Nuclear Spin Basis Functions

In Section 2.3 we described the properties of a and j8, the basis functions for a spin y2.The definitions given there and the relationships developed in that section are essential for the presentation in this chapter and might profitably be reviewed at this point. 

For two or more noninteracting spins, the correct wave function is simply the product of the functions for the individual spins. For N spins, each with I = /2, there are 2N such product functions. For N spins that may interact with each other, it makes sense to start with the 2N products of a and / as basis functions and to see how the spin interactions cause them to be modified. 

---

To use the method, we need a complete set of well-behaved nuclear-spin basis functions spin-spin coupling term is omitted from (8.41), we get the Hamiltonian... 

The way we combine the nuclear spin basis functions with the rotation-vibration-electronic basis functions in H2 follows the same type of argument using the nuclear permutation group Rovibronic states of symmetry... 

It would appear that identical particle permutation groups are not of help in providing distinguishing symmetry labels on molecular energy levels as are the other groups we have considered. However, they do provide very useful restrictions on the way we can build up the complete molecular wavefimction from basis functions. Molecular wavefunctions are usually built up from basis functions that are products of electronic and nuclear parts. Each of these parts is further built up from products of separate uncoupled coordinate (or orbital) and spin basis functions. When we combine these separate functions, the final overall product states must conform to the permutation symmetry mles that we stated above. This leads to restrictions in the way that we can combine the uncoupled basis functions. 

This equation is fully equivalent to the time-independent Schrodinger equation and its roots give the exact energies. Since the set (2.50) generally has an infinite number of functions, the secular determinant generally is of infinite order. In certain cases, however, one need only deal with a finite-order secular determinant. For example, in electron-spin (and nuclear-spin) problems, the number of basis spin functions is finite. Applications will be found in later chapters. 

The solution of the NMR secular equation is simplified by the usual method (Section 1.10) of using basis functions that are eigenfunctions of operators that commute with the Hamiltonian. The operator Tz for the total z component of nuclear spin is... 

For the calculation of the EPR spectra, the basis functions were extended to include the four spin functions of copper nuclei, and the following expression for the nuclear hyperfine interaction added to the Hamiltonian ... 

We must therefore examine the possible nuclear spin states and classify them according to (6.257) or (6.258). Since each 14N nucleus has spin 7 = 1, and three spatial orientations with M/ = +1, 0, — 1, there are nine basis spin functions, which are... 

The magnetic hyperfine interaction terms were given in equation (8.351) and the electric quadrupole interaction in equation (8.352). We extend the basis functions by inclusion of the 7Li nuclear spin I, coupled to J to form F the value of / is 3/2. We deal with each term in turn, first deriving expressions for the matrix elements in the primitive basis set (8.353), and then extending these results to the parity-conserved basis. All matrix elements are diagonal in F, and any elements off-diagonal in S and / can of course be ignored. 

The simplest approach to the book-keeping problem of calculating all the nuclear spin-dependent matrix elements is first to evaluate all the terms in the primitive basis set, leaving only J, I and F as variables. We therefore construct the following 6x6 matrix, using the functions listed in (8.353). The rotational levels are widely spaced compared with the hyperfine terms, so that we also confine attention to matrix elements diagonal in J. The required 6x6 matrix is as follows. 

The presence of two nuclear spins means that there is considerable choice in the selection of basis functions the reader who wishes to practice virtuosity in irreducible tensor algebra is invited to calculate the matrix elements in the different coupled representations that are possible In fact the sensible choice, particularly when a strong magnetic field is to be applied, is the nuclear spin-decoupled basis set t], A N, S, J, Mj /N, MN /H, MH). Again note the possible source of confusion here MN is the space-fixed component of the nitrogen nuclear spin /N, not the space-fixed component of N. This nuclear spin-decoupled basis set was the one chosen by Wayne and Radford in their analysis of the NH spectrum. 

The analysis of the spectrum was accomplished using a case (a) basis, with the addition of two nuclear spins, I and h, for 63Cu and 19F respectively. The basis functions therefore take the form A S, E J, 72, I, F, h- F, Mp), and leaving aside nuclear spin interactions, the theory follows closely the same path as that already described for 3n CO in chapters 9 and 10. The effective Hamiltonian is the sum of terms representing the spin orbit, rotational, spin-rotation, spin-spin and centrifugal distortion contributions and is written [56] ... 

Bally and Rablen ° followed up their important study of the appropriate basis sets and density functional needed to compute NMR chemical shifts with an examination of procedures for computing proton-proton coupling constants." They performed a comparison of 165 experimental with computed proton-proton coupling constants from 66 small, rigid molecules. They tested a variety of basis sets and functionals, along with questioning whether all four components that lead to nuclear-nuclear spin coupling constants are required, or if just the Fermi contact term would suffice. 

Symmetry properties of the nuclear wavefunction are different in the diabatic and adiabatic representations. The pair of adiabatic electronic states (see (1)) belong to the Al and A2 irreducible representations of the double group of S3. The diabatic states obtained from the adiabatic ones by applying the U matrix form a basis for the two dimensional irreducible representation E of S3. For quartet nuclear spin states, the electronuclear wavefunction, nuclear spin part excluded, must belong to the A2 irreducible representation. This requires the nuclear wavefunction (without nuclear spin) to be of the same E symmetry as the electronic one, because of the identity E x E = Ai + A2 + E. For doublet spin states, the E electronuclear wave-function (nuclear spin excluded) is obtained with an Ai or A2 nuclear wavefunction, combined with the E electronic ones. 

---