Journal of Electron Sp~et.roscopy and Related Phenomw, 41 (lQ66) X31-166 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherl~~

THE PHOTOELECTRON SPECTRUM OF PHOSPHABENZENE, AND MATCHING MODEL CALCULATIONS, TRANSCRIBED INTO MUSIC*

SYDNEY LEACH

Dt!partement d’Astrophysique Fondamentale, Obsermtoire de Paris-Meudon, 92190 Me&m (France) and ~bora~~re a’s Photophysiq~ Mo~a~ire du C.N.R.S .**, ~ni~rsi~ Pad-So, 31405 Orsuy (France)

GIUSEPPE G. ENGLERT

D~partement de Musique, Universite Paris VIII, 93625 Saint-Denis (France)

(Received 12 August 1966)

ABSTRACT

A musical transcription technique for amplifying differences between experimental and cal- culated photoelectron spectra is described and applied to the spectra of phosphabensene. This new diagnostic procedure includes the development of the transcribed mus~~~a into a piano com- position. The results reveal a number of features of both musical and scientific interest.

INTRODUCTION

The dialectic between mathematics and music has proved to be fruitful since early historical times, Classical mathematical physics is considered to have begun with the observation by Pythagoras of the simple relation between the lengths of struck metal rods and known musical intervals in the ensuing sounds [l]. From physics to metaphysics [2] was but a short step that led to the conclusion that physical harmony and order reign in nature, expressed for example at the dawn of the modern world by Kepler’s “music of the spheres” [3]. A blending of these approaches can be found in our time by research into the music of elementary particles [4] and of gene structure [5, S], To these motivations can be added the use of mathematical relationships as a priori determinants of musical construction and development. This is found among great composers of the past, e.g., J.S. Bach and A. SchGnberg and flourishes in our day and age with composers such as Messiaen, Boulez, Xenakis and Stock-

* This fantasia is dedicated to Prof. E. Heilbronner on the occasion of his 66th birthday. **Laboratory associated Universitb Paris-Sud.

~~I~~~.~ @ 1986 Elsevier Science Publishers B.V.

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hausen. For example, Fibonacci series have been used as musical determinants by recent composers as diverse as Scriabin, Mahler, Bartok and Debussy.

The use of random number selection in assigning notes to particular digits, has been used for compositional purposes, with special rules to reinforce or create harmony [7]. There is perhaps more than an analogy here with the work of Heilbronner and Schmelzer [8] who used a set of random numbers to generate predicted energy values of the photoelectron spectrum of phosphabenzene (MORON procedure). They showed, using least-squares techniques, that the MORON calculated PES band energies matched the observed ones with cor- relation coefficients in excess of 0.98. This was found to be almost as good as the results of calculations by von Niessen et al. [9], using many-body Green’s function treatment, including electron correlation and reorganization, which gave an excellent match to experiment with a correlation coefficient greater than 0.99.

We were thus inspired to transcribe to music the three photoelectron spectra of phosphabenzene (A = MORON calculated spectrum, B = many-body Green’s function calculation, C = experimental photoelectron spectrum) in order to amplify differences in the three spectra for diagnostic purposes. We first describe the method of transcription and then analyze the principal feat- ures revealed by the results. This is followed by remarks on the development into a piano composition of the three musical sequences generated by the spectra and by a presentation of the composition. Finally, conclusions are given on the musical and scientific aspects of the work.

METHOD OF TRANSCRIPTION

The transcription to music of the photoelectron spectra was devised by one of us (S.L.) and carried out as follows. The two musical parameters to be transcribed from the PES were pitch and duration. The twelve photoelectron spectral bands of phosphabenzene given in ref. 8 were first ordered in frequency I(i) and the intervals AI(i) = I(i + 1) - I(i) were determined. The AI values in hertz were converted into audiofrequencies v,,~ by means of a scaling factor, Iz. = 4.454 x 10m5, chosen such that if AI = y(meV) then V.~ = y(Hz)

v,,~(Hz) = kAI(Hz) (1)

A sequence of eleven audio frequencies v,,&) was obtained from eqn. 1 corresponding to AI(i), i = 1,2, . . . 11, which were listed as a succession of musical notes m(i) by aSSigning each v.,d to the nearest frequency v(i) in the equal temperament scale. Since the AI(i) values differ for the spectra A, B, C, three different note sequences (musispectra) were obtained.

The duration of note m(i) was taken as being proportional to the intensity of the experimental photoelectron band i [8, lo]. No equivalent intensity data were available for the MORON and many-body Green’s function calculated spectra. Therefore the duration of corresponding notes m(i) in spectra A and B was taken as that given by the experimental spectrum C. Calibration of the

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duration scale was obtained by equating the intensities of PES bands i = 1 and i = 6 with a minim (half-note in North-American musical language).

RESULTS AND DISCUSSION

Transcription and composition

The crude musical transcriptions of the three phosphabenzene photoelec- tron spectra are shown in Fig. 1. Examination of the three musical sequences reveals some striking features.

(1) From the melodic point of view, musispectra A and B are very similar. In fact sequence A could be a variation on sequence B and vice versa. On the other hand, musispectrum C has no resemblance to the other versions.

(2) Musispectrum A is harmonically centred around the note D and its octave. The first two notes (octave G-G) are the farthest from D, and seem to impose a tonal centre G. The sequence therefore divides itself into two groups; first two notes (G-G) and the last nine notes, which represent small deviations around D, in increasing order (+ 1, - 1, + 2, + 3).

(3) Musispectrum B is the least rewarding musically; a hypothetical tonal centre D is not convincingly confirmed, and the octave F-F in the middle sounds awkward.

(4) Musispectrum C, based on the experimental photoelectron spectrum of phosphabenzene, is the most surprising of the three sequences. All of its notes are part of the same tonal spectrum (fundamental G of double bass x 5OHz). The sequence moves inside one constant resonance. Each note has its place in a harmonic Fourier series, the order of each term being shown in Fig. 1 below musispectrum C.

PHOTOELECTRON SPECTRUM OF PHOSPHABENZENE MUSICAL TRANscR~PTIC+JS

SYDNEY LEACH

A

B

C

1

_ FOURIER SERIES - TERMS

Fig. 1. Photoelectron spectrum of phoephabenzene; musical transcriptions of the MORON model calculated spectrum (A), the many-body Green’s function calculated spectrum (B), and the experi- mental spectrum (C).

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The three musical sequences were developed by one of us (G.G.E.) into the piano composition given in Fig. 2. The composition sought to ensure that the above observations would be audible in a piano performance of the three little pieces. The order C-B-A-C was chosen by one of us (S.L.) to satisfy his sense of scientific appropriateness. No gratuitous ornamentation has been included. The composer’s purpose was not an aesthetic one, but rather to achieve clarity in the musical observation of scientific data.

MUSICAL DEVELWMENT AND INTERPRETATION FOR PIANO

GIUSEWE G ENGLERT

05 07 1996

s&?o Ped Ped -

Fig. 2. Experimental and calculated photoelectron spectra of phosphabenzene; musical develop. ment and interpretation for piano.

CONCLUSIONS

From the musical point of view it should be stated that any transcription into musical events of occurrences outside the domain of music has to accept some arbitrary decisions during the process of realisation. It is considered that the ~a~~~ption in the present work, based on experimental and calculated photoel~on spectra of phospha~~ene, is as good as any other possible transcription.

It would be of interest to invent further rules- of transcription in order to derive other musical parameters. Furthermore, within the context of transcrip- tion of photoelectron spectra, it could prove fascinating to extend the analysis so as to include vibrational features and even vibronic and jahn-Teller coupl- ing effects [ll]. It is expected that the ensuing musispectra would need to be transcribed in finely-divided musical scales, analogous to those current in much non-Western music.

From the scientific viewpoint, whether the Fourier series term relation is a

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general property of experimental photoelectron spectra requires further re- search. If so, it could give new insight into the physics of ionization and the structure of molecular ions and provide new constraints for testing the quality of calculated photoelectron spectra.

The differences in the experimental and calculated photoelectron spectra are clearly enhanced by their musical transcription and in the developed musical composition. It is suggested that the musical transcription method may provide a diagnostic means whereby a fine ear can tell which of a series of calculated spectra is the best. We recommend that every serious laboratory of photoelectron spectroscopy should recruit a composer. Conversely, musical conservatories should become aware of the potential of photoelectron spectro- scopy. They should consider investing in a He(I) photoelectron spectrometer and eventually develop-a good He(I1) lamp in order to extend the musispectral range. The financial investment involved is of the order of a very good concert piano. In the initial stages it is not recommended to invest in a variable photon energy source such as synchrotron radiation [12, 131 which, although more flexible and capable of very extensive musical research over vast octave ranges, would require the technical skill, creative imagination and financial wizardry associated with the creation and development of a first-rate symphony orchestra [14].

REFERENCES

1 There is some confusion in the tradition as to the exact experimental observations supposedly made by Pythagoras. See, for example, Anicus Manlius Torquatus Severinus Boethius, De Institutione Musica, I. 10-11 (ca. SOOA.D.), G. Friedlein, Leipxig, 1867; R.P. Feynmann, R.B. Leighton and M. Sands, The Feynmann Lectures on Physics, Vol. I, Calif. Inst. Technol., 1963, pp. 50-51.

2 For an anti-Pythagorean view, Bee C. Fruttero and F. Lucentini, L’ldraulico non Verra, Mario Spagnol, Milan, 1971, p. 24.

3 J. Kepler, Harmonices Mundi, Linx, 1619. This work was dedicated to King James I of England, great patron of the arte and learning.

4 J. Sternheimer, C. R. Acad. Sci. Paris, 297 (1983) 829. 5 K. Hayashi and N. Munakata, Nature (London), 310 (1984) 96. 6 A. L. Lecocq and A. Danchin, La Recherche, 17 (1986) 977. 7 J.A. Morrison, Faraday Discuss. Chem. Sot., 80 (1986) 7. 8 E. Heilbronner and A. Schmelxer, Nouv. J. Chim., 4 (1980) 23. 9 W. von Niessen, G.H.F. Diercksen and L.S. Cederbaum, Chem. Phys., 10 (1976) 345.

10 C. Batich, E. Heilbronner, V. Hornung, A.J. Ashe III, D.T. Clark, U.T. Cobley, D. Kilcast and I. Scanlan, J. Am. Chem. Sot., 96 (1973) 928; J. Bastide, E. Heilbronner and J.P. Maier, Tetrahedron Lett., (1976) 411; A.J. Ashe III, F. Burger, M.Y. El-Sheik, E. Heilbronner, J.P. Maier and J.-F. Muller, Helv. Chim. Acta, 69 (1976) 1944.

11 C. Cossart-Magos, D. Coseart and S. Leach, Mol. Phys., 37 (1979) 793. 12 P. Dagneaux, C. Depautex, P. Dhez, J. Durup, Y. Farge, R. Fourme, P.M. Guyon, P. Jaegle,

S. Leach, R. Lopez-Delgado, G. Morel, R. Pinchaux, C. Vermeil and F. Wuilleumier, Ann. Phys., 14 i&me drie, 9 (1976) 9.

13 S. Leach, in K.L. Kompa and J. Wanner (Eds.), Laser Applications in Chemistry, Plenum, New York, 1984, pp. 36 and 47.

14 N. Cardus, Sir Thomas Beecham, Collins, London, 1961