Triplet function

R12R21 ] Evaluating this sum for each of the three symmetries E = Ai, A2, and E, one obtains values of 0, 2, and 0, respectively. That is, the projection of the each of the original triplet determinants gives zero except for A2 symmetry. This allows one to conclude that there are no Ai or E triplet functions in this case the triplet functions are of pure A2 symmetry. 

To make any progress, a relation between the pair and triplet function must be specified. Unfortunately, no accurate relation exists. Kirkwood [8,9]... 

The singlet function corresponds to zero total electron spin angular momentum, S = 0 the triplet functions correspond to S = 1. Operating on these functions with the spin Hamiltonian, we get ... 

One sees immediately that the singlet functions have space parts that are symmetrical and spin parts that are antisymmetrical with respect to interchange of the electron coordinates, while the converse is true for the triplet functions. Since the Hamilton operator H[. .. n) operates only on space parts which for and are identical, it is apparent that these three functions... 

Fig. 6. A) Orbital representation B) Electron densities q ) C) Electron pair densities Q ri, ) of the singlet and triplet functions of two electrons in a one-dimensional box. The two axes are n and j-2...

Fig. 8. Block form of the triplet configuration interaction problem involving 25 singly excited configurations Ag 6 Big 6 B2n 8 B3a 5). The symmetry classification refers to the space part of the triplet functions...

Fig. 10. Types of configurations to be considered in the calculation of Tm - T spectra, a) Excitation number with respect to the ground state A. b) Excitation number with respect to the main component B of the lowest triplet state Ti. c) Number of triplet functions with a specific Ms value...

For simpUcity, let us return to the true isoenergetic triplet functions given by Eq. (23) involving just the two electrons outside the closed-shell part. 

First-order perturbation theory is then apphed to derive the nominal "singlet ground state and first excited triplet functions. Pure spin states are no longer possible. 

The calculated state energies, the transition moments, and the symmetry classification are given in Table 3. The symmetry species of the triplet functions is obtained by taking the direct product of irreducible representation of the space and the spin functions Fx, Fy, Fz, which transform as the rotations Rx, Ry, and Rz-... 

Turning first to the three triplet functions, one notes that the effect of the symmetry projector acting on each of these three was the following multiple of the respective function Sr Xr(R) [R11R22... 

These four wave functions, or linear combinations of them, represent the lowest excited orbital states for the molecule. They can be used both as spin-singlet and spin-triplet functions since the single orbitals are different. 

The spatial parts of the a, b XA states can couple to X 3Bi via the y component of the spin-orbit operator. The operator Sy couples the singlet spin function So (Ai) to the Bi triplet function. 

The configurational functions of all three components of the triplet state are listed on the three lines of Equation (1.17). From top to bottom, they correspond to the occupation of the MOs 0, and 0, with two electrons with an a spin, with one electron each with a and jS spin, and with two electrons with a p spin. The z component of the total spin is equal to M5 = 1,0, and — 1, respectively. The three triplet functions are degenerate (i.e., have the same energy) in the absence of external fields and ignoring relativistic effects (i.e., with a spin-free Hamiltonian). For our purposes, it is therefore sufficient to consider only one of the components, e.g., the one corresponding to Ms = 0. 

For an evaluation of the matrix element < (n,.n ) soP(- )> of Ihe operator given in Equation (1.42), it is convenient to first determine the effect of the spin part of the three components t Ij, and /x of the scalar product t Son the triplet function and to perform the spin integration, which reduces the number of terms considerably. In this way one obtains the following ... 

For Q spin-independent — which will usually be the case — the xpp will be automatically either singlet or triplet functions. The singlet and triplet blocks can be diagonalized independently. 

Triple bond, bending, 205 Triplet, 12, 14, 180 impure, 28. 220-21, 227 Triplet energy of sensitizer, 372, 407 Triplet function, 222 Triplet state, aromatics, 76 calculation, 55 ethylene, 64... 

The SOC operator contains the product of a term that acts on the spin part of a wavefunction, converting a triplet function to a singlet function and a term that acts on the space part of the wavefunction, changing one electronic configuration to another. Each component of the SOC vector, < So /z so T >, u = x, y or z, is a sum over all atoms and each atom contributes a sum over pairs of all basis set orbitals in the molecule. The terms in the sum are multiplied by numerical coefficients obtained from the Cl expansion and from the coefficients of the valence (primarily p-) orbitals in the two singly occupied MOs ofTj. The dominant contributors to the sum are those in which both basis set orbitals are located on the same atom ( one-centre terms ). Each term reflects the degree to which a 90° rotation around the axis u through the atom converts one member of the orbital pair into the other. If the space part of the wavefunctions differs only by the occupation number of two MOs ij/i and ij/j, we need to consider only the three matrix elements <... 

The exchange integral J is one-half the vertical separation of the singlet and triplet functions. The inset shows the splitting of the triplet levels in a magnetic field. Further explanation, see text. 

The use of group theory in the evaluation of SOC in molecules [6] in general and in biradicaloids [55] in particular has been common. The overall symmetry of the three possible triplet functions can be derived by considering the space part of Tj and the three possible spin parts 0, 0, 0, which transform like the rotations R, Ry and R. Only those of the three triplet functions that belong to the same irreducible representation as Sq can be mixed with it by the action of the totally symmetric spin-orbit coupling operators. For others, SOC vanishes. 

The dissociation energy of O2 is 554 KJmoF for the triplet and 398 KJrnol for the singlet dioxygen with the DFT approximation (the experimental value is 493.7 KJmol ). The calculated gas phase AE for the reaction 2NO2 -I- O2 = 2NO3 is -402.2 KJmol using a spin-unpolarized functional, rising to -246.2 KJmol when a spin-polarized (triplet) functional is used. 

The upper and lower signs refer to the triplet and singlet states, respectively. One can verify by inspection that both functions are antisymmetric to electron interchange in the singlet state the spatial part is symmetric and the spin part is antisymmetric. The remaining triplet functions are obtained more simply. 

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