## Niels E. Henriksen and Flemming Y. Hansen

Print publication date: 2008

Print ISBN-13: 9780199203864

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780199203864.001.0001

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# (p.304) B Microscopic reversibility and detailed balance

Source:
Theories of Molecular Reaction Dynamics
Publisher:
Oxford University Press

Both Newton’s equation of motion for a classical system and Schrödinger’s equation for a quantum system are unchanged by time reversal, i.e., when the sign of the time is changed. Due to this symmetry under time reversal, the transition probability for a forward and the reverse reaction is the same, and consequently a definite relationship exists between the cross-sections for forward and reverse reactions. This relationship, based on the reversibility of the equations of motion, is known as the principle of microscopic reversibility, sometimes also referred to as the reciprocity theorem. The statistical relationship between rate constants for forward and reverse reactions at equilibrium is known as the principle of detailed balance, and we will show that this principle is a consequence of microscopic reversibility. These relations are very useful for obtaining information about reverse reactions once the forward rate constants or cross-sections are known. Let us begin with a discussion of microscopic reversibility.

# B.1 Microscopic reversibility

## B.1.1 Transition probability

The trajectory of a classical particle may be found by integrating Newton’s equation of motion

(B.1)
from say time t 0 to time t 1. Let us then introduce the substitution
(B.2)
into Eq. (B.1). We find
(B.3)

As t varies from t 0 to t 1, τ is seen from Eq. (B.2) to vary from t 1 to t 0, so the substitution is equivalent to a time reversal. Since the equations of motion are identical, the system will follow exactly the same trajectory; the only difference will be that in one (p.305) case it is followed forward in time from t 0 to t 1, and in the other case backward in time from t 1 to t 0. In other words, it is not possible from the equation of motion to decide whether it describes a forward or a reverse propagation in time. This is summarized by saying that the classical equation of motion has time-reversal symmetry. As a consequence, the probability for a forward scattering process must equal the probability for the reverse scattering process, where all velocities dr/dt and time have changed sign (see Eq. (B.3)).

The Schrödinger equation for a quantum system also has time-reversal symmetry.

The solution to the time-dependent Schrödinger equation

(B.4)
may be formally written as
(B.5)
where the propagator
(B.6)
(formally defined by its Taylor expansion) propagates the wave function from time t 0 to time t 1. From this definition it is clear that
(B.7)
where † indicates Hermitian conjugation. Suppose now that we propagate the wave function ψ(t 1) at time t 1 backward in time to time t 0; then we have
(B.8)
or
(B.9)
where we have used Eq. (B.6). The propagators in Eqs (B.5) and (B.9) for, respectively, a forward and a backward propagation in time are identical, and if the wave functions are real, it is immediately obvious that the same equation may be used for forward and backward propagation in time as in a classical system; hence the Schrödinger equation has time-reversal symmetry. In general, however, the wave functions are complex, so the two equations of motion differ in the sense that it is the wave function itself that is propagated in the forward direction of time, whereas it is the complex conjugate wave function that is propagated in the reverse direction of time. The complex conjugation of the wave packet is equivalent to a change in sign of the momentum of the wave packet (the Gaussian wave packet, Eq. (4.113), has this property), just like time reversal in a classical system, where the velocities change sign.

(p.306) Then it is remembered that the wave function itself does not have a physical meaning. So let us, for example, determine the transition probability from say state |k(t 0)> at time t 0 to state |m(t 1)> at time t 1. Then we have

(B.10)
and the transition probability is
(B.11)

This transition probability is now compared to the transition probability for the reverse process, where we consider the transition from |m(t1)> to |k(t 0)>. We find

(B.12)
where we have used the relation $< k ( t 0 ) | U ^ ( t 1 − t 0 ) | m ( t 1 ) > * = < m ( t 1 ) | U ^ † ( t 0 − t 1 ) | k ( t 0 ) >$ and Eq. (B.7). In other words, the transition probability from state |k(t 0) at time t 0 to state |m(t 1)> at time t1 is equal to the transition probability for the reverse process, that is, from |m(t 1)> to |k(t0)>. This is a manifestation of the time-reversal symmetry of the equation of motion.

The transition probabilities per unit time and per scattering center in Eqs (B.11) and (B.12) are for a collision process, often written in a more explicit form to emphasize the particular transition considered:

(B.13)
where υi and υf are the initial and final relative speeds, respectively, of the colliding particles, ij are the initial internal states of the two colliding molecules, ml are the final internal states of the product molecules, and Ω is the solid angle into which the particle of interest is scattered. This angle, as measured from the direction of the initial relative velocities, respectively, υi and υf, is the same for both the forward and the reverse scattering process. Finally, let us emphasize that the transition probability P refers to a well-specified initial and final state and that it gives us the probability per unit time and per scattering center for the scattering process indicated.

## B.1.2 Cross-sections

From the fundamental relation in Eqs (B.12) or (B.13), we are now going to derive a relation between the differential cross-section for the forward and the reverse scattering process.

As defined previously in Eq. (2.7), differential cross-sections are defined as the flux of particles scattered into a range of solid angle dΩ around the solid angle Ω per unit initial flux and per scattering center. The state-to-state transition probabilities are already per unit scattering center and, since we do not scatter into one single final state but a range of final states as given by dΩ, we need to determine how many final states are consistent with that uncertainty in the final solid angle. It is then assumed that the transition probability to that small range of final states is the (p.307) same, irrespective of the state; then the product of the transition probability and the number of final states will be the total transition probability. The scattering into the solid angle dΩ around Ω with a relative final velocity vf is equivalent to scattering into a final state with momentum pf, pf + dpf. The number of final states in this range may be determined in the following way. We consider a free particle in one dimension where the momentum eigenfunction is given by ψp(x) = c(p) exp (ipx/ħ). In a (macroscopic) box of length L, periodic boundary conditions on the wave function ψp(x + L) = ψ p(x) imply that exp(ipL/ħ) = 1 or p/ħ = 2nπ/L, where n is an integer. That is, the spacing between momentum values is h/L, which implies that the density is L/h. This argument can be extended to three dimensions, and it can be shown that the density, ρ(p), is (L/h)3 and

(B.14)
where we have changed from Cartesian coordinates to spherical coordinates and used the definition of the solid angle dΩ.

If the density of particles in the incident beam of A(i) molecules with relative velocity vi with respect to the B molecules is na(i), then the magnitude of the incoming flux is

(B.15)
and the number of scattered molecules in the solid angle dΩ per unit time is
(B.16)

From the definition of the differential cross-section, we obtain1

(B.17)

The factor δ(EfEi)dE f on the left-hand side of the equation has been introduced to emphasize energy conservation for the process indicated in the argument to σ R . The transition probability P on the right-hand side is determined directly from the equation of motion, so it is therefore not necessary to include the delta function on the right-hand side.

(p.308) The cross-section for the reverse reaction may be written analogously:

(B.18)

When we use Eq. (B.13) and cancel common factors, we find from Eqs (B.17) and (B.18) that

(B.19)

From the energy balance

(B.20)
where μi is the reduced mass for the relative motion of the reactants and μf for the products, we find the following relation between pi and pf for fixed internal states:
(B.21)

The desired relation between cross-sections for the forward and the reverse reactions is thus found to be

(B.22)

This relation expresses the principle of microscopic reversibility for the differential cross-sections.

Alternatively, using Eq. (4.173) and the symmetry of the S-matrix element, using Eq. (B.12), we immediately obtain

(B.23)
where, on the left-hand side, n and m refer to the quantum state of the reactant and product and, on the right-hand side, the reverse reaction is considered, i.e., the initial and final quantum states are m and n, respectively. E is the total energy.

Equation (B.23) is the desired relation (at the level of integrated cross-sections), and it is equivalent to Eq. (B.22), since p = ħk.

If the system has an internal angular momentum (associated with rotational states of molecules) there will, in the absence of an external field, be degeneracies in the system that will be practical to display explicitly in the expression for microscopic reversibility in Eq. (B.22). For systems with angular momenta, time reversal of the quantum equations of motion reverses the signs of both the momenta and their projections on a given direction, just like in a classical system. To express this explicitly Eq. (B.13) is written as

(B.24)

The ‘starred’ quantum states differ from the ‘unstarred’ ones by the sign of the projection of their angular momenta, so if the projection of the angular momentum of (p.309) state i on the z-axis is m i then the projection of i* on the z-axis is -m i, etc. By an analysis similar to that used to derive Eq. (B.22), we obtain

(B.25)

Experimental and calculated cross-sections are usually averaged over the degenerate quantum states associated with the angular momenta when there is no external field. A more useful statement of microscopic reversibility can therefore be obtained by working with average cross-sections $σ ¯ R$, which refer to transitions between sets of degenerate levels in the initial and final states. Let Ji denote the total angular momentum of state i and mi its projection on the z-axis. Then by summing Eq. (B.24) over all degenerate states of the reactants and products one obtains

(B.26)
where the summations over the m k are from – J k to J k. Note that we have dropped the star on the quantum states because the range of the m k extends over both positive and negative values.

Since the pi and pf are constant when summed over degenerate states, we may now use Eqs (B.17) and (B.26) to introduce a cross-section, σ R (s), where we have summed over all degenerate states:

(B.27)

So the principle of microscopic reversibility may be rewritten as

(B.28)

The σR(s) refers to the sum of cross-sections for the transition from the degenerate set of states $i ¯ , j ¯$ to $l ¯ , m ¯$ defined for unit flux from each initial state (see Eqs (B.15)–(B.17)). We may instead operate with an average cross-section $σ ¯ R$ based on an initial flux for the total set of degenerate states:

(B.29)
where gi is the number of degenerate states, gi = 2Ji + 1. Substitution of Eq. (B.29) into Eq. (B.28) leads to yet another expression of microscopic reversibility, based here on the average cross-sections for the degenerate states:
(B.30)

It is noted that this relation only holds in the absence of an external field, when the sets of states are degenerate. Other degeneracies may also occur in the internal molecular states. In this case, the g factors should include those degeneracies.

# (p.310) B.2 Detailed balance

Detailed balance provides a relation between the macroscopic rate constants k f and k r for the forward and reverse reactions, respectively. On a macroscopic level, the relation is derived by equating the rates of the forward and reverse reactions at equilibrium. Here it will be shown that the principle of detailed balance can be readily obtained as a direct consequence of the microscopic reversibility of the fundamental equations of motion.

On a macroscopic level in the absence of an external field, we cannot distinguish between sets of degenerate states in the reactants and products; so the most detailed relation between macroscopic rate constants and microscopic cross-sections will be one where we have summed over all degenerate states as in Eq. (B.27). The macroscopic rate constant for a particular transition between degenerate states is then given by

(B.31)
analogous to the relation in Eq. (2.18). This relation is general in the sense that it applies to both equilibrium and non-equilibrium systems. We now assume that the velocity distributions for υa and υb(or momenta distributions p A and pB) are given by the equilibrium Maxwell–Boltzmann distribution at temperature T. We obtain, as in Section 2.2 (see Eq. (2.29)), the following result:
(B.32)
where μi is the reduced mass and pi = μ iυi is the momentum associated with the relative translation of the reactants. Similarly, the expression for the rate constant for the reverse reaction is
(B.33)
where μf refers to the reduced mass and pf = μfυf is the relative momentum of the products. Note that $σ R ( s ) ( m ¯ , l ¯ | i j ¯ ; υ i )$ is the integrated cross-section (see Eq. (2.12)) where$σ R ( s ) ( m ¯ , l ¯ | i j ¯ ; υ i , Ω )$ is integrated over all space angles.

To relate the rate constants in Eqs (B.32) and (B.33), we substitute Eq. (B.28) and use the conservation of energy to relate the differentials and limits of integration. Conservation of energy requires

(B.34)
where
(B.35)
is the change in the internal energies associated with the reaction. E 0,p is the zero-point energy of the products and E 0,r is the zero-point energy of the reactants. We then have for the reverse rate constant, using Eqs (B.21), (B.28), and dp = 4πp 2 dp, (p.311)
(B.36)

From Eq. (B.32), we then obtain the following relation between the forward and reverse rate constants:

(B.37)

It is noted that the right-hand side is the ratio of the translational partition functions of products and reactants times the Boltzmann factor for the internal energy change. In the derivation of this expression we have only used that the translational degrees of freedom have been equilibrated at T through the use of the Maxwell–Boltzmann velocity distribution. No assumption about the internal degrees of freedom has been used, so they may or may not be equilibrated at the temperature T. The quantity $K ( m ¯ l ¯ , i ¯ j ¯ )$ may therefore be considered as a partial equilibrium constant for reactions in which the reactants and products are in translational but not necessarily internal equilibrium.

To obtain the statement of detailed balance for complete equilibrium, with both translational and internal degrees of freedom in thermal equilibrium, we must sum over the rate constants in Eqs (B.32) and (B.33), weighting each by its equilibrium Boltzmann distribution; that is (as in Eq. (2.18)),

(B.38)
and
(B.39)

Here kf is the rate constant for the forward reaction and k r for the reverse reaction. $p A( i ¯ )$ is the probability (mole fraction) of A in any of the set of states $i ¯$, which is given by statistical mechanics according to Eq. (A.4), and similarly for the other constituents. If we substitute these relations into Eqs (B.38) and (B.39), we obtain

(B.40)
and for the reverse reaction
(B.41)

Note that the degeneracies gi do not appear explicitly in the equations, because we use ‘barred’ quantities as indices in the sums. They imply a sum over degenerate states; had we used ‘unbarred’ indices, then the degeneracy factor gi should be included (p.312) explicitly. From Eqs (B.40) and (B.41) and using Eq. (B.37), we find the following relation between the rate constants:

(B.42)
where the subscript ‘int’ has been added in order to emphasize that the partition functions refer to internal (non-translational) degrees of freedom. Furthermore, in the second line, following Eq. (A.10), the partition functions are evaluated with the energies measured relative to the zero-point levels of the reactants and products, respectively. Rearranged, the general statement of detailed balance at equilibrium may be written
(B.43)
where K(T) is the equilibrium constant for the reaction. This is the usual statistical mechanical expression for the equilibrium constant in terms of the molecular partition functions.

In summary, we have seen that the application of microscopic reversibility for the forward and reverse cross-sections and the use of complete equilibrium distributions for the evaluation of the statistical rate constant lead to the usual results known from equilibrium statistical mechanics. If one knows the cross-section for a forward reaction, one can always determine the inverse cross-section through the principle of microscopic reversibility. Also, if one knows the cross-section for the forward reaction, and in addition one knows that the translational and internal distribution functions of reactants and products have reached equilibrium, one can calculate the rate constant. Detailed balance then permits the calculation of the reverse rate constant.

(1) Note that, in this appendix, we use a notation for the differential cross-sections that differs somewhat from the one in Chapter 2. Thus, σR(ml|ij;υi, Ω) $d σ R d Ω ( i j , υ | m l , Ω )$ in the notation of Chapter 2.