Two-center Two-electron Integrals Research Articles

The eXact integral simplified time-dependent density functional theory (XsTD-DFT).

In the framework of simplified quantum chemistry methods, we introduce the eXact integral simplified time-dependent density functional theory (XsTD-DFT). This method is based on the simplified time-dependent density functional theory (sTD-DFT), where all semi-empirical two-electron integrals are replaced by exact one- and two-center two-electron integrals, while other approximations from sTD-DFT are kept. The performance of this new parameter-free XsTD-DFT method was benchmarked on excited state and (non)linear response properties, including ultra-violet/visible absorption, first hyperpolarizability, and two-photon absorption (2PA). For a set of 77 molecules, the results from the XsTDA approach were compared to the TDA data. XsTDA/B3LYP excitation energies only deviate on average by 0.14eV from TDA while drastically cutting computational costs by a factor of 20 or more depending on the energy threshold chosen. The absolute deviations of excitation energies with respect to the full scheme are decreasing with increasing system size, showing the suitability of XsTDA/XsTD-DFT to treat large systems. Comparing XsTDA and its predecessor sTDA, the new scheme generally improves excitation energies and oscillator strengths, in particular, for charge transfer states. TD-DFT first hyperpolarizability frequency dispersions for a set of push-pull π-conjugated molecules are faithfully reproduced by XsTD-DFT, while the previous sTD-DFT method provides redshifted resonance energy positions. Excellent performance with respect to the experiment is observed for the 2PA spectrum of the enhanced green fluorescent protein. The obtained robust accuracy similar to TD-DFT at a fraction of the computational cost opens the way for a plethora of applications for large systems and in high throughput screening studies.

H2SOLV: Fortran solver for diatomic molecules in explicitly correlated exponential basis

We present the Fortran package H2SOLV for an efficient computation of the nonrelativistic energy levels and the wave functions of diatomic two-electron molecules within the Born–Oppenheimer approximation. The wave function is obtained as a linear combination of the explicitly correlated exponential (Kołos–Wolniewicz) functions. The computations of H2SOLV are performed within the arbitrary-precision arithmetics, where the number of working digits can be adjusted by the user. The key part of H2SOLV is the implementation of the algorithm of an efficient computation of the two-center two-electron integrals for arbitrary values of internuclear distances developed by one of us (Pachucki, 2013). This have been one of the long-standing problems of quantum chemistry. The code is parallelized, suitable for large-scale computations limited only by the computer resources available and can produce highly accurate results. As an example, we report several benchmark results obtained with H2SOLV, including the energy value accurate to 18 decimal digits. Program summaryProgram title: H2SOLVCatalogue identifier: AFBA_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AFBA_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 8740No. of bytes in distributed program, including test data, etc.: 43089Distribution format: tar.gzProgramming language: Fortran 90.Computer: PCs or higher performance computers.Operating system: Linux / Unix.Has the code been vectorized or parallelized?: YesRAM: From several Mbytes to several Gbytes, depending on the size of the basis.Classification: 2.1.External routines: MPFR library [2] and GMP library [3] need to be pre-installed on the computer.Nature of problem:Numerical solution of the two-center two-electron Schrödinger equation within the Born–Oppenheimer approximation using the explicitly correlated basis set of Kołos–Wolniewicz functions.Solution method:The method of solution is based on the algorithm developed in Ref. [1].Running time:From seconds to days, depending on the size of the basis. The tests provided each take between 2 and 15 minutes to run.

Benchmark values for molecular two-electron integrals arising from the Dirac equation.

The two-center two-electron Coulomb and hybrid integrals arising in relativistic and nonrelativistic ab initio calculations on molecules are evaluated. Compact, arbitrarily accurate expressions are obtained. They are expressed through molecular auxiliary functions and evaluated with the numerical Global-adaptive method for arbitrary values of parameters in the noninteger Slater-type orbitals. Highly accurate benchmark values are presented for these integrals. The convergence properties of new molecular auxiliary functions are investigated. The comparison for two-center two-electron integrals is made with results obtained from single center expansions by translation of the wave function to a single center with integer principal quantum numbers and results obtained from the Cuba numerical integration algorithm, respectively. The procedures discussed in this work are capable of yielding highly accurate two-center two-electron integrals for all ranges of orbital parameters.

Analytical two-center integrals over Slater geminal functions

We present analytical formulas for the calculation of the two-center two-electron integrals in the basis of Slater geminals and products of Slater orbitals. Our derivation starts with establishing a inhomogeneous fourth-order ordinary differential equation that is obeyed by the master integral, the simplest integral with inverse powers of all interparticle distances. To solve this equation it was necessary to introduce a new family of special functions which are defined through their series expansions around regular singular points of the differential equation. To increase the power of the interparticle distances under the sign of the integral we developed a family of open-ended recursion relations. A handful of special cases of the integrals is also analysed with some remarks on simplifications that occur. Additionally, we present some numerical examples of the master integral that validate the usefulness and correctness of the key equations derived in this paper. In particular, we compare our results with the calculations based on the series expansion of the exp(-\gamma r12) term in the master integral.

Born-Oppenheimer potential for HeH+

We demonstrate high accuracy calculations for the HeH$^+$ molecule using newly developed analytic formulas for two-center two-electron integrals with exponential functions. The Born-Oppenheimer potential for the ground electronic $^1\Sigma^+$ state is obtained in the range of 0.1 -- 60 au with precision of about $10^{-12}$ au. As an example at the equilibrium distance $r=1.463\,283$ au the Born-Oppenheimer potential amounts to $-2.978\,708\,310\,771(1)$. Obtained results lay the ground for theoretical predictions in HeH$^+$ with spectroscopic precision.

Born-Oppenheimer potential forH2

The Born-Oppenheimer potential for the $^1\Sigma_g^+$ state of H$_2$ is obtained in the range of 0.1 -- 20 au, using analytic formulas and recursion relations for two-center two-electron integrals with exponential functions. For small distances James-Coolidge basis is used, while for large distances the Heitler-London functions with arbitrary polynomial in electron variables. In the whole range of internuclear distance about $10^{-15}$ precision is achieved, as an example at the equilibrium distance $r=1.4011$ au the Born-Oppenheimer potential amounts to $-1.174\,475\,931\,400\,216\,7(3)$. Results for the exchange energy verify the formula of Herring and Flicker [Phys. Rev. {\bf 134}, A362 (1964)] for the large internuclear distance asymptotics. The presented analytic approach to Slater integrals opens a window for the high precision calculations in an arbitrary diatomic molecule.

Two-center two-electron integrals with exponential functions

We present an efficient approach to evaluate two-center two-electron integrals with exponential functions and with an arbitrary polynomial in electron-nucleus and electron-electron distances. We show that the master integral with the single negative power of all distances can be obtained from the second-order differential equation in $r$, the distance between nuclei. For particular values of nonlinear parameters corresponding to the James-Coolidge basis, we find a fully analytic expression. For integrals with arbitrary powers of all distances, we construct recursion relations which starts from the master integral. The presented approach opens a window for the high-precision calculations of relativistic effects in diatomic molecules.

Rapid evaluation of two-center two-electron integrals

We present optimal formulae for two-center two-electron replusion integrals (ERIs) over Cartesian Gaussian basis functions. Floating-point operations (flop) counts reveal that two-center ERIs are often an order of magnitude cheaper than their four-center analogues. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1505–1510, 2000

New translation method for STOs and its application to calculation of two-center two-electron integrals

The new translation method for Slater-type orbitals (STOs) previously tested in the case of the overlap integral is extended to the calculation of two-center two-electron molecular integrals. The method is based on the exact translation of the regular solid harmonic part of the orbital followed by the series expansion of the residual spherical part in powers of the radial variable. Fair uniform convergence and stability under wide changes in molecular parameters are obtained for all studied two-center hybrid, Coulomb, and exchange repulsion integrals. Ten-digit accuracy in the final numerical results is achieved through multiple precision arithmetic calculation of common angular coefficients and Gaussian numerical integration of some of the analytical formulas resulting for the radial integrals. c 2000 John Wiley & Sons, Inc. Int J Quantum Chem 79: 91-100, 2000

Extension of the Neglect of Diatomic Differential Overlap Method to Spectroscopy. NDDO-G Parametrization and Results for Organic Molecules

A new semiempirical scheme, referred to as NDDO-G, for calculating geometries and spectroscopic properties of molecules, is described. The method is based on the NDDO (neglect of diatomic differential overlap) approximation. It uses the point-charge model and the Mataga−Nishimoto formula to evaluate two-center two-electron integrals. The NDDO-G model has been parametrized for the elements H, C, N, and O. Molecular geometries of organic molecules are well predicted by NDDO-G; for 60 molecules, the mean absolute error of bond lengths is 0.014 A and that of bond angles is 1.9°. The spectroscopic variant of the NDDO-G scheme provides electronic excitations using configuration interaction of singly excited states (CIS). It has been applied to calculate absorption spectra (vertical transitions) of several dozens organic molecules and photoelectron spectra within Koopmans' approximation. These NDDO-G results are compared with experimental results and with results of high-level ab initio calculations. The mean ab...

Calculation of two-center integrals involving a rapidly oscillating free electron wave function

Optical potentials are used in a quantum mechanical treatment of loss processes, e.g., ionization, where the loss of flux is described by the imaginary part. We present a numerical method for calculating two-center two-electron integrals necessary to construct the imaginary part of the optical potential. By introducing Slater-type orbitals with complex-valued exponents (CSTOs), we are able to represent the free electron wave with a limited number of CSTOs. For the representation of free electron wave functions with many oscillations, i.e., in a large r range or for a high kinetic energy, these new CSTOs form a more natural set of basis functions. The introduction of CSTOs is inevitable for the calculation of integrals concerning collisions in the mK energy range, where the interaction acts over large internuclear distances. Extensive numerical checks show that the final imaginary part of the optical potentials can be calculated with an accuracy better than 2%.

Extension of the MNDO formalism tod orbitals: Integral approximations and preliminary numerical results

The point charge model for calculating the two-center two-electron integrals in MNDO and related methods is extended tod orbitals. It is suggested to expand these integrals in terms of semiempirical multipole-multipole interactions where all monopoles, dipoles and quadrupoles are included, and all higher multipoles are neglected. The proposed scheme has been implemented, and numerical results for the integrals are reported. A preliminary MNDO parametrization for chlorine indicates that the inclusion ofd orbitals improves the results significantly, compared with the original MNDO and related methods.

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.

Comparison of AB-initio methods for calculating two-center two-electron integrals in a Slater basis

Comparison of AB-initio methods for calculating two-center two-electron integrals in a Slater basis

Autoionization widths for Ne*(3s)-Ar and Ne**(3p)-Ar collisions.

Following Jones and Dahler [Phys. Rev. A 37, 2916 (1988)] we study the theory of the process of ionization within the framework of the Feshbach projection-operator formalism, with the discrete fine-structure states (before ionization) and the continuum states (after ionization) as the two separate subspaces of Hilbert space. For the Ne*(3s)-Ar and Ne**(3p)-Ar systems we have calculated ab initio autoionization widths. In the approximation in which only two electrons play an active role, the coupling matrix element between the two subspaces is a linear combination of two-center two-electron integrals for the exchange and the radiative mechanisms. In general, our calculations support the semiempirical results of Morgner [J. Phys. B 18, 251 (1985)] for the coupling matrix elements as derived from experimental data. The autoionization widths \ensuremath{\Gamma}J\ensuremath{'},J\ensuremath{\Omega} are presented for initial states |J,\ensuremath{\Omega}〉 of the Ne*(*) atom and final states |J\ensuremath{'}〉 of the Ar+ ion. They show a pronounced \ensuremath{\Omega} dependence and a strong correlation of initial and final states, e.g., |0,0〉\ensuremath{\rightarrow}|3/2〉 and |2,2〉\ensuremath{\rightarrow}|3/2〉. For the Ne*(3s) states the two-center two-electron calculation is in good agreement with the two-state basis of Driessen et al. [Phys. Rev. A 42, 4058 (1990)], based on a one-electron overlap approximation. In a semiclassical model the polarized-atom ionization cross sections are calculated. For the Ne**(3p)-Ar system the energy dependence of both the polarization effect and the average cross-section value is reproduced very well, in contrast to the one-electron approximation. For the Ne*(3s)-Ar system we have to conclude that the ab initio results cannot explain all experimental observed features, such as the large cross-section ratio Q(P30)/Q(P32).

A possible solution to the rotational invariance and silicon parameter problem in theMNDO approximation

In the original formulation of the MNDO method, the approximation of two-center two-electron integrals over p orbitals is not rotationally invariant, and the coulombtype integrals do not match their one-center counterparts. In recent computer codes the rotational invariance has been restored by dropping the original approximation for two-center exchange integrals over π orbitals, which at the same time disconnected these too from the one-center values. It is shown in this paper that the rotational invariance problem can be overcome in the framework of the original approximation. It is also demonstrated that the mismatch between the one- and two-center two-electron integrals may be the reason for the too short Si-Si bond lengths predicted by MNDO.

Combined analytical expressions for two-center two-electron integrals with slater-type orbitals

Combined analytical expressions for two-center two-electron integrals with slater-type orbitals

Approximate Self-Consistent-Field Study of the(NiF6)4−Cluster

An approximate Hartree-Fock self-consistent-field (SCF) calculation has been carried out for the ${(\mathrm{N}\mathrm{i}{\mathrm{F}}_{6})}^{4\ensuremath{-}}$ cluster. Exponential-type orbitals with four independent radial functions were employed for the nickel $3d$ electrons, two radial functions for the fluorine $1s$ and $2s$ electrons, and one for the fluorine $2p$ electrons. A core potential was used to represent the metal ($1s\ensuremath{-}3p$) electrons. All one-electron integrals and all one- and two-center two-electron integrals were evaluated. A closed-shell SCF program was used with orbital occupation numbers adjusted to approximate the open shell. The results of comparisons at different stages in the convergence of the SCF iteration show large differences in $10Dq$ and the degree of delocalization. This implies that some of the earlier simplifications of the SCF treatment are not valid; that is, significant ligand ($2s\ensuremath{-}2{p}_{\ensuremath{\sigma}}$) mixing is found and the metal $3d$ functions make an important contribution to the bonding orbitals.

Molecular-Orbital Study of the Geometry of HO2

Determination of the energy of HO2 as a function of the HO1O2 angle for fixed values of R(H–O1) and R(H–O2) has been carried out using LCAO—MO—SCF calculations on closed-shell ions. All one-electron and one- and two-center two-electron integrals used were evaluated accurately while three-center integrals were approximated. An isosceles-triangle configuration is favored by these calculations and gives a value of the ionization potential of 9.5 eV and dissociation energy of 4.4 eV, consonant with experimental results. Approximation by means of ionic calculations used here is shown to be valid for the purposes of this discussion.