Hartree–Fock wave functions with a modified GTO basis for atoms

We have solved the atomic Hartree–Fock equations by using the algebraic approach, expanding the single-particle radial wave function in terms of a modified Gaussian type orbitals (GTOs) basis. Several atomic properties such as Kato's cusp condition for the electron density or the correct asymptotic behavior of the electron momentum density distribution are accurately verified. Additionally the energy of the atomic ground state can be obtained by using a smaller number of basis functions than in standard GTO expansions. This study has been performed for several atoms of the first three rows. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 59–64, 1997     

Kohn-Sham potentials corresponding to Slater and Gaussian basis set densities

A new method based on linear response theory is proposed for the determination of the Kohn-Sham potential corresponding to a given electron density. The method is very precise and affords a comparison between Kohn-Sham potentials calculated from correlated reference densities expressed in Slater-(STO) and Gaussian-type orbitals (GTO). In the latter case the KS potential exhibits large oscillations that are not present in the exact potential. These oscillations are related to similar oscillations in the local error function δ _i (r)=(−ɛ _i )ϕ _i (r) when SCF orbitals (either Kohn-Sham or Hartree-Fock) are expressed in terms of Gaussian basis functions. Even when using very large Gaussian basis sets, the oscillations are such that extreme care has to be exercised in order to distinguish genuine characteristics of the KS potential, such as intershell peaks in atoms, from the spurious oscillations. For a density expressed in GTOs, the Laplacian of the density will exhibit similar spurious oscillations. A previously proposed iterative local updating method for generating the Kohn-Sham potential is evaluated by comparison with the present accurate scheme. For a density expressed in GTOs, it is found to yield a smooth “average” potential after a limited number of cycles. The oscillations that are peculiar to the GTO density are constructed in a slow process requiring very many cycles. Received: 24 February 1997 / Accepted: 18 June 1997     

Minimal-basis-adapted pseudopotentials for transition metal atoms

A pseudopotential technique is presented to be used with a single function constructed from the three primitive GTOs for the representation of d orbitals of transition-metal atoms. Applied to Cu, CU₂, and CuF, it has permitted a reduction in the number of 3d-type GTO primitives without reducing the quality of the results.     

Evaluation of one-electron integrals for arbitrary operators V(r) over Cartesian Gaussians: Application to inverse-square distance and Yukawa operators

A general procedure is presented for generating one-electron integrals over any arbitrary potential operator that is a function of radial distance only. The procedure outlines that for a nucleus centered at point C integrals over Cartesian Gaussians can be written as linear combinations of 1-D integrals. These Cartesian Gaussian functions are expressed in a compact form involving easily computed auxiliary functions. It is well known that integrals over the Coulomb operator can be expressed in terms of F_n(T) integrals, where By means of a substitution for F_n(T) by other simple functions, algorithms that form integrals over an arbitrary function can be generated. Formation of such integrals is accomplished with minor editing of existing code based on the McMurchie–Davidson formalism. Further, the method is applied using the inverse-square distance and Yukawa potential operators V(r) over Cartesian Gaussian functions. Thus, the proposed methodology covers a large class of one-electron integrals necessary for theoretical studies of molecular systems by ab initio calculations. Finally, by virtue of the procedure's recursive nature it provides us with an efficient scheme of computing the proposed class of one-electron integrals. © 1993 John Wiley & Sons, Inc.     

Matrix elements for Ĵ2 and Ĵz operators over explicitly correlated Cartesian Gaussian functions

General formalism for evaluation of multiparticle integrals involving J?² and J?_z operators over explicitly correlated Cartesian Gaussian functions is presented. The integrals are expressed in terms of the general overlap integrals. An explicitly correlated Cartesian Gaussian function is a product of spherical orbital Gaussian functions, powers of the Cartesian coordinates of the particle, and exponential Gaussian factors, which depend on interparticular distances. This development is relevant to both adiabatic and nonadiabatic calculations of energy and properties of multiparticle systems. © 1995 John Wiley & Sons, Inc.     

Function extended spaces

Extended function spaces defined over the real field are defined as vector spaces made by the Cartesian product of a real Euclidian space and a real function space. This construct is related to the Holographic Electronic Density Theorem and to the stereographic projection of quantum chemically related and well behaved functions in general. It permits to establish the basis for the Holographic General Function Theorem.     

Contracted auxiliary Gaussian basis integral and derivative evaluation

The rapid evaluation of two-center Coulomb and overlap integrals between contracted auxiliary solid harmonic Gaussian functions is examined. Integral expressions are derived from the application of Hobson's theorem and Dunlap's product and differentiation rules of the spherical tensor gradient operator. It is shown that inclusion of the primitive normalization constants greatly simplifies the calculation of contracted functions corresponding to a Gaussian multipole expansion of a diffuse charge density. Derivative expressions are presented and it is shown that chain rules are avoided by expressing the derivatives as a linear combination of auxiliary integrals involving no more than five terms. Calculation of integrals and derivatives requires the contraction of a single vector corresponding to the monopolar result and its scalar derivatives. Implementation of the method is discussed and comparison is made with a Cartesian Gaussian-based method. The current method is superior for the evaluation of both integrals and derivatives using either primitive or contracted functions.     

Half-numerical evaluation of pseudopotential integrals

A half-numeric algorithm for the evaluation of effective core potential integrals over Cartesian Gaussian functions is described. Local and semilocal integrals are separated into two-dimensional angular and one-dimensional radial integrals. The angular integrals are evaluated analytically using a general approach that has no limitation for the l-quantum number. The radial integrals are calculated by an adaptive one-dimensional numerical quadrature. For the semilocal radial part a pretabulation scheme is used. This pretabulation simplifies the handling of radial integrals, makes their calculation much faster, and allows their easy reuse for different integrals within a given shell combination. The implementation of this new algorithm is described and its performance is analyzed.     

A first-principle study of the adsorption of 1-amino-3-cyclopentene on the (100) silicon surface

The adsorption of 1-amino-3-cyclopentene on the (100) silicon surface has been studied by methods rooted in the density-functional theory using both delocalized (plane waves, PWs) and localized (Gaussian-type orbitals, GTOs) basis functions. The results obtained by modeling the surface by silicon clusters of different sizes are quite similar, thus confirming that the reaction is quite localized. Furthermore, PW and GTO computations give comparable results, provided that the same density functional and carefully chosen computational parameters (contraction of GTO, pseudopotentials, etc.) are used. Slab computations performed in the PW framework show that the cluster results are retrieved when low-coverage adsorption on the surface is considered. On these grounds, we are quite confident that reaction parameters obtained by the more reliable hybrid density functional (PBE0) are essentially converged, our best estimates of reaction and activation free energies are thus -40 and 6 kcal/mol, respectively.     

Four‐center integrals for Gaussian and exponential functions

The shift operator technique is used for deriving, in a unified manner, the master formulas for the four‐center repulsion integrals involving Gaussian (GTO), Slater (STO), and Bessel (BTO) basis functions. Moreover, for the two classes of exponential‐type functions (ETO), i.e., STO and BTO, we give the expressions corresponding to both the Gauss and Fourier transforms. From the comparison of the master formulas of GTO and ETO, we conclude that ETO can perform more efficiently than GTO, and we remark the points where the effort must be focused to carry out this possibility. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 16–28, 2001     

Multicenter and multiparticle integrals for explicitly correlated cartesian gaussian-type functions

An analytical derivation of multicenter and multiparticle integrals for explicitly correlated Cartesian Gaussian-type cluster functions is demonstrated. The evaluation method is based on the application of raising operators that transform spherical cluster Gaussian functions into Cartesian Gaussian functions.     

Calculation of 4-Center Coulomb Repulsion Matrix Elements in a Basis of Exponential Type Spherical AO Using 9-Dimensional Polyspherical Harmonics

A method for calculating 4-center Coulomb repulsion integrals in a basis of exponential type AO with regular sectorial harmonics as angular terms is proposed. The initial integrals are represented as a partial differentiation operator with respect to the Cartesian coordinates of the centers of AO, acting on the scalar function which is a 4-center integral of s functions. Differentiation is performed by calculating the Fourier transform of this scalar function in 9-dimensional Euclidean space with the help of the sectorial harmonic argument summing theorem. Thus compact representation of quantum-chemical multicenter integrals is obtained in a basis of exponential type functions with arbitrary angular parts.     

Gaussian vs. Slater representations of d orbitals: An information theoretic appraisal based on both position and momentum space properties

A detailed appraisal of Gaussian-type orbital (GTO) and Slater-type orbital (STO) expansions of 3d orbitals is carried out for the ²S state of copper—a case that should be maximally unfavorable for STOs. The appraisal is based on a wide variety of both position and momentum space properties and utilizes an information theoretic quality assessment technique. It is found that GTO expansions are not as useful as STO expansions for the prediction of 〈p⁸〉, 〈p⁷〉, and 〈r^?6〉 because these properties probe the functional deficiencies of GTOs at small r and large p. On the other hand, GTO expansions can predict accurate values of large r properties like 〈r⁸〉 despite the fact that their position space asymptotic decay is too fast. Unlike the case of s orbitals in helium, there does not seem to be any consistent ordering between accuracy in position space and accuracy in momentum space. The quality measures are found to be very useful for pinpointing the deficiencies of various expansions. This information enables us to construct easily a new GTO and a new STO expansion that are more accurate than any of the others in the literature. It is suggested that one STO is worth no more than two GTOs in the case of d orbitals.     

Quantum dynamics of solid Ne upon photo-excitation of a NO impurity: A Gaussian wave packet approach

A high-dimensional quantum wave packet approach based on Gaussian wave packets in Cartesian coordinates is presented. In this method, the high-dimensional wave packet is expressed as a product of time-dependent complex Gaussian functions, which describe the motion of individual atoms. It is applied to the ultrafast geometrical rearrangement dynamics of NO doped cryogenic Ne matrices after femtosecond laser pulse excitation. The static deformation of the solid due to the impurity as well as the dynamical response after femtosecond excitation are analyzed and compared to reduced dimensionality studies. The advantages and limitations of this method are analyzed in the perspective of future applications to other quantum solids.     

Density-functional generalized-gradient and hybrid calculations of electromagnetic properties using Slater basis sets

In this paper we extend our density-functional theory calculations, with generalized gradient approximation and hybrid functionals, using Slater-type orbitals (STOs), to the determination of second-order molecular properties. The key to the entire methodology involves the fitting of all STO basis function products to an auxiliary STO basis, through the minimization of electron-repulsion integrals. The selected properties are (i) dipole polarizabilities, (ii) nuclear magnetic shielding constants, and (iii) nuclear spin-spin coupling constants. In all cases the one-electron integrals involving STOs were evaluated by quadrature. The implementation for (ii) involved some complexity because we used gauge-including atomic orbitals. The presence of two-electron integrals on the right-hand side of the coupled equations meant that the fitting procedure had to be implemented. For (iii) in the hybrid case, fitting procedures were again required for the exchange contributions. For each property we studied a number of small molecules. We first obtained an estimate of the basis set limit using Gaussian-type orbitals (GTOs). We then showed how it is possible to reproduce these values using a STO basis set. For (ii) a regular TZ2P quality STO basis was adequate; for (i) the addition of one set of diffuse functions (determined by Slater's rules) gave the required accuracy; for (iii) it was necessary to add a set of 1s functions, including one very tight function, to give the desired result. In summary, we show that it is possible to predict second-order molecular properties using STO basis sets with an accuracy comparable with large GTO basis sets. We did not encounter any major difficulties with either the selection of the bases or the implementation of the procedures. Although the energy code (especially in the hybrid case) may not be competitive with a regular GTO code, for properties we find that STOs are more attractive.     

Propagation with distributed Gaussians as a sparse, adaptive basis for higher-dimensional quantum dynamics

A simple quantum wavepacket propagation algorithm is presented, designed to produce a very compact, non-direct product representation in higher-dimensional cases. Instead of moving basis functions around, localized basis functions at pre-defined centers are added to and deleted from the representation, generating an active basis function set strictly localized to the region where the moving wavepacket has significantly non-zero values. Simple one-dimensional examples prove this property, as well as the ability of the algorithm to accommodate splitting and rejoining of an arbitrary number of wavefunction pieces, and tunnelling through potential energy barriers. It is argued that future applications to higher-dimensional examples will be less expensive than with traditional direct-product bases, since making the basis adaptive has a lower scaling than the elementary steps necessary for any propagation algorithm itself.     

Position-momentum Heisenberg uncertainty in Gaussian enfoldments of Euclidian space

Gaussian enfoldments have been derived from the form of general Gaussian functions centered at arbitrary positions in Euclidian space. Every point of Euclidian space acts in this way as the center of a Gaussian function defined in position space. Using the position-momentum Fourier transform in the quantum mechanical way and applied into the position functions of the Gaussian Euclidian enfoldment, the transform result provides a unique momentum Gaussian function, centered at the origin. In this way, the Euclidian enfoldment disappears in momentum space. Further analysis of the position-momentum relationship indicates that the product of the variances of the enfoldment in position and the corresponding momentum Fourier transform produces some kind of Heisenberg’s uncertainty relation.     

Molecular integrals for Gaussian and exponential‐type functions: Shift operators

Basis functions with arbitrary quantum numbers can be attained from those with the lowest numbers by applying shift operators. We derive the general expressions and the recurrence relations of these operators for Cartesian basis sets with Gaussian and exponential radial factors. In correspondence, the expressions of molecular integrals involving functions with arbitrary quantum numbers can be obtained by applying these operators on the integrals with the lowest quantum numbers. Since the original form of the shift operators is not appropriate to deal with integrals, we give their representation in terms of derivatives with respect to the parameters on which these integrals explicitly depend. Moreover, we translate the recurrence relations to the new representation and, finally, we analyze the general expressions ot the molecular integrals. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 137–145, 2000     

On the origin of Turing instability

A reaction–diffusion system consisting of one, two or three chemical species and taking place in an arbitrary number of spatial dimensions cannot exhibit Turing instability if none of the reaction steps express cross‐inhibition. A corollary of this result – obtained by elementary calculations – underlines the importance of nonlinearity in the formation of stationary structures, a kind of self‐organization on a chemical basis. Relations to global stability of reaction–diffusion systems, and results on multispecies systems are also mentioned. The statements are not restricted to mass action type models. As a by‐product, the solution of a basic inverse problem of formal kinetics is also presented which extends a previous result by Hárs and Tóth (1981) to models with arbitrary – including rational – functions as reaction rates so often occurring, e.g., in enzyme kinetics.