Spinless theory

Since the transfer of protons is truly contact, this reaction is best suited for comparison with the contact and spinless theory given above. However, the authors who first monitored it in the time domain tried to fit the fluorescence signal as a biexponential one [59]. The similar reaction but of a more reactive photoacid (2-naphthol-6-sulfonate) with an acetate anion has been studied, and its kinetics, which is neither exponential nor biexponential, was fitted to the tme theory of contact quenching [60]. It is especially important that the fluorescence... 

The spinless theory that we are using here is approximate for electron transfer but absolutely correct for the proton reactions. The widely studied example is the reversible proton transfer from an excited photoacid to the... 

The irreversible quenching of the triplet Ruthenium complex by methylviologen [148] represented by Eq. (3.329) was considered within the spinless theory in Section VII. The more detailed reaction scheme accounting for the spin states and their incoherent conversion is as follows ... 

Only the result for the fast conversion (x —> oo) is identical to the spinless theory equation (3.208), provided k%/4 is substituted for kc in Eq. (3.205). Such a substitution conditioned by a full equilibration of the spin state populations results in a proper redefinition of the EM parameter z in highly polar solvents (C = 0) ... 

With the constraint (3.601) at any spin conversion (p(r) is exactly the same as in the equation of the spinless theory, (3.211). This is because the transitions between the singlet and triplet RIPs do not modulate their recombination rates, leaving z = kc/4na as well as product yields, 1 [Pg.318]

Here pss is the Laplace transformation of the solution to the coupled equations (3.593), where ka + kc is substituted for kc in the boundary conditions. In the spinless theory they reduce to a single equation that is the contact analog of Eq. (3.234) for G = pSs-... 

It is instructive to note that neither of these quantities depends on x if k r = kt. In this particular case the spin conversion does not modulate the rate of recombination so that cp = (1 + ksr/kD) l as in the spinless theory and w j is the weight of the singlet in the uncorrelated pairs composed in the bulk. 

These quantities are found exactly the same as in EM or contact spinless theory. Using them together with w from Eq. (3.643) in the general definition (3.629), we have... 

The shape of the kernels (3.651) is rather obvious from the physical point of view. The kernels R and R are exactly the same as in Eq. (3.369) of the spinless theory. When spin conversion during encounter is negligible, the backward electron transfer to the ground state remains the single channel of geminate ion recombination. Therefore, the kernel R is 4 times smaller than... 

All other results obtained previously with the spinless theory are also generalized in the same way. In particular, using the appropriate kernels in Eq. (3.822), we obtain the geminate quantum yield of triplets in the following form ... 

In non-relativistic Schrodinger theory every component of the orbital angular momentum L = r x p, as well as L2, commutes with the Hamiltonian H = p2/2m + V of a spinless particle in a central field. As a result, simultaneous eigenstates of the operators H, L2 and Lz exist in Schrodinger theory, with respective eigenvalues of E, l(l + l)h2 and mh. In Dirac s theory, however, neither the components of L, nor L2, commute with the Hamiltonian 10. 

It is well known that the exact electronic energy can also be given explicitly in terms of the spinless 1-RDM and the two-particle charge density (2-CD). This suggests an alternative viewpoint regarding D-functional theory. One could employ the exact functional but with an approximate 2-CD that is built from... 

The solution of (2.3.69) is a purely mathematical problem well known in the theory of diffusion-controlled processes of classical particles. However, a particular form of writing down (2.3.69) allows us to use a certain mathematical analogy of this equation with quantum mechanics. Say, many-dimensional diffusion equation (2.3.69) is an analog to the Schrodinger equation for a system of N spinless particles B, interacting with the central particle A placed... 

However, the theory of exciplex dissociation cannot be made spinless like that for photoacids (Section V.D). The dissociation products are radical ions and the spin conversion in RIPs essentially affects

other quantities listed in Eq. (3.589). To illustrate this phenomenon, let us concentrate on the fluorescence yield, which is affected through %E(ks) and the charge separation quantum yield cp(cr). We will consider the general solution obtained for these two quantities in Ref. 31 only in the simplest case of highly polar solvents for which the Green functions are well known. 

The spinless variant of the present theory was already discussed in Section V.D and its interrelationship with IET and a number of other theories of exciplexes or stable complexes was disclosed. In the next Section XI.D we also consider not an excited-state but a ground-state particle. It is subjected to thermal dissociation to radicals followed by their geminate and subsequent bimolecular recombination into the fluorescent product. 

If all the oscillators have the same frequency, the energy of a state simply depends on the number of vibrational quanta E = nhv, and the density of states in the classical theory is replaced by the degeneracy of this energy level. Since vibrational quanta are indistinguishable (and spinless) the rule for counting their states is the rule that applies to bosons. If there are n quanta in s oscillators, then the degeneracy is equal to... 

However, simple band theory is not sufficient to explain the electrical behavior of conducting polymers. For example, simple band theory cannot explain why the charge carriers, usually electrons and holes, in polyacetylene and polypyrrole are spinless. In order to overcome these and other difficulties, the concepts of solitons, polarons, and bipolarons have been used since the 1980s to explain the electronic behavior of conductive polymers [21]. 

Regardless of the comments above, we need to speak about nuclear spins, since they are everywhere (at least in the form of protons) and because they also can be connected with quantum effects in bio-systems. Recent experiments (Buchachenko and Kouznetsov 2008 Buchachenko et al. 2005) demonstrate that intramitochondrial nucleotide phosphorylation is a nuclear spin controlled process because the magnetic magnesium isotope Mg(ll) increases the rate of mitochondrial ATP synthesis in comparison with the spinless nonmagnetic Mg(ll), Mg(lI) ions. Such nuclear spin isotope effect is usually interpreted in terms of radical pair theory for separated spins in solvent cage (Buchachenko 1977), but an alternative explanation based on... 

In the development of the Slater method (Section 3.1) it was noted that the Pauli principle in the form (1.2.27) could always be satisfied by constructing the electronic wavefunction from determinants (i.e. antisymmetrized products) of spin-orbitals. In an earlier section, however, it was shown that for a two-electron system the antisymmetry principle could also be satisfied by writing the wavefunction as a product of individually symmetric or antisymmetric factors—one for spatial variables and the other for spin variables. Since, in the usual first approximation the Hamiltonian does not contain spin variables, it is natural to enquire whether a corresponding exact N-electron wavefunction might be written as a space-spin product in which the spatial factor is an exact eigenfunction of the spinless Hamiltonian (1.2.1). To investigate this possibility, we need a few basic ideas from group theory (Appendix 3). 

Another approach to GUHF theory has been given by Seeger and Pople (1977). In practice, however, with a spinless Hamiltonian, it is seldom necessary to admit the 2-component functions since the UHF forms are usually stable (see e.g. Cook, 1981, 1984). The question of stability in the general case, where the wavefunction is not even an eigenfunction of S, has been discussed by Fukutome and others (e.g. Fukutome, 1981 Calais, 1985) but the value and physical significance of the wavefunction then remains open to question (cf. Problem 6.12). 

One attractive aspect of the soliton theory of charge transport is that the carriers (cations or anions) carry no spin, i,e, the conducting compositions do not contain unpaired electrons, ESR experiments on the doping of PA(49) show that in certain intermediate doping regimes the spin concentration is much lower than expected from the observed conductivity values this phenomenon is referred to as spinless conductivity. If the conduction involved a normal process of defect-induced hole or electron transport, there would be a direct correlation between ESR determined spin concentration and conductivity. The same conclusion of spinless conduction is obtained from ESR experiments on doped PPP(61) however the soliton theory is not applicable to the PPP system(25). In Section VII, we present an alternate transport mechanism based on bipolarons (dications or dianions) which is applicable to all conducting polymer systems(26),... 

Angeli C, Cimiraglia R, Malrieu JP. n-electron valence state perturhation theory A spinless formulation and an efficient implementation of the strongly contracted and of the partially contracted variants. J Chem Phys. 2002 117 9138. 

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