The closed shell atom

The use of atomic symmetry block-diagonalizes the matrix representation of the DHFB Hamiltonian of (95) giving the generalized matrix eigenvalue equations 

The matrix representation of the Fock operator for closed-shell systems may be separated into one-body and two body parts 

The matrices contain contributions from electrons of other symmetries, and can be expanded in the general form 

The one-body operators appearing in H - can be reduced to simple radial integrals by inserting the definition (142) in the relevant formulae. Thus, from (100) we find that the elements of the Gram matrices reduce to radial integrals of the form  

This orthonormality of the spin-angular functions, (H8), ensures 

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We write for the ground state wave function of the closed shell atom or molecule a Slater determinant for the N electrons... 

Even for overlap integrals as large as 0.3, the second term in equation (97) is of the order 0. and the third term 0.01 Afk, so that except for very close interactions, this series may be safely cut off after the third term. This may be useful for evaluating the interaction energy between the closed-shell atomic cores within a molecule, for which just the first term may well be sufficient. 

In 1978, Ludena [102] carried out a Hartree-Fock calculation by using a wave function consisting of a single Slater determinant for the closed-shell atoms, whereas he used a linear combination of the Slater determinants for the open-shell atoms. Each Slater-type orbital times a cut-off function of the form (1 — r/R) to satisfy the boundary conditions. Ludena studied pressure effects on the electronic structure of the He, Li, Be, B, C and Ne neutral atoms. The energies he obtained for the confined helium atom are slightly lower than those Gimarc obtained, especially for box radii in the range R > 1.6 au. 

A.O.Williams Jr. noted in his Hartree-calculations on the closed shell atom Cu as early as 1940 (Phys. Rev. 58, 723) The charge density of each single electron turns out to resemble that for the nonrelativistic case, but with the maxima "pulled in " and raised.. .. The size of the relativistic corrections appear to be just too small to produce important corrections in atomic form factors or other secondary characteristics of the whole atom.. .. However, it must be noticed that copper is a relatively light ion, and the corrections for such an ion as mercury would be enormously greater. S.Cohen in 1955 and... 

The asymptotic behavior of the second-order energy of the M0ller-Plesset perturbation theory, especially adapted to take advantage of the closed-shell atomic structure (MP2/CA), is studied. Special attention is paid to problems related to the derivation of formulae for the asymptotic expansion coefficients (AECs) for two-particle partial-wave expansions in powers... 

An embedding curve with a minimum is often constructed by adding an attractive square root-dependent term to the repulsive linear term of the closed shell atoms, namely [3],... 

This time, only some of the closed shell atoms appear twice in the table, namely, nickel and palladium, but not the noble gases,... 

In a later paper, Kromhout and Linder have reexamined the problem of calculating 0 and present a theory for calculating the London -van der Waals effect on nuclear magnetic shielding of closed shell atoms [205]. They also attempt to extend the theory to polyatomic molecules and derive two explicit expressions for the factor B in Eq. (3.24). The value of B is found to be proportional to the polarizability, a, of the atom under study, the total diamagnetic shielding constant for the closed shell atom and a function of the ionization... 

The first and third examples illnstrate a characteristic property of electron movanent If an electron pair moves toward an atom, that atom must have a place to put that electron pair, so to speak. In nucleophilic substitution, the carbon atom in a haloalkane has a filled outer shell another electron pair cannot be added without displacement of the electron pair bonding carbon to halogen. The two electron pairs can be viewed as flowing in a synchronous manner As one pair arrives at the closed-shell atom, the other departs, thereby preventing violation of the octet rule at carbon. When you depict electron movement with curved arrows, it is absolutely essential to keep in mind the rules for drawing Lewis structures. Correct use of electron-pushing arrows helps in drawing such structures, because aU electrons are moved to their proper destinations. 

One may argue that the Dirac formula describes the trends in relativistic effects only for one-electron atoms. For neutral atoms the Pauli exclusion principle pushes valence density out of the inner region therefore, relativistic effects may be small. In 1940 A. O. Williams noted in analyzing his Hartree calculations on the closed shell atom Cu ... 

Scheme 3). The qualitative energy levels (Scheme 4) show the number of valence electrons necessary to obtain closed-shell electronic structures. Each orbital in the. y-orbital set is assumed to be occupied by a pair of electrons since the 5-orbital energies are low and separate from those of the p-orbital ones, especially for heavy atoms. The total number of valence electrons for the closed-shell structures... 

As an example of the interest to scrutinise the UHF solution, one may quote the Bea problem [19]. The bond is weak but it takes plaee at short interatomic distance and is definitely not the dispersion well which one might expect from two closed shell atoms (and which occurs in Mga and heavier eompounds). Quantum chemical calculations only reproduce this bond when using large basis sets and extensive Cl calculations [20]. It is amazing to notice that the UHF solution gives a qualitatively correct behaviour, and suggests a physical interpretation of this bond since in... 

SCF-CI calculations were performed at 20 different intemuclear separations, from 1.2 bohr to 4-00. The lowest separate atom states are, B( P,2p) and H( S) therefore, in order to have a homolytic dissociation and three degenerate 2p orbitals on B we have adopted the closed shell Fock hamiltonian with fractional occupation [23] one electron was placed in the 3(t orbital, correlating with H(ls) at infinite separation, and 1/3 each in the 4it and Itr orbitals correlating with B(2p). 

For most molecules studied, modest Hartree-Fock calculations yield remarkably accurate barriers that allow confident prediction of the lowest energy conformer in the S0 and D0 states. The simplest level of theory that predicts barriers in good agreement with experiment is HF/6-31G for the closed-shell S0 state (Hartree-Fock theory) and UHF/6-31G for the open-shell D0 state (unrestricted Hartree-Fock theory). The 6-31G basis set has double-zeta quality, with split valence plus d-type polarization on heavy atoms. This is quite modest by current standards. Nevertheless, such calculations reproduce experimental barrier heights within 10%. 

The peculiar behavior of H might be relevant to understand the hydrogen bond, which deforms the electronic cloud of the proton. On the other hand, it is surprising to discover an anomalous behavior for a closed-shell atom like He. However, it has been demonstrated in helium-atom-scattering that interactions between He atoms... 

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