Hydrogen atom normal state

In 1971 a paper was published that applied the normalized variation function N exp(—brV o cr/ao) to the hydrogen atom and stated that minimization of the variational integral with respect to the parameters b and c yielded an energy 0.7% above the true ground-state energy for infinite nuclear mass. Without doing any calculations, state why this result must be in error. 

The quantity p2 as a function of the coordinates is interpreted as the probability of the corresponding microscopic state of the system in this case the probability that the electron occupies a certain position relative to the nucleus. It is seen from equation 6 that in the normal state the hydrogen atom is spherically symmetrical, for p1M is a function of r alone. The atom is furthermore not bounded, but extends to infinity the major portion is, however, within a radius of about 2a0 or lA. In figure 3 are represented the eigenfunction pm, the average electron density p = p]m and the radial electron distribution D = 4ir r p for the normal state of the hydrogen atom. 

As an example we may calculate the energy of the helium atom in its normal state (24). Neglecting the interaction of the two electrons, each electron is in a hydrogen-like orbit, represented by equation 6 the eigenfunction of the whole atom is then lt, (1) (2), where (1) and (2) signify the first and the second electron. 

By bringing the nuclei into coincidence a helium atom in the normal state is formed and a value for its energy can be obtained from the expression for the hydrogen molecule by neglecting the internuclear energy and by putting p = 0. It is found that Wb. 19... 

Excited states of the hydrogen molecule may be formed from a normal hydrogen atom and a hydrogen atom in various excited states.2 For these the interelectronic interaction will be small, and the Burrau eigenfunction will represent the molecule in part with considerable accuracy. The properties of the molecule, in particular the equilibrium distance, should then approximate those of the molecule-ion for the molecule will be essentially a molecule-ion with an added electron in an outer orbit. This is observed in general the equilibrium distances for all known excited states but one (the second state in table 1) deviate by less than 10 per cent from that for the molecule-ion. It is hence probable that states 3,4, 5, and 6 are formed from a normal and an excited atom with n = 2, and that higher states are similarly formed. 

The interaction of two alkali metal atoms is to be expected to be similar to that of two hydrogen atoms, for the completed shells of the ions will produce forces similar to the van der Waals forces of a rare gas. The two valence electrons, combined symmetrically, will then be shared between the two ions, the resonance phenomenon producing a molecule-forming attractive force. This is, in fact, observed in band spectra. The normal state of the Na2 molecule, for example, has an energy of dissociation of 1 v.e. (44). The first two excited states are similar, as is to be expected they have dissociation energies of 1.25 and 0.6 v.e. respectively. 

Each of these tautomers in its normal state is represented not by the conventional valence-bond structure shown above, but by a resonance hybrid of this structure and others. For tautomer A, with the hydrogen atom attached to the nitrogen atom 1, the principal resonance is between structures A I and A II, with A I the more important smaller contributions are made also by other structures such as A III. Similar resonance occurs for tautomer B. Thus for both tautomers the principal resonance... 

At this point let us briefly consider the relationship between the carbonyl triplet state and another system capable of hydrogen atom abstraction alkoxy radicals. A comparison of the differences and/or similarities between the reactivity of the carbonyl triplet and that of an alkoxy radical should indicate whether the triplet state behaves as a normal ground state radical or if electronic excitation imparts unique properties leading to reactions not characteristic of ground state radicals. 

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