Spherical tensor theory

Stone AJ, Tough RJA (1984) Spherical tensor theory of long-range intermolecular forces. Chem Phys Lett 110 123-129... 

Gray CG, Lo BWN (1976) Spherical tensor theory of molecular multipole moments and polarizabilities. Chem Phys 14 73—87... 

A. J. Stone and R. J. A. Tough, Chem. Phys. Lett., 110,123 (1984). Spherical Tensor Theory of Long-Range Intermolecular Forces. 

When induction operators of high-order multipoles are taken into account intensity calculations tend to become very cumbersome [30,31]. We propose a relatively easy way of performing these calculations using the irreducible spherical tensor theory of multipole light scattering [e.g., Eqs. (6) and (7)] together with symbolic calculations of the Wigner coefficients by computer. 

In both of the above treatments, spherical tensor and cartesian, we have factored the quadrupole interaction into the product of two terms, one of which operates only on functions of proton coordinates within the nucleus and the other only on functions of coordinates of electrons and protons outside the nucleus. We shall see in subsequent chapters that the spherical tensor form is rather more convenient for the calculation of matrix elements of 3Cq. However, we shall find this easier to appreciate once we have considered some of the theory of angular momentum in chapter 5 so we defer discussion until later. 

Angular momentum theory and spherical tensor algebra... 

Thus we see that the operator g is not strictly an angular momentum operator in the quantum mechanical sense, which is why we have assigned it a different symbol. More importantly for the present purposes, we cannot use the armoury of angular momentum theory and spherical tensor methods to construct representations of the molecular Hamiltonian. In addition, the rotational kinetic energy operator, equation (7.89), takes a more complicated form than it has for a nonlinear molecule where there are three Euler angles (rotational coordinates). 

We will not pursue the analysis in detail any further. It does illustrate the power of spherical tensor methods, and one can only shudder at the possibility of developing the theory in a cartesian coordinate system, with direction cosines. We list the final values of the molecular parameters for 14N35C1 in table 10.12. The values of the hyperfine constants may be interpreted semi-empirically in the following way. The outmost pair of electrons occupy a 3 /rT molecular orbital and the Fermi contact constants, given in table 10.12, may be compared with the atomic values [144] of 1811 and 5723 MHz for the nitrogen and chlorine atoms respectively one concludes that the s electron character... 

We now calculate the perturbation to the Zeeman field due to the quadrupolar interaction by means of average Hamiltonian theory.This is accomplished by transforming TYq to the Zeeman interaction frame and then applying the spherical tensor rotation properties to the spin elements 72,The resulting quadrupolar Hamiltonian TTq in the rotating frame is given by ... 

Some time ago we contributed to the development of the irreducible spherical tensor multipolar theory of light scattering [13,28], According to Ref. 13, the M component of the A th-rank dipole-arbitrary order multipole linear polarizability of a pair of interacting molecules A and B reads as... 

This book is for graduate students just beginning research, for theorists curious about what experimentalists actually measure, for experimentalists bewildered by theory, and for potential users of spectroscopic data in need of a user s guide. We have minimized recourse to abstract and elegant treatments (e.g., spherical tensors) wherever a simpler one (e.g., ladder operators) would suffice. We have worked through many examples rather than attempt to provide formulas and literature examples for all conceivable cases. 

---