Hamiltonian rotating diatom

The Hamiltonian in this problem contains only the kinetic energy of rotation no potential energy is present because the molecule is undergoing unhindered "free rotation". The angles 0 and (j) describe the orientation of the diatomic molecule s axis relative to a laboratory-fixed coordinate system, and p is the reduced mass of the diatomic molecule p=mim2/(mi+m2). 

The rotational Hamiltonian for a diatomic molecule as given in Chapter 3 is... 

We next apply these classical relationships to the rigid diatomic molecule. Since the molecule is rotating freely about its center of mass, the potential energy is zero and the classical-mechanical Hamiltonian function H is just the kinetic energy of the two particles,... 

Here eR is the rotational energy of a rigid rotor and r0 is the equilibrium ground state bond length of the diatomic molecule. The total Hamiltonian is thus... 

The vibration-rotation hamiltonian of a polyatomic molecule is more complicated than that of a diatomic molecule, both because of the increased number of co-ordinates, and because of the presence of Coriolis terms which are absent from the diatomic hamiltonian. These differences lead to many more terms in the formulae for a and x values obtained from the contact transformation, and they also lead to various kinds of vibrational and rotational resonance situations in which two or more vibrational levels are separated by so small an energy that interaction terms in the hamiltonian between these levels cannot easily be handled by perturbation theory. It is then necessary to obtain an effective hamiltonian over these two or more vibrational levels, and to use special techniques to relate the coefficients in this hamiltonian to the observed spectrum. 

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 ... 

It is instructive to consider two of the formulae for the spectroscopic constants in more detail, and for this we choose — and xrs for an asymmetric top, these being respectively the coefficients of (vT + )J, the vibrational dependence of the rotational constant, and (vr + i)(fs + i), the vibrational anharmonic constant quadratic in the vibrational quantum numbers. As for diatomic molecules these two types of spectroscopic constant provide the most important source of information on cubic and quartic anharmonicity, respectively. The formulae obtained from the perturbation treatment for these two coefficients in the effective hamiltonian are as follows ... 

We require only three nuclear coordinates to define the nuclear motion and we choose these to be R, the internuclear distance, and 0 the third Euler angle x is a redundant coordinate. In fact, because there are no nuclei lying off-axis in a diatomic molecule, X is undefineable it is, however, expedient to retain it because of simplification in the final form of the rotational Hamiltonian. We shall examine this point in more detail in... 

In conclusion we summarise the total Hamiltonian (excluding nuclear spin effects), written in a molecule-fixed rotating coordinate system with origin at the nuclear centre of mass, for a diatomic molecule with electron spin quantised in the molecular axis system. We number the terms sequentially, and then describe their physical significance. The Hamiltonian is as follows ... 

We recall also that the y l i,, (< >) are proportional to the eigenfunctions of a symmetric top rotational Hamiltonian (section 5.3.4). Realising that a diatomic molecule behaves as a symmetric top (albeit a rather special one), we write the rotational part of the wave function as... 

In this chapter we introduce and derive the effective Hamiltonian for a diatomic molecule. The effective Hamiltonian operates only within the levels (rotational, spin and hyperfine) of a single vibrational level of the particular electronic state of interest. It is derived from the Ml Hamiltonian described in the previous chapters by absorbing the effects of off-diagonal matrix elements, which link the vibronic level of interest to other vibrational and electronic states, by a perturbation procedure. It has the same eigenvalues as the Ml Hamiltonian, at least to within some prescribed accuracy. 

It is well known from the Bom-Oppenheimer separation [1] that the pattern of energy levels for a typical diatomic molecule consists first of widely separated electronic states (A eiec 20000 cm-1). Each of these states then supports a set of more closely spaced vibrational levels (AEvib 1000 cm-1). Each of these vibrational levels in turn is spanned by closely spaced rotational levels ( A Emt 1 cm-1) and, in the case of open shell molecules, by fine and hyperfine states (A Efs 100 cm-1 and AEhts 0.01 cm-1). The objective is to construct an effective Hamiltonian which is capable of describing the detailed energy levels of the molecule in a single vibrational level of a particular electronic state. It is usual to derive this Hamiltonian in two stages because of the different nature of the electronic and nuclear coordinates. In the first step, which we describe in the present section, we derive a Hamiltonian which acts on all the vibrational states of a single electronic state. The operators thus remain explicitly dependent on the vibrational coordinate R (the intemuclear separation). In the second step, described in section 7.55, we remove the effects of terms in this intermediate Hamiltonian which couple different vibrational levels. The result is an effective Hamiltonian for each vibronic state. 

There is a particular difficulty in the formulation of the full Hamiltonian for a linear molecule (and hence for a diatomic molecule) which was first identified by Hougen [15], The source of this difficulty is the fact that only two rotational coordinates are required to define the orientation of a diatomic molecule in laboratory space, the Euler... 

To obtain the isomorphic Hamiltonian for a diatomic molecule, x is introduced as an independent variable and the coordinates of the particles which make up the molecule are measured in an axis system (x, y, z) whose orientation is described by the Euler angles (<-/>, 0, x) in the (X, Y, Z) axis system. We recall that we chose x to be zero in constructing the true Hamiltonian. The (x, y, z) axes are therefore obtained by rotation of the (x7, y, z ) axis system about the z (= z) axis through the angle x As a result, we have... 

Thus far, we have investigated the various contributions to the effective Hamiltonian for a diatomic molecule in a particular electronic state which arise from the spin-orbit and rotational kinetic energy terms treated up to second order in degenerate perturbation theory. Higher-order effects of such mixing will also contribute and we now consider some of their characteristics. 

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. 

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... 

An applied electric field (E) interacts with the electric dipole moment (p,e) of a polar diatomic molecule, which lies along the direction of the intemuclear axis. The applied field defines the space-fixed p = 0 direction, or Z direction, whilst the molecule-fixed q = 0 direction corresponds to the intemuclear axis. Transformation from one axis system to the other is accomplished by means of a first-rank rotation matrix, so that the interaction may be represented by the effective Hamiltonian as follows ... 

The system Hamiltonian can be approximated, as in the ARRKM theory, by decoupling the diatom vibrational motion from overall rotational motion of the molecule and from the van der Waals bond stretching. With this approximation. 

For a diatomic molecule (AB) there is only one degree of freedom namely the bond length between the two atoms denoted r. The Hamiltonian without rotational contributions is given by... 

If only one reaction channel is relevant, e.g. ABC + hv AB + C, Jacobi coordinates are suitable since one bond (AB) dominates and is not broken. The Jacobi coordinates are the set R, r,9), where r is the diatomic intemuclear distance (A-B) and R is the distance between the atom (C) and the centre of the mass of the diatom (AB). 9 is the Jacobi angle between R and r. Following Balint-Kurti et al. the non-rotating Hamiltonian in Jacobi coordinates is given by... 

F. Effective Hamiltonian for Rotational Excitations in Diatomic Molecules... 

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