Bounds to average interelectronic angles in Hartree-Fock theory of atoms

The average interelectronic is the expectation value of the angle thetaij (0 < or = thetaij < or = pi) subtended by the position vectors ri and rj of a pair of electrons i and j. In the Hartree-Fock theory of atoms, we point out that the angle and its subshell-pair components nl,n'l' are bounded from above and below, where n and l are the principal and azimuthal quantum numbers. The upper bounds for nl,n'l' with 0 < or = l, l' < or = 3 are 9pi/16 (=101.25 degrees), 135pi/256 (approximately 94.922 degrees), 265pi/512 (approximately 93.164 degrees), and 129pi/256 (approximately 90.703 degrees) for sp, pd, df, and sf pairs, respectively, while they are pi/2 (=90 degrees ) for the other ll' pairs, independent of n and n'. A weighted sum of these subshell-pair bounds gives an upper bound to . The lower bounds are pi/2 in all the cases.     

Optimal Kinoshita wave functions for heliumlike atoms

An extensive optimization has been performed for the composition ofN terms, as well as the exponent and the mixing coefficients, of Kinoshita wave functions for heliumlike atoms with atomic numberZ. The optimalN-term Kinoshita functions have been constructed forN=1–10, 20, 30, 40, 50, 100 andZ=1(H^–)–10(Ne⁸⁺). The present results demonstrate that the optimal term selection dramatically improves the accuracy of the Kinoshita function: In the case of He, for example, the optimal 100-term Kinoshita function gives – 2.903 724 376 95 hartrees, which is only 8×10^–11 hartrees higher than the most accurate literature value.     

The Hellmann-Feynman theorem applied to long-range forces

The relations between the Hellmann-Feynman forces in laboratory fixed (L-) and relative (R-) coordinate systems are clarified. In the usualL-coordinate system, the force is interpreted as force on nucleus, while in theR-coordinate system, it means force on whole particles consisting of the electrons and nuclei of each interacting subsystem. From a perturbation theoretical viewpoint, the concept of the force on whole particles correctly corresponds to the perturbation energy and is superior to the force on the nucleus.     

Momentum density interpretation of the H(1s)–H+ long-range force

The recently proposed method of using momentum densities for interatomic interactions is applied to the long-range force between the ground-state hydrogen atom and the proton, and the results are compared with those from using the position density based on the electrostatic Hellmann–Feynaman theorem. A new physical interpretation of the long-range force is obtained, which is complementary to that in position space. It is found that some perturbative changes in the position density do not accompany changes in the momentum density.     

Atomic one- and two-electron subshell moments in position and momentum spaces

For 357 subshells of the 53 neutral atoms He through Xe in their ground states, the two-electron intracule (relative motion) <u ^k > _nl and extracule (center-of-mass motion) <R ^k > _nl subshell moments in position space are examined as well as their counterparts <v ^k > _nl and <P ^k > _nl in momentum space, where n and l are the principal and azimuthal quantum numbers of the atomic subshell, respectively. It is clarified that between the intracule and extracule moments the “2 ^k -rule” is strictly valid, which means <u ^k > _nl = 2 ^k <R ^k > _nl and <v ^k > _nl = 2 ^k <P ^k > _nl for any nl subshell. Theoretical analysis also proves that for a particular case of k = +2, two relations <u ²> _nl = (N _nl −1)<r ²> _nl and <v ²> _nl = (N _nl −1)<p ²> _nl hold exactly, where N _nl (≥2) is the number of electrons in the subshell nl, and <r ^k > _nl and <p ^k > _nl are the familiar one-electron subshell moments in position and momentum spaces, respectively. The latter equality establishes a new and rigorous relation between the second electron-pair moments in momentum space and the total energy of an atom through the virial theorem. For k=+1, −1, and −2, the numerical Hartree-Fock results for the 357 subshells show that there are approximate but accurate linear relations between <u ^k > _nl and <r ^k > _nl and between <v ^k > _nl and <p ^k > _nl , in which the proportionality constant in each space depends on n,l, and k. Received: 27 April 1998 / Accepted: 29 May 1998 / Published online: 28 August 1998     

The synthesis of hexahydrooxoepithiopyridinedicarboximides by the reaction of thioamides with N-substituted maleimides

The synthesis of hexahydrooxoepithiopyridinedicarboxyimide (5: X₂ = N-Ph) by the reaction of thioamides 1 with N-substituted maleimide ( 2a ) was examined. The reaction of primary thioamides, such as thiobenzamide and p-toluthioamide with N-phenylmaleimide gives compounds 5 together with corresponding 4-hydroxy-1,3-thiazoles 4 . However, a similar reaction of secondary thioamides, such as N-methylthioacetamide, thiobenzanilide, with N-phenylmaleimide did not provide compounds 5 without addition of acid. The reaction pathway and the configuration of 5 were also investigated.     

Metalloporphyrin Cr(TPP)Cl-catalyzed Claisen rearrangement of simple aliphatic allyl vinyl ethers and its unique stereoselectivity

The catalytic Z-selective Claisen rearrangement of simple aliphatic allyl vinyl ethers can be achieved using a chromium(III) porphyrin complex, Cr(TPP)Cl, as a catalyst: Cr(TPP)Cl significantly enhances reversal of E-Z selectivity in the thermal Claisen rearrangement of allyl vinyl ethers, especially, 4,5- and 4,6-disubstituted derivatives, at low catalyst loading.     

Medium-size Gaussian basis sets for hydrogen through argon

Summary Medium-sized Gaussian basis sets are reoptimized for the ground states of the atoms from hydrogen through argon. The composition of these basis sets is (4s), (5s), and (6s) for H and He, (9s5p) and (12s7p) for the atoms Li to Ne, and (12s8p) and (12s9p) for the atoms Na to Ar. Basis sets for the² P states of Li and Na, and the³ P states of Be and Mg are also constructed since they are useful in molecular calculations. In all cases, our energies are lower than those obtained previously with Gaussian basis sets of the same size.     

Two-electron atomic wave functions which satisfy proportionality relations between one-and two-electron moments

Based on momentum- and position-space analyses of the moment operators for two-electron atoms, it is shown that there exists a family of two-electron wave functions which satisfy a proportionality relation, r/ ₁ ^v /r ₁₂ ^v =p/ ₁ ^v /p ₁₂ ^v =2^–v/2, between the one and two-electron moments in position and momentum spaces, where v is an arbitrary number for which the moments are well-defined.