Orbital separated-atom

United atom Molecular orbital Separated atoms Wave function... 

For two and three dimensions, it provides a erude but useful pieture for eleetronie states on surfaees or in erystals, respeetively. Free motion within a spherieal volume gives rise to eigenfunetions that are used in nuelear physies to deseribe the motions of neutrons and protons in nuelei. In the so-ealled shell model of nuelei, the neutrons and protons fill separate s, p, d, ete orbitals with eaeh type of nueleon foreed to obey the Pauli prineiple. These orbitals are not the same in their radial shapes as the s, p, d, ete orbitals of atoms beeause, in atoms, there is an additional radial potential V(r) = -Ze /r present. However, their angular shapes are the same as in atomie strueture beeause, in both eases, the potential is independent of 0 and (j). This same spherieal box model has been used to deseribe the orbitals of valenee eleetrons in elusters of mono-valent metal atoms sueh as Csn, Cun, Nan and their positive and negative ions. Beeause of the metallie nature of these speeies, their valenee eleetrons are suffieiently deloealized to render this simple model rather effeetive (see T. P. Martin, T. Bergmann, H. Gohlieh, and T. Lange, J. Phys. Chem. 6421 (1991)). 

The two sets of coefficients result in two sets of Fock matrices (and their associated density matrices), and ultimately to a solution producing two sets of orbitals. These separate orbitals produce proper dissociation to separate atoms, correct delocalized orbitals for resonant systems, and other attributes characteristic of open shell systems. However, the eigenfunctions are not pure spin states, but contain some amount of spin contamination from higher states (for example, doublets are contaminated to some degree by functions corresponding to quartets and higher states). 

In the conventional MO-LCAO theory, the function u is approximated by a Is orbital, but better approximations may be obtained by including higher orbitals. The total wave function is such that, for separated atoms, there is a fifty per cent chance that the mole-... 

In conclusion we note that the method of alternant molecular orbitals leads to a correct behavior of the energy curve for separated atoms, which is of essential importance in considering correlation effects (see Section II.D(4c)) and in studying magnetic phenomena. 

With the single exchange integrals negative, as they usually are for orbits on separate atoms, this leads to attraction between a and b and between c and d, and repulsion between other pairs. 

P n s p hybridized atom has three coplanar hybrid orbitals separated by 120° angles. One unchanged p orbital is perpendicular to the plane of the hybrids. 

An sp-hybridized atom has two hybrid orbitals separated by 180°. The remaining two atomic p orbitals are perpendicular to the hybrids and perpendicular to each other. 

Consider two well-separated atoms A and B with electron wave functions and which are eigen functions of the atoms, with energies and ei. If we bring these atoms closer, the wave functions start to overlap and form combinations that describe the chemical bonding of the atoms to form a molecule. We will neglect the spin of the electrons. The procedure is to construct a new wave function as a linear combination of atomic orbitals (LCAO), which for one electron has the form... 

In fact, it turns out that the orbitals resulting from SCF or valence MCSCF calculations in molecules ean be described in extremely simple terms by comparing them with the RHF orbitals of the separated atoms. 

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). 

Figure 1.10 Energy diagram for the hydrogen molecule. Combination of two atomic orbitals, is, gives two molecular orbitals, 0iec and iF moiec- The energy of Iffnoiec is lower than that of the separate atomic orbitals, and in the lowest electronic state of molecular hydrogen it contains both electrons.

It means, for example, that atomic data can only rarely be used as a substitute for molecular integrals since the atom-in-molecule orbitals are not the same as the separate atom orbitals — worse, they are no longer equivalent among themselves. An atomic self-repulsion integral (0j0, 0j0j) is different if 0j is the lone-pair hybrid of NH3 or the bond-pair hybrid as the Gillespie-Nyholm rules suggest. 

If we now consider the numerical results quoted in Table 1 for the optimum exponents, three conclusions follow immediately. Firstly, the 1 s orbital on the heavy atom is unchanged by molecule formation this is to be expected. Second, the sp3 orbitals involved in the X—H bond are all contracted with respect to their free-atom values. Finally, the sp3 orbitals containing Tone pairs of electrons are largely unchanged or expanded slightly on molecule formation. In fact, of course, the optimum separate atoms minimal basis functions do not have the same orbital exponent for the 2 s and 2 p AOs. To facilitate comparisons therefore in Table 1 the optimum n = 2 exponent is given for the atoms when such a constraint is imposed (the qualitative conclusions are, in any event, unchanged by use of these exponents for comparison or a notional exponent of 1/4 (fs + 3 fp) or any reasonable choice). 

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