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Internal Degrees of Freedom of Molecules
Adiabatic Theory of Molecule
For the internal dynamics of molecules it is of crucial importance that nuclei are much heavier than electrons. This leads to the adiabatic picture of the nuclei motion with respect to the motion of electrons. In the adiabatic picture, the coordinates of nuclei play the role of external parameters with respect to the state of the electronic sub-system, that is they play the same role as the adiabatic parameter λ in our consideration of the adiabatic process. Here, however, we are dealing with more subtle situation where nuclei coordinates behave as external parameters with respect to the electrons only, not to each other. Mathematically, the adiabatic picture implies the following structure of the molecule wavefunction Ψ(r, R) = ψ(R) ΦR(r) , (1)
where r and R stand for all the electron and nuclei coordinates, respectively. The function ψ(R) is the wavefunction of nuclei. The function ΦR(r) is the wavefunction of electrons at fixed positions of all nuclei. It obeys the Schr¨odinger equation
Hel(R )Φ = E(elR )Φ. (2)
Here Hel(R )is the electronic Hamiltonian in which the nuclei at fixed positions play the role of external potentials. Both the wavefunction ψ and the eigenvalue Eel thus depend on R. The nuclei motion is described by the effective Schr¨odinger equation
Hnuc(kin) ψ + Veff ψ = E ψ , (3)
where Hnuc(kin) is the standard kinetic energy term and
Veff (R) = V (R) + Eel(R ), (4)
is the effective potential energy equal to the sum of bare potential energy, V (R), (Coulomb repulsion between nuclei) and the electronic energy as a function of R. Note that it is entirely due to the
attractive term Eel(R )that nuclei bind into a molecule. The effective adiabatic potential Veff (R) is very rigid in the sense that the typical inter-nuclei separation turns out to be much smaller than its quantum uncertainty. Hence, the inter-nuclei separations are good quantum numbers characterizing the structure of the molecule. Consider some estimates proving this fact. First we estimate the electronic energy as
Eel(R ) ∼ ¯h^2 me R^2
¯h^2 me a^2 B
where me is the electron mass and aB is the Bohr radius. Then we estimate the strength of the effective adiabatic potential Veff (R). To this end we replace it with a harmonic potential with respect to the relative coordinates of the nuclei. This is a quite accurate approximation because relative coordinates of nuvlei do not change significantly with respect to their equilibrium values and we can expand Veff (R) in the Taylor series in the vicinity of the equilibrium configuration. The equivalent harmonic potential with respect to one of the relative coordinates, R, then reads
Veff (R) = const − (k/2)(R − R 0 )^2 , (6)
where R 0 is the equilibrium position, and k is the spring constant, which we estimate as
k ∼
∂^2 E(elR) ∂R^2
Eel R^20
Eel a^2 B
Given the spring constant, we readily estimate the frequency of the vibrations
ωvib ∼
√ k/m∗ (8)
(m∗ is the typical nuclei mass) and the typical inter-level separation for the atomic vibrations
Evib = ¯hωvib ∼ Eel
( me m∗
) 1 / 2
. (9)
To estimate the quantum mechanical uncertainty, ∆R, of the coordinate R, we just need to recall the result for the ground-state uncertainty of the coordinate of the harmonic oscillator of the mass m∗ and the spring constant k. Using the relation between ∆R and Evib,
¯h^2 m∗(∆R)^2 ∼ Evib , (10)
we find (^) ( ∆R R
) 2 ∼
( me m∗
) 1 / 2 ø 1. (11)
Let us compare inter-level separations of vibrational and rotational degrees of freedom. For the typical inter-level separation of the rotational levels we have (only the kinetic part of the Hamiltonian is relevant for the rotations)
Erot ∼
¯h^2 m∗R^2 ∼ Evib
( me m∗
) 1 / 2 ∼ Eel
me m∗
We see that Erot ø Evib ø Eel. (13) Finally, we establish the orders of magnitudes of realistic values. As is known, Eel ∼ (1 ÷ 10) eV , where 1 eV = 1. 16 × 104 K. (14)
Note that for our purposes it is most convenient to measure inter-level separations in Kelvins, since this immediately yields the ranges of the temperatures when corresponding degrees of freedom get excited. So, Eel ∼ (10^4 ÷ 105 ) K. (15)
A reasonable estimate for the parameter
√ me/m∗ is ∼ 10 −^2 leads us to
Evib ∼ (10^2 ÷ 103 ) K , (16)
Erot ∼ (1 ÷ 10) K. (17)
S = ln(T /θ) + 1 , (27) E = T , (28) C = 1. (29)
Ultra-quantum regime. In this regime the temperature is much smaller than the separation between the ground state and the first excited state. Hence, with the exponential accuracy in the parameter θ/T the molecule does not rotate—rotational degrees of freedom are frozen out. We can easily take into account the main exponential correction. To this end we keep the second term in the series (35), neglecting the rest of the terms [which are exponentially smaller]. We have
Z = 1 + 3e−2(θ/T^ )^ , (30)
and then find F = −T ln
[ 1 + 3e−2(θ/T^ )
] ≈ − 3 T e−2(θ/T^ )^ , (31)
S = [3 + 6(θ/T )] e−2(θ/T^ )^ , (32) E = 6θ e−2(θ/T^ )^ , (33) C = 12(θ/T )^2 e−2(θ/T^ )^. (34)
Gas of Molecules
The statistics of internal degrees of freedom of molecules is decoupled from the statistics of the translational degrees of freedom, simply because the translational motion—the motion of the center of mass of a molecule—is dynamically independent from the internal motions. Hence, for a dilute (= non-interacting) gas of molecules the partition function factorizes into the product of the partition functions of translational and internal motions:
Z(V, T ) = Ztr(V, T ) Zint(T ). (35)
Note that Zint is V -independent. Correspondingly, we have
F (V, T ) = Ftr(V, T ) + Fint(T ) , (36)
and see that the equation of state for the gas of molecules is the same as for the mono-atomic gas, since Fint(T ) is V -independent and thus cannot change the expression for the pressure:
P = −
( ∂F ∂V
)
T
( ∂Ftr ∂V
)
T
= N T /V. (37)
Similarly, for the entropy we have
S(V, T ) = Str(V, T ) + Sint(T ) , (38)
and from the fact that Sint depends only on T conclude that for any heat capacity Cξ [ξ is either V , or P , or any other fixed quantity] we get
Cξ = C ξ(tr) + C(int)^ , (39)
where
C(int)^ = T dSint dT
The contribution of the internal degrees of freedom to Cξ is one and the same for any ξ.
Classical limits for internal motions. The vibrational degrees of freedom are just harmonic oscillators. Hence, we conclude that each vibrational degree contributes at T ¿ Evib the amount T = (T /2 + T /2) to the energy and, correspondingly, 1 = (1/2 + 1/2) to the heat capacity (recall that each half comes independently from kinetic and potential energies). So, if νvib is the total number of vibrational degrees of freedom, we have
Evib = νvib T (T ¿ Evib) , (41) cvib = νvib (T ¿ Evib). (42)
(Here we use a small letter c to emphasize that we are dealing with the heat capacity of a single molecule rather than that of the whole gas.) Let us find the total contribution to the energy of all the internal degrees of freedom in the classical limit. The total number of degrees of freedom is 3ν, where ν is the total number of nuclei. As we remember, in the classical statistics the kinetic and potential energies are statistically independent. Moreover, the total kinetic energy is always equal to (3/2)νT. In our case, the potential energy consists only of νvib harmonic potentials (in terms of relative distances), each of these potentials contributing T /2 to the energy [1/2 to the heat capacity]. Subtracting (3/2)T associated with the translational motion of the molecule, we arrive at the energy and heat capacity of the internal degrees of freedom:
Eint = 3(ν − 1) 2
T +
νvib 2 T (T ¿ Evib, Erot) , (43)
cint =
3(ν − 1) 2
νvib 2 (T ¿ Evib, Erot). (44)
The number νvib can be related to ν by subtracting from 3ν [the total number of degrees of freedom] the 3 translational degrees and the rotational degrees. In a general case, there are 3 rotational degrees of freedom (3 independent angles), while in the axially symmetric case there are only 2 rotational degrees, since the rotation around the symmetry axis does not change the configuration of particles and thus is not a degree of freedom. We thus have
νvib =
{ 3 ν − 5 (axial symmetry) , 3 ν − 6 (no axial symmetry).
In the temperature regime Erot ø T ø Evib the vibrational degrees of freedom are still frozen out, while the rotational degrees of freedom are already in their classical limit. Hence, it is reasonable to separate in the equations (43) and (44) vibrational contributions from the rotational ones, so that we could use their classical limits independently. Subtracting (41) and (42) from (43) and (44), respectively, and taking into account (45), we get
Erot =
{ T (axial symmetry) (3/2) T (no axial symmetry) (T ¿ Erot) , (46)
