Watson Hamiltonians

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 ... 

Here, Aftot is the total mass of the nuclei. Note that when eliminating the motion of the center-of-mass we arbitrarily eliminated the first nuclear coordinate (we could eliminate any single coordinate y). Equation (1-13) can further be transformed to the body-fixed frame. When applying the so-called Eckart conditions35 one would get the standard Watson s Hamiltonian describing the nuclear motions in molecules36. 

We wish to add that Martin and Bratoz330 also considered a case (c) corresponding to almost rigid complex. The treatment of the dynamics in the case (c) does not differ from the standard treatment of rotations and vibrations in rigid molecules with the Watson s Hamiltonian for nuclear motions. 

Watson JKG (1968) Simplification of themolecular vibration-rotation Hamiltonian. Mol Phys 15 479-490... 

It took several decades for the effective Hamiltonian to evolve to its modem form. It will come as no surprise to learn that Van Vleck played an important part in this development for example, he was the first to describe the form of the operator for a polyatomic molecule with quantised orbital angular momentum [2], The present formulation owes much to the derivation of the effective spin Hamiltonian by Pryce [3] and Griffith [4], Miller published a pivotal paper in 1969 [5] in which he built on these ideas to show how a general effective Hamiltonian for a diatomic molecule can be constructed. He has applied his approach in a number of specific situations, for example, to the description of N2 in its A 3 + state [6], described in chapter 8. In this book, we follow the treatment of Brown, Colbourn, Watson and Wayne [7], except that we incorporate spherical tensor methods where advantageous. It is a strange fact that the standard form of the effective Hamiltonian for a polyatomic molecule [2] was established many years before that for a diatomic molecule [7]. 

Brown, Colbourn, Watson and Wayne [7] have shown that this type of indeterminacy occurs for any 2,s 1 A state which conforms to Hund s case (a) coupling. This result can be established most easily by applying a contact transformation to the effective Hamiltonian. Let us divide 3Q,V into a principal part 3C and a remainder 3( ... 

Another complication arises from the fact that the rotational constants are usually obtained as effective fitting parameters of a reduced rotational Hamiltonian [14]. Not only do the numerical values of the rotational constants depend on the exact form of the reduced Hamiltonian, they also contain small contributions from quartic and higher order centrifugal distortion terms. Watson [14] has proposed to always determine the so-called determinable combinations of these constants. The values of these combinations are independent of the form of the reduction, although they still contain small contributions from the distortion terms. Up to the quartic centrifugal distortion terms, the determinable combinations of the rotational constants are... 

We return now to the coupled-channels approach based on operator equations. The formalism is adequately covered in several books (see e.g. Goldberger and Watson, 1964 Newton, 1966 Levine, 1969) and we shall only present the main equations. Assume for simplicity that only two reaction channels a (for A + BC)and 6(AB + C) exist. The total Hamiltonian H may be split into two terms, a channel Hamiltonian Hc for free motion and a channel interaction Vc, with c a, b. If a is the initial free state and we want the scattering states in channel a, i.e. those in the absence of rearrangement, then the Lippmann-Schwinger equation gives... 

There are two versions of MM. One, that we refer to as single-reference MM is based on the exact Watson Hamiltonian, which is the Hamiltonian in rectilinear... 

Previous fully quantum mechanical studies of predissociation phenomena in triatomic molecules do not, to our knowledge, use a Hamiltonian that has a non-zero total angular momentum. Tennyson et al[43, 44, 45, 46, 47, 48, 49, 50, 51] solve the same equations as we do but have not yet, to our knowledge, treated any predissociation problems. The adiabatic rotation approximation method of Carter and Bowman[52] plus a complex C2 modification have, on the other hand, been used to compute rovibrational energies and widths in the HCO[53, 54] and HOCl[55, 56, 57] molecules. This method is based upon the the Wilson and Howard[58], Darling and Dennison[59] and Watson[60] formalism. It is less transparent but the exact formalism in refs.[58, 59, 60] is equivalent to the one presented here and in ref [43]. While both we and Tennyson et al[43] include the exact Hamiltonian in our formalism the latter authors 152] use an approximate method which they have analysed and motivated. 

State calculations. With the extensions provided, the method can be applied to the full Watson Hamiltonian [51] for the vibrational problem. The efficiency of the method depends greatly on the nature of the anharmonic potential that represents couphng between different vibrational modes. In favorable cases, the latter can be represented as a low-order polynomial in the normal-mode displacements. When this is not the case, the computational effort increases rapidly. The Cl-VSCF is expected to scale as or worse with the number N of vibrational modes. The most favorable situation is obtained when only pairs of normal modes are coupled in the terms of the polynomial representation of the potential. The VSCF-Cl method was implemented in MULTIMODE [47,52], a code for anharmonic vibrational spectra that has been used extensively. MULTIMODE has been successfully applied to relatively large molecules such as benzene [53]. Applications to much larger systems could be difficult in view of the unfavorable scalability trend mentioned above. 

In our approach, we use the Watson Hamiltonian [14]. For the general case of nonlinear molecules this Hamiltonian is given (in atomic units) by... 

The effective form for the spin-rotation Hamiltonian is given by Brown and Watson (1977) as... 

With the above approach we can combine the use of curvilinear normal coordinates with the Eckart frame. When we do so, the harmonic oscillator, rigid rotor, and, to lowest order, the Coriolis and centrifugal coupling contributions to H have exactly the same form as those found for the more commonly used Watson Hamiltonian (58). 

Along these lines an efficient protocol, called DEWE has been developed [155] which is based on the DVR of the Eckart-Watson Hamiltonians involving an exact inclusion of potentials expressed in an arbitrary set of coordinates. The DEWE procedure has been tested both for nonlinear (H2O, H3, and CH4) and linear (CO2, HCN, and HNC) molecules. 

Watson, J.K.G. Simplification of the molecular vibration-rotation Hamiltonian, Mol. Phys. 1968, 5, 479-90. 

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