This problem seemed especially intriguing to me for two reasons:

• It is a Nuclear Magnetic Resonance problem.
• It is a vintage problem, waiting for a solution for 19 years.

A non-vanishing asymmetry parameter h is an essential prerequisite for non-elementary singularity patterns of exchange spectra.

If Euler angles a and g are equal one obtains singularity patterns which are symmetrical with respect to the diagonal. (The same symmetry occurs if the sum of a and g is an integer multiple of 180°.) If I failed to solve this special case g = a, there would be no chance at all to cope with the more difficult case of arbitrary Euler angles.

Initially, even this constricted problem looked completely hopeless. Two ideas, however, made a big difference, and gradually I got the impression that an analytical description could be within my reach. The solution struck me on a Friday evening at 10:15 pm. Now I had laid the first cornerstone of my singularity project. I had calculated an analytical description of symmetrical singularity patterns which did not consist of elementary geometrical objects, and probably no one had done that before. Three days later I plotted the patterns on a computer screen and saw they were correct.

Of course, I hoped a modification or generalization of my procedure could solve the case of arbitrary Euler angles, but whatever idea I had, there were always some unwanted non-vanishing terms which prevented an exact solution.

There was a second problem which attracted my interest during the summer of 1999. I had discovered that symmetrical exchange spectra are apparently C₃ symmetrical if h, a and b are chosen appropriately. Certainly, my analytical description of the symmetrical singularity patterns contained conditions for C₃ symmetrical singularity patterns implicitly. At first I hoped to extract them within a few days, but this task proved to be surprisingly difficult.

At the end of September, after an interval with only minor progress, I made the decision to focus my time and effort for a while on a presentation of my research results and to postpone further investigations of the singularities.

By mid-November 1999 I had derived an equation which made me really happy. It was nothing more than an alternative formulation of the singularity problem. But it promised to be enormously powerful.

By the help of this formulation I succeeded to describe exactly the singularity patterns of the most general exchange spectrum. For several reasons, however, I could not accept this description as a perfect final one.

The equation found on November 13 did not make the remaining parts of the problem easy, but it made them tractable. It was the starting point of all subsequent investigations, a point already halfway to the solution.

In March 2000, nine months after my calculation of exact singularity patterns for the special case g = a, I achieved a breakthrough when I derived a necessary condition for C₃ symmetrical singularity patterns:

Three days later I found an independent second condition.

(1) and (2) must be fulfilled simultaneously to get C₃ symmetrical singularity patterns.

f and g are by no means simple functions of the asymmetry parameter h and the Euler angles a and b. To check whether I'm on the right track I used a numerical standard procedure and determined pairs of a and b for given values of h. Indeed each of these parameter triplets led to apparently C₃ symmetrical singularity patterns, and I could aim at my next goal, the transformation of (1) and (2) into analytical expressions for a and b. This job was done after three weeks:

b is a function of h alone and must be calculated first. Then a can be obtained as a rather simple function of h and b.

The derivation of equations (1) through (4) may have been sped up by an event which attracted more attention than mathematics can do: The EXPO 2000 in Hannover, Germany.
In March 2000 I undertook the task of calculating series of 2D NMR exchange spectra which were to be assembled to a movie and presented at the EXPO in the Science Tunnel of the Max-Planck-Society. One year ago I had already developed a program which calculated numerically high quality 2D exchange spectra at moderate computation times. What remained to be done was to find a trajectory in the parameter space which resulted in an attractive series of spectra.
I knew that, in principle, it was possible to create a series of C₃ symmetrical spectra by varying the asymmetry parameter h and adjusting Euler angles a and b in some way. But in which way? Certainly, it was impossible to set the angles by trial and error for hundreds of spectra. Either I had to forget about a C₃ symmetrical series of spectra or I had to calculate a and b as functions of h, and that's what I finally did.

Equations (3) and (4) are necessary conditions for C₃ symmetrical singularity patterns and therefore necessary conditions for C₃ symmetrical spectra. I've never investigated whether these conditions are also sufficient in order to get C₃ symmetrical spectra, but apparently they are.
The existence of a C₃ symmetrical series of spectra has been quite surprising even for experienced researchers in this field.

Click on the icons below to see a small selection of the thousands of frames which were assembled to the movie presented at EXPO 2000. Note that all these spectra have been broadened by convolution with the same Gauß function.

Until July 2000 I had tried to describe singularity patterns of exchange spectra. Those are obtained if both interaction tensors under consideration are characterized by the same asymmetry parameter h. Lifting this constraint leads to the more general class of correlation spectra.

At the end of July I discovered two facts which opened the way for further investigations:

• a) Dealing with two different asymmetry parameters does not increase the complexity of the problem significantly.
• b) On the other hand, setting one of the asymmetry parameters to zero eliminates the problem's toughest complications.

Indeed, it was possible to solve this simplified version of the problem by the help of the alternative characterization found nine month before. I figured out an exact and complete description of the singularity patterns.

The special case h₁ = 0 is not only of theoretical interest. It describes the correlation of the ¹³C chemical shift anisotropy with the ¹³C-¹H dipolar interaction. The first correlation spectra of this kind were obtained by Linder et al. in an experiment which is considered the first 2-dimensional NMR powder experiment:

[1]  M. Linder, A. Höhener and R. R. Ernst:
      Orientation of Tensorial Interactions from 2D NMR Powder Spectra.
      J. Chem. Phys. 73, 4959 (1980).

The point patterns found in this publication on page 4967 were created by determining singular points on straight lines parallel to the chemical shift axis. This method allows to calculate the coordinates of such points exactly, but it provides no information about how these points are to be connected. Singularity patterns obtained in this way are always made up of isolated points.

A comparison of these patterns with those calculated by my program seems at first disappointing. But don't worry! That's only because in [1] the polar angles a and b, which describe how the dipolar interaction tensor is oriented in the principal axis system of the chemical shift tensor, are chosen in a different way than I've done it. If the angles a and b as chosen in [1] are renamed to a' and b', then these angles are related to a and b as chosen by me in the following way:

To facilitate a comparison of my singularity patterns with those published 21 years ago I've created an additional controls panel which uses a' and b' as input parameters.

My program deals with traceless interaction tensors which are normalized in such a way that the principal value of largest absolute value is 1. Then the principal values expressed by asymmetry parameter h₂ are

The singularity patterns in [1], however, were calculated for a chemical shift tensor with principal values

To extract the traceless part, a third of the trace is subtracted from each principal value:

Division by -90.53 ppm and sorting in ascending order results in the normalized tensor

As seen from (5), the asymmetry parameter is simply the difference of the second and first diagonal element: h₂ = 0.358. Choose an h₂ value of 0.36 on the Dip-Shift controls panel to get a close match with the patterns presented in [1].

The normalization of the tensor has changed the sign of all its components which would cause a reflection of the singularity pattern about the ordinate. In order to keep the comparison of the patterns as easy as possible I have inverted the abscissa. In contrast to all other panels, the abscissa values range from +1 (left limit) to -1 (right limit). As usual, the ordinate values range from -1 (lower limit) to +1 (upper limit).

At the beginning of September 2000 only one case remained to be solved: The most general one. All other cases were more or less simplified by special choices of asymmetry parameters or Euler angles which made one or the other term vanish. Now all these terms were present at the same time, and I had to find some way to the final goal, the exact description of the corresponding singularity patterns.

As usual, I started at my alternative description of the problem. Of course I tried to use the strategy which was capable of solving all special cases. This way would lead me also to the solution of the general case if I was able to overcome the following obstacles:

• a) Decompose an expression consisting of trigonometrical functions into factors.
• b) Calculate the (usually) four different zeros of another expression which depends on the solution of part a).

By the end of September I had found different solutions of part a). Part b) was the crucial one and had to show which solution of a), if any, was appropriate.

Meanwhile, I was totally convinced that I would solve the singularity problem completely, and the time had come to prepare a presentation of these new results. One year ago, exactly on October 21, 1999, I had already decided to present them on my private website. Now I bought the book "Java in a Nutshell" by David Flanagan, an excellent introduction to the Java programming language and a quick-reference for the core Java packages. This book contains everything I needed to write the mathematical part of my program. The book "Java Foundation Classes in a Nutshell", written by the same author, enabled me to write the graphical part.

The 15th of December 2000 ended with a final farewell to the singularity problem! I solved the last open part of this longstanding problem by analytically calculating the four zeros of the expression mentioned above.

Included in this controls panel is a pull-down menu which enables you to switch between the different panels.

Basically, my program has to perform three tasks:

• Draw a controls panel which displays the actual values of the input parameters and which allows the user to change them.
• Calculate the associated singularity pattern.
• Plot the calculated pattern.

Since the number or type of input parameters is different for the different constraints laid upon them, I had to create six different controls panels altogether.

Use the left pull-down menu to switch between the different controls panels. Choosing another controls panel will preserve the values of input parameters if possible or will set them in a reasonable way. Check it out! The simple rules followed by the parameter values are easily recognized.

It is not especially funny if one clicks a button and the singularity pattern remains the same. Therefore, the color of a button informs you whether clicking on it will change the pattern (green) or not (red). If a button is colored in red because a boundary of a parameter interval is reached, nothing at all will happen if you try to leave this interval.

But if buttons are red because the actual pattern is not sensitive to a change of a certain parameter, clicking on such a button will change the parameter as intended, and the corresponding pattern is calculated and displayed.

The constituents of the patterns are drawn in characteristical colors to provide the user with additional information. That's possible because the program knows from the values of the input parameters which subroutine is appropriate to calculate the singularity patterns and which type of pattern will result.

If an object is drawn in orange you can be sure it is exactly the outline of an ellipse. An object drawn in red is exactly a straight line, an object drawn in black is exactly a point. Since a single black pixel is nearly invisible, a point is represented by a black cross consisting of five pixels.
The outline of the convex shaped area is drawn in green, the outline of the concave shaped one in blue.

Since for every kind of singularity pattern an exact mathematical description is available, the calculation of a single pattern is done in no time, and it is very tempting to display also series of frames. Of course the number of frames per second is not limited by the time the processor needs to calculate a pattern, but by the time the graphics hardware needs to plot a new frame on the screen.