Quadrupole terms

The range of systems that have been studied by force field methods is extremely varied. Some force fields liave been developed to study just one atomic or molecular sp>ecies under a wider range of conditions. For example, the chlorine model of Rodger, Stone and TUdesley [Rodger et al 1988] can be used to study the solid, liquid and gaseous phases. This is an anisotropic site model, in which the interaction between a pair of sites on two molecules dep>ends not only upon the separation between the sites (as in an isotropic model such as the Lennard-Jones model) but also upon the orientation of the site-site vector with resp>ect to the bond vectors of the two molecules. The model includes an electrostatic component which contciins dipwle-dipole, dipole-quadrupole and quadrupole-quadrupole terms, and the van der Waals contribution is modelled using a Buckingham-like function. 

Developed into a power series in R 1, where R is the intermolecular separation, H exhibits the dipole-dipole, dipole-quadrupole terms in increasing order. When nonvanishing, the dipole-dipole term is the most important, leading to the Forster process. When the dipole transition is forbidden, higher-order transitions come into play (Dexter, 1953). For the Forster process, H is well known, but 0. and 0, are still not known accurately enough to make an a priori calculation with Eq. (4.2). Instead, Forster (1947) makes a simplification based on the relative slowness of the transfer process. Under this condition, energy is transferred between molecules that are thermally equilibriated. The transfer rate then contains the same combination of Franck-Condon factors and vibrational distribution as are involved in the vibrionic transitions for the emission of the donor and the adsorptions of the acceptor. Forster (1947) thus obtains... 

For high accuracy, it is necessary to add the nuclear-Zeeman and electric-quadrupole terms... 

Finally, the interaction between the dipole and quadrupole of donor and acceptor molecules [13] is generally much weaker than the dipole-dipole interaction. The dipole—quadrupole term [/ (r) r-8] is typically 10—100 times weaker than the dipole—dipole term, though if the acceptor absorption spectrum is symmetry-forbidden (and so weak) but not spin-forbidden, the dipole transition moment for the acceptor is small [127]. Such is the case for energy transfer between rare-earth ions in tungstates typically separated by 1.7 nm [146]. The kinetics of dipole—quadrupole energy transfer are discussed in Chap. 4, Sect. 2.6. 

One of the simplest orientational-dependent potentials that has been used for polar molecules is the Stockmayer potential.48 It consists of a spherically symmetric Lennard-Jones potential plus a term representing the interaction between two point dipoles. This latter term contains the orientational dependence. Carbon monoxide and nitrogen both have permanent quadrupole moments. Therefore, an obvious generalization of Stockmayer potential is a Lennard-Jones potential plus terms involving quadrupole-quadrupole, dipole-dipole interactions. That is, the orientational part of the potential is derived from a multipole expansion of the electrostatic interaction between the charge distributions on two different molecules and only permanent (not induced) multipoles are considered. Further, the expansion is truncated at the quadrupole-quadrupole term. In all of the simulations discussed here, we have used potentials of this type. The components of the intermolecular potentials we considered are given by ... 

The quadrupole term of H, cannot couple the 6pni states to the 6s and 5d continua, and we ignore the octopole term which couples the 6pn state to the 5dei" continua. 

Fig. 9. Torsional potentials for HS-SH molecule calculated in 6-31G basis set (a) ab initio SCF results, lab initio SCF (b) CMMM estimates (c = 6) up to quadrupole-quadrupole term, + CMM (Q-Q) (c) PCM results (c = 6, s = 9, R = 0.1 au), PCM (R = 0.1 au) (Reproduced from [90] copyright-Springer-Verlag)...

The third expressions are the quadrupole terms, with nine components, that is, a second rank tensor ... 

No variation with time was observed and we concluded Am/m < 5 x 10-23. Also, a limit on the quadrupole term in the mass tensor was evaluated. 

A Refined Application. Avgul et ah (1) have shown that the dipole-quad-rupole and quadrupole-quadrupole terms in Equation 2 are not negligible, as was assumed by Crowell nor are they cancelled by the repulsion term, as has been stated by de Boer (5). A more accurate result could, therefore be obtained by using all the terms in Equation 2 in conjunction with Crowell s expression for the lattice sums, Equation 11. Instead of Equation 13 we would now have ... 

The electrostatic quadrupole term in this sum is that for which ( = 2, so that the quadrupole Hamiltonian is given by... 

We are now in a position to write down the Hamiltonian operator for all nuclear spin and quadrupole terms for a diatomic molecule we allow for the possibility that both nuclei are involved and therefore sum over the nuclear index a. The terms are expressed in a molecule-fixed rotating coordinate system with origin at the nuclear centre of mass, except that we retain Ia as being quantised in a space-fixed axis system. We number the terms sequentially and then describe their physical significance. 

Additional terms involving the scalar and tensor interactions between the two nuclear spins were found to be too small to be significant in the first study, but we will meet them later. We encountered the quadrupole term in (8.171) in our earlier discussion of the D2 molecule, and obtained the following results for the matrix elements in the coupled representation ... 

The subscripts 1 and 2 refer to the Br and Li nuclei respectively. This Hamiltonian differs from that used previously for CsF, equation (8.282), only through the additional quadrupole term and the explicit addition of a Stark effect term. Although the weak electric fields (a few V cm ) used in this work were employed mainly to transfer electric dipole intensity into the resonance transitions, the resulting Stark shifts were measurable because of the extremely small linewidths obtained (about 300 Hz). 

HF is similar to CsF, discussed earlier, except that both nuclei have spins / of 1 /2 and therefore no quadrupole moments. The effective Hamiltonian for HF employed by Weiss [86] and by Muenter and Klemperer [87] was the same as that used for CsF in equation (8.281), without the quadrupole term, but with the addition of an electric field term. In the case of DF, described later, the quadrupole term is present. In irreducible tensor form, therefore, our Hamiltonian follows equation (8.282), and is written ... 

The sum over k represents the terms for both nuclei, whereas the quadrupole term exists only for the 14N nucleus. Equation (9.134) also recognises implicitly that only terms diagonal in the ground vibronic state will be included. For the magnetic field interactions the effective Hamiltonian used by Wayne and Radford was... 

One form of the effective Hamiltonian for the CN radical in its 2 + ground state was given in chapter 9. Omitting the quadrupole term, which is not required for 13CO+, our starting point might be... 

The CN radical in its 21 ground state shows fine and hyperfine structure of the rotational levels which is more conventional than that of CO+, in that the largest interaction is the electron spin rotation coupling../ is once more a good quantum number, and the effective Hamiltonian is that given in equation (10.45), with the addition of the nuclear electric quadrupole term given in chapter 9. The matrix elements in the conventional hyperfine-coupled case (b) basis set were derived in detail in chapter 9,... 

The fourth and last terms in (11.49) are spin-orbit distortion corrections to the spin-rotation and Fermi contact interactions. The hyperfine and quadrupole terms in this Hamiltonian refer to the 14N nucleus. 

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