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The display of rotational transitions and their intensities by computer graphics
JOURNAL
OF MOLECULAR
SPECTROSCOPY
%,87-93
(1982)
The Display of Rotational Transitions and Their Intensities by Computer Graphics GRAHAM
H. KIRBY
Department of Computer Studies, University of Hull, Hull HU6 7RX, England
Fortrat diagrams have the disadvantage of giving no indication of the intensities of the transitions plotted on them. A new diagram drawn with the aid of computer graphics is demonstrated which indicates the intensities of rotational transitions. It is an isometric projection of a surface which represents the intensities of rotational transitions by the height of the surface above a plane describing their wavenumber and J quantum number values. The diagram is of particular use for displaying all the significant rotational transitions contributing to a computed band contour and for illustrating the effect of changes in the rotational constants. Illustrations of the new diagram are given with computed band contours for bands in the electronic spectra of aniline, phenol, and formyl fluoride. INTRODUCTION
Band contours observed in infrared, Raman, and electronic spectroscopy result from unresolved rotational lines. In the case of large molecules the density of rotational lines compared with the Doppler linewidth is such that individual lines can never be resolved unless the effect of Doppler broadening can be avoided. Fortrat diagrams have been used to show the positions of lines in branches within such band contours and it has been shown (1) that plotting as many as 10 000 transitions from one band contour does lead to a meaningful display of the positions of rotational lines. Such diagrams have been used for the assignment of both intense and weak features in band contours, especially in the electronic spectra of large molecules. Fortrat diagrams originated from the analysis of resolved rotational structure of small molecules where individual lines could be resolved and represented on a graph of the Fortrat parabola (e.g., p. 47 of (2)). The Fortrat diagram has a major drawback in that it gives no indication of the intensities of the transitions. Some workers have attempted to overcome this by indicating the position of the most intense line in a branch on their Fortrat diagrams (3). This paper describes a new type of diagram produced with the aid of computer graphics in which both the J quantum number and the line intensity are plotted against wavenumber. The line intensity is that which would be summed into a band contour and includes Boltzmann and statistical weight factors. COMPUTER GRAPHICS
When the intensity of each transition is to be represented in a diagram as well as the J value of the lower level and the frequency of the transition, the result is a 87
C!O22-28521821 I 10087-07$02.00/O Copyright
Q 1982 by Academic
All rights al reproduction
Press, Inc.
in any form reserved.
88
GRAHAM
H. KIRBY
three-dimensional surface. The display of this is most meaningful to spectroscopists if the transition intensities are represented by the z axis rising above the x, y plane of the conventional Fortrat diagram, which represents wavenumber plotted against J value. Computer graphics software has become available in recent years which facilitates the display of three-dimensional surfaces by isometric and perspective projections onto the two-dimensional surface of a plotter or screen, examples being ASPEX (4), used in the work described here, and SYMVU (5). These programs were initially used in geography for mapping, but can be applied to the problem of displaying three-dimensional scientific data (6). The programs generate projections of a surface which is described by an input data matrix of values giving the z coordinate for a regular grid of x and y coordinates which are implied by the row and column of each z value. One row of the input matrix in this application represents the total intensity of rotational transitions with the same J quantum number in the ground state. The number of columns is determined by the resolution and the range of wavenumbers to be plotted. In the illustrations used here the resolution was 0.125 cm-’ resulting in some 240 columns to cover the wavenumber range from -20 to + 10 cm-‘. The ASPEX program has an advantage in this respect over SYMVU in that the input matrix can be up to 500 X 500 in dimension whereas in the current SYMVU program it is restricted to 130 X 130. It follows that, in the preparation of the input matrix, where two or more rotational lines lie closer together than the resolution of the diagram their intensities must be summed. A particular point on the surface ultimately plotted does not therefore necessarily represent one transition. This does have an advantage in that the intensities of coincident lines are correctly represented whereas in the Fortrat diagram two or more marks superimposed on one another may be indistinguishable from one mark. Since computer programs which calculate rotational transitions do so for each J value in turn, it is a simple matter to add the computation of each row of the ASPEX input matrix to existing software. The projection diagram is drawn in the form of continuous lines along either the rows, the columns, or the diagonals of the surface defined by the input matrix. Lines hidden from the viewpoint are not drawn. Drawing lines along the rows has more significance in this application since each row represents one J value. It also permits the display at the top of the diagram, effectively behind the surface, of the computed band contour which is, of course, obtained by summing the input matrix by columns. This requires the intensities representing the band contour to be summed into a further row of the input matrix, following that of highest J. The contour is plotted at the top of the projection by the drawing of this last row. Isometric projections are preferable to perspective projections in this application since the scale of the diagram is then the same for all rows. The viewpoint, in the case of isometric projections, is described by two angles specifying altitude and azimuth. The altitude used for the work described here was typically about 45” though, with contours where there are more intense transitions of low J, greater angles may be needed. The azimuth normal to the rows (0 or 180” depending upon the ordering of the rows) is that most suitable for display of all the transitions, though azimuths varying by up to 260” from this may be helpful for viewing certain
DISPLAY OF ROTATIONAL TRANSITIONS
89
branches. Some initial investigation into the most satisfactory altitude and azimuth is recommended for a new contour. Both the ASPEX and SYMVU programs can smooth the z values of the input matrix by a binomial redistribution amongst adjacent elements of the x, y grid. Smoothing helps to remove the spiky appearance of the surface. The zero regions of the input matrix need not be drawn and this does improve the clarity of the diagram. Computer graphics hardware has also developed rapidly in recent years and graphics terminals are now readily available with the necessary resolution to display the projections illustrated here in a matter of minutes. Pen plotters have some advantages, though, in that a permanent copy of the display can be produced at a scale to suit the user. The drawing on a pen plotter of a projection of the type illustrated here is very much quicker than the drawing of the corresponding Fortrat diagram, which would involve plotting tens of thousands of separate points. RESULTS
Illustrations produced by the ASPEX program are given in Figs. 1 and 2 which correspond to the type B O-O bands in the A’&-2 ‘A, electronic transitions of aniline and phenol. These may be compared with the Fortrat diagrams (1) and the detailed assignments (7, 8) given elsewhere. The resolution used in Figs. 1 and 2 is rather less than in the previous calculations and cutoffs have been applied to save computation time by eliminating very weak transitions and those with high K, values. Each projection diagram provides a complete record of how the rotational transitions contribute to the computed band contour, which of course can be compared with the observed contour. The relative importance of different transitions to the contour are made clear by the relative heights of the projected surface which are on a linear scale. In Figs. 1 and 2 very strong R- and Q-branch lines are responsible for the peaks which dominate the band centers with Q- and stronger P-branch lines causing the regular fine structure to lower wavenumbers. The effect on the rotational transitions, and hence on the contour, of small differences in the changes from ground to excited state of rotational constants B and C (7, 8) for these two isoelectronic molecules is made very clear by a comparison of Figs, 1 and 2. In Fig. 3 is a projection diagram for the transitions in a type C band of a smaller molecule, formyl fluoride (HCOF). Well-resolved rotational structure is observed to low wavenumber of the band center and the published analysis includes a Fortrat diagram of some of the Q branches (9). With a greater wavenumber range to be covered, the resolution was reduced to 0.2 cm-’ giving an input matrix with some 450 columns. It was found beneficial to the appearance of this diagram to use smoothing and to raise the altitude from which the projection was viewed to about 80”. This diagram shows that, apart from two or three more intense features just to low wavenumber of the band origin which result from Q branches, the resolved structure at lower wavenumbers is of P lines which are of greater intensity than the Q lines in the same region. Of particular interest is the way that individual transitions stand out in the projection. The diagram can be used to assign features in the contour to individual lines or groups of lines.
GRAHAM
_)
0 0
H. KIRBY
DISPLAY OF ROTATIONAL TRANSITIONS
92
GRAHAM H. KIRBY
-
-
c
-
DISPLAY OF ROTATIONAL
TRANSITIONS
93
CONCLUSION
A new diagram has been illustrated in which the intensities of rotational transitions are represented by the heights of a surface above the plane represented by their wavenumber and J values. The data required for the drawing of such diagrams can be easily obtained from band contour programs and the ASPEX program which draws the isometric projection performs its task in a fraction of the time required to draw a complete Fortrat diagram for a large molecule. The resulting diagrams should be an aid to the understanding and analysis of band contours and resolved rotational structure in molecular spectra. ACKNOWLEDGMENTS I am indebted to Mr. A. Rixon who first used the ASPEX program on data for rotational transitions and to Dr. J. M. Hollas for his comments on the manuscript. RECEIVED:
April
30, 1982
Note added in prool: The author’s attention has heen drawn to the use of isometric projections of Locmis-Woods plots by F. Winther (J. Mol. Spectrosc. 62, 232-246 (1976)) in the assignment of weak lines in the far-infrared spectrum of HCNO.
REFERENCES I. 2. S. 4. 5. 6. 7.
8. 9.
G. H. KIRBY, Mol. Whys. l&371-382 (1970). G. HERZBERG,‘Spectra of Diatomic Molecules,” p. 47, Van Nostrand, New York, 1950. A. J. MCHUGH, D. A. RAMSAY,AND I. G. Ross, Aust. J. Chem. 21, 2835-2845 (1968). M. HANSON,“ASPEX User’s Reference Manual,” Laboratory for Computer Graphics and Spatial Analysis, Harvard University, Cambridge, Mass., 1978. F. J. RENS, “SYMVU Manual,” 3rd ed., Laboratory for Computer Graphics and Spatial Analysis, Harvard University, Cambridge, Mass., 1977. G. H. KIRBY AND A. RIXON, Comput. Phys. Commun. 21,287-291 (1981). J. CHRISTOFFERSEN,J. M. HOLLAS, AND G. H. KIRBY, Mol. Phys. 16,441-452 (1969). J. CHRISTOFFERSEN,J. M. HOLLAS, AND G. H. KIRBY, Proc. Roy. Sac. A 307,97-l 10 (1968). J. E. PARKIN AND K. K. INNES,J. Mol. Spectrosc. 16, 93-99 (1965).
OF MOLECULAR
SPECTROSCOPY
%,87-93
(1982)
The Display of Rotational Transitions and Their Intensities by Computer Graphics GRAHAM
H. KIRBY
Department of Computer Studies, University of Hull, Hull HU6 7RX, England
Fortrat diagrams have the disadvantage of giving no indication of the intensities of the transitions plotted on them. A new diagram drawn with the aid of computer graphics is demonstrated which indicates the intensities of rotational transitions. It is an isometric projection of a surface which represents the intensities of rotational transitions by the height of the surface above a plane describing their wavenumber and J quantum number values. The diagram is of particular use for displaying all the significant rotational transitions contributing to a computed band contour and for illustrating the effect of changes in the rotational constants. Illustrations of the new diagram are given with computed band contours for bands in the electronic spectra of aniline, phenol, and formyl fluoride. INTRODUCTION
Band contours observed in infrared, Raman, and electronic spectroscopy result from unresolved rotational lines. In the case of large molecules the density of rotational lines compared with the Doppler linewidth is such that individual lines can never be resolved unless the effect of Doppler broadening can be avoided. Fortrat diagrams have been used to show the positions of lines in branches within such band contours and it has been shown (1) that plotting as many as 10 000 transitions from one band contour does lead to a meaningful display of the positions of rotational lines. Such diagrams have been used for the assignment of both intense and weak features in band contours, especially in the electronic spectra of large molecules. Fortrat diagrams originated from the analysis of resolved rotational structure of small molecules where individual lines could be resolved and represented on a graph of the Fortrat parabola (e.g., p. 47 of (2)). The Fortrat diagram has a major drawback in that it gives no indication of the intensities of the transitions. Some workers have attempted to overcome this by indicating the position of the most intense line in a branch on their Fortrat diagrams (3). This paper describes a new type of diagram produced with the aid of computer graphics in which both the J quantum number and the line intensity are plotted against wavenumber. The line intensity is that which would be summed into a band contour and includes Boltzmann and statistical weight factors. COMPUTER GRAPHICS
When the intensity of each transition is to be represented in a diagram as well as the J value of the lower level and the frequency of the transition, the result is a 87
C!O22-28521821 I 10087-07$02.00/O Copyright
Q 1982 by Academic
All rights al reproduction
Press, Inc.
in any form reserved.
88
GRAHAM
H. KIRBY
three-dimensional surface. The display of this is most meaningful to spectroscopists if the transition intensities are represented by the z axis rising above the x, y plane of the conventional Fortrat diagram, which represents wavenumber plotted against J value. Computer graphics software has become available in recent years which facilitates the display of three-dimensional surfaces by isometric and perspective projections onto the two-dimensional surface of a plotter or screen, examples being ASPEX (4), used in the work described here, and SYMVU (5). These programs were initially used in geography for mapping, but can be applied to the problem of displaying three-dimensional scientific data (6). The programs generate projections of a surface which is described by an input data matrix of values giving the z coordinate for a regular grid of x and y coordinates which are implied by the row and column of each z value. One row of the input matrix in this application represents the total intensity of rotational transitions with the same J quantum number in the ground state. The number of columns is determined by the resolution and the range of wavenumbers to be plotted. In the illustrations used here the resolution was 0.125 cm-’ resulting in some 240 columns to cover the wavenumber range from -20 to + 10 cm-‘. The ASPEX program has an advantage in this respect over SYMVU in that the input matrix can be up to 500 X 500 in dimension whereas in the current SYMVU program it is restricted to 130 X 130. It follows that, in the preparation of the input matrix, where two or more rotational lines lie closer together than the resolution of the diagram their intensities must be summed. A particular point on the surface ultimately plotted does not therefore necessarily represent one transition. This does have an advantage in that the intensities of coincident lines are correctly represented whereas in the Fortrat diagram two or more marks superimposed on one another may be indistinguishable from one mark. Since computer programs which calculate rotational transitions do so for each J value in turn, it is a simple matter to add the computation of each row of the ASPEX input matrix to existing software. The projection diagram is drawn in the form of continuous lines along either the rows, the columns, or the diagonals of the surface defined by the input matrix. Lines hidden from the viewpoint are not drawn. Drawing lines along the rows has more significance in this application since each row represents one J value. It also permits the display at the top of the diagram, effectively behind the surface, of the computed band contour which is, of course, obtained by summing the input matrix by columns. This requires the intensities representing the band contour to be summed into a further row of the input matrix, following that of highest J. The contour is plotted at the top of the projection by the drawing of this last row. Isometric projections are preferable to perspective projections in this application since the scale of the diagram is then the same for all rows. The viewpoint, in the case of isometric projections, is described by two angles specifying altitude and azimuth. The altitude used for the work described here was typically about 45” though, with contours where there are more intense transitions of low J, greater angles may be needed. The azimuth normal to the rows (0 or 180” depending upon the ordering of the rows) is that most suitable for display of all the transitions, though azimuths varying by up to 260” from this may be helpful for viewing certain
DISPLAY OF ROTATIONAL TRANSITIONS
89
branches. Some initial investigation into the most satisfactory altitude and azimuth is recommended for a new contour. Both the ASPEX and SYMVU programs can smooth the z values of the input matrix by a binomial redistribution amongst adjacent elements of the x, y grid. Smoothing helps to remove the spiky appearance of the surface. The zero regions of the input matrix need not be drawn and this does improve the clarity of the diagram. Computer graphics hardware has also developed rapidly in recent years and graphics terminals are now readily available with the necessary resolution to display the projections illustrated here in a matter of minutes. Pen plotters have some advantages, though, in that a permanent copy of the display can be produced at a scale to suit the user. The drawing on a pen plotter of a projection of the type illustrated here is very much quicker than the drawing of the corresponding Fortrat diagram, which would involve plotting tens of thousands of separate points. RESULTS
Illustrations produced by the ASPEX program are given in Figs. 1 and 2 which correspond to the type B O-O bands in the A’&-2 ‘A, electronic transitions of aniline and phenol. These may be compared with the Fortrat diagrams (1) and the detailed assignments (7, 8) given elsewhere. The resolution used in Figs. 1 and 2 is rather less than in the previous calculations and cutoffs have been applied to save computation time by eliminating very weak transitions and those with high K, values. Each projection diagram provides a complete record of how the rotational transitions contribute to the computed band contour, which of course can be compared with the observed contour. The relative importance of different transitions to the contour are made clear by the relative heights of the projected surface which are on a linear scale. In Figs. 1 and 2 very strong R- and Q-branch lines are responsible for the peaks which dominate the band centers with Q- and stronger P-branch lines causing the regular fine structure to lower wavenumbers. The effect on the rotational transitions, and hence on the contour, of small differences in the changes from ground to excited state of rotational constants B and C (7, 8) for these two isoelectronic molecules is made very clear by a comparison of Figs, 1 and 2. In Fig. 3 is a projection diagram for the transitions in a type C band of a smaller molecule, formyl fluoride (HCOF). Well-resolved rotational structure is observed to low wavenumber of the band center and the published analysis includes a Fortrat diagram of some of the Q branches (9). With a greater wavenumber range to be covered, the resolution was reduced to 0.2 cm-’ giving an input matrix with some 450 columns. It was found beneficial to the appearance of this diagram to use smoothing and to raise the altitude from which the projection was viewed to about 80”. This diagram shows that, apart from two or three more intense features just to low wavenumber of the band origin which result from Q branches, the resolved structure at lower wavenumbers is of P lines which are of greater intensity than the Q lines in the same region. Of particular interest is the way that individual transitions stand out in the projection. The diagram can be used to assign features in the contour to individual lines or groups of lines.
GRAHAM
_)
0 0
H. KIRBY
DISPLAY OF ROTATIONAL TRANSITIONS
92
GRAHAM H. KIRBY
-
-
c
-
DISPLAY OF ROTATIONAL
TRANSITIONS
93
CONCLUSION
A new diagram has been illustrated in which the intensities of rotational transitions are represented by the heights of a surface above the plane represented by their wavenumber and J values. The data required for the drawing of such diagrams can be easily obtained from band contour programs and the ASPEX program which draws the isometric projection performs its task in a fraction of the time required to draw a complete Fortrat diagram for a large molecule. The resulting diagrams should be an aid to the understanding and analysis of band contours and resolved rotational structure in molecular spectra. ACKNOWLEDGMENTS I am indebted to Mr. A. Rixon who first used the ASPEX program on data for rotational transitions and to Dr. J. M. Hollas for his comments on the manuscript. RECEIVED:
April
30, 1982
Note added in prool: The author’s attention has heen drawn to the use of isometric projections of Locmis-Woods plots by F. Winther (J. Mol. Spectrosc. 62, 232-246 (1976)) in the assignment of weak lines in the far-infrared spectrum of HCNO.
REFERENCES I. 2. S. 4. 5. 6. 7.
8. 9.
G. H. KIRBY, Mol. Whys. l&371-382 (1970). G. HERZBERG,‘Spectra of Diatomic Molecules,” p. 47, Van Nostrand, New York, 1950. A. J. MCHUGH, D. A. RAMSAY,AND I. G. Ross, Aust. J. Chem. 21, 2835-2845 (1968). M. HANSON,“ASPEX User’s Reference Manual,” Laboratory for Computer Graphics and Spatial Analysis, Harvard University, Cambridge, Mass., 1978. F. J. RENS, “SYMVU Manual,” 3rd ed., Laboratory for Computer Graphics and Spatial Analysis, Harvard University, Cambridge, Mass., 1977. G. H. KIRBY AND A. RIXON, Comput. Phys. Commun. 21,287-291 (1981). J. CHRISTOFFERSEN,J. M. HOLLAS, AND G. H. KIRBY, Mol. Phys. 16,441-452 (1969). J. CHRISTOFFERSEN,J. M. HOLLAS, AND G. H. KIRBY, Proc. Roy. Sac. A 307,97-l 10 (1968). J. E. PARKIN AND K. K. INNES,J. Mol. Spectrosc. 16, 93-99 (1965).