Singlet excitation operators

The singlet excitation operators play an important role in the second-quantization treatment of molecular electronic structure. More generally, they are known as the generators of the unitary group, satisfying the same commutation relations 

Rank reduction occurs in all cases since the singlet excitation operator is a rank-1 operator. These commutators are proved in Exercise 2.3. 

To gain some familiarity with the singlet excitation operator, we shall consider its effect on a simple two-electron system. By substitution and expansion, the following commutator is seen to hold for the singlet excitation operator and the two-body creation operators  

We consider the excitation of electrons from orbital 0, to an orbital 0p different from tpg. Both orbitals are assumed to be nondegenerate. The initial closed-shell state is given by 

Applying the excitation operator to this state, we arrive at an open-shell singlet state 

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The singlet excitation operator 2 1. + 2f If gives rise to the overlap... 

Singlet excitation operators may be generated by the genealogical coupling of doublets of creation operators a p, aj, and annihilation operators —Opp, Opa] as described in Section 2.6.7. In the spin-orbital basis, the excitation operators Xp, contain the inactive annihilation doublets —a,, fl,a and the virtual creation doublets a , a p). To obtain excitation operators of singlet symmetry, we must, for the active orbitals, include the pairs a p and —Ovp, a . However, for the high-spin... 

Linear independence is ensured by requiring a>b and i > j. The equality is included only for and not for which vanishes for identical indices - in accordance with the fact that, for two identical indices, spin coupling gives only one singlet excitation operator. 

Building on our results for the two-body creation operators in Section 2.3.3, we replace by —cigp, Oga) in (2.3.16)-(2.3.19). We thus obtain the singlet excitation operator... 

Except for a scaling factor, the singlet excitation operator (2.3.21) is identical to the orbital excitation operator in (2.2.7) ... 

The spin-free one- and two-electron operators (2.2.6) and (2.2.15) are thus expressed entirely in terms of the singlet excitation operator and we may, for example, write the electronic Hamiltonian (2.2.18) in the form... 

Evaluating the coefficients and expressing the result in terms of the singlet excitation operators (2.3.21) and (2.3.29), we obtain ... 

J.R. Bolton In solution most photochemical electron transfer reactions occur from the triplet state because in the collision complex there is a spin inhibition for back electron transfer to the ground state of the dye. Electron transfer from the singlet excited state probably occurs in such systems but the back electron transfer is too effective to allow separation of the electron transfer products from the solvent cage. In our linked compound, the quinone cannot get as close to the porphyrin as in a collision complex, yet it is still close enough for electron transfer to occur from the excited singlet state of the porphyrin Now the back electron transfer is inhibited by the distance and molecular structure between the two ends. Our future work will focus on how to design the linking structure to obtain the most favourable operation as a molecular "photodiode . 

The photolysis of the diazobicyclo[2.2.2]heptene derivative (142) was studied at different temperatures and was found to give mixtures of syn (143) and anti (144) products. The experimental data support the homolytic (Xe2) pathway as the prevalent reaction channel at elevated temperatures for the generation of the sterically encumbered syn product, whereas at low temperatures the triplet pathway operates and loss of the syn selectivity is observed. The loss of syn selectivity at low temperatures is due to efficient intersystem crossing in the singlet-excited azoalkane to afford the planar, nitrogen-free triplet diradical which unselectively ring closes. 

A more novel calculation of excitation energies by Truhlar490 uses excitation operator methods and other methods that are described in detail by McKoy and coworkers.491 Comparison with the results obtained by other workers shows that the singlet excited states cannot be adequately represented by a valence-like basis set it is necessary to include diffuse orbitals in the basis, as were used by Hunt et a/.488 487 Finally, we should mention calculations on several molecules, including H2O, using localized orbitals, both SCF and SCF-CI.492... 

For singlet excitations a = —1 andb = 2 for tritlet excitations a = —, b = 0 h, J, and K are, respectively, the one-electron, the Coulomb, and the exchange operators. They have the same meaning as in the usual Hartree-Fock operator. Here Ff1 3 is the N-electron operator with the hole in the ith occupied MO. Thus, Hunt and Goddard have replaced a single Hartree-Fock operator by a whole set of operators [Eq. (28)] differing in the position of the vacancy. The spectrum of each of these operators is an orthonormal set of MOs ... 

Now, the R operator is not only singlet excitations but also triplet, doublet (ionized and electron-attached), and higher-spin multiplicities. Thus, the SAC-CI method can calculate the ground and excited states in various spin-multiplicities. 

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