Spin wavefunctions

Electrons and most other fiindamental particles have two distinct spin wavefunctions that are degenerate in the absence of an external magnetic field. Associated with these are two abstract states which are eigenfiinctions of the intrinsic spin angular momentum operator S... 

In spin relaxation theory (see, e.g., Zweers and Brom[1977]) this quantity is equal to the correlation time of two-level Zeeman system (r,). The states A and E have total spins of protons f and 2, respectively. The diagram of Zeeman splitting of the lowest tunneling AE octet n = 0 is shown in fig. 51. Since the spin wavefunction belongs to the same symmetry group as that of the hindered rotation, the spin and rotational states are fully correlated, and the transitions observed in the NMR spectra Am = + 1 and Am = 2 include, aside from the Zeeman frequencies, sidebands shifted by A. The special technique of dipole-dipole driven low-field NMR in the time and frequency domain [Weitenkamp et al. 1983 Clough et al. 1985] has allowed one to detect these sidebands directly. 

The spin Hamiltonian operates only on spin wavefunctions, and all details of the electronic wavefunction are absorbed into the coupling constant a. If we treat the Fermi contact term as a perturbation on the wavefunction theR use of standard perturbation theory gives a first-order energy... 

It is possible to construct a HF method for open-shell molecules that does maintain the proper spin symmetry. It is known as the restricted open-shell HF (ROHF) method. Rather than dividing the electrons into spin-up and spin-down classes, the ROHF method partitions the electrons into closed- and open-shell. In the easiest case of the high-spin wavefunction ( op = — electrons in op... 

Before going on to calculate the energy levels it is necessary to digress and briefly describe the wavefunction. The spin Hamiltonian only operates on the spin part of the wavefunction. Every unpaired electron has a spin vector /S = with spin quantum numbers ms = + and mB = — f. The wavefunctions for these two spin states are denoted by ae) and d ), respectively. The proton likewise has I = with spin wavefunctions an) and dn)- In the present example these will be used as the basis functions in our calculation of energy levels, although it is sometimes convenient to use a linear combination of these spin states. 

Our task is now to write out the spin Hamiltonian Hs, to calculate all the energy-matrix elements in Equation 7.11 using the spin wavefunctions of Equation 7.14 and the definitions in Equations 7.15-7.17, and to diagonalize the complete E matrix to get the energies and the intensities of the transitions. We will now look at a few examples of increasing complexity to obtain energies and resonance conditions, and we defer a look at intensities to the next chapter. 

These coefficients (Equation 7.30) are required to calculate the transition probability or spectral amplitude (cf. Chapter 8). Note that for systems with more than two spin wavefunctions (S > 1/2) the energy eigenvalue problem is usually not solvable analytically (unless the matrix can be reduced to one of lower dimensionality because it has sufficient off-diagonal elements equal to zero) and numerical diagonalization is the only option. 

Thus far in this chapter we have considered single-spin systems only. The zero-field interaction that we worked out in considerable detail was understood to describe interaction between unpaired electrons localized all on a single paramagnetic site with spin S and with associated spin wavefunctions defined in terms of its m5-values, that is, (j) = I ms) or a linear combination of these. However, many systems of potential interest are defined by two or more different spins (cf. Figure 5.2). By means of two relatively simple examples we will now illustrate how to deal with these systems in situations where the strength of the interaction between two spins is comparable to the Zeeman interaction of at least one of them S Sh B Sa. 

The spin wavefunctions are compounded one partrefers to the electron spinandanother part to the nuclear spin, Ims m) (an alternative name is product wavefunctions) ... 

In this chapter we continue our journey into the quantum mechanics of paramagnetic molecules, while increasing our focus on aspects of relevance to biological systems. For each and every system of whatever complexity and symmetry (or the lack of it) we can, in principle, write out the appropriate spin Hamiltonian and the associated (simple or compounded) spin wavefunctions. Subsequently, we can always deduce the full energy matrix, and we can numerically diagonalize this matrix to obtain the stable energy levels of the system (and therefore all the resonance conditions), and also the coefficients of the new basis set (linear combinations of the original spin wavefunctions), which in turn can be used to calculate the transition probability, and thus the EPR amplitude of all transitions. 

We have seen that a spin Hamiltonian in combination with its associated spin wavefunctions defines an energy matrix, which can always be diagonalized to obtain all the real energy sublevels of the spin manifold. Furthermore, the diagonaliza-tion also affords a new set of spin wavefunctions that are a basis for the diagonal matrix, and which are linear combinations of the initial set of spin functions. The coefficients in these linear combinations can be used to calculate the transition probabilities of all transitions within the spin manifold. 

The spin wavefunctions Ip) and Iq) are those obtained after diagonalization of the complete energy matrix. 

Upon diagonalization, the basis set of spin wavefunctions in Equation 8.28 change into... 

Physically, it means that it is possible to know simultaneously the square of the intensity of the spin angular momentum and its component along z. Since the spin wavefunctions are not eigenfunctions of the operators S or /, it is impossible to... 

In quantum mechanics, spin is described by an operator which acts on a spin wavefunction of the electron. In the present case this operator describes an angular momentum with two possible eigenvalues along a reference axis. The first requirement fixes commutation rules for the spin components, and the second one leads to a representation of the spin operator by 2 x 2 matrices (Pauli matrices [Pau27]). One has... 

Since the total (spatial x spin) wavefunction must be antisymmetric, four acceptable functions can be written from expressions (2.2.41) to (2.2.46) ... 

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... 

It follows that, if there are na and nb SCF-MI active orbitals on fragments A and B, we obtain a total of na nb optimised virtual orbitals. The spin space is described by the spin wavefunction... 

SINGLET AND TRIPLET SPIN WAVEFUNCTIONS FOR FOUR SINGLY-OCCUPIED ACTIVE SPACE ORBITALS... 

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