Polarization functions for gaussian basis sets for the first row atoms

Exponent optimization was performed for a single set ofd-type Gaussians on the first row atoms C, N, and O in fifteen small molecules. The hydrogenp-exponents were kept at the fixed value of 1.0. For the underlying valence shell basis sets, Dunning's double zeta basis sets were used. Standard exponents of polarization functions are suggested for the most common valence states of the C, N, and O atoms.     

Valence bond functions for the hydrogen molecule

Valence bond wavefunctions were constructed for H₂. Use of Slater exponents resulted in very slow convergence to the ground state energy. Convergence was improved by optimizing exponents which were found to increase as the principal quantum number n. However, this gave problems of linear dependence since optimum orbitals were strikingly similar for all n. The best function without angular correlation contained 27 terms constructed from 1s, 3s, 2p ₀, 3d ₀, 4f ₀, and 5g ₀ orbitals and gave an energy of –1.1594a.u. The best function with angular correlation gave E=-1.1656 a.u.

---

Zusammenfassung Für das H₂-Molekül werden Wellenfunktionen nach der Valenzstrukturmethode konstruiert. Die Benutzung von Exponenten nach Slater führt zu einer sehr langsamen Konvergenz zur Grundzustandsenergie. Die Konvergenz wurde durch Optimierung der Exponenten verbessert, wobei diese mit der Hauptquantenzahl ansteigen. Dabei ergab sich jedoch das Problem linearer Abhängigkeit der Funktionen, da die optimalen Orbitale sehr ähnlich für alle n waren. Die beste Funktion ohne Winkelkorrelation enthielt 27 Terme, die aus den Orbitalen 1s, 3s, 2p ₀, 3d ₀, 4f ₀ und 5g ₀ konstruiert waren, und ergab eine Energie von –1,1594A.E. Die beste Funktion mit Winkelkorrelation ergab E=–1,1656 A.E.

---

Résumé Des fonctions d'onde de liaison de valence sont construites pour H₂. L'emploi d'exposants de Slater entraîne une très faible convergence vers l'énergie de l'état fondamental. La convergence a été améliorée par optimisation des exposants qui croissent comme le nombre quantique principal n. Cependant, ceci crée des problèmes de dépendance linéaire car les orbitales optimales sont étonnement similaires pour tous les n. La meilleure fonction sans corrélation angulaire contient 27 termes construits à partir d'orbitales 1s, 3s, 2p ₀,3d ₀,4f ₀ et 5g ₀ et donne une énergie de –1,1594 u.a. La meilleure fonction avec corrélation angulaire donne E= –1,1656 u.a.

---

---

Presented in part at the Symposium on Molecular Structure and Spectroscopy, The Ohio State University, Sept., 1968.

---

N.I.H. Predoctoral Fellow 1964–1967.     

A mixed basis set of slater and gaussian functions for molecular calculations

A procedure for using simultaneously Slater and Gaussian basis functions in molecular calculations is presented here. The analytic expressions of the integral prototypes involving both Slater and Gaussian functions are explicitly derived.     

An algorithm for calculating atomic D states with explicitly correlated gaussian functions

An algorithm for the variational calculation of atomic D states employing n-electron explicitly correlated gaussians is developed and implemented. The algorithm includes formulas for the first derivatives of the hamiltonian and overlap matrix elements determined with respect to the gaussian nonlinear exponential parameters. The derivatives are used to form the energy gradient which is employed in the variational energy minimization. The algorithm is tested in the calculations of the two lowest D states of the lithium and beryllium atoms. For the lowest D state of Li the present result is lower than the best previously reported result.     

Formulas for evaluation of expectation values of monoelectric operators with modified gaussian functions

Formulas are presented for the evaluation of the expectation values of various monoelectronic operators. The integrals are based on ?Hermite-Gausian”? or ?Modified Gaussian Functions”? and are expressed in suitable form for a computer programming. It is pointed out that the final expresions are simpler than the analogou omes obtained from the usual Gausian functions and can be written as linear combinations of a few baic integrals.     

Non-empirical LCAO-MO-SCF-CI calculations on organic molecules with gaussian type functions

This part of the series presents the results obtained by open shell SCF computations on the G, N, O, and F Atoms.

---

Zusammenfassung Dieser Teil der Reihe gibt die Ergebnisse von SCF-Rechnungen für offene Schalen am C, N, O und F an.

---

Résumé Cette partie de la suite d'articles présente les résultats obtenus par des calculs SCF à couches ouvertes sur les atomet C, N, O et F.

     

Calculations using a set of evenly spaced S-type gaussian functions

A basis set of evenly spaced S-type Gaussian functions with common exponents is examined. Formulas for common one- and two-electron integrals are derived. Because of thesymmetry of this basis set, a very compact two-electron integral list is produced. The number of two-electron integrals that must be stored is approximately eight times the number of basis functions. Use of this basis set in an SCF calculation is examined. Numerical results show that this approach works well for molecules containing only small atoms such as hydrogen, helium, or lithium, but that the method has problems with the core orbitals of heavier atoms. Procedures for augementing this basis set in calculations involving heavier atoms are examined.     

The calculation of matrix elements for valence bond functions

Pauling's formulas for the calculation of matrix elements for valence bond functions are derived using a simple substitutional process. The results generalize and simplify the formulas. In particular, the formulas do not depend upon orthogonality of atomic orbitals nor upon the nature of the choice of bond structures (canonical or not). The results are particularly adaptable to automatic computation.     

Theory of the expansion of wave functions in a gaussian basis

The convergence properties of the expansions of (a) the function 1/r and (b) the function exp(-αr) in an even-tempered basis of Gaussians are studied analytically. The starting points are the Gaussian integral representations of 1/r and exp(?αr). One arrives at an expansion in a finite number of Gaussians in three steps: (1) a restriction of the integration domain, (2) a variable transformation, and (3) discretization of the integral. The cutoff error goes in both cases essentially as exp(?ah), and the discretization error, as exp(?b/h). The minimum overall error is reached for the β-parameter of an even-tempered basis β ∽ exp(c/√n), where n is the dimension of the basis, and the error itself decreases as ? ∽ exp(?d√n). Different optimum basis parameters are obtained depending on which quantity one wants to minimize, e.g., the error of the energy expectation value, the distance in Hilbert space, the variance of the energy, or the density at the nucleus. © 1994 John Wiley & Sons, Inc.     

Basis sets for transition metals: Optimized outer p functions

Although the (n + 1)p orbital is unoccupied in transition-metal ground-state configurations which are all nd^x(n + 1)s^y, these (n + 1)p functions play a crucial role in the structure of transition metal complexes. As we show here, the usual solution, adding one or more diffuse functions, can be insufficient to create an orbital of the correct energy. The major problem appears to be due to the incorrect placement of the (n + 1)p orbital's node. Even splitting the most diffuse component of the np orbital and adding a second diffuse function does not completely solve this problem. Although one can usually solve this deficiency by further uncontracting of the np function, here we offer a set of properly optimized (n + 1)p functions that offer a more compact and satisfactory solution to the proper placements of the node. We show an example of the common deficiencies seen in typical basis sets, including standard basis sets in GAUSSIAN94, and show that the new optimized (n + 1)p function performs well compared to a fully uncontracted basis set. © 1996 by John Wiley & Sons, Inc.     

Integrals of gaussian and continuum functions for polyatomic molecules. An addition theorem for solid harmonic gaussians

An addition theorem for solid harmonic gaussian functions in terms of surface spherical harmonics is derived and presented in a form particularly useful for evaluating molecular integrals involving gaussian and continuum functions on array processors.     

Rovibrational structures in floppy triatomics: distributed gaussian functions treatment for the Ne2H- system

The full sequence of the bound states for a very floppy triatomic complex, Ne2H- in its ground electronic state, are initially computed for the rotationless situation and employing a variational approach that expands the total nuclear wave function over a large set of symmetry-adapted, distributed Gaussian functions and employs accurate atom-atom potential energy data. The results are tested for numerical convergence, compared with the behavior of both its diatomic fragments, Ne2 and NeH-, and further compared with the results for the Ne3 case. The computational analysis is extended to the production of the rotational constants for the very nonclassical ground state vibrational configuration by making use of the previous findings. The method is shown to provide us with several illuminating details on the nanoscopic internal dynamics of this very weakly bound quantum aggregate.     

On the choice of gaussian 4f functions for use in calculations on transition metal compounds

One uncontracted 4f gaussian function is found to perform well as a polarization function at the SCF level When correlation is included, it is found that a two-gaussian fit to a Slater 4f function is much superior to a single gaussian, but that a three-term GTO fit is only slightly better than the two-term GTO 4f.     

Optimized chimeragenesis; creating diverse p450 functions

A drawback to generating chimeric proteins by chimeragenesis, especially when the "parent" proteins share low sequence identity, is that unfolded proteins frequently result. In this issue of Chemistry & Biology, Arnold and coworkers report their use of the SCHEMA algorithm to effectively predict ideal hybrids of cytochromes p450.     

On the functional representation of bond energy functions

Ab initio calculations are used to test the ability of various representations to reproduce bond energies. It is found that expansion in 1/R, where R is the bond length, is remarkably efficient and is consistently better than the usual R expansion. A quadratic form in 1/R is better than a cubic representation in R and sometimes even as good as a quartic representation. A cubic function in 1/R is, in all cases studied, better performing than the quartic expansion in R. It is also found that parameters derived with the 1/R expansion are defined more sharply than those derived for the R expansion. It is suggested that the 1/R expansion may be computationally more efficient for simulations of large biomolecules and for constructions of reactive force fields than the standard bond functions. © 1994 by John Wiley & Sons, Inc.     

The effect of bond functions on dissociation energies

The procedure employing bond functions recently suggested by Wright and Buenker has been applied to the N₂ X ¹Σ_g⁺ potential curve within the CAS SCF+MRSD Ci treatment of electron correlation. The basis set used herein is identical to that employed by these authors in their SCF+CI calculations. The D_e and the shape of the resulting potential curve, as judged by the computed vibrational levels, is not so accurate as would be expected from the results reported by Wright and Buenker. Our results indicate that using the CI superposition errors associated with bond functions to cancel basis set incompleteness depends on the treatment of the electron correlation.