Hamiltonian equations diatomic molecules

The energy levels are described by quantum theory and may be found by solving the time-independent Schroedinger equation by using the vibrational Hamiltonian for a diatomic molecule [13,14]. 

Here R, ru...,rN are respectively the-C.M. position of the diatomic molecule and the position vectors of the fluid atoms, P, PU...,PN are the conjugate momenta, n and Q are the momentum and coordinate characterizing the oscillatory degree of freedom, r is the vector representing the orientation and length of the bond in the diatomic molecule, and L is the angular momentum of the molecule about the C.M. The interaction Hamiltonian has already been linearized in the oscillatory coordinate Q in the last equation. 

Contact Transformation for the Effective Hamiltonian.—The vibration-rotation hamiltonian of a polyatomic molecule, expressed in terms of normal co-ordinates, has been discussed in particular by Wilson, Decius, and Cross,24 and by Watson.27- 28 It is given by the following expression for a non-linearf polyatomic molecule, to be compared with equation (17) for a diatomic molecule ... 

In practice, the result of the perturbation treatment may be expressed as a series of formulae for the spectroscopic constants, i.e. the coefficients in the transformed or effective hamiltonian, in terms of the parameters appearing in the original hamiltonian, i.e. the wavenumbers tor, the anharmonic force constants , the moments of inertia Ia, their derivatives eft , and the zeta constants These formulae are analogous to equations (23)—(27) for a diatomic molecule. They are too numerous and too complicated to quote all of them here, but the various spectroscopic constants are listed in Table 3, with their approximate relative orders of magnitude, an indication of which parameters occur in the formula for each spectroscopic constant, and a reference to an appropriate source for the perturbation theory formula for that constant. 

In this section, we shall use the degenerate perturbation theory approach to derive the form of the effective Hamiltonian for a diatomic molecule in a given electronic state. Exactly the same result can be obtained by use of the Van Vleck or contact transformations [12, 13]. The general expression for the operator up to fourth order in perturbation theory is given in equation (7.43). Fourth order can be considered as the practical limit to this type of approach. Indeed, even its implementation is very laborious and has only been used to investigate the form of certain special terms in the effective Hamiltonian. We shall consider some of these terms later in this chapter. For the moment we confine our attention to first- and second-order effects only. 

The form of the nuclear electric quadrupole interaction in the effective Hamiltonian for a diatomic molecule is given in equations (7.158) and (7.161), with the latter applying only to molecules in n electronic states. The two parameters which can be determined from a fit of the experimental data are eqo Q and et/i Q respectively. Since the electric quadrupole moment eQ is known for most nuclei, an experimental observation gives information on q0 (and perhaps qi), the electric field gradient at the nucleus. This quantity depends on the electronic structure of the molecule according to the expression... 

The nuclear spin-rotation interaction becomes very simple for a diatomic molecule. The principal components of the tensor a for a polyatomic molecule were described in equation (8.163) this expression reveals that for a diatomic system the axial component (c/)zz is zero and, of course, the two perpendicular components are equal. The nuclear spin rotation interaction for a diatomic molecule is therefore described by a single parameter c/. The appropriate term in the effective Hamiltonian, first presented in equation (8.7), is... 

After a general introduction, the methods used to separate nuclear and electronic motions are described. Brown and Carrington then show how the fundamental Dirac and Breit equations may be developed to provide comprehensive descriptions of the kinetic and potential energy terms which govern the behaviour of the electrons. One chapter is devoted solely to angular momentum theory and another describes the development of the so-called effective Hamiltonian used to analyse and understand the experimental spectra of diatomic molecules. The remainder of the book concentrates on experimental methods. 

An alternative approach widely used in polyatomic molecule studies is based on the Golden Rule and a perturbative treatment of the anharmonic coupling (57,62). This approach is not much used for diatomic molecules. In the liquid O2 example cited above, the Hamiltonian must be expanded to 30th order or so to calculate the multiphonon emission rate. But for vibrations of polyatomic molecules, which can always find relatively low-order VER pathways for each VER step, perturbation theory is very useful. In the perturbation approach, the molecule s entire ladder of vibrational excitations is the system and the phonons are the bath. Only lower-order processes are ordinarily needed (57) because polyatomic molecules have many vibrations ranging from higher to lower frequencies and only a small number of phonons, usually one or two, are excited in each VER step. The usual practice is to expand the interaction Hamiltonian (qn, Q) in Equation (2) in powers of normal coordinates (57,62) ... 

In the first part of this section, the relationship between the solution of the Schrodinger equation and the hamiltonian in the space generated by a given basis set is discussed in some detail. Since basis set limitations appear to be one of the largest sources of error in most present day molecular calculations, the concept of a universal even-tempered basis set is discussed in the second part of this section. This concept represents an attempt to overcome the incomplete basis set problem, at least for diatomic molecules. Further aspects of the basis set truncation problem are discussed in the final part of this section. 

The form of the spin-orbit Hamiltonian has been given by Van Vleck (1951) and is an extension to diatomic molecules of the solution for the relativistic equation originally derived for a two-electron atom ... 

Bryce and Autschbach performed the accurate calculation of the isotropic and anisotropic (AT) parts of indirect nuclear spin spin coupling tensors for diatomic alkali metal halides (MX M = Li, Na, K, Rb, Cs X = F, Cl, Br, I) with the relativistic hybrid DFT approach. The calculated coupling tensor components were compared with experimental values obtained from molecular-beam measurements on diatomic molecules in the gas phase. Molecular-beam experiments offer ideal data for testing the success of computational approaches, since the data are essentially free from intermolecular effects. The hyperfine Hamiltonian used in analyzing molecular-beam data contains Hc IkDIi and //C4/a /l terms. The relationships between the parameters C3 and C4, used in molecular-beam experiments, and Rdd, A/, and used in NMR spectroscopy, are summarized in the following equations ... 

This results from the mathematical fact that an exponential of a sum is a product of the exponentials of each of the terms in the sum, e.g., 6 + " = e e .) We should keep in mind that Equations 11.41 and 11.42 correspond to an approximation. For instance, vibrational-rotational coupling prevents strict separation of the vibrational and rotational parts of a molecular Hamiltonian. For now, we will use Equation 11.42 to obtain q via obtaining each of the elements in the product. We shall also rely on model problems presented in Chapters 7 and 8 as idealizations of rotation and vibration of a diatomic molecule. 

Each vibration a molecule exhibits corresponds to two quadratic terms in the Hamiltonian as in Equation 7.35. There is a quadratic momentum term and a quadratic term in the associated displacement coordinate. Thus, each vibrational mode contributes 2 X NkT/2 to U. We can collect this information concisely by saying that for a molecular gas, U is 3NkT/2 (translation), plus either NkT/2 for linear molecules or 3NkT/2 for nonlinear molecules (rotation), plus NkT times the number of modes of vibration. The gas of an ideal diatomic molecule has U = 7NkT/2 as in Equation 11.59. A linear triatomic molecule has U = 13NkT/2 because there are three vibrational modes, one of which is twofold degenerate. 

Consider the vibration and rotation of a diatomic molecule. Since the molecule is rotating in space, the Hamiltonian is best written in terms of spherical coordinates. The potential V(r) depends only on the separation of the atoms, and it develops from the electrons and the chemical bonding that occurs between the atoms. The Schroedinger equation for a rotating and vibrating diatomic molecule is... 

In diatomic VER, the frequency Q is often much greater than so VER requires a high-order multiphonon process (see example C3.5.6.1). Because polyatomic molecules have several vibrations ranging from higher to lower frequencies, only lower-order phonon processes are ordinarily needed [34]- The usual practice is to expand the interaction Hamiltonian > in equation (03.5.2) in powers of nonnal coordinates [34, 631,... 

Just as for an atom, the hamiltonian H for a diatomic or polyatomic molecule is the sum of the kinetic energy T, or its quantum mechanical equivalent, and the potential energy V, as in Equation (1.20). In a molecule the kinetic energy T consists of contributions and from the motions of the electrons and nuclei, respectively. The potential energy comprises two terms, and F , due to coulombic repulsions between the electrons and between the nuclei, respectively, and a third term Fg , due to attractive forces between the electrons and nuclei, giving... 

The direct variational solution of the Schrddinger equation after separation of the center of mass motion is in general possible and can be performed very accurately for three- and four- body systems such as (Kolos, 1969) and H2 (Kolos and Wolniewicz, 1963 Bishop and Cheung, 1978). For larger systems it is unlikely to perform such calculations in the near future. Therefore the usual way in quantum chemistry is to introduce the adiabatic approximation. The nonrelativistic hamiltonian for a diatomic N-electron molecule in the center of mass system has the following form (in atomic units). 

The London equation has in addition been the progenitor of semi-empirical (or semi-theoretical) valence-bond methods of which Moffitt s method of atoms in molecules (95) and Ellison s method of diatomics in molecules(96), the latter not in fact being a direct generalisation of the former, are the most important. It is beyond the scope of this article to give the details of these methods and their modifications and the reader is referred to other reviews that encompass them (85, 97). It is however important to emphasize that they work by using relatively simple polyatomic wavefunctions and introducing corrections to the resulting Hamiltonian matrix... 

Only spatially degenerate states exhibit a first-order zero-field splitting. This condition restricts the phenomenon to atoms, diatomics, and highly symmetric polyatomic molecules. For a comparison with experiment, computed matrix elements of one or the other microscopic spin-orbit Hamiltonian have to be equated with those of a phenomenological operator. One has to be aware of the fact, however, that experimentally determined parameters are effective ones and may contain second-order contributions. Second-order SOC may be large, particularly in heavy element compounds. As discussed in the next section, it is not always distinguishable from first-order effects. 

The rotational and Zeeman perturbation Hamiltonian (X) to the electronic eigenstates was given in equation (8.105). It did not, however, contain terms which describe the interaction effects arising from nuclear spin. These are of primary importance in molecular beam magnetic resonance studies, so we must now extend our treatment and, in particular, demonstrate the origin of the terms in the effective Hamiltonian already employed to analyse the spectra. Again the treatment will apply to any molecule, but we shall subsequently restrict attention to diatomic systems. 

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