Exchange energy density definitions from the optimized exchange-force,exemplified for non-relativistic Ne- and Ar-like atomic ions in the limit of large nuclear charge

An alternative exchange energy density (eed) in DFT is first proposed, depending on the optimized exchange force (OEF). It is evaluated using a numerical form of OEF for the non-relativistic Ne-like atomic ion series for atomic number Z = 92. Comparison is made with the Dirac form of eed available analytically. Related results are also presented for Ar-like atomic ions.     

A proposed family of variationally correlated first-order density matrices for spin-polarized three-electron model atoms

In early work of March and Young (Phil Mag 4:384, 1959), it was pointed out for spin-free fermions that a first-order density matrix (1DM) for $N-1$ particles could be constructed from a 2DM ( $\Gamma $ ) for $N$ fermions divided by the diagonal of the 1DM, the density $n(\mathbf{r}_1)$ , as $2\Gamma (\mathbf{r}_1,\mathbf{r}^{\prime }_2;\mathbf{r}_1,\mathbf{r}_2)/n(\mathbf{r}_1)$ for any arbitrary fixed $\mathbf{r}_1$ . Here, we thereby set up a family of variationally valid 1DMS constructed via the above proposal, from an exact 2DM we have recently obtained for four electrons in a quintet state without confining potential, but with pairwise interparticle interactions which are harmonic. As an indication of the utility of this proposal, we apply it first to the two-electron (but spin-compensated) Moshinsky atom, for which the exact 1DM can be calculated. Then the 1DM is found for spin-polarized three-electron model atoms. The equation of motion of this correlated 1DM is exhibited and discussed, together with the correlated kinetic energy density, which is shown explicitly to be determined by the electron density.     

Explicit form of Pauli potential for direct derivation of pair density from a two-particle differential equation for the quintet state of four electrons with harmonic interparticle interactions

Recently, an analytic two-particle density matrix (2DM) has been derived for the quintet state of four electrons interacting via two-body harmonic forces. Here we use this 2DM to extract the exact pair density $\Gamma (\mathbf{r}_1,\mathbf{r}_2)$ . This is then employed in the known two-particle partial differential equation for the pair density amplitude to extract the Pauli potential $v_P(\mathbf{r}_1,\mathbf{r}_2)$ for this quintet state.     

An efficient protocol for the enantioselective preparation of a key polyfunctionalized cyclohexane. New access to (R)- and (S)-4-Hydroxy-2-cyclohexenone and (R)- and (S)-trans-cyclohex-2-ene-1,4-diol

Starting from very accessible raw materials such as p-methoxyphenol, ethylene glycol, and thiophenol, a protocol has been developed to prepare multigram quantities of the polyfunctionalized cyclohexane (+/-)- 7. A highly efficient resolution of (+/-)- 7 has been achieved through enantioselective acetylation catalyzed by Candida antarctica lipase B. Straightforward and enantioselective syntheses of 4-hydroxy-2-cyclohexenone, 1, trans-cyclohex-2-ene-1,4-diol, 2, and their O-protected derivatives 18 and 19 have been readily accomplished from 7.     

Empirically predicted 13C NMR chemical shifts for 8-hydroxyflavone starting from 7,8,4'-trihydroxyflavone and from 7,8-dihydroxyflavone

8-Hydroxyflavone is not found in nature. While the (13)C chemical shifts of 8-hydroxyflavone have been reported previously, the observed (13)C chemical shifts were not assigned. A previously reported empirical predictive tool has been applied in reverse in order to deconvolute the (13)C chemical shifts for 8-hydroxyflavone from each of those of 7,8,4'-trihydroxyflavone and 7,8-dihydroxyflavone together with those of 7-hydroxyflavone, 4'-hydroxyflavone, and flavone. The two sets of calculated (13)C chemical shifts for 8-hydroxyflavone are in good agreement with each other in that the average absolute difference is 0.4 ppm. The previously reported but unassigned (13)C chemical shifts for 8-hydroxyflavone have been assigned by matching them with the averages of the two sets of calculated (13)C chemical shifts for 8-hydroxyflavone such that the minimum average absolute difference is 0.63 ppm. The assigned (13)C chemical shifts of 8-hydroxyflavone may be used, along with the (13)C chemical shifts of the remaining monohydroxyflavones, as part of a predictive tool to rapidly assess the (13)C NMR spectra of C8-hydroxylated flavonoids.     

Diastereoselective synthesis of aliphatic α,α-difluoro-β3-amino esters via a sonocatalyzed Reformatsky reaction

(R)-2-Phenylglycine ethyl ester was found to be a cheap and effective auxiliary for the preparation of aliphatic α,α-difluoro-β(3)-amino esters via a Reformatsky reaction performed under sonication conditions. The products were obtained in good to high yield and ≥96:4 dr, thus providing a new stereoselective route to this under-represented class of compounds. A facile one-pot removal of the phenylglycine moiety and concomitant Boc protection subsequently afforded the corresponding Boc-protected β(3)-amino esters in excellent yield.     

Statistical–mechanical models with separable many-body interactions: especially partition functions and thermodynamic consequences

We start from a classical statistical–mechanical theory for the internal energy in terms of three- and four-body correlation functions g ₃ and g ₄ for homogeneous atomic liquids like argon, with assumed central pair interactions f(r_ij){\phi(r_{ij})} . The importance of constructing the partition function (pf) as spatial integrals over g ₃, g ₄ and f{\phi} is stressed, together with some basic thermodynamic consequences of such a pf. A second classical example taken for two-body interactions is the so-called one-component plasma in two dimensions, for a particular coupling strength treated by Alastuey and Jancovici (J Phys (France) 42:1, 1981) and by Fantoni and Tellez (J Stat Phys 133:449, 2008). Again thermodynamic consequences provide a particular focus. Then quantum–mechanical assemblies are treated, again with separable many-body interactions. The example chosen is that of an N-body inhomogeneous extended system generated by a one-body potential energy V(r). The focus here is on the diagonal element of the canonical density matrix: the so-called Slater sum S(r, β), related to the pf by pf(b) = òS(r, b)d[(r)\vec]{{\rm pf}(\beta) = \int {S({\bf r}, \beta)}d\vec {r}}, β = (k _B T)⁻¹. The Slater sum S(r, β) can be related exactly, via a partial differential equation, to the one-body potential V(r), for specific choices of V which are cited. The work of Green (J Chem Phys 18:1123, 1950), is referred to for a generalization, but now perturbative, to two-body forces. Finally, to avoid perturbation series, the work concludes with some proposals to allow the treatment of extended assemblies in which regions of long-range ordered magnetism exist in the phase diagram. One of us (Z.D.Z.) has recently proposed a putative pf for a three-dimensional (3D) Ising model, based on two, as yet unproved, conjectures and has pointed out some important thermodynamic consequences of this pf. It would obviously be of considerable interest if such a pf, together with conjectures, could be rigorously proved.     

Free energy in relation to order parameter in magnets and pyroelectrics

Theoretical models will be presented in which the internal energy of (a) a ferromagnet and (b) a pyroelectric material is expressed in terms of magnetization and electric polarization, respectively. For the ferromagnet, simple models of elementary excitations (e.g., spin wave theory in an insulator, to which Stoner excitations must be added in a metal) lead to formulas for the internal energy at low temperature as power laws in the change of magnetization from its saturation value. An unconventional use of two order parameters, the sublattice magnetization plus the metallic discontinuity in momentum distribution at the Fermi surface, allows the phase transition between metallic and insulating states of antiferromagnets to be treated at T = 0, the three-dimensional transition-metal dichalcogenides being an example here. The treatment of the internal energy of the ferromagnet is then extended to include the entropy also, using specifically the Ising model in nonzero external magnetic field. Its relevance to the Landau theory of phase transitions will be emphasized. Some comments will finally be made about the analogue for pyroelectrics. © 1996 John Wiley & Sons, Inc.