Angular momentum term symbols

Seetion treats the spatial, angular momentum, and spin symmetries of the many-eleetron wavefunetions that are formed as anti symmetrized produets of atomie or moleeular orbitals. Proper eoupling of angular momenta (orbital and spin) is eovered here, and atomie and moleeular term symbols are treated. The need to inelude Configuration Interaetion to aehieve qualitatively eorreet deseriptions of eertain speeies eleetronie struetures is treated here. The role of the resultant Configuration Correlation Diagrams in the Woodward-Hoffmann theory of ehemieal reaetivity is also developed. 

D i,i,o - D i,-i,0, and the tools of angular momentum coupling allows these integrals to be expressed, as above, in terms of products of the following 3-j symbols ... 

In a manner similar to that by which the atomic states were designated as s, p, d, or /, the letters S, P, D, and F correspond to the values of 0, 1, 2, and 3, respectively, for the angular momentum vector, L. After the values of the vectors L, S, and / have been determined, the overall angular momentum is described by a symbol known as a term symbol or spectroscopic state. This symbol is constructed as Ps+1)Lj where the appropriate letter is used for the L value as listed earlier, and the quantity (2S + 1) is known as the multiplicity. For one unpaired electron, (2S + 1) = 2, and a multiplicity of 2 gives rise to a doublet. For two unpaired electrons, the multiplicity is 3, and the state is called a triplet state. 

The problem is not simplified by Eq. (15), since there exists a closed-form expression for the multi-scattering matrix for n spheres in terms of spherical Bessel and Hankel functions, spherical harmonics and 3j-symbols, where l, l and to, m are total angular momentum and z-projection quantum numbers, respectively (Henseler, Wirzba and Guhr, 1997) ... 

The term symbol summarizes the properties of any state and also permits a concise representation of spectral transitions. It consists of an upper case letter (S, P, D. ..) to represent the net orbital angular momentum (L) and a number written as a superscript on the upper left to indicate spin multiplicity (i. e. the number of possible orientations of total spin of the atom). L is zero for Fe " (no angular momentum) and 2 for Fe ". The spin multiplicity is defined as (2S -i- 1) S = 5/2 and 2 for Fe " and Fe ", respectively. The ground state term symbol for Fe " is, therefore, 85 2 and for Fe " it is 04. The subscript on the right is the value J. 

A.12-1 for a brief review of atomic-spectroscopic notation). In a given term-symbol, T will be S, P, D, F, or G etc. depending on whether the total electronic orbital angular-momentum quantum number L is 0, 1,... 

It has been pointed out above that two electrons in the Is orbital must have their spins opposed, and hence give rise to the singlet state So, with no spin or orbital angular momentum, and hence with no magnetic moment. Similarly it is found that a completed subshell of electrons, such as six electrons occupying the three 2p orbitals, must have S — 0 and L = 0, corresponding to the Russell-Saunders term symbol lS0 such a completed subshell has spherical symmetry and zero magnetic moment. The application of the Pauli exclusion prin-... 

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. 

Here A2 symbolizes a pseudo-scalar of A2 symmetry, normalized to unity. The actual form of this pseudoscalar need not bother us. The only property we will have to use later on is that even powers of A2 are equal to +1. Now we can proceed by defining rotation generators f x, y,t 2 in the standard way, as indicated in Table 1 [10]. Note that primed symbols are used here to distinguish the pseudo-operators from their true counterparts in real coordinate space. Evidently the action of the true angular momentum operators t y, (z on the basis functions is ill defined since these functions contain small ligand terms. 

As in configuration A, an examination of the angular momentum attributes of these differential equations provides insight into the effects of averaging over Afi, In this case, however, we can show that in the ten-level case B the M and — A terms contribute equally, so that control is maintained after M averaging. To sefl this, consider how each M dependent 3-j symbol changes when M — — M Given Eqs. (51) and (53), we have for the dipole transition matrix element associated with ] that... 

With light elements (e.g. first-row and second-row transition metals), to which principal consideration is given in this book, the Russell-Saunders (or IS) coupling scheme suffices. Every microstate is designated with a term symbol of the general form where 25 -I- 1 is the spin multiplicity (5 = 5,), L is the total orbital angular momentum... 

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