Hamiltonian diatomic molecule

As our first model problem, we take the motion of a diatomic molecule under an external force field. For simplicity, it is assumed that (i) the motion is pla nar, (ii) the two atoms have equal mass m = 1, and (iii) the chemical bond is modeled by a stiff harmonic spring with equilibrium length ro = 1. Denoting the positions of the two atoms hy e 71, i = 1,2, the corresponding Hamiltonian function is of type... 

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... 

Let us suppose that the system of interest does not possess a dipole moment as in the case of a homonuclear diatomic molecule. In this case, the leading term in the electric field-molecule interaction involves the polarizability, a, and the Hamiltonian is of the form ... 

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,... 

In the general case R denotes a set of coordinates, and Ui(R) and Uf (R) are potential energy surfaces with a high dimension. However, the essential features can be understood from the simplest case, which is that of a diatomic molecule that loses one electron. Then Ui(R) is the potential energy curve for the ground state of the molecule, and Uf(R) that of the ion (see Fig. 19.2). If the ion is stable, which will be true for outer-sphere electron-transfer reactions, Uf(R) has a stable minimum, and its general shape will be similar to that of Ui(R). We can then apply the harmonic approximation to both states, so that the nuclear Hamiltonians Hi and Hf that correspond to Ui and Uf are sums of harmonic oscillator terms. To simplify the mathematics further, we make two additional assumptions ... 

A first insight into a different description of a chemical process can be obtained from an analysis of a (diatomic) dissociation process. Consider the standard treatment of a stable diatomic molecule. The word stable implies already the existence of a measurable characteristic size around which the electro-nuclear system fluctuates in its ground electronic state (i.e. a stationary molecular Hamiltonian with ground state). In standard quantum chemistry, this is the nuclear equilibrium distance. 

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]. 

In diatomic molecules, T2 = 0, and thus the expectation value of C vanishes. This is the reason why this operator was not considered in Chapter 2. However, for linear triatomic molecules, t2 = / / 0, and the expectation value of C does not vanish. We note, however, that D J is a pseudoscalar operator. Since the Hamiltonian is a scalar, one must take either the absolute value of C [i.e., IC(0(4 2))I or its square IC(0(412))I2. We consider here its square, and add to either the local or the normal Hamiltonians (4.51) or (4.56) a term /412IC(0(412))I2. We thus consider, for the local-mode limit,... 

To illustrate this point, consider a composite system composed of two noninteracting subsystems, one with p electrons (subsystem A) and the other with q = N — p electrons (subsystem B). This would be the case, for example, in the limit that a diatomic molecule A—B is stretched to infinite bond distance. Because subsystems A and B are noninteracting, there must exist disjoint sets Ba and Bb of orthonormal spin orbitals, one set associated with each subsystem, such that the composite system s Hamiltonian matrix can be written as a direct sum. 

Adiabatic passage schemes are particularly suited to control population transfer between states, since the adiabatic following condition assesses the stability of the dynamics to fluctuations in the pulse duration and intensity [3]. The time evolution of the wave function does not depend on the dynamical phase, and is therefore slow in comparison with the vibrational time scale. This fact guarantees that the time variation of the observables during the controlled dynamics will be slow. Adiabatic methods can therefore be of great utility to control dynamic observables that do not commute with the Hamiltonian. We are interested in the control of the bond length of a diatomic molecule [4]. 

In Section 1.19 we classified the electronic wave functions of homonuclear diatomic molecules as g or u, according to whether they were even or odd with respect to inversion g and u refer to inversion of the electronic coordinates with respect to the molecule-fixed axes. This is to be distinguished from the inversion of electronic and nuclear coordinates with respect to space-fixed axes, which was discussed in this section. The electronic Hamiltonian for a diatomic molecule is... 

The complete, nonrelativistic Hamiltonian for a diatomic molecule is given by (1.272). If one inverts the Cartesian coordinates of all particles (nuclei and electrons), then H in (1.272) is unchanged, since all interparticle distances are unchanged. Thus the parity operator IT commutes with this Hamiltonian, and we can characterize the overall wave function of a diatomic molecule by its parity. (This statement applies to both homonuclear and heteronuclear diatomics.)... 

Relaxation times can be expressed in terms of time-correlation functions. Consider, for example, the case of a diatomic molecule relaxing from the vibrationally excited state n + 1> to the vibrational state /i> due to its interactions with a bath of solvent molecules. The Hamiltonian for the system is... 

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. 

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... 

This follows from the invariance of the Hamiltonian to the interchange of the two atoms in a homonuclear diatomic molecule. 

To summarize, therefore, it is reasonable to say that ab initio calculations of spin-orbit coupling constants may be successfully performed on atoms (although relativistic wavefunctions will be necessary for the heavier ones) and diatomic molecules (especially hydrides). For larger molecules, such methods may be too time-consuming and resort to semi-empirical techniques will be necessary. The atoms-in-molecules approach has proved extremely successful, but it should be possible to use semi-empirical wavefunctions with the full hamiltonian before long. This will be probably more useful with very large molecules. 

Atoms in Molecules.—In this approach, which was first proposed by Moffitt,105 a wavefunction for a particular electronic state of a molecule is constructed from products of atomic wavefunctions, these, moreover, being taken to be exact eigenfunctions of their respective atomic hamiltonians. We confine our attention to the case of diatomic molecules AB so that, according to this procedure, the wavefunction is written as... 

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 ... 

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