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In
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, a rotational transition is an abrupt change in
angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
. Like all other properties of a quantum
particle In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...
, angular momentum is quantized, meaning it can only equal certain discrete values, which correspond to different
rotational energy Rotational energy or angular kinetic energy is kinetic energy due to the rotation of an object and is part of its total kinetic energy. Looking at rotational energy separately around an object's axis of rotation, the following dependence on the ob ...
states. When a particle loses angular momentum, it is said to have transitioned to a lower rotational energy state. Likewise, when a particle gains angular momentum, a positive rotational transition is said to have occurred. Rotational transitions are important in physics due to the unique
spectral lines A spectral line is a weaker or stronger region in an otherwise uniform and continuous spectrum. It may result from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used ...
that result. Because there is a net gain or loss of energy during a transition,
electromagnetic radiation In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength ...
of a particular
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
must be absorbed or emitted. This forms
spectral line A spectral line is a weaker or stronger region in an otherwise uniform and continuous spectrum. It may result from emission (electromagnetic radiation), emission or absorption (electromagnetic radiation), absorption of light in a narrow frequency ...
s at that frequency which can be detected with a
spectrometer A spectrometer () is a scientific instrument used to separate and measure Spectrum, spectral components of a physical phenomenon. Spectrometer is a broad term often used to describe instruments that measure a continuous variable of a phenomeno ...
, as in
rotational spectroscopy Rotational spectroscopy is concerned with the measurement of the energies of transitions between quantized rotational states of molecules in the gas phase. The rotational spectrum (power spectral density vs. rotational frequency) of chemical pola ...
or
Raman spectroscopy Raman spectroscopy () (named after physicist C. V. Raman) is a Spectroscopy, spectroscopic technique typically used to determine vibrational modes of molecules, although rotational and other low-frequency modes of systems may also be observed. Ra ...
.


Diatomic molecules

Molecules have
rotational energy Rotational energy or angular kinetic energy is kinetic energy due to the rotation of an object and is part of its total kinetic energy. Looking at rotational energy separately around an object's axis of rotation, the following dependence on the ob ...
owing to rotational motion of the nuclei about their
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point) is the unique point at any given time where the weight function, weighted relative position (vector), position of the d ...
. Due to quantization, these energies can take only certain discrete values. Rotational transition thus corresponds to transition of the molecule from one rotational energy level to the other through gain or loss of a
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
. Analysis is simple in the case of
diatomic molecules Diatomic molecules () are molecules composed of only two atoms, of the same or different chemical elements. If a diatomic molecule consists of two atoms of the same element, such as hydrogen () or oxygen (), then it is said to be homonuclear. Ot ...
.


Nuclear wave function

Quantum theoretical analysis of a molecule is simplified by use of
Born–Oppenheimer approximation In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the assumption that the wave functions of atomic nuclei and electrons in a molecule can be treated separately, based on the fact that the nuclei are much h ...
. Typically, rotational energies of molecules are smaller than
electronic transition In atomic physics and chemistry, an atomic electron transition (also called an atomic transition, quantum jump, or quantum leap) is an electron changing from one energy level to another within an atom or artificial atom. The time scale of a qua ...
energies by a factor of ≈ 10−3–10−5, where is electronic mass and is typical nuclear mass.Chapter 10, ''Physics of Atoms and Molecules'', B.H. Bransden and C.J. Jochain, Pearson education, 2nd edition. From
uncertainty principle The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...
, period of motion is of the order of the
Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
divided by its energy. Hence nuclear rotational periods are much longer than the electronic periods. So electronic and nuclear motions can be treated separately. In the simple case of a diatomic molecule, the radial part of the
Schrödinger Equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
for a nuclear wave function , in an electronic state , is written as (neglecting spin interactions) \left \frac \frac \left(R^2 \frac\right)+ \frac+ E_s(R)-E\right_s(\mathbf R) = 0 where is
reduced mass In physics, reduced mass is a measure of the effective inertial mass of a system with two or more particles when the particles are interacting with each other. Reduced mass allows the two-body problem to be solved as if it were a one-body probl ...
of two nuclei, is vector joining the two nuclei, is energy
eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
of electronic wave function representing electronic state and is orbital
momentum operator In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimensio ...
for the relative motion of the two nuclei given by N^2 = -\hbar^2 \left \frac \frac\left(\sin \Theta \frac\right)+ \frac \frac \right The total
wave function In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
for the molecule is \Psi_s = F_s(\mathbf R)\Phi_s(\mathbf R,\mathbf r_1, \mathbf r_2, \dots, \mathbf r_N) where are position vectors from center of mass of molecule to th electron. As a consequence of the Born-Oppenheimer approximation, the electronic wave functions is considered to vary very slowly with . Thus the Schrödinger equation for an electronic wave function is first solved to obtain for different values of . then plays role of a
potential well A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy ( kinetic energy in the case of a gravitational potential well) because it is cap ...
in analysis of nuclear wave functions .


Rotational energy levels

The first term in the above nuclear wave function equation corresponds to
kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
of nuclei due to their radial motion. Term represents rotational kinetic energy of the two nuclei, about their center of mass, in a given electronic state . Possible values of the same are different rotational energy levels for the molecule. Orbital angular momentum for the rotational motion of nuclei can be written as \mathbf N = \mathbf J - \mathbf L where is the total orbital angular momentum of the whole molecule and is the orbital angular momentum of the electrons. If internuclear vector is taken along z-axis, component of along z-axis – – becomes zero as \mathbf N = \mathbf R \times \mathbf P Hence J_z = L_z Since molecular wave function Ψs is a simultaneous
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of and , J^2 \Psi_s = J(J+1) \hbar^2 \Psi_s where J is called rotational quantum number and can be a positive integer or zero. J_z \Psi_s = M_j\hbar \Psi_s where . Also since electronic wave function is an eigenfunction of , L_z \Phi_s = \pm \Lambda\hbar \Phi_s Hence molecular wave function is also an eigenfunction of with eigenvalue . Since and are equal, is an eigenfunction of with same eigenvalue . As , we have . So possible values of rotational quantum number are J = \Lambda, \Lambda +1, \Lambda+2, \dots Thus molecular wave function is simultaneous eigenfunction of , and . Since molecule is in eigenstate of , expectation value of components perpendicular to the direction of z-axis (internuclear line) is zero. Hence \langle \Psi_s, L_x, \Psi_s\rangle = \langle L_x \rangle = 0 and \langle \Psi_s, L_y, \Psi_s\rangle = \langle L_y \rangle = 0 Thus \langle \mathbf J . \mathbf L \rangle = \langle J_z L_z \rangle = \langle ^2 \rangle Putting all these results together, \begin \langle \Phi_s , N^2, \Phi_s \rangle F_s(\mathbf R) &= \langle \Phi_s , \left(J^2 + L^2 - 2 \mathbf J \cdot \mathbf L\right) , \Phi_s \rangle F_s(\mathbf R) \\ &= \hbar^2 \left (J+1)-\Lambda^2\rightF_s(\mathbf R) + \langle \Phi_s , \left(^2 + ^2\right) , \Phi_s \rangle F_s(\mathbf R) \end The Schrödinger equation for the nuclear wave function can now be rewritten as - \frac\left \frac \left(R^2 \frac\right)- J(J+1)\right_s(\mathbf R)+ s(R)-E_s(\mathbf R) = 0 where _s(R) = E_s(R) - \frac + \frac \langle \Phi_s , \left(^2 + ^2\right), \Phi_s \rangle E′s now serves as effective potential in radial nuclear wave function equation.


Sigma states

Molecular states in which the total orbital momentum of electrons is zero are called sigma states. In sigma states . Thus . As nuclear motion for a stable molecule is generally confined to a small interval around where corresponds to internuclear distance for minimum value of potential , rotational energies are given by, E_r = \frac J(J+1) = \frac J(J+1) = BJ(J+1) with J = \Lambda, \Lambda +1, \Lambda+2, \dots is
moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular/rotational mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is defined relatively to a rotational axis. It is the ratio between ...
of the molecule corresponding to
equilibrium Equilibrium may refer to: Film and television * ''Equilibrium'' (film), a 2002 science fiction film * '' The Story of Three Loves'', also known as ''Equilibrium'', a 1953 romantic anthology film * "Equilibrium" (''seaQuest 2032'') * ''Equilibr ...
distance and is called rotational constant for a given electronic state . Since reduced mass is much greater than electronic mass, last two terms in the expression of are small compared to . Hence even for states other than sigma states, rotational energy is approximately given by above expression.


Rotational spectrum

When a rotational transition occurs, there is a change in the value of rotational quantum number . Selection rules for rotational transition are, when , and when , as absorbed or emitted photon can make equal and opposite change in total nuclear angular momentum and total electronic angular momentum without changing value of . The pure rotational spectrum of a diatomic molecule consists of lines in the far
infrared Infrared (IR; sometimes called infrared light) is electromagnetic radiation (EMR) with wavelengths longer than that of visible light but shorter than microwaves. The infrared spectral band begins with the waves that are just longer than those ...
or
microwave Microwave is a form of electromagnetic radiation with wavelengths shorter than other radio waves but longer than infrared waves. Its wavelength ranges from about one meter to one millimeter, corresponding to frequency, frequencies between 300&n ...
region. The frequency of these lines is given by \hbar \omega = E_r(J+1)-E_r(J) = 2B(J+1) Thus values of , and of a substance can be determined from observed rotational spectrum.


See also

* Vibrational transition


Notes


References

* * {{DEFAULTSORT:Rotational Transition Chemical physics