CHAPTER 1
Calculations of the Vibration-Rotation Spectra of Small Molecules
BY B.T. SUTCLIFFE
1 Introduction
It would, I believe, be widely agreed that the modern theory of molecular spectra began with publication by Carl Eckart in 1935 of his paper Some Studies Concerning Rotating Axes and Polyatomic Molecules. It would probably also be widely agreed that the apogee of this work occurred in 1968 when James K. G. Watson2 published Simplification of the molecular vibration-rotation hamiltonian which put Eckart's classical mechanical form into a proper quantum mechanical one. This leads to the wave mechanical problem for molecular vibration-rotational motion specified by what we shall call the Eckart-Watson Hamiltonian.
This report begins with an account of the theories of molecular spectra that preceded the work of Eckart and the interpretation of spectra that followed his paper during the nineteen forties and fifties. This discussion will involve some consideration of diatomic molecules but they will not subsequently be discussed. So this article is concerned entirely with polyatomic molecules and, in particular those that become linear, only in somewhat excited states. The initial historical discussion, it is hoped, will put the computational work that began in the nineteen seventies into a proper context. During the nineteen eighties and nineties it will be seen that two strands develop in the computational study of molecular spectra. The first is an essentially perturbation theoretic approach, confined almost entirely to the Eckart formulation. The second is a variation theoretic approach which, although sometimes using the Eckart formulation, has found greatest use in formulations using Hamiltonians specifically constructed to describe particular molecules. In referring to such a class of Hamiltonians, they will be called tailor-made.
The aim of this this report is to provide an informative context in which relevant examples of computational work on the spectra of small molecules can be presented in a way that, it is hoped, is balanced, fair and comprehensible to the non-expert reader. It is not aimed to provide a comprehensive survey of the literature, since that can nowadays be done in an effective and timely fashion with the aid of facilities on the Internet. Rather it is aimed to provide representative examples of work so that the reader can gain some feeling for what has been done, what is being done and, perhaps, what might be done.
2 History
In December 1926 the US National Research Council published in its Bulletin a Report of the Committee on Radiation in Gases entitled Molecular Spectra in Gases. The members of the Committee were Edwin C. Kemble, Raymond T. Birge, Walter F. Colby, F. Wheeler Loomis and Leigh Page. The coordinating editor seems to have been Kemble who, in his Preface thanks Professor R. S. Mulliken "whose suggestions and criticisms have been numerous and invaluable".
To put this report in context. Heisenberg's first paper on "the new quantum mechanics" had appeared late in 1925, as had Dirac's first paper, too. Schrodinger's first paper on wave mechanics appeared during February of 1926 and others followed throughout the year. The report was thus written a time of real flux in the underlying theory and its theoretical aspects give testimony to a somewhat uneasy co-existence between the old and new quantum theories, with a strong overlay of classical mechanics, in the theory of molecular spectra.
In his introductory chapter Kemble says that
the foundation for the present theory of band spectra was laid in 1892 by the older Lord Rayleigh when he pointed out that if an oscillator which at rest emits and absorbs light of frequency v0 is caused to rotate with a frequency v, about an axis perpendicular to the axis of vibration, then it should emit and absorb in about equal proportions the two frequencies v0+vr and v0-vr
and he goes on to say:
As early as 1904 Drude from the study of the dispersion of various crystals was led to the conclusion that the infrared absorption spectra and emission bands of most substances, including gases, must be due to the vibrations of electrically charged atoms and molecules rather than to the oscillations of electrons inside the atoms, and in 1912 Niels Bjerrum called attention to the fact that the breadth of the as yet unresolved infrared absorption bands of gases was of the order of magnitude to be expected from the superposition of molecular rotations on molecular vibrations.
These considerations were, at the time being described here, effective only for diatomic molecules and in this case it follows from the Maxwell-Boltzmann law and the classical mechanics of a rigid rotor that a group of molecules in thermal equilibrium at the temperature T the number having rotational frequencies between vr and vr+dv, is
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where I is the moment of inertia and k is Boltzmann's constant. The quantity dn/dv, should be proportional to the absorption coefficient for either of the two frequencies v0± vr and so the band should be a doublet with a splitting
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By 1913, however, experimental techniques had advanced sufficiently for the predicted doublet to be observed and the calculated moments of inertia led to what Kemble justly observed, were "plausible values" for bond lengths. For example the bond length of CO was estimated to be l.14A while that of HCl was put at l.34A. A portion of the near infrared spectrum of of HCl at this level of resolution is given as Figure 30 in Chapter II 2 of ref. 8. By 1914 Bjerrum had developed a theory for CO2, that treated the vibrations of the system in terms of atoms moving in a potential with a minimum at an isosceles triangle geometry. Among the potentials that he tried were a central field one and a valence field one, this last expressed in terms of a pair of bond oscillators coupled to a bond angle bending oscillator. This was an extremely important step for it introduced a molecular model into molecular spectroscopy. The model idea was that the infrared spectrum of a molecule could be understood if the molecule was looked upon as a vibrating-rotating entity whose vibrations could be interpreted in terms of a collection of point masses moving in a potential with a minimum at a particular geometry with the whole system undergoing free rotation. If a way could be found of attributing particular spectral features to the rotational motion, then it would be possible to establish the moments of inertia of the molecule as a nearly rigid body and from these moments of inertia to determine the geometry of the potential minimum.
Almost simultaneously with these developments however and at about the same time that Bohr's quantum theory came on the scene, further instrumental advances led to the discovery that the diatomic infrared bands were not really continuous but were resolvable into fine structure. A diagram of a portion of the near infrared spectrum of HCl showing this fine structure can be found as Figure 32 in Chapter II 2 of ref. 8. As Kemble remarks:
It was immediately evident that the existence of this fine structure must be regarded as conclusive evidence for the of the quantization of the rotational motion of the molecules.
In fact the possibility of such quantization had been suggested in 1911 by Nernst and in 1912 by Lorentz but it was Ehrenfest in 1913 who suggested that diatomic quantization should be in multiples of ½hvr with
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and this gave rise to a re-interpretation of the diatomic spectra which, although it yielded a moments of inertia really quite close to those obtained from the older theory, was by 1916 providing the generally accepted way of interpreting diatomic spectra. Although the molecular model developed by Bjerrum was still central to the understanding of molecular spectroscopy, by 1920, the model had been incorporated into the old quantum theory and it is discussed, chiefly in relation to diatomics, in Page's contribution (Chapter II) to ref. 3 and somewhat more generally in Kemble's Chapter VII in the same volume. It is also discussed in the context of band spectra in a textbook by Baly 10 in 1927. In his chapter on emission band spectra, Baly pays particular tribute to the work of Kratzer in describing a rotating non-harmonic oscillator and also to his recognition, simultaneously with that of Loomis in 1920, that the infrared spectrum of HCl could be interpreted as due to the vibrations of two distinct molecular species, corresponding to the 35 and 37 isotopes of chlorine.
The new quantum theory proved immediately attractive to workers in the field and in August 1926, Dennison 11 published a paper called The Rotation of Molecules in which Heisenberg's form of quantum mechanics was used to describe the motions of a rigid rotor and of a symmetric top. In later developments however, it was Schrodinger's form of quantum mechanics, generally called wave mechanics, that was the preferred basis for theoretical descriptions. The molecular model in its wave mechanical form is described with great elegance and economy in a 1931 review by Dennison. In this review the wave mechanical form of the molecular model is realised by taking the Hamiltonian for the whole system as a sum of one for the internal motions and one to describe the rotations. The internal motions are assumed to be those of s atomic nuclei which have a possible equilibrium position which is non-linear. There will be n= 3s-6 independent displacement coordinates chosen so that they are infinitesimal when compared with the normal distances between nuclei. It is then possible to choose a linear combination of these coordinates as normal coordinates so that both the kinetic and potential energy operators are quadratic forms and so that the solution to the internal motion problem can be written as a product of simple harmonic oscillator functions, one for each normal coordinate. The rotational motion is described by treating the molecule as a rigid body and in terms of its principal moments of inertia. The rotational Hamiltonian consists therefore simply of a kinetic energy term of the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)
where the Lα are the components of the total angular momentum operator and I0αα are the principal moments of inertia and are regarded as constants. If the moments of inertia are all different, the most general case, this is the Hamiltonian for an asymmetric top and its energy levels as functions of the principal moments of inertia were already well known by the time the review was written, as were the selection rules for electric dipole transitions between the levels. And spectroscopists were able to identify segments of spectra with rotational motion and from the observed lines were often able to determine moments of inertia. By using the model of a rigid framework of point masses, it was then possible to determine a molecular geometry yielding bond lengths and angles. The structures so determined made very good sense in classical chemical terms. It should be remembered that spectra that can be associated solely with rotational transitions did not really become accessible until after the development of microwave sources during World War II. And so what is being spoken of here actually arises from the analysis of rotational fine structure on vibrational bands.
The Hamiltonian for the molecular model described in Dennison's review is sui generis and is not derived from any more general model. We shall call it the Bjerrum model when a short designation is required. In 1934 however, Eckart in a paper entitled The Kinetic Energy of Polyatomic Molecules attempted to describe the motion of a non-rigid assembly of particles in such a way that the rotational, vibrational and coupling terms could be distinguished. Eckart worked in classical mechanics. At nearly the same time, Hirschfelder and Wigner were trying to do the same in wave mechanics. Eckart actually concerned himself only with the kinetic energy and Hirschfelder and Wigner did not explicitly specify their potential choice. However their arguments are valid for any potential that is invariant under uniform translations and rotation-reflections of all the particle coordinates. When a brief designation of such a potential is required, it will be called geometrical for it depends only on the geometry of the particles and not on their position or orientation.
What they both did was to choose a frame, fixed or embedded in the molecule whose orientation is described by the handedness of the frame and three Eulerian angles. They actually chose the Eulerian angles with which the rotational motion is described, to define an orthogonal matrix that diagonalises the molecular inertia tensor. This yields moments of inertia and puts the Lagrangian into principal axis form, just the form appropriate to describe a rigid rotator in classical mechanics. However doing this yields a rotational Hamiltonian of the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)
Here no rigid body assumptions have been made and the moments of inertia are not constants but functions of the internal coordinates. The operator is obviously not at all like the rigid rotor operator given above. Here the operator is divergent whenever two moments of inertia are the same. It is thus quite impossible to describe a symmetric top molecule in this formulation. It seemed to pose such a severe problem that Eckart observed in the abstract of his paper that:
The ordinary moments of inertia appear in the Lagrangian kinetic energy but these are replaced by other functions of the radii of gyration in the Hamiltonian. This throws doubt upon all molecular configurations assigned on the basis of empirical values of moments of inertia.
Indeed it turns out more generally, inspite of some heroic efforts by Van Vleck, that the Hamiltonian so derived is largely ineffective in describing molecules in terms of their traditional geometrical structures and so it found no use in the elucidation of molecular spectra.
In his second paper, 1 which has already been spoken of above, Eckart still continued to use classical mechanics but began his development by considering the potential in the problem. In order to choose a set of axes fixed in the molecule, he took his potential to be such that, when described in the chosen axis system (frame), it could be expressed in terms of the (dependent) cartesian coordinates Zi referred to the centre of mass of the system and which took constant values ai at the minimum of the potential. The bold face is used to denote a column matrix of three cartesian components. The coordinates must satisfy the three constraints
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to be displacement independent. By expanding the potential about the minimum and considering how the the rotation-reflection invariance requirements may be satisfied in the expanded form while maintaining the displacement invariance requirements, he showed that his frame choice could be achieved if the zi satisfied the further three constraints, using an obvious vector notation,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)
provided that the ai do not define a straight line.
The two conditions (1.3) and (1.4) are usually called the Eckart conditions (though sometimes also the Sayvetz conditions). They can therefore, simply be regarded as specifying a reference geometry or framework for the system. They are also often referred to as the conditions for fixing or embedding a coordinate frame in the body and a set of six such conditions must always be chosen even for a tailor-made Hamiltonian.
Although the derivation of the second Eckart condition arises by considering a potential expandable up to quadratic terms about a minimum potential energy geometry specified by the ai, the condition can be imposed on any Hamiltonian with a geometrical potential. It can thus be chosen with a potential defined by the coulomb interaction between charged particles, for example. The minimum energy requirement on the ai geometry is of no consequence in the expression of the kinetic energy operator and is important in the potential energy only if it is wished to expand it in a Taylor series about the minimum. Of course to get normal coordinates such an expansion is essential, so the physical relevance of the choice is vitally important.
