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.     

Exact-exchange Hartree–Fock calculations for periodic systems. I. Illustration of the method

A scheme is presented for performing linear-combination-of-atomic-orbitals (LCAO ) self-consistent-field (SCF ) ab initio Hartree–Fock calculations of the electronic structure of periodic systems. The main aspects which characterize the present method are (i) a thorough discussion of both translational and local symmetry properties and the derivation of general formulas for the transformation of all the relevant monoelectronic and bielectronic terms under symmetry operators. (ii) The use of general yet powerful criteria for the truncation of infinite sums; in particular, the Coulomb electron–electron interactions are subdivided into terms corresponding to intersecting or nonintersecting charge distributions; the latter are grouped into shell contributions and the interaction is evaluated by multipolar expansions; the exchange interaction may be evaluated with great precision by retaining a relatively small number of two-electron integrals according to a truncation criterion which fully preserves its nonlocal character. (iii) The use of a procedure for performing integrals over k , as needed in the evaluation of the Fermi energy and in the reconstruction of the Fock matrix, which is particularly simple because it employs partially intersecting small spheres as integration subdomains where linear extrapolation is admitted. A comparison is finally made of our fundamental equations in the critical SCF stage with those obtainable by a recent proposal which uses Fourier transforms to express Coulomb and exchange integrals.     

General recurrence-relation generation scheme for molecular integral evaluation

We develop a new scheme for evaluating different molecular integrals using Gaussian type orbitals. In this new scheme, the evaluation of integrals is performed in two steps during runtime. The first step is a top-down procedure that maps each recurrence relation into a jagged array (array of arrays), where each element of a member array represents either the final results or some intermediate integrals that are stored in our developed data structure “coarse-grained circular buffer”. This step is the same for all different one- and two-electron operators so that the same algorithm and source codes can be used. In the second step, a bottom-up procedure is carried out that computes all the intermediate and the final molecular integrals by backtracking elements from the last member array of each jagged array. Different source codes should in principle be used for different electron operators in the second step, but which can be generated automatically by our developed recurrence-relation compiler. The currently proposed general recurrence-relation generation scheme provides a new, generic and automatic programming way for various one- and two-electron integrals needed in computational chemistry. Users can even introduce new electron operators and evaluate their integrals during runtime by combining the implementation of the proposed new scheme and the just-in-time compilation technique.     

Author index to volume 73

A series of test calculations on diatomic oxides and hydrides of Sc, Ti, Cr, Ni and Zn have been carried out in order to test the reliability of some pseudopotential methods. Several different forms of some pseudopotential operators were used. Only the highest valence orbitals of each atomic symmetry were explicitly included in the calculations. The results indicate that there are problems associated with all the investigated operators particularly for the lighter transition elements. It is suggested that more reliable results may be obtained with pseudopotential methods using smaller cores.     

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     

Efficient evaluation of nonlocal pseudopotentials via Euler exponential spline interpolation.

An Euler exponential spline (EES) based formalism is employed to derive new expressions for the electron-atom nonlocal pseudopotential interaction (NL) in electronic structure calculations performed using a plane wave basis set that can be computed more rapidly than standard techniques. Two methods, one that is evaluated by switching between real and reciprocal space via fast Fourier transforms, and another that is evaluated completely in real space, are described. The reciprocal-space or g-space-based technique, NLEES-G, scales as NMlogM approximately N2logN, where N is the number of electronic orbitals and M is the number of plane waves. The real-space based technique, NLEES-R, scales as N2. The latter can potentially be used within a maximally spatially localized orbital method to yield linear scaling, while the former could be employed within a maximally delocalized orbital method to yield NlogN scaling. This behavior is to be contrasted with standard techniques, which scale as N3. The two new approaches are validated using several examples, including solid silicon and liquid water, and demonstrated to be improvements on other, reduced-order nonlocal techniques. Indeed, the new methods have a low overhead and become more efficient than the standard technique for systems with roughly 20 or more atoms. Both NLEES methods are shown to work stably and efficiently within the Car-Parrinello ab initio molecular dynamics framework, owing to the existence of p-2 continuous derivatives of a pth-order spline.     

A computationally efficient exact pseudopotential method. I. Analytic reformulation of the Phillips-Kleinman theory

Even with modern computers, it is still not possible to solve the Schrodinger equation exactly for systems with more than a handful of electrons. For many systems, the deeply bound core electrons serve merely as placeholders and only a few valence electrons participate in the chemical process of interest. Pseudopotential theory takes advantage of this fact to reduce the dimensionality of a multielectron chemical problem: the Schrodinger equation is solved only for the valence electrons, and the effects of the core electrons are included implicitly via an extra term in the Hamiltonian known as the pseudopotential. Phillips and Kleinman (PK) [Phys. Rev. 116, 287 (1959)]. demonstrated that it is possible to derive a pseudopotential that guarantees that the valence electron wave function is orthogonal to the (implicitly included) core electron wave functions. The PK theory, however, is expensive to implement since the pseudopotential is nonlocal and its computation involves iterative evaluation of the full Hamiltonian. In this paper, we present an analytically exact reformulation of the PK pseudopotential theory. Our reformulation has the advantage that it greatly simplifies the expressions that need to be evaluated during the iterative determination of the pseudopotential, greatly increasing the computational efficiency. We demonstrate our new formalism by calculating the pseudopotential for the 3s valence electron of the Na atom, and in the subsequent paper, we show that pseudopotentials for molecules as complex as tetrahydrofuran can be calculated with our formalism in only a few seconds. Our reformulation also provides a clear geometric interpretation of how the constraint equations in the PK theory, which are required to obtain a unique solution, are themselves sufficient to calculate the pseudopotential.     

Symmetrization of operator matrix elements

The method of Dupuis and King for generating matrix elements of a totally symmetric one-electron operator in terms of symmetry-distinct integrals only is generalized to the case of nontotally symmetric operators. For operators constructed from two-electron integrals, explicit reduction of integral processing to permutationally inequivalent symmetry-distinct integrals only is described, while for one-electron operators further reductions are derived using double coset decompositions. Finally, some computational consequences of this approach are briefly discussed.     

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.     

Hardness conservation as a new transferability criterion: Application to fully nonlocal pseudopotentials

The concept of chemical hardness has been recently adopted in the framework of Kohn-Sham theory as a faithful ab initio measure of pseudopotential transferability. A fully self-consistent hardness theory was developed and employed to evaluate the transferability of semilocal pseudopotentials. Hardness contains most of the relevant physical information determining the transferability of pseudopotentials and is an important step forward with respect to the logarithmic derivative analysis. We discuss the main features of chemical hardness and the relations between chemical hardness and the original definitions of absolute and local hardness. We then apply the new criterion to investigate the transferability of fully nonlocal Kleinman-Bylander pseudopotentials. Hardness conservation allows us to obtain a meaningful comparison between them and the conventional norm-conserving ones and gives us a criterion to improve the pseudopotential transferability of fully nonlocal pseudopotentials by suitably resetting their local part. © 1997 John Wiley & Sons, Inc.     

Quantum and semiclassical theories for nonadiabatic transitions based on overlap integrals related to fast degrees of freedom

Alternative treatments of quantum and semiclassical theories for nonadiabatic dynamics are presented. These treatments require no derivative couplings and instead are based on overlap integrals between eigenstates corresponding to fast degrees of freedom, such as electronic states. Derived from mathematical transformations of the Schr?dinger equation, the theories describe nonlocal characteristics of nonadiabatic transitions. The idea that overlap integrals can be used for nonadiabatic transitions stems from an article by Johnson and Levine [Chem. Phys. Lett. 13, 168 (1972)]. Furthermore, overlap integrals in path-integral form have been recently made available by Schmidt and Tully [J. Chem. Phys. 127, 094103 (2007)] to analyze nonadiabatic effects in thermal equilibrium systems. The present paper expands this idea to dynamic problems presented in path-integral form that involve nonadiabatic semiclassical propagators. Applications to one-dimensional nonadiabatic transitions have provided excellent results, thereby verifying the procedure. In principle these theories that are presented can be applied to multidimensional systems, although numerical costs could be quite expensive.     

Nonadiabatic couplings from time-dependent density functional theory. II. Successes and challenges of the pseudopotential approximation

We present extensive calculations of nonadiabatic couplings (NACs) between the electronically ground and excited states of molecules, using time-dependent density functional theory (TDDFT) within (modified) linear response [C. Hu et al. J. Chem. Phys. 127, 064103 (2007)]. Our approach is implemented in the pseudopotential framework, with the consideration of nonlinear core corrections. The features of either the ordinary Jahn-Teller conical intersections in X(3) (X=Li, Na, K, Cu, Ag, Au) trimers, or the elliptic Jahn-Teller conical intersections in NaH(2), have been well reproduced. In particular, anticipated results for the H-H(2) collision near the avoided crossing are obtained, showing appealing improvement over the first, real-time, TDDFT calculation. The other important type of intersections, Renner-Teller glancing intersection, has also been studied for several typical molecular systems (BH(2), AlH(2), CH(2)(+), SiH(2)(+)), giving results in reasonable agreement with the theoretical model. Despite these successes, it is found that for some systems, including both Jahn-Teller and Renner-Teller systems, the pseudopotential scheme might give inaccurate results for some NAC components on nonhydrogen atoms. By trying different construction schemes of pseudopotentials, e.g., using local pseudopotentials, the results of NACs are found scheme-dependent and show improvement for some cases. Since there is much freedom in constructing ab initio nonlocal pseudopotentials, our findings on TDDFT calculation of NACs in the pseudopotential scheme might be helpful to give clues for constructing more "realistic" pseudopotentials.     

Slater-type geminals in explicitly-correlated perturbation theory: application to n-alkanols and analysis of errors and basis-set requirements

In the recent years, Slater-type geminals (STGs) have been used with great success to expand the first-order wave function in an explicitly-correlated perturbation theory. The present work reports on this theory's implementation in the framework of the Turbomole suite of programs. A formalism is presented for evaluating all of the necessary molecular two-electron integrals by means of the Obara-Saika recurrence relations, which can be applied when the STG is expressed as a linear combination of a small number (n) of Gaussians (STG-nG geminal basis). In the Turbomole implementation of the theory, density fitting is employed and a complementary auxiliary basis set (CABS) is used for the resolution-of-the-identity (RI) approximation of explicitly-correlated theory. By virtue of this RI approximation, the calculation of molecular three- and four-electron integrals is avoided. An approximation is invoked to avoid the two-electron integrals over the commutator between the operators of kinetic energy and the STG. This approximation consists of computing commutators between matrices in place of operators. Integrals over commutators between operators would have occurred if the theory had been formulated and implemented as proposed originally. The new implementation in Turbomole was tested by performing a series of calculations on rotational conformers of the alkanols n-propanol through n-pentanol. Basis-set requirements concerning the orbital basis, the auxiliary basis set for density fitting and the CABS were investigated. Furthermore, various (constrained) optimizations of the amplitudes of the explicitly-correlated double excitations were studied. These amplitudes can be optimized in orbital-variant and orbital-invariant manners, or they can be kept fixed at the values governed by the rational generator approach, that is, by the electron cusp conditions. Electron-correlation effects beyond the level of second-order perturbation theory were accounted for by conventional coupled-cluster calculations with single, double and perturbative triple excitations [CCSD(T)]. The explicitly-correlated perturbation theory results were combined with CCSD(T) results and compared with literature data obtained by basis-set extrapolation.     

Self-consistent second-order screening in many-body theory

Self-consistent screening arises in all many-body problems where one-particle equations containing functionals of the solutions are considered. In the theory of simple metals, e.g., these one-particle equations are usually solved to second-order perturbation in the crystal pseudopotential and to first order in the screening. However, higher-order screening terms have a sizable effect on the form factors, the electron charge density, and the screened pseudopotential. The effects of second order are presented for the bcc simple metals Na and Ba. A nonlocal exchange-correlation potential has been used since local density approximations such as ρ^1/3 cause difficulties in the electron–electron dielectric function. © 1994 John Wiley & Sons, Inc.     

An efficient local coupled cluster method for accurate thermochemistry of large systems

An efficient local coupled cluster method with single and double excitation operators and perturbative treatment of triple excitations [DF-LCCSD(T)] is described. All required two-electron integrals are evaluated using density fitting approximations. These have a negligible effect on the accuracy but reduce the computational effort by 1-2 orders of magnitude, as compared to standard integral-direct methods. Excitations are restricted to local subsets of non-orthogonal virtual orbitals (domain approximation). Depending on distance criteria, the correlated electron pairs are classified into strong, close, weak, and very distant pairs. Only strong pairs, which typically account for more than 90% of the correlation energy, are optimized in the LCCSD treatment. The remaining close and weak pairs are approximated by LMP2 (local second-order Mo?ller-Plesset perturbation theory); very distant pairs are neglected. It is demonstrated that the accuracy of this scheme can be significantly improved by including the close pair LMP2 amplitudes in the LCCSD equations, as well as in the perturbative treatment of the triples excitations. Using this ansatz for the wavefunction, the evaluation and transformation of the two-electron integrals scale cubically with molecular size. If local density fitting approximations are activated, this is reduced to linear scaling. The LCCSD iterations scale quadratically, but linear scaling can be achieved by neglecting some terms involving contractions of single excitations. The accuracy and efficiency of the method is systematically tested using various approximations, and calculations for molecules with up to 90 atoms and 2636 basis functions are presented.     

Analysis of the long range potentials of the X 1Σ+ and a 3Σ+ states of NaK

The difference in energy between the singlet and triplet states of NaK dissociating to Na(3s) and K(4s) is found from experimental data and from pseudopotential calculations. The contributions of various one- and two-electron integrals are evaluated, illustrating the present ambiguity over the exchange integral and the exchange interaction terms.     

Nonempirical parameters of the nonlocal core pseudopotential for the second and third row elements

For the core pseudopotential (CP) model constructed in terms of Bonifacic-Huzinaga nonlocal CP theory, parameters of the local component of CP are calculated for the second-and third-row elements. The resulting CP are associated with the Coulomb, exchange, and correlation potentials created by the nuclear charge and electron density of the core electrons. The electronic structure and potential energy surface are calculated for the hydrides of the second-row elements (LiH, CH₄, NH₃, H₂O, HF); the calculations are performed by the nonempirical nonlocal CP method. The results of these calculations agree well with those of SCF MO LCAO ab initio calculations and with experimental data.     

Locality of exchange matrices for common Gaussian basis sets

This article investigates a new approach to local approximations to exchange. When a finite basis set is introduced, any operator may be regarded as equivalent to a local operator if its matrix can be reproduced as the matrix of a multiplicative function operator. The expectation values of such operators can be determined from the electron density. Any matrix can be divided into local and nonlocal components, in a way dependent on linear dependencies among basis-set products. A measure of locality is provided by the ratio of the norm of the local component to that of the whole matrix. The local contribution to an expectation value can also be compared with the total. The self-consistent field exchange matrices and their expectation values for atoms Li through Ne and for LiH and HF with several Cartesian Gaussian basis sets were investigated in this way, and for the atoms, the exchange supermatrix was also examined. It has been found in all cases that the matrices and expectation values are more than 92% local; most are more than 99% local. © 1997 John Wiley & Sons, Inc.     

Three-dimensional numerical integration for electronic structure calculations

Two three-dimensional numerical schemes are presented for molecular integrands such as matrix alements of one-electron operators occuring in the Fock operator and expectation values of one-electron operators describing molecular properties. The schemes are based on a judicious partitioning of space so that product-Gauss integration rules can be used in each region. Convergence with the number of integration points is such that very high accuracy (8–10 digits) may be obtained with obtained with a modest number of points. The use of point group symmetry to reduce the required number of points is discussed. Examples are given for overlap, nuclear potential, and electric field gradient integrals.     

Grid-free DFT implementation of local and gradient-corrected XC functionals

Following an approach to density functional theory calculations based on the matrix representation of operators, we implemented a scheme as an alternative to traditional grid-based methods. These techniques allow integrals over exchange-correlation operators to be evaluated through matrix manipulations. Both local and gradient-corrected functionals can be treated in a similar way. After deriving all the required expressions, selected examples with various functionals are given. Received: 7 March 1998 / Accepted: 21 May 1998 / Published on line: 6 August 1998