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Theories of Molecular Reaction DynamicsThe Microscopic Foundation of Chemical Kinetics$

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.337) E Small-amplitude vibrations, normal-mode coordinates

(p.337) E Small-amplitude vibrations, normal-mode coordinates

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

In the following we show that a simple description of the (quantum or classical) dynamics can be obtained in a multidimensional system close to a stationary point. Thus, the system can be described by a set of uncoupled harmonic oscillators. The formalism is related to the generalization of the harmonic expansion in Eq. (1.7) to multidimensional systems.

E.1 Diagonalization of the potential energy

We consider a potential energy surface expressed in Cartesian laboratory coordinates, q i (i = 1, …, n), for a system of n = 3N (where N is the number of nuclei) degrees of freedom. A Taylor expansion of the potential V around the point (q10,,qn0) gives

(E.1)E Small-amplitude vibrations, normal-mode coordinates
where ηi=qiqi0 is the displacement from the point of expansion.

We now assume that the expansion of the potential is around a stationary point (stable or unstable, depending on the sign of the second-order derivatives), that is, all the first-order derivatives vanish. The energy is measured relative to the value at equilibrium, and we obtain

(E.2)E Small-amplitude vibrations, normal-mode coordinates
where (η')T (‘T’ for transpose) and η' are row and column vectors, respectively, and V is an n × n matrix.

We introduce mass-weighted displacement coordinates

(E.3)E Small-amplitude vibrations, normal-mode coordinates

In these coordinates, the potential takes the form (p.338)

(E.4)E Small-amplitude vibrations, normal-mode coordinates
where Fij=Vij/(mimj) is the symmetric mass-weighted force constant matrix.

In the potential of Eq. (E.4), we still find that all the coordinates are coupled, i.e., it contains off-diagonal terms of the form ηiηj with i ≠ j. However, since the potential is a quadratic form, we know from mathematics that it is possible to introduce a linear transformation of the coordinates such that the potential takes a diagonal form in the new coordinates. To that end, normal-mode coordinates, Q, are introduced by the following linear transformation of the mass-weighted displacement coordinates:

(E.5)E Small-amplitude vibrations, normal-mode coordinates
where L is an n × n matrix. The potential can now be written in the form
(E.6)E Small-amplitude vibrations, normal-mode coordinates
and we are going to determine L such that the matrix LTFL becomes diagonal. The columns of L are now chosen as eigenvectors of the matrix F. Thus,
(E.7)E Small-amplitude vibrations, normal-mode coordinates
where ωj2(j=1,,n) (j = 1, …, n) are the corresponding eigenvalues, which are determined as roots to the equation |F − ω2 I| =0. Since F is real and symmetric, we know from matrix theory that the column vectors of L are mutually orthogonal. If the column vectors are normalized to unit length, then L becomes an orthogonal matrix. The inverse of an orthogonal matrix is obtained by transponation, L−1 = L T, that is, LTL = I or equivalently ∑i(L)il(L)ik= δlk.

Equation (E.7) can be written in the form

(E.8)E Small-amplitude vibrations, normal-mode coordinates
where ω2 is a diagonal matrix with the n eigenvalues along the diagonal (note that Lω2 ≠ ω2L) and
(E.9)E Small-amplitude vibrations, normal-mode coordinates
since the inverse of L is obtained by transponation. Using Eqs (E.6) and (E.9), we obtain
(E.10)E Small-amplitude vibrations, normal-mode coordinates

We have now obtained the desired diagonal form of the potential energy. If all the frequencies, ωs2, are positive then the stationary point represents a minimum on (p.339) the potential energy surface. If, on the other hand, one or more frequencies, ωk2, are negative (which implies that ωk is imaginary) then the potential corresponds to an inverted harmonic potential in that mode and the associated motion is not oscillatory but unbound. A saddle point is an example of such an unstable point.

E Small-amplitude vibrations, normal-mode coordinates

Fig. E.1.1 (a) Symmetric and (b) anti-symmetric stretch in a symmetric molecule like CO2. Note that the C atom does not participate in the symmetric stretch motion.

E Small-amplitude vibrations, normal-mode coordinates

Fig. E.1.2 One of the two stretch modes in OCS. Note that this mode, essentially, corresponds to a pure CO vibration.

In practice, from a given potential we first calculate the mass-weighted force constant matrix F (Eq. (E.4)). The eigenvalues of this matrix give the normal-mode frequencies (Eq. (E.7)). The corresponding eigenvectors give, according to Eq. (E.5), the normal-mode coordinates expressed as a linear combination of atomic (mass-weighted) displacement coordinates; thus, Q= LTη. The normal modes are often presented in graphical form by ‘arrows’ that represent (the magnitude and sign of) the coefficients in the linear combinations of the atomic displacement coordinates.

Some examples are given in Fig. E.1.1 and E.1.2, for triatomic molecules. A linear triatomic molecule has four (3 × 3 − 5) vibrational modes: two bond-stretching modes and two (degenerate) bending modes. Fig. E.1.2 shows one of the four normal modes in OCS.

E.2 Transformation of the kinetic energy

As shown above, the potential energy can be expressed as a sum of harmonic potentials. We now consider the expression for the kinetic energy in classical as well as quantum mechanical form. In classical mechanics,

(E.11)E Small-amplitude vibrations, normal-mode coordinates
where we have introduced mass-weighted coordinates according to Eq. (E.3). Expressed in normal-mode coordinates, the kinetic energy takes the form (p.340)
(E.12)E Small-amplitude vibrations, normal-mode coordinates

Thus, from Eqs (E.12) and (E.10) we see that the classical dynamics of the normal modes is just the dynamics of n uncoupled harmonic oscillators.

In quantum mechanics the kinetic energy is represented by the operator

(E.13)E Small-amplitude vibrations, normal-mode coordinates
where the relation /qi=mi/ηi ∂/∂ηi was used in order to derive the second line. Since Q l = ∑k(LT)lkηk, the chain rule gives ∂/∂ηi = ∑ l(∂Ql/∂ηi)(∂/∂Q l)= ∑ l(L)il∂/∂Q land
(E.14)E Small-amplitude vibrations, normal-mode coordinates

Thus, with this result for the kinetic energy and Eq. (E.10) for the potential energy, we conclude that the quantum dynamics of the normal modes is just the dynamics of n uncoupled harmonic oscillators; that is,

(E.15)E Small-amplitude vibrations, normal-mode coordinates

The total wave function can, accordingly, be written as a product of wave functions corresponding to each mode. The energy eigenfunctions corresponding to each mode are, in particular, just the well-known eigenfunctions for a harmonic oscillator.

E.3 Transformation of phase-space volumes

We consider here the relation between volume elements in phase space; in particular, the relation between dqdp and dQdP, where dq = dq 1 … dq n refers to Cartesian coordinates in a laboratory fixed coordinate system, dQ = dQ 1dQ n refers to normal-mode coordinates, and p and P are the associated generalized conjugate momenta.

(p.341) For coordinate transformations we generally have the following relation between the volume elements: dq = |J|dQ, where |J| is the absolute value of the Jacobian J, which is given by the determinant

(E.16)E Small-amplitude vibrations, normal-mode coordinates
From Eq. (E.5), we get q i = qi=qi0+1mij L ij Q j, and therefore
(E.17)E Small-amplitude vibrations, normal-mode coordinates
so
(E.18)E Small-amplitude vibrations, normal-mode coordinates

Since L is an orthogonal matrix, |LLT| = |I|, i.e., |L| · |LT| = 1, which implies that |L| = ± 1. That is,

(E.19)E Small-amplitude vibrations, normal-mode coordinates
is the relation between the volume elements in configuration space.

In momentum space, we have the similar relation dp= |J|dP. The Jacobian is given by a determinant similar to the one in Eq. (E.16), now with the elements ∂pi/∂Pj. The momenta are defined by Eq. (4.61), that is, pi = ∂L/∂q i and P j= ∂L/∂Q˙j, where L is the Lagrange function. The relation between the two sets of momenta is

(E.20)E Small-amplitude vibrations, normal-mode coordinates

(p.342) From Eq. (E.5), we get Q˙ = LT η˙, that is, Q˙j=iLijmiq˙i Thus,

(E.21)E Small-amplitude vibrations, normal-mode coordinates
and
(E.22)E Small-amplitude vibrations, normal-mode coordinates

Thus,

(E.23)E Small-amplitude vibrations, normal-mode coordinates
and finally,
(E.24)E Small-amplitude vibrations, normal-mode coordinates
demonstrating the invariance of the volume element in phase space. It can be shown that this invariance of the volume element in phase space holds in general, i.e., for any two sets of coordinates and their conjugate momenta. That is, q and Q can be any two sets of coordinates that describe the same point.

Further reading/references

Bibliography references:

H. Goldstein, Classical mechanics, second edition (Addison–Wesley, 1980)

E.B. Wilson, J.C. Decius and P.C. Cross, Molecular vibrations (Dover, 1980)