One-center Integrals Research Articles

Expansion of multicenter Coulomb integrals in terms of two-center integrals

Three- and four-center Coulomb integrals in the solid spherical harmonic Gaussian basis are solved by expansion in terms of two-center integrals. The two-electron Gaussian product rule, coupled with the addition theorem for solid spherical harmonics, reduces four-center Coulomb integrals into a linear combination of two-center Coulomb integrals and one-center overlap integrals. With this approach, three- and four-center Coulomb integrals can be reduced to the same form of two-center integrals. Resulting two-center Coulomb integrals can be further simplified into a simpler form, which can be related to the Boys function. Multi-center Coulomb integrals are solved hierarchically: simple two-center Coulomb integrals are used for calculation of more complicated two-center Coulomb integrals, which are used in the calculation of multicenter integrals.

Determination of consistent semiempirical one-centre integrals based on coupled-cluster theory

ABSTRACTWe examine the one-centre integrals used in semiempirical molecular orbital theory, for the elements H–Ne. The currently used parameters do not provide good estimates for the relative energies of ionised states of atoms. Directly calculating the one-electron integrals Uss and Upp with coupled-cluster theory and fitting the two-electron repulsion integrals Gss and Gpp to accurate coupled-cluster ionisation curves improves this behaviour. Since all the remaining parameters can be derived from these, the number of fitted variables is reduced from seven to two. The two-parameter model provides qualitative agreement with coupled-cluster theory for all ionisation potentials (IPs) and the principal electron affinity (EA). To obtain quantitative agreement for the principal IP and EA, Uss and Upp are included as variables in a four-parameter model. We discuss the new parameters and implications for the development of new, consistent semiempirical Hamiltonians.

Analytical evaluations of exponentially correlated unlinked one-center, three- and four-electron integrals

We discuss an extension of the Hylleraas configuration-interaction method by incorporation of an exponentially correlated term. The necessary unlinked one-center, three- and four-electron integrals are derived in a closed form. No three- or four-electron auxiliary integrals appear in the final expressions. The evaluation scheme is an extension of the recent development of Ruiz [J. Math. Chem. 46, 1322 (2009)]. The prescription of combining diverged integrals to retrieve a finite value is avoided in the present approach.

Range-Separated Exchange Functionals with Slater-Type Functions.

An implementation of range-separated density functionals utilizing the Yukawa potential and Slater-type functions is described. The density-functional part of the range-separated regime is straightforward. The exact exchange part makes use of established methods for evaluating exchange integrals over Slater-type functions but still requires new one- and two-center integrals. Equations for the one-center integrals are derived. The two-center integrals are evaluated through a combination of new equations and techniques taken from procedures for evaluating two-center Coulomb integrals over Slater-type functions. In a first application, the performance of range-separated functionals in the prediction of transition metal thermochemistry is evaluated using a database of average ligand removal energies. The range-separated functionals perform better than a GGA parent and similarly to commonly used hybrid and meta-hybrid functionals. The results were relatively insensitive to the chosen value of the attenuation parameter.

Donor−Acceptor Biradicals as Ground State Analogues of Photoinduced Charge Separated States

A Valence Bond Configuration Interaction (VBCI) model is used to relate the intraligand magnetic exchange interaction (J) to the electronic coupling matrix element (HAB) in Tp(Cum,MeZn)(SQNN), a compound that possesses a Donor-Acceptor (D-A) SemiQuinone-NitronylNitroxide (SQNN) biradical ligand. Within this framework, an SQ --> NN charge transfer state mixes with the ground state and stabilizes the spin triplet (S = 1). This charge-transfer transition is observed spectroscopically and probed using resonance Raman spectroscopy. In addition, the temperature-dependent electronic absorption spectrum of the Ni(II) complex, Tp(Cum,MeNi)(SQNN), has been studied. Exchange coupling between the S = 1 Ni(II) ion and S = 1 SQNN provides a mechanism for observing the formally spin-forbidden, ligand-based 3GC --> 1CTC transition. This provides a means of determining U, the mean GC --> CTC energy, and a one-center exchange integral, K(0). The experimental determination of J, U, and K(0) permits facile calculation of HAB, and we show that this methodology can be extended to determine the electronic coupling matrix element in related SQ-Bridge-NN molecules. As magnetic susceptibility measurements are easily acquired in the solid state, H(AB) may be effectively determined for single molecules in a known geometry, provided a crystal structure exists for the biradical complex. Thus, SQ-Bridge-NN molecules possess considerable potential for probing both geometric and electronic structure contributions to the magnitude of the electronic coupling matrix element associated with a given bridge fragment.

Expansion Formulae for Two-center Integer and Noninteger n STO Charge Densities and their Use in Evaluation of Multi-center Integrals

Using complete orthonormal sets of Ψα-exponential type orbitals (Ψα-ETOs, α =1, 0, −1, −2, ...) introduced by the author, the series expansion formulae are derived for the two-center integer and noninteger n STO (ISTO and NISTO) charge densities in terms of integer n STOs at a third center. The expansion coefficients occurring in these relations are presented through the two-center overlap integrals between STOs with integer and noninteger principal quantum numbers. The general formulae obtained for the STO charge densities are utilized for the evaluation of two-center Coulomb and hybrid integrals of NISTOs appearing in the Hartee–Fock–Roothaan approximation. The final results are expressed in terms of both the overlap integrals and the one-center basic integrals over integer n STOs. It should be noted that the result for the multi-center multielectron integrals with two-center noninteger n STO charge densities presented in this paper were not appeared in our past publications.

Calculation of molecular integrals over Slater-type orbitals using recurrence relations for overlap integrals and basic one-center Coulomb integrals.

The recurrence relations are established for the basic one-center Coulomb integrals over Slater-type orbitals (STOs). These formulae and the recurrence relations for basic overlap integrals are utilized for the calculation of multicenter electron-repulsion integrals. The calculations of multicenter electron-repulsion integrals are performed by the use of translation formulae for STOs obtained from the Lambda and Coulomb Sturmian exponential-type functions (ETFs). It is shown that these integrals show a faster convergence rate in the case of Coulomb Sturmian ETFs. The accuracy of the results is quite high for the quantum numbers of STOs and for the arbitrary values of internuclear distances and screening constants of atomic orbitals.

Multicenter Integration Scheme for Electronic Structure Calculations of Periodic and Nonperiodic Polyatomic Systems

The authors present a numerical integration scheme designed to treat the type of multicenter integrals encountered in electronic structure calculations. By developing a notation that differentiates between those atomic centers where integrands have significant amplitudes and those where they do not, they find a way to decompose multicenter integrals (into sums over one-center integrals) such that the number of operations needed for a given matrix element does not increase with increasing system size. In addition, a new adaptive one-center grid is presented that accounts for the shell structures of core electrons while allowing for the vastly different behavior of integrands in the valence and tail regions. Through the use of model integrands the necessary grid points are automatically generated for a given system based on the accuracy requested. The new multicenter decomposition scheme and one-center grid have been tested separately and in conjunction with each other. Results of such tests demonstrate that their decomposition scheme combined with their one-center grid provides significant improvements over existing multicenter integration schemes. In addition to demonstrating the efficiency of the method for any size system, the authors will show that the CPU cost of an integral remains constant for systems larger than some easilymore » achievable threshold size. In general comparison shows that the larger the system, the higher is the percent gain in efficiency over previously published methods. In addition, the higher the accuracy targeted, the higher percentage the gain. Also, the higher the accuracy required for a given system, the higher is the gain in efficiency. The method is therefore of great use for large polyatomic molecules and periodic systems.« less

Scaling the Coulomb interaction in calculations of electron spectra of transition metal complexes

UDC 541.6 + 539.194 A simple technique of scaling two-electron integrals in ab initio calculations of the electronically excited states of transition metal complexes is proposed. This technique uses the fact that one-center two-electron integrals depend linearly on the scaling factor when Slater type functions are subjected to scaling transformation. This leads to a linear dependence of the d-d transition energy on the "scale" of Coulomb interaction, which allows one to affect the calculation result by varying the Slater exponential. To test the technique, ab initio configuration interaction and full active space calculations of the low excited states of the CrF63-, MnF 2-, and VF 3- complexes are performed. For transition elements, a basis of Slater type effective functions chosen from the optical spectra of the atoms and ions of transition elements is used. It is shown that in the STO-6G basis with effective exponentials, experimental transitions are reproduced with an accuracy of about 2000 cm -! even with the use of small active space determined by the orbitals localized on the central atom of the complex.

NMR-K reduced coupling constants within the CLOPPA-PM3 approach. II: shortcomings and how to overcome them

The shortcomings encountered in the CLOPPA-PM3-RPA coupling constant calculations shown in part I of this work are analyzed. These are: (i) the Hartree–Fock quasi-instabilities of the non-singlet type in PM3 wavefunctions; and (ii) the erroneous PM3-derived description of the hybridization of some localized molecular orbitals. It is shown that these problems arise from: (a) the parameter's optimization procedure used within the PM3 scheme, which apparently takes into account some effects at the nuclear sites in an inappropriate way; and (b) the selected set of properties chosen to be reproduced by the parameter's optimization procedure. Properties do not critically depend on the electronic surroundings of the nuclei. Hence, some effects at nuclear sites are badly produced when the PM3 wavefunction is used. We have found three parameters which are mainly involved: the one-center two-electron integrals; the β s or β p parameters; and the U ss one-electron energies. The FC electronic mechanism is the most affected by the problems analyzed here.

Molecular Orbitals Calculation on LiH with Algebraic Treatment of the Integrals

In this work we employ an algebraic computational method to solve the integrals which arise in the study of diatomic molecules. Using the Slater-type orbitals (STO), we obtain analytical solutions for the one-center and two-center hybrid and Coulomb integrals. The exchange integrals are considered, as much as possible, in a similar manner. These results are used to calculate the electronic properties of the ground state of the LiH molecule, by means of a variational calculation using a basis of molecular orbitals. The behavior of the Slater exponents of the atomic orbitals are studied as functions of internuclear distance. We show that the STO undergo a lengthening of their shape when the nuclear separation increases.

Spin-polarization effects in (AB2)n polymeric chains.

The techniques of renormalization of interactions and transfer matrices are used to investigate spin-polarization effects in (AB${)}_{2}$ polymeric chains, within a Hartree-Fock solution to a diagonal Hubbard Hamiltonian approximation. The electronic structure of the infinite chain is obtained and the corresponding magnetization is examined for different values of the one-center repulsion integrals J. It is observed that one-third of the total electronic states available are associated to nondispersive states localized on the B sites; for half-filled systems the conductive and magnetic properties of the chain are obtained as a function of J. The damped spin density wave resulting from a substitutional impurity on an A center is also calculated. \textcopyright{} 1996 The American Physical Society.

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.).

Magnetic impurities in ( AB2) n polymeric chains

The techniques of renormalization of interactions and transfer matrices are used to investigate spin polarization effects in ( AB 2) n chains, within a Hartree-Fock solution to a Hubbard diagonal hamiltonian approximation. The electronic structure of infinite chains is obtained and the magnetization of the system is examined for different values of the one-center interaction integrals U. It is observed that one third of the total available electronic states is associated to non-dispersive states localized on the B sites and that both the position in energy of these localized states and the gap width are controlled by the strength of the parameter U. The damped spin density wave resulting from the introduction of a substitutional impurity on an A center can then be calculated for different regions of the parameter space of the problem.

Improved recursion formulas for the calculation of two-center central potential integrals

A commutator algebra procedure is used to get improved recurrence relations for the calculation of any f(r) off-diagonal two-center matrix elements in the general case of displaced arbitrary central potential wave functions. As expected, the proposed formulas reduce properly to the generalized recursion equations for the calculation of one-center integrals. Besides, when f(r) = rk, one obtains the equivalent of the Kramer rule for two-center multipolar matrix elements for arbitrary potentials, and when f(r) is constant, the Wu formula (for the calculation of Franck–Condon factors) is ameliorated in the sense that the new relation not necessarily considers equal mass. Furthermore, all diagonal matrix elements as well as all off-diagonal integrals between nondisplaced potentials appear, in our treatment, as particular cases in good agreement with already published results. As a useful application, the corresponding recurrence relations for the calculation of one-center hydrogenic matrix elements and two-center Kratzer potential integrals are given as examples. However, our approach is general and can be easily extended to obtain recursion formulae for other potential wave functions. © 1995 John Wiley & Sons, Inc.

Analysis of some integrals arising in the atomic four-electron problem

An analysis is presented for the evaluation of the one-center integrals of the form ∫ri1rj2rk3rl4rm12 rn13rp14rq23rs24 rt34e−ar1−br2−cr3−dr4 dr1 dr2 dr3 dr4 which arise in the solution of the atomic four-electron problem when a Hylleraas expansion is employed. A majority of the cases that arise can be solved by a reduction of the four-electron integrals to integrals arising in the three-electron problem. A second general approach is developed which is able to handle almost all cases that arise in practical calculations.

The evaluation of the hybrid one-center integrals used in the semiempiricalNDDO-type methods ford-elements from spectra

AbstractFor semiempirical SCF MO methods based on the NDDO assumption, the so-called one-center hy-brid integrals of the type (ab/cd) are required, These integrals are expressed in terms of theSlater-Condon parameters (scp) Rk(abcd) with different radial functions. The SCPsfrom atomicspectra are evaluated by taking into account configurational interaction (CI).As an example, the R1(sdpp),R2(sdpp), and R2(sddd) sCPsfor Ti I, Ti II, and Ti III have been evaluated and the valuesof these parameters are compared with those calculated with Slater-type orbitals. 1. Introduction One of the main features of the modern semiempirical SCFMOtheories is theapproximation of one-center integrals by certain quantities estimated from spec-troscopic data [1]. These parameters are evaluated in the following way: Thespectroscopic energy term of an atom according to Slater-Condon theory basedon the central-field approximation is expressed as a linear function of the Fk and Gk Slater-Condon parameters (scp) [2]. The sCPsare the radial parts of theCoulomb and exchange integrals of electronic interaction. Hence, one can obtainthe semiempirical one-center integrals by least-squares' fitting of the availablespectroscopic data to the theoretical equations.The semiempirical integrals obtained in [3] are sufficient to construct thesemiempirical methods in CNDO,INDO,and NDDOapproximations [4]for

On the Coulomb repulsion integrals used in the parametrized equations-of-motion method for π → π* transitions

The effective equations of motion (EOM) for the π → π* transitions of conjugated molecules explicitly show the effect of the σ–π correlations. Writing the EOM in terms of the one- and two-center Coulomb repulsion integrals in the ZDO approximation for the ethylene molecule, we show that only the one-center integral is significantly reduced due to the σ–π correlations but the two-center integral remains unaffected.

Classical Coulomb interaction between undistorted atoms of arbitrary symmetry

Abstract A general expression is obtained for the classical electrostatic interaction between undistorted atomic charge distributions constructed through one-electron Hartree-Fock wave functions. The original two-center problem is uncoupled by working in Fourier space leading to one-center integrals. The results of this work are suitable for applications of the independent atom model (IAM) in the study of molecule formation.

Parametrized Equations-of-Motion Method at the (1p-1h)+(2h-2p) Level and Its Application to Linear Polyenes

Abstract The previously proposed idea of parametrizing the equations-of-motion method [Int. J. Quantum Chem., 18, 205 (1980)] has been worked out for the π→π* transitions of linear polyenes. The excitation operator consists of not only 1p-1h creation and annihilation operators but also 2p-2h creation and annihilation operators. [Bull. Chem. Soc. Jpn., 61, 1103 (1988)]. The proposed scheme accounts for the effect of σ→σ* mixings in contrast with the traditional pi-electron theories. In the present work, the two-center Coulomb integrals are evaluated by 1/R, and all the one-center integrals on the C atom are assumed to be the same. Values of the three parameters (a, b, ζ) introduced in the scheme are so determined that the computed excitation energies of ethylene and s-trans-butadiene fit the observed values. The scheme has been applied to a series of all-s-trans linear polyenes through dodecahexaene. Some notable results are: (1) The σ→σ* mixings play a significant role in lowering the excitation energies of the allowed transition to 11Bu; (2) the 21Ag state is of Ag− type and almost free from this effect; (3) double excitations (2p-2h states) are dominant in the 21Ag state but practically negligible in the 11Bu state.