With recent advances in electronic structure methods, first-principles calculations of electronic response properties, such as linear and nonlinear polarizabilities, have become possible for molecules with more than 100 atoms. Basis set incompleteness is typically the main source of error in such calculations since traditional diffuse augmented basis sets are too costly to use or suffer from near linear dependence. To address this problem, we construct the first comprehensive set of property-optimized augmented basis sets for elements H–Rn except lanthanides. The new basis sets build on the Karlsruhe segmented contracted basis sets of split-valence to quadruple-zeta valence quality and add a small number of moderately diffuse basis functions. The exponents are determined variationally by maximization of atomic Hartree–Fock polarizabilities using analytical derivative methods. The performance of the resulting basis sets is assessed using a set of 313 molecular static Hartree–Fock polarizabilities. The mean...

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Property-optimized gaussian basis sets for molecular response calculations.

Ab Initio Methods for the Calculation of NMR Shielding and Indirect Spin-Spin Coupling Constants

The theory of molecular structure determined by the gradient vector field of the charge density ρ identifies the set of atomic interactions present in a molecule. The interactions so defined are characterized in terms of the properties of the Laplacian of the charge density ∇2ρ(r). A scalar field is concentrated in those regions of space where its Laplacian is negative and depleted in those where it is positive. An expression derived from the quantum mechanical stress tensor relates the sign of the Laplacian of ρ to the relative magnitudes of the local contributions of the potential and kinetic energy densities to their virial theorem averages. By obtaining a map of those regions where ∇2ρ(r) 0. The mechanics are characterized by the relatively large value of the kinetic energy, particularly the component parallel to the interaction line. In the closed‐shell interactions, the regions of dominant potential energy contributions are separately localized within the boundaries of each of the interacting atoms or molecules. In the shared interactions, a region of low potential energy is contiguous over the basins of both of the interacting atoms. The problem of further classifying a given interaction as belonging to a bound or unbound state of a system is also considered, first from the electrostatic point of view wherein the regions of charge concentration as determined by the Laplacian of ρ are related to the forces acting on the nuclei. This is followed by and linked to a discussion of the energetics of interactions in terms of the regions of dominant potential and kinetic energy contributions to the virial as again determined by the Laplacian of ρ. The properties of the Laplacian of the electronic charge thus yield a unified view of atomic interactions, one which incorporates the understandings afforded by both the Hellmann–Feynman and virial theorems.

The characterization of atomic interactions

Hartree‐Fock perturbation theory of magnetic susceptibility and magnetic shielding is developed using a basis set of gauge invariant atomic orbitals. The theory is used to calculate magnetic shielding and spin‐rotation constants associated with the nuclei in LiH and HF giving results in good agreement with experimental values.

Molecular Orbital Theory of Magnetic Shielding and Magnetic Susceptibility

Multiconfiguration perturbation theory with imaginary level shift

Publisher Summary The purpose of this chapter is to provide information on the recent developments in perturbation theory. In recent years, there is a great increase of interest in the application of perturbation theory to the fundamental problems of quantum chemistry. Perturbation theory is designed to deal systematically with the effects of small perturbations on physical systems when the effects of the perturbations are mathematically too difficult to calculate exactly, and the properties of the unperturbed system are known. The new applications have been mainly to atoms where the reciprocal of the atomic number, l/Z, provides a natural perturbation parameter. These may be divided into two groups. The first consists of calculations of energy levels, and is a natural outgrowth of Hylleraas's classic work on the 1/Z expansion for two-electron atoms. The applications in the second group are to the calculation of expectation values and other properties of atoms and molecules, and are of much more recent origin. There are two principal reasons for the success of these new applications: (1) sufficient accuracy is frequently obtained from knowledge of a first-order perturbed wave function, and (2) a great advantage of perturbation theory is that the functional form of the perturbed wave function is shaped by the perturbation itself.

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Recent Developments in Perturbation Theory

The well‐known relationship between gauge invariance and current conservation is exhibited within the usual quantum mechanical formalism. It is then shown that the use of Gauge Invariant Atomic Orbitals does not necessarily lead to the expected current conservation. The reason is found to lie in the constrained nature of the gauge invariance which is provided by the use of GIAO's. It is concluded that this invariance is, of itself, no argument in favor of their use.

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Gauge invariance, current conservation, and GIAO's

The most important contributions to the expansion of the exact long‐range retarded interaction energy, between two nondegenerate atoms, are evaluated accurately for the H–H, H–He, and He–He interactions.

Evaluation of long-range retarded interaction energies.

For an isoelectronic molecular process X→Y, the energy change ΔW = WY—WX may be computed from any one of three equivalent formulas if the exact molecular wavefunctions, ψX and ψY, are known: the expectation‐value difference, the integrated Hellmann—Feynman formula, and the integral Hellmann—Feynman formula. Should only approximate wavefunctions be available, ΨX and ΨY, these formulas give different estimates of the energy change. If the Hamiltonians for X and Y differ in values of some parameter or parameters λ, say λ = 0 for X and λ = 1 for Y, one has HX=H(0), HY=H(1), H=H(λ), ΨX=Ψ(0), ΨY=Ψ(1), Ψ=Ψ(λ), and the three formulas are as follows: ΔWed=〈Ψ(1) |H(1) |Ψ(1)〉/〈Ψ(1) |Ψ(1)〉−〈Ψ(0) |H(0) |Ψ(0)〉/〈Ψ(0) |Ψ(0)〉,ΔWd= ∫ 01dλ〈Ψ(λ) |∂H(λ)/∂λ |Ψ(λ)〉〈Ψ(λ) |Ψ(λ)〉,ΔWl=〈Ψ(0) |H(1)−H(0) |Ψ(1)〉/〈Ψ(0) |Ψ(1)〉. Relative advantages and disadvantages of these formulas are discussed, and illustrations are given of their use. Conditions for the equivalence of the formulas are established. It is shown...

Integrated and Integral Hellmann—Feynman Formulas

(i) Gauge invariance in the presence of nonlocal potentials is reviewed. It is shown that the uncoupled Hartree‐Fock method is not gauge invariant. (ii) Gauge invariance of variational calculations is reviewed and it is shown that the coupled Hartree‐Fock approximation is gauge invariant. (iii) A procedure for choosing a best gauge is suggested and in a special case is shown to reduce to a method recently proposed by Karplus and Kolker.

Saul T. Epstein

Papers

Recent Developments in Perturbation Theory

Gauge invariance, current conservation, and GIAO's

Evaluation of long-range retarded interaction energies.

Integrated and Integral Hellmann—Feynman Formulas

Gauge invariance of the hartree-fock approximation