Electron spin quantum numbers

MOs around them - rather as we construct atomic orbitals (AOs) around a single bare nucleus. Electrons are then fed into the MOs in pairs (with the electron spin quantum number = 5) in order of increasing energy using the aufbau principle, just as for atoms (Section 7.1.1), to give the ground configuration of the molecule. 

In non-linear polyatomic molecules the process of deterioration of quantum numbers continues to such an extent that only the total electron spin quantum number S remains. The selection rule... 

In the above derivation, we have made no explicit assumption about the total electron spin quantum number S so that the results should be correct for S = 1 /2 as well as higher values. However, the fine structure term is not usually included in spin Hamiltonians for 5=1/2 systems. The fine structure term can be ignored since in that case the results of operating on a spin-1/2 wave function is always zero ... 

In an atomic term symbol the value of L = 1,2,..., etc. is coded according to spectroscopic notation as 5, P, D,F,G,... etc. (alphabetically for L > 3). Likewise, each atomic state can be characterized by a total electronic spin quantum number S,... 

The valence electron for the cesium atom is in the 6s orbital. In assigning quantum numbers, n = principal energy level = 6. The quantum number l represents the angular momentum (type of orbital) with s orbitals = 0, p orbitals = 1, d orbitals = 2, and so forth. In this case, l = 0. The quantum number m is known as the magnetic quantum number and describes the orientation of the orbital in space. For, v orbitals (as in this case), mt always equals 0. For p orbitals, mt can take on the values of -1, 0, and +1. For d orbitals, can take on the values -2, -1, 0, +1, and +2. The quantum number ms is known as the electron spin quantum number and can take only two values, +1/2 and -1/2, depending on the spin of the electron. 

Singlet and triplet states refer to those states that have a total electron spin quantum number equal to 0 and 1, respectively. The spin multiplicity calculated as 25 - - 1, where S corresponds to the spin angular momentum, corresponds to 1 and 3 for the singlet and triplet states, respectively. 

Fig. 7. NMRD profiles calculated for slightly asymmetric, weakly deformable complexes with different electron spin quantum numbers (a) cylindrically-symmetric ZFS, E = 0 (b) maximum rhombicity E = DjS. Reprinted from J. Magn. Reson. vol. 146, Nilsson, T. Kowalewski, J., Slow-motion theory of nuclear spin relaxation in paramagnetic low-symmetry complexes A generalization to high electron spin , pp. 345-358, Copyright 2000, with permission from Elsevier.

An analytical theory of the outer-sphere PRE for slowly rotating systems with an arbitrary electron spin quantum number S, appropriate at the limit of low field, has been proposed by Kruk et al. (144). The theory deals with the case of axial as well as rhombic static ZFS. In analogy to the inner sphere case (95), the PRE for the low field limit could be expressed in terms of the electron spin spectral densities s ... 

The Florence NMRD program (8) (available at www.postgenomicnmr.net) has been developed to calculate the paramagnetic enhancement to the NMRD profiles due to contact and dipolar nuclear relaxation rate in the slow rotation limit (see Section V.B of Chapter 2). It includes the hyperfine coupling of any rhombicity between electron-spin and metal nuclear-spin, for any metal-nucleus spin quantum number, any electron-spin quantum number and any g tensor anisotropy. In case measurements are available at several temperatures, it includes the possibility to consider an Arrhenius relationship for the electron relaxation time, if the latter is field independent. 

Metal ions like copper(II), oxovanadium(IV), titanium(III), silver(II) have electron-spin quantum number equal to 1/2, like radicals. Therefore, they have no ZFS. These systems can be divided into three classes, according to the spin orbit interaction of the paramagnetic center ... 

Fig. 23. Effect of static ZFS on the NMRD profiles for integer (S = 1) electron spin quantum numbers (panels A, C, and E) and half-integer (S = 3/2) electron spin quantum numbers (panels B, D, and F). Conditions Aj = 0.1 cm, = 5 x 10 s (A) and...

In Eqs. (4)-(7) S is the electron spin quantum number, jh the proton nuclear magnetogyric ratio, g and p the electronic g factor and Bohr magneton, respectively. r//is the distance between the metal ion and the protons of the coordinated water molecules, (Oh and cos the proton and electron Larmor frequencies, respectively, and Xr is the reorientational correlation time. The longitudinal and transverse electron spin relaxation times, Tig and T2g, are frequency dependent according to Eqs. (6) and (7), and characterized by the correlation time of the modulation of the zero-field splitting (x ) and the mean-square zero-field-splitting energy (A. The limits and the approximations inherent to the equations above are discussed in detail in the previous two chapters. 

For a given total electron spin quantum number (S), the multiplicity is the number of possible orientations of the spin angular momentum for the same spatial electronic wavefunction. Thus, the multiphcity equals 25 -F 1. For... 

Any state which has a total electron spin quantum number of zero. 

S is the electron spin quantum number and E = gfiSiH. Nuclear resonance field shifts AH then are observed which are given by the expression... 

The electron spin quantum number, m t dctfcnniijcs the magnetic field generated by the electron and has a value of... 

For atoms with more than one electron, we must take account of a fourth quantum number, ms, the electron spin quantum number, which has only two values, ms = 1/2. An electron has a magnetic moment which can be rationalized by imagining that electrons spin about an... 

For isolated electron spins the magnetic moment, p, is equal to —gHoS, where Po is the Bohr magneton, the electron spin quantum number, S, is equal to %, and g is a constant (known as the g-value). The y-value is characteristic of the... 

A. Mn(II) EPR. The five unpaired 3d electrons and the relatively long electron spin relaxation time of the divalent manganese ion result in readily observable EPR spectra for Mn2+ solutions at room temperature. The Mn2+ (S = 5/2) ion exhibits six possible spin-energy levels when placed in an external magnetic field. These six levels correspond to the six values of the electron spin quantum number, Ms, which has the values 5/2, 3/2, 1/2, -1/2, -3/2 and -5/2. The manganese nucleus has a nuclear spin quantum number of 5/2, which splits each electronic fine structure transition into six components. Under these conditions the selection rules for allowed EPR transitions are AMS = + 1, Amj = 0 (where Ms and mj are the electron and nuclear spin quantum numbers) resulting in 30 allowed transitions. The spin Hamiltonian describing such a system is... 

FJectron correlations are intimately associated with two assumptions (1) a fourth quantum number, the electron-spin quantum number s, and ( ) the Pauli exclusion principle. In order to account for spectral data, it is necessary to postulate that electrons spin about their own axis to create a magnetic moment (G2o). Whereas the magnetic moment associated with the angular momentum may have (2Z + 1) components mi in the direction of an external magnetic field H, the spin moment may have only two components corresponding to s = ms = 1/2. Classically the magnitude of the moment fia associated with an angular momentum p is... 

What do quantum numbers mean As we shall see in Sections 3.5, the three spatial quantum numbers (n, l, m ) for the H atom identify the allowed eigenstates for the solution of the Schrodinger89 equation, with certain characteristic energies and spatial features (e.g., the angular momentum quantum number describes how much angular momentum the atom has). But then, what is the meaning of the "electron spin" quantum number ... 

Electron spin quantum component Electron spin quantum number Hyperhne coupling constant Larmor angular frequency Larmor frequency Magnetogyric ratio Nuclear magneton Nuclear spin quantum component Nuclear spin quantum number Orbital quantum number Orbital quantum number component Principal quantum number Quadrupole moment Relaxation time longitudinal transverse Shielding constant... 

Before we delve further into the properties of the nucleus, let us momentarily shift our attention back to one of the electrons zooming around the nucleus. Just like photons, electrons exhibit both wave and particle properties. Each electron wave in an atom is characterized by four quantum numbers. The first three of these numbers can be taken as the electron s address and describe the energy, shape, and orientation of the volume the electron occupies in the atom. This volume is called an orbital. The fourth quantum number is the electron spin quantum number s, which can assume only two values, or - f. (Why J was selected rather than, say, 1 will be described a little later.) The Pauli exclusion principle tells us that no two electrons in an atom can have exactly the same set of four quantum numbers. Therefore, if two electrons occupy the same orbital (and thus possess the same first three quantum numbers), they must have different spin quantum numbers. Therefore, no orbital can possess more than two electrons, and then only if their spins are paired (opposite). 

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