Electron-pair radii and relative sizes of atoms

The electron-pair intracule (relative motion) h(u) and extracule (center-of-mass motion) d(R) densities represent probability densities for the interelectronic distance and the center-of-mass radius of any pairs of electrons, respectively. For 102 atoms from He (atomic number Z = 2) to Lr (Z = 103), we report that electron-pair radii R_2i and R_2e, defined by h(R_2i) = c_2i and d(R_2e) = c_2e, have good linear correlations with the relative sizes R_1 of atoms introduced based on the single-electron density ρ(r) such that ρ(R_1) = c_1, where c_1, c_2i, and c_2e are constants common to the 102 atoms. It is also shown that an interesting relation R_2e ≅ R_2i/2 holds, if c_2e is set equal to 8_c2i

Local-scaling density-functional theory for excited states

The local-scaling density-functional method enables us to determine the ground-state electron density directly and variationally through the generation of a parent wave function of a given density. A generalization of the method to excited states is developed by the use of a configuration-interaction-type reference wave function. From a given density which approximates the nth-state density, all the mth-state wave functions (m≤n) are generated in such a manner that they satisfy the wave function and Hamiltonian orthogonalities. The nth-state electron density is determined so as to minimize the Hamiltonian expectation value over the generated nth-state wave function. An illustrative application is presented for the 2 1S state of the helium atom, and simple electron-density functions which compare well with the near-exact density are reported

Relative sizes of atoms observed in electron momentum densities

The radial electron momentum densities I(p) of atoms are known to reveal several local maxima and minima. For the 103 atoms from H to Lr in their ground states, we report that the reciprocal momenta 1/p_max and 1/p_min, where p_max and p_min are the locations of the maxima and minima in I(p), respectively, have good linear correlations with the relative sizes R of atoms, defined based on the spherically averaged densities ρ(r) in position space

Correlated electron-pair properties of the Be atom in position and momentum spaces

Based on multiconfiguration Hartree–Fock calculations, correlated electron-pair intracule (relative motion) and extracule (center-of-mass motion) properties are reported for the Be atom in position and momentum spaces. Particularly in the latter space, the present results are more accurate and consistent than those in the literature

Sum rules for generalized electron-pair moments

For many-electron atoms, the generalized electron-pair density function g(q;a,b) represents the probability density function for the magnitude ∣ari+brj∣ of two-electron vector ari+brj to be q, where a and b are real-valued parameters. It is pointed out that the second moments 〈q2〉(a,b), associated with g(q;a,b), satisfy a rigorous sum rule which connects one- and two-electron properties of atoms and molecules for any exact and approximate wave functions. The same is also true in momentum space

Position moments linearly averaged over the wave function

It is shown that there exists a set of relations between position moments r^k (with k being a nonnegative integer) linearly averaged over the wave function. The true wave function must satisfy all of these relations, and therefore they can be used as criteria to assess the accuracy of approximate wave functions. The zero momentum energy formula proposed previously is the simplest case of the present results

Zero potential energy criterion applied to Hartree–Fock wave functionｓ

The zero potential energy criterion, which is a necessary condition for the exact wave function, is applied to Hartree–Fock wave functions for a closed‐shell system with doubly occupied spatial orbitals. With the help of the known long‐range asymptotic behavior of Hartree–Fock orbitals, we first derive a single‐electron zero potential energy criterion to be satisfied by Hartree–Fock orbitals. The Hartree–Fock wave function is then shown to never satisfy the zero potential energy criterion, which implies that the Hartree–Fock approximation cannot describe the correct long‐range asymptotic behavior of many‐electron wave functions. Some numerical illustrations are given

Zero potential energy criterion for approximate wave functions

A zero potential energy expression, which is a possible partner to the zero momentum energy expression presented previously, is proposed and discussed as a criterion for assessing the accuracy of approximate wave functions. Applicability of these criteria is illustrated and compared for several approximate wave functions for the 1sσ_g and 2pσ_u states of the {H^+}_{2} molecule

Accurate leading term of the relativistic dispersion force between two ground‐state hydrogen atoms

Using a special form of the nonrelativistic first order perturbation wave function reported very recently for the van der Waals interaction between two ground‐state hydrogen atoms, we show that an accurate leading term (proportional to α^2R^{−4} of the relativistic dispersion force of this system can be obtained in an extremely simple manner