The Spin Paradigm

The spin paradigm, widely-used by the chemical community, is the replacement of fermion dynamics by spin dynamics in the discussion of spectra, structure and mechanism. The average chemist feels very comfortable with this spin paradigm and resists passionately any attempt to break it. The spin paradigm does have a number attractive features  

Freeon dynamics provides a dynamically-correct replacement for the faulty spin paradigm. In particular its freeon Gel fand diagrams are a dynamically correct replacement for spin arrows as a primitive pattern of understanding . 

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The use of spin to label the freeon spaces, which carry the dynamics of the system, has confused those chemists who are locked into the spin paradigm. 

From all this one must conclude that the determinantal and second-quantized formulations should be regarded as a poor man s group theory which, while convenient, hides the basic freeon dynamics. These fermion methods have the additional disadvantage that their antisymmetric fermion functions are not normally pure spin (freeon) states so that spin-projection may be required. A method for avoiding (approximately) spin projection is the employment of the variation principle to approximate the ground state e. g., unrestricted Hartree-Fock theory. Finally the use of the fermion formulations has lead to the spin paradigm as a replacement for the more fundamental freeon dynamics. 

In this, my third contribution, I apply freeon dynamics to problems of interest in chemistry and physics and compare with the results obtained by the spin paradigm. In particular I will apply freeon dynamics to the following "spin phenomena" i) spin exchange, ii) spin superexchange, iii) spin polarization, iii) spin density, iv) high-and low-spin states of the transition-metal ions, v) the periodic table, vi) ferromagnetism, vii) spin waves and viii) high-Tc superconductivity. 

In this section the simplicity of freeon dynamics is illustrated by its application to polyenes [6]. Here I relate molecular-orbital Gel fand states to atomic-orbital Gel fand states and then relate the atomic-orbital Gel fand states to valence bond states. Note that this construction provides a theoretical basis for the Rumer rule i.e., for the number of linearly independent valence bond states. We then use this freeon dynamics to explicate the spin paradigm. 

The spin paradigm employs spin arrows to indicate electron pairing and antipairing. Here I compare the spin-arrow diagrams for ethylene to their Gel fand and the valence bond counterparts ... 

The spin paradigm can lead to serious inconsistencies. Consider for example the Hund and the Heitler-London rules discussed in Section 4.2. The spin-arrow description of the states assigns parallel spins to triplet states and antiparallel spins to singlet states as follows ... 

In the usual preparatioii-evohition-detection paradigm, neither the preparation nor the detection depend on the details of the Hamiltonian, except hi special cases. Starthig from equilibrium, a hard pulse gives a density matrix that is just proportional to F. The detector picks up only the unweighted sum of the spin operators,... 

To be distinguished, the use of HDVV paradigm in systems with firm covalent bonds may really reveal the extension to which the model itself is applicable and to suggest the further necessary terms. This is indeed the case encountered in our approach. We found that for an improved description of selected set of states, the spin Hamiltonian has to be expanded with new terms, proposed here as the intercentric generalization of the traditional biquadratic terms. We propose and use here the following extended form ... 

Slater determinants and second-quantization constitute a poor-man s group theory which is "ungodly" and leads to the nefarious spin paradigm. 

In the usual preparation-evolution-detection paradigm, neither the preparation nor the detection depend on the details of the Hamiltonian, except in special cases. Starting from equilibrium, a hard pulse gives a density matrix that is just proportional to F. The detector picks up only the unweighted sum of the spin operators, /. It is only during an evolution (perhaps between sampling points in an FID) that these totals need be divided amongst the various lines in the spectrum. Therefore, one of the factors in the transition probability represents the conversion from preparation to evolution the other factor represents the conversion back from evolution to detection. 

In fact the concept of electronic configuration as a causally explanatory feature has become very much the domain of chemistry or to be more precise it is the dominant paradigm in modem chemistry. Conversely, physicists are only too aware of the limitations of the electronic configuration model and they only draw upon it as a zero order approximation. Hettema and Kuipers further state that Bohr s theory of the atom, despite its level of approximation, is to be regarded as a physical theory because the explanation of the periodic table was only a spin-off from its development. But given Heilbron and Kuhn s detailed version of the historical development, it was precisely the explanation of the periodic table which provided the initial impetus for Bohr s famous theory of the atom, whereas the explanation of the hydrogen spectmm only arose later. (Heilbron and Kuhn, 1969). 

Quantum systems of any kind can in principle be candidates for quantum hardware, including different kinds of spin qubits we briefly review some of these in the next section. Much effort has been expended on the question of which physical systems are best suited for use in QIP, but no ultimate answer has been found so far. A much quoted list of conditions to build computers was established by DiVincenzo [35], but one has to note that some of these restrictions are specific to the quantum circuit paradigm. 

Considering any of these paradigms, a minimal goal for toy models would be to manipulate the quantum dynamics of a small number of spin levels , and that requires a known and controlled composition of the wavefunction, sufficient isolation and a method for coherent manipulation. As illustrated in Figure 2.13, the first few magnetic states of the system are labelled and thus assigned qubit values. The rest of the spectrum is outside of the computational basis, so one needs to ensure that these levels are not populated during the coherent manipulation. 

Since the two-spin state forms can lead to different products, the products obtained will be a mixture that reflects the initial fractionation of the reaction between the two-spin states. The fractionation in turn is a reflection of the interplay and the probability of cross-over between the two-spin states (8). Thus, the two-state reactivity paradigm resolves the dilemma of whether a radical recombination or a direct insertion mechanism governs cytochrome P450-catalyzed hydroxylation actually they are both involved and the degree to which either is expressed depends upon the specific substrate hydroxylated and the specific enzyme. 

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