Slater Exponents Research Articles

Quantum chemistry with Coulomb Sturmians: Construction and convergence of Coulomb Sturmian basis sets at the Hartree-Fock level

The first discussion of basis sets consisting of exponentially decaying Coulomb Sturmian functions for modelling electronic structures is presented. The proposed basis set construction selects Coulomb Sturmian functions using separate upper limits to their principle, angular momentum and magnetic quantum numbers. Their common Coulomb Sturmian exponent is taken as a fourth parameter. The convergence properties of such basis sets are investigated for second and third row atoms at the Hartree-Fock level. Thereby important relations between the values of the basis set parameters and the physical properties of the electronic structure are recognised. For example, an unusually large limit for the angular momentum quantum number in unrestricted Hartree-Fock calculations can be linked to the breaking of spherical symmetry in such cases. Furthermore, a connection between the optimal, i.e. minimum-energy, Coulomb Sturmian exponent and the average Slater exponents values obtained by Clementi and Raimondi (E. Clementi and D. L. Raimondi, J. Chem. Phys. 38, 2686 (1963)) is made. These features of Coulomb Sturmian basis sets emphasise their ability to correctly reproduce the physical features of Hartree-Fock wave functions.

Franck-Condon simulation, including anharmonicity, of the photodetachment spectrum of P2H−: Restricted-spin coupled-cluster single-double plus perturbative triple and unrestricted-spin coupled-cluster single-double plus perturbative triple -F12x potential energy functions of P2H and P2H−

Geometry optimization and harmonic vibrational frequency calculations have been carried out on the X̃(2)A(') state of P(2)H and the X̃(1)A(') state of P(2)H(-) using the restricted-spin coupled-cluster single-double plus perturbative triple excitation [RCCSD(T)] and explicitly correlated unrestricted-spin coupled-cluster single-double plus perturbative triple excitation [UCCSD(T)-F12x] methods. For RCCSD(T) calculations, basis sets of up to the augmented correlation-consistent polarized valence quintuple-zeta (aug-cc-pV5Z) quality were employed, and contributions from extrapolation to the complete basis set limit and from core correlation of the P 2s(2)2p(6) electrons were also included. For UCCSD(T)-F12x calculations, different atomic orbital basis sets of triple-zeta quality with different associated complementary auxiliary basis sets and different geminal Slater exponents were used. When the P 2s(2)2p(6) core electrons were correlated in these F12x calculations, appropriate core-valence basis sets were employed. In addition, potential energy functions (PEFs) of the X̃(2)A(') state of P(2)H and the X̃(1)A(') state of P(2)H(-) were computed at different RCCSD(T) and UCCSD(T)-F12x levels, and were used in variational calculations of anharmonic vibrational wavefunctions, which were then utilized to calculate Franck-Condon factors (FCFs) between these two states, employing a method which includes allowance for anharmonicity and Duschinsky rotation. The photodetachment spectrum of P(2)H(-) was then simulated using the computed FCFs. Simulated spectra obtained using the RCCSD(T)/aug-cc-pV5Z and UCCSD(T)-F12x(x = a or b)/aug-cc-pCVTZ PEFs are compared and found to be essentially identical. Based on the computed FCFs, a more detailed assignment of the observed vibrational structure than previously reported, which includes "hot bands," has been proposed. Comparison between simulated and available experimental spectra has been made, and the currently most reliable sets of equilibrium geometrical parameters for P(2)H and its anion have been derived. The photodetachment spectrum of P(2)D, yet to be recorded, has also been simulated.

Correlation consistent basis sets for molecular core-valence effects with explicitly correlated wave functions: The atoms B–Ne and Al–Ar

Correlation consistent basis sets have been optimized for accurately describing core-core and core-valence correlation effects with explicitly correlated F12 methods. The new sets, denoted cc-pCVnZ-F12 (n=D, T, Q) and aug-cc-pC(F12)VnZ (n=D, T, Q, 5), were developed by augmenting the cc-pVnZ-F12 and aug-cc-pVnZ families of basis sets with additional functions whose exponents were optimized based on the difference between all-electron and valence-electron correlation energies. The number of augmented functions added is fewer, in general, than in the standard cc-pCVnZ and cc-pwCVnZ families of basis sets. Optimal values of the geminal Slater exponent for use with these basis sets in MP2-F12 calculations are presented and are also recommended for CCSD-F12b calculations. Auxiliary basis sets for use in the resolution of the identity approximation in explicitly correlated calculations have also been optimized and matched to the new cc-pCVnZ-F12 series of orbital basis sets. The cc-pCVnZ-F12 basis sets, along with the new auxiliary sets, were benchmarked in CCSD(T)-F12b calculations of spectroscopic properties on a series of homo- and heteronuclear first and second row diatomic molecules. Comparing the effects of correlating the outer core electrons in these molecules with those from conventional CCSD(T) at the complete basis set limit, which involved calculations with new cc-pCV6Z basis sets for the second row elements that were also developed in the course of this work, it is observed that the F12 values are reasonably well converged already at just the triple-zeta level.

Extrapolating MP2 and CCSD explicitly correlated correlation energies to the complete basis set limit with first and second row correlation consistent basis sets

Accurate extrapolation to the complete basis set (CBS) limit of valence correlation energies calculated with explicitly correlated MP2-F12 and CCSD(T)-F12b methods have been investigated using a Schwenke-style approach for molecules containing both first and second row atoms. Extrapolation coefficients that are optimal for molecular systems containing first row elements differ from those optimized for second row analogs, hence values optimized for a combined set of first and second row systems are also presented. The new coefficients are shown to produce excellent results in both Schwenke-style and equivalent power-law-based two-point CBS extrapolations, with the MP2-F12/cc-pV(D,T)Z-F12 extrapolations producing an average error of just 0.17 mE(h) with a maximum error of 0.49 for a collection of 23 small molecules. The use of larger basis sets, i.e., cc-pV(T,Q)Z-F12 and aug-cc-pV(Q,5)Z, in extrapolations of the MP2-F12 correlation energy leads to average errors that are smaller than the degree of confidence in the reference data (approximately 0.1 mE(h)). The latter were obtained through use of very large basis sets in MP2-F12 calculations on small molecules containing both first and second row elements. CBS limits obtained from optimized coefficients for conventional MP2 are only comparable to the accuracy of the MP2-F12/cc-pV(D,T)Z-F12 extrapolation when the aug-cc-pV(5+d)Z and aug-cc-pV(6+d)Z basis sets are used. The CCSD(T)-F12b correlation energy is extrapolated as two distinct parts: CCSD-F12b and (T). While the CCSD-F12b extrapolations with smaller basis sets are statistically less accurate than those of the MP2-F12 correlation energies, this is presumably due to the slower basis set convergence of the CCSD-F12b method compared to MP2-F12. The use of larger basis sets in the CCSD-F12b extrapolations produces correlation energies with accuracies exceeding the confidence in the reference data (also obtained in large basis set F12 calculations). It is demonstrated that the use of the 3C(D) Ansatz is preferred for MP2-F12 CBS extrapolations. Optimal values of the geminal Slater exponent are presented for the diagonal, fixed amplitude Ansatz in MP2-F12 calculations, and these are also recommended for CCSD-F12b calculations.

The accuracy of hyperfine integrals in relativistic NMR computations based on the zeroth-order regular approximation

The accuracy of the hyperfine integrals obtained in relativistic NMR computations based on the zeroth–order regular approximation (ZORA) is investigated. The matrix elements of the Fermi contact operator and its relativistic analogs for s orbitals obtained from numerical nonrelativistic, ZORA, and four–component Hartree–Fock–Slater calculations on atoms are compared. It is found that the ZORA yields very accurate hyperfine integrals for the valence shells of heavy atoms, but performs rather poorly for the innermost core shells. Because the important observables of the NMR experiment—chemical shifts and spin–spin coupling constants—can be understood as “valence properties” it is concluded that ZORA computations represent a reliable tool for the investigations of these properties. On the other hand, absolute shieldings calculated with the ZORA might be substantially in error. Because applications to molecules have so far exclusively been based on basis set expansions of the molecular orbitals, ZORA hyperfine integrals obtained from atomic Slater-type basis set computations for mercury are compared with the accurate numerical values. It is demonstrated that the core part of the basis set requires functions with Slater exponents only up to 104 in the case where errors in the hyperfine integrals of a few percent are acceptable.

Core ionization energies of atoms and ions calculated using the generalized Sturmian method

The generalized Sturmian method for solving the many-electron Schrodinger equation is reviewed. The method is illustrated with calculations of the core ionization energies of a series of atoms and ions. It is shown that when the “basis potential” is chosen to be the actual attractive potential of the nuclei in the system being studied, convergence is rapid, and a correlated solution can be obtained without the use of the self-consistent field approximation. Furthermore, when many-electron basis functions of this type are used, the kinetic energy term disappears from the secular equation, the nuclear attraction potential is diagonal, and the Slater exponents of the basis functions are automatically optimized. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 1–7, 2000

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.

A charge transfer process in o– p H 2 conversion induced by 3d impurities on a perovskite

We consider an ABO 3 perovskite surface in which a transition metal ion I is substituted to the cation B. An H 2 molecule is assumed to sit on the top of the metal ion at a distance d = 4.5 b. We consider the perturbation of the ionic electron by the molecular electrons and protons and calculate the resulting molecule → ion charge transfer admixtures. All non-diagonal, one and two electron, matrix elements have been calculated analytically and are represented as functions of Slater exponents Z i . The o– p conversion process is completed by a non-diagonal and charge-transfer hyperfine contact. This process, denoted XY, is found more efficient than the dipolar one, for small Z i . We conclude that the o– p rates could give some information on the long-range part of the 3d eigenstates in an oxide substrate.

A new semiempirical electronic structure and total energy calculation method for solids and large molecules

In the framework of tight-binding approximation a new self-consistent method is presented for a quantitative description of both configurational and spectroscopic energies and related properties of solids with imperfections as well as large molecules. This method in respect of its capabilities is rather close to the semiempirical methods based on the neglecting differential overlap (NDO) approximation than the empirical tight-binding methods (TBM) and can, in principle, provide more precise results than NDO methods. This is due to the suggested total binding energy expression consisting of well-shared and self-reduced terms inside of which all one-center energies and interatomic electron-electron repulsive energies are correctly involved or cancelled out. In this method, unlike existing “standard” TBM, no absolute values of the band structure energy are needed for the total energy calculations, and the parameters of the method are to be fitted not to bulk properties but to the properties of simple molecules. Moreover, the energies and Slater exponents of AO of individual atoms and screened nuclear charges are used to describe both the intra- and interatomic energies. The advantages of the method are illustrated applying it to calculating diatomic molecules and the silicon vacancy which is to be known as “Anderson negative-effective-U” system.

Basis‐Modified hydrogen atoms as embedding atoms in ab initio chemisorption cluster model calculations on Si surfaces

AbstractA comparative and systematic ab initio study of different models simulating the Si (111) surface has been carried out for a variety of embedding hydrogen atoms including unmodified hydrogen atoms and modified hydrogen atoms described with a STO‐4G basis set and a Slater exponent optimized to have the cluster atoms as neutral as possible. The study has been extended to some chemisorption processes as Ag and Al on Si (111). The main conclusion of the present work is that neither the electronic structure of the isolated cluster models nor the nature of the chemisorption bond depend on the kind of embedding hydrogen atoms used to saturate the free valences of the cluster edge atoms. © John Wiley & Sons, Inc.

Molecular Geometries by the Extended‐Hückel Molecular Orbital method II: Hydrocarbons and organic molecules containing O, N, and S

AbstractBond lengths and bond angles of hydrocarbons and of organic molecules containing oxygen, nitrogen, and sulfur have been investigated by the extended‐Hückel method in its improved ASED (atom superposition and electron delocalization) form. We have examined in detail bond lengths and bond angles of hydrocarbon ‐aliphatic, conjugated, rings, and aromatic – and we have also studied reaction enthalpies. Both properties can be calculated resonably well, if a small adjustment of the parameter k in the distance dependent Wolfsberg‐Helmholz formula is accepted. We have also found that moderate contraction of the usually applied 2s‐oxygen, the 2s‐nitrogen, and the 3s‐sulfur Slater exponents is sufficient to obtain astonishingly good geometries for important classes of organic molecules containing these elements. The three‐membered rings (CH2)2X(X  CH2, NH, O, and S) which have attracted much interest and stimulated theoretical studies have been investigated, and we have found that their geometries can be reproduced nicely. It is important that geometry calculation of the investigated molecules can be carried out without losing transparency and well established prediction capabilities of the original EHMO procedure. The extended‐Hückel method in its improved ASED form is, therefore, a useful tool for combining the information of EHMO results with good geometry calculation.

Selected applications of hyperspherical harmonics in quantum theory

Hyperspherical harmonics are the d-dimensional generalization of ordinary d-dimensional spherical harmonics. In this paper, we review some properties of hyperspherical harmonics. We also discuss the generalization of Fock's reciprocal-space treatment of hydrogen-like atoms and show how it leads naturally to Sturmian basis sets. The hyperspherical Sturmian method yields a matrix whose roots are half the square root of the binding energy of the system. Diagonalization of the matrix determines not only the energy levels of the system, but also appropriate Slater exponents for the basis functions

From PPP‐MO theory to all‐valence electron calculations of ionic and excited states in organic molecules

AbstractPPP MO theory initiated in 1953 a reliable SCF approach to excited states in multielectronic systems. Further developments, such as Del Bene‐Jaffé's CNDO/S method, modified Pople's CNDO all‐valence electron (AVE) approach to improve the results but maintained some basic troubles of this theory which were not in the PPP method. A new approach to SCF‐MO‐NDO calculations with a priori chosen parameters and configuration interaction of singly excited states is developed in this paper from the ground considerations of the PPP method and it is called CNDOL. Thus, AVE NDO approximations have been modified in order to take into account independent monocentric terms in the Fock matrix elements for each kind of azimuthal quantum number atomic orbitals in the basis set. The only input data are Slater exponents, valence state ionization potentials, and electron affinities of each kind of atomic orbital. No parameters are empirically adjusted to fit results to experimental or previous accurate theoretical data. CNDOL is suitable for obtaining theoretically consistent results in polyatomic systems which are too large to be treated by more sophisticated procedures. Applications to hydrocarbons are shown which compare PPP results with the new AVE approach derived from it here.

Two‐center molecular repulsion integrals over slater functions

AbstractFor the general two‐electron two‐center integral over Slater functions, use of the Neumann expansion for the electron‐electron interaction term yields the standard auxiliary functions. These are expanded and integrated explicitly by two independent methods. The resulting simple analytic formula for the total integral is completely general, requiring only the Slater function quantum numbers and exponents and the internuclear separation. Hence all two‐electron hydrid, coulomb, exchange, and one‐center integrals are considered. The efficiency of calculation of this expression is compared with those of other methods, indicating an order of magnitude improvement in speed over recursion for the exchange integral.

Theoretical studies on organic transition-metal complexes—I. Parameterization and calibration of an extended CNDO/2 method for the calculation of stability constants

An extended CNDO/2 method has been developed for the calculation of stability constants of organic transition-metal complexes. An atom parameterization scheme is described for Mn to Zn using Slater exponents which is found to give a reasonable correlation with the ionization potential of the respective metal. The corresponding bonding parameters have been evaluated by initially deriving the best correlation between the calculated and experimental stability constants of a series of substituted copper bis(salicylaldehydes) and then relating these values to the other transition metals using the bond dissociation energies of simple diatomics as the essential criteria.

Quantum chemical studies on electrophilic addition: Part IV. Reaction of bromine with ethylene

Non-empirical SCF MO calculations have been carried out on the Br + + ethylene electrophilic addition process for the first time. Geometry optimizations were performed with a standard STO—3G basis set, except for bromine. For this atom the STO—3G basis was formed from 1 s, 2 sp, 3 sp, 3 d, 4 s, 4 p STO expansions with atomic Slater exponents. Geometry optimizations were also performed with a large Gaussian basis set for H, C and Br. Although the larger basis consistently gives shorter CC and CH distances and larger CBr distances than the STO—3G basis set, both calculations indicate that the cyclic intermediate cation is much more stable than the open structure (by 15–30 kcal). In fact the existence of a stable open structure is doubtful, since both calculations indicate a very flat segment along the open → cyclic structure interconversion potential.

Calculations of zero-field splittings in pyridine derivatives

Zero-field splittings were calculated for benzene, pyridine, and a number of simple derivatives. The experimental decrease of the zfs parameter D upon addition of substituents on the benzene and pyridine rings and upon protonation of pyridine was reproduced theoretically without invoking either aplanarity or nπ* interaction. For a lowest ππ* triplet state of pyridine, D = 0.165 cm−1 and E = −0.020 cm−1 are predicted. Zfs integrals were calculated from a wave function formed from CNDO/2 orbitals with two-configuration CI, using a five-term Gaussian expansion with the best Slater exponents for atoms in molecules.

Generalized scattering factors and generalized Fourier transforms

A method is proposed for evaluating generalized X-ray scattering factors (Fourier transforms of products of atomic orbitals) in the two-center case with Slater-type orbitals. This method is especially appropriate if one of the Slater exponents is considerably larger than the other.

Theoretical nuclear spin–spin coupling constants using atom–atom polarizabilities. 13CH and HH′ coupling constants of some [2.2.1] bicyclic compounds using the INDO approximation

AbstractTheoretical nuclear spin–spin coupling constants are calculated using mutual and self atom–atom polarizabilities according to a theory where no semi‐empirical parameters are used, except Slater exponents which can be obtained from other sources. As an application, the 13CH and HH′ couplings of some [2.2.1] bicyclic compounds are calculated with the aid of INDO molecular orbitals and compared with the experimentally obtained coupling constants.

Second row molecular orbital calculations. Minimal and minimal + 3d basis calculations on second row diatomics

The optimal valence orbital exponents of a series of 24 diatomic molecules containing second row atoms are determined, with and without 3d orbitals in the basis set. Equilibrium bond lengths and dipole moments are calculated using Slater exponents as well as the optimal values. It is found that optimization of valence s and p exponents is as important as the addition of d orbitals to the basis set for bond length predictions and that separate optimization of d exponents does not significantly improve calculated bond lengths. The computed dipole moments are less satisfactory for all basis sets, although the optimal exponent basis including d orbitals are in best overall agreement with experiment.