Software to obtain spatially localized functions from different radial functions

We have developed an algorithm that enables simplified box orbital functions (SBO) to be obtained with optimized coefficients by fitting them to functions of many types. SBOs are a linear combination of radial functions useful in quantum chemistry calculations which can be spatially restricted (defined in $$0 \le r < r_{0}$$ interval, and zero for $$r \ge r_{0}$$). The algorithm proposed makes it possible to obtain the optimal radius $$r_{0}$$ and the coefficients of the SBOs of any number of terms from the functions to be fitted, but also allows the user to define a particular radius r and calculate the coefficients of the combination of terms of the SBOs. SBOs have proved to be useful in the calculation of molecular properties, and can reduce the complexity of the integral calculations, especially in huge chemical systems such as atomic clusters. These types of functions are also adequate for studying confined systems such as molecules in solution or big chemical systems such as atomic clusters. In addition, while carrying out the examples presented in this study we have tested the suitability of SBO functions to calculate molecular reactivity, showing that the basis functions provide results as good as the basis sets typically used for this kind of calculations.     

Power moments of hydrogenic Green's functions and Green's functions of the second kind

Power moments for all of the reduced hydrogenic Green's functions have been computed. These moments have the same form as the corresponding moments of the free-particle Green's functions. Green's functions of the second kind are defined, and uses for these objects in model potential theory and the theory of many-body Green's functions are pointed out. In the case of the ground state of the hydrogenic atom, the Green's function of the second kind has been given.     

Cuboidal basis functions

Summary An evaluation of the coulomb integral for a cuboid with uniform density is presented in analytic form, leading to the development of non-overlapping cube basis functions. The coulomb energy of the hydrogen molecule is determined with these functions fitted to the molecular orbital, and this result is compared with theab initio coulomb energy.     

Concerning the theory of chirality functions

Within the scope of the theory of chirality functions, qualitatively complete chirality functions are subject to restrictions concerning both generality and applicability. In contrast thereto, the concept of qualitative supercompleteness results in less restrictive requirements for chirality functions. Consequently, the applicability of qualitatively supercomplete chirality functions is unlimited with respect to the number of ligand kinds. Given this concept, a group theoretical treatment is performed supplying the formal conditions of qualitative supercompleteness. Subsequently a construction rule for qualitiatively supercomplete chirality functions is presented, which is elaborated in detail in the appendix. On combining physical considerations with the requirement of qualitative supercompleteness the resulting chirality functions appear to include all the possible interactions within and/or between ligands and skeleton. From both a mathematical and a physical point of view these chirality functions should be adequate for describing the chiroptical properties of molecules belonging to a given skeletal class. Nevertheless, all the other critical objections to the theory of chirality functions remain.     

Localized functions on a circle

The derivation of hybrids as localized equivalent functions in the plane is discussed using the simultaneous eigenfunctions of the x and y position operators, as represented in a finite basis. It proves helpful, initially, to use complex exponentials as basis functions, but the transformation to a real basis is made later. The introduction of alias functions to produce commuting matrices is described. Full results are obtained for any number of functions in the plane. © 1995 John Wiley & Sons, Inc.     

Thermodynamic functions for methylhalosilanes

Thermodynamic functions (heat capacity, enthalpy, entropy and free energy) are calculated for methylhalosilanes, dimethylhalosilanes, and dimethyldihalosilanes in the ideal gas state from 298.16 to 1200 K at 1 atm pressure. Statistical thermodynamic methods have been used in the calculations, with functions corrected for internal rotation by the method of Pitzer. Agreement with other literature data, where available, is satisfactory.     

Overlap integrals of B functions

Two different methods for the evaluation of overlap integrals of B functions with different scaling parameters are analyzed critically. The first method consists of an infinite series expansion in terms of overlap integrals with equal scaling parameters [14]. The second method consists of an integral representation for the overlap integral which has to be evaluated numerically. Bhattacharya and Dhabal [13] recommend the use of Gauss-Legendre quadrature for this purpose. However, we show that Gauss-Jacobi quadrature gives better results, in particular for larger quantum number. We also show that the convergence of the infinite series can be improved if suitable convergence accelerators are applied. Since an internal error analysis can be done quite easily in the case of an infinite series even if it is accelerated, whereas it is very costly in the case of Gauss quadratures, the infinite series is probably more efficient than the integral representation. Overlap integrals of all commonly occurring exponentially declining basis functions such as Slater-type functions, can be expressed by finite sums of overlap integrals of B functions, because these basis functions can be represented by linear combinations of B functions.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

Integral transform functions and separable potentials

General forms for asymptotic wave functions are derived from properties of the relevant Green's function. The use of separable potentials constructed from the asymptotic functions is described and the relation with integral transform functions is discussed.     

Response theory for vibrational wave functions

A formalism for deriving and implementing response functions for vibrational wave functions is described. The formalism utilizes a recently developed second-quantization formulation of many-mode dynamics to define nonredundant parameterizations for different types of approximate vibrational wave functions. The derived response functions cover the cases of an exact state, a vibrational self-consistent field state, and a vibrational configuration interaction state. Sample calculations are presented for the linear-response function and response excitation energies for a two-mode model system and for formaldehyde employing a quartic force field. The advantages and disadvantages of the response theoretical approach for describing excitation energies using different parameterizations are discussed.     

The excess thermodynamic functions of three-component Lennard-Jones systems based on binary distribution functions

Integral equation theory was used to study the binary distribution functions of the excess properties of three-component Lennard-Jones mixtures. The results obtained for the behavior of the excess functions of caloric properties (internal energy, enthalpy, and isochoric heat capacity) are reported. The influence of the third component with various potential parameters on excess functions was studied. Calculations were performed for systems under super-and subcritical conditions with different sizes of molecules, σ₁/σ₂ = 1–4, and attraction energies between them, ?₁/?₂ = 1–4. The results were compared with numerical experiment data to find that the approach used was fairly accurate     

Ring functions,as polarization functions,forab initio calculations on small rings: Dioxirane

The exponents and relative positions have been optimized for bond functions to be used as polarization functions in the theoretical treatment of small cycles. The transferability of the conventional bond functions to this type of “compact” systems is analyzed for the particular case of dioxirane and it is shown that they can be replaced by a single function located inside the ring (ring function) which describes equally well the polarization effects on the equilibrium geometry, charge distribution and dipole moment. These conclusions are corroborated by the characteristics exhibited by the corresponding localized molecular orbitals.     

Thermodynamic functions for halogenated naphthalenes

Thermodynamic functions (heat capacity, enthalpy, entropy and free energy) have been calculated for naphthalene and 11 halogenated naphthalenes in the ideal gas state from 273.15 to 1200 K at 1 atm pressure. All the functions were obtained by statistical—mechanical calculation methods. Agreement with experimental results, where such are available, is satisfactory.     

A mixed basis set of Slater and Gaussian functions. Integrals with modified Gaussian functions

The analytic expressions of the integral prototypes involving both Slater and s-type Gaussian functions, explicity derived in Ref. 1, are generalized to the case of higher order modified Gaussian functions [2].     

Nonorthogonal ultralocalized functions and fitted Wannier functions for local electron correlation methods for solids

Despite the formal exponential decay behavior of Wannier functions (WFs), their spatial extent, which is a key parameter determining the computational cost of local correlation calculations for solids, is still rather large. The problems with the localization of the WFs can partly be attributed to their mutual orthogonality. Possibilities of reduction of the spatial extent of the WFs without losing the accuracy of the calculations are investigated. A method for generation of nonorthogonal ultralocalized functions based on maximization of their Löwdin populations is developed. A scheme for fitting of the WFs and nonorthogonal localized functions with a limited support is proposed. The calculations show that by combining both techniques one can obtain quite compact linearly independent localized functions, which may significantly decrease the computational cost in post-HF calculations.     

Two-electron distribution functions and intracules

 Two-electron distribution functions and intracules are functions of electronic coordinates and occupy an important, and frequently overlooked, middle ground between the beguiling simplicity of electron densities and the bewildering complexity of wavefunctions. We survey the functions that have been considered by earlier workers and introduce two new ones, the Wigner intracule and the action intracule, that have not previously been discussed. To illustrate their usefulness, we consider the intracules of jellium, a few small atoms and the dissociating hydrogen molecule. Received: 26 July 2002 / Accepted: 20 October 2002 / Published online: 30 January 2003 Correspondence to: P.M.W. Gill e-mail: peter.gill@nott.ac.uk     

Quantum Monte Carlo with Jastrow-valence-bond wave functions

We consider the use in quantum Monte Carlo calculations of two types of valence bond wave functions based on strictly localized active orbitals, namely valence bond self-consistent-field and breathing-orbital valence bond wave functions. Complemented by a Jastrow factor, these Jastrow-valence-bond wave functions are tested by computing the equilibrium well depths of the four diatomic molecules C(2), N(2), O(2), and F(2) in both variational Monte Carlo and diffusion Monte Carlo. We show that it is possible to design compact wave functions based on chemical grounds that are capable of describing both static and dynamic electron correlations. These wave functions can be systematically improved by inclusion of valence bond structures corresponding to additional bonding patterns.     

Localization of electron group functions

The requirements of variational freedom of electron group functions and their spatial separation are basic statements of the group functions method as they are necessary for taking into account the intragroup electron correlation while neglecting the intergroup one. But these requirements seem to be inconsistent with one another. This contradiction can be removed using the notion of antisymmetrical annulment of many-electron functions introduced in the present work. The transformation of group functions (GF ) by means of functions antisymmetrically annulling (ASA ) other GF 's is proposed that does not affect the whole system's wave function but can be used for localizing GF 's. The problem of construction of a function $ \tilde \Phi $ ASA a given Φ can be reduced to solving a system of linear algebraic equations. A sufficient condition of existing of nontrivial $ \tilde \Phi $ is obtained.     

Reorientational correlation functions and memory functions in the J-diffusion limit of the extended rotational diffusion model

New methods for calculating free rotor memory functions and reorientational correlation functions with the J-diffusion model are presented. The methods are much more efficient than previous methods, and have applications in other areas.     

Automated establishment and plotting of multivariate functions

The earlier package for univariate functions is extended to multivariate functions in the QQN series software, which contains millions of models. The models are characterized by numbered basic and compound functions which are selected by the user. Plots for a selected model are displayed in rotatablethree-dimensional coordinates. The package can be run on an IBM-PC or compatible computer.