Nuclear Attraction Integrals Research Articles

Self-Consistent Molecular-Orbital Methods. II. Projection of Diatomic Differential Overlap (PDDO)

A method is developed for approximating 1- and 2-electron integrals over Slater-type orbitals (STO's) involving 2-center charge distributions. The latter are projected by least-squares, with overlap and dipole constraints, onto STO's with appropriate exponents, located only on the relevant atomic centers. The approach is applicable to any AO basis set, and is illustrated for first-row atom minimal STO basis sets. Error analysis is given for the projection per se, and for the 1- and 2-electron integrals obtained from the projected distributions. For integrals with magnitude > 10−2 a.u., the rms relative error is ∼1%. Integral errors are sytematic, and balance of errors in nuclear attraction and hybrid 2-electron integrals is examined. Comparison is made with earlier results of Boys and Shavitt for 1s orbitals.

Evaluation of Multicenter Integrals by Polished Brute-Force Techniques. II. Accuracy, Timing, Integral Values, and General Computational Considerations

The analysis presented recently by the authors for the general evaluation of the multicenter electron-repulsion integrals 〈χa(1)χb*(1)(1 / r12)χc(2)χd*(2)〉 and the nuclear attraction integrals 〈χa(1)(1 / rc)χb*(1)〉 is briefly reviewed followed by a characterization of the method's accuracy, speed, and generality. Our method, which integrates a one-electron potential Vab with a one-electron charge distribution Ωcd, is operative on an IBM 360/75 computer for Slater-type orbitals with quantum numbers n ≤ 8,l ≤ n − 1,|m| ≤ l, arbitrary geometry, and scale factors. A precision of 10−6 hartree is routinely achieved at an average time per integral of between 100 and 200 msec depending upon the total number of integrals evaluated.

Molecular Three-Center Nuclear-Attraction Integrals via Ellipsoidal Coordinates

Formulas are presented for a series expansion method for molecular three-center nuclear-attraction integrals, using a modified Laplace expansion due to Sack. It is shown how to extend the technique to two-electron-exchange integrals and to one type of three-center electron-repulsion integral. Electronic integrations are conducted in ellipsoidal coordinates using two centers as foci. The auxiliary functions arising from the analysis of the nuclear-attraction integrals are simpler than those occurring with conventional expansion methods.

New Aspects of the Bipolar Expansion and Molecular Multicenter Integrals

Standard bipolar expansions contain radial functions Jl1l2 L(r1, r2, R) which depend on three variables, r1, r2, R. Using representations of Bessel functions as series of Laguerre polynomials, the J's are expanded in terms of functions of the individual variables. As a result, the inverse distance between two points is brought into the form r12−1 = ∑ nlllml ∑ n2l2m2 Fn1l1m1, n2l2m2(R, θ, Ψ) Yl1m1(θ1, φ1)fn1l1(r1)Yl2m2(θ2, θ2)fn2l2(r2). This modified bipolar expansion is used to derive expressions for multicenter electron repulsion and nuclear attraction integrals. The method is particularly suitable for Gaussian orbitals expressed with spherical harmonics and yields compact expressions directly. For Slater-type orbitals, the multicenter energy integrals appear as series involving only integrals of the two-center overlap type. The one-center and multipole limits of the bipolar expansion are examined.

Effects of Charge Separation in a Hetronuclear Molecule: I, Changes in H11Due To Neighboring Charge

Abstract Ionization energies are commonly used to estimate atomic orbital matrix elements (H11′s) used in semiempirical molecular orbital calculations, but no adequate method is available to predict the change in atomic ionization energies resulting from charges on neighboring atoms. Using as a model the one-electron nuclear attraction integral, this paper develops a function to predict the effective potential produced by neighboring charges. The potential function is of the form

About evaluation of many-center molecular integrals

Studies of truncated expansions of Slater s-orbitals, in terms of associated Laguerre functions and spherical harmonics at another center, are carried out. The possibility of using such an expansion to calculate many-center molecular integrals is considered. The procedure is applied to solve three-center nuclear attraction integrals; it is shown that in some cases this expansion can provide relatively good results.

A new approach to the molecular orbital theory of bonding

The Moffitt ‘atoms in Molecules’ approach [8] and the Mulliken approximation [11] to two-centre integrals have been applied to M. O. Theory to give a simple equation for the energy of a diatomic molecule in terms of atomic spectral parameters and nuclear attraction integrals. The equation has been used to calculate the potential energy curves of 24 electronic states of the diatomic hydrides of second row elements and 24 states of the corresponding iso-electronic singly charged positive molecular ions. The agreement between the calculated and experimental values of the energies, the equilibrium internuclear separations, the force constants and the dipole moments is satisfactory.

One-Center Integrals of Extraordinary Functions

One-center integrals required in wave mechanical computations with the individual lobes of real spherical harmonics are discussed. Convenient formulas and tables are given for overlap, nuclear attraction, kinetic energy, and Coulomb repulsion integrals employed in the variational computation of a helium-atom ground-state wavefunction by the interaction of configurations built from Slater orbitals of the angular types s, px, py, and pz; and the extraordinary functions |px|, |py|, and |pz|.

Protonation Effects on the Electronic Spectra of Pyrazine

Abstract The protonation effect on the electronic structure and spectra of pyrazine has been studied theoretically employing quantum mechanical calculation. Semi-empirical Hartree-Fock SCF MO’s for pyrazine interacting with two protons have been computed with two different methods, respectively. i) Electrostatic potential due to proton has been added to the core potential in the total electronic Hamiltonian and nuclear attraction integrals which arise owing to the proton potential have been computed theoretically employing the usual Slater 2pπ AO’s. ii) The doubly protonated pyrazine has been replaced by doubly charged pyrazinium ion and molecular integrals have been evaluated semi-empirically using valence state ionization potential and electron affinity of the N+ atom. The calculated spectra with method i are in a very good agreement with the experiment. The calculation with method ii has led to a result which is in rather poor agreement with the experimental observation. On the basis of these results, some discussions about the protonation effect on the electronic structure of this molecule have been given.

Molecular Calculations. II. Methods of Approximation of Molecular Integrals

A description is given of the integral approximations used in the LCAO MO self-consistent field treatment of H2O reported in the preceding paper. The estimation of electronic repulsion integrals by replacing one of the electron charge distributions by a point charge is considered in detail and found to be reliable in certain general cases. A modification of the Sklar approximation is developed which is applicable to integrals involving 1saχb, in which μ(1sa) (effective Z) is small and μ(χb) is large. A method is described for securing a better ``corrected'' Mulliken approximation of exchange integrals. The estimation of unsymmetrical nuclear attraction integrals is carried out by using uniformly charged spheres of finite size to represent AO distributions. This method is compared with the Mulliken and Löwdin approximations.

Use of Molecular Quantum-Mechanical Approximations Exemplified in the Energy Calculation of the H3— System

Certain approximations of molecular quantum mechanics are examined. The size of the resultant error due to approximate methods for specific multicenter integral evaluations are cited. Examination of the Sklar approximate multicenter electron repulsion formula together with the generalizations and extensions of Vroeland and Löwdin allow application of the underlying approximation principle to three center nuclear attraction integrals. Specific approximate formulation of these integrals is given. The accumulation of computational errors arising from the many individual integral approximations in the energy evaluation of the linear H3— system is specifically examined. The ground state energy of this system found from the Heitler-London and Wang-like treatments were — 70 kcal/mole and — 84 kcal/mole for the Sklar type approximations, — 1 kcal/mole and — 19 kcal/mole for the Mulliken type approximations, and — 26 kcal/mole and — 41 kcal/mole for exact integral evaluations. It is concluded that the Sklar type approximations, while preferable to the Mulliken type in a few specific evaluations lead like that formulation to considerable discrepancy between the over-all energy obtained for a polyatomic system. The Sklar approximations have the further disadvantage of requiring two-center integral evaluations not needed in the over-all energy calculations. However, both the Sklar and the Mulliken approximations do give a rapid method for evaluating ``best'' distance and effective charge parameters for use with more extensive exact integral variational method calculations.

Use of Nuclear Attraction Integral Approximations in Molecular Quantum Mechanics

Most molecular quantum-mechanical studies of today make use of eigenfunctions composed of atomic orbitals. Such eigenfunctions give rise to difficult multicenter integrals which are usually evaluated by approximate methods. A study of the accuracy of such methods and of their derived results has long been desirable. Certain approximations for three-center nuclear attraction integrals are here examined. Complete sample calculations are made with the approximate integral values for the triatomic activated complex molecule, H3. The use of the Mulliken approximate evaluation of integrals in the Heitler-London treatment for H3 gives a binding energy 7.6 kcal/mole less than does the use of exact integrals. On modifying the Heitler-London treatment by inclusion of screening constants for the outer and central atomic orbitals the Mulliken type approximate integrals gave for a trial calculation a binding energy 5.3 kcal/mole better than the best previous determination. This trial calculation (using the parametric values R = 1.8, Z0 = 1, Zm = 1.08) while the best of several tried is not likely the best that can be found. However, the binding energy of 72.7 kcal/mole calculated is 26 kcal/mole better than the Heitler-London value (53.4 kcal/mole) and 5 kcal/mole better than the Wang-like covalent-bond-plus-ionic-terms determination using a single screening parameter. The cited trial calculation indicates that similar determinations now in progress using exact integral evaluations will give a substantial increase in the calculated binding energy of H3. This would be, for this simplest polyatomic molecule, the first major advance since 1936. That the approximate calculation is essentially correct is substantiated by the increase in binding energy obtained with the proper screening constant variation for the diatomic molecule. In the H2 case Wang obtained an improvement from 72.4 kcal/mole to 86.7 kcal/mole over the best covalent-plus-ionic term calculations.

A Study of Three-Center Integrals Useful in Molecular Quantum Mechanics

An integrated equation for the three-center nuclear attraction integral has been formulated in a general way for Slater-type atomic s orbitals arising from any integral principle quantum number. No restrictions are placed on the effective nuclear charges or the interatomic distances. The formulation is expressed linearly in terms of certain auxiliary functions which are discussed in detail. Numerical values of some of the auxiliary functions for the triatomic hydrogen molecule are cited as examples of this exact formulation. An alternative approximate formulation is also given.

A Study of Two-Center Integrals Useful in Calculations on Molecular Structure. I

Convenient formulas have been obtained for the overlap integrals ∫χaχbdv, kinetic energy integrals —½∫χaΔχbdv, nuclear attraction integrals Z∫χa(1/ra)χbdv and Z∫χb(1/ra)χb′dv, and coulomb repulsion integrals ∫ ∫ χa(1)χb(2)(1/r12)χa′(1)χb′(2)dv1dv2, where χa, χa′, χb, χb′ are Slater-type AO's on the centers a and b. Explicit formulas are given for all the integrals arising from the principal quantum numbers 1 and 2, for arbitrary values of the effective nuclear charges and the interatomic distance.