Molecular vibrations

Molecular vibrations are one of three types of molecular motion, which also include translational motion (when all atoms of the molecule are displaced in the same direction) and rotational motion (when the molecule rotates by a certain angle). In contrast to the last two cases, when the geometry of the molecule does not change, during vibrations, the position of atoms relative to each other changes.

In general, a molecule of N atoms has 3 N - 6 normal vibrations , with the exception of linear molecules with 3 N - 5 vibrations. A diatomic molecule , as a special case of a linear one, has only one vibration, in which the distance between the two atoms of the molecule changes.

Content

Types of Oscillations

In the case of polyatomic molecules, the vibrations are quite complex, and they are usually described as a combination of vibrations of different fragments of the molecule. Often these are triatomic fragments of a molecule, for example, a methylene group (−CH ₂ -) in organic molecules. Six types of vibrations of a triatomic fragment of a molecule can be distinguished: symmetric and antisymmetric stretching vibrations, scissor, pendulum, fan and torsional vibrations. For molecules containing only three atoms, for example, water molecules, the last three types of vibrations do not exist, since they simply correspond to rotations of the molecule relative to three mutually perpendicular axes (for these vibrations, the distances between the three atoms of the fragment do not change).

Stretching		Scissoring
Symmetrical	Antisymmetric	Scissoring

Pendulum (Rocking)	Fan Wagging	Twisting

Oscillation Energy

Classical Mechanics

H H molecule as an example of an anharmonic oscillator oscillating with an energy of E ₃ . D ₀ is the dissociation energy , r ₀ is the bond length , U is the potential energy . Energy is expressed in wave numbers .

In classical mechanics , molecular vibrations are considered from the perspective that bonds between atoms behave like springs. In the harmonic approximation, the oscillations obey Hooke's law : force $F,$ {\ displaystyle F,} which you need to apply to stretch the spring is directly proportional to the amount of stretching $Q$ ${\ displaystyle Q}$ . The proportionality constant in the case of molecular vibrations is called the force constant $k .$ ${\ displaystyle k.}$

\mathrm {F} =-kQ.

{\ displaystyle \ mathrm {F} = -kQ.}

From Newton’s second law, this force is also equal to the product of the reduced mass $\mu$ ${\ displaystyle \ mu}$ and acceleration:

\mathrm {F} =\mu {\frac {d^{2}Q}{dt^{2}}}.

{\ displaystyle \ mathrm {F} = \ mu {\ frac {d ^ {2} Q} {dt ^ {2}}}.}

From this we obtain the ordinary differential equation :

\mu {\frac {d^{2}Q}{dt^{2}}}+kQ=0.

{\ displaystyle \ mu {\ frac {d ^ {2} Q} {dt ^ {2}}} + kQ = 0.}

His solution is harmonic oscillations :

Q(t)=A\cos(2\pi \nu t);\ \ \nu ={1 \over {2\pi }}{\sqrt {k \over \mu }},

{\ displaystyle Q (t) = A \ cos (2 \ pi \ nu t); \ \ \ nu = {1 \ over {2 \ pi}} {\ sqrt {k \ over \ mu}},}

Where $A$ ${\ displaystyle A}$ - amplitude of the oscillation coordinate $Q .$ ${\ displaystyle Q.}$ For a diatomic molecule AB, the reduced mass $\mu$ ${\ displaystyle \ mu}$ equals:

{\frac {1}{\mu }}={\frac {1}{m_{A}}}+{\frac {1}{m_{B}}},

{\ displaystyle {\ frac {1} {\ mu}} = {\ frac {1} {m_ {A}}} + {\ frac {1} {m_ {B}}},}

where m _A and m _B are the masses of atoms A and B.

In the harmonic approximation, the potential energy of the molecule $V$ ${\ displaystyle V}$ is a quadratic function of the normal coordinate. In this case, the force constant is equal to the second derivative of the potential energy:

k={\frac {\partial ^{2}V}{\partial Q^{2}}}.

{\ displaystyle k = {\ frac {\ partial ^ {2} V} {\ partial Q ^ {2}}}.}

Quantum Mechanics

In quantum mechanics , as well as in classical mechanics , the potential energy of a harmonic oscillator is a quadratic function of the normal coordinate. The following values of the vibrational energy are possible from the solution of the Schrödinger equation :

E_{n}=h\left(n+{1 \over 2}\right)\nu =h\left(n+{1 \over 2}\right){1 \over {2\pi }}{\sqrt {k \over m}}\!,

{\ displaystyle E_ {n} = h \ left (n + {1 \ over 2} \ right) \ nu = h \ left (n + {1 \ over 2} \ right) {1 \ over {2 \ pi}} { \ sqrt {k \ over m}} \ !,}

where n is a quantum number that takes values 0, 1, 2 ... In molecular spectroscopy, this vibrational quantum number is often denoted as v ^[1] ^[2] , since other types of molecular energy are possible, which correspond to other quantum numbers.

Notes

↑ JM Hollas, Modern Spectroscopy (3rd ed., John Wiley 1996), p21
↑ PW Atkins and J. de Paula, Physical Chemistry (8th ed., WH Freeman 2006), p. 291 and p. 453

[1] JM Hollas, Modern Spectroscopy (3rd ed., John Wiley 1996), p21

[2] PW Atkins and J. de Paula, Physical Chemistry (8th ed., WH Freeman 2006), p. 291 and p. 453