Hydrogen atom observables

Our next task is to combine the wavelike properties of electrons with the nuclear model of the atom and hope to explain the strange ladder of energy levels in a hydrogen atom observed experimentally. In doing so, we shall greatly enrich our model of the internal structure of an atom. 

An aliphatic hydrocarbon has a formula C H2 + 2. Each ring or unsaturation that is present in a hydrocarbon decreases the number of hydrogen atoms by two units. Let x be the number of hydrogen atoms observed then, if N, is the number of rings and unsaturations, we have... 

Fig. 3.1. The probability of finding an electron with absolute momentum p in a hydrogen atom, observed by measuring the complete kinematics of ionisation events at the total energies shown (Lohmann and Weigold, 1981). The curve shows the square of the momentum-space wave function.

Figure Bl.4.9. Top rotation-tunnelling hyperfine structure in one of the flipping inodes of (020)3 near 3 THz. The small splittings seen in the Q-branch transitions are induced by the bound-free hydrogen atom tiiimelling by the water monomers. Bottom the low-frequency torsional mode structure of the water duner spectrum, includmg a detailed comparison of theoretical calculations of the dynamics with those observed experimentally [ ]. The symbols next to the arrows depict the parallel (A k= 0) versus perpendicular (A = 1) nature of the selection rules in the pseudorotation manifold.

This discussion may well leave one wondering what role reality plays in computation chemistry. Only some things are known exactly. For example, the quantum mechanical description of the hydrogen atom matches the observed spectrum as accurately as any experiment ever done. If an approximation is used, one must ask how accurate an answer should be. Computations of the energetics of molecules and reactions often attempt to attain what is called chemical accuracy, meaning an error of less than about 1 kcal/mol. This is suf-hcient to describe van der Waals interactions, the weakest interaction considered to affect most chemistry. Most chemists have no use for answers more accurate than this. 

Figure 1.1 Energy levels (vertical lines) and observed transitions (horizontal lines) of the hydrogen atom, including the Lyman, Balmer, Paschen, Brackett and Pfund series...

The explanation of the hydrogen atom spectmm and the photoelectric effect, together with other anomalous observations such as the behaviour of the molar heat capacity Q of a solid at temperatures close to 0 K and the frequency distribution of black body radiation, originated with Planck. In 1900 he proposed that the microscopic oscillators, of which a black body is made up, have an oscillation frequency v related to the energy E of the emitted radiation by... 

Whereas the emission spectrum of the hydrogen atom shows only one series, the Balmer series (see Figure 1.1), in the visible region the alkali metals show at least three. The spectra can be excited in a discharge lamp containing a sample of the appropriate metal. One series was called the principal series because it could also be observed in absorption through a column of the vapour. The other two were called sharp and diffuse because of their general appearance. A part of a fourth series, called the fundamental series, can sometimes be observed. 

Liquid ethylene oxide under adiabatic conditions requires about 200°C before a self-heating rate of 0.02°C/min is observed (190,191). However, in the presence of contaminants such as acids and bases, or reactants possessing a labile hydrogen atom, the self-heating temperature can be much lower (190). In large containers, mnaway reaction can occur from ambient temperature, and destmctive explosions may occur (268,269). 

The mass spectrum of 2-pyrone shows an abundant molecular ion and a very prominent ion due to loss of CO and formation of the furan radical cation. Loss of CO from 4-pyrone, on the other hand, is almost negligible, and the retro-Diels-Alder fragmentation pathway dominates. In alkyl-substituted 2-pyrones loss of CO is followed by loss of a hydrogen atom from the alkyl substituent and ring expansion of the resultant cation to the very stable pyrylium cation. Similar trends are observed with the benzo analogues of the pyrones, although in some cases both modes of fragmentation are observed. Thus, coumarins. 

The acidic character of the hydrogen atoms of C-methyl groups linked to the pyrazolium ring (Figure 22 Section 4.04.2.1.1(11)) facilitates a number of reactions difficult to carry out with neutral pyrazoles. Since efficient methods of dealkylation have been described (Section 4.04.2.3.lO(ii)), the synthesis via the pyrazolium salt is a useful alternative. The same behaviour is observed for indazolium salts, for example, nucleophilic addition to aromatic aldehydes (78JOC1233). 

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