Molecular orbital studies on small molecules using H2+‐type elliptical basis orbitals. Application to H2+, H2, He2++ and H3+

AbstractsWe investigated various nonstandard 1 s basis functions (generalized Slater-Gaussian, ellipsoidal Gaussian, floating spherical and ellipsoidal Gaussian, rational function, HulthSn approximation, two-Slater-type orbital, generalized Guillemin-Zener function, and various noninteger-n elliptical orbitals) for approximating the ' Z: ground state of H, and He: +.A CI trial wave-function including u,-type MO'S is adopted and. molecular integrals are evaluated numerically. The energy improvement on the 1s STO is small except for noninteger-n orbitals which closely approach the "SCF limit". Diverses fonctions de base non usuelles (forme gCnCralisie d'orbitale de Slater ouGaussienne, Gaussienne ellipsoidale, Gaussienne sphkrique ou ellipsoidale flottante, fonction rationelle, approximation de HulthCn, double orbitale de Slater, fonction de Guillemin-Zener gCnCralisCe, diverses orbitales a valeurs non entieres de n exprimks en coordonnCes elliptiques) sont appliquks en tant que orbitale atomique de type 1s a l'etat fondamental ' Z: de H, et He:+. On adopte une fonction d'onde incluant des orbitales moliculaires de type ug dans le cadre d'une interaction de configuration restreinte. Les intigrales molCculaires sont bvaluies numeriquement. Le gain en Cnergie est faible par rapport a l'orbitale 1s de Slater hormis le cas des orbitales a valeurs non entiires de n qui approchent fortement la "limite XF".Verschiedene nicht ubliche 1 s Grundfunktionen (VeraQgemeinertes Slater oder Gaussches Orbital, Ellipsoid-Gaussches Orbital, schwebendes spharisch-oder EllipsoidGaussches Orbital, rationelle Funktion, Hulthkn-Naherung, Doppelorbital von Slater, verallgemeinerte Funktion von Guillemin-Zener, verschiedene n-nicht ganzzahlige elliptische Orbitale) wurden zur Naherung des '2: Grundzustandes von H, und He:+ verwandt. Eine Wellenfunktion mit Molekulorbitalen des ug Types wurde als VersuchsKonfigurationswechselwirkung angenommen. Dabei wurden die Molekulintegrale numerisch geschatzt. Der Energie-Zuwachs ist gering dem Slater-1 s Orbital gegenuber, mit Ausnahme fur die n-nicht ganzzahligen Orbitalwerte, die an die "SCF Grenze" sehr nah herankommen.

Section: Resultsmentioning

confidence: 71%

“…[46] and a LCETO-MO with two H:-type ETO'S ( -1.13360 and -3.6091 a.u.) [40]. Fraga and Ransil [42] also reported an energy of -3.60928 a.u.…”

Section: Resultsmentioning

confidence: 92%

Comparative study of unconventional 1s basis functions for the ¹Σ ground state of H₂ and He

Leclerc

1976

“…Excluding the off-diagonal Lagrange multipliers there are a total of N ( N l + N , + 5) unknowns in (9). There are an equivalent number of equations, not counting the orthogonality conditions (9h).…”

Section: Methods Of Solutionmentioning

confidence: 99%

A finite‐difference solution of the Hartree–Fock equations for diatomic molecules

Fimple

White

1975

AbstractsA more general application of the self-consistent field iteration is coupled with a finitedifference Newton-Raphson algorithm to solve the set of coupled second-order integrodifferential equations with split boundary conditions which constitutes the Hartree-Fock problem for diatomic molecules. The N orbitals are assumed to be of the form ya = La(1)Ma(~)eima~(2n)-1/2, ( x = 1, -. . , N ) , where 1, p, and 4 are the usual confocal elliptical coordinates. Requiring the expectation value of the electronic Hamiltonian to be stationary with respect to independent variations of the functions L, and M u , subject to constraints of orthonormality, leads to a set of coupled one-dimensional differential equations for the functions La and Ma . I n the new method a corresponding set of finite-difference equations including the split boundary conditions for each function, as well as the Lagrange multipliers and associated constraints on normalization and orthogonality, are incorporated into a large system of nonlinear algebraic equations which is solved by means of a coupled self-consistent field-generalized Newton-Raphson iteration. FIMPLE AND WHITEEine allgemeinere Anwendung der SCF-Iteration wird rnit einen Newton-Raphson'schen Algorithmus fur endliche Differenzen gekoppelt, urn den Satz von gekoppelten Integrodifferentialgleichungen zweiter Ordnung mit aufge-spalteten Randbedingungen zu h e n , die das Hartree-Fock-Problem fur zweiatomige Molekule bilden. Die N orbitale werden mit der Form y, = La(I)Ma (~)eima4(2.rr)-1'2, ( a = 1, . . . , 1 % '

“…I n Equations (2) and (3), 6 is the radial and q is the angular elliptical variables, and R is the internuclear separation.…”

Section: (3)mentioning

confidence: 99%

“…For the linear variational wave function y N given in Equation ( l ) , the appropriate basis set is one of Ht-type ETOS [2], which are defined in Equations (2) and…”

Section: Introductionmentioning

confidence: 99%

Error analysis of variationally optimized upper‐ and lower‐bound wave functions for H

Ley

Thorpe

Rothstein

1974

Self Cite

AbstractsWave functions which are a linear combination of HZ-type elliptical orbitals are optimized to provide either an upper bound or a lower bound to the H$ ground state. For the latter, Temple's formula is used. Three criteria are considered to determine the relative accuracy of these wave functions: ( i ) energy (calculated versus exact eigenvalue) ; (ii) average error; and (iii) local energy. Although the lower-bound optimized wave functions obtained are the most accurate available for H$ from approximate wave functions, they are still inferior to the corresponding upper-bound wave functions by criteria (i) and (ii). I n particular, using criterion (ii), it is shown numerically that the upper-bound functions are "correct to second order," while the lower-bound functions are almost, but not quite, "correct to second order." Despite this, the local energy analysis, criterion (iii), reveals that the lower-bound wave functions can be more accurate than the upper-bound functions in some regions of space, and hence give more accurate values for physical properties sensitive to these regions. Examples considered are the dipole-dipole and Fermi contact interactions. Des fonctions d'ondes