 Rigid rotor

The rigid rotor is a mechanical model that is used to explain rotating systems. An arbitrary rigid rotor is a 3dimensional rigid object, such as a top. To orient such an object in space three angles are required. A special rigid rotor is the linear rotor which is a 2dimensional object, requiring two angles to describe its orientation. An example of a linear rotor is a diatomic molecule. More general molecules like water (asymmetric rotor), ammonia (symmetric rotor), or methane (spherical rotor) are 3dimensional, see classification of molecules.
Contents
The linear rotor
The linear rigid rotor model consists of two point masses located at fixed distances from their center of mass. The fixed distance between the two masses and the values of the masses are the only characteristics of the rigid model. However, for many actual diatomics this model is too restrictive since distances are usually not completely fixed. Corrections on the rigid model can be made to compensate for small variations in the distance. Even in such a case the rigid rotor model is a useful point of departure (zerothorder model).
The classical linear rigid rotor
The classical linear rotor consists of two point masses m_{1} and m_{2} (with reduced mass ) each at a distance R. The rotor is rigid if R is independent of time. The kinematics of a linear rigid rotor is usually described by means of spherical polar coordinates, which form a coordinate system of R^{3}. In the physics convention the coordinates are the colatitude (zenith) angle , the longitudinal (azimuth) angle and the distance R. The angles specify the orientation of the rotor in space. The kinetic energy T of the linear rigid rotor is given by
where and are scale (or Lamé) factors.
Scale factors are of importance for quantum mechanical applications since they enter the Laplacian expressed in curvilinear coordinates. In the case at hand (constant R)
The classical Hamiltonian function of the linear rigid rotor is
The quantum mechanical linear rigid rotor
The linear rigid rotor model can be used in quantum mechanics to predict the rotational energy of a diatomic molecule. The rotational energy depends on the moment of inertia for the system, I. In the center of mass reference frame, the moment of inertia is equal to:
 I = μR^{2}
where μ is the reduced mass of the molecule and R is the distance between the two atoms.
According to quantum mechanics, the energy levels of a system can be determined by solving the Schrödinger equation:
where Ψ is the wave function and is the energy (Hamiltonian) operator. For the rigid rotor in a fieldfree space, the energy operator corresponds to the kinetic energy^{[1]} of the system:
where is Planck's constant divided by 2π and is the Laplacian. The Laplacian is given above in terms of spherical polar coordinates. The energy operator written in terms of these coordinates is:
This operator appears also in the Schrödinger equation of the hydrogen atom after the radial part is separated off. The eigenvalue equation becomes
The symbol represents a set of functions known as the spherical harmonics. Note that the energy does not depend on . The energy
is fold degenerate: the functions with fixed and have the same energy.
Introducing the rotational constant B, we write,
In the units of reciprocal length the rotational constant is,
with c the speed of light. If cgs units are used for h, c, and I, is expressed in wave numbers, cm^{−1}, a unit that is often used for rotationalvibrational spectroscopy. The rotational constant depends on the distance R. Often one writes where R_{e} is the equilibrium value of R (the value for which the interaction energy of the atoms in the rotor has a minimum).
A typical rotational spectrum consists of a series of peaks that correspond to transitions between levels with different values of the angular momentum quantum number (). Consequently, rotational peaks appear at energies corresponding to an integer multiple of .
Selection rules
Rotational transitions of a molecule occur when the molecule absorbs a photon [a particle of a quantized electromagnetic (em) field]. Depending on the energy of the photon (i.e., the wavelength of the em field) this transition may be seen as a sideband of a vibrational and/or electronic transition. Pure rotational transitions, in which the vibronic (= vibrational plus electronic) wave function does not change, occur in the microwave region of the electromagnetic spectrum.
Typically, rotational transitions can only be observed when the angular momentum quantum number changes by 1 (). This selection rule arises from a firstorder perturbation theory approximation of the timedependent Schrödinger equation. According to this treatment, rotational transitions can only be observed when one or more components of the dipole operator have a nonvanishing transition moment. If z is the direction of the electric field component of the incoming em wave, the transition moment is,
A transition occurs if this integral is nonzero. By separating the rotational part of the molecular wavefunction from the vibronic part, one can show that this means that the molecule must have a permanent dipole moment. After integration over the vibronic coordinates the following rotational part of the transition moment remains,
Here is the z component of the permanent dipole moment. The moment μ is the vibronically averaged component of the dipole operator. Only the component of the permanent dipole along the axis of a heteronuclear molecule is nonvanishing. By the use of the orthogonality of the spherical harmonics it is possible to determine which values of l, m, l', and m' will result in nonzero values for the dipole transition moment integral. This constraint results in the observed selection rules for the rigid rotor:
Nonrigid linear rotor
The rigid rotor is commonly used to describe the rotational energy of diatomic molecules but it is not a completely accurate description of such molecules. This is because molecular bonds (and therefore the interatomic distance R) are not completely fixed; the bond between the atoms stretches out as the molecule rotates faster (higher values of the rotational quantum number l). This effect can be accounted for by introducing a correction factor known as the centrifugal distortion constant (bars on top of various quantities indicate that these quantities are expressed in cm^{−1}):
where
 is the fundamental vibrational frequency of the bond (in cm^{1}). This frequency is related to the reduced mass and the force constant (bond strength) of the molecule according to
The nonrigid rotor is an acceptably accurate model for diatomic molecules but is still somewhat imperfect. This is because, although the model does account for bond stretching due to rotation, it ignores any bond stretching due to vibrational energy in the bond (anharmonicity in the potential).
Arbitrarily shaped rigid rotor
An arbitrarily shaped rigid rotor is a rigid body of arbitrary shape with its center of mass fixed (or in uniform rectilinear motion) in fieldfree space R^{3}, so that its energy consists only of rotational kinetic energy (and possibly constant translational energy that can be ignored). A rigid body can be (partially) characterized by the three eigenvalues of its moment of inertia tensor, which are real nonnegative values known as principal moments of inertia. In microwave spectroscopy—the spectroscopy based on rotational transitions—one usually classifies molecules (seen as rigid rotors) as follows:
 spherical rotors
 symmetric rotors
 oblate symmetric rotors
 prolate symmetric rotors
 asymmetric rotors
This classification depends on the relative magnitudes of the principal moments of inertia.
Coordinates of the rigid rotor
Different branches of physics and engineering use different coordinates for the description of the kinematics of a rigid rotor. In molecular physics Euler angles are used almost exclusively. In quantum mechanical applications it is advantageous to use Euler angles in a convention that is a simple extension of the physical convention of spherical polar coordinates.
The first step is the attachment of a righthanded orthonormal frame (3dimensional system of orthogonal axes) to the rotor (a bodyfixed frame) . This frame can be attached arbitrarily to the body, but often one uses the principal axes frame—the normalized eigenvectors of the inertia tensor, which always can be chosen orthonormal, since the tensor is Hermitian. When the rotor possesses a symmetryaxis, it usually coincides with one of the principal axes. It is convenient to choose as bodyfixed zaxis the highestorder symmetry axis.
One starts by aligning the bodyfixed frame with a spacefixed frame (laboratory axes), so that the bodyfixed x, y, and z axes coincide with the spacefixed X, Y, and Z axis. Secondly, the body and its frame are rotated actively over a positive angle around the zaxis (by the righthand rule), which moves the y to the y'axis. Thirdly, one rotates the body and its frame over a positive angle around the y'axis. The zaxis of the bodyfixed frame has after these two rotations the longitudinal angle (commonly designated by ) and the colatitude angle (commonly designated by ), both with respect to the spacefixed frame. If the rotor were cylindrical symmetric around its zaxis, like the linear rigid rotor, its orientation in space would be unambiguously specified at this point.
If the body lacks cylinder (axial) symmetry, a last rotation around its zaxis (which has polar coordinates and ) is necessary to specify its orientation completely. Traditionally the last rotation angle is called .
The convention for Euler angles described here is known as the z'' − y' − z convention; it can be shown (in the same manner as in this article) that it is equivalent to the z − y − z convention in which the order of rotations is reversed.
The total matrix of the three consecutive rotations is the product
Let be the coordinate vector of an arbitrary point in the body with respect to the bodyfixed frame. The elements of are the 'bodyfixed coordinates' of . Initially is also the spacefixed coordinate vector of . Upon rotation of the body, the bodyfixed coordinates of do not change, but the spacefixed coordinate vector of becomes,
In particular, if is initially on the spacefixed Zaxis, it has the spacefixed coordinates
which shows the correspondence with the spherical polar coordinates (in the physical convention).
Knowledge of the Euler angles as function of time t and the initial coordinates determine the kinematics of the rigid rotor.
Classical kinetic energy
The following text forms a generalization of the wellknown special case of the rotational energy of an object that rotates around one axis.
It will be assumed from here on that the bodyfixed frame is a principal axes frame; it diagonalizes the instantaneous inertia tensor (expressed with respect to the spacefixed frame), i.e.,
where the Euler angles are timedependent and in fact determine the time dependence of by the inverse of this equation. This notation implies that at t = 0 the Euler angles are zero, so that at t = 0 the bodyfixed frame coincides with the spacefixed frame.
The classical kinetic energy T of the rigid rotor can be expressed in different ways:
 as a function of angular velocity
 in Lagrangian form
 as a function of angular momentum
 in Hamiltonian form.
Since each of these forms has its use and can be found in textbooks we will present all of them.
Angular velocity form
As a function of angular velocity T reads,
with
The vector contains the components of the angular velocity of the rotor expressed with respect to the bodyfixed frame. It can be shown that is not the time derivative of any vector, in contrast to the usual definition of velocity^{[2]}. The dots over the timedependent Euler angles indicate time derivatives. The angular velocity satisfies equations of motion known as Euler's equations (with zero applied torque, since by assumption the rotor is in fieldfree space).
Lagrange form
Backsubstitution of the expression of into T gives the kinetic energy in Lagrange form (as a function of the time derivatives of the Euler angles). In matrixvector notation,
where is the metric tensor expressed in Euler angles—a nonorthogonal system of curvilinear coordinates—
Angular momentum form
Often the kinetic energy is written as a function of the angular momentum of the rigid rotor. This vector is a conserved (timeindependent) quantity. With respect to the bodyfixed frame it has the components , which can be shown to be related to the angular velocity,
Since the bodyfixed frame moves (depends on time) these components are not time independent. If we were to represent with respect to the stationary spacefixed frame, we would find time independent expressions for its components. The kinetic energy is given by
Hamilton form
The Hamilton form of the kinetic energy is written in terms of generalized momenta
where it is used that the is symmetric. In Hamilton form the kinetic energy is,
with the inverse metric tensor given by
This inverse tensor is needed to obtain the LaplaceBeltrami operator, which (multiplied by ) gives the quantum mechanical energy operator of the rigid rotor.
The classical Hamiltonian given above can be rewritten to the following expression, which is needed in the phase integral arising in the classical statistical mechanics of rigid rotors,
Quantum mechanical rigid rotor
See also: Rotational spectroscopyAs usual quantization is performed by the replacement of the generalized momenta by operators that give first derivatives with respect to its canonically conjugate variables (positions). Thus,
and similarly for p_{β} and p_{γ}. It is remarkable that this rule replaces the fairly complicated function p_{α} of all three Euler angles, time derivatives of Euler angles, and inertia moments (characterizing the rigid rotor) by a simple differential operator that does not depend on time or inertia moments and differentiates to one Euler angle only.
The quantization rule is sufficient to obtain the operators that correspond with the classical angular momenta. There are two kinds: spacefixed and bodyfixed angular momentum operators. Both are vector operators, i.e., both have three components that transform as vector components among themselves upon rotation of the spacefixed and the bodyfixed frame, respectively. The explicit form of the rigid rotor angular momentum operators is given here (but beware, they must be multiplied with ). The bodyfixed angular momentum operators are written as . They satisfy anomalous commutation relations.
The quantization rule is not sufficient to obtain the kinetic energy operator from the classical Hamiltonian. Since classically p_{β} commutes with cos β and sin β and the inverses of these functions, the position of these trigonometric functions in the classical Hamiltonian is arbitrary. After quantization the commutation does no longer hold and the order of operators and functions in the Hamiltonian (energy operator) becomes a point of concern. Podolsky^{[1]} proposed in 1928 that the LaplaceBeltrami operator (times ) has the appropriate form for the quantum mechanical kinetic energy operator. This operator has the general form (summation convention: sum over repeated indices—in this case over the three Euler angles ):
where  g  is the determinant of the gtensor:
Given the inverse of the metric tensor above, the explicit form of the kinetic energy operator in terms of Euler angles follows by simple substitution. (Note: The corresponding eigenvalue equation gives the Schrödinger equation for the rigid rotor in the form that it was solved for the first time by Kronig and Rabi^{[3]} (for the special case of the symmetric rotor). This is one of the few cases where the Schrödinger equation can be solved analytically. All these cases were solved within a year of the formulation of the Schrödinger equation.)
Nowadays it is common to proceed as follows. It can be shown that can be expressed in bodyfixed angular momentum operators (in this proof one must carefully commute differential operators with trigonometric functions). The result has the same appearance as the classical formula expressed in bodyfixed coordinates,
The action of the on the Wigner Dmatrix is simple. In particular
so that the Schrödinger equation for the spherical rotor (I = I_{1} = I_{2} = I_{3}) is solved with the (2j + 1)^{2} degenerate energy equal to .
The symmetric top (= symmetric rotor) is characterized by I_{1} = I_{2}. It is a prolate (cigar shaped) top if I_{3} < I_{1} = I_{2}. In the latter case we write the Hamiltonian as
and use that
Hence
The eigenvalue E_{j0} is 2j + 1fold degenerate, for all eigenfunctions with have the same eigenvalue. The energies with k > 0 are 2(2j + 1)fold degenerate. This exact solution of the Schrödinger equation of the symmetric top was first found in 1927.^{[3]}
The asymmetric top problem () is not exactly soluble.
See also
 Balancing machine
 Gyroscope
 Infrared spectroscopy
 Rigid body
 Rotational spectroscopy
 Spectroscopy
 Vibrational spectroscopy
 Quantum rotor model
References
 ^ ^{a} ^{b} Podolsky, B. (1928). Phys. Rev. 32: 12. Bibcode 1928PhRv...32...12J. doi:10.1103/PhysRev.32.12.
 ^ Chapter 4.9 of Goldstein, H.; Poole, C. P.; Safko, J. L. (2001). Classical Mechanics (Third ed.). San Francisco: Addison Wesley Publishing Company. ISBN 0201657023.
 ^ ^{a} ^{b} R. de L. Kronig and I. I. Rabi (1927). "The Symmetrical Top in the Undulatory Mechanics". Phys. Rev. 29: 262–269. Bibcode 1927PhRv...29..262K. doi:10.1103/PhysRev.29.262.
General references
 D. M. Dennison (1931). "The Infrared Spectra of Polyatomic Molecules Part I". Rev. Mod. Physics 3: 280–345. Bibcode 1931RvMP....3..280D. doi:10.1103/RevModPhys.3.280. (Especially Section 2: The Rotation of Polyatomic Molecules).
 Van Vleck, J. H. (1951). "The Coupling of Angular Momentum Vectors in Molecules". Rev. Mod. Physics 23: 213–227. Bibcode 1951RvMP...23..213V. doi:10.1103/RevModPhys.23.213.
 McQuarrie, Donald A (1983). Quantum Chemistry. Mill Valley, Calif.: University Science Books. ISBN 093570213X.
 Goldstein, H.; Poole, C. P.; Safko, J. L. (2001). Classical Mechanics (Third ed.). San Francisco: Addison Wesley Publishing Company. ISBN 0201657023. (Chapters 4 and 5)
 Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics. SpringerVerlag. ISBN 0387968903. (Chapter 6).
 Kroto, H. W. (1992). Molecular Rotation Spectra. New York: Dover.
 Gordy, W.; Cook, R. L. (1984). Microwave Molecular Spectra (Third ed.). New York: Wiley. ISBN 0471086819.
 Papoušek, D.; Aliev, M. T. (1982). Molecular VibrationalRotational Spectra. Amsterdam: Elsevier. ISBN 0444997377.
Categories: Molecular physics
 Rigid bodies
 Rotation
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