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8.2 Molecular vibrations 285

Fig. 8.7 Coordinates of the triangular molecule H

of the dipole moment takes place along or perpendicular to the axis of symmetry

of the molecule which is the internuclear axis. The antisymmetric longitudinal or

the ν

-vibration induces a dipole moment parallel to the axis of symmetry. Thus

- and ν

-vibrations are active in the infrared spectrum. The most important CO

band in the infrared spectral range is centered at the wave number ˜ν = 667.40 cm

−1

(λ = 15 µm) which results from the ν

-vibration. In contrast to the CO

molecule,

the diatomic gases N

and O

, abundantly occurring in the Earth’s atmosphere,

do not possess permanent electric dipole moments and, therefore, do not exhibit

infrared absorption bands.

We have now examined in some detail the normal vibrations of a linear triatomic

molecule. The actual internal vibrations of a semi-rigid system (small amplitude

vibrations) may be very complicated. However, the motion can always be decom-

posed into a sum of elementary motions described by the normal vibrational modes.

The frequency equations (8.22) and (8.29) are in exact agreement with the results

shown in the standard reference book Infrared and Raman Spectra by Herzberg

(1964b) where many details can be found.

8.2.4 Nonlinear triatomic molecules

It is not always as simple as in the case of the linear triatomic molecule to ﬁnd the

normal coordinates. This will become apparent in the discussion of the nonlinear

triatomic H

O, see Figure 8.7. The three masses are labelled as 1, 2 and 3. The

equilibrium distance from the central mass m

to the masses m

is l.

Employing the conservation of the center of mass according to (8.15) we ﬁnd

+ x

) + m

= 0, m

+ y

) + m

= 0 (8.31)

If we place ourselves in the resting position of the atom m

so that r

0,2

=0, noting

that |r

0,1

|=|r

0,3

|, we ﬁnd that the conservation of angular momentum is given by

− y

) sin α = (x

+ x

) cos α (8.32)

286 Absorption by gases

Details of the calculation will be left up to the exercises. Now we must ﬁnd the

change of l due to stretching between the masses m

(mass point 1) and m

(mass

point 2) and between m

(2) and m

(3). The changes l

and l

are found by

projecting the vectors x

− x

and x

− x

on the directions of the lines connecting

the masses m

(point 1) and m

and m

(point 3). A simple calculation

gives

δl

= (x

− x

) sin α + (y

− y

) cos α

δl

=−(x

− x

) sin α + (y

− y

) cos α

(8.33)

The change of the angle 2α is found by projecting the vectors x

− x

and x

− x

on the directions perpendicular to the lines connecting the points 1 and 2 and 2 and

3. The result is given by

δ =

[

− x

) cos α − (y

− y

) sin α − (x

− x

) cos α − (y

− y

) sin α

]

(8.34)

Details of the calculations are left to the exercises.

The Lagrangian function is found as

L =





−



(δl

)

+ (δl

)



−

(lδ)

(8.35)

where

i +

j, etc. The third and fourth right-hand side terms describe the

potential energy of the extension of the springs connecting the masses and of the

bending of the molecule.

As motivated by the previous example, we introduce the new coordinates

= x

+ x

, q

s,1

= x

− x

, q

s,2

= y

+ y

=⇒

+ q

s,1

), x

=−

, x

− q

s,1

)

s,2

+ q

cot α), y

=−

s,2

, y

s,2

− q

cot α)

(8.36)

where we have employed the conservation of the center of mass. A simple but

tedious calculation gives the Lagrangian function of the system

L =



sin



s,1

s,2

−q



sin



1 +

sin



−

s,1

sin

α + 2k

cos

α)

−q

s,2

cos

α + 2k

sin

α) + q

s,1

s,2

(2k

− k

) sin α cos α

(8.37)

8.2 Molecular vibrations 287

We immediately recognize that q

is a normal coordinate since no cross-term occurs

with the other coordinates. The coordinates q

s,1

and q

s,2

are not normal coordinates

due to the appearance of a cross-term. As we will see later, the sufﬁxes a and s

stand for antisymmetric and symmetric vibrations. We omit the simple details in

the calculation of the equation of motion involving the coordinates

and q

which

is decoupled from the remainder of the system. The solution of the q

equation of

motion is given by



1 +

sin



(8.38)

Using again the Lagrangian form of the equation of motion, after a few steps we

obtain

s,1

+ A

s,1

+ A

s,2

= 0

s,2

+ B

s,2

+ B

s,1

= 0

(8.39)

with

sin

α + 2k

cos

α), A

=−

(2k

− k

) sin α cos α

cos

α + 2k

sin

α), B

=−

(2k

− k

) sin α cos α

(8.40)

and m

= 2m

+ m

. It is seen that (8.39) is a coupled system of two second-order

linear differential equations. The solution to this system can be found by any one

of the standard methods. The operator method is particularly easy to apply since

the two equations are decoupled almost immediately. We leave it to the exercises

to verify that the characteristic equation is given by

− ω

+ B

) + (A

− A

) = 0 (8.41)

permitting us to determine the eigenfrequencies ω

s,1

and ω

s,2

of the normal vibra-

tions q

s,1

and q

s,2

We will now brieﬂy discuss the normal modes of vibration. First we consider the

antisymmetric vibration described by (8.38) of the H

O molecule. Pure q

vibrations

exist if x

= x

and y

=−y

. Thus q

describes antisymmetric vibrations with

respect to the y-axis. This case is shown in Figure 8.8(a).

Inspection reveals that the vibrations corresponding to the coordinate q

s,1

and

s,2

are symmetric with respect to the y-axis as shown in parts (b) and (c) of the

ﬁgure. We set q

= 0 and ﬁnd x

=−x

and from (8.36) follows that y

= y

A more exact but also more involved analysis is described in Herzberg (1964a,b)

who introduces an additional interaction coefﬁcient. This changes slightly the fre-

quency equations (8.38) and (8.41). Setting this small interaction coefﬁcient equal

288 Absorption by gases

(a)

(b)

(c)

-vibration

Fig. 8.8 Normal modes of vibration of the H

O molecule.

to zero results in the equations we have derived above. The inclusion of the inter-

action coefﬁcient implies that the normal coordinates shown in Figure 8.8 also

change slightly. Particularly the ν

-vibration (Figure 8.8(a)), must be modiﬁed in

such a way that the arrows extending from the masses m

are nearly parallel to the

lines connecting them with m

. The remaining parts of the ﬁgure are qualitatively

correct. Part (b) is called the ν

-vibration and part (c) the ν

-vibration. None of

the normal modes of vibration we have shown is drawn to scale. The most impor-

tant absorption band in the infrared spectral range is centered at the wave number

˜ν = 1594.78 cm

−1

(λ = 6.27 µm) which results from the ν

-vibration.

The famous books Molecular Vibrations by Wilson et al. (1955) and Infrared

and Raman Spectra by Herzberg (1964b) describe in great detail the theory of

vibrational spectra. Discussion on normal coordinates are given in any textbook on

theoretical mechanics. Our reference goes to Greiner (1989), Mechanik, Volume 2,

who gives a number of examples on normal vibrations. A wealth of information

relevant to radiative transfer in the atmosphere is summarized in Goody (1964a).

8.3 Some basic principles from quantum mechanics

We shall not attempt a rigorous development of quantum mechanics, but we shall

merely state some of the basic results as they apply to the individual atom or atomic

systems. The quantum mechanical description of atomic or molecular systems

is carried out with the help of the wave function or the state function . This

function, in general, is a complex number and considered to be a function of all of

the conﬁgurational coordinates including time.

8.3 Some basic principles from quantum mechanics 289

According to the basic postulates of quantum mechanics, the square of the abso-

lute value ||

of the wave function is a measure of the probability that the consid-

ered system is located at the conﬁguration corresponding to the particular values of

the coordinates. Sometimes the product 

∗

 or ||

is called the probability dis-

tribution function or the probability density. For example, if the system consists of a

single electron, then the probability that the electron is located somewhere between

x, y, z and x + dx, y + dy, z + dz is given by 

∗

(x, y, z)(x, y, z) dxdydz.

From the interpretation of the wave function it follows that we cannot be certain

that the electron is located at any particular place. Only the probability of being

there within certain limits can be known. This interpretation is consistent with

Heisenberg’s uncertainty principle. Since the electron has to be somewhere in

space the total probability has to be unity as stated in



∞

−∞



∞

−∞



∞

−∞



∗

 dx dy dz = 1 (8.42)

Functions satisfying (8.42) are classiﬁed as quadratically integrable normalized

functions.

8.3.1 Stationary and coherent states

We will now brieﬂy discuss two particular states which are known as stationary

and coherent states.

(i) Stationary states

An eigenstate or characteristic state corresponds to a perfectly deﬁned energy. A

given system may have many eigenstates each possessing, in general, a different

energy. If E

denotes the particular energy of one of its eigenstates, the complete

wave function can be written as



(x, y, z, t) = ψ

(x, y, z)exp



−

¯h



with ¯h = h/(2π ) (8.43)

The ﬁrst factor ψ

(x, y, z, ) depends on the space coordinates only while the

second factor gives the time dependency. The parameter h is Planck’s constant.

Multiplication of the wave function by its conjugate yields



∗



= ψ

∗

(8.44)

which indicates that the probability density is constant in time or stationary in the

sense that no changes at all are taking place with respect to the external surroundings.

Thus an eigenstate is also a stationary state. In this situation the system does not

radiate.

290 Absorption by gases

(ii) Coherent states

Suppose the system is in the process of changing from eigenstate 

to 

. During

the transition the state function is a linear combination of the two state functions

as shown in

 = C

exp



−

¯h



+ C

exp



−

¯h



(8.45)

The time variation of the parameters C

and C

is slow in comparison with

the time variation of the exponential factors. A state of the type (8.45) is called

a coherent state. Inspection of this formula shows that the energy of a coherent

state is not well deﬁned since two energies are involved. In contrast to the coherent

state, the energy of a stationary state is well deﬁned. The probability density of the

coherent state is given by



∗

 = C

∗

+ C

∗

+ C

∗

exp(iωt)

+ C

∗

exp(−iωt)

with ω = 2πν = (E

− E

)/¯h

(8.46)

The quantum mechanical description of a radiating atom may be stated in the

following way. During the change from one quantum state to another, the probabil-

ity distribution of the electron becomes coherent and oscillates sinusoidally. This

oscillation is associated with an oscillating electromagnetic ﬁeld which constitutes

the radiation.

8.3.2 The Schr¨odinger equation

So far we have not given any information in which way we might ﬁnd the wave

function . From classical physics we know that the time-dependent wave equation

can be written in the form

∂



∂t

= v

∇

 (8.47)

where  represents any wave function. Substituting the trial solution

 = ψ(x, y, z) exp(iωt) (8.48)

assuming the usual sinusoidal time dependency of  into (8.47) we obtain the

time-independent wave equation

∇

ψ +



2π



ψ = 0 (8.49)

where λ is the wavelength.

8.3 Some basic principles from quantum mechanics 291

In order to ﬁnd Schr

odinger’s equation, we make use of Einstein’s mass–energy

relation

hν = mc

(8.50)

where c is the speed of light in empty space and m is the mass of the photon. The

linear momentum of the photon is denoted by

p = mc =

hν

(8.51)

Just as light exhibits both wavelike and particle properties, De Broglie assumed that

a material particle m moving with speed v, also possesses a wave-like character as

stated in

λ =

(8.52)

This assumption was later conﬁrmed experimentally.

Replacing in (8.49) the wavelength λ by means of (8.52) we ﬁnd

∇

ψ +



2π p



ψ = 0 (8.53)

If E stands for the total energy of the particle as given by E = mv

/2 + V , V is

the potential energy, the linear momentum may be expressed by

= 2m(E − V ) (8.54)

Substitution of (8.54) into (8.53) yields the famous Schr¨odinger equation

∇

ψ +

8π

(E − V )ψ = 0

(8.55)

permitting us to ﬁnd the wave function ψ. In case that the physical system consists

of N particles, (8.55) must be replaced by



k=1



∂

∂x

∂

∂y

∂

∂z



8π

(E − V )ψ = 0

(8.56)

if Cartesian coordinates are used.

To obtain the wave function ψ from the Schr¨odinger equation can be very dif-

ﬁcult. The solution procedure requires the speciﬁcation of the potential function

V (x, y, z) of the physical system. Not all mathematical solutions of the Schr¨odinger

equation are acceptable since they may not be physically meaningful. To be an

acceptable solution, the function ψ must tend to zero for inﬁnite values of the

coordinates in such a way that it is quadratically integrable. This requirement leads

292 Absorption by gases

Table 8.1 Examples of quantum mechanical operators

Variable Operator

Position

Linear momentum

¯h

∂

∂x

¯h

∂

∂y

¯h

∂

∂z

Kinetic energy



+ v



−

¯h



∂

∂x

∂

∂y

∂

∂z



Potential energy

V (x, y, z) V (x, y, z)

Total energy

E −

¯h

∂

∂t

to the result that acceptable functions can exist only if the energy E has deﬁnite

values. These allowed values of E, known as eigenvalues, are characteristic energy

levels of the system. The corresponding solutions are the eigensolutions. Simple

examples will be given later.

In quantum mechanics every variable, such as position or momentum is associ-

ated with an operator. If one of these variables is denoted by g and the corresponding

operator by G, then the operation on the wave function of the system by G gives,

in some cases, the value for the variable g times ψ, i.e.

Gψ = gψ (8.57)

Examples of some quantum mechanical operators are given in Table 8.1.

It is often convenient to employ a form of the Schr¨odinger equation that makes

use of a formal analogy between classical and quantum mechanics. From classical

mechanics (see Appendix 8.8.1) we know that for a conservative dynamical system

the sum of the kinetic energy K and the potential energy V is equal to the constant

E, i.e.

H = K + V = E (8.58)

8.3 Some basic principles from quantum mechanics 293

The sum of K and V is called the Hamilton function H.If(x

, y

, z

) are the

coordinates of particle k and ( p

, p

) the components of the linear momentum

of this particle, the Hamiltonian function H can be written in the form

H =



k=1



+ p



+ V (x

, y

, z

,...x

, y

, z

,...) (8.59)

Replacing the momenta and E according to Table 8.1 and introducing the wave

function  on which the operator is applied, we ﬁnd the quantum mechanical

analogy. This gives the time-dependent Schr

odinger equation

−

¯h



k=1



∂



∂x

∂



∂y

∂



∂z



+ V  =−

¯h

∂

∂t

(8.60)

Utilizing the information listed in Table 8.1, the Schr¨odinger equation can also be

written as

H = E =−

¯h

∂

∂t

(8.61)

While we denote the Hamiltonian function by the symbol H we will use the

calligraphic print H to designate the analogous quantum mechanical Hamilton

operator.

The general solution to (8.61) is given by

 =







exp



−

¯h



(8.62)

In textbooks on quantum mechanics it is shown that any two eigenfunctions of an

atomic system belonging to different eigenvalues are orthogonal. Assuming that

the eigenfunctions are normalized we may write



∞

−∞

∗

dτ = δ

m,n

with dτ = dx dy dz (8.63)

If the wave function  is normalized to 1, then the following relation must be valid



∗



= 1 (8.64)

The latter equation implies that the product |a

= a

∗

represents the probability

of ﬁnding the system in a state of energy E

at time t.

In order to discuss radiation theory we need to ﬁnd a suitable expression for the

Hamiltonian operator for a charged particle in an electromagnetic ﬁeld. As before,

we begin our discussion with the classical Hamiltonian function.

294 Absorption by gases

8.3.3 Hamilton operator for a charged particle in an electromagnetic ﬁeld

The reader may wish to refer to Appendix 2 of this chapter where we have brieﬂy

summarized those relationships from electromagnetic theory which are needed

later. The interaction of a charged particle of mass m with an electromagnetic ﬁeld

is described by the well-known Lorentz force equation. If E and B represent the

electric and the magnetic ﬁeld vector, e and v the charge and the velocity of the

particle, then the Lorentz force equation can be written in the form

F = eE + ev × B

(8.65)

In deriving the classical Hamiltonian function it is more convenient to use the vector

potential A and the scalar potential φ rather than the ﬁeld vectors themselves. The

basic relationships are

E =−∇φ −

∂A

∂t

, B =∇×A (8.66)

Hence the force equation assumes the form

F =−e

∂A

∂t

− e∇φ + ev × (∇×A) (8.67)

We will now brieﬂy show that equation (8.67) can also be derived with the help

of Lagrange’s equation of motion if L is given by

L =

+ e(v · A) −eφ

) + e(

) − eφ

(8.68)

where we have used Cartesian coordinates. The use of the Lagrangian equation

requires that we treat (x, y, z) and (

z) as independent variables. For the x-

component we obtain

∂ L

∂x

= e



∂ A

∂x

∂ A

∂x

∂ A

∂x



− e

∂φ

∂x

∂ L

∂

= m

x + eA

= p

(8.69)

so that Lagrange’s equation of motion can be generalized to

(mv + eA) =∇L (8.70)

Using the vector identity

v × (∇×A) = (∇A) · v − v · (∇A) (8.71)