Coordinate system and Hamiltonian

Let us consider a diatomic molecule AB plus a weakly interacting atom C (e.g., 

Ar or CO. .. He), the total system in its electronic ground state. Let us centre the origin of the body-fixed coordinate system (with the axes oriented as in the space-fixed coordinate intern, see Appendix I, p. 971) in the centre of mass of AB. The problem involves therefore 3 x 3 — 3 = 6 dimensions. 

However strange it may sound, six is too much for contemporary (otherwise impressive) computer techniques. Let us subtract one dimension by assuming that no vibrations of AB occur (rigid rotator). The five-dimensional problem becomes manageable. The assumption about the stiffness of AB now also pays off because we exclude right away two possible chemical reactions C -l- AB  

CA -I- B and C -I- AB CB - - A, and admit therefore only some limited set of nuclear configurations - only those that correspond to a weakly bound complex C -I- AB. This approximation is expected to work better when the AB molecule is 

We will introduce the Jacobi coordinates (Fig. 12, cf. p. 776) three components Jacobi of vector R pointing to C from the origin of the coordinate system (the length R coordinates 

---

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). 

Finally, in section 2.10 we considered the alternative transformation scheme in which the electron spin remained quantised in the space-fixed coordinate system. The Hamiltonian for this situation is easily derived from (3.267), (3.268) and (3.269) by making the substitutions... 

Here we would like to add some comment of a general character. Dirac (1958) argued that the transformation from classical to quantum mechanics should be made, first by constructing the classical Hamiltonian in the Cartesian coordinate system and then by replacing the positions and momenta by their quantum-mechanical operator equivalents, which are determined by the particular representation chosen. The important point is that this transformation should be performed in the Cartesian coordinate system, for it is only in this system that the Heisenberg uncertainty principle for the positions and momenta is usually enunciated. In this connection, notice that some momentum wave functions such as those obtained by Podolsky and Pauling (1929) are correct wave functions that are useful in calculations of the expectation value of any observable, but at the same time they have a drawback in that the momentum variables used there are not conjugate to any relevant position variables (see also, Lombardi, 1980). 

The conformers give spectra as if they were separate molecules. Molecular motions that are fast on the NMR time scale but slow relative to molecular tumbling are considered to be between conformers. The observed NMR spectra correspond to a spin Hamiltonian that is a weighted mean of those of the individual conformers. In general, each such con-former will be characterized by its own structure, molecule-fixed coordinate system, and motional constants. 

Being the exact quantum medianical treatment of the dynamics of systems containing more than four atoms presently computationally out of question, only particular cases or approximate methods are nowadays being developed in order to extend the hyperspherical method to complex reactions. Proper formulations of the coordinate systems and relevant hamiltonians have already been referred to [27,28,45], see also [51]. 

Spin-Hamiltonian parameters were measured at the temperature given in the How produced column unless otherwise noted. Consult original reference for relative orientation of spin-Hamiltonian coordinate system and crystallographic axes. Only one g value or A value quoted indicates parameter is isotropic. Absolute magnitudes are given for ), and A unless indicated by (+) or (-). 

Actually, the actions (/i, J2 are most generally line integrals and are constants of the motion for any system with a zeroth-order Hamiltonian that is decomposable into the sum of two one-dimensional Hamiltonians in some coordinate system, and this coordinate system need not be explicitly known. A more general way to write down the actions that refleas these facts is... 

The Schrddinger equation for the one-electron atom (Chapter 6) is exactly solvable. However, because of the interelectronic repulsion terms in the Hamiltonian, the Schrbdinger equation for many-electron atoms and molecules is not separable in any coordinate system and cannot be solved exactly. Hence we must seek approximate methods of solution. The two main approximation methods, the variation method and perturbation theory, will be presented in Chapters 8 and 9. To derive these methods, we must develop further the theory of quantum mechanics, which is what is done in this chapter. 

Because of the Vryi term, the Schrodinger equation for helium cannot be separated in any coordinate system, and we must use approximation methods. To use the perturbation method, we must separate the Hamiltonian (9.39) into two parts, and i, where is the Hamiltonian of an exactly solvable problem. If we choose... 

The following is an example of the SCF method in practice in the treatment of the two electrons present in a helium atom. Because helium has two electrons orbiting a -1-2 nucleus, it presents itself as the three-body problem shown in Figure 4.9, where the nucleus is presumed to be at rest and therefore sits at the origin of the coordinate system. The Hamiltonian for the helium atom includes three potential energy terms an attractive force between electron I and the nucleus (r ), an attractive force between electron 2 and the nucleus (r2), and the electron-electron repulsion between the two electrons (r 2), as shown in Equation (4.8). 

In an isotropic liquid the mean value of is zero and the quadrupole interaction contributes only to relaxation. In an anisotropic medium, on the other hand, the mean value of is no longer zero and a quadrupole splitting appears in the NMR spectrum. While the quadrupole hamiltonian in Eq. (7.3) may be evaluated in any coordinate system it is convenient to express the spin operators in a laboratory-fixed coordinate system and the electric field gradients in a principal axes coordinate system fixed at the nucleus. It is then suitable to rewrite the hamiltonian as... 

For triatomic molecules many coordinate systems have been used to represent the vibrational motions, see ref. 12 for example. For polyatomic clusters it is possible to imagine a large number of possible coordinate systems and a similar proliferation of Hamiltonians. In this context it should be noted that the derivation and application of Hamiltonians in arbitrary coordinates is far from simple. The choice of an objectively inferior coordinate system for technical reasons is thus a common occur-ance. For polyatomic Van der Waals dimers however,so-called scattering coordinates based upon the interaction coordinate of the two monomers and associated angles of orientation would appear a natural choice. A general, body-fixed Hamiltonian for these coordinates has already been derived... 

The plan of this paper is as follows. Section 2 describes the coordinate system and the representation of the Hamiltonian together with computational details for the spectra simulations. In Sect. 3, we present the results obtained and discuss their comparison with previous experimental and theoretical data, while Sect. 4 summarizes our conclusions. 

Coordinate system, model Hamiltonian operator, and computational details... 

Another very common case is a potential that behaves as l/ r, known as the Coulomb potential, from the electrostatic interaction between particles with electrical charges at distance r. We will discuss this case for the simplest physical system where it applies, the hydrogen atom. For simplicity, we take the proton fixed at the origin of the coordinate system and the electron at r. The hamiltonian for this system takes the form... 

To try to deal with the rotational motion it is possible to reformulate the diatomic problem to exhibit explicitly the angular symmetry of the Hamiltonian. As shown in Kotos and Wolniewicz (1963) and, in a somewhat more general way in Sutcliffe (2007), it is possible to define an internal CO ordinate system by a transformation that makes the internuclear vector t the z-axis in a right-handed coordinate system and in this system the electronic Hamiltonian (O Eq. 2.24) becomes... 

One might hope to get somewhere with the idea of a geometrical structure by considering the development of the definition to deal with a system described in a frame fixed in the body. The idea here is to somehow fix a coordinate frame in the system and to define its orientation by means of 3 Eulerian angles, (j>m defined entirely in terms oi A — translationally invariant position variables associated with the nuclei. In such a coordinate system the Hamiltonian takes the general form... 

In addition, there could be a mechanical or electromagnetic interaction of a system with an external entity which may do work on an otherwise isolated system. Such a contact with a work source can be represented by the Hamiltonian U p, q, x) where x is the coordinate (for example, the position of a piston in a box containing a gas, or the magnetic moment if an external magnetic field is present, or the electric dipole moment in the presence of an external electric field) describing the interaction between the system and the external work source. Then the force, canonically conjugate to x, which the system exerts on the outside world is... 

Kuppermann A 1996 Reactive scattering with row-orthonormal hyperspherical coordinates. I. Transformation properties and Hamiltonian for triatomic systems J. Phys. Chem. 100 2621... 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

---