Coordinate system nuclear

The conceptually simplest approach to solve for the -matrix elements is to require the wavefimction to have the fonn of equation (B3.4.4). supplemented by a bound function which vanishes in the asymptote [32, 33, 34 and 35] This approach is analogous to the fiill configuration-mteraction (Cl) expansion in electronic structure calculations, except that now one is expanding the nuclear wavefimction. While successfiti for intennediate size problems, the resulting matrices are not very sparse because of the use of multiple coordinate systems, so that this type of method is prohibitively expensive for diatom-diatom reactions at high energies. 

The standard orientation is the coordinate system used internally by the program as it performs the calcula on, chosen to optimize performance. The origin is placed at the molecule s center of nuclear charge. Here, the oxygen atom sits on the Y-axis above the origin, and the two hydrogen atoms are f ced below it in die XY plane. 

The variable p (r) denotes the nuclear charge density at a point r with coordinates r = xi,X2,x ), and V r) is the Coulomb potential set up at that point by all other charges (the Coulomb constant k = l/(47t o) is dropped in this description). The integration variable in (4.1) is the volume element dr = Ax dr2dx3. The origin of the coordinate system is chosen to coincide with the center of the nuclear charge. A more convenient expression can be obtained by expanding V r) at f = (0,0,0) in a Taylor series, that is,... 

In eqn (4.1), g and A-t are 3x3 matrices representing the anisotropic Zeeman and nuclear hyperfine interactions. In general, a coordinate system can be found - the g-matrix principal axes - in which g is diagonal. If g and A, are diagonal in the same coordinate system, we say that their principal axes are coincident. 

This is not as useful as Eq. (19.4) because products of different coordinates appear in the second term. However, the symmetry properties of this term ensure the existence of a coordinate system in which the cross-terms can be eliminated and the nuclear Hamiltonian reduces to a sum of harmonic oscillator terms ... 

This is referred as BO ansatz. This ansatz is taken as a variational trial function. Terms beyond the leading order in m/M are neglected m is the electronic and M is nuclear mass, respectively). The problem with expansion (4) is that functions /(r, R) contain except bound states also continuum function since it includes the centre of mass (COM) motion. Variation principle does not apply to continuum states. To avoid this problem we can separate COM motion. The remaining Hamiltonian for the relative motion of nuclei and electrons has then bound state solution. But there is a problem, because this separation mixes electronic with nuclear coordinates and also there is a question how to define molecule-fixed coordinate system. This is in detail discussed by Sutcliffe [5]. In the recent paper by Kutzelnigg [8] this problem is also discussed and it is shown how to derive adiabatic corrections using, as he called it, the Bom-Handy ansatz. There are few important steps to arrive at formula for a diabatic corrections. Firstly, one separates off COM motion. Secondly, (very important step) one does not specify the relative coordinates (which are to some extent arbitrary). In this way one arrives at relative Hamiltonian Hrd [8] with trial wavefunction If we make BO ansatz... 

In physics and chemistry there are two different forms of spatial symmetry operators the direct and the indirect. In the direct transformation, a rotation by jr/3 radians, e.g., causes all vectors to be rotated around the rotation axis by this angle with respect to the coordinate axes. The indirect transformation, on the other hand, involves rotating the coordinate axes to arrive at new components for the same vector in a new coordinate system. The latter procedure is not appropriate in dealing with the electronic factors of Born-Oppenheimer wave functions, since we do not want to have to express the nuclear positions in a new coordinate system for each operation. 

In addition to g tensor anisotropy, EPR spectra are often strongly affected by hyperfine interactions between the nuclear spin I and the electron spin S. These interactions take the form T A S, where A is the hyperfine coupling tensor. Like the g tensor, the A tensor is a second-order third-rank tensor that expresses orientation dependence, in this case, of the hyperfine coupling. The A and g tensors need not be colinear in other words, A is not necessarily diagonal in the coordinate systems which diagonalize g. 

Fig. 4.1 Space-fixed coordinate system for internal nuclear motion.

In Section 5.1, we noted that to a good approximation the nuclear motion of a polyatomic molecule can be separated into translational, vibrational, and rotational motions. If the molecule has N nuclei, then the nuclear wave function is a function of 3/V coordinates. The translational wave function depends on the three coordinates of the molecular center of mass in a space-fixed coordinate system. For a nonlinear molecule, the rotational wave function depends on the three Eulerian angles 9, principal axes a, b, and c with respect to a nonrotating set of axes with origin at the center of mass. For a linear molecule, the rotational quantum number K must be zero, and the wave function (5.68) is a function of 6 and only only two angles are needed to specify the orientation of a linear molecule. Thus the vibrational wave function will depend on 3N — 5 or 3N — 6 coordinates, according to whether the molecule is linear or nonlinear we say there are 3N — 5 or 3N — 6 vibrational degrees of freedom. 

To understand these distributions, one needs to consider the fission transition nucleus. Figure 11.22 shows a coordinate system for describing this nucleus in terms of its quantum numbers J, the total angular momentum M, the projection of J upon a space-fixed axis, usually taken to be the direction of motion of the fissioning system, and K, the projection of J upon the nuclear symmetry axis. 

Many problems in nuclear physics and chemistry involve potentials, such as the Coulomb potential, that are spherically symmetric. In these cases, it is advantageous to express the time-independent Schrodinger equation in spherical coordinates (Fig. E.6). The familiar transformations from a Cartesian coordinate system (x, y, z) to spherical coordinates (r, 0, tp) are (Fig. E.6)... 

SCF and MSCF electric field dependence of the magnetizability and nuclear magnetic shielding have been studied by Rizzo et al (17-19) within GIAO basis sets. The use of London orbitals guarantees invariance of theoretical estimates in a change of coordinate system, which is a basic requirement in the computation of magnetic response properties. 

By a nuclear configuration (NC) we understand the set of informations NC Xk, Zk, Mk consisting of the coordinates Xk, the masses Mk and charge numbers Zk of the nuclei 1, 2,..., K of a molecular system. The coordinate vectors will be referred to a coordinate system, which will be defined when required. Important coordinate systems will be the laboratory system (LS, basis e 1) and the frame system (FS, basis e"f). The latter is attached to the nuclear configuration by a prescription to be defined in each case. The relation between e1 and may be expressed by... 

Whereas the group jr and its representations are relevant and sufficient for problems which are completely defined by relative nuclear configurations (RNCs) of a SRM, primitive period isometric transformations have to be considered as nontrivial symmetry operations in all those applications where the orientation of the NC w.r.t. the frame and laboratory coordinate system is relevant, e.g. the rotation-internal motion energy eigenvalue problem of a SRM. Inclusion of such primitive period operations leads to the internal isometric group ( ) represented faithfully by... 

This shows that the representation of the group on the nuclear position vectors referred to the laboratory fjxed coordinate system is not a faithful representation of the isometric group ( ) = r[Pg.25]

An alternative would be the hyperspherical coordinate system, introduced into the study of the ground-state helium by Gronwall [86], developed for nuclear reactions by Delves [87, 88], adopted in molecular reactive collisions by Smith [89], and initiated applications to two-electron excited QBSs by Macek [90]. The hyperradius p and one of the hyperangles, the radial hyperangle a, are defined by... 

For energies below the dissociation threshold we can use various coordinate systems to solve the nuclear Schrodinger equation (2.32). If the displacement from equilibrium is small, normal coordinates are most appropriate (Wilson, Decius, and, Cross 1955 ch.2 Weissbluth 1978 ch.27 Daudel et al. 1983 ch.7 Atkins 1983 ch.ll). However, if the vibrational amplitudes increase so-called local coordinates become more advantageous (Child and Halonen 1984 Child 1985 Halonen 1989). Eventually, the molecular vibration becomes unbound and the molecule dissociates. Under such circumstances, Jacobi or so-called scattering coordinates are the most suitable coordinates they facilitate the definition of the boundary conditions of the continuum wavefunctions at infinite distances which we need to determine scattering or dissociation cross sections (Child 1991 ch.l0). Normal coordinates become less and less appropriate if the vibrational amplitudes increase they are completely impractical for the description of unbound motion in the continuum. 

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