4.2 Mathematics of Structural Fitting (χ 2 -Fitting)
4.2.4 Choice of Structural Coordinates for Fitting
So far, we have not specified a coordinate basis, in which the molecular structure is defined. An intuitive way to define a molecular structure is to specify the
N
3 Cartesian coordinates of the individual positions of all N atoms atom in the molecule. However, internal coordinates are a better choice for the fitting parameters, because it eliminates 3 translational and 3 rotational degrees of freedom from the problem. The choice of which internal coordinates to use is, however, not unique.
Previously, structural refinement in the UED lab at Caltech was conducted using a minimal set of 3N−6 internal coordinates defined through a z-matrix formalism, using
1
N− bond lengths, N−2 bond angles and N−3 dihedral angles.9,18 A drawback of using such a minimal set of internal coordinates is that they are often strongly coupled, that is a change in one coordinate produces a change in another coordinate, e.g., changing a bond angle in a ring structure necessarily changes other bond lengths as well.
Mathematically, coupling between structural coordinates manifest itself as large off- diagonal elements in the Hessian matrix α defined in Equation 4.32. Furthermore, the simultaneous fitting of two correlated coordinates could lead to a large change in both coordinates, but the cumulative effect results only in a negligible reduction in the
χ2-value, such that the values of the two coordinates cannot be determined reliably based on the diffraction data. The use of 3N−6 z-matrix coordinates was sufficient for studies on very small molecules and molecules of high internal symmetry, where correlations between fitting parameters could be easily identified by manually fitting each coordinate separately. For larger molecules, however, and for molecules of low symmetry, which are reported in this thesis (see Chapter 5), this was no longer possible, such that an automated fitting procedure had to be developed.
Redundant internal coordinates, which use more than 3N −6 coordinates, are advantageous, because they allow for a more complete and physically reasonable description of the molecular structure, especially ring structures.103 Additionally, they provide significant improvements in efficiency, when used in optimization/minimization problems, compared to nonredundant internal coordinates (e.g., z-matrix coordinates) or Cartesian coordinates.104 To avoid the problem of coupling between coordinates, we have implemented a fitting routine based in singular value decomposition (SVD) in ueda.x (see below) that identifies, from an initial set of redundant internal coordinates, a new set of fitting coordinates, in the direction of which the χ2-surface curvature is most pronounced.
The redundant internal coordinate system s is related to the Cartesian coordinate system through
, Bx
s= (4.39)
where the transformation matrix B is evaluated at the equilibrium geometry xe and the equality is only valid for infinitesimal Cartesian displacements. Since x is a 3N-vector
and s is an M -vector (M >3N), the dimension of B is M×3N, i.e., B is not a square matrix and cannot be readily inverted. A recipe for calculating the matrix B is given in many textbooks.105,106
4.2.4.1 Singular Value Decomposition
For automated analysis of electron diffraction data, the fitting algorithm has to have two important capabilities. First, the coordinate, which reduces the χ2-value to the largest extent, has to be identified among all specified coordinates. Second, all the other coordinates need to be identified, which have changed simultaneously with the chosen coordinate. By employing new coordinates, which consist of an appropriate linear combination of the initially specified internal coordinates, this can be accomplished. In fact, the second issue can be avoided altogether, if the new internal coordinates can be defined to be orthogonal to each other. According to the spectral theorem,107 for any real and symmetric matrix (such as α), there exists a real orthogonal matrix Q, such that
1 ,
T QΛQ
Q QΛ
α= = − (4.40)
where Λ is a diagonal matrix containing the eigenvalues (or in this context the singular values) of α in ascending order and we have used the property QT =Q−1 for orthogonal matrices. The columns of the matrix Q consist of the eigenvectors of α and these eigenvectors together form an orthonormal basis, in which this problem can be solved.
We can see this by substituting Equation 4.40 into Equation 4.33 to obtain .
β Q a Q
Λ T∆ = T (4.41)
Since the same linear transformation, QT, is being applied to both ∆a and β, we can rewrite this equation as
. ' ' β a
Λ∆ = (4.42)
Thus, the new fitting parameters, ∆aj', are composed of a linear combination of the originally defined structural coordinates with coefficients given by the columns of Q (or the rows of QT).
∑
∆=
∆
i ji j
j Q a
a ' (4.43)
Since this new basis is orthogonal, the coupling between individual fitting parameters has been eliminated.
The parameters with the lowest associated fitting error (the directions of highest χ2-curvature and therefore highest confidence) are automatically identified through SVD. It was already pointed out that the variances of the estimated fitting parameters can be read off the covariance matrix C≡α−1, which in this new basis can be evaluated trivially as the diagonal matrix C'≡Λ−1. Thus, when fitting in the direction of parameter
j' a
∆ , a large eigenvalue λj corresponds to a small variance λj
1 and a small eigenvalue
λj corresponds to a large variance λj
1 . In other words, the χ2-surface is very shallow
along the coordinates corresponding to small eigenvalues. The minimum position obtained along these directions is likely to be affected by noise in the experimental data.
For this reason, it would be prudent not to try to fit these parameters, but rather freeze
them at their initial value. This can be accomplished by setting to zero all values λj
1 ,
where λj falls below a certain threshold value. As a consequence, Equation 4.42 is solved using the pseudo-inverse of Λ, namely Λ'Thresh−1 , with the result that fitting occurs only in the (orthogonal) directions, in which the curvature of the χ2-surface is most pronounced, and to which the experimental data is most sensitive.
' '
' Λ β
a= −1
∆ Thresh (4.44)
In practice, fitting of the diffraction data is best accomplished by the stepwise unfreezing of fixed parameters, while monitoring the molecular structure at each point. Thus, after finding the χ2-minimum in the first direction, the χ2-minima in the second, third, … orthogonal direction are found. If the quadratic approximation to the χ2-surface is poor, then the method of steepest descent can be employed, as before, in this structural parameter basis by transforming Equation 4.35,
. ' constant
' β
a= ⋅
∆ (4.45)
The values determined for the fitted parameters and their associated covariance matrix can then be projected back onto the initially defined structural coordinates basis using
' a Q a= ∆
∆ and C=QΛ'−Thresh1 Q−1, respectively.