Nucleus quantum numbers

Complete wave function Electron I at nucleus Electron 2 at nucleus Quantum number of Orientation of spin... 

In principle, every nucleus in a molecule, with spm quantum number /, splits every other resonance in the molecule into 2/ -t 1 equal peaks, i.e. one for each of its allowed values of m. This could make the NMR spectra of most molecules very complex indeed. Fortunately, many simplifications exist. 

The negative sign in equation (b 1.15.26) implies that, unlike the case for electron spins, states with larger magnetic quantum number have smaller energy for g O. In contrast to the g-value in EPR experiments, g is an inlierent property of the nucleus. NMR resonances are not easily detected in paramagnetic systems because of sensitivity problems and increased linewidths caused by the presence of unpaired electron spins. 

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

The hydrogen atom is a three-dimensional problem in which the attractive force of the nucleus has spherical symmetr7. Therefore, it is advantageous to set up and solve the problem in spherical polar coordinates r, 0, and three parts, one a function of r only, one a function of 0 only, and one a function of [Pg.171]

Orbitals are described by specifying their size shape and directional properties Spherically symmetrical ones such as shown m Figure 1 1 are called y orbitals The let ter s IS preceded by the principal quantum number n n = 2 3 etc ) which speci ties the shell and is related to the energy of the orbital An electron m a Is orbital is likely to be found closer to the nucleus is lower m energy and is more strongly held than an electron m a 2s orbital... 

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. 

The nuclei of many isotopes possess an angular momentum, called spin, whose magnitude is described by the spin quantum number / (also called the nuclear spin). This quantity, which is characteristic of the nucleus, may have integral or halfvalues thus / = 0, 5, 1, f,. . . The isotopes C and 0 both have / = 0 hence, they have no magnetic properties. H, C, F, and P are important nuclei having / = 5, whereas and N have / = 1. 

According to quantum mechanics, the maximum observable component of the angular momentum is Ih/lir, where h is Planck s constant. A nucleus can assume only 21+1 energy states. Associated with each of these states is a magnetic quantum number m. where m has the values I, I — I, I —2,, —1+ 1, —I. 

All have zero nuclear spin except (33.8% abundance) which has a nuclear spin quantum number this isotope finds much use in nmr spectroscopy both via direct observation of the Pt resonance and even more by the observation of Pt satellites . Thus, a given nucleus coupled to Pt will be split into a doublet symmetrically placed about the central unsplit resonance arising from those species containing any of the other 5 isotopes of Pt. The relative intensity of the three resonances will be (i X 33.8) 66.2 ( x 33.8), i.e. 1 4 1. 

The energy of any one-electron species in its nth state (n = principal quantum number) is given by E = —BZVn2, where Z is the charge on the nucleus and B is 2.180 X 10 18 J. Find the ionization energy of the Li2+ ion in its first excited state in kilojoules per mole. 

For example, if the first quantum number is 3 the second quantum number can take values of 2, 1, or 0. Each of these values of will generate a number of possible values of mt and each of these values will be multiplied by a factor of two since the fourth quantum number can adopt values of 1/2 or -1/2. As a result there will be a total of 2n2 or 18 electrons in the third shell. This scheme thus explains why there will be a maximum total of 2, 8, 18, 32, etc., electrons in successive shells as one moves further away from the nucleus. 

It has been found possible to evaluate s0 theoretically by means of the following treatment (1) Each electron shell within the atom is idealised as a uniform surface charge of electricity of amount — zte on a sphere whose radius is equal to the average value of the electron-nucleus distance of the electrons in the shell. (2) The motion of the electron under consideration is then determined by the use of the old quantum theory, the azimuthal quantum number being chosen so as to produce the closest approximation to the quantum... 

The first set of screening constants was obtained from the discussion of the motion of an electron in the field of the nucleus and its surrounding electron shells, idealized as electrical charges uniformly distributed over spherical surfaces of suitably chosen radii. This idealization of electron shells was first used by Schrodinger3), and later by Heisenberg4) and Unsold5), who pointed out that it is justified to a considerable extent by the quantum mechanics. The radius of a shell of electrons with principal quantum number nt is taken as... 

As an example, Figure 7-21 shows that the — 3 orbitals of the copper atom have their maximum electron densities at similar distances from the nucleus. The same regularity holds for all other atoms. The quantum numbers other than tt affect orbital size only slightly. We describe these small effects in the context of orbital energies in Chapter 8. 

The quantum number / — 1 corresponds to a p orbital. A p electron can have any of three values for Jitt/, so for each value of tt there are three different p orbitals. The p orbitals, which are not spherical, can be shown in various ways. The most convenient representation shows the three orbitals with identical shapes but pointing in three different directions. Figure 7-22 shows electron contour drawings of the 2p orbitals. Each p orbital has high electron density in one particular direction, perpendicular to the other two orbitals, with the nucleus at the center of the system. The three different orbitals can be represented so that each has its electron density concentrated on both sides of the nucleus along a preferred axis. We can write subscripts on the orbitals to distinguish the three distinct orientations Px, Py, and Pz Each p orbital also has a nodal plane that passes through the nucleus. The nodal plane for the p orbital is the J z plane, for the Py orbital the nodal plane is the X Z plane, and for the Pz orbital it is the Jt plane. 

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