Singlet and triplet states

Pericyclic reactions may be ring-opening, ring closing or rearrangements and are classified in three major categories which are  

Transitions described in Jablonski diagram take place within chetnical molecules and therefore they concern the atoms that constitute these molecules. The electrons of these molecules are responsible for the different transitions shown in the Jablonski diagram. For this reason, the Jablonski diagram is also called the electronic transitions diagram. 

Since localization of an electron is difficult, one referred to four quantum numbers (n, I, m and s) to characterize an eiectron and to differentiate it from the others. The princ pal quantum number n determines the energy of any one-electron atom of nuclear charge Z. n can assume any positive integral value, excluding zero. 

The ai ular momentum quantum number 1 determines the angular momentum of the eiectron. ft may assume all integral values from 0 to n-1 inciusive. 

The magnetic quantum number m characterizes the magnetic fieid generated by the eiectric current of the eiectron, circulating in a loop, m can assume all integral values between -4 and i-f including zero. 

The spin quantum number s is Ihe result of the electron spinning about its own axis. Thus, a local smali magnet is generated with a spin s. 

For a singlet state the spin quantum number S equals zero, and for a triplet state S equals unity. Since the triplet state has two electrons with parallel spins, such a state contrasts with a singlet state in that it is paramagnetic. This feature can be verified by showing that both the excited state paramagnetism and the phosphorescence decay at identical rates. The three components of the triplet state result in three different combinations of the magnetic quantum number Ms, which have the values 1,0, and-1. 

The state energies of the So, Si, and Ti states of a metal complex (ML) are given by Eqs. (1.6)-(l. 8), where they are defined as the sum of a zero-order energy, plus electron repulsion energies  

In these equations J is the matrix element for electron repulsion due to electron exchange, K is the matrix element for coulombic interactions, and (ML ) is the metal complex in the excited state. Since both J and K are positive, the difference between E(S ) and (Ti) is 27, which results in E(S ) being greater than E(Ti). This higher energy of the singlet state over that of the triplet states is an important feature in understanding photochemical reactions. 

Internal conversion, thermal reversion of the excited singlet to the ground state with the release of heat to surrounding molecules such as the solvent (Sj So + heat). Because these two states are of like multiplicity, the transformation is allowed in terms of quantum theory and is often a very favorable process with a rate constant close to diffusion control Fluorescence, emission of visible or ultraviolet radiation (Sj Sq -l- hi ). The emitted photon is always of a longer wavelength than the absorbed photon. This is also a quantum mechanically allowed process and usually occurs rapidly. 

Intersystem crossing, a quantum-forbidden transition that produces a new 

Such reactions may be photochemically productive if the new triplet possesses sufficient energy to take part in other processes. 

Conversion of a triplet to the corresponding ground-state singlet (without the intervention of a quencher) is another quantum-mechanically forbidden and usually inefficient ISC process, called phosphorescence when it is accompanied by emission of radiation. Phosphorescence and fluorescence are examples of luminescence phenomena. 

In a normal molecule, where all the MOs are either doubly occupied or empty, the electrons all have to appear in pairs with opposite spin because of the Pauli principle. This is not the case in an excited state (see Fig. 6.1b), because there are now two singly occupied MOs. The electrons in these may or may not retain their opposite spins (Fig. 6.7). States in which they do are called singlet states and those in which they do not are called triplet 

FIGURE 6.6. Relation between absorption and emission bands corresponding to the same electronic transition. 

FIGURE 6.7. Ground state Sq and the singlet Sj and triplet Tj states arising from the first electronic transition. 

Since a given pair of singlet and triplet states differ only in the spins of the electrons in the two single occupied MOs, it follows from Hund s rule that the triplet must be lower in energy. The order of the states in Fig. 6.7 is therefore Sq Tj Sj. 

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The perturbations in this case are between a singlet and a triplet state. The perturbation Hamiltonian, H, of the second-order perturbation theory is spin-orbital coupling, which has the effect of mixing singlet and triplet states. 

We substitute these expressions into the Slater determinants that form the singlet and triplet states and eolleet terms and throw out terms for whieh the determinants vanish. 

This then gives the singlet and triplet states in terms of atomie-orbital oeeupaneies where it is easier to see the energy equivalenees and differenees. 

The latter rule is rigidly obeyed in the observed spectrum of helium. From the accurately known energy levels it is known precisely where to look for transitions between singlet and triplet states but none has been found. 

Figure 4.9 Mixing of pure singlet and triplet states may generate artificial minima on the UHF energy surface...

At the dissociation limit the UHF wave function is essentially an equal mixture of a singlet and a triplet state, as discussed in Section 4.4. Removal of the triplet state by projection (PUHF) lowers the energy in the intermediate range, but has no effect when the bond is completely broken, since the singlet and triplet states are degenerate here. 

These singlet and triplet state species exhibit the important differences in chemical behavior to be expected. The former species, with their analogy to carbonium ions, are powerful electrophiles and the relative rates of their reaction with a series of substrates increases with the availability of electrons at the reaction center their addition reactions with olefins are stereospecific. Triplet state species are expected to show the characteristics of radicals i.e., the relative rates of additions to olefins do not follow the same pattern as those of electrophilic species and the additions are not stereospecific. 

Both CSs and CSs were also successfully generated by the fragmentation of ionized 4,5-dioxo-2-thioxo-l,3-dithione (65) and 2-thioxo-l,3-dithiole (66) (90JA3750). Tire three sulfur atoms in the anion and cation radicals were chemically equivalent, suggesting that they take the D h (or C2u) form (67 or 68). On the other hand, under similar conditions, 3-thioxo-1,2-dithiole (69) yielded two isomeric cation radicals the (or 2 ) form and the carbon disulfide 5-sulfide form (70). Ab initio calculations on three electronic states of CS3 at the 6-31G -l-ZPVE level indicated that the C21, form (68) was more stable than the carbon disulfide 5-sulfide form (70) in the neutral (both singlet and triplet states) and the anion radical states, but 68 was less stable than 70 in the radical cation state. 

Fluorescence and phosphorescence spectra of poly(propynoic acid)(FPA), polyphenylene (PP), and DPAcN show that the difference of energies between the lower excited singlet and triplet states, as observed in the case of PP (583 nm) and DPAcN (528 nm), is considerably greater than that of poly(propynoic acid) (270—300 nm) which besides transitions may undergo rr - transitions. PCSs showing only... 

Since two electrons with symmetric space wavefunctions and antisymmetric space wavefunctions represent singlet and triplet states respectively, then obviously the triplet state (E ) is of lower energy than the singlet state E+) by an amount Had an attractive force... 

Jablonski (48-49) developed a theory in 1935 in which he presented the now standard Jablonski diagram" of singlet and triplet state energy levels that is used to explain excitation and emission processes in luminescence. He also related the fluorescence lifetimes of the perpendicular and parallel polarization components of emission to the fluorophore emission lifetime and rate of rotation. In the same year, Szymanowski (50) measured apparent lifetimes for the perpendicular and parallel polarization components of fluorescein in viscous solutions with a phase fluorometer. It was shown later by Spencer and Weber (51) that phase shift methods do not give correct values for polarized lifetimes because the theory does not include the dependence on modulation frequency. 

In 1982 the present author discovered cyclic orbital interactions in acyclic conjugation, and showed that the orbital phase continuity controls acyclic systems as well as the cyclic systems [23]. The orbital phase theory has thus far expanded and is still expanding the scope of its applications. Among some typical examples are included relative stabilities of cross vs linear polyenes and conjugated diradicals in the singlet and triplet states, spin preference of diradicals, regioselectivities, conformational stabilities, acute coordination angle in metal complexes, and so on. 

S-T gap Energy gap between the lowest singlet and triplet states... 

Fig. 1 A schematic illustration of the in-phase and out-of-phase combinations of the atomic orbitals into the bonding and antibonding molecular orbitals, respectively. The dissociation limit of a H molecule corresponds to a pure diradical with degenerate singlet and triplet states...

The delocalization of excessive a- (or P-) spins and the bond polarization can take place among radical orbitals, p and q, and the central n (or o) and n (or o ) orbitals, resulting in the electron transferred configurations (T) and locally excited configurations (E), respectively (Fig. 5a). The delocalization-polarization mechanisms are different between singlet and triplet states, as addressed in the following subsections. 

It is interesting that the sign of the left side of inequality (11) is opposite to that of inequality (6) for the triplet state. That means the phase continuity properties of the singlet and triplet states of a given diradical are opposite to each other. It should be... 

Like TME, the diradical 15 was shown to have nearly degenerate singlet and triplet states by magnetic susceptibility [60, 61], although the early works by Dowd identified a triplet ground state on the basis of ESR spectrum [62, 63], The UCCSD(T) calculations predicted a singlet ground state with a small S-T gap of... 

Such an orbital phase picture in Fig. 14 is also applicable to rationalize the relative S-T gaps of hetero diradicals 19 and 20. hi comparison with their parent system, 1,3-dimethylenecyclobutadiene (DMCBD, 10), the introduction of oxygen atoms does destabilize the triplet state. The calculated energy gap between singlet and triplet states, AE deaeases in the order 10 (18.2 kcal moF ) > 19 (7.7 kcal moF ) > 20 (-20.7 kcal moF ) [64]. These results supported the orbital phase predictions. 

Table 2 Energy differences between the lowest singlet and triplet states of the trimethylene-based 1,3-diradicals calculated by (6,6)CASSCF and (6,6) CAS-MP2 methods with the 6-3IG basis sets...

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