Electronic spin angular momentum

Figure 1.10 Space quantization of electron spin angular momentum...

The component of the total (orbital plus electron spin) angular momentum along the intemuclear axis is Qfi, shown in Figure 7.16(a), where the quantum number Q is given by... 

The singlet function corresponds to zero total electron spin angular momentum, S = 0 the triplet functions correspond to S = 1. Operating on these functions with the spin Hamiltonian, we get ... 

The quantum number S defines the total electronic spin angular momentum and its allowed values are... 

Most molecules with A>0 obey Hund s case (a). Here, the axial components of electronic orbital angular momentum (Ah) and of electronic spin angular momentum (2fc) combine to give a resultant axial component of total electronic angular momentum (A + 2)fi. The quantum number ft is defined as (Section 1.19)... 

The older literature uses K instead of N.) For the most common case-(b) case, we have A = 0 here N has the possible values 0,1,2,... and represents just rotational angular momentum. The angular momentum N then adds to the electronic spin angular momentum S to give a total angular momentum apart from nuclear spin, which, as usual, is called J. The quantum number J has the possible values [Equation (1.265)]... 

For case (b) coupling, if we first neglect electronic spin angular momentum, then N is analogous to J and A is analogous to /f, so that (5.67) becomes... 

Considering a A2 as part of the electronic energy and adding a term to take account of the electronic spin angular momentum, we get the expression (4.149). The condition J > K becomes N > A [Equation (4.147)]. 

Polyatomic molecules. The same term classifications hold for linear polyatomic molecules as for diatomic molecules. We now consider nonlinear polyatomics. With spin-orbit interaction neglected, the total electronic spin angular momentum operator 5 commutes with //el, and polyatomic-molecule terms are classified according to the multiplicity 25+1. For nonlinear molecules, the electronic orbital angular momentum operators do not commute with HeV The symmetry operators Or, Os,. .. (corresponding to the molecular symmetry operations R, 5,. ..) commute... 

Now the total angular momentum of an electron is the resultant of the orbital angular momentum vector and the electron-spin angular momentum vector- Both of these are quantized, and we can define a new quantum number, j ... 

The dynamic state is defined by the values of certain observables associated with orbilal and spin motions of the electrons and with vibration and rotation of [lie nuclei, and also by symmetry properties of the corresponding stationary-state wave functions. Except when heavy nuclei ate present, the total electron spin angular momentum of a molecule is separately conserved with magnitude Sh. and molecular slates are classified as singlet, doublet, triplet., . according to the value of the multiplicity (25 + I). This is shown by a prefix superscript lo the term symbol, as in atoms. 

Rather than detail these effects here, however, we can illustrate the effects of angular momentum in a simpler case by examining the electronic spin angular momentum S. Experimentally, it is found that all electrons have angular momentum, and the length of the angular momentum vector is always S = a/37 /2. This angular momentum... 

From our present standpoint, we know that the deflection of a silver atom in the Stern-Gerlach experiment is caused by the interaction of its electronic spin angular momentum S with an inhomogeneous magnetic field. The projection of S on the direction of this field, Ms, is quantized. For a silver atom, Ms can take two values +vjh and — h, where h is Planck s constant h over 2ji and adopts a value of 1.054571596 x 10-34 Js in cgi units. 

Let us assume, in analogy to Eq. (3.20.6), that the conversion factor between the electron spin angular momentum S and the concomitant spin magnetic moment pu is... 

Many free radicals in their electronic ground states, and also many excited electronic states of molecules with closed shell ground states, have electronic structures in which both electronic orbital and electronic spin angular momentum is present. The precession of electronic angular momentum, L, around the intemuclear axis in a diatomic molecule usually leads to defined components, A, along the axis, and states with A =0, 1, 2, 3, etc., are called , n, A, , etc., states. In most cases there is also spin angular momentum S, and the electronic state is then labelled 2,s+1 Id, 2,s+1 A, etc. 

We now consider some aspects of the theory of electronic spin angular momentum. What follows here is a relatively brief and simple exposition we will return to a comprehensive description of the details of electron spin theory in chapter 3. For a single electron, the spin vector. S is set equal to (1 /2)[Pg.54]

To take a specific example, let us consider P = S, the electron spin angular momentum for a diatomic molecule in a Hund s case (a) coupling scheme where the basis functions are simple products of orbital, rotational and spin functions. Using standard... 

In a case (a) basis set, the electron spin angular momentum is quantised along the linear axis, the quantum number E labelling the allowed components along this axis. Because we have chosen this axis of quantisation, the wave function is an implicit function of the three Euler angles and so is affected by the space-fixed inversion operator E. An electron spin wave function which is quantised in an arbitrary space-fixed axis system,. V. Ms), is not affected by E, however. This is because E operates on functions of coordinates in ordinary three-dimensional space, not on functions in spin space. The analogous operator to E in spin space is the time reversal operator. 

In section 7.4.2 we dealt with perhaps the simplest contribution to the effective Hamiltonian for a particular electronic state, that of the rotational kinetic energy. We now turn our attention to contributions which are only slightly more complicated, the so-called fine structure interactions involving the electron spin angular momentum S. Obviously for S to be non-zero, the molecule must be in an open shell electronic state with a general multiplicity (2S +1). 

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