Reliability of atomic natural orbital basis sets in calculations involving pseudopotentials

The performance of Atomic Natural Orbital (ANO) basis sets for calculations involving nonempirical core pseudopotentials has been studied by comparing the results for atomic and molecular nitrogen obtained using contracted ANO basis sets with those obtained using both the primitive set and a segmented one. The primitive set has been optimized at the SCF level for atomic N treated as a five-electron pseudo-atom, and consists of 7s and 7p primitive GTOs supplemented by 2d and 1f GTOs optimized at the CI level. From this primitive set three contracted [3s 3p 2d 1f] sets have been obtained. The first one has been derived from the ANOs of the neutral atom, the second has been obtained from an averaged density matrix and the third one is a segmented set. For the atom, the segmented set gives a zero contraction error at the SCF level as it must be in valence-only calculations. The ANO basis sets show some small contraction error at the SCF level but perform better in CI calculations. However, for the diatomic N₂ molecule the ANO basis sets exhibit a rather large contraction error in the calculated SCF energy. A detailed analysis of the origin of this error is reported, which shows that the conventional strategy used to derive ANO basis sets does not work very well when pseudopotentials are involved.     

A simple orbital basis set for π electron calculations of the polarizabilities and hyperpolarizabilities of conjugated systems

Within the theory of coupled clusters, it is proposed to use the orbital basis set of ethylene molecules in the π electron calculations of the polarizabilities and hyperpolarizabilities of conjugated systems. Test calculations show high accuracy of the method in comparison with the full configuration interaction method.     

On the importance of scale and polarization of the atomic orbital basis for LCAO calculations of electronic intensities

The importance of scale and polarization of the atomic basis set for LCAO calculations of the intensities of electronic transitions is investigated using the molecular hydrogen ion, H ₂ ⁺ , as a model. The transitions under consideration are the 1 _g–1 _u the 1 _u–1 _g and the 1 _g–1_uit excitations of this ion, and the electric dipole transition moments are calculated for a range of nuclear separations using the dipole length, the dipole velocity, the dipole acceleration and the time-derivative of the dipole acceleration formulations. For the 1 _g–1 _u and the 1 _u–1 _g excitations scaling and/or polarization are found very efficient for the improvement of the calculated transition moments for small internuclear separations for all the dipole formulations, whereas only the dipole length and the dipole velocity results are well-behaved for large separations. For the 1 _g–1 _u excitation scaling is found to be more important than polarization for all internuclear separations.

---

Zusammenfassung Die Bedeutung der Skalierung und der Polarisation atomarer Basisfunktionen für LCAO-Berechnungen der Intensitäten elektronischer Übergänge wird am Wasserstoffmolekülion, H ₂ ⁺ , als Modell untersucht. Die betrachteten Übergänge sind der 1 _g–1 _u der 1 _u–1 _g und der 1 _g –;1 _u-Übergang dieses Ions. Die zugehörigen Dipolübergangsmomente werden für eine Reihe von Kernabständen berechnet, wobei die Dipollänge, die Dipolgeschwindigkeit, die Dipolbeschleunigung und die zeitliche Ableitung der Dipolbeschleunigung zu Grunde gelegt werden. Für die 1 _g–1 _u und die 1 _u–1 _g-Anregung findet man, daß Skalierung und/oder Polarisation sehr wirksam für eine Verbesserung der berechneten Übergangsmomente bei kleinen Kernabständen sind. Dies gilt für alle Dipolformulierungen, während sich für große Kernabstände nur die Dipollänge und die Dipolgeschwindigkeit richtig verhalten. Für den 1 _g–1 _u-Übergang ist die Skalierung bei allen Kernabständen wichtiger als die Polarisation.

---

Résumé Etude de l'influence de l'échelle et de la polarisation de la base atomique sur les calculs LCAO des intensités des transitions électroniques, en utilisant l'ion moléculaire H ₂ ⁺ comme modèle. On considère les transitions 1 _g–1 _u,1 _u–1 _get 1 _g–1 _u; les moments dipolaires de transition sont calculés pour un éventail de séparations nucléaires en utilisant les différentes formulations: longueur dipolaire, vitesse dipolaire, accélération dipolaire et dérivée par rapport au temps de l'accélération dipolaire. Pour les excitations 1 _g–1 _u-et 1 _u–1 _gles facteurs précités ont une grande importance pour l'amélioration des moments de transition calculés à faible séparation nucléaire dans toutes les formulations, alors qu'à grande séparation nucléaire seules la longueur et la vitesse donnent des résultats convenables. Pour l'excitation 1 _g–1 _ul'échelle est un facteur plus important que la polarisation à toutes distances internucléaires.

     

Coupled-cluster theory in a projected atomic orbital basis

We present a biorthogonal formulation of coupled-cluster (CC) theory using a redundant projected atomic orbital (PAO) basis. The biorthogonal formulation provides simple equations, where the projectors involved in the definition of the PAO basis are absorbed in the integrals. Explicit expressions for the coupled-cluster singles and doubles equations are derived in the PAO basis. The PAO CC equations can be written in a form identical to the standard molecular orbital CC equations, only with integrals that are related to the atomic orbital integrals through different transformation matrices. The dependence of cluster amplitudes, integrals, and correlation energy contributions on the distance between the participating atomic centers and on the number of involved atomic centers is illustrated in numerical case studies. It is also discussed how the present reformulation of the CC equations opens new possibilities for reducing the number of involved parameters and thereby the computational cost.     

On the use of spatial symmetry in ab initio calculations. Transformation of the two-electron integrals from atomic orbital to localized molecular orbital basis

Anm ⁵-dependent integral transformation procedure from atomic orbital basis to localized molecular orbitals is described for spatially extended systems with some Abelian symmetry groups. It is shown that exploiting spatial symmetry, the number of non-redundant integrals for normal saturated hydrocarbons can be reduced by a factor of 2.5-3.5, depending on the size of the system and on the basis. Starting from a list of integrals over basis functions in canonical order, the number of multiplications of the four-index transformation is reduced by a factor of 2.8-3.5 as compared to that of Diercksen's algorithm. It is pointed out that even larger reduction can be achieved if negligible integrals over localized molecular orbitals are omitted from the transformation in advance.     

Two-parameter exponential-type basis functions for atomic calculations

It is found that ordinary STO s fall off too fast in the atomic region in many cases. A new type of basis set, which is more adaptable to the rather different requirements of the various atomic orbitals in an atom, is developed. The suggested functional form χ(r) = Nr^n-1 exp{?α[(βr + 1)^1/2 ? 1]} contains the STO s as a limiting case. Calculations on a series of atoms from H to Zn show that the new basis gives better results than STO s for equal basis set size. The necessary integrals do not present any problem to evaluate.     

Coupled Hartree-Fock calculations of molecular hyperpolarizabilities

The second-order SCF perturbation method for the calculation of electrical polarizabilities is extended to the third order for the calculation of hyperpolarizabilities. Exploratory calculations for the hyperpolarizability of FH suggest that the convergence of the third-order theory is comparable to that of the second, and confirm the sensitivity of β to choice of basis set as found by other workers.     

Interpretation of the vibrational spectra of nitramines on the basis ofab initio calculations

It is shown that, unlike conventional methods of vibrational spectroscopy, the use ofab initia harmonic force fields, IR and Raman intensities, and depolarizations makes possible a rigorous interpretation of the experimental spectra of the simplest aliphatic nitramines (CH₃)₂NNO₂, CH₃NHNO₂, H₂NNO₂, and their isotopomers. The scale factors, which were introduced for each compound to remove the systematic errors of the SCF MO LCAO calculation by fitting the parameters to the observed frequencies, were mutually adjusted during the solution of the inverse vibrational problem. The set of transferable scale factors established in this work can be used directly to analyze spectra of larger molecules. Some common patterns of the force fields and vibrational spectra of nitramines are discussed.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya No. 12, pp. 2106–2117, December, 1994.The authors are grateful to the Russian Foundation for Basic Research (Project No. 93-03-4410) and to the Robert A. Welch Foundation for financial support of the parts of this work performed at the Department of Chemistry of Moscow State University and at the University of Texas at Austin, USA. The authors also acknowledge the support by the Scientific Technical Program Universities of Russia.     

On the use of small atomic basis sets in non-empirical molecular calculations

The quality of an atomic basis set in molecular calculations can be characterized by the deviation of the valence-shell orbital energies from their values in a Hartree-Fock limit calculation. For atoms of a certain row of the periodic system, this quality appears to depend not only on the number of basis functions used in the calculations, as is usually accepted, but also on the number of valence shell electrons of the various atoms.As a consequence of this result, rules can be formulated for the combination of small atomic basis sets to a basis set for a molecular calculation. In fact, the best results are obtained with basis sets in which the deviations of all atomic valence shell orbital-energies from their Hartree-Fock limit values are of the same order of magnitude.

---

Zusammenfassung Die Qualität eines Atombasissatzes zur Rechnung an Molekülen kann durch die Abweichung der Energien der Orbitale der Valenzschalen von den Werten einer Hartree-Fock-Grenzrechnung charakterisiert werden. Im Falle der Atome einer bestimmten Periode im Periodensystem scheint diese Qualität nicht nur von der Zahl der bei der Rechnung benutzten Basisfunktionen abzuhängen, wie allgemein akzeptiert ist, sondern auch von der Anzahl der Elektronen der Valenzschale der verschiedenen Atome.Aufgrund dieses Resultats können Regeln zur Kombination kleiner Atombasissätze zu einem Basissatz für Rechnungen an Molekülen formuliert werden. In der Tat werden die besten Resultate mit Basissätzen erhalten, bei denen die Abweichungen aller Orbitalenergien der Atomvalenzschalen von ihrem Hartree-Fock-Grenzwert in der gleichen Größenordnung sind.

     

Variational calculations with a hyperspherical basis on atomic helium

We have recently formulated an expansion of the N electron wavefunction in an appropriate set of harmonics on the 3 N-dimensional hypersphere. Angular correlation appears in the usual way, while radial correlation appears as a generalized angular correlation. Calculations on¹S helium have been performed to explore the convergence of this expansion. Energies for various angular approximations have been compared with Bunge's angular limits and show a fractional error <3.5× 10^–4. A theoretical contraction procedure is shown to usefully reduce basis size without forfeiting accuracy.Supported in part by a research grant to the Johns Hopkins University from the National Science Foundation.     

Molecular integrals with an arbitrary many-electron operator in the slater atomic orbital basis

Formulas are derived for molecular integrals with an arbitrary many-electron operator. Final results are expressed in terms of overlap integrals and one-center many-electron integrals.Baku University. Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 27, No. 4, pp. 446–449, July–August, 1991. Original article submitted October 12, 1990.     

Large atomic natural orbital basis sets for the first transition row atoms

Large atomic natural orbital (ANO) basis sets are tabulated for the Sc to Cu atoms. The primitive sets are taken from the large sets optimized by Partridge, namely (21s13p8d) for Sc and Ti and (20s12p9d) for V to Cu. These primitive sets are supplemented with threep, oned, sixf, and fourg functions. The ANO sets are derived from configuration interaction density matrices constructed as the average of the lowest states derived from the 3d ⁿ 4s ² and 3d ⁿ⁺¹4s ¹ occupations. For Ni, the¹ S(3d ¹⁰) state is included in the averaging. The choice of basis sets for molecular calculations is discussed.     

Finite electric field valence-shell calculations of molecular hyperpolarizabilities

Using the Finite Electric Field CNDO/II method discussed previously (Chem. Physics Letters 5, 507, (1970)), components of the first hyperpolarizability tensor _ijk are calculated for a series of linear molecules, for H₂O and NH₃ and for methane and its fluorinated derivatives. These are compared, where possible, with results of previous calculations and with experimental results.

---

Zusammenfassung Mit Hilfe der zuvor diskutierten CNDO/II-Methode, bei der endliche Werte des störenden elektrischen Feldes Verwendung finden (Chem. Physics Letters 5, 507 (1970)), werden die Komponenten des ersten Hyperpolarisierbarkeitstensors _ijk für eine Reihe von linearen Molekülen, für H₂O, NH₃ und für Methan und seine fluorierten Derivate berechnet. Soweit möglich, werden die Ergebnisse mit früheren Rechnungen und experimentellen Werten verglichen.

     

Resolution of molecular electric hyperpolarizabilities into atomic terms

Using the dipole acceleration formalism, a general theory for the resolution of molecular electric hyperpolarizabilities, at an arbitrary frequency, into atomic terms is presented. Test calculations for the molecules NH₃, H₂O and HF, at the random phase approximation level of accuracy, are reported.     

Momentum space properties of various orbital basis sets used in quantum chemical calculations

Momentum expectation values (p^k), values of the momentum density P(p) at selected points, and coefficients in the MacLaurin and asymptotic expansions of P(p) are used to test the quality of various orbital basis sets previously used for the atomic helium Hartree-Fock problem. The wellknown position-space defects of Gaussian-type orbital expansions are shown to have their momentum-space counterparts. Expansions of even-tempered Slater-type orbitals are found to be rather accurate. The exponentially damped rational function is found to be the outstanding two-parameter unconventional orbital.     

Quantum calculations of the third order hyperpolarizabilities for inert gases

For inert gases third order hyperpolarizabilities are calculated with a method based on variation-perturbation theory.Curves showing the variation of those coefficients with the pulsations of the incident perturbing wave are given.     

Systematic approach to extended even-tempered orbital bases for atomic and molecular calculations

Explicit formulas are established for simply generating arbitrarily large basis sets of optimal even-tempered Gaussian primitives which systematically approach complete bases for the entire function space. These bases, moreover, reproduce the corresponding optimal atomic SCF wavefunctions extremely closely and permit an extrapolation of the SCF energies to the Hartree-Fock limit. On the basis of the detailed quantitative information available from these calculations a simple general procedure is formulated for generating optimal even-tempered basis sets for molecular calculations.Dedicated to Professor Dr. Hermann Hartmann on the occasion of his 65th birthday.     

Electronic and vibrational hyperpolarizabilities of alkali- and alkaline-earth-doped boron nitride nanotubes

This work reports the results of the vibrational corrections and frequency dependence to the first hyperpolarizabilities of the alkali- and alkaline-earth-doped boron nitride nanotubes. The electronic contributions were computed by means of the density functional theory with the M06-2X functional, and the vibrational corrections were calculated using the perturbation theoretical method and the field-induced coordinates methodology. The results for the electronic contribution show that such materials exhibit large first hyperpolarizabilities and electride characteristic. We also show that the distribution of the excess electron, which originates from the doping atoms, plays an important role in the large electronic hyperpolarizabilities (β^el). Moreover, our findings strongly indicate that the effect of vibrations on the hyperpolarizabilities can be quite important and can even be much larger than the electronic counterpart.