P. 1. Goodfriend
Texaco Experiment Incorporated Richmond, Virginio
Concepts of Species and State in Chemistry and Molecular Physics
T h e idea of atomic and molecular species is basic to chemistry and forms the theoretical connection between the laws governing physical systems and the existence and behavior of chemical substances with definite properties. Beyond the realm of atomic species there is the idea of "fundamental particle" in terms of which the structure of atoms and molecules are described. These structured atoms and molecules, considered as single species, can exist in a number of states. At times, different states of the same collection of fundamental particles are considered as separate entities, while in other situations the entire set of accessible states is considered to he a single species. I t seems of value to examine the concepts of species and state to see to what extent they are equivalent and to what extent they are different concepts, and to unravel the variations in usage for each. The value of such an examination lies in the insight it gives into the nature of the theory of chemistry and molecular physics. There may also be a practical value, in that previously overlookedaspectsof molecularproblemsmay be brought to light. Since in this discussion we are focusing our attention on atomic and molecular species, we will not treat the problem of the nature of fundamental particles, but will use electrons and nuclei as our fundamental ' c elements." The universe of chemistry and molecular physics is then considered as a collection of nuclei spaced at various distances from each other with different (or the same) masses, charges, and nuclear spins, over which electrons have been poured. Molecular events then unfold as the system approaches the state of minimum potential energy and maximum randomness [see for example THIS JOURNAL 38, 334 (1961)l. Included in this scheme of things are all gases, liquids, solids, and plasmas. A quantum mechanical system is characterized by wave functions which contain all of the information that nature allows one to have about the system. In principle, the wave functions can be found by solving a Scbrodinger equation containing a term for each interaction between the particles which make up the system. Complete solution of such an equation yields a set of wave functions representing the different possible states of the system and their corresponding energies. The separation of nuclei and electrons into groups which are to constitute single species is based upon the magnitude of interactions between the particles. Strictly speaking one must consider the entire universe and, for example, solve the Scbrodinger equation for the system of all particles and all interactions; but because some groups of particles interact more strongly with each other than with all other particles, they may be
considered a separate entity or a set of separate entities. Thus an electron and a proton fairly close to each other and "far" from all other particles is considered a hydrogen atom or one of the states of the hydrogen atom. It should be pointed out that not only stationary states, but also nonstationary states such as "colliding" atoms may be considered as single entities in the time interval during which they are interacting strongly (- one vibration, lo-" sec). This is also true of the activated complex of absolute reaction rate theory (1). An ionic crystal (e.g., NaC1) or a metallic crystal is then a separate entity, no matter how large the crystal, by virtue of the magnitude of the interactions between different parts; whereas a gas approaching ideal behavior is a collection of species. I n other cases hierarchies of interactions introduce the idea of subspecies. Thus a crystal of methane is composed of methane molecules which are distinct n~olecules by virtue of the strong valence forces bet,ween the atoms; yet the crystal is also a species by virtue of the van der Waals forces between the molecules. It was pointed out above that colliding particles may be considered a single entity at the time of the collision. Such an entity is generally unstable in the sense that, as time increases, the interactions vanish. Usually one prefers to consider as entities those collections of particles which are in bound, stationary states and stick together for an extended period of time, or would do so if left alone. Even in such a case the lifetime may be small enough so that one resorts to arbitrary definitions such as "it is a molecule if it lives long enough t o vibrate" (-10-l4 sec). Let us now turn our attention to individual entities in the sense described above. We must consider the fact that this system may exist in a number of states. I n an ideal gas we have the case of a collection of molecules characterized by the fact that their fundamental particles are interacting strongly with each other and almost not at all with the other molecules. In equilibrium at any given temperature the system will have a Boltzmann distribution of energies. The gas considered as a single substance composed of one kind of molecule may then he spoken of in terms of the population of energy levels. If, on the other hand, a molecule in a particular quantum state is considered as a different species from the same collection of fundamental particles in a different quantum state, then the sample of gas may be considered as a mixture of species. Clearly the distinction between these points of vie\. is nebulous and is determined by whether attention is focused upon which fundamental particles are strongly interacting or upon the detailed properties of a collection of fundamental particles. Thus a cylinder of helium is entirely Volume 43, Number 2, February 1966
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composed of helium atoms, but it is He(%&)which is involved in the chemical reaction He(%)
+ 2 He(%)
-
Hes(%)
+ He('So)
I n the case of ortho and para hydrogen, we have another interesting aspect of the relationship between state and species. As a consequence of the Pauli exclusion principle and the existence of nuclear spin, molecular hydrogen states of even rotational quantum number may undergo transitions among themselves, optically or in collisions (as may states of odd quantum number), but intercombinations are forbidden. Thus molecular hydrogen consists of two distinct substances which can be separated and can be interconverted only by breaking apart the molecules or by using special catalysts (8). Here we have two sets of accessible states, such that those in one set are not accessible to those in the other set. The fact that ortho and para hydrogen molecules both consist of bound systems of two protons and two electrons is not enough to specify the properties of macroscopic samples. Information concerning the quantum states of the molecules in the sample is also necessary. Therefore a molecule of hydrogen may be considered as belonging to the set of all possible bound states of two protons and two electrons or as belonging to the set of even or of odd rotational quantum number, or it may be considered an entity in a particular quantum state. As before, the particular situation will dictate the choice of the point of view. Calculating the weight of water obtained by burning a sample of hydrogen needs only the first point of view, a prediction of heat capacity requires the second, and the detailed mechanism of a reaction may require hydrogen molecules in a particular state, e.g., the reaction OH(?Z+)
+ H2('X.+)
-
OH(%)
+ 2H(B)
Up to this point in our discussion no really novel facets of the relation between state and species have been presented. As we turn to considerations of polyatomic molecules, however, we are able to see certain molecular phenomena in an unusual light. Let us consider a set of nuclei and electrons which interact strongly so that a polyatomic molecule results. The exact Schrodinger equation for the stationary states of this system includes terms for the motions and interactions of all the particles. The resulting equation cannot be solved exactly. As a first step, the motion of the nuclei is separated from the motion of the electrons, which can he done because electrons are so much lighter and more mobile than nuclei. This is the adiabatic or Born-Oppenheimer approximation (S,4), resulting in the conventional description of the energy levels of molecules in terms of electronic, vibrational, and rotational levels. Using this starting point, a quantum mechanical treatment of a molecule of interest assumes a configuration for the nuclei and treats the motion of the electrons in the field of the "clamped" nuclei. Consider the case of the following cis and trans isomers X
\
Y
/"
Y /c=c\y Cis iwmer
96
/
x\C=C / /
Trans isomer
Journol o f Chemical Educofion
Using the Born-Oppenheimer approximation and the usual tractable approximations (4), these two molecules are separate quantum mechanical problems. The symmetry properties of the field due to the nuclei are different; the molecules belong to two separate point groups (5). But if the Schrodinger equation is considered without recourse to approximations, it is clear that the wave functions of the cis isomer and those of the trans isomer are all solutions of the same equation. They represent different states of the same system of nuclei and electrons. The ground state of one of the isomers is lower in energy than that of the other; the higher energy isomer is a metastable excited state. Optical isomers may be treated by a similar argument. However, in this case the approximate approach would yield identical wave functions for the dextro- and levoenantiomorphs, but an exact treatment must yield a difference between them since they are different. I n fact they are degenerate states of the same system, i.e., states of the same energy but different wave functions. The validity of this view of isomers can be seen in cases where cis and trans isomers have a common triplet excited state. Such excited states have been assigned as intermediates in cis-trans isomerieation reactions (6). Changes in the geometry of molecules in the course of electronic excitation are common (7). The use of correlation diagrams such as those of Walsh (8) to predict changes in geometry has been very successful. I n any case, the point group of the electronic ground state is not necessarily the point group of even the lowest excited electronic state. What has been pointed out for cis-trans and optical isomers may be extended to all kinds of isomers. The idea of localized chemical bonds, of course, offers distinctions between isomers. The use of this idea (chemical bond) in the definition of molecular species involves separation of the states into classes in a manner somewhat analogous to separation of the states of molecular hydrogen into two classes (ortho and para) and depends upon the low probability of transitions between classes. The nature of chemical bonding between pairs of atoms has been extensively discussed elsewhere (9). On a superficial level the argument may be based on the forces between pairs of nuclei. Nevertheless, the argument holds that isomers are diierent states of the same system. Tautomerism diiers from other forms of isomerism only in the rapidity of the isomerization reaction. If it were a matter of a direct jump of a proton from one position to another, then we would have a "protonic" transition strictly analogous to electronic transitions. The evidence (10) indicates, however, that separation of the proton from the molecule is involved and that a role is played by the other substances present. On the other hand, "tunneling" from one configuration to another may, in a sense, be considered a "protonic" transition, analogous to electronic transitions. In conclusion it might be worthwhile to suggest the examination of problems from more than one side of the species-state dichotomy. Like the postman who, after 20 years of front door deliveries, started delivering mail to back doors, it might open a whole new world for us.
References (1) GLASSTONE, G., LAIDLER,K., AND EYRING,H., "The
Theory of Rate Processes," McGraw-Hill Book Co., New York, 1941. G., "Spectra of Diatomic Molecules," D. Van (2) HERZBERG, Nostrand Company, Princeton, New Jersey, 1950, p. 140. (3) BORN,1\1., AND OPPENAEIMER, J. R., Ann. Physik, 84, 457 119271. ,~ (4) SLATER, J. C., "Quantum Theory of Molecules and Solids," McGraw-Hill Book Co., New York, 1963, "01. 1,p. 9.
( 5 ) HERZBERG, G., "Inirared and Raman Spectra," D. Van Nostrand Co., Princeton, N. J., 1945, Chap. 2.
(6) REID, C., "Excited States in Chemistry and Biology," Butterworths, London, 1957, p. 78. (7) ROBINSON, G. W., in "Methods of Experimental Physics," Vol. 3, "Molecular Physics," Academic Press, New York, 1962, Table VI, p. 208. (8)W a ~ s nA. , D., J. Chem. Sac., 2260 (1953). K., Rev. Mod. Phys., 34,326 (1962). (9) RUEDENBERG, (10) HINE, J., "Physical Organic Chemistry," MeGraw-Hill Book Co., New York, 1956.
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