Alpha-spin state

The two equations couple because the alpha Fock matrix depends on both the alpha and the beta solutions, C and cP (and sim ilarly for the beta Fock matrix). The self-consistent dependence of the Fock matrix on molecular orbital coefficients is best represen ted, as before, via the den sity matrices an d pP, wh ich essen -tially state the probability of describing an electron of alpha spin, and the probability of finding one of beta spin ... 

According to quantum mechanics, an electron has two spin states, represented by the arrows T (up) and l(down) or the Greek letters a (alpha) and P (beta) We can think of an electron as being able to spin counterclockwise at a certain rate... 

As the MSADs have the largest coefficients in the ground-state wave function, one may expect that for alternant free radicals or polyradicals, the dominant positive spin density will be largest on the atoms that bear an alpha spin in the MSAD. An example that was analyzed in Chapter 7 is the allyl radical, where the MSAD predicts positive spin densities at positions 1, 3 and a negative density at position 2. Similarly, the MSAD of benzyl radical predicts positive spin densities on the benzylic carbon and on the ortho and para positions. 

Andersson P, Annila A, Otting G, An alpha/beta-HSQC-alpha/beta experiment for spin-state selective editing of IS cross peaks, J. Magn. Resort., 133 364-367, 1998. 

Cilice 0 < c < 1 the first term shows tliat UHF orbitals reduce the ionic coutribution relative to the covalent structures, compared to the RHF case, eq. (4.18). This is the same effect as for the Cl procedure (eq. (4.19)), i.e. the first term shows that the UHF wave function partly includes electron correlation The fii t term can he written as a linear combination of the 0 and 1 determinants, and describes a pure singlet. state. The last part of the UHF determinant, however, has terms identical to two of those in the triplet combination (4.22). If we had chosen the alternative set of UHF orbitals A ith the alpha spin being primarily on centre B in eq. (4.20), we would have obtained the other two terms in i.e. the last term in (4.24) breaks the symmetry. The UHF determinant is therefore not a pure spin state, it contains both singlet and triplet spin ates. This feature is known as spin contamination. Foi c — 1 the UHF wave function i... 

No levels are known for but the B levels are very well established by the dp) and oLp) reactions and by resonances in total cross section for neutrons. Angular distributions have been measured in both reactions and spin values obtained. The beta transitions from B and g,re both allowed and proceed partly to excited states alpha emitting states are observed for ot... 

Figure 49 shows the band structure of the F-center in LiF for the Sig-supercell model obtained at the UlTF level along with the band structure of bulk LiF. Alpha and beta electrons are described by different sets of orbitals. Two band structures are obtained for the a- and p-spin states. The shape of the bands is similar to those of the perfect system, but a new band appears in the... 

The energy gap between the alpha and beta spin states as the result of an applied, external magnetic field. 

The general principles of NBO donor-acceptor interactions (Chapter 5) suggest the strong delocalizations to be expected for the triplet-state Lewis structures (11.22). In the alpha spin set, the allylic-like C=C I pattern immediately suggests strong nc n cc delocalization, and the anionic -O oxo pattern suggests enhanced... 

To use HyperChem for calculations, you specify the total molecular charge and spin multiplicity (see Charge, Spin, and Excited State on page 119). The calculation selects the appropriate many-electron wave function with the correct number of alpha or beta electrons. You don t need to specify the spin function of each orbital. 

Symbol, mass number nuclear spin I, nuclear parity -Half-life (s = seconds, ps = microseconds, d = days, m = months, or y = years) -Decay mode (emission), and energy (MeV( if to ground state)), separated by / if several modes if in parentheses, mode produces a shortlived daughter, or occurs <10% Emissions a = alpha = 2He4++ P"= electron ... 

Bismuth (Z = 83) is the heaviest stable element in group 15 (VA) of the periodic table (see Periodic Table Trends in the Properties of the Elements). The Bi isotope, which is 100% abundant, has a 9/2 nuclear spin. Bi, an alpha emitter is used in nuclear medicine as a radiotherapeutic agent. Bismuth has two stable oxidation states Bi(V), corresponding to complete loss of the valence electrons, and Bi(III), a lower oxidation state that retains two valence electrons. Both oxidation states are diamagnetic. The latter is more stable and more common since Bi(V) has a large reduction potential ... 

In the above equation, the first factor in square brackets on the RHS is the probability of finding the nucleus in the excited state Ex and spin Jr (with Jo being the ground state spin), while the second factor ra/% is the decay rate of the excited state with an alpha particle emission. Now since Ex = Er + Q, we have ... 

During the late 1950 s and first part of the 60 s Ingvar made major contributions to our knowledge of spins and electromagnetic moments of nuclear ground and excited states. In the 1965 edition of the Bible of nuclear spectroscopy, Alpha-, Beta- and Gamma-ray Spectroscopy , Editor K.Siegbahn, he contributed an exhaustive table of nuclear spins and moments, obtained with various techniques. 

Let us say a little more about these methods. The HF methods are divided into spin-restricted HF (RHF) and spin-unrestricted HF (UHF) methods. Closed-shell systems are almost always calculated using RHF. In this procedure, one set of molecular orbitals is calculated and pairs of electrons are entered to the lowest-energy orbitals. If the molecule has an odd number of electrons, one orbital will be singly occupied and the species is a radical (spin = 0.5, expectation value of the spin-squared operator =0.75). Inmost cases, however, radicals are calculated using the UHF formalism. UHF calculations determine two sets of molecular orbitals, one for each type of spin named alpha and beta. These MO sets are similar but not identical. A radical, for instance, has one more a than P electron. The UHF procedure is more flexible than RHF because the paired a and P orbitals, which correspond to doubly occupied MOs in the RHF formalism, need not be identical. So UHF allows for spin polarization but, on the other hand, spin-contamination occurs (i.e., states of higher spin are mixed into the wave function). 

The two-stage nature of the triple-alpha reaction was recognized by Salpeter in the 1950s. Neverttieless, shortly after Salpeter s insight, Sir Fred Hoyle recognized the triple-alpha reaction would be too slow to explain the abundances of the heavy elements unless the reaction Be + He occurred resonantly in stars. Hoyle pointed out that this required a 0 excited state (that is, a state with spin zero and positive parity) in C at 7.6 MeV, and Cook, Fowler, and collaborators demonstrated the existence of this state experimentally (6)- The prediction of the... 

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