Triplet wavefunction

For the excited configuration <7 aj 1, there are two states, one singlet and one triplet. This situation is similar to that found in the excited configuration ls12s1 of the helium atom. The singlet and triplet wavefunctions for these excited configurations are ... 

Symmetry dictates that the representations of the direct product of the factors in the integral (3 /T Hso 1 l/s2) transform under the group operations according to the totally symmetric representation, Aj. The spin part of the Hso spin-orbit operator converts triplet spin to singlet spin wavefunctions and singlet functions to triplet wavefunctions. As such, the spin function does not have a bearing on the symmetry properties of Hso- Rather, the control is embedded in the orbital part. The components of the orbital angular momentum, (Lx, Ly, and Lz) of Hso have symmetry properties of rotations about the x, y, and z symmetry axes, Rx, Ry, and Rz. Thus, from Table 2.1, the possible symmetry... 

In order to calculate the matrix elements, we can use the following formulae for the effect of various spin operators on the triplet wavefunctions [18] ... 

The initial impetus which led to the formulation of the Nuclear Magnetic Resonance Triplet Wavefunction Model (NMRTWM)2 came from the... 

How would this picture change when the hydrogen bond is a symmetric, proton-shared hydrogen bond Figure 12 illustrates three types of lower-energy excited-state triplet wavefunctions. Types a and... 

Among the cutting-edge methods and studies reviewed in this decennial volume of the series are the Density Functional Theory DFT method, vibrational electron energy loss spectroscopy EELS), computational models of the reaction rate theory, the nuclear magnetic resonance triplet wavefunction model (NMRTWM) and biological reactions that benefit from computational studies. 

To see that Eq. (34) needs at least to be extended, consider a triplet wavefunction of two electrons. The three degenerate triplet components can be written as combinations of Slater determinants of orbitals a(r) and b(r), which for the triplet case can be assumed orthogonal without loss of generality ... 

The determination of D therefore places an upper limit r h on the extent of the triplet wavefunction. 

Figure 7.8. Simulated fijll- and half-field powder pattern ESR spectra of triplet states for different values of E/D (see text) [59] (a) = 0 the triplet wavefunction is axially symmetric. The pattern contains singularities at // (hv D/2 - /8hv)/gli and steps at D)/gP. (b) 0 E <. D/i the axial symmetry of the wavefunction is broken. The singularities occur at...

The effect of spin-orbit coupling on mixing of singlet and triplet states can be seen by considering first-order non-degenerate perturbation theory [8). Let us consider the perturbed triplet wavefunction for the state where represents the unperturbed triplet wavefunction for the... 

These two states transform according to two different irreducible representations of the point group of the dimer as the spatial part of the triplet wavefunction is antisymmetric with respect to interchange of the two electrons (i.e., halves of the dimer), and that of the singlet is symmetric. The energies of these states are evaluated from the Hamiltonian... 

Returning to other methods for computing the purely electronic contribution to AEst in square CB, we come to those based on single-determinantal UHF wavefunctions. It should be recalled that a single-determinantal wavefunc-tion for the singlet is an approximately 1 1 mixture of singlet and triplet wavefunctions. In fact, (S ) = 1.25 for the singlet UHF wavefunction, which is actually closer to the value for a pure triplet. In contrast, because the triplet UHF wavefunction cannot benefit from spin polarization of the electrons in itj. 

Table 3 shows that, as in the case of square CB, UHF and MBPT methods based on one-configuration UHF wavefunctions for the singlet states do not work well for TMM, which is not surprising in view of the inherent inappropriateness of the zero-order wavefunction. Following projection of the first high-spin contaminants from the singlet and triplet wavefunctions, Bj is erroneously predicted to lie lower than the triplet at both the PUHF and PUMP2 levels of theory. 

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