Extremal electron pairs

Extremal pair functions for an n-electron wave function of a closed-shell state are defined as linear combinations of spin-orbital-product pair functions that make some functionals (e.g., r²₁₂ or r⁻¹₁₂) extremal. They are related to the natural spin geminals in the uncorrelated limit and are useful both for an analysis of wave functions in view of an understanding of the chemical bond and for the treatment of electron correlation. Numerical examples are shown and discussed for He₂ as well as the 10-electron systems Ne, HF, H₂O, NH₃, and CH₄. © 1996 John Wiley & Sons, Inc.     

Approximate ansatz for the expansion of the spherically averaged wave function in terms of interelectronic separation r12 for the Hookean atom,atomic ions,and the H2 molecule

For the two‐electron Hookean atom, it is first emphasized that, for a specific force constant k = 1/4, the ground‐state wave function has a simple dependence on the interelectronic separation r₁₂, namely, (1 + ½r₁₂)exp(??r). For this two‐electron model, therefore, the study of Rassolov and Chipman on the electron–electron cusp conditions on the spherically averaged wave function for the N electron atomic ions can be generalized to all orders in the interelectronic separation r₁₂. This Hookean model has therefore been used to give some justification for an ansatz for the spherically averaged wave function in atomic ions with N electrons for N ≥ 2. Several approximate two‐electron wave functions satisfying the Rassolov and Chipman conditions were tested and found to give excellent results. Another ansatz has been tested numerically on the ground state of two‐electron atomic ions and the H₂ molecule. Finally, for the Hookean atom a partial differential equation that is essentially for the pair correlation density is given in the Appendix . © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 95: 21–29, 2003     

Radial limit of lithium revisited

Configuration interaction (CI) calculations are carried out for the ground state of lithium using a thoroughly optimized basis set of s-type Slater functions. They establish that the radial limit of the nonrelativistic energy of the ground ²S state of lithium is no higher than −7.448666443E_h. Thus, radial correlation accounts for 35.2% of the total correlation energy. The radial CI wave function predicts a significantly more accurate Fermi contact parameter than the Hartree-Fock wave function. However, the imbalanced treatment of electron correlation in the radial CI wave function leads to an excessively diffuse electron density that is worse than that of the Hartree-Fock wave function. © 1997 John Wiley & Sons, Inc.     

Boson and fermion many-body assemblies: fingerprints of excitations in the ground-state wave functions,with examples of superfluid 4He and the homogeneous correlated electron liquid

After a brief summary of some basic properties of ideal gases of bosons and of fermions, two many-body Hamiltonians are cited for which ground-state wave functions allow the generation of excited states. Because of the complexity of ground-state many-body wave functions, we then consider properties of reduced density matrices, and in particular, the diagonal element of the second-order density matrix. For both the homogeneous correlated electron liquid and for an assembly of charged bosons, the ground-state pair correlation function g(r) has fingerprints of the zero-point energy of the plasmon modes. These affect crucially the static structure factor, S(k), in the long wavelength limit. This is best understood by means of the Ornstein–Zernike direct correlation function c(r), which plays an important role throughout this article. Turning from such charged liquids, both boson and fermion, to superfluid ⁴He, the elevated temperature (T) structure factor S(k, T) is related, albeit approximately, to its zero-temperature counterpart, via the velocity of sound, reflecting the collective phonon excitations and the superfluid density. Finally, some future directions are pointed.     

Pair density related to one‐electron information for the ground state of spin‐compensated two‐electron systems

The recent study by Joubert on effects of Coulomb repulsions in a many‐electron system has focused attention on an integral identity involving the pair density. This has motivated the derivation presented here of a vectorial differential form related to this integral result. Our differential identity is then illustrated explicitly by using (i) an exact ground‐state wave function for the so‐called Hookean atom having external potential energy (1/2)kr², with k = 1/4, and (ii) Moshinsky's model in which both the interparticle interaction and the external potential are of harmonic type. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Electron‐pair density decomposition for core–valence separable systems

The electron pair density of a core‐valence separable system can be decomposed into three parts: core‐core, core‐valence, and valence‐valence. The core‐core part has a Hartree‐Fock like structure. The core‐valence part can be written as Γ_cv (1,2) = γ_c (1,1)γ_v (2,2) ? γ_c (1,2)γ_v (2,1) + γ_c (2,2)γ_v (1,1) ? γ_c (2,1)γ_v (1,2), where only the 1‐matrices from the core and valence orbitals contribute. The valence‐valence part is left to be determined from the reduced frozen‐core type wave function, which often contains the essential information on the electron correlation and the chemical bond. We demonstrate the analysis to the ground state of negative ion Li^? and 2¹Σ_u⁺ excited state of the Li₂ molecule. © 2012 Wiley Periodicals, Inc.     

Fermi and Coulomb holes of molecules in excited states

Correlation holes of electrons with the same (Fermi hole) and different (Coulomb hole) spins in the ground (X¹Σ⁺), first (A¹Σ⁺) and second (B¹II) excited states of LiH were constructed from full configuration interaction (CI ) wave functions. It was found that the shapes of both the Fermi and Coulomb holes in these states are dependent on the location of the reference electron. When the reference electron is chosen to be close to the Li nucleus, the Fermi correlation results in a large negative hole for all three states. However, the A¹Σ⁺ excited state is further characterized by displaying a second hole around the H nucleus, and in the B¹II state, the hole is elongated along the molecular axis. Coulomb correlation shows up strongly in the A¹Σ⁺ state and, in addition, there is clearly correlation of electrons at the two nuclei. These features of the correlation holes were compared with those from a two-Slater-determinant model wave function. The Hartree, Fermi, and Coulomb screening potentials in these states were also studied in the light of possible modeling of the correlation functionals for the excited states. © 1995 John Wiley & Sons, Inc.     

Ab initio calculations including electron correlation for the minimum energy path of the (1A1) CH2+(1Σ)H2→ (1A1) CH4 insertion reaction

Ab initio calculations on the SCF level and with the inclusion of valence shell electron correlation in the IEPA–PNO (independent electron pair approximation with pair natural orbitals), the PNO–CI (pair-natural-orbital configuration interaction) and the CEPA–PNO (coupled electron pair approximation with pair natural orbitals) schemes with Gaussian lobe functions of “double zeta quality” have been performed for the minimum energy path of the insertion of singlet (¹A₁) methylene to the (¹Σ)H₂ molecule to yield methane. The energy was minimized on the SCF level to all geometrical parameters for various values of the “approximate” reaction coordinate. The energy along the reaction path decreases monotonically without a barrier and the curves representing the total energy of the system as a function of approximate reaction coordinates obtained at different levels of approximations have the same shape. From the physical point of view three phases of the reaction can be distinguished (chemically two steps) with different geometrical arrangements and different internal geometries of the partners.     

One‐particle density matrix functional for correlation in molecular systems

Based on the analysis of the general properties for the one‐ and two‐particle reduced density matrices, a new natural orbital functional is obtained. It is shown that by partitioning the two‐particle reduced density matrix in an antisymmeterized product of one‐particle reduced density matrices and a correction Γ^c we can derive a corrected Hartree–Fock theory. The spin structure of the correction term from the improved Bardeen–Cooper–Schrieffer theory is considered to take into account the correlation between pairs of electrons with antiparallel spins. The analysis affords a nonidempotent condition for the one‐particle reduced density matrix. Test calculations of the correlation energy and the dipole moment of several molecules in the ground state demonstrate the reliability of the formalism. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 317–323, 2003     

Molecular formation and electron correlation in HeH+

A one-centre CI wave function for HeH⁺ reported by Stuart and Matsen for 0.1 ? R ? 5.0 has been analysed in detail from the viewpoint of molecular formation. Further, by means of a natural orbital analysis, it was possible to obtain some measure of the electron correlation contained within such wave functions for various R values. These effects were illustrated by means of a series of difference maps for the electron density. One- and two-particle expectation values were obtained as a function of R. Thus, it was possible to study several aspects of the influence of the proton on the electron charge cloud as we pass from He through to the united atom Li⁺. The occupation numbers within the natural expansions were compared with those which arise from a similar analysis of a two-centre wave function for HeH⁺. The “character” of such wave functions for HeH⁺, and also for He and Li⁺, were analysed and compared.     

Asymptotic behavior of the primitive function of different “symmetry-adapted” perturbation schemes for the H ground state

The leading terms in the asymptotic 1/R expansion of the wave functions and energies of various “symmetry-adapted” perturbation schemes for intermolecular forces, as well as for the “polarization approximation” (PA ) are derived for the H ground state, both exactly (i.e., to infinite order in λ) and in the first two orders of the λ expansion. It is pointed out that only in the PA and the Hirschfelder–Silbey scheme is the formal primitive function Φ genuinely primitive, whereas Φ in the other schemes is asymmetric in a rather strange way. In this lack of genuine primitivity lies the reason why in these schemes the leading term of the 1/R expansion is only recovered to infinite order in λ and requires knowledge of the R^?4 term of the wave function, provided that one uses the “internal” energy expression. Four different energy expressions are compared that behave differently in the different schemes. The results obtained here are basis independent, but the implications of the basis completeness problem are discussed as well.     

Correlation factor approach to the correlation energy functional

A global survey of the correlation factor energy functionals and its application to atomic and molecular properties is made. Its performances are compared with those of the density functional theory (DFT) correlation energy functionals, and some interesting conclusions from previous publications are reinforced here; namely, after removing the one-Slater-determinant hypothesis from the Kohn–Sham method, all DFT correlation functionals are able to provide reasonable results in any circumstance, with an additional restriction, for systems having a quasi-degenerate wave function, the DFT correlation functionals must depend explicitly on the on-top density. Acknowledgement.This work has been done under the support of the Spain DGICYT, project n⁰ BQU2001-0883.     

A survey of wave function effects on theoretical calculation of gas phase F NMR chemical shifts using factorial design

The wave functions for calculating gas phase ¹⁹F chemical shifts were optimally selected using the factorial design as a multivariate technique. The effects of electron correlation, triple-ξ valance shell, diffuse function, and polarization function on calculated ¹⁹F chemical shifts were discussed. It is shown that of the four factors, electron correlation and the polarization functions affect the results significantly. B3LYP/6-31 + G(df,p) wave functions have been proposed as the best and the most efficient level of theory for calculating ¹⁹F chemical shifts. An additional series of fluoro compounds were used as a test set and their predicted ¹⁹F chemical shifts values confirmed the validity of the approaches.     

Direct evaluation of one-electron properties in coupled cluster methods

Summary An analysis of a method for approximate calculations of expectation values for one-electron operators from available coupled cluster amplitudes is presented and illustrated numerically for the polarizability of the Be atom. The one-particle density matrix resulting from the present approach is accurate through the fourth order in the electron correlation perturbation. It has been found that, in order to obtain quantitative agreement between the energy derivative results and the approximate expectation value formalism, the third orderT ₁ T ₂⁽⁰⁾ wave function term must be included into the calculation of the one-particle density matrix. The present method is also considered as a promising tool for calculations of higher-order atomic and molecular properties from high level correlated wave functions.     

Diatomic interactions in momentum space. The 1sσg and 2pσ u states of H 2 + system

The method of momentum electron density for interatomic interactions has been applied to the two lowest σ states of the H ₂ ⁺ system. For attractive (1sσ_g) and repulsive (2pσ _u ) interactions, the behaviour of momentum density and its effect on the stabilization energy of the system are examined quantitatively. The concept of contraction and expansion of the momentum density is shown to form an important guiding principle in this approach. The origin of covalent bonding is discussed based on the energy partitioning proposed previously.