applied to 3D linewidth-resolved 15N HSQC experiments to evaluate 15N T1 and T2 relaxation times of ubiquitin. The resulting spectrum is a true 3D spectrum, where the signals are separated by their 1H and 15N chemical shifts (HSQC correlations) and by their relaxation rate values (R1 or R2). From this spectrum, the relaxation times can be obtained directly with a simple peak-picking procedure.     

The determination of the Förster distance (R0) for phenanthrene and anthracene derivatives in poly(methyl methacrylate) films

Experiments that employ direct resonance energy transfer (DET) to obtain information about distances or domain sizes in polymer systems require independent information about the magnitude of the characteristic (F?rster) energy transfer distance R(0). Values of R(0) are relatively straightforward to obtain by the traditional spectral overlap method (R(0)(SO)) for dyes in fluid solution, but are much more difficult to obtain for dyes in rigid polymer films. Here one can obtain a value for R(0) as a fitting parameter (R(0)(FF)) for donor fluorescence decay experiments for samples containing a random distribution of donor and acceptor dyes in the polymer film. In previous experiments from our group, we needed values of R(0) for various phenanthrene (Phe, donor) and anthracene (An, acceptor) derivatives. In this paper, we describe experiments which determine R(0) values by both methods for a series of Phe-An donor-acceptor pairs in poly(methyl methacrylate) and polystyrene films. Both the location of substituents on the donor and acceptor as well as the choice of the medium had an effect on the measured R(0), which varied between 2.0 and 2.6 nm. We also ascertained that there is some unknown factor, also prevalent in the work of others, which results in the F?rster radius being larger when determined by the F?rster fit method than by the method of spectral overlap.     

Low-cost evaluation of the exchange Fock matrix from Cholesky and density fitting representations of the electron repulsion integrals

The authors propose a new algorithm, "local K" (LK), for fast evaluation of the exchange Fock matrix in case the Cholesky decomposition of the electron repulsion integrals is used. The novelty lies in the fact that rigorous upper bounds to the contribution from each occupied orbital to the exchange Fock matrix are employed. By formulating these inequalities in terms of localized orbitals, the scaling of computing the exchange Fock matrix is reduced from quartic to quadratic with only negligible prescreening overhead and strict error control. Compared to the unscreened Cholesky algorithm, the computational saving is substantial for systems of medium and large sizes. By virtue of its general formulation, the LK algorithm can be used also within the class of methods that employ auxiliary basis set expansions for representing the electron repulsion integrals.     

Tensor decomposition in post-Hartree-Fock methods. I. Two-electron integrals and MP2

A new approximation for post-Hartree-Fock (HF) methods is presented applying tensor decomposition techniques in the canonical product tensor format. In this ansatz, multidimensional tensors like integrals or wavefunction parameters are processed as an expansion in one-dimensional representing vectors. This approach has the potential to decrease the computational effort and the storage requirements of conventional algorithms drastically while allowing for rigorous truncation and error estimation. For post-HF ab initio methods, for example, storage is reduced to O(d·R·n) with d being the number of dimensions of the full tensor, R being the expansion length (rank) of the tensor decomposition, and n being the number of entries in each dimension (i.e., the orbital index). If all tensors are expressed in the canonical format, the computational effort for any subsequent tensor contraction can be reduced to O(R(2)·n). We discuss details of the implementation, especially the decomposition of the two-electron integrals, the AO-MO transformation, the M?ller-Plesset perturbation theory (MP2) energy expression and the perspective for coupled cluster methods. An algorithm for rank reduction is presented that parallelizes trivially. For a set of representative examples, the scaling of the decomposition rank with system and basis set size is found to be O(N(1.8)) for the AO integrals, O(N(1.4)) for the MO integrals, and O(N(1.2)) for the MP2 t(2)-amplitudes (N denotes a measure of system size) if the upper bound of the error in the l(2)-norm is chosen as ε = 10(-2). This leads to an error in the MP2 energy in the order of mHartree.     

Implementation of screened hybrid density functional for periodic systems with numerical atomic orbitals: basis function fitting and integral screening

We present an efficient O(N) implementation of screened hybrid density functional for periodic systems with numerical atomic orbitals (NAOs). NAOs of valence electrons are fitted with gaussian-type orbitals, which is convenient for the calculation of electron repulsion integrals and the construction of Hartree-Fock exchange matrix elements. All other parts of Hamiltonian matrix elements are constructed directly with NAOs. The strict locality of NAOs is adopted as an efficient two-electron integral screening technique to speed up calculations.     

A study of the valence interaction integrals in effective core potential applications

The valence interactions in two effective core potential (ECP) methods, the frozen orbital ECP and the method of Sakai and Huzinaga, are shown to yield atomic valence orbital Coulomb and exchange interaction integrals closely approximating all-electron calculations. The ECP approximation is studied in some detail with special application to the ScO molecule. The too short bond distance in ScO when the 3s. 3p orbitals are included in the core is shown to stem from a long-range attraction of the ECP.     

Second order coalescence conditions of molecular wave functions

Kato's cusp condition gives the exact first order dependence of molecular wave functions on interparticle separation near the coalescence of two charged particles. We derive conditions correct to second order in interparticle separation, which concern second order derivatives of the wave function at the coalescence point. For identical particle coalescence, we give equations correct to third order. In addition to a universal, particle dependent term, a system and state dependent term arises in the higher order conditions, which we interpret as an effect of Coulombic screening. We apply our analysis to the standard orbital-based methods of quantum chemistry and discuss the implications for Jastrow- and R12-type correlation factors.