The Calculation of Spectroscopic Constants

The answer to the question why calculate spectroscopic constants is not merely to find the value . There are in fact two possible reasons why one should use theoretical methods to compute spectroscopic constants. Firstly, there are cases where an experimental value has been measured for some constant, but the observed magnitude or even sign is not comprehensible in terms of the idea of electronic structure which the spectroscopist has in mind. In such cases a relatively crude calculation which only reproduces the observation to an order of magnitude may offer explanations in terms of perturbations by unobserved states or the atomic constitution of molecular orbitals. 

These crude calculations which assist the spectroscopist in his interpretation have been possible for some time and are increasingly used almost as an experimental tool. Recently, however, the power of electronic computers has advanced to the stage where, for small molecules, wavefunctions of a very accurate nature can be calculated. For very good wavefunctions, energies and, more especially, other expectation values can be computed very accurately. Indeed in some instances spectroscopic constants can now be computed to greater precision than they can be measured. 

Typically such an improvement is counterbalanced by some corresponding loss. In this case the loss is a loss of understanding in terms of simple pictorial concepts. It is possible to calculate a number very accurately but it may be no longer possible to explain its magnitude in terms of major components. 

With these more recent calculations the work is done with the intention of finding a numerical value. As a result there is a change of emphasis in the type of problem attempted. Simple calculations which go hand in glove with experiment remain in the area of problems where both experiment and computation are possible. The accurate work is different. If there is only a number resulting, then it is perhaps pointless doing the work if that number is experimentally accessible, save for testing the method. The accurate work attempts to extend spectroscopy by computing properties which cannot be observed, or have eluded the experimentalist. Frequently this will mean calculations on excited electronic states and on the electronic levels of molecular ions. From the 

Although some accurate work has been attempted for polyatomic molecules, the majority of published papers1 deals with diatomic molecules. This is not only for the reason that they are simpler but also because the level of understanding of diatomic molecule spectra is so much higher than that for polyatomic species. In the latter case relatively few spectra have been analysed to the level where all the rotational and fine structure is assigned. 

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In this chapter, we therefore consider whether it is possible to eliminate spin-orbit coupling from four-component relativistic calculations. This is a situation quite different from that of more approximate relativistic methods where a considerable effort is required for the inclusion of spin-orbit coupling. We have previously shown that it is indeed possible to eliminate spin-orbit coupling from the calculation of spectroscopic constants [12,13]. In this chapter, we consider the extension of the previous result to the calculation of second-order electric and magnetic properties, i.e., linear response functions. Although the central question of this article may seem somewhat technical, it will be seen that its consideration throws considerable light on the fundamental interactions in molecular systems. We will even claim that four-component relativistic theory is the optimal framework for the understanding of such interactions since they are inherently relativistic. 

This Report deals with the calculation of spectroscopic constants both for diatomic and for polyatomic molecules, while concentrating on the former. The spectroscopic constants included are restricted to those measured in high-resolution gas-phase work. We cover the whole range of complexity in computation since this is determined not by the method used in computing the expectation value, but by the quality of the wavefunction used. Generally the wavefunctions used are of the ab initio type, but their quality will depend on the size and type of basis set employed as well as the method. 

In the previous sections, we have discussed the application of the most accurate theoretical methods for the calculation of spectroscopic constants. Another important area of computational chemistry is dynamics. A variety of... 

Recently, Stoll et al [94] used a very similar approach to EPCISO. One minor difference is the use of the DGCI Pitzer s code which works with CSFs basis functions instead of determinants. Apparently another difference here is the absence of a selection process of the spin-orbit matrix elements. In this study small-core and large-core energy-consistent pseudopotentials were combined for the calculation of spectroscopic constants of lead and bismuth compounds (BiH, BiO, PbX, BiX, (X=F, Cl, Br, I)). 

J. F. Gaw and N. C. Handy, Chem. Phys. Lett., 121,321 (1985). Ab initio Quadratic, Cubic and Quartic Force Constants for the Calculation of Spectroscopic Constants. 

For gas-phase reactions, Eq. (5-40) offers a route to the calculation of rate constants from nonkinetic data (such as spectroscopic measurements). There is evidence, from such calculations, that in some reactions not every transition state species proceeds on to product some fraction of transition state molecules may return to the initial state. In such a case the calculated rate will be greater than the observed rate, and it is customaiy to insert a correction factor k, called the transmission coefficient, in the expression. We will not make use of the transmission coefficient. 

On matrix form the non-unitary transformations (27) and (30) of the previous section are easily extended to the complete Hamiltonian and have therefore allowed relativistic and non-relativistic spin-free calculations of spectroscopic constants and first-order properties at the four-component level (see, for instance. Refs. [45 7]). In this section, we consider the elimination of spin-orbit interaction in four-component calculations of second-order electric and magnetic properties. Formulas are restricted to the Hartree-Fock [48] or Kohn-Sham [49] level of theory, but are straightforwardly generalized. 

The cubic force field of pyramidal XYg-type molecules contains 14 independent parameters. Usually, however, the number of spectroscopic constants dependent on the enharmonic force field, such as rotation-vibration constants, l-type doubling constants, or anharmonicity constants, is smaller than the number of parameters to be determined. Their number thus has to be reduced by introducing model potentials and imposing certain constraints. Some of the possible routes were presented by Morino et al. [50]. Cubic force constants for NF3 pertinent to model potentials (mostly Morse potentials) have been calculated by several groups of workers [11, 12, 19, 51, 52], Some of the principal quartic constants have been estimated as well [12, 51]. 

At this point it is worth mentioning a special case. If analytic second derivatives are available and the force field is to be used only in a VFT2 calculation of spectroscopic constants. 

Section BT1.2 provides a brief summary of experimental methods and instmmentation, including definitions of some of the standard measured spectroscopic quantities. Section BT1.3 reviews some of the theory of spectroscopic transitions, especially the relationships between transition moments calculated from wavefiinctions and integrated absorption intensities or radiative rate constants. Because units can be so confusing, numerical factors with their units are included in some of the equations to make them easier to use. Vibrational effects, die Franck-Condon principle and selection mles are also discussed briefly. In the final section, BT1.4. a few applications are mentioned to particular aspects of electronic spectroscopy. 

Terms representing these interactions essentially make up the difference between the traditional force fields of vibrational spectroscopy and those described here. They are therefore responsible for the fact that in many cases spectroscopic force constants cannot be transferred to the calculation of geometries and enthalpies (Section 2.3.). As an example, angle deformation potential constants derived for force fields which involve nonbonded interactions often deviate considerably from the respective spectroscopic constants (7, 7 9, 21, 22). Nonbonded interactions strongly influence molecular geometries, vibrational frequencies, and enthalpies. They are a decisive factor for the transferability of force fields between systems of different strain (Section 2.3.). 

Isotope effects on anharmonic corrections to ZPE drop off rapidly with mass and are usually neglected. The ideas presented above obviously carry over to exchange equilibria involving polyatomic molecules. Unfortunately, however, there are very few polyatomics on which spectroscopic vibrational analysis has been carried in enough detail to furnish spectroscopic values for Go and o)exe. For that reason anharmonic corrections to ZPE s of polyatomics have been generally ignored, but see Section 5.6.3.2 for a discussion of an exception also theoretical (quantum package) calculations of anharmonic constants are now practical (see above), and in the future one can expect more attention to anharmonic corrections of ZPE s. 

For a spectroscopic observation to be understood, a theoretical model must exist on which the interpretation of a spectrum is based. Ideally one would like to be able to record a spectrum and then to compare it with a spectrum computed theoretically. As is shown in the next section, the model based on the harmonic oscillator approximation was developed for interpreting IR spectra. However, in order to use this model, a complete force-constant matrix is needed, involving the calculation of numerous second derivatives of the electronic energy which is a function of nuclear coordinates. This model was used extensively by spectroscopists in interpreting vibrational spectra. However, because of the inability (lack of a viable computational method) to obtain the force constants in an accurate way, the model was not initially used to directly compute IR spectra. This situation was to change because of significant advances in computational chemistry. 

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