Population isotropic

Gaussian computes isotropic hyperfine coupling constants as part of the population analysis, given in the section labeled "Fermi contact analysis the values are in atomic-units. It is necessary to convert these values to other units in order to compare with experiment we will be converting from atomic units to MHz, using the following expressions ri6ltYg ... 

Compute the isotropic hyperfine coupling constant for each of the atoms in HNCN with the HF, MP2, MP4(SDQ) and QCISD methods, using the D95(d,p) basis set Make sure that the population analysis for each job uses the proper electron density by including the Density=Current keyword in the route section. Also, include the 5D keyword in each job s route sectionfas was done in the original study). 

The geometry and structure of a bone consist of a mineralised tissue populated with cells. This bone tissue has two distinct structural forms dense cortical and lattice-like cancellous bone, see Figure 7.2(a). Cortical bone is a nearly transversely isotropic material, made up of osteons, longitudinal cylinders of bone centred around blood vessels. Cancellous bone is an orthotropic material, with a porous architecture formed by individual struts or trabeculae. This high surface area structure represents only 20 per cent of the skeletal mass but has 50 per cent of the metabolic activity. The density of cancellous bone varies significantly, and its mechanical behaviour is influenced by density and architecture. The elastic modulus and strength of both tissue structures are functions of the apparent density. 

Movement through the body of a solid is called volume, lattice, or bulk diffusion. In a gas or liquid, bulk diffusion is usually the same in all directions and the material is described as isotropic. This is also true in amorphous or glassy solids and in cubic crystals. In all other crystals, the rate of bulk diffusion depends upon the direction taken and is anisotropic. Bulk diffusion through a perfect single crystal is dominated by point defects, with both impurity and intrinsic defect populations playing a part. 

The principle of the ENDOR method is illustrated in Fig. 1. It refers to the most simple spin system with an electron spin S = 1/2 and a nuclear spin I = 1/2 for which an isotropic hf interaction, aiso, is considered. In a steady state ENDOR experiment4, an EPR transition (A, D), called the observer, is partly saturated by microwave radiation of amplitude B while a driving rf field of amplitude B2, called the pump, induces nuclear transitions. At frequencies vj and v2, the rf field tends to equalize the populations within the ms-states. This alters the degree of saturation of the observer so that, in the display of the EPR signal height versus the radio frequency, two ENDOR lines at transition frequencies vj = aiso/2 - vn (A, B) and v2 = ais0/2 + v (C, D) will be observed (v = / NgnBo denotes the nuclear Zeeman frequency for a static field B0). 

Fig. 1. Energy level diagram for an S = 1/2,1 = 1/2 spin system with an isotropic hfs a, > 2 Ng B0 > 0. The steady state populations shown are for the case where the (A, D) (EPR observer) and the (A, B) (rf pump) transitions are saturated...

In an alternative formulation of the Redfield theory, one expresses the density operator by expansion in a suitable operator basis set and formulates the equation of motion directly in terms of the expectation values of the operators (18,20,50). Consider a system of two nuclear spins with the spin quantum number of 1/2,1, and N, interacting with each other through the scalar J-coupling and dipolar interaction. In an isotropic liquid, the former interaction gives rise to J-split doublets, while the dipolar interaction acts as a relaxation mechanism. For the discussion of such a system, the appropriate sixteen-dimensional basis set can for example consist of the unit operator, E, the operators corresponding to the Cartesian components of the two spins, Ix, ly, Iz, Nx, Ny, Nz and the products of the components of I and the components of N (49). These sixteen operators span the Liouville space for our two-spin system. If we concentrate on the longitudinal relaxation (the relaxation connected to the distribution of populations), the Redfield theory predicts the relaxation to follow a set of three coupled differential equations ... 

Table 14. Unpaired electron distributions in diatomic halide radical anions. Sa, Pa indicate the spin population of the valence s and p orbitals of atom A, derived from the isotropic and anisotropic hyperfine coupling constants 67)...

Spiering et al. (1982) have developed a model where the high-spin and low-spin states of the complex are treated as hard spheres of volume and respectively and the crystal is taken as an isotropic elastic medium characterized by bulk modulus and Poisson constant. The complex is regarded as an inelastic inclusion embedded in spherical volume V. The decrease in the elastic self-energy of the incompressible sphere in an expanding crystal leads to a deviation of the high-spin fraction from the Boltzmann population. Pressure and temperature effects on spin-state transitions in Fe(II) complexes have been explained based on such models (Usha et al., 1985). 

At the liquid helium temperature, we can safely assume that all para-hydrogen molecules are populated in the t = 0, /=0 ground state. The characteristic of this ground-state para-hydrogen molecule is that it has an isotropic distribution of its molecular wave function. Therefore, the electrostatic intermolecular interaction... 

In this equation, the energies , and / of the initial and final states, i) and I/), and the dipole moment all refer to a pair of diatomic molecules hcvij = Ef — Ei is Bohr s frequency condition. With isotropic interaction, rotation and translation may be assumed to be independent so that the rotational and translational wavefuntions, population factors, etc., factorize. Furthermore, we express the position coordinates of the pair in terms of center-of-mass and relative coordinates as this was done in Chapter 5. 

Since the binding constants of Mn2 + to the tight and weak metal ion sites of unadenylylated enzyme are 5.0 x 10 7 and 4.5 x 10 5 M, respectively, the tight site can be selectively populated under conditions where [enzyme] > [Mn2 +]. Figure 24 shows EPR spectra obtained with a solution of 0.79 mil/ enzyme subunit concentration and 0.7 mAf Mn2+ concentration (113). This spectrum represents Mn2+ bound only at the tight sites with no free Mn2 + present. It shows that bound Mn2 + is in a relatively isotropic environment (i.e., the zero-field splitting is small). 

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