Big data analysis of ab Initio molecular integrals in the neglect of diatomic differential overlap approximation

Most modern semiempirical quantum-chemical (SQC) methods are based on the neglect of diatomic differential overlap (NDDO) approximation to ab initio molecular integrals. Here, we check the validity of this approximation by computing all relevant integrals for 32 typical organic molecules using Gaussian-type orbitals and various basis sets (from valence-only minimal to all-electron triple-ζ basis sets) covering in total more than 15.6 million one-electron (1-e) and 10.3 billion two-electron (2-e) integrals. The integrals are calculated in the nonorthogonal atomic basis and then transformed by symmetric orthogonalization to the Löwdin basis. In the case of the 1-e integrals, we find strong orthogonalization effects that need to be included in SQC models, for example, by strategies such as those adopted in the available OMx methods. For the valence-only minimal basis, we confirm that the 2-e Coulomb integrals in the Löwdin basis are quantitatively close to their counterparts in the atomic basis and that the 2-e exchange integrals can be safely neglected in line with the NDDO approximation. For larger all-electron basis sets, there are strong multishell orthogonalization effects that lead to more irregular patterns in the transformed 2-e integrals and thus cast doubt on the validity of the NDDO approximation for extended basis sets. Focusing on the valence-only minimal basis, we find that some of the NDDO-neglected integrals are reduced but remain sizable after the transformation to the Löwdin basis; this is true for the two-center 2-e hybrid integrals, the three-center 1-e nuclear attraction integrals, and the corresponding three-center 2-e hybrid integrals. We consider a scheme with a valence-only minimal basis that includes such terms as a possible strategy to go beyond the NDDO integral approximation in attempts to improve SQC methods. © 2018 Wiley Periodicals, Inc.     

A Crystal Orbital approach for two- and three-dimensional solids on the basis of CNDO/INDO Hamiltonians. Basis equations

A Hartree–Fock (HF ) self-consistent field (SCF ) crystal orbital (CO ) formalism for two- and three-dimensional (2D/3D) solids on the basis of semiempirical CNDO /INDO (complete neglect of differential overlap; intermediate neglect of differential overlap) Hamiltonians is presented. The employed SCF variants allow for the treatment of atomic species up to bromine under the inclusion of the first (i.e., 3d) transition metal series. Band structure investigations of 2D and 3D materials containing more than 30 atoms per unit cell are feasible by the present SCF HF CO formalism. The theoretical background of the computational scheme is given in this contribution. Special emphasis is placed on physically reliable truncation criteria for the lattice sums, the adaptation of the crystal symmetry in k space, as well as the suitable choice of domains in Brillouin zone (BZ ) integrations required in the determination of charge-density matrices. The capability and limitations of the semiempirical SCF HF CO approach is demonstrated for some simpler solids by comparing the present computational results with those of ab initio CO schemes as well as conventional numerical methods in soid-state theory. The employed model solids are graphite and BN (2D and 3D networks for both solids) as well as diamond, silicon, germanium, and TiS₂.     

Implementation of an NDDO/CI/SOS approach for second‐order hyperpolarizabilities

We report the implementation of the sum‐over‐states (SOS) formalism for neglect of diatomic differential overlap (NDDO) Hamiltonians Austin model 1 (AM1), parametric method 3 (PM3), and modified neglect of differential overlap (MNDO) for calculation of third‐order nonlinear optical properties into the program package VAMP and its application to third harmonic generation (THG). Extensive comparisons between THG experimental data and published MOPAC/FF to VAMP/PECI/SOS and AMPAC/FF calculated values were carried out in gas phase and in solvent for a data set of 236 compounds of the general type DπA and conjugated π systems. Great care was taken to derive the global minimum conformers, yielding significant deviations of the geometries derived by the three Hamiltonians. The data set therefore gives an overview of the shortcomings and strengths of the semiempirical methods. Here, the implementations of solvent effects in both semiempirical packages are especially problematic in the case of elongated molecules, so a threshold for molecular globularity had to be defined to eliminate erroneous data. The correlation statistics presented for γ are in acceptable agreement for the whole data set as for all experimentally well‐defined substance classes with scalable correlation slopes smaller than unity. The data become more reliable for large γ, probably due to more precise experimental values. Inclusion of solvent effects raises the polarizabilities of the molecules consistently. These results enable us to qualitatively predict trends for small as well as large second‐order polarizabilities, to derive scaling functions for quantitative predictions, and to calculate experimentally nonaccessible tensor elements of γ. The SOS formalism even allows us to obtain insights into the frequency dependence of second‐order hyperpolarizability effects beyond THG. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 727–760, 2000     

An optimum strategy for solution chemistry using semiempirical molecular orbital method: Importance of description of charge distribution

For the purpose to execute direct dynamics calculation in solution chemistry, we propose an optimum strategy for solution chemistry using semiempirical molecular orbital (MO) method with neglect of diatomic differential overlap (NDDO) approximation with specific solution reaction parameters (SSRP), i.e., the NDDO‐SSRP method. In this strategy, the empirical parameters of the semi‐empirical MO method were optimized individually for target molecule or ion by reference to the ab initio MO calculation data for many configurations on the potential energy surface near the reaction path. For demonstration, the NDDO‐SSRP method was applied to two molecules and two ions (OH^?, H₂O, NH₃, NH₄⁺) at their equilibrium states in aqueous solution, respectively. Accordingly, it was verified that both the potential energy surface and the charge distribution of these solutes in aqueous solution are dramatically improved to reproduce themselves accurately at ab initio MO calculation level. In conclusion, it is expected that the NDDO‐SSRP method should become quite useful for dynamic and statistical applications to chemical reaction systems in solution. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Neglect of four- and approximation of one-, two-, and three-center two-electron integrals in a symmetrically orthogonalized basis

A new integral approximation for use in molecular electronic structure calculations is proposed as an alternative to the traditional neglect of diatomic differential overlap models. The similarity between the symmetrically orthogonalized and the original basis functions (assumed orthonormal within each atomic set but nonorthogonal between different centers) is used to construct a robust approximation for the two-electron integrals, with the error being quadratic in the deviation between the products of the functions. Invariance properties of this procedure are rigorously proved. Numerical studies on a representative set of molecules at valence-only minimal basis Hartree-Fock level show that the approximation introduces relatively small errors, encouraging its future application in the semiempirical field.     

Comparison of the RHF,NDDO, and MOM molecular one-electron expectation values calculated using minimum basis sets with Slater,Burns, Clementi,and BLMO exponents

The effect of different exponents, Slater, Burns, Clementi, and Best Limited Molecular Orbital (BLMO ) on the approximate one-electron property expectation values from minimum basis-set calculations is reported for Roothaan–Hartree–Fock (RHF ), neglect of diatomic differential overlap (NDDO ), and maximum overlap method (MOM ) calculations on FH, CO, and LiH.     

Semiempirical NDDO calculations with STO-3G and 4-31G basis sets

Possible refinements of semiempirical methods include the use of larger basis sets and of correlated wave functions. These possibilities are investigated in semiempirical NDDO SCF calculations with the STO-3G and 4-31G basis sets, and in correlated calculations at the STO-3G level. The present approach is characterized by the analytical evaluation of all one-center terms and two-electron integrals, and the semiempirical adjustment of the remaining one-electron integrals and the nuclear repulsions. The NDDO SCF results tend to reproduce the correspondingab initio results more closely than experimental data, even if they are parametrized with respect to experiment. The explicit inclusion of electron correlation at the STO-3G level improves the calculated results only slightly.     

Optimization of parameters for semiempirical methods I. Method

A new method for obtaining optimized parameters for semiempirical methods has been developed and applied to the modified neglect of diatomic overlap (MNDO) method. The method uses derivatives of calculated values for properties with respect to adjustable parameters to obtain the optimized values of parameters. The large increase in speed is a result of using a simple series expression for calculated values of properties rather than employing full semiempirical calculations. With this optimization procedure, the rate-determining step for parameterizing elements changes from the mechanics of parameterization to the assembling of experimental reference data.     

Gaussian and finite-element Coulomb method for the fast evaluation of Coulomb integrals

The authors propose a new linear-scaling method for the fast evaluation of Coulomb integrals with Gaussian basis functions called the Gaussian and finite-element Coulomb (GFC) method. In this method, the Coulomb potential is expanded in a basis of mixed Gaussian and finite-element auxiliary functions that express the core and smooth Coulomb potentials, respectively. Coulomb integrals can be evaluated by three-center one-electron overlap integrals among two Gaussian basis functions and one mixed auxiliary function. Thus, the computational cost and scaling for large molecules are drastically reduced. Several applications to molecular systems show that the GFC method is more efficient than the analytical integration approach that requires four-center two-electron repulsion integrals. The GFC method realizes a near linear scaling for both one-dimensional alanine alpha-helix chains and three-dimensional diamond pieces.     

Semiempirical calculation of two-electron integrals using bipolar expansion of the Ohno potential

Bipolar expansion of the Ohno potential as a method of calculating two-center Coulomb integrals that appear in the NDDO approximation is generalized to one-center two-electron integrals. A unified semiempirical scheme is suggested for estimating two-electron interactions in molecules. This scheme can be readily extended to arbitrary Slater basis sets (including the s,p,d-orbitals) and involves no a priori data on the valent states of atoms. In this work, the scheme is employed to extend the semiempirical PM3 method to the s,p,d-basis set. The efficiency of the method is proven by test calculations of 24 chromium compounds (π-complexes, carbonyls, isocyanides, etc.). Scientific Research Institute of Chemistry at N. I. Lobachevskii Nizhnii Novgorod State University. Translated fromZhurnal Struktumoi Khimii, Vol. 36, No. 4, pp. 593–599, July–August, 1995. Translated by I. Izvekova.     

Calculation of two-center two-electron integrals in the slaters,p,d-basis set for the Ono electron-electron potential

This paper deals with the problems of extending semiempirical MNDO methods to compounds with d-elements. The problem is solved by estimating two-center two-electron integrals (TTI) with the Ono potential modeling interaction between two electrons in a molecule. A scheme for calculating TTI is suggested which uses the expansion of the Ono potential in a series of pairwise products of spherical harmonics centered on two atoms in the molecule. The scheme is stable and efficient for calculations in arbitrary Slater basis sets (including the s,p,d-basis set) and seems to be useful for development of NDDO methods. Scientific Research Institute of Chemistry at N. I. Lobachevskii Nizhnii Novgorod State University. Translated fromZhurnul Strukturnoi Khimii Vol. 35, No. 4, pp. 24–27, July–August, 1994. Translated by L. Smolina     

Direct INDO/SCI method for excited state calculations

Intermediate neglect of differential overlap (INDO) is the most commonly utilized semiempirical technique for performing excited state calculations on large organic systems such as organic semiconductors and fluorescent dyes. The calculations are typically done at the singles-configuration interaction (SCI) level. Direct methods provide a more efficient means of performing configuration interaction (CI) calculations, and the computational trade offs associated with various approaches to direct-CI theory have been well characterized for ab initio Hamiltonians and high-order CI. However, the INDO and SCI approximations lead to a new set of trade offs. In particular, application of the electron-electron interactions in the atomic basis leads to savings in computational time that scale as the number of atomic orbitals, which for a large organic system can be two to three orders of magnitude. These savings are largest when only a few low-lying excited states are generated and when a full SCI basis, which includes excitations between all filled and empty molecular orbitals, is used. In addition, substantial memory savings are achieved in the direct method by avoiding the evaluation of the two electron integrals in the molecular orbital basis. The method is demonstrated by calculating the absorption spectrum of a poly(paraphenylenevinylene) oligomer containing 16 phenyl rings.     

Testing electronic structure methods for describing intermolecular H...H interactions in supramolecular chemistry

In this article a wide variety of computational approaches (molecular mechanics force fields, semiempirical formalisms, and hybrid methods, namely ONIOM calculations) have been used to calculate the energy and geometry of the supramolecular system 2-(2'-hydroxyphenyl)-4-methyloxazole (HPMO) encapsulated in beta-cyclodextrin (beta-CD). The main objective of the present study has been to examine the performance of these computational methods when describing the short range H. H intermolecular interactions between guest (HPMO) and host (beta-CD) molecules. The analyzed molecular mechanics methods do not provide unphysical short H...H contacts, but it is obvious that their applicability to the study of supramolecular systems is rather limited. For the semiempirical methods, MNDO is found to generate more reliable geometries than AM1, PM3 and the two recently developed schemes PDDG/MNDO and PDDG/PM3. MNDO results only give one slightly short H...H distance, whereas the NDDO formalisms with modifications of the Core Repulsion Function (CRF) via Gaussians exhibit a large number of short to very short and unphysical H...H intermolecular distances. In contrast, the PM5 method, which is the successor to PM3, gives very promising results. Our ONIOM calculations indicate that the unphysical optimized geometries from PM3 are retained when this semiempirical method is used as the low level layer in a QM:QM formulation. On the other hand, ab initio methods involving good enough basis sets, at least for the high level layer in a hybrid ONIOM calculation, behave well, but they may be too expensive in practice for most supramolecular chemistry applications. Finally, the performance of the evaluated computational methods has also been tested by evaluating the energetic difference between the two most stable conformations of the host(beta-CD)-guest(HPMO) system.     

A combined SAMO-ZDO method for the evaluation of ionization potentials

A scheme for the ab initio calculation of vertical ionization potentials without the necessity to compute two-electron repulsion integrals is discussed. The method employs the simulated ab initio molecular orbital (SAMO) method to generate Koopman's theorem eigenvalues. These are then corrected for the change in relaxation and correlation effects due to ionization by a Green's function perturbation scheme, in which all necessary integrals are evaluated using the zero differential overlap (ZDO) approximations, in this case the complete neglect of differential overlap (CNDO) method.     

生物大分子体系量子化学计算方法新进展

本文就近年来报道的4 种研究生物大分子体系的量子化学计算方法(计算显微镜方法、定域分子轨道方法、线性标度半经验量子化学方法和并行算法) 作了较为详细的介绍, 并展望了该领域的研究前景。     

A re-examination of the justification of neglect of differential overlap approximations in terms of a power series expansion inS

Neglect of differential overlap methods are treated as approximations to calculations in a symmetrically orthogonalized basis. The accuracy of this approximation is investigated in terms of a power series expansion of the overlap matrix. TheS-matrix can be transformed into a matrix which will give a convergent series, and this series is used in the examination. The only approximation having any justification from this point of view is the NDDO method and even this neglects certain important three-electron integrals. Corrected expressions for the repulsion integral scaling factors introduced by Chandrasekharet al. are also derived. On leave from The Chemistry School, University of Western Australia.