Hamiltonian for triatomic molecules

We find it convenient to reverse the historical ordering and to stait with (neatly) exact nonrelativistic vibration-rotation Hamiltonians for triatomic molecules. From the point of view of molecular spectroscopy, the optimal Hamiltonian is that which maximally decouples from each other vibrational and rotational motions (as well different vibrational modes from one another). It is obtained by employing a molecule-bound frame that takes over the rotations of the complete molecule as much as possible. Ideally, the only remaining motion observable in this system would be displacements of the nuclei with respect to one another, that is, molecular vibrations. It is well known, however, that such a program can be realized only approximately by introducing the Eckart conditions [38]. 

An alternative form of exact nonrelativistic vibration-rotation Hamiltonian for triatomic molecules (ABC) is that used by Handy, Carter (HC), and... 

Jensen, P. (1988), A New Morse Oscillator-Rigid Bender Internal Dynamics (MORBID) Hamiltonian for Triatomic Molecules, J. Mol. Sped. 128,478. 

As an aside, we note that the n-mode representation of the potential can be made in any set of coordinates and for use in the corresponding Hamiltonian. For triatomic and tetraatomic molecules it is possible to avoid this representation because the dimensionality of the internal space is 3 or 6 and exact treatments are possible in both normal [30,31 ] and various curvilinear coordinates which have been used in exact formulations and calculations for triatomic and tetraatomic molecules [32-34]. However, for larger molecules it could prove quite useful. 

The anharmonic potential energy is usually easier to represent in internal coordinates than in normal mode coordinates. However, what restricts the use of internal coordinates is the complicated expression for the vibrational/rotational kinetic energy in these coordinates (Pickett, 1972). It is difficult to write a general expression for the vibrational/rotational kinetic energy in internal coordinates and, instead, one usually considers Hamiltonians for specific molecules. For a bent triatomic molecule confined to rotate in a plane, the internal coordinate Hamiltonian is (Blais and Bunker, 1962) ... 

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... 

As discussed in Section 8.2.1, when nonadiabatic couplings cannot be neglected, the BO approximation is not reliable and coupled electronic states must be considered simultaneously with their interactions. For small systems, several full-dimensional approaches based on the vibronic or spin-rovibronic wavefunctions and taking into account simultaneously at least two electronic states have been developed [2, 100-104]. To quote some examples, the full vibronic Hamiltonians have been derived and employed for linear tetra-atomic molecules showing Renner-Teller interactions [103] or CXaY-like molecules of Csv symmetry showing Jahn-Teller interactions [104]. In the following, we will present the computational approaches based on the full rovibronic Carter-Handy Hamiltonian [100], developed for triatomic molecules and expressed in internal coordinates, which allows us to take into account up to three interacting electronic states [2, 100, 101]. 

Other Hamiltonians, which also do not require knowledge of the full three-dimensional potential curve for triatomic molecules but use only their bending part in addition to various sections of the stretching potential, are the so-called rigid bender,semi-rigid, and nonrigid bender Hamiltonians. ... 

The expressions for the rotational energy levels (i.e., also involving the end-over-end rotations, not considered in the previous works) of linear triatomic molecules in doublet and triplet II electronic states that take into account a spin orbit interaction and a vibronic coupling were derived in two milestone studies by Hougen [72,32]. In them, the isomorfic Hamiltonian was inboduced, which has later been widely used in treating linear molecules (see, e.g., [55]). 

Local-mode Hamiltonian for linear triatomic molecules... 

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,... 

For bent triatomic molecules one can easily construct a local mode Hamiltonian whose eigenvalues reproduce the spectrum ... 

Terms involving Majorana operators are nondiagonal, but their matrix elements can be simply constructed using the formulas discussed in the preceding sections. The total number of parameters to this order is 15 in addition to the vibron numbers, N and N2- This has to be compared with 4 for the first-order Hamiltonian (4.91). For XY2 molecules, some of the parameters are equal, Xi,i = X2,2 XU2 = X2,12, Y112 = Y2 U, A] = A2, reducing the total number to 11 plus the vibron number N = Aj = N2. Calculation of vibrational spectra of linear triatomic molecules with second-order Hamiltonians produce results with accuracies of the order of 1-5 cm-1. An example is shown in Table 4.8. 

Calculations of vibrational spectra of bent triatomic molecules with second order Hamiltonians produce results with accuracies of the order of 1-5 cm-1. An example is shown in Table 4.9. These results should again be compared with those of a Dunham expansion with cubic terms [Eq. (0.1)]. An example of such an expansion for the bent S02 molecule is given in Table 0.1. Note that because the Hamiltonian (4.96) has fewer parameters, it establishes definite numerical relations between the many Dunham coefficients similar to the so-called x — K relations (Mills and Robiette, 1985). For example, to the lowest order in l/N one has for the symmetric XY2 case the energies E(vu v2, V3) given by... 

The procedure for studying tetratomic molecules is identical to that followed in the study of diatomic and triatomic molecules. One begins with a local-mode Hamiltonian... 

---