Molecular Hamiltonian - Watson's Nuclear Motion Hamiltonian

Watson's Nuclear Motion Hamiltonian

In order to obtain a Hamiltonian for external (translation and rotation) motions coupled to the internal (vibrational) motions, it is common to return at this point to classical mechanics and to formulate the classical kinetic energy corresponding to these motions of the nuclei. Classically it is easy to separate the translational—center of mass—motion from the other motions. However, the separation of the rotational from the vibrational motion is more difficult and is not completely possible. This ro-vibrational separation was first achieved by Eckart in 1935 by imposing by what is now known as Eckart conditions. Since the problem is described in a frame (an "Eckart" frame) that rotates with the molecule, and hence is a non-inertial frame, energies associated with the fictitious forces: centrifugal and Coriolis force appear in the kinetic energy.

In general, the classical kinetic energy T defines the metric tensor g = (g_ij) associated with the curvilinear coordinates s = (s_i) through

The quantization step is the transformation of this classical kinetic energy into a quantum mechanical operator. It is common to follow Podolsky by writing down the Laplace–Beltrami operator in the same (generalized, curvilinear) coordinates s as used for the classical form. The equation for this operator requires the inverse of the metric tensor g and its determinant. Multiplication of the Laplace–Beltrami operator by gives the required quantum mechanical kinetic energy operator. When we apply this recipe to Cartesian coordinates, which have unit metric, the same kinetic energy is obtained as by application of the quantization rules.

The nuclear motion Hamiltonian was obtained by Wilson and Howard in 1936, who followed this procedure, and further refined by Darling and Dennison in 1940. It remained the standard until 1968, when Watson was able to simplify it drastically by commuting through the derivatives the determinant of the metric tensor. We will give the ro-vibrational Hamiltonian obtained by Watson, which often is referred to as the Watson Hamiltonian. Before we do this we must mention that a derivation of this Hamiltonian is also possible by starting from the Laplace operator in Cartesian form, application of coordinate transformations, and use of the chain rule. The Watson Hamiltonian, describing all motions of the N nuclei, is

$\hat{H} = -\frac{\hbar^2}{2M_\mathrm{tot}} \sum_{\alpha=1}^3 \frac{\partial^2}{\partial X_\alpha^2} +\frac{1}{2} \sum_{\alpha,\beta=1}^3 \mu_{\alpha\beta} (\mathcal{P}_\alpha - \Pi_\alpha)(\mathcal{P}_\beta - \Pi_\beta) +U -\frac{\hbar^2}{2} \sum_{s=1}^{3N-6} \frac{\partial^2}{\partial q_s^2} + V .$

The first term is the center of mass term

$\mathbf{X} \equiv \frac{1}{M_\mathrm{tot}} \sum_{i=1}^N M_i \mathbf{R}_i \quad\mathrm{with}\quad M_\mathrm{tot} \equiv \sum_{i=1}^N M_i.$

The second term is the rotational term akin to the kinetic energy of the rigid rotor. Here is the α component of the body-fixed rigid rotor angular momentum operator, see this article for its expression in terms of Euler angles. The operator is a component of an operator known as the vibrational angular momentum operator (although it does not satisfy angular momentum commutation relations),

$\Pi_\alpha = -i\hbar \sum_{s,t=1}^{3N-6} \zeta^{\alpha}_{st} \; q_s \frac{\partial}{\partial q_t}$

with the Coriolis coupling constant:

$\zeta^{\alpha}_{st} = \sum_{i=1}^N \sum_{\beta,\gamma=1}^3 \epsilon_{\alpha\beta\gamma} Q_{s, i\beta}\,Q_{t,i\gamma} \;\; \mathrm{and}\quad\alpha=1,2,3.$

Here ε_αβγ is the Levi-Civita symbol. The terms quadratic in the are centrifugal terms, those bilinear in and are Coriolis terms. The quantities Q _{s, iγ} are the components of the normal coordinates introduced above. Alternatively, normal coordinates may be obtained by application of Wilson's GF method. The 3 × 3 symmetric matrix is called the effective reciprocal inertia tensor. If all q _s were zero (rigid molecule) the Eckart frame would coincide with a principal axes frame (see rigid rotor) and would be diagonal, with the equilibrium reciprocal moments of inertia on the diagonal. If all q _s would be zero, only the kinetic energies of translation and rigid rotation would survive.

The potential-like term U is the Watson term:

$U = -\frac{1}{8} \sum_{\alpha=1}^3 \mu_{\alpha\alpha}$

proportional to the trace of the effective reciprocal inertia tensor.

The fourth term in the Watson Hamiltonian is the kinetic energy associated with the vibrations of the atoms (nuclei) expressed in normal coordinates q_s, which as stated above, are given in terms of nuclear displacements ρ_iα by

$q_s = \sum_{i=1}^N \sum_{\alpha=1}^3 Q_{s, i\alpha} \rho_{i\alpha}\quad\mathrm{for}\quad s=1,\ldots, 3N-6.$

Finally V is the unexpanded potential energy by definition depending on internal coordinates only. In the harmonic approximation it takes the form

$V \approx \frac{1}{2} \sum_{s=1}^{3N-6} f_s q_s^2.$

Read more about this topic: Molecular Hamiltonian

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