Schrodinger equation rigid-rotor

The simplest approach to modeling rotational spectroscopy is the so-called rigid-rotor approximation. In this approximation, the geometry of the molecule is assumed to be constant at the equilibrium geometry qeq. In that case, V(qeq) in Eq. (9.37) becomes simply a multiplicative constant, so that we may write the rigid-rotor rotational Schrodinger equation as... 

As was also previously noted in Section 9.3.1, the completely general rigid-rotor Schrodinger equation for a molecule characterized by three unique axes and associated moments of inertia does not lend itself to easy solution. However, by pursuing a generalization of the classical mechanical rigid-rotor problem, one can derive a quantum mechanical approximation that is typically quite good. Within that approximation, the rotational partition function becomes... 

Solution of the Schrodinger equation for a rigid rotor shows that the rotational energy Er is quantized with values... 

Two points should be made about this expression. First it has exactly the same form as that introduced in our discussion of the hydrogen atom, equation (6.7). Second, for the rigid rotor the first term in (6.157) disappears, and the Schrodinger equation becomes... 

We recall from our discussion (section 6.8.1) of the rigid rotor model of a diatomic molecule that the Schrodinger equation is... 

From Eqs. (3.40) and (3.35) it is obvious that the inversion—rotation wave functions i//°. (0,, X, p) of NH3 which are the eigenfunctions of the operator, , can be written as a product of the rigid-rotor symmetric top wave functions depending on the Euler angles 0,4>, x and the inversion wave functions, depending on the variable p. Integration of the Schrodinger equation... 

Figure 21.1 Eigenstates and energy eigenvalues of the Schrodinger equation of a rigid Dj rotor in a harmonic twofold potential for two different depths of the potential barrier (adapted from Ref [40]). Upper panel ...

The construction of a Hamiltonian is normally an easy problem. The solution of the Schrodinger equation, on the contrary, represents a serious problem. It can be solved exactly for several model cases a particle in a box (one-, two- or three-dimensional), harmonic oscillator, rigid rotor, a particle passing through a potential barrier, hydrogen atom, etc. In most applications only an approximate solution of the Schrodinger equation is attainable. 

In the remainder of this chapter, we assume that the Born-Oppenheimer approximation is good, and that Eq. 3.16 holds. In this section we consider the rotational motion of a idealized rigid diatomic rotor, in which R is fixed at Rq. Then U,f R) = U Rq) is a constant that we may set to zero for convenience. Using Eq. 3.15 for V in the Schrodinger equation (3.16), we immediately have... 

According to Schrodinger s equation, the energy of the rigid rotor of the moment of inertia 1 (see Appendix A. 2) is given by ... 

It may not be surprising to learn that the 0 and parts of the wavefunction I are the spherical harmonics, discussed earlier for the 3-D rigid rotor. These solutions impose two integers called quantum numbers, and m, which determine the exact mathematical expression. Because the Schrodinger equation can be written in terms of the total angular momentum operator L, we can substitute the solutions for that part of the operator into the Schrodinger equation and get a differential equation in terms of r and R alone ... 

There is a polynomial solution for the radial part of the H atom solution of the Schrodinger equation. The functions are related to previously studied Laguerre polynomials. The total solution for the H atom is the product of the rigid rotor (0, < ) wave functions with the Laguerre radial functions. The eigenvalues for the orbitals are exactly the same as for the... 

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