By PROFESSOR J. E. LENNARD-JONES, The University of Bristol.
Transactions of the Faraday Society
Volume 25, p. 668-686 (1929)
Received 5th September, 1929.
A knowledge of the electronic structure of molecules is of the greatest importance, both to the physicist and the chemist ; to the physicist because the facts of molecular spectra can only be interpreted in terms of electronic structure, to the chemist because a detailed knowledge of electronic structure is necessary to indicate the exact conditions (of excitation, etc.), under which atoms will combine to form molecules. A number of methods, both experimental and theoretical, have recently been converging to a solution of this problem. On the experimental side there is the important work of Franck1 and his collaborators, who have shown how a molecule splits up by light absorption into its component atoms (or ions). This work shows that in most cases a normal molecule splits up (adiabatically) into two unexcited atoms or ions. This work must now be supplemented by the important results of Herzberg,2 who has shown that in some cases a molecule in its normal state splits up into one normal and one excited component. In these cases it is an excited molecule which leads on dissociation to two normal components. The converse must be true, so that, for example, an excited C atom and a normal N atom are necessary to produce a normal CN molecule.
The recent advances on the theoretical side are the direct outcome of the new quantum mechanics, as the older theory was incapable of dealing with the simplest molecules. The first new and important concept was Heisenberg's idea of resonance in atoms.3 This led, as is well known, to a satisfying explanation of the ortho- and para-system of helium, and to a general explanation of multiplet spectra. Of the possible spectral terms, which can arise from a given outer group of electrons, the one of
1 Franck, Trans. Farad. Soc., 21, 536 (1925) ; Franck, Kuhn, and Rolefson, Z. Physik, 43, 155 (1927) ; H. Sponer, Ergebnisse d. exakt. Naturwissen., 6, 75 (1927).
2 Herzberg, Ann. Physik, 86, 189 (1928.); Heitler and Herzberg, Z. Physik, 53, 52 (1929).
3 Heisenberg, Z. Physik, 38, 411 ; 39, 499 (1926) ; 41, 239 (1927)
greatest multiplicity has the lowest energy and of the terms with the same multiplicity the one with the greatest "l" value is lowest. The same result is doubtless true also of molecular terms.
The natural extension of the resonance principle of Heisenberg to an "Austausch" principle was made by Heitler and London.4 Although the underlying ideas are somewhat different, the mathematical analysis is practically identical. It was shown that the exchange between the electrons of two hydrogen atoms led to an energy lower than that of the separated atoms ; in other words, it led to the formation of a molecule. This important result has since been extended by Heitler and London.5 The latter has shown that the new theory gives a quantum mechanical explanation of the empirical ideas of Lewis and others concerning chemical valency and correlates the idea of valency with that of electron spin. Although the general conditions under which binding can occur in a quantum mechanical sense, have now been well developed, only one case has been worked out quantitatively, viz., that of the hydrogen molecule. Certain difficulties have appeared in more complicated cases, as, for instance, in O2 and F2, where the theory is unable to account for molecular binding, and at the same time to account for the 3S ground level of O2. We shall refer to this point later and shall suggest a possible explanation. Of the possible molecular states which can arise from atoms in given atomic states, the theory as developed at present indicates that the one of lowest multiplicity has the lowest energy, though this is not always true (as in O2, for example). The theory is not yet able to say what is the relative energy of terms of like multiplicity.
An exhaustive treatment of the relation between atomic spectral terms and the molecular terms which are possible on bringing two atoms together has been made by Wigner and Witmer.6 This theory gives all the molecular terms which are theoretically possible to satisfy the requirements of group theory, but it gives no indication which of the possible terms correspond to stable molecular states and which to unstable. Thus, two F atoms in 2P states could give rise to one D molecular state, two P states and three S states, and each could be a singlet or a triplet. The theory gives no idea which of them actually occur or what is the relative depth of the energy levels.
The most important work on the problem of the detailed electronic structure of molecules has been done by Hund7 and Mulliken,8 who have ascribed to each electron in a molecule a definite and unique quantum notation. In the first place, each electron is regarded as having a definite quantised angular momentum about the molecular axis. When this is zero, the electron is described as a s electron ; similarly, when it has the values, 1, 2, . . . , it is said to be a p, d.... electron, in analogy with the s, p, d. . . . notation in atoms. In the second place, the electron is given those atomic quantum numbers which it would have, if the atoms of the molecule were pushed together to form one united atom. This defines each electron uniquely. Thus, we may have 2ss, 2ps, 2pp. . . . electrons. The Pauli Principle is assumed to hold in molecules as in atoms, so that it is assumed that there are not more than two electrons
4 Heitler and London, Z. Physik, 44, 455 (1927).
5 Heitler, Z. Physik, 46, 47, 47, 835, 51, 805 (1928); London, Z. Physik, 46, 455, 50, 24 (11928)
6 Wigner and Witmer, Z. Physik, 51, 859 (1928).
7 Hund, Z. Physik, 51, 759 (1928).
8 Mulliken, Physical Rev., 32, 186 (1928).
in the same s-state, but as a p-electron may have an angular momentum of +1 or - 1, there can be four, and not more than four, electrons in the same p-state. The electronic structure of a molecule may then be given unambiguously, as, for example, 1ss2, 2ps2, 2ss2, 2pp3, 3ps, the upper index indicating in each case the number of "equivalent" electrons.
The next problem considered by Hund7 is the relative energies of these various electron levels. This is known, of course, in the limiting case when the internuclear distance is zero and molecular states become atomic states. Thus, we have the series 1s2, 2s2, 2p2, 3s2, . . . When the nuclei are pulled slightly apart, Hund gives reasons for supposing that the series of corresponding molecular states is 1ss2, 2ss2, 2ps2, 2pp4, 3ss2, . . . . Again, the energy levels of another limiting case is known, viz., that in which the component atoms are infinitely distant. By considering the splitting of the degenerate states which the work of Heitler and London indicates, another series of energy levels can be written down. For the intermediate stages which actual molecules occupy, a series of energy levels is assumed by interpolation, and is adjusted so as to fit in with many known facts and to give a smooth transition from the one extreme limiting case to the other.
The character of a molecular term is determined in the first place by the resultant angular momentum of all the electrons about the nuclear axis. When this is zero, the term is described as a S term; similarly, when it is one, two, the terms are denotedby P, D, . . ., in analogy with the S, P, D, notation of atoms. The multiplicity is denoted as usual by an index on the left-hand side, as in 2P, 3S, and so on. A closed molecular group, whether s or p, must give a 1S term, as the resultant angular momentum and the resultant electron must both be zero.
Both Hund and Mulliken have found it necessary to supplement the table of energy levels by certain empirical rules. The former uses a working hypothesis to limit the intersections of the possible transitions from a given state of the separated atoms to the corresponding states of the united atom. The latter uses the hypothesis that the number of s, p . . . . electrons remains constant during possible transitions. Both methods have met with considerable success, and have led to a better understanding of the structure of the more complicated diatomic molecules, like N2. Further, they have obtained analogous electronic structures for certain series of diatomic molecules with a like number of electrons and are thus able to explain why these molecules should have similar physical properties, a well-known but otherwise unexplained fact.
There are, however, certain difficulties in the theories of Hund and Mulliken, as has been pointed out by Herzberg.9 In particular, they are not capable of explaining the experimental results obtained by Herzberg,10 and interpreted by Heitler and Herzberg,11 viz., that N2+ in the normal state, splits up into a normal N atom and an excited N ion. Similar results hold also in the cases of CN, CO+, BO and SiN. Herzberg9 has proposed a modification of the Hund-Mulliken scheme to bring it more in line with the Heitler and London ideas, and has used as a working hypothesis that each electron in the transition from atom to molecule
9 Z. Physik, to appear shortly. I am greatly obliged to Dr. Herzberg for letting me see an advance copy of his paper and for many interesting discussions of the points at issue.
10 Herzberg, Ann. Physik, 86, 189, 1928.
11 Heitler and Herzberg, Z. Physik, 53, 52, 1929.
retains its own characteristic quantum numbers (in the Hund-Mulliken notation).
It seems to the writer that there is a great disadvantage in relating the notation of electrons in molecules to a condition which cannot be realised. It is more important to know what happens to a molecule on dissociation than to know what happens when it is compressed to form a single atom. Further, the Hund-Mulliken notation would become somewhat complicated if extended to heavy atoms for there is a progressive increase in the quantum numbers required. Even for atoms with principal quantum number 2, it is necessary to use the symbols 3dp and 4ps for the molecular states, and such states are not readily interpreted. Further, the notation gives no indication as to whether the various electronic states in a molecule correspond to binding or repulsion. These difficulties are due to the attempt to preserve the Pauli Principle in molecules so as to have not more than two electrons with given quantum numbers. There are thus only two electrons in each molecule with principal quantum 1. There must, however, be a limit for practical purposes to such an application of the Pauli Principle, otherwise it would be necessary to give a different notation for every pair of K electrons in a solid (say, a lump of lead). It seems more reasonable and practicable in heavy molecules to allot two K electrons to each nucleus. We shall regard the Pauli Principle as fulfilled when we say that two K electrons " belong" to a nucleus A and two to a nucleus B. Similarly, there must be a stage when it is more convenient to assign a complete group of L electrons to each nucleus. A distinction is therefore drawn in this paper between atomic levels and molecular levels, even in a molecule. The latter are those which arise from an "Austausch" phenomenon in the Heitler and London sense, and give binding. There is also a set of levels, which give rise to repulsion, the corresponding eigenfunctions having a certain asymmetry not possessed by the " binding" or "molecular" levels. The main purpose of this paper is to show that these levels should not be considered in the process of building molecules in their normal states.
An "Aufbau" Principle is adopted which assigns the electrons one by one to a molecule, and places them in the lowest available molecular and atomic levels subject to certain rules which have to be observed in order not to conflict with the Heitler-London requirements.
The assigning of electrons to molecular levels leads to the idea of resonance between molecular states when two electrons are in different molecular levels. In such a case, that state is held to be lowest which has the greatest multiplicity, as is the case in atoms. Examples occur in Bo2, [sic] and O2, which should both be 3S states in the normal level, and therefore paramagnetic, while the lowest state of N2 is a 1S state, and is therefore diamagnetic.
In the older quantum theory, one was guided in the assignment of quantum numbers to electrons in atoms by the quantum numbers necessary to specify the different orbits in the simple hydrogen atom. Owing to the effect of the core, the energy levels of electrons in the outer shells were different from those of hydrogen, but nevertheless "effective" quantum numbers were deduced for the various levels, and it was in general easy to infer the correct "true" quantum numbers. Except in
the more complicated atoms, it was possible to use an Aufbau Principle, whereby the electrons were added one at a time to the nucleus, and were placed in the orbits corresponding to the progressive changes in the hydrogen quantum numbers. The Bohr (or, later, the Stoner) subgroups were successively filled by electrons. It was found permissible to assume as a first approximation that the energy of electrons in outer incomplete groups was not seriously modified by the other outer electrons, that is, electrons with the same n, but different l.
As a second approximation, it was possible to combine the "l" (or angular momentum) values of incompleted groups vectorially to give different effective or resultant L values. This resultant L determined the spectral type, which was denoted by S, P, D . . . according as it had the values 0, 1, 2. . . .
Next, the coupling between spin momenta of the electrons gave a "resultant" spin vector S, which determined the multiplicity. The multiplicity was given by R = 2S + 1, if S was measured in units of h/2p. Finally, the components of a multiplet were regarded as determined by the coupling between the resultant L vector and the resultant S vector.
These vectors methods have proved not to be of general applicability, but the results obtained thereby are in most cases confirmed by the more well-founded "resonance" methods of Heisenberg.3 In particular, both lead to the same relative energy values of the various multiplet terms, which can arise from a given number of outer electrons with prescribed n and l values.
We seek a similar and analogous method for molecule-building. First we consider the possible energy levels of one electron in the presence of a two-nuclear system. An accurate analytical solution of the two centre-problern is not yet possible, but certain approximate methods can now be used to indicate the relative position of the various energy levels -- in each case as a function of internuclear distance. We use this system of energy levels for the many-electron problem just as was done in atoms. We then suppose a molecule built up in the following way. We add one charge at a time to each nucleus and then, supposing the nuclei held fixed, add to the system two electrons successively. The system is then allowed to take up its equilibrium value adiabatically.
Next, we add the components of the angular momenta about the nuclear axis. This determines the spectral type, S, P, or D, etc. Then as in atoms, we add the electron spins to determine the multiplicity.
In most molecules, the electronic energy is much greater than that due to vibration and so, when the nuclei are no longer held fixed, a series of energy levels arise due to the same electron configuration but different quantised vibrations. We assume that the vibration does not seriously modify the electron configuration. Again, when rotation takes place, it may in general be assumed that neither the electronic configuration nor the vibration energy are greatly perturbed. This is not true of the light molecule He2, where an uncoupling of the "l" values from the nuclear axis takes place when the rotation is large.12 We shall not enter this problem, but shall mostly confine ourselves to ground levels or to excited states of the heavier molecules.
12 Weizel, Z. Physik, 52, 175 (1929) ; 56, 727 (1929).
May 30, 2002 - to be continued.