Calculating the Root-Mean-Squared Amplitudes of Vibration

4.3 Defining the Theoretical Model

4.3.2 Calculating the Root-Mean-Squared Amplitudes of Vibration

There exist several methods to calculate the rms amplitudes of vibration, either based on experimental spectroscopic data,¹¹³ empirical equations,^114,115 or quantum chemical force constants.19,22,116,117 The last two methods have been used for the studies contained in this thesis and their use will be described in detail here.

4.3.2.1 Empirical Equations

For UED3 studies, the calculation of the rms amplitudes of vibration has traditionally been carried out by using the following empirically determined formulae for C–C distances in the range of 1.217 Å ≤ r_e ≤ 5.618 Å,¹¹⁴

, 000147 .

0 023398 .

0 013837 .

0 e e²

h r r

l = + − (4.46)

and for C–H distances in the range of 1.080 Å ≤ r_e ≤ 4.677 Å,¹¹⁵ . 001805 .

0 027368 .

0 050134 .

0 e e²

h r r

l = + − (4.47)

These values, which were experimentally determined for molecules at 298 K, are then scaled to any arbitrary temperature T using the relation for normal mode vibrations based on a harmonic model.^28,106

2 , 8 ² coth

2 



 



= 

kT h

l_h h ν

µν

π ^(4.48)

where h is Planck’s constant, k is the Boltzmann constant, μ is the reduced mass of the internuclear pair, ν is the harmonic vibrational frequency. Equation 4.48 is used to solve for ν at 298 K and then the rms vibrational amplitudes lh at the desired T can be obtained by assuming that ν is independent of temperature. These empirically determined rms

amplitudes can be used for small molecules, whose size falls within in the ranges stated above, e.g., nitrobenzene, described in Section 5.3. However, it is difficult to justify the physical basis of extrapolating these empirical results for the calculation of the rms amplitudes of vibration to larger nonbonded internuclear distances of large molecules.

Therefore, a new method was implemented using the calculation output from the quantum chemical software packages mentioned above.

4.3.2.2 Quantum Chemical Force Constants

Quantum chemical calculations also allow for the calculation of vibrational frequencies of a molecular structure and the associated harmonic force constants, which in turn can be used to evaluate the harmonic rms amplitudes of vibration. In the following, the steps to obtain these parameters from the calculation output will be described. The described routines have been implemented in the program amplitude.x (initially written by Dr. Sang Tae Park here at Caltech), which is available in the online supporting material.⁴²

After running a quantum chemical geometry optimization and frequency calculation, the following output is obtained: The equilibrium geometry in Cartesian coordinates, represented by the 3N-vector x and the 3N×3N matrix of second derivatives of the potential energy ₂

2

H_x x d

V

= d (Cartesian Hessian matrix or force constant

matrix). (The Cartesian gradient is equal to zero at the equilibrium geometry, which is located at the potential energy minimum.)

To describe the distortion of the molecular frame due to molecular vibrations, the Cartesian force constants are projected onto the redundant internal coordinate basis described in Section 4.2.4. Using Equation 4.39, the Cartesian Hessian matrix can be transformed into the redundant internal coordinate basis by

( )

^{1 T} _x ¹ ,

s B H B

H = ^{− −} (4.49)

where B⁻¹ is the generalized inverse of the matrix B, defined as¹⁰⁶

(

^{1 T}

)

^{1 1 T 1 .}

T 1

1 M B BM B M B G

B⁻ = ^{− − −} = ^{− −} (4.50)

The Hessian matrix in redundant internal coordinates is then further transformed into mass-weighted redundant internal coordinates according to

s , G

ms = ⁻¹² (4.51)

and

.

12 2 s 1

ms G H G

H = ^(4.52)

Since the Hessian matrix H_ms is real and symmetric it can be diagonalized by a real orthogonal matrix L_ms according to (cf. Equation 4.40)

, F L H

L^T_{ms ms ms} = (4.53)

where F is a diagonal matrix containing the eigenvalues of H_ms in ascending order and the columns of the matrix L_ms are the corresponding eigenvectors of H_ms. Because we are using mass-weighted coordinates, the eigenvalue matrix F contains the mass- weighted force constants of the normal modes of vibration, such that the frequency of the kth normal mode is obtained through

2 . 1 kk

k π F

ν = (4.54)

Since F is a matrix of dimension M×M, the first M −

(

3N−6

)

eigenvalues will be (numerically) zero. It follows from the above discussion that the eigenvector matrix L_ms can be expressed in redundant internal coordinates as

ms ,

12

s G L

L = ^(4.55)

or Cartesian coordinates as

ms .

12 s 1

x B 1L B G L

L = ⁻ = ^{− (4.56)}

The eigenvectors of the L matrices allow us to determine the distortion to the molecular frame in 3N−6 orthogonal directions, which are the normal coordinates Q, defined as

x , L

Q_x = ^T_x (4.57)

s , L

Q_s = ^T_s (4.58)

, ms L

Q_ms = ^T_ms (4.59)

and we see that the columns of L (or the rows of L^T) contain the coefficients of the linear combinations of respective coordinates that make up the normal coordinates.

Under the harmonic approximation, the mean-squared amplitude of vibration along the kth normal mode can then be obtained using the equivalent of Equation 4.48

2 , 8 ² coth

2

, 



 



= 

kT h

l h ^k

k k

h ν

µν

π ^(4.60)

where the equilibrated internal temperature of the molecule under the experimental conditions is used. Another consequence of the harmonic approximation is that the probability density of internuclear distance is given as (cf. Equation 2.20)

2 . 2 exp

) 1

( ₂ ²₂



 



 ∆−

=

∆

h lh

P l

π ^(4.61)

To determine the molecular structure due vibrational distortion along the normal coordinates Q_s, the molecular frame is iteratively displaced by an amount

[

−4lh_,k,4lh_,k

]

∈

∆ along each direction k. The corresponding changes in the values of the individual redundant internal coordinates are then calculated as

, )

( _{, ,}

,k ie sik

i s L

s ∆ = +∆⋅ (4.62)

where Ls_,ik are the coefficients in the matrix L_s that corresponds to the ith redundant internal coordinate and the kth normal mode.

As stated earlier, the transformation matrix B is only defined in an infinitesimal region around the equilibrium geometry, such that the relation between the correspondingly displaced Cartesian coordinates x_k(∆) and the displaced redundant internal coordinates s_k(∆) is linear only at ∆=0 (equilibrium geometry).¹⁹ However, because the displacement ^∆ can take on finite values, the coordinates x_k(∆) have to be found iteratively in a series of n Cartesian displacement steps and through the recalculation of an instantaneous matrix B_n obtained from the resulting molecular geometry after each step. Once the Cartesian coordinates x_k(∆) corresponding to s_k(∆) are determined, the Cartesian distances between internuclear pairs, r_ij,k(∆), are readily found.

The distortion of an internuclear distance between atom i and atom j from the equilibrium value, rij_,e, due to overall vibrational motion is obtained by

, )

( )

( _,

4

4 ,

,

,

, ije

k l l ijk

e ij ij ij

r d P r

r r r

k h

k h

−

∆

∆

∆

=

−

=

∂

∑∫

^(4.63)

where the brackets denote average quantities. Finally, the overall mean-squared amplitude of vibration between atom i and atom j is obtained by⁸

(

( )

)

( ) ₄⁴ _, ( ) ( ) ² .

4 4

2 , 2 2

2,

,

, ,

∑

^_^

∫

, ^{∆ ∆ ∆−}_^

∫

^{∆ ∆ ∆}_^{ }_^

=

−

=

k

l l ijk l

l ijk ij ij ij h

k h

k h k

h

k

h r P d r P d

r r l

(4.64)