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

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

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 $({q}_{1}^{0},\mathrm{\dots},{q}_{n}^{0})$ gives

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

^{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

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

*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 η_{i}η_{j} 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:

*L*is an

*n*×

*n*matrix. The potential can now be written in the form

^{T}FL becomes diagonal. The columns of L are now chosen as eigenvectors of the matrix F. Thus,

*j*= 1, …,

*n*) are the corresponding eigenvalues, which are determined as roots to the equation |F − ω

^{2}

**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, L

^{T}L = I or equivalently ∑

_{i}(L)

_{il}(L)

_{ik}= δ

_{lk}.

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

^{2}is a diagonal matrix with the

*n*eigenvalues along the diagonal (note that Lω

^{2}≠ ω

^{2}L) and

We have now obtained the desired diagonal form of the potential energy. If all the frequencies, ${\omega}_{s}^{2}$, 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, ${\omega}_{k}^{2}$, are negative (which implies that ω_{k} 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.

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= L^{T}η. 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,

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

_{i}was used in order to derive the second line. Since

*Q*

_{l}= ∑

_{k}(L

^{T})

_{lk}η

_{k}, the chain rule gives ∂/∂η

_{i}

*= ∑*

_{l}(∂

*Ql*/∂η

_{i})(∂/∂

*Q*

_{l})

*= ∑*

_{l}(L)

_{il}∂/∂

*Q*

_{l}and

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,

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 *d*q*d*p and *d*Q*d*P, where *dq = dq* _{1} *… dq* _{n} refers to Cartesian coordinates in a laboratory fixed coordinate system, *dQ = dQ* _{1} … *dQ* _{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|d*Q, where |*J|* is the absolute value of the Jacobian *J*, which is given by the determinant

*q*

_{i}= ${q}_{i}={q}_{i}^{0}+\frac{1}{\sqrt{{m}_{i}}}$ ∑

_{j}

*L*

_{ij}

*Q*

_{j}, and therefore

Since L is an orthogonal matrix, |LL^{T}| = |**I**|, i.e., |L| · |L^{T}| = 1, which implies that |L| = ± 1. That is,

In momentum space, we have the similar relation *d*p*= |J|d*P. 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/∂${\dot{Q}}_{j}$, where *L* is the Lagrange function. The relation between the two sets of momenta is

(p.342)
From Eq. (E.5), we get $\dot{Q}$
= L^{T} $\dot{\eta}$, that is, ${\dot{Q}}_{j}={\displaystyle {\sum}_{{i}^{\prime}}{L}_{{i}^{\prime}j}\sqrt{{m}_{{i}^{\prime}}}{\dot{q}}_{{i}^{\prime}}}$ Thus,

Thus,

*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.