Orbitals polarization

Flere two electrons occupy the 1 s orbital (with opposite, a and p spins) while the other electron pair resides in 2s-2p polarized orbitals in a maimer that instantaneously correlates their motions. These polarized orbital... 

A set of polarized orbital pairs is described pictorially in figure B3.1.6. In each of the tln-ee equivalent temis in the above wavefunction, one of the valence electrons moves in a 2s+a2p orbital polarized in one direction while the other valence electron moves in the 2s - a2p orbital polarized in the opposite direction. For example, the first temi (2s - a2p )a(2s+a2p )P - (2s-a2p )P(2s+a2p )a describes one electron occupying a 2s-a2p polarized orbital while the other electron occupies the 2s+a2p orbital. The electrons thus reduce their... 

Figure B3.1.6. Polarized orbital pairs involving 2s and 2p orbitals.

In this example, tlie two non-orthogonal polarized orbital pairs involve mixing the k and k orbitals to produce two left-right polarized orbitals as depicted in figure B3.1.7. Here one says that the n electron pair undergoes left-right correlation when the (n configuration is introduced. 

Figure B3.1.7. Left- and right-polarized orbital pairs involving ti and ji orbitals.

It should be stressed that these polarized orbital pairs are not the same as hybrid orbitals. The latter are used to deseribe direeted bonding, but polarized orbital pairs are eaeh a mixture of two mean-field orbitals with... 

In most ab initio quantum ehemieal methods, the eorrelation ealeulation is aetiially earried out by fonning a linear eombination of the mean-field eonfiguration state funetions and detemiining the C ] amplitudes by some proeediire. The identities diseussed hr some detail above are then introdueed merely to pemrit one to interpret the presenee of eonfigurations that are doubly exeited relative to the dominant mean-field eonfiguration in temrs of polarized orbital pairs. 

Functions with higher / values and with sizes like those of lower-/ valence orbitals are also used to introduce additional angular correlation by pemiitting angularly polarized orbital pairs to be fomied. Optunal polarization functions for first- and second-row atoms have been tabulated and are included in the PNNL Gaussian orbital web site data base [45]. 

The need for such large CSF expansions should not be surprising considering (i) that each electron pair requires at least two CSFs to fomi polarized orbital pairs, (ii) there are of the order of N N - 1 )/2 = X electron pairs for N electrons, hence (iii) the number of temis in the Cl wavefiinction scales as 1. For a molecule containing ten electrons, there could be 2 =3.5x10 temis in the Cl expansion. This may be an overestimate of the number of CSFs needed, but it demonstrates how rapidly the number of CSFs can grow with the number of electrons. 

There is a fundamental difference, however, between the polarized orbital pairs introduced earlier ( )+ = (2s a2px,y,or z) and the corresponding functions (j) + = (2s ia2px,y,or z)... 

These densities are identieal to one another and do not deseribe polarized orbital densities. Therefore, the Cl wavefunetion whieh mixes the two eonfigurations with like sign, when analyzed in terms of orbital pairs, plaees the eleetrons into orbitals ([) +=(2s ia2px,y,or z)... 

In the olefin example mentioned above, the two non-orthogonal "polarized orbital pairs" involve mixing the n and 7i orbitals to produee two left-right polarized orbitals as depleted below ... 

In the alkaline earth atom ease, the polarized orbital pairs are formed by mixing the ns and np orbitals (aetually, one must mix in equal amounts of pi, p.i, and po orbitals to preserve overall S symmetry in this ease), and give rise to angular eorrelation of the eleetron pair. Use of an (n+l)s2 CSF for the alkaline earth ealeulation would eontribute in-out or radial eorrelation beeause, in this ease, the polarized orbital pair formed from the ns and (n+l)s orbitals would be radially polarized. 

Funetions with higher 1-values and with sizes more in line with those of the lower-1 orbitals are also used to introduee additional angular eorrelation into the ealeulation by permitting polarized orbital pairs (see Chapter 10) involving higher angular eorrelations to be formed. Optimal polarization funetions for first and seeond row atoms have been tabulated (B. Roos and P. Siegbahn, Theoret. Chim. Aeta (Berl.) 17, 199 (1970) M. J. Friseh, J. A. Pople, and J. S. Binkley, J. Chem. Phys., 3265 (1984)). 

In summary, doubly exeited CSFs are often employed to permit polarized orbital pair formation and henee to allow for eleetron eorrelations. Singly exeited CSFs are ineluded to permit orbital relaxation (i.e., orbital reoptimization) to oeeur. 

The high sensitivity of atomic Sagnac interferometers to rotation rates will enable HYPER to measure the modulation of the precession due to the Lense-Thirring effect while the satellite orbits around the Earth. In a Sun-synchronous, circular orbit at 700 km altitude, HYPER will detect how the direction of the Earth s drag varies over the course of the near-polar orbit as a function of the latitudinal position 9 ... 

The Advanced-Very-High-Resolution Radiometer (AVHRR) carried on board the NOAA-7 satellite has been collecting radiance data from the earth s surface since 1978. The polar-orbiting satellite records global data on a neardaily basis. The 4-kilometer data have been remapped by NO A A into monthly composites. The data are collected in 2 bands-one visible (VIS), the other near infrared (NIR). The Normalized Difference Vegetation Index, or NDVI, defined... 

Keywords Orbital mixing. Orbital amplitude. Orbital phase. Orbital polarization. Orbital deformation, Regioselectivity, Stereoselectivity, n Facial selectivity... 

In all the variational methods, the choice of trial function is the basic problem. Here we are concerned with the choice of the trial function for the polarization orbitals in the calculation of polarizabilities or hyperpolarizabilities. Basis sets are usually energy optimized but recently we can find in literature a growing interest in the research of adequate polarization functions (27). 

Each of the eight hyperfine resonances is an unresolved quadmpole doublet, due to the quadmpole interaction of Os in the hexagonal Os metal source. The authors have interpreted the hyperfine fields in terms of core polarization, orbital and spin-dipolar contributions. 

The Darwin mission will not be carried out until the middle of the next decade. However, the COROT (Convection, Rotation and Planetary Transits) telescope was launched from the Baikonur cosmodrome in December 2006. The satellite, which weighs 630 kg, circles the Earth at a height of about 900 km in a polar orbit. The mission is planned to last 21/2 years, and more than 120,000 stars are to be observed. 

The SCF, SCVB, full valence tv, and full valence tt + S results of using a 6-3IG basis on benzene are given in Table 15.6. The geometry used is that of the minimum SCF energy of the basis. In this case the SCVB energy is lower by 0.4 eV than the full valence it energy. This is principally due to the 3d polarization orbitals present in the SCVB orbital, but absent in the valence calculation. The SCVB orbital is... 

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