ABSTRACT QUANTUM MECHANICAL ANALYSIS OF HIGH-RESOLUTION NUCLEAR MAGNETIC RESONANCE SPECTRA by Yuh Kang Pan The quantum mechanical and algebraic procedures involved in the analysis of high-resolution nuclear magnetic resonance spectra were investigated. The formulation of the quantum mechanical problem for analysis of high-resolution nuclear magnetic resonance spectra was first discussed. Particular attention has been given to developing a convenient computer program for calculating matrix elements of the high-resolution nuclear magnetic resonance spin-coupling Hamiltonian. The derivations of general equations for the chemical shifts and spin coupling constants of a number of systemSof nuclei with spin 1/2 in terms of the values of the experimental energy levels have been developed and presented. A computer technique for assigning the observed spectral lines to transitions within the energy-level diagram in a manner consistent with equal-spacing and intensity-sum rules has been described. It has been shown that the analysis of many complex NMR spectra can be reduced to the problem of assigning observed spectral lines to the appropriate transitions within schematic energy level diagrams, followed by direct calculation of the desired spin parameters. This computer assignment technique was then applied to examples of two, three, four and five-spin systems to illustrate the procedure. QUANTUM MECHANICAL ANALYSIS OF HIGH-RESOLUTION NUCLEAR MAGNETIC RESONANCE SPECTRA BU Yuh-Kang Pan A THESIS Submitted to Michigan State University in partial fulfullment of the requirements for the degree of ' DOCTOR OF PHILOSOPHY Department of Chemistry 1966 ACKNOWLEDGEMENTS The author wishes to express his sincere appreciation to Professor Max T. Rogers for his counsel and encourage- ment throughout this work. The author also wishes to thank Professor D. Whitman of the Case Institute of Technology and Professor Y. N. Chiu of the Catholic University of America for helpful discussions. Thanks are also expressed to the staff of the computer Laboratory of Michigan State University for their valuable cooperation. He also wishes to express his gratitude to the National Science Foundation and the Atomic Energy Comission for research assistantships and to the Department of Chemistry, Michigan State University for the teaching assistantships. Special thanks are also due to my wife for her help in some of the calculations and for her encouragement. ************ ii TABLE OF CONTENTS I. PRINCIPLES OF HIGH-RESOLUTION NMR SPECTROSCOPY................ I. A. I. B. I. C. I. D. I. E. I. F. I. G. I. H. Introduction............................................ Nuclear Magnetic Moments................................ The Nucleus in a Magnetic Field......................... Relaxation.............................................. The Nuclear Magnetic Resonance Experiment............... Chemical Shift.......................................... Spin-Spin Interactions.................................. 0183.1f1cat10n Of Kaela” GrOUpS...........o...........o II. FORMULATION OF THE QUANTUM MECHANICAL PROBLEM................. II. A. II. B. II. C. BuiltonimOOOOOOO0000000000900.00.00.0000.000.00.000... Spin Functions, Basic Product Functions and Basic State iwavemctionsoO9900000000OOOOOOOOOOOOOOOOOO0.00.00.00.00 Matrix ElementBOOOOOO000000.0000°OOOOOOOOOOOOOOOOOOOOOO. II. C-a. Glossary of Definition and Notations........... II. C-b. Logical Argument of the Computer Program....... . II. C-c. Description of the Program..................... II. D. II. E. The secul” EqmtionOOQOOOOO0000000000..0.00....00...... Selection Rules and Intensities of Transitions.......... III. A SURVEY OF THE VARIOUS METHODS CURRENTLY USED FOR ANALYSIS OF m SPEMRAOOOOOOOOOOOOOOOOOOO00000000000000.0000...0.0.00.0... III. A. III. B. III. C. III. D. Itcr‘tive ApproaChOOOOOOOOOOOOOOO0°00.000000000000000... Sub-spectral Analysis Approach.......................... Direct Calculation Approach............................. Miscellaneous Approaches................................ .iii Page ‘OO\UDN 11 12 '18 20 21 22 26 28 3O 32 3h 35 37 3? ho hh h? IV. COMPUTER ASSIGNMENT TECRNIQUE FOR ANALYSIS OF NMR SPECTRA.. IV. A. General Equations for the Chemical Shifts and Spin Coupling Constants.................................. IV. D. The Computer Assignment Technique................... IV. B-a. IV. B-b. IV. B-c. Intensity S‘m Rules.OOOOOOOOOOOOOOOOOOOOOOO Line Spacing Rules......................... Description of the Computer Program........ IV. C. ExampleSeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee IV. IV. IV. IV. IV. C-a. C-b. C-c. C-d. C-e. Two Spin System............................ Three Spin System.......................... Four Spin System........................... Five Spin System........................... Conclusion................................. BibliographyOOOOO000......0..O..0.0......OOOOOOOOOOOOOOOOOO Appendix I. Program "Matrele".............................. Appendix II. Program "Assign".............................. iv Page h8 h8 60 61 62 63 68 68 70 71 75 77 81 89 92 LIST OF TABLES Table Page I. Typical Values of v0 for Ho - lO4 Gauss................. 5 II. 0.00.00.00.00...00......0.0.0.....0.00.0.00000000000000031 III. 0.0.0.0000...OCOCOOOOOOOOOOOOO0.00.000.000.000...0.0.0.032 IV. Basic Functions and Diagonal Matrix Elements of the Hamiltonian for ABB'CC'OOOOOOOOOOOOOOOO0.0.0.000000051 V. Off-diagonal Matrix Elements of the Hamiltonian for ABB'CC'0..OOOQOOOOOOOOOOOOOOOOOO0.0.0.0000000000000053 Table II. III. IV. LIST OF TABLES Page Typical Values of v0 for Ho - 104 Gauss................. 5 .OOOOOOOOOOOOOOOOOOOOOOO0.0.0.0....OOOOOOOOOOOOOOOOOO0.031 O0.0.00.00...OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO00.0.32 Basic Functions and Diagonal Matrix Elements of the Hamiltonian for ABB'CC'O00.0.0.0...0.0.0.0000000000051 Off-diagonal Matrix Elements of the Hamiltonian for ABB'CC'OOOOOOOOOOOO...OOOOOOOOOOOOOOOOOOOO0000......53 LIST OF TABLES Table Page I. Typical Values of v0 for Ho . 104 Gauss................. 5 II. 0 O O O O O O O O O O O O O O. O O O O O O O. O O O O O O O O O O O O O O O O O O O O O O O. O O O O O O .031 III. OOOOOOOIOOOOOOOOOOOOCI00......0.0...00.0.00000000000000032 IV. Basic Functions and Diagonal Matrix Elements of the Hamiltonian for ABB'CC'.OOOOOOOOOOOOOOOOOO0.00.00.0051 V. Off-diagonal Matrix Elements of the Hamiltonian for ABB'CC'...OOOOOOOOOOOOIOOOOOI.0.0.0.000000900000000053 Table II. III. IV. LIST OF TABLES Page Typical Values of v0 for Ho - 104 Gauss................. 5 OOOOOOOOOOOOOOOOOOOOOOOO00.0.0.0...00.000.000.000000000031 0.0.00.0.0...OOOOOOOOOOOOOOOOOO0.00...00.00.00.00000000032 Basic Functions and Diagonal Matrix Elements of the Hamiltonian for ABB'CC'OOOOOOOOOOOOOO0.0.0.00000000051 Off-diagonal Matrix Elements of the Hamiltonian for ABB'CC'OOOOOOOOOOOOOOOOOOOO..00...0.0.0000900000000053 Figure LIST OF FIGURES Page Energy Levels For Nuclei With I = %' And I - 1 In The Magnetic FieldOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 4 00......0.0...OOOOOOOOOOOOOOOOOO0.00......0000000000000016 Schemic Energy-level Diagram For a General Four— Spin System, Illustrating Equal-spacing Conditions......62 Spectra 0f 2-Bromo-5-Chlorothiophene, Styrene 0x1de And 0-DiChlorObenzeneO0.0.0.0.0.0.000000000000000079 Spectrum of Cyclopropyl Cyanide.........................80 vi I. PRINCIPLES OF HIGH-RESOLUTION NMR SPECTROSCOPY I. A. Introduction The introduction of the molecular beam resonance method (1), which permits a direct measurement of nuclear gyromagnetic ratios, was the first successful applicationof the nuclear magnetic resonance ‘ technique, but it was pointed out at an early stage that it should be possible to observe resonance absorption in other forms of’matter (2). The phenomenon of nuclear resonance was first discovered in the ‘condensed phase in l9h5 by Purcell, Torrey and Pound (3) at Harvard*fi University and by Bloch, Hansen and Packard (h) at Stanfbrd University. The growth and scope of this new field has been enormous, its uses now extend into many fundamentalrresearch branches of physics and chemistry including'the structure of solids, study of large local magnetic fields in single crystals, the internal structure of molecules in the liquid and gaseous states, the intermolecular structure of liquids and asso- ciated electric and magnetic field effects, chemical.kinetics and. still others, and its use as an analytical tool has been extensive. This research will investigate the quantum mechanical and algebraic procedures involved in the analysis of high-resolution NMR spectra. The computer assignment—technique and the direct calculation method for analysis of NMR spectra will be discussed. In this chapter an attempt is made to give a very brief dis-. cussion of the terminology and principles of nuclear magnetic reso- nance spectroscopy which will be important in following chapters. .For further details on any-point, a number of excellent references are available which have covered general theoretical and experimental advances in detail (5-26). I. B. Nuclear Magnetic Moments In order to explain the hyperfine structure of some optical spectra, Pauli (27) suggested in l92h that some nuclei possess an intrinsic angular-momentum and magnetic moment. It has been found that only the atomic nuclei with odd“atomic number or odd mass number “or with both odd atomic number‘and odd mass number have a nuclear spin and these have integral or odd halfbintegral spin quantum number. Nuclei with even mass number and even atomic number such as 12C or 160 do not have a nuclear spin-so the spin quantum number 180. The spin angular momentum.§ of an atomic nucleus may be characterized by a spin quantum.number I, such that the angular momentum.is I’in units h. The spin of the positively charged nucleus confers on it a magnetic 'moment 3', which is proportional (28) to I, and.the proportionality constant is y, the nuclear gyromagnetic ratio (or nuclear magnetogyric ratio): zflth'yp ‘(I.1) For a simple particle of mass M and charge e, the value of y 1880......— 2Mc nance spectroscopy which will be important in following chapters. .For further details on any point, a number of excellent references are available which have covered general theoretical and experimental advances in detail (5-26). I. B. Nuclear Magnetic Moments In order to explain the hyperfine structure of some optical spectra, Pauli (27) suggested in 192% that some nuclei possess an intrinsic angular momentum and magnetic~moment. It has been found that only the atomic nuclei with odd atomic number or odd mass number -or-with both odd atomic number and odd mass number have a nuclear spin and these have integral or odd halfrintegral spin quantum.number. Nuclei with even mass number and even atomic number such as 12C or 160 do not have a nuclear spin sothe spin quantum number ISO. The spin angular momentum.i'of an atomic nucleus may be characterized by a spin quantum.number I, such that the angular momentum is I in units 5. The spin of the positively charged nucleus confers on it a magnetic 'moment 3', which is proportional (28) to I, and.the proportionality constant is y,the nuclear germagnet c ratio (or nuclear magnetogyric ratio): z'vffi'v'fi ' (L1) For a simple particle of mass M’a d charge e, the value of y 1380......— 2Mc + e " and 11830 'hI (1.2) 2Mc where g is called the nuclear g factor. From the theory oflquantum mechanics (28) we know that I has the magnitude [ I (1+1)]3 but that the only measurable values of this vector are given by the magnetic quantum number m, which may take on any of the (2I+l) values: eI, -(I-l), ~(I-2),.....,+(I-l),...., +1. The maximum Observable value ofbfi is , therefore: eh 2Mc where 8N - eH/2Mc, and is known as the nuclear magneton. It has the 8° . I 3 u . 8 BNeI (10 3) value 5.050 x 10-27ergs/gauss. The maximum.observable value of ‘K (the above equation) isocalled "the magnetic moment" of the nucleus and is denoted by the letter u . I. C. The Nucleus in a Magnetic Field In the absence of a magnetic field, all orientations of a nuclear magnet possess the same energy. But when a strong magnetic field, Ho, is supplied, this degeneracy is removed. The energy of a nucleus of magnetic moment i, in the magnetic field no is .y Em ' ;3 ° Ho '- -Ho x component of‘fi along Ho "Ho°m34. (I. 16) There are (21+l) energy levels, corresponding to the (21+l) values of m. These energy levels are illustrated in Fig. l for the cases of I - éand I- 1. The selection rule for transitions among these nuclear energy levels is mez.l, so that a quantum of radiation could induce a tran- (1.5) sition 1f Ern+l,m 3 h “08 Em+1 .- Em = 8 HO BN 2 or using the definition of the magnetogyric ratio of equation (1.1), n 10- y 5 no or v03 F Ho (I.6) 21! \ 301‘ .Ho’) _- - "7r" ’“HO ”Ho --1----1-- --.._ l o----d Fig. 1. Energy levels for nuclei with I-jfi and I=l in the magnetic field. When transitions are induced between energy levels of this kind, the phenomenon is known as nuclear resonance. If more nuclei are excited from the lower level to the upper than the reverse, then a net amount of energy is absorbed and an NMR spectrum may be observed., From equation (I.6) we note that the frequency is prOportional to the applied field. For HO - 10“ gauss, typical values of we are given in Table I. The frequencies lie in the radiofrequency region of the spectrum; the experimental procedure is discussed in section (I.E). The theory of transition probabilities shows that the Einstein transition probability coefficient for the absorption of energy is equal to the probability coefficient for stimulated emission (29). For 'this reason, any detectable net absorption of energy in a bulk sample Table I. Typical Values of v0 for Ho 8 10“ gauss 1:a====se— - v0 Mc/sec. Nucleus I - 4 u when Ho-IO“ gauss 1 1H 4 w 2' 2.79270 h2.577 2H 1 0.85738 6.536 12C 0 0.00000 --- 13C %. 0.70216 10.705 inn 1 0.h0357 3.076 160 0 0.00000 ' -- S 170 '2' .10 8930 50772 19F %. 2.6273 no.055 7 requires the existence of a population difference between adjacent energy levels. If there is weak coupling among a group of identical nuclei, and-between the nuclei and the remainder of the system, the "lattice", thermal equilibrium may exist, and the energy levels will be populated according to the Boltzmann factor. That is, the ratio of the population of a lower energy level to that-of the next higher level YfiHO/kTs will be the factor e where TS is the temperature of the spin system. As a typical example, we consider the proton, the ratio of 1 the population of a low energy level m a 3- to that of the next . 1 higher level m I - 5- at temperature 27°C and Ho 8 10,000 gauss is N l . me+5- ZUHo/RTB l = e g 2" Ho + 1 = 1.000007. So, under typical- Nmp-g' . kT ' "‘ ' '" conditions the fractional excess in the lower energy state will be only about 10'5, but it is this-small excess which gives rise to an observable net absorption of energy. As a transition-inducing electromagnetic radiation is applied to a system.of nuclear magnetic moments at thermal equilibrium, net absorption will occur, and the excess population in the lower energy" levels will rapidly diminish. The net absorption will then disappear, unless some mechanism exists by which nuclei "relax" from the higher to the lower energy levels, thus maintaining thermal equilibrium and the Boltzmann distribution of populations. I . D. Relaxation If there is to be a net absorption of energy in the nuclear magnetic resonance experiment, then some mechanisms must exist to - promote thermal equilibrium between the papulations of the energy levels and the surrounding matter (the lattice) which may be liquid, solid or gas. There exist various possibilities for radiationless transitions by means of which the nuclei can exchange energy with 'their environment and it can be shown (6) that such transitions are more likely to occur from an upper to a lower state than in the reverse direction, We therefore have the situation in which the applied radiofrequency field is trying to equalize the spin-state equilibrium while radiationless transitions are counteracting this process. In the type of systems of interest to us a steady state .is usually reached such that the original Boltzmann excess of nuclei in the lower states is somewhat decreased but not to zero so that a net absorption can still be registered. The various types of radia- tionless transitions, by means of which nuclei in an upper spin state return to a lower state, are called relaxation processes. We may divide relaxation processes into two categories, namely,spin-lattice relaxation and spin-spin relaxation. ‘ Spin-lattice relaxation (30) is sometimes called longitudinal relaxation (31).»This process is responsible for the establishment and maintenance of the absorption condition. The magnetic nuclei are usually part of an assembly of molecules which constitute a sample under investigation and the entire molecular system is referred to as the lattice irrespective of the physical state of the sample. For the moment we will confine our attention to liquids and gases in which the atoms and molecules constituting the lattice will be undergoing random translational and rotational motion. Since some or all of these atoms and molecules contain the magnetic nuclei such motions will be associated with fluctuating magnetic fields. Now, any given magnetic nucleus will be precessing about the direction of the applied field Ho and at the same time it will experience the fluctuating magnetic fields associated with nearby lattice components. The fluctuap ting lattice fields can be regarded as being built up of a number of oscillating components so that there will be a component which will Just match the precessional frequency of the magnetic nuclei. In other ‘words, the lattice motions, by virtue of the magnetic nuclei contained in the lattice, can from time to time generate in the neighborhood of a nucleus in an excited spin state, a field, which like the applied radiofrequency field H1, is correctly oriented and phased to induce spin-state transitions. In these circumstances a nucleus in an upper spin state can relax to the lower state and the energy lost is given to the lattice as extra translational or rotational energy. The same process is responsible for producing the Boltzmann excess of nuclei in lower states when the sample is first placed in the magnetic field. Since the exchange of energy between nuclei and lattice leaves the total energy of the sample unchanged,it follows that the process must always operate so as to establish the most probable distribution of energy or, in other words, so as to establish the Boltzmann excess of nuclei in lower states. The so called spin-lattice relaxation time T1 is a measure of the rate at which the spin system comes into thermal equilibrium with the other degree of freedom. It, in effect, is the half-life required for a perturbed system of nuclei to reach an equilibrium condition. The value of T1 will depend on the magneto- gyric ratio (or ratios) of nuclei in the lattice and on the nature and rapidity of the molecular motions which produce the fluctuating fields. Because of the great restriction of molecular motions in the crystal lattice, most highly purified solids exhibit very long spin- lattice relaxation times, often of the order of hours. For liquids the value of T1 usually lies between 10‘2 and 102 sec., although in the presence of paramagnetic ions it may be as low as 10'“ second. Spin-spin relaxation or transverse relaxation (3), is a process in which a nucleus in its upper state transfers its energy to a neigh- bouring nucleus of the same isotope by a mutual exchange of spin. This relaxation process therefore does nothing to offset the equalizing of the spin state populations caused by radiofrequency absorption and is not directly responsible for maintaining the absorption condition. This relaxation process occurs with a characteristic time T2 called the spin-spin relaxation time or transverse relaxation time. Both spine lattice relaxation and spin-spin relaxation processes may control the natural line width of a spectral line. We have seen that adequate spin-lattice relaxation is a necessary condition for the continued observation of radiofrequency absorption. In practice this condition is not always fulfilled and in such circums- tances the observed absorption signal diminishes with time and may, in extreme cases vanish. For example, if the relaxation process is a slow one, or if the perturbing radiofrequency field is strong, the observed' absorption signal may vanish. This behaviour is called saturation. This occurs when the papulations of all the energy levels are nearly equal, in which case no net absorption of energy occurs. I. E. The Nuclear Magnetic Resonance Experiment The apparatus for observing nuclear magnetic resonance absorption of energy consists essentially of four parts: (1) A magnet capable of producing a very strong homogeneous field. (2) A means of continuously varying the magnetic field over a very small range. (3) A radiofrequency oscillator. (h) A radiofrequency receiver. I The magnet is necessary to produce the condition fOr the absorption of radiofrequency radiation. The remaining components then have analogues in other method of absorption spectroscopy. Thus the radiofrequency oscillator is the source of radiant energy. The device for varying the 10 magnetic field over a small range corresponds to a prism or grating in as much as it permits us to scan the spectrum and determine the positions of absorption lines in terms of frequency or field strength. The radiofrequency receiver is the device which tell us when energy. from the source is being absorbed by the sample. A sample containing nuclei which possess magnetic moments is placed-between the poles of a magnet of magnetic field strength Ho. The magnetic moments of the nuclei in the sample tend to orient in the direction of the field, giving rise to a resultant macroscopic magnetic moment. The effect of the magnetic field is to cause a precession of the macroscopic moment about the direction of the field with an angular frequency 7 Ho. If now a small coil.connected to an rf signal generator, is wound around the sample so that the axis of the coil is at right angles to the direction of the applied field, there is introduced a small alternating magnetic field of strength H, which rotates about the H6 direction with the particular radio fre- quency used. The field H1 tends to tilt the direction of the macros- copic moment away from the H6 direction as the radiofrequency approaches the precession frequency; at the resonant frequency, - transitions are induced between the nuclear Zeeman levels. These ”transitions correspond to some of the nuclear magnets.changing their orientation in the field. The energy absorbed in this process produces a drop in rf voltage in the tuned circuit containing the transmitter coil; the voltage drop may be;detected, amplified, and fed into the vertical deflection plates of an oscilloscope. In practice, the radio frequency of the signal generator is usually fixed and the applied 11 field H0 is varied near the value at which resonance occurs. This is accomplished by mounting, on the pole faces of the magnet, coils which can be used to sweep the field with an amplitude of a few gauss at some low frequency (about 50 cps). The same sweep signal can be fed into the horizontal deflection.plates an oscilloscope, and the recur- ring absorption signal is displayed on the screen. I. F. Chemical Shift The nuclear resonance frequency of a particular nucleus occurs .. at different values of a given applied magnetic field, according to the' nature of the chemical compound containing the nucleus. These frequency ' differences have therefore been called "chemical shifts". In equation (1.5), Ho refers to the magnetic field actually experienced by the nucleus and this is not equal to the applied magnetic field, H, when the nucleus is ‘ present in a chemical compound. The reason is that when any chemical substance is placed in a magnetic field, weak currents are induced in the electron clouds surrounding the nuclei. These induced currents flow . according to Lenz's law in such a way as to set up a magnetic moment which opposes the applied fields, and it is this effect which is respohsible‘for the bulk diamagnetism of all matter. These weak induced diamagnetic moments reduce the field experienced by the nucleus to a value smaller than the “ applied field, and it is usual to express this effect as 30-11(1-6) (1.7) where 0 represents the factor by which H is modified by the induced diamagnetism. Equation (1.5) is then written.: “H (1-.) (La I hv 0 12 and the parameter a is known as the "chemical shift" or magnetic shielding constant. The chemical shift is dependent upon the externally applied field, becoming larger with increasing field strength. I. G. Spin-Spin Interactions In NMR spectra the absorption lines are usually affected by inter- actions between the nuclei and between the nuclei and their environment.“ These interactions can be classified as dipole-dipole interactions, electric quadrupole coupling, and electron coupled spin-spin interactions. Dipole-dipole interactions between neighboring nuclear magnetic moments are dependent upon separation and relative orientations of the nuclear moments. Thus, instead of all nuclei experiencing the.same uniform magnetic field Ho, different nuclei in a specimen will experience various fields spread over a range of frequencies and the spectral line will be broadened. These considerations are effective, however, only if the nuclei maintain the same orientation relative to one another and to the external field, as in solids. In liquids and gases, where the molecules are rotap' ting and tumbling about rapidly, the magnetic field at any one nucleus. due to the others effectively averages out to zero. The cause of the magnetic dipole broadening (dipole-dipole interaction) is removed by "" this averaging and the resonance signals become much sharper. In solids, ' since nuclei for which I I‘% have no electric quadrupole moment, the dipole-dipole coupling is usually the dominant mechanism for line broap dening. In a solid containing nuclei (I ..% ) grouped in relatively isolated pairs, each nucleus experiences a magnetic field whose direction is taken as the z-axis Ho 2 Hlocal , where “local is the local magnetic 13 field set up by one nuclear magnet in the region of the other. For dipoles of moment u at a distance r apart and with the internuclear vector making an angle“. with Ho (which is parallel to the z-axis) Hlocal - 23(3 c0826 - 1) (1.9) r so that H = no t “a (3 c0826 - 1). (1.10) r The nuclear resonance spectrum would therefore be expected to be a pair of lines at frequencies separated by 2 u 3 29 - 1 . ‘ I.ll _;3.( cos ) ( ) In fact, however, there is an additional quantum mechanical interaction which in this case leads to a pair of lines separated by i g (3 cosze - 1): for pairs of identical nuclei. (1.12) The detailed shape of this pair of lines has been obtained by Pake (32) by_a simple quantum mechanical perturbation calculation using the ‘ dipole-dipole interaction Hamiltonian: )6: r-3[ 31-352 - 3:"? (it.- a) (32-3)], (1.13) where r is the internuclear distance and r is the unit vector. For two protons at a distance of l: , the doublet separation is of the order of lO gauss or h2 kc/sec at 10,000 gauss field. If the pair of nuclei arennot,identical the line separation is then 2 u 2 . “;r ( 3 cos ex 1) (1.1h) In a single crystal containing pairs of nuclei with the internuclear vectors all pointing in the same direction, the doublet separation '*varies from 3 u when 6 is “ to 6 u when 0 - 0. From the r3 7 .3 1h variation of the doublet separation with the orientation of the crystal in the magnetic field, the directions of the H-H vectors in the crystal can be found, and from the spacing of the doublet r can be deduced. When the nuclei are grouped in a crystal in more complicated arrangements than those described above, the absorption line is often a broad and shapeless hump. Van Vleck (33) showed that useful information can still be derived. Rigorous expressions were obtained for the second and fourth' moments of the absorption line in terms of the internuclear distances in the crystals. Nuclei for which Izi ‘usually have an electric quadrupole'moment 2 from spherical symmetry (7). This is a measure of the deviation of the electric charge dis- tribution within the nucleus. If the positive charge is spread over a prolate spheroid,the quadrupole moment is said to be positive; if the charge is spread over an oblate spheroid,the quadrupole moment is taken to be negative. Nuclei with I -'%Phave no electric quadrupole moment, therefOre NMR experiments on these nuclei are not complicated by direct interactions of the nuclear spin with the electrical environ- ment. ‘ When a number of liquids were examined by NMR spectroscopy, it 'was feund that certain substances showed more lines than could be explained by means of the chemical shift alone. For example, Gutowsxy, ‘ McCall and Slichter (3h) found that the fluorine resonance spectrum of - POCle consists of two lines of equal intensity, although there is only one fluorine atom in the molecule. Other molecules gave symme- trical multiplet signals. These multiplicities were attributed by ‘Gutowsky, McCall and Slichter (3h), and by Hahn and Maxwell (35), to 15 an interaction between the nuclear spins which is proportional to the scalar product IioIJ where Ti and 13 are the nuclear spin vectors. 'Unlike the direct interaction of magnetic dipoles (dipole-dipole interaction) an energy of this sort does not average to zero when the molecules are in rapid random motion, so its effect is still observable in the spectra of liquids and solutions. Furthermore, the splitting of the lines which results from this interaction is independent of the applied magnetic field, in contrast to the sepap ration of chemically shifted lines which is proportional to it. The pinterpretation of these interactions was first given by Ramsey and Purcell (36) and by Ramsey (37). They showed that they arise from an indirect coupling mechanism via the electrons in the molecule. Thus a nuclear spin tends to orient the spins of’ the) electrons and consequently spins of other nuclei. The magnitudes of the spin- interaction energies are usually expressed in cycles per second (cps). Observed interaction energies vary from about 1,000 cps to small values at the limit of experimental detection ( <1 cps ). The way in which the spin-spin coupling affects the NMR spectrum can be seen easily for the simple case of a pair of unlike nuclei, A and B, coupled together. From equations (1.1) and (I.h) the energy of interaction of the nucleus with the strong field Ho,taken to be along the z-axis, is 45» Yomfio(l- 0) ' (1.15) For a system of two particles with no spin coupling, the Hamiltonian -for interaction with a static field Ho in the z-direction is therefore 3‘6,- -fi[YAHo(l- °A) IzA + yBH°(1- GB) 123 . (1.16) If we take‘vo as the mean of the two resonance frequencies, and. 5 as 15 an interaction between the nuclear spins which is prOportional to the' scalar product IioIJ where Ti and I3 are the nuclear spin vectors. Unlike the direct interaction of’magnetic dipoles (dipole-dipole interaction) an energy of this sort does not average to zero when the molecules are in rapid random motion, so its effect is still observable in the spectra of liquids and solutions. Furthermore, the splitting of the lines which results from this interaction is independent of the applied magnetic field, in contrast to the sepap ration of chemically shifted lines which is proportional to it. The ’interpretation of these interactions was first given by Ramsey and Purcell (36) and by Ramsey (37). They showed that they arise from an indirect coupling mechanism via the electrons in the molecule. Thus a nuclear spin tends to orient the spins of' the. electrons and consequently spins of other nuclei. The magnitudes of the spin- interaction energies are usually expressed in cycles per second (cps). Observed interaction energies vary from about 1,000 cps to small values at the limit of experimental detection ( <1 cps ). The way in which the spin-spin coupling affects the NMR spectrum can be seen easily for the simple case of a pair of unlike nuclei, A 3 and B, coupled together. From equations (1.1) and (I.h) the energy of interaction of the nucleus with the strong field Ho,taken to be along the z-axis, is 46» YomHo(l- 0) (1.15) For a system of two particles with no spin coupling, the Hamiltonian for interaction with a static field Ho in the z-direction is therefore )6,- -6[1Ano(1- 0A) 12A + ”3110(1- 03) IzB . (1.16) If we take‘vo as the mean of the two resonance frequencies, and 5 as 15 an interaction between the nuclear spins which is proportional to the scalar product Iiofj where Ti and I: are the nuclear spin vectors. Unlike the direct interaction of magnetic dipoles (dipole-dipole interaction) an energy of this sort does not average to zero when the molecules are in rapid random motion, so its effect is still observable in the spectra of liquids and solutions. Furthermore, the splitting of the lines which results from this interaction is independent of the applied magnetic field, in contrast to the sepap ration of chemically shifted lines which is proportional to it. The {interpretation of these interactions was first given by Ramsey and Purcell (36) and by Ramsey (37). They showed that they arise from an indirect coupling mechanism via the electrons in the molecule. Thus a nuclear spin tends to orient the spins of’ the electrons and consequently spins of other nuclei. The magnitudes of the spin- interaction energies are usually expressed in cycles per second (cps). . Observed interaction energies vary from about 1,000 cps to small values at the limit of experimental detection ( <1 cps ). The way in which the spin-spin coupling affects the NMR spectrum can be seen easily for the simple case of a pair of unlike nuclei, A and B, coupled together. From equations (1.1) and (1.h) the energy of interaction of the nucleus with the strong field Ho,taken to be along the z-axis, is .11. YomHo(l- a) (1.15) .For‘a.system of two particles with no spin coupling, the Hamiltonian ‘for-interaction with a static field H0 in the z-direction is therefore )6; -munou- 0A) 1.1 + mucu- .3) 123 . (1.16) ' If we take v0 as the mean of the two resonance frequencies, and 5 as 16 h the chemical shift between them, this can be re-written: M- -h[(v° - -:- 5) IM + ( v0 + -:- a) 123] . (1.17) The energy levels are therefore, Emma - 4.11.0 - 1.) “A + < .0 + 5 ) m3] . (1.18) The energy levels are shown in Fig. 2 (a),and if the allowed transitions are Am.A a l and AmB - l, we obtain two lines in the spectrum, sepa» rated by the chemical shift 5 , each of the two lines being doubly degenerate. TA ":3 -i ‘2’ T-jF-‘~~---- A B I l A -5 +2 5““.M.“ 1% ”‘""““"'""t:‘" 1, 1 -1. -——"" J +2 2 T—""""“"“"‘f; A B | t *‘2‘ *3: (a) (b) Fig. 2. If the two nuclei are coupled together, the Hamiltonian now becomes: 1 1 36- -h[( v0-35) 1M + (”0‘1“ IzB]-£J1A.TB (1.19) where J is the coupling constant in cps. If J is very much smaller than.5 , the last term in equation (1.19) can be treated as a small perturbation onJ‘e , so that mA and mB remain good quantum numbers, and we can write E -h[(vo - .56) mA + (v0 *‘36 ) mg + JmAmB] (I.20) The energy levels are now as in Fig. 2 (b) and transitions of nuclei A and 17 B give energy changes of ABA-‘h(vo -.%5 + JmB) and AEBIh(v° +'%5 +Jmm), respectively. The perturbation has lifted the degeneracy of the two lines which are split by J cps. Note that the resonance due to the A nuclei is split according to the values m3 of the B nuclei and vice- versa. If the two nuclei A and B are entirely equivalent (5 I 0), then no splitting of the lines can be observed and only a single resonance occurs. This is because the now indistinguishable nuclei must be described by writing wave functions which are either symmetric or antisymmetric in the spin, as in describing the ortho and para states of hydrogen. The singlet state with 1- O has no magnetic sublevels and the triplet state has m I +1, 0 and -1. However, all the sublevels of the triplet have IAf TB . +-% , so that the interaction shifts them all equally and pro- duces no Observable splitting. In the case of the proton resonance spectrum of acetaldehyde (CH3CH0), there are;two sets of equivalent nuclei, one in the CHO group (A) and the other in the CH3 group (B). Since the protons of the CH3 group are indistinguishable we must treat them as a group with MEI 2mB. The energy changes are then feund to be: A i e - l e e . l CHO group. AE - h (v0 25 + JMB), CH3 group. 5E3 h(v° +m56 +JmA). Now mAcan take on the values t g. , so that the CH3 resonance is a doublet, and since the two values of mAare equally probable, the two components of the doublet are of equal intensity. ”B can take on the values ;. , %.,-é_ , -3 by combination of the three B spins: 1 1 1 1 1 1 1 1 1 +-1—-.l.+i . -l+-L-l ’ 2 2 2 2 2 2 2 2 2 2 2 2 18 The CHO resonance is therefOre a quartet, and since the values +3, 2 and -%. of MB can be achieved in three times as many ways as the values 3 +3 and - % , the intensities of the quartet lines are in the ratio 133:3:1- In the general case for a set of NA equivalent nuclei of type A inter- acting with Nx equivalent nuclei of type X, the A signal has 2lex + 1 components and the X signal has 2NAIA + 1 components. The relative intensities of each group of signals are in the ratio of the corres- ponding binomial coefficients. The A and X nuclei may belong to different species, or they may be of the same species if the chemical shift between their resonance signals is large. When the value of J is not small comp~ pared with 6 , then it can no longer be considered as a simple pertur- "bation and an exact calculation must be made. The methods for the analysis of complex spectra will be discussed in Chapters III (and IV.. I. H. Classification of Nuclear Group; It is convenient to introduce a notation for typical groups of nuclei which may appear in molecules and which will possess character- istic NMR spectra. First of all we should distinguish between iso- chronous nuclei and equivalent nuclei (38). Isochronous nuclei are "those which have exactly the same chemical shift; while equivalent "nuclei not only have the same chemical shift but are also identically ‘ coupled to all other nuclei in the system. A pair of nuclei can be equivalent only if they are isochronous. There is a well known theorem stating that scalar couplings between equivalent nuclei are unobservable in an NMR experiment (18, 23). The proof of this theorem does not depend on symmetry in any way. The importance of the distinction between iso- 19 chronous and equivalent nuclei lies in the fact that the total spin angular momentum of a group of equivalent nuclei is a good quantum number. As a consequence, the NMR spectrum of a molecule containing a group of equivalent nuclei consists of a superposition of spectra arising from the various "spin particles" formed by the equivalent nuclei. This observation makes direct analysis of spectra of this type much simpler, once assignment of spectral lines to the corres- ponding transitions within the appropriate energy-level diagrams is accomplished. We shall use the symbols A, B, ... for nonequi- valent nuclei of the same species whose relative chemical shifts are of the same order of magnitudes as the spin couplings between them. X, Y, ... will be used for another such set whose signals are not close to those of the set A, B, ... The nuclei in the set X, Y, °-- may or may not be of the same species as those in the set A, B, °-°, the only feature that is important in the theory is that the chemical shift between the groups A, B, ... and X, Y, --- is large compared with any of the spin couplings. Equivalent nuclei will be described by the same symbol. Thus, 1,1,1-trifluoroethane (CH3CF3) is an example of an A3X3 system since the carbon nuclei have no magnetic moment and may be ignored. The protons in l-bromoethane (CHBCHZBr) form two groups of equivalent protons and are described as an A B system, for the chemical shift between the three protons in 3 2 CH3-group and the two protons in CH2 small. o-Dichlorobenzene protons, on the other hand, constitute a -group is observed to be relatively system of two groups of isochronous protons and would be represented as AA'BB'. Here we notice that the primes on A and B are used to describe nuclei that are isochronous but not equivalent. II. FORMULATION OF THE QUANTUMAMECHANICAL PROBLEM The fundamental procedure involved in the analysis of NMR spectra ' consists of finding the energies and transition intensities corresponding' to the stationary states of the nuclear spin system. The basic quantum-—‘ mechanical method of finding expressions for the nuclear energy levels of the system of interest, together with expressions for the relative transition probabilities between these levels, is quite similar to the methods that have been extensively employed in other field of spectros- copy (e.g” infrared, ultraviolet etc.). This requires deriving or hypo- thesizing a satisfactory Hamiltonian for the system of nuclear magnetic‘ moments in magnetic field and solving the SchrBdinger equation fer the‘ eigenvalues of this Hamiltonian, which are the desired levels of the system. Usually, the exact Hamiltonian for the energy of a molecule is ' simplified in-some way so as to give an approximate Hamiltonian: the "‘- approximate Hamiltonian is chosen to be as accurate as possible and, at ' the same time, to be such that the Schr8dinger equation can be solved exactly and conveniently. The zero-order energies and wavefunctions thus obtained are then used as a starting point for a more accurate calculation of energies and wave functions using, in so far as possible, a complete exact Hamiltonian. This latter calculation is, in many cases, greatly simplified by selecting the zero-order wave functions to be eigen- functions of all those molecular properties that commute with the complete Hamiltonian. Molecular symmetry and spin or rotational angular momenta are typical of such molecular prOperties. The exact wave functions and energies are also commonly classified according to such molecular pro- perties-inasmuch as the spectroscopic selection rules are generally conveniently formulated in terms of such classifications. Also the 20 21 classification of energy states according to these molecular properties often enables one to predict how transition energies and intensities will behave under specified perturbations. II. A. Hamiltonian It has been well established that high—resolution NMR spectra (omitting relaxation effects) are fully accounted for by the following spin Hamiltonian (18, 3h, 36, 37): M '}(€0 ) +a»€_( 1 )' (11.1) SHE? ) , the external-field Hamiltonian, corresponds to the interaction of the nuclear moments with the external field. If the direction of the- strong magnetic field H is the "negative" z-direction, the energy of a nucleus in this field will be thIz, measured in ergs if H is measured‘ in gauss. The more convenient unit for measurement will be cps. With ' this unit, the interaction becomes yHIz/2n . For a set of nuclei with' 'magnetogyric ratios Y1 and acted on by field Hi’ the external-field Hamiltonian will be 94: °’ - (2 ”"15? 71111141) - (11.2) Where 11 will depend only on the species of nucleus and 12(‘) is the angular momentum component in the z-direction (in units of 2b ). For s 1 nuclei of spin;- , 12(i) can take values 4- .. or - }_ . The sign 2 2 convention is such that the external field is in the negative z-direction so that nuelei with positive spins have high energies. The magnetic field H1 will differ from the external field HD because of electronic screening. Thus,we write Hi I Ho (1- Oi): where oi is the appropriate screening "constant. Because the theoretical presentation is simpler, we shall 22 discuss the set of energy levels when the external field H0 is held constant. although.as we have already mentioned1the experiment is usually perfbrmed by varying Ho to get the resonance at a fixed frequency. The other part of the Hamiltonian corresponds to the indirect spin coupling and can be written: (1) -> + 96 - 1: .1 1(1)~1(1) (11.3) .. w ‘3 where 1(i) is the spin angular momentum vector (in units _2_.) and 21 J1J is the coupling constant between nuclei 1 and J and will have the dimensions of energy(cpal 'In the presence of a perturbing rf field, HI I 2H1cos mt, along the x-axis, it is necessary to include a third term.in the Hamiltonian: (2) “ ag— - - 11‘) + + (IIJ-t) 2" 2 1(1) 1; 1 _ , .ial IX However, to avoid saturation Hx is kept very small in practice and this term.may be neglected in the complete spin Hamiltonian. So the complete spin Hamiltonian is: 0 1 as ..a Mac} ’ -1 + 0-) - (2x) :11 1.111201%~ 153 JiJIU) 1(1) (11.5) II. B. Spin Functions, Basic Product Functions and Basic State Wavefunctions We write Z + 0+ <. (op |1 where the up and sq are single products of spin functions such as oBca ......(basic product functions). These integrals are easily eva- luated-by the equations . , z + 1 . _ 1 2 ‘ "(P '1‘.) JiJHi) it)” "’p> To- 1‘.) JiJTiJ (11.12) - where TU. +1 if spins iand J are parallel, and T I -1 if spine 1 i.) and J are antiparallel, and < .1) I133 J13I(i)°I(J)| .q , ..§. (111J (11.13) where U I 1 if Hp only differs from *q in the permutation of spins' i and J, and U I O in all other cases. If the basic set consists of some linear combinations of products, corresponding matrix elements are easily evaluated by expansion. The evaluation of these matrix elements of the spin-coupling Hamiltonian is a relatively simple pro- blem with but three or four nuclei; with five or more spins, however there is considerable labor and tedious work involved. Corio (ks) has proposed a method to simplify the calculations, but the calculation still.must be done manually. A new method which employs a digital com- puter to evaluate the matrix elements of the spin-coupling Hamiltonian is proposed-in the following sections. 28 II. C-a. Glossary of Definition and Notations J: M(J): The number of basic state wavefunctions, e.g.,the two spin system has four states (22 Ih), so the maximum value of J is h. *1 a as, J I 1; $2 I]; (08 +80 ), J I 2;...etc. The number of basic product functions in a basic state wavefunction.due to degeneracy. K I l, ...., M(J), e.g.,in #2.: AM 68 +801), (yé-WB has J - 2, K- 1; (5-)... has J - 2, K - 2. The number of spin functions, L I l, ..., N in a basic product function. e.g.,in #2 I.;5.( a8 + Bo ) the spin function a in the first term of the right side of the equation has J I 2, XhI l, L I l. The total number of states (or basic state wavefunctions), e.g.,two spin system has a total of four states (22 I h) (or four basic state wavefunctions); J I l, 2, ...... I. The total number“ of spins; e.g”.for the two spin system, N - 2. The total number of basic product functions in the state J; e.g., for the two spin system in state J I 1, ((1 has only one basic product function on, so M(J) I M(l) : 1; while in state J I 2, $2 has two basic product functions, namely, (.dl')“ and(f§-)Ba ,soM(J)IM(2)I2, and (,1)... is the first basic product function in state J I 2 for which H I 1 while (1%.)31 -is the second basic product function in state J - 2 for which K - 2; x -'1, 2, ...., M(J).- 29 CD(J, K): The coefficient for each of the basic product wavefunctions: e.g., wz-j%((ae + Bo), CD(2, 1) - CD(2, 2) - Z; . QI(K, L): Coefficients for the coupling constants between nuclei K and L. JL; The number of the last basic state wavefunction in the system. JF: Label for final basic state wavefunction in the matrix 1 element of¢}€£ ) connecting initial and final states as: < final state|3f£l)| initial state >. J1: Label for initial basic state wavefunction in the matrix 1) element °fdd under computation, e.g.,dllaéul *3 > JFIl,JII3. KF: Final basic product function. KI: Initial basic product function. LS: Comparing two basic product functions the first pair of different spin is labeled LS. LT: The second pair of different spin. L1: Label for the thh spin under consideration. MD: --- The total number of pairs of different spins. .The definitions of ND(S), ND(t), NABS, NDD, NSUM, .... etc. are given in Table 111. Other variables in the flow chart are dummy variables. From the above definitions and notations, we can use a three- dimension array 1D(J, X, L) to denote gpig_functions, Eggigiproduct functions and 23.1%.! _s_t_at_e_ wavefunctions, e. g. , in a two spin system, ID(l, l, 1) means the first spin function in the first product function of the first state wavefunction- Thus, 3. ID(2, 2, 1) means the first 29 CD(J, X): The coefficient for each of the basic product wavefunctions; 1 l2 QI(K, L): Coefficients for the coupling constants between nuclei K e.g., was]; ((18 + Bo), CD(2, 1) - CD(2, 2) . and L. JL; The number of the last basic state wavefunction in the system. JF: Label for final basic state wavefunction in the matrix 1 element of‘3€£_) connecting initial and final states as: < final state|3f£1)| initial state >. J1: Label for initial basic state wavefunction in the matrix (1) 1 element of3.Q_ under computation, e.g., , JF I l, JI I 3. HF: Final basic product function. KI: Initial basic product function. LS: Comparing two basic product functions the first pair of different spin is labeled LS. LT: The second pair of different spin. L1: Label for the LIth spin under consideration. MD: ~~- The total number of pairs of different spine. .The definitions of ND(S), ND(t), NABS, NDD, NSUM, .... etc. are given in Table 111. Other variables in the flow chart are dummy variables. From the above definitions and notations, we can use a three- dimension array ID(J, K, L) to denote gpin_functions, begig_product functions and w'm wavefunctions, e.g. , in a two spin system, ID(l, l, l) means the first spin function in the first product function of the first state wavefunction- Thus. a. ID(2, 2, 1) means the first 29 CD(J, K): The coefficient for each of the basic product wavefunctions; 1 l e.g., )2-15.(ae + Bo), CD(2, 1) - CD(2, 2) . If . QI(K, L): Coefficients for the coupling constants between nuclei K and L. JLi The number of the last basic state wavefunction in the system. JF: Label for final basic state wavefunction in the matrix 1 element of¢}€£ ) connecting initial and final states as: < final state|3{51)| initial state >. J1: Label for initial basic state wavefunction in the matrix (1) (1) element of&Q_ under computation, e.g., , JF - 1, J1 - 3. HF: Final basic product function. KI: Initial basic product function. LS: Comparing two basic product functions the first pair of different spin is labeled LS. LT: The second pair of different spin. L1: Label for the LIth spin under consideration. MD: -.~ The total number of pairs of different spine. .The definitions of ND(S), ND(t), NABS, NDD, NSUM, .... etc. are given in Table 111. Other variables in the flow chart are dummy variables. From the above definitions and notations, we can use a three- dimension array 1D(J, X, L) to denote 32i2_functions, begig_product functions and m'm wavefunctions, e.g. , in a two spin system, ID(l, l, 1) means the first spin function in the first product function of the first state wavefunction- Thus, 3. ID(2, 2. 1) means the first 3O spin function in the second product function of the second state wave function; i.e., B . So in the two spin system, plI as can be repre- sented by ,1. ID(l, 1, 1) ID(l, 1, 2); (12' (153(3); +3., ) by *2' 1 {ID(2, 1, 1) ID(2, 1, 2) + ID(2, 2, 1) ID(2, 2, 2)} .....etc. '7? The coefficients are introduced as CD(J, K). II. C-b. Logical‘Argument of the Computer Program From the last section and the simple rules for evaluating the matrix elements (h2), we know that matrix elements of Ts°ft ' Igx°Itx * Isy°Ity I Isz°Itz exist only when the two basic product functions 1D(J, K, L),L I l, ....,N; 1....(IDJK11DJK20.°°ooIDJKN). and ID(J'. K'. L')‘ L. ‘ 1.0eeee,N; is... (IDJ'H'l computer calculation,we let these ID's have numerical values 1 or 2 IDJ.K.2.....IDJoKoN)differ by no more than two of the ID's. For according as ID's are a or B spins; e.g., if a basic product function, is (IDJK11DJK21DJKBIDJKh) I aBaB , then it has the numerical value l212. If another basic product function is (IDJ'K'11DJ'K'ZIDJ'K'3IDJ'K'h) I sang, then it has the numerical value 1121. These two wavefunctions differ by three of the ID's (IDJK2 I IDJ.K12 3 IDJK3 # IDJIKI3 ; IDJKh f IDJ'K'h) so there are no matrix elements between these two product functions. Also.if the two product functions differ by one of the ID's only, there are still no matrix element between them; e.g., (IDJKIIDszIDJK3IDJKh) I eBaB I 1212, and (IDJOKIllpJ.K.21DJOKOBIDJOKIh) case I 1211 differ by one 1D only(IDJKh I IDJ'K'Q" the other ID's in these two product functions are the same , so there are no matrix elements between them. Matrix elements exist between two wavefunctions :for even differences up to two (namely zero and two) of the spin func- 31 JKl J'K'IIDJ'K'2 IDJ'K'3IQJ'K'h) I ease I 1212 have zero difference, so matrix elements tions, e.g., (1D IDJKQIDJKBIDJKH) - aBoB - 1212 and (ID exist between them. (IDJKlIDszanK3IDJKh) I aBoBI 1212 and (IDJchl IDJ.K.21DJ,K.3IDJ,K.h) . 0880 -1221 differ by two ID's (1DJK3 i 1DJ1K13 and IDJKh I IDJOKOh), so matrix elements also exist between these two basic product functions. From the above, we know that the typical term of the matrix elements of the spin coupling Hamiltonian is J1JI(i)°I(J), in which only two nuclei are involved. So when we calculate the matrix elements, we need only consider two nuclei each time and Table II can be easily~ obtained. Using the numerical characteristics of each two pairs of spin functions, Table 111 can be constructed. Based on Table III, the flow chart and the Fortran computer program MATREL have been written for the Control Data 3600 computer (see Appendix I). Table II. a ‘ Coeff. for Coeff. for Coeff. for Total coeff. for _a_._ " __L ' a a a a O + 0 + l/h I l/h a B a B 0 + 0 + .(-l/h) I -l/h' s o a a (o + o + (-1/h) - A -1/h B 8 B B 0 + 0 4- l/h --- l/h a a B 8 l/h + (-1/h) + ,0 I 0 B B c a 1/h + (-l/h) + 0 I 0 a B B a l/h + l/h + 0 I l/2 B a a B 1/h + 1/h + O I 1/2 31 JKl J'KOIIQJcKIZ IDJ'K'3IQJ'K'h) I dads I 1212 have zero difference, so matrix elements tions, e.g., (ID IDJK21DJK3IDJKh) - aBaB . 1212 and (ID exist between them. (IDJKlIDJKZIDJK3IDJKh) I 0868' 1212 and (IDJchl IDJ.K.21DJ,K.3IDJ,K.h) . asaa I1221 differ by two ID's (IDUK3 I IDJ.K.3 and IDJKh # IDgoKth), so matrix elements also exist between these two basic product functions. From the above, we know that the typical term of the matrix elements of the spin coupling Hamiltonian is Jidf(i)°f(J), in which only two nuclei are involved. So when we calculate the matrix elements, we need only consider two nuclei each time and Table II can be easily"~ obtained. Using the numerical characteristics of each two pairs of spin functions, Table III can be constructed. Based on Table III, the flow chart and the Fortran computer program MATREL have been written for the Control Data 3600 computer (see Appendix I). Table II. . _.i_-A"" _ Coeff. for Coeff. for Coeff. for Tota% coeff. for (IDJKSIDJKt) (IDJchSIDJoKvt) JBtIBXItx +J8tI3yIty *JstIBZItz I Jot ,- t c a a a 0 + 0 + l/h I l/h a a a B o + yo + .(-1/h) . -1/h- B a ‘ a a .o + o + (-1/u) . - -1/u B a B B o + o + 1/h --' 1/h a a B 8 l/h + (-1/h) + .0 I 0 B B a a l/h + (Illh) + 0 I 0 a B B a l/h + 1/h + 0 I l/2 B a a B l/h + 1/h + 0 I ll? 32 Table III. ” Coeff. ND(S)§ NDIt)- NABSI NDD NEUMI "“ JKt‘ JKt IND(8}+ ND(s)-ND(B)*ID Both IDJ'K'S IDJ'K't IND(t |_ND(t) ND(t) II Pairs JKt 1/b o o a a 1-1-0 1-1-0 0 o o 0 same -1/h o a o 8 1-1-0 2-2.0 o o o 71 same -1/h a a e a 2-2-0 1-1-0 0 o o .1 same 1/h 'e a a a 2-2-0 2-2Io o o o 0 same 0 o o s a 1-2-1 1-2-1 2 o -2 0 diff. o . a B o a 2-1-1 2-1-1 2 o 2 0 diff. 1/2 o a a a 1-2-1 ~2-1-1 2 -2 o -1 diff. 1/2 8 o o a 2-1-1 1-2I-1 2 2 o 1 diff. II. C-c. Description of the Program The program used for calculation of spin-coupling Hamiltonian matrix*‘ elements is quite straightforward. The basic state wavefunctions, the basic product functions and the spin functions are numbered by J's, K's and L‘s ~ respectively. The program steps systematically through all of the basic ‘ wavefunctions, attempting to calculate-the matrix element between each" different pair of the basic state wavefiynctions. First.it calculates the‘ e " 4 matrix element between W1 and h , then “’1 and 11:2 , then m and $3 ,....to ‘*JL which is the last of the basic state wavefunctions; then it comes back 'to calculate vzand *1, $2 and $2 , and so forth. The computer picks up- two basic state wavefunctions Vland.-wl, then compares the basic product -' functions in the basic state wavefunctions. (e;g., it compares the basic product function of J -'1,~x --1 with that of J - 1, K I 1; then J - 1, K I l with J I l, K I 2; then J I l, K I 2 with J I l, K I 1; then J I l, 33 K I 2 with J I l, K I 2). If these two basic product wavefunctions have matrix elements between them, then the computer goes further to pick up each different pair of spin functions and performs various tesuias listed in Table III, classifies it into one of the eight cases‘ . in Table III, and gives a value fer the coefficient of the spin coup- ling constant of these two nuclei. If these two basic product wave- functions do not differ-(i.e., have the same spin wavefunction) the computer also assigns a value according to Table III for the coeffi-‘ cients of the spin coupling constants of any two nuclei. If these two basic product do not have matrix elements, then the computer picks up another basic product function for comparison. These procedures "g won and on until it has calculated all the possible matrix elements. 'The-output will be all the coefficients of all the spin coupling cons- tents, e.g., in five-spin systemm,the matrix element between *1 and *1 will be represented by coefficients of J1J in J11 J12 J13 J1“ J15 o J2.1000000000J25’ J31 OOOOOOOJas. JH1°°°°°°°J‘95’ J51 OOOOOOOJSS 0 'Here obviously, we know that J11,J 33 ,...etc. are zero and that 22 ’J J13 I J J ,.....etc..ao from the output we can easily obtain 31 ’ J12 ' 21 _ the matrix element between any two basic state wavefunctions. In order to handle larger spin systems, the input basic state wavefunctions of aysystem can be broken down into many subgroups, and reach time we may take one subgroup as the input basic state wavefunc- ' Ations; e.g., in a five-spin system, there are a total of thirty-two basic state wavefunctions; each time we can take eight or sixteen basic 'state-wavefunctions as input. The size of the subgroup (the total nump her-of basic state wavefunctions of the subgroup) depends entirely on 3b the size of the system (total number of nuclei) and the capacity of the computer. One can also use the basic state functions with the same projection of spin angular momentum M8 whichever is more convenient for the computer . II. D. The Secular Equation 0 1 0 . andad ), respectively, and let E1 and Ego) be the corresponding 0 eigenenergies. Our quantum mechanical problem will then be to solve We let 01 and 0 be the stationary state eigenfunctions of 29, the secular equation I)? - s 6 l - 0 (11.11;) , J mn mn for the‘energies, whereae I «2%) 92> (11.15) ‘ mm and. Gmn 'I 1 if m I n and Gmn I 0 if m I n. The exact eigenfunctions of . . O the complete Hamiltonian are expressed in terms of the On by the equations 0 0mg”)? a O , [131 mn n ( 11.16) where .the' 5m are obtained from the solutions of the P simultaneous equations “£1139“ - Gmn Em] 8m I 0 ' - . (11.17) The order of secular equation (II.lh) is 2", where N is the total number of nuclei in the molecule with spin % . This 2Nx2N secular determinant can only be useful for obtainingthe energies En for molecules with N>,5 when it is easily factored. It will be very important to obtain any possible factorization of the secular determinant. The secular equation (II.1h) can be factorized into a number of equations of lower degree in E if we classify the basic state wavefunctions by total spin and synetry i.e., we make use of the following mixing rules: (1) There is no mixing between states of different total spin F2, 35 where Fz . z I (1) (11.18) 1 z ' (2) There is no mixing between states of different symmetry. The factorization of the secular equation arises because there are no off-diagonal matrix elements of the Hamiltonian between the basic pro- duct functions corresponding to different values of F and there are z' no matrix elements of the Hamiltonian between functions belonging to different irreducible representations. This is because the operator Fz commutes with the Hamiltonianae and the Hamiltonian is totally symme- tric with respect to permutations of equivalent nuclei. As a result we can divide the basic state wavefunctions into classes according to their values of Fz and their symmetries, and then it is only necessary to evaluate the submatrices of functions in one such class. In other words, for these cases, the basic symmetry functions are themselves stationary-state functions. The set of functions ae,/%.( cB+ Be). '/%( e6- 80). BB , for example, all differ from one another either in spin or symmetry. There is no mixing and these are the correct stationary -state wavefunctions for the symmetrical two-nuclear system.A2. II. E. Selection Rules and Intensities of Transitions The probability P(m+n)(in sec“) that a nuclear system undergoes the transition m+n (i.e., ¢fi+ °n) is given (h6)-by the equation: 3 P . 1' . (m) '1?— (fom 9» (11.20) . where Mx I 2 vi 1x(i) (11:21) 1 and flax)“ . ( ml "xl on) ; (11.22) MI is the x-component of the magnetic moment operator andpv is the 36 energy density per unit frequency range arising from the oscillatory radiofrequency field in the x-direction and 6m and in are the stationary state wavefunctions. McConnell et al. (h2) derive the following two selection’rules: (1) In any allowed transition, the change of total spin is AF I t 1 - (11.23) (2) All allowed transitions must be between two states of the same symmetry. So (Mi)mn’ the transition moment, in equation (II.22) is different from zero only when these two selection rules are obeyed and the inten- sity of the transition between two states m and n is proportional to the square of the transition moment. III. A SURVEY OF THE VARIOUS METHODS CURRENTLY USED FOR ANALYSIS OF NMR SPECTRA In analysis of simple high-resolution NMR spectra, in which the "ispin-spin couplings are much smaller than the difference between the ' chemical shifts, the simple rules based on a simple interaction Hamil- tonian and first-order perturbation theory can be generally used (3%, h2,-h7). The NMR spectra become complicated in substances where there are nonequivalent nuclei of the same species whose relative chemical' shifts are of the same order of magnitude as the splittings due to spin coupling. If the chemical shifts are still moderately large, higher order perturbation methods can be used with some success (h8, 39), but eventually. individual multiplets become merged in a general mixed group of lines which may have few features of regularity. One is then faced‘ with the problem of interpreting such a band system, assigning each line"‘ to a definite transition, and finally extracting numerical values fer the chemical shifts and spin-coupling constants. A spectrum is considered- 'analyzed when the chemical shifts153 and the spin couplingconstants~513~' of the system have been completely determined. Various methods have been- proposed fer the analysis of complex NMR spectra. These methods can be ‘roughly~classified into three different approaches, namely, the iterative approach (50-78), the subspectral analysis approach (79-102) and the dir- ect calculation approach (103-113). 111. A. Iterative Approach Analysis of complex spectra has most often been performed using the iterative approach. In iterative procedures, Judicious estimates of chemi- 37 38 cal shifts and spin coupling constants are inserted into the spin Hamil— tonian and the eigenspectrum problem is solved to obtain transition energies and relative transition probabilities which can be used to plot a calcu- lated line spectrum. This calculated spectrum is compared with the experi- mental spectrum and any differences are used as a basis for readjustment 'of the initial estimates of the chemical shifts and spin coupling cons- tants. The initial estimates of the chemical shifts and spin coupling 'constants can frequently be obtained from known values in similar cases or sometimes by use of the moment method of Anderson and McConnell or by use of double resonance techniques. Swslen and Reilly's, Hoffman's,'Arata,"‘ Shimizu and Fujiwara's and Castellano andeothner-By's methods are all based on the iterative principle. In Swalen and_Reilly's method (SO-Sh), the experimental energy levels are derived from the observed spectrum by making use of the trace invariance property of the Hamiltonian matrix. They use derived levels for iterative purposes. An approximate HamiltonianaLo is chosen and is brought into diagonal form by a similarity transformation 8-968 I A0. From.the experimental spectrum an energy level scheme is constructed, and the reverse similarity transformation is then applied to the experi- mental energy level matrix to obtain an improved Hamiltonian SAéxptls-1' ”imp. From” imp new values of 51 and J i J are deduced, and the'process repeated until a consistent set emerges. Two Fortran computer programs based on this method have been written. These two programs are used in three stageswin the analysis of a given spectrum, Systems up to and including eight nuclei of spin-% can be analyzed. Recently, Ferguson and Marquardt use magnetic equivalence factoring as a means of removing 38 cal shifts and spin coupling constants are inserted into the spin Hamil— tonian and the eigenspectrum problem is solved to obtain transition energies and relative transition probabilities which can be used to plot a calcu- lated line spectrum. This calculated spectrum is compared with the experi- mental spectrum and any differences are used as a basis for readjustment 'of the initial estimates of the chemical shifts and spin coupling cons- tants. The initial estimates of the chemical shifts and spin coupling ‘constants can frequently be obtained from known values in similar cases ‘or sometimes by use of the moment method of Anderson and McConnell or by use of double resonance techniques. Swalen and Reilly's, Hoffman's, Arata,- 'Shimizu and Fujiwara's and Castellano and Bothner-By's methods are all based on the iterative principle. In Swalen anleeilly's method (SO-Sh), the experimental energy levels are derived from the observed spectrum by making use of the trace invariance property of the Hamiltonian matrix. They use derived levels for iterative purposes. An approximate Hamiltonian){0 is chosen and is brought into diagonal form by a similarity transformation 8-1»: 8 I Ito. From the experimental spectrum an energy level scheme is constructed, and the reverse similarity transformation is then applied to the experi- . .. ~1.x exptls mental energy level matrix to obtain an improved Hamiltonian 8A ”imp. Fromm imp new values of 51 and J i j are deduced, and the'process repeated until a consistent set emerges. Two Fortran computer programs based on this method have been written.*These two programs are used in three stageshin‘the analysis of a given spectrum» Systems up to and including eight nuclei of spin %-can be analyzed. Recently, Ferguson and Marquardt use magnetic equivalence factoring as a means of removing 39 the eight spin limitation of Swalen and Reilly's method and have worked out the case of ten nuclei of spin ‘/2 . Arata, Shimizu and Fujiwara (59) use the observed frequencies and relative intensities simultaneously for the iterative procedures. Both‘ frequencies and intensities are reduced to a dimensionless representa- tion. The appropriate differentials are feund by equating terms in the power series expansion of the correct parameters in terms of the dimena sionless trial constants with the corresponding terms in the-perturba- tion expansion of the Hamiltonian and Ix matrices. In Hoffman's method (56, 57). first order perturbation theory is used to determine the correction to an approximate set of parameters fitted to the observed line positions. The line positions instead of the derived experimental energy levels are used for iterative purposes. An approximate Hamiltonian matrix is diagonalized. For selected experi- mental transition frequencies, the approximation eigenvectors' ($568)!“- 0 (S’iJQjS)nn -‘vmn I*w - (vexptl) are evaluated, using the approxi- mn mn mate eigenvectors $.34L] is an estimate of the correction to the appro- ximate Hamiltonian. If it is expressed algebraically as an array of linear functions of the fundamental‘parameters, the set of approxima- tions is obtained in the form of a set of linear simultaneous equations in the corrections to the parameters. These may be solved by various methods to yield the corrections and the iteration may be repeated until a consistent set is obtained. Castellano and Bothner-By's method (61, 62, 67, 68) is very closely related to that of Hofflman. This method is appli- Icable even if not all lines are assignable, is not affected by symmetry :in the Hamiltonian, yields an estimate of the ellipsoid of error, and ho converges relatively rapidly to a predetermined assignment. A Fortran computer program based on this method has been written systems up to and including seven nuclei of spin %-are acceptable. The most serious disadvantage of the iterative procedures is that they are unsystemmatic and tedious where there are more than one or.two -variables. In addition, these procedures still require identification of the corresponding experimental and calculated spectral lines with one another. For spectra containing many lines or with some closely-spaced lines, such an assignment is particularly difficult. As a consequence" ambiguity may exist in the derived values for the chemical shifts 6i and~ spin-spin coupling constants J13 and these procedures do not always give 'unique results. If only the transition frequencies (or experimental energy levels) are used as the criteria for satisfactory agreement between - calculated and experimental spectra ambiguities always exist since more than one set of parameters give results consistent with the experimental data~(10h). The iterative approach is unsatisfactory for the reasons mentioned“ - above; but once an approximate set of the NMR parameters is given, this "‘ method can then be applied to any general spin system up to eight nuclei ' 'of spin é. III. B. Sub-eppctral Analysis‘Approach-" The composite "particle" method of Waughrandwbobbs-(80)~and Whitman, Onsager, Saunders and Dubb (79) is a special case of the subspectral ana- llysis approach. This method has been applied to complex systems made up of a' number of groups of magnetically equivalent nuclei. The spectra of hl such complex systems have been attributed to the superposition of sim— pler spectra. The method considers each group of identical nuclei as a composite "particle" with fixed total spin. When the symmetry of the Hamiltonian (but not necessarily of the molecule) is very high, it is often possible to handle the entire problem without any explicit refer- ence to the zero-order spin eigenfunctions, thereby greatly simplifying the mechanical details of calculation. This method offers no particular advantages in dealing with cases of high molecular symmetry but low sym- metry of the Hamiltonian, several examples of which have been previously discussed (5. 6). A more recent and powerful method is the effective frequencies method which also can be considered as a special case of sub-spectral analysis. Alexander (60) first recognized that complicated spectra can be considered as being composed of two or more simpler spectra when one interacting group is greatly chemically shifted from the other nuclei . (or is of different nuclear species). This is the concept of "effective frequencies". This concept has been first used by Narasimhan and Rogers - (81, 82) in the interpretation of the proton portion (i.e., the ethyl group portion) of the spectra of some organometallic compounds. The use of effective frequencies was first put on firm theoretical ground and applied to the calculations of ABR3X and ABZXq type spectra by Pople and Schaefer (83) and Diehl and Pople (8h). In Pople, Schaefer and Diehl's “effective frequencies method, if a group of n magnetically equivalent nuclei, In, are greatly chemically shifted from any group of strongly coupled nuclei, for example, ABC, then ABCXh spectrum can be considered ‘to be composed of n+1 ABC type subspectra. They define “A! v3, we as R2 the chemical shifts that the A, B and C nuclei would have in the absence“ of spin coupling to the Kn group and they define x -‘Fz(xn) as the z- component of the total spin of the group In. Then the apparent "internal ' chemical shifts" or "effective Larmor frequencies" fer each ABC subspectrum, vA'(x), vB*(x) and vc*(x) are given by the equations: vA’(X) I “A + x JAx e vB (X) I VB + x JBX * I + J 111.1 we (I) “C X CX ( ) The statistical weights of the various subspectra are given by the binomial coefficients of n. Each ABC subspectrum must have the same value of JAB’ JAc and JBC’ but will have different internal chemical shifts (if JAx I JBx I ch). Hence frequency and intensity sum rules are directly applicable to the job of dividing the ABC transitions into their apprOpriate subspectra.*' ABC subspectra are then solved by previously developed methods, and equaI tion (111.1) gives directly the magnitudes of J J and ch, and their AX’ Bx signs relative to each other, but not relative to the couplings within the ABC group. This method has been limited to the analysis of spectra of O O the type AA ....BB ....Rp....xq non-equivalence. In those case it leads to an impressive simplification where the prime symbol denotes magnetic of the analysis, and it has been possible to derive all the coupling cons- ‘tants and chemical shift data from the analysis of the subspectra'of the " system. Diehl, et. a1. and Bernstein (.859 87s 83) expamhthe effective fre- <1uencies method and preposeda sub-spectral analysis which is‘applicable 'to all-possible combinations of magnetically equivalent and non-equivalent, h3 strongly and weakly coupled, groups of nuclei and includes the special ' case of the composite particle model as well as the effective frequency-~ method. In this method, the number and type of sub-spectra can be Obtained*' without knowledge of the Hamiltonian simply from group theory and good' quantum numbers. In order to derive the relations between the parameters“"‘ of the complex problem and the parameters of the simple sub-spectra it‘ is necessary,however,to compare corresponding parts of the Hamiltonian and to find transformations which leave transitions unchanged. As these transitions may be non-analytical in terms of the molecular NMR para- meters the transformations have to be obtained by a study of invariants.- Diehl et al. (87) have given the rules for the general breakdown of NMR ‘ spectra into simpler sub-spectral problems as follows: (1) Construct the basic local symmetry wave functions for the chemically equivalent groups. (2) Determine the total molecular symmetry. (3) Reclassify the basic local symmetry functions according to their stransformation properties under the covering operations of the total molecular symmetry and rewrite them in the abbreviated notation. (h) Construct the 2n (n is number of nuclei in the system. We consider only systems of spinémuclei here) molecular basic symmetry product, - - wave functions. Use these to form the 2n possible basic group symmetry functions. (5) Derive the symmetry species of the products of basic groupasymmetry functions. (6) Regroup the molecular wave functions into the molecular symmetry species. A3 strongly and weakly coupled, groups of nuclei and includes the special case of the composite particle model as well as the effective frequency-~ method. In this method, the number and type of sub-spectra can be obtained*' without knowledge of the Hamiltonian simply from group theory and good' quantum numbers. In order to derive the relations between the parameters“" of the complex problem and the parameters of the simple sub-spectra it‘ is necessary,however,to compare corresponding parts of the Hamiltonian and to find transformations which leave transitions unchanged. As these transitions may be non-analytical in terms of the molecular NMR para- meters the transformations have to be obtained by a study of invariants.- Diehl et al. (87) have given the rules for the general breakdown of NMR ' spectra into simpler sub-spectral problems as follows: (1) Construct the basic local symmetry wave functions for the chemically equivalent groups. (2) Determine the total molecular symmetry. (3) Reclassify the basic local symmetry functions according to their stransformation properties under the covering operations of the total molecular symmetry and rewrite them in the abbreviated notation. (h) Construct the 2n (n is number of nuclei in the system. We consider only systems of spin.:..nuclei here) molecular basic symmetry product - - wave functions. Use these to form the 2n possible basic group symmetry functions. (5) Derive the symmetry species of the products of basic groupssymmetry functions. (6) Regroup the molecular wave functions into the molecular symmetry species. Ah (7) Sort out the transitions of the contributing species (e.g., ABB' transitions of ABB'XX' : AFz(XX') I 0; AFz(ABB') I I l). (8) Isolate the sub-spectral patterns and analyze them. (9) Using the sub-spectra of maximum lel derive the parameters of the strongly coupled parts. (10) In those cases where the Hamiltonian matrix elements of the NMR problem are given in the literature in detail, steps 1 to 5 can be deleted and step 7 can be performed on the existing tables. In general, sub-spectral transformations considerably simplify 'the treatment of complex systems containing at least one pair of weakly~ coupled nuclei and help the analysis. However, in sorting out of sub- spectra, one needs a lot of experience and possibly the help of double resonance experiments or tickling techniques. If the structure of a given system is not very well known, it is difficult to sort out sub- spectra. Sometimes, even if the sorting out of sub-spectra can be easily' achieved, the sub-spectra may be still very complicated. If we use the 'conventional methods to analyze these sub-spectra, sometimes clear cut solutions can not be obtained. So this method also has the disadvantages “of the conventional methods as mentioned previously. III. C. Direct Calculation Approach' The moment method was worked out by Anderson and McConnell (103) on a basis laid down by Van Vleck (ll5) and makes use of the experimen- ~tally determined moments of the spectrum. This method provides a tech— nique for direct calculation of chemical shifts and spin coupling con- stants from the observed line positions and intensities of the experi- hS mental spectra in principle, but it is seriously handicapped in practice- by its sensitivity to the relatively large errors present in most inten- sity measurements. Considering this point, Castellano and Waugh (10h) have developed a new method for calculating the chemical shifts and spin- spin coupling constants dtrectly from the observed spectrum which does-" not suffer from this limitation. Their method consists in astigning each experimental spectral line to one of the possible transitions between spin energy levels, utilizing the trace invariance property of the Hamil- tonian matrix and its square in a manner similar to that of Swalen and Reilly (sh), of Alexander (60) and of Banwell' and Sheppard (70). The use -- “of experimental intensities is kept separate from that of experimental I frequencies, and the former may be omitted entirely when experimental values of sufficient accuracy are not available by using intensity rules- for transitions. In this method, trial-and-error adjustment of the chemi-~ “cal shifts-and spin-spin coupling constants is avoided entirely, and the- values of these parameters obtained are exactly consistent with‘the input information. From the theoretical point of view this method is an ideal‘ one. In order to apply this method completely, however, one needs consi- derable practice and one must be able to resolve and measure most of the lines that are theoretically present. This presents practical difficulties when the lines overlap each other or the line intensity is very weak. In addition, the amount of manual labor involved is such as to make exhaus- £ tion of all possibilities impractical in many cases. In practice this :method can be applied only to an analysis of the nonequivalent'three spin system. Whitman (107) has developed a similar technique for calculating NMR h6 ' spectra. He uses frequency sum rules and intensity sum rules to Obtain an assignment of spectral lines to an energy-level diagram for the system. He instructs the computer to construct all possible energy-level diagrams consistent with the sum rules and a set of estimated experimental errors.“ When a satisfactory spectral line assignment has been found, it is used " to calculate the experimental eigenvalues of the spin Hamiltonian. Whitman“ (108) also has developed the general equations for chemical shifts and spin coupling constants of a number of proton systems in terms of the ex4 perimental energy eigenvalues. In many cases formulas explicit in the NMR . parameters are obtained while in other cases the equations are implicit and must be solved numerically.So the chemical shifts and spin coupling constants can then be derived from the observed experimental energy eigen- values. This method has the advantage of eliminating any bias, but1requires -considerable computer time trying the many possibilities.~The-assignment~ 'technique is limited by the resolution of the experimental spectrum. If ‘ two real lines are*unresolved in an experimental spectrum,then any assign-. ‘ment based on this pair considered as a single line is doomed to failure. ' Unfortunately, the more complex the spectrum, the greater the probability of unresolved pairs of lines. However, this shortcoming can be overcome by examining high-resolution spectra at two different frequencies and by choosing a relatively large validity limit on the intensity sum rules in the assignment program. Hence, this method is a very promising one. ~Primas and Banwell (l09, 110) have developed another method for direct calculation of NMR spectra which gives the resonance frequencies and intensities directly as solutions ofta new eigenvalue problem, invol- ' ving the derivation superoperator of the Hamiltonian. From this direct h? 'method the more elegant and compact correlation function method is developed, yielding the complete spectrum as a single entity. The two- nucleus AB system has been worked out by this method. It seems very pro- mising. However, its applicability to larger spin systems is still un- known. BY’Starting from the energy diagram of the spectrum and without guessing initial values, the perturbation method proposed by Granger (11h) yields an iterative procedure leading to 6 and J. The ABC system is worked out as an example. However, this method can not handle large spin systems. 111. D. Miscellaneous Approaches -~- . ~ There are many miscellaneous methods (116-158). Some of the. methods have -'combinedudifferent approaches. Some methods are proposed for special aysq 'tems. However, there is still no general perfect method. All-thermethods ~~ proposed for calculating NMR spectra have their advantages and disadvanI‘ tages. It is suggested that by combining different methods for special systems, when appropriate, better results could be obtained. IV. COMPUTER ASSIGNMENT TECHNIQUE FOR ANALYSIS OF NMR SPECTRA The present method is based on that of Whitman for the direct ana- lysis of spectra. The aim of the present work is to utilize more fully the computer and more general computer language in order to handle bigger spin systems. This investigation consists of two parts, namely, the derivations of general equations for the chemical shifts and spin coupling constants, and the computer assignment of experimental spectral lines to transi-. tions between spin-energy levels. The first part gives the general equa- tions for the chemical shifts and spin coupling constants of a number of nuclei with spin-2)- systems in terms of the values of the experimental energy levels.From the second part we can obtain the unambiguous experimental 'eigenvalues. When the experimental eigenvalues are inserted into these equations derived in the first part, they can be solved numerically for the chemical shifts and the spin coupling constants. Thus, the calcula- - tion of these NMR parameters from the experimental spectrum is reduced "in the-most general case to the solution of a system of nonlinear simul- taneous equations. IV. A. General Equations for the Chemical Shifts and Spin Coupling 8 Constants The method of derivation of equations for the calculation of chemi- cal shifts and spin coupling constants of a molecule directly from the epin energy levels derived from its experimental nuclear magnetic reso- nance spectrum (nuclei with é-spin) consists of several essential steps. -First of all, we have to write out the Hamiltonian for the system and hR ”9 Obtain the basic state wavefunctions for the system, then evaluate die-- gonal and off-diagonal matrix elements of the Hamiltonian between the basic state wavefunctions. We reduce the original forms of the matrix elements into standardized-forms by choosing the center of gravity of the spectrum as the origin for measuring the transition energies and choosing the energy of the state with the highest spin quantum number as zero energy level. The secular determinant obtained with the Hamil- tonian, using basic product functions or basic state wavefunctions as a basis set, factor into subdeterminants corresponding to the different eigenvalues of 12° If the nuclear system is symmetric or if groups of equivalent nuclei exist, additional factoring results. The factoring of the analytical secular determinants necessarily corresponds with - the factoring of experimentally-observed energy eigenvalue diagrams. Thus, the secular subdeterminants in analytical form.must be isomor- - phic with the experimentally-observed subdeterminants in diagonal form3" Exploitation of this equivalence permits derivation of equations for the* spin—coupling constants and the chemical shifts in terms of observed energy eigenvalues. The eigenvalue problem is essentially worked "back--- wards" by a;method similar to the one proposed by Parker and Brown (159). The five spin ABB'CC' system will be discussed in detail as an illustrap tive example. The spin Hamiltonian of the ABB'CC' system is 3Q. _- ZvBIBz(l) + 2.0102(3) + VAIAZG) + JBB.IB(1)IB,(2) + Ic(h)IC.(3) + 2JABIA(5)IB(1)’+ 2JACIA(5)IC(h) + BC,IB(1)ICo(3) (IV.l) After we obtain the basic state wavefunctions by the method des- J cc' 2JBCIB(1)IC(h) + 2J cribed in section II. B, the diagonal and off-diagonal matrix elements 50 of the Hamiltonian (equation IV. 1) between the basic state wavefunctions' can be evaluated easily by the proposed computer program MATREL (see sec- tion 11. C-a to II. C-d). In Table IV, the basic state wavefunctions, the“ ' diagonal matrix elements and their standardized forms (last column in the'" table) are given. In Table V, the off-diagonal matrix elements and their standardized forms are shown. Following Pople, Schneider and Bernstein's notations (18), we can use the following convenient variables: K.JBC+JBC'; M I J LiI J -J AC N I J - J P I J + J ‘BC BC'; AB; AC AB; cc‘ BB'3” BB.JCC' Reducing the diagonal and off-diagonal matrix elements into standar- + J -QIJ dized forms is a rather important step. It will be illustrated in the following examples. As mentioned before, the transition energies can be "measured relative to anyidesired origin, but for the present purposes-it' is most convenient to choose the center of gravity of the spectrum as the - origin. In the ABB'CC' system this corresponds to setting vB+vc+‘%»A.I‘0*-‘ in the analytical form of the matrix elements. A plane of symmetry exists~~ '-in this system and consequently the basic state wavefunctions canwbewcho----' sen to-be either symmetric or antisymmetric. The symmetric functions lead~' "to secular subdeterminants of orders 1, 3, 6, 6, 3 and 1 corresponding to the values +5, ti, ti. I1, I2.and -5 for Iz,respectively. The antisymme-' 2' 2 2 2 2 '5 tric functions yield secular subdeterminants of orders 2, h, h and 2 cor- responding to 1z I egg +%, -%-and ‘§~ We identify experimental energy eigenvalues by a subscript indicating twice the corresponding Izvaiue and- a superscript s or a indicating symmetric or antisymmetric. 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AAAAAA Aemieveemx o MS oemmem . flange. «i-e., .o... . AeAA.+~AAdmin.”A.+mAA.+A~A.+AAuww+<9luo9um9cno3~m an: a mum ATEIdAmuuemuAA ow. ..AAAJAAMXAAQ 9... Aum9unA..AAv~AA.AI.AAA.A..< amum 9-....AAAAA Asaiovanmx a «.8 AEAAAA 3.3353 8.855 .33.: 389.3 Fofloooé < oesvoooo 19H oases 53 Table V. Off-«11:30:19.1 matrix elements of‘ the Hamiltonian for ABB'CC' Hz 3 = %K ' %(J23+Jla) Hz.» - (4-) I? (Mm) - 4- (J35) (4-) (51.) (Mi) - f; (J15) , 85,9 .- I/ZL )(.:.)(L+x) =7;- (J23) H7,9 -(,21)(.21.)(K-L)-',§.(J13) (4. Hymn.) ”2‘ (.135) 5.9 ' H5,10 ' 38.9 a 0 H6,7 '%P " J12 u”- “m' (gym-u) - 5%.) a: :13 n: u u: on C q as +- I I I II: a: a: I as,” - 37’... - guinm) fin“) H13,10 ' ix ' '%(J23+J13) 39,10 - ( éHiHM-N) 5,;- (J...) an ’12 - (,4- )(éHM—N) 1,;- (J15) H11,13 ‘é‘x ‘%(J23*J13) an.“ - 1111,15 - g) (.21) (m) ‘§‘(J35) “11,16 ' “12,13 ' “12,16 ' ° “12.... - “u... - (A. ) Ii.) (K-L) - (é. ) (J13) 312,15 - 315’", - (,5. ) (.i.) am.) = (é) (J23) H13,1» ' “13,15 " ('3') L? ”H” ' -;-(J15) 1113,15 - (J;— ) (é) (um) - (é. )(Jas) l 3:10.15 . 3P 3 J12 Continued Sh Table V- Continued 3.7,... - (é— )(é-IIM-N) 4,4401”) 317.19 - (,ZL )(;.)-J”-%‘°J°c>-JH- JW’ J.-:- .. J»J»J6.-6.» J- JJWH J6.J.» J J— W; <6.-6.» J- J- JJ-MJ %(“'N)2 + 7333+ %(M+N)2} +{[- -§1°B‘5c)‘“][‘ £153.50» #0. .31:- Ttxmnmé (63- 5C)+ é‘s‘% K- %(M-N)]+ %(M2_N2)K_ gin-NFL gun-ac)». €50- ix- lawn]- gxzt- ggwcbul- élw’nfl gag-ac» g3- éx- gmm - -s3+sz(sgl+sgz+ 3 8 3 " E33)*E(E§1E32? EglEga+ 332333) *’E§1E§2E:3 (IV‘Z) 57 IfI I-.3.,then' z 2 35 +35 -M-E 1- K .741 mm E" -E o '0 5350 2 22 ) '31 1 3 2 1 1 3 —K ._6+_6-1-x M-N-E M-N - o E as o 2 sascz-T‘ ) m‘ ) '32 .374“) £74144) Efi-kfimM-E o o 3' -E 22 22 sBscz 1. 43 -3215 2.5--3A{256- 25-5A1A1A1 -25- 3W JWJJJJJJ. .> J J JJJ J 5 c 5 B 1 l l 2 1 ‘1 2 l 1 5c” Ed];- 5" :(M-NHJJI 3133-3c)- :50- Ex-fimmll-g- (sB-Gc)- :33- '2'“ r(M-N)]+ 1 2 2 1 2 2 F(M—N) + EEK + _8_(M+n)}+{[ .5A(5B+5C)-M][ gala-ac)- é‘c" ix. J1: M+N>][- $63-63- 1-_1 12-2-122_-1_-1 _ 253 ix T{ma-.101». 31cm a) 3414.11) [#513 so) 356 2 Amen] fixfigaf cc)-M]- gmmzt- .§.(éB-6C)- g8- 12x. gmm - 23.32 (12:331 .3132 mi” )- H s s s s s 9 s s s 4' . E(E_31 _32 E_31 _33 E_32E_33 ) E_313_32E_33 (IV 3) Adding the coefficients of E2 on both sides cf equations (IV. 3) and (IV.2), we have - 3.5 - is -x— 4&3 1.6 + is -K- 314 - -2K- I 3' +3“ +38 +13" + SB 50 2 5‘Bsc 2 3M 3132 33-31 8 8 3 1 8 8 8 8 8 I 3.32 +E_33 or K - - ;M» .24331 +£32+E33+E41 +E_32 +E_33 ) (IVA) Substracting the coefficients of E on both sides of equations (IV.2) and (IV.3) we have {-2u[ gonna”- gchwcnm. $.14]- gsnmcnm- £424.10» get-1c- i-(M-uH-I ix. 5mm %“B“c’* .3631} - (GBJGCH .3)» 55-19 gm 5-x. ‘5“) 1+ Nl- $63-60)?ng é‘63'60)+3%5c] - (aBJGCH §KT%M]+IGB-5C][0]+N[-l-§-(6B-GC)] - (Eg1EgtE.1E; *Enga 4:313:32‘3131313 3'3132‘3133) (3mm sync? . 3 3 s s 8 s 3 o, N A 5“ -6 ) + {2(E31332*Ea1E33*332333‘E-3lE-32‘ -31 -33" -32 -33 B C (Iv.5) Subtracting the coefficients of E0 (the constant term) on both sides of equations (IV. 2) and (IV. 3) we consider the first term: 6-26+6-M-26-6+.1.5A1A}_M+n $5-5 +16Ale-LMARJA £54. - 2 _ __ _ l -1 _ _ -1 _ l -1 . cc) 1411ch 60) 154C 3x :(MJNm $6363 3.63 3K T(rd-m} 58 2 3 i i 2 2 2 1 2 2 2 2 2'( §-) (53+5c)(§B-5C) +2.( 3;) (53-50) ( 3353-24 .5.) (513-50 )( 966-- 2 1 2 2 ' 1 1 1 2~( 'g')(63+6c)( 3) 636C -2«( .5.)(53+GC)[- 7K— 11.4mm“- '2'“ ?(14.11)]- 2 l l l 1 2 1 2M( §)(3B-5c)[- fx- gnu-N) 1-2MI- 2K" ?(mm) It r“n"c’* 3.6 (Iv.6) B] To the above equation (IV.6), add the following terms: . l 2 3. 6 6 1.6 .1. 2 2 5 l 2 2 1 -2 304-1!) [- 5( 3' C)+ 5 C1-2» “K [- 34 Been-2. 3"”) [- 1:“:"c" .63] therefore, the total I EglEngga dialing” Then this is reduced to the following form: (5 +60) 5(23 E“ E33 - E' E“ E' ) (GB-6C)2 . :5 + 31 32 33 -31 -32 .33 (IV.7) 1 (53“c) when I2 I 5 115 5-3 0 o o o o Efl-E o o o o o o as 5-}: o o o o o Efz-E o o o o o o 11., 7-3 0 o o o o Ef3-E o o o o o o Hag-E o o . o o o Elan-E o o o o o o 119 9-13 0 o o o o sfs-E o o o o 0 H10 1ans o o o o o Q‘s-s 5 S h l l 3 l l h l 6 5 B ' . 3 E'E (- g5+54- -:M*~4*‘SJ-—5-J+eeeeoeeee).E-E (3611+312*000+E15*E ) B C 2 B s c 2 16 therefore, -h1<- gn- P - (231+.......+E:5 4-13.16 ) P - Auc- gu-(s'il +232 +313 +33“ +E'15 +236) (IV.8) For the antisymmetric case, when I2 I -:- , 323 23-1=.: o o o E’l‘l-E o o o o 112,. “-163 o o - o E'fz-s o o o o 1125 25-}: o o 0 31‘s.}: 0 o o o “25 26-1: 0 o o E?.,-E( 2}. 1 1 1 . a a 15.21%“ avaC-P- 314-- E.vA-JJ- EM 531 +312 +313 +E” (IV.9) 59 When Iz I - %.fOr the antisymmetric case, 327 27-3 0 o o Bill-E o o o 0 H28 ze-E o o o sflz-E -o o A o o 329 29-3 0 o o tha-E o o o 0 H30 ao-E ' L o o o Effie-E :0 1 a a 8 8» 1-27311 ‘ 'vn'vc’P’ 5“ ' L11 “5-12 ”-13 “L13 ("'10) Adding equations (IV.9) and (IV.lO) and using the results given in equations (IV.h) and (IV.8) for K and P, we have: 1 a 8 a a a a a a M -- 634E“ +1212 +1313 +1.3” +£1.11 +£1.12 +3.13 +2.” ) (IV.ll) Subtracting equation (IV.10) from equation (IV.9) and using the results 'given in equations (lV.h) and (IV.8), we have: S 8 8‘ a 8. ' a a I. a, 63+6C - II E-(Ell +E12 +E13 +El‘! -E—ll -E_12 -E_13 -E_1“ ) (IVel2) 'WhenJIz I'i-and I2 I - %.in the anti-symmetric case: H -3 11 .15 +25 -1-P-—1-K n+1: -12 l L 2121 2122 SBSCZZ-éQ-‘llfl ) 2 8 J ' 1 3 2 1 1 1 1 “22, 21 H22 22"E '2' L 7:: GBTGC'Ip‘Ix*NM'N)‘E £21 - E o 3 0 E32 -E (Iv.13) 3 ~ 2 1 1 3 1 1 H31 31-3 331 32 eff-5507? T(M‘N)"E - 7L . 1 2 3 1 1 1 1 “32 31 “32 32"” " TL 's'Gs’E'Gc ‘Iq‘r(“‘")‘E_ § L31 '3 0 o 3232 -3 '(Iv.1h) 60 Multiplying out the above two equations (IV.13) andfl(iv,14),‘Zfiflflsimilifying we have: Q I e%N-[(E:1E§é - 3:312:32)/(5B-50)] (IV.15) 2 2 and 1. - A(eBAcc)2_.n1_(N+2q) -2(E§'IE§‘2 mini”) (IV.16) By successive use of equations (IV.h), (IV.5). (IV.7). (IV.8), (IV.ll), (IV.12), (IV.15) and (111.16) it is possible to solve directly for both of the chemical shifts and all of the spin coupling constants of this five-spin system in terms of the experimental energy values obtained from the observed spectrum. Using similar procedures to those described above, the general equations for the chemical shifts and the spin coupling constants for other spin systems can be derived. The equations for A3, A32, A232, AA'BB', A BC, ABC and ABCD are given by Whitman (108). , 2 61V. B. The Cgmput r Assignment Technique- 'The most.difficult step in any analysis of a complex NMR spectrum is usually that of making an assignment of the observed lines to the many possible energy transitions. The prOper assignment must be sorted out from many thousands of possibilities. For example, in the general fiveqspin system a total.of 210 transitions which obey the selection rule AFzI -l are possible, while a typical high-resolution spectrum.might contain about he'lines. The total number of ways of distributing ho lines among V 210 transitions is about ’22}. . So, all possibilities can be exhausted only by the utilization of the speed of modern computers. A correct assign- ment of experimental spectral lines to the permitted transitions within the ‘energy-level diagram must be consistent with two sets of rules, namely, the "6intensity suerules and the line spacing rules. 60 Multiplying out the above two equations (IV.13) andfl(IV,14),‘and"simili£ying we have: Q I q%N-[(E:1E§§ - E231E:BZ)/(GB-GC)] (IV.15) 2 2 8 and _ L - -(GB-éc)2-.117(N+2Q) -2(E§1E§2 Jar-33311;”) (Iv.16) By successive use of equations (IV.h), (IV.5). (IV.T). (IV.8), (IV.ll), (IV.12), (IV.lS) and (IV.l6) it is possible to solve directly for both of the chemical shifts and all of the spin coupling constants of this five-spin system in terms of the experimental energy values obtained from the observed spectrum. Using similar procedures to those described above, the general equations for the-chemical shifts and the spin coupling constants for other spin systems can be derived. The equations for AB, ABZ, A232, AAfBB', AzBC, ABC and ABCD are given by Whitman (108). -1V. B. The ngputer Assignment Technique- The most.difficult step in any analysis of a complex NMR spectrum is usually that of making an assignment of the observed lines to the many possible energy transitions. The proper assignment must be sorted out from many thousands of possibilities. For example, in the general fivegspin system a total.of 210 transitions which obey the selection rule AFzI -l are possible, while a typical high-resolution spectrum might contain about he‘lines. The total number of ways of distributing ho lines among 210 transitions is about-jaLgl- . So, all possibilities can be exhausted only by the utilization of the speed of modern computers. A correct assign- ment of experimental spectral lines to the permitted transitions within the 'energy-level diagram.must be consistent with two sets of rules, namely, the “—intensity suerules and the line spacing rules. 60 Multiplying out the above two equations (IV.13) ane%(IV,lh){'3nddsimilifiying we have: q - -%N-[(E§1E§2 - Egalnfnwwn-GCH (H.165) 2 2 2 a and g L - 468-60) -.117(N+2q) -2(E§1E§2 +£1.11”) (IV.16) By successive use of equations (IV.h), (IV.S), (IV.7). (IV.8), (IV.ll), (IV.12), (IV.lS) and (IV.l6) it is possible to solve directly fer both of the chemical shifts and all of the spin coupling constants of this five-spin system in terms of the experimental energy values obtained from the observed spectrum. Using similar procedures to those described above, the general equations for the chemical shifts and the spin coupling constants for other spin systems can be derived. The equations for AB, ABZ, A232, AAfBB', AZBC, ABC and ABCD are given by Whitman (108). -‘-iV. B. The ngputer Assignment Technique The most.difficult step in any analysis of a complex NMR spectrum is usually that of making an assignment of the observed lines to the many possible energy transitions. The prOper assignment must be sorted out from many thousands of possibilities. For example, in the general fiveqspin system a total of 210 transitions which obey the selection rule AFzI -l are possible, while a typical high-resolution spectrum might contain about he'lines. The total number of ways of distributing ho lines among 210 transitions is about‘ingl- . So, all possibilities can be exhausted only by the utilization of the speed of modern computers. A correct assign- ment of experimental spectral lines to the permitted transitions within the ‘energy-level diagram.must be consistent with two sets of rules, namely, the ~—intensity sum rules and the line spacing rules. 61 IV. B-a. Intensity Sum Rules The principle of spectroscopic stability states that the sum of the intensities of all the transitions between two sets of nearly degen- erate energy levels is independent of the strength of a perturbation. This principle was originally applied to multiplet structures in atomic 'and molecular spectra. It applies in the NMR case even more rigorously. When a small perturbation is applied to a system.so as to break up a line into a number of components, the sum of the intensities must be equal to the intensity of the unsplit line. In NMR spectroscopy the small per- turbations are the various chemical shift differences and the spin-spin coupling constants. These perturbations are clearly small when compared with a resonant frequency in the megacycle-per-second range. Castellano and Waugh (10%) have derived the intensity-sum rules for a three proton system. If the total spectral intensity is normalized to H2N'1, then -a general sum.rule~of this sort for an N-spin system is: i133 - 1113+ 21’2”) (IV.17) for any level J ( JJis the upper level on the left of equation (IV.lT) and the lower one on the right). The rule states that the sum.of the ~ "intensities of all transitions from a given energy level is simply related to the sum of the intensities of all transitions tg_that level. The proof of this general rule as stated above has been given by Gioumousis and Swalen (160) and by Whitman (107). The considerable experimental error involved in measurements of intensities of spectral lines means that it is futile to require of the proper assignment exact adherence to the intensity sum rules. It is necessary to introduce some -~validity limit on intensity sums by which the correct assignment may 62 differ from exact agreement with the intensity sum rules. This parameter is an estimate of the experimental error in intensity measurements. IV. B-b. Line Spacing Rules The line spacing rules (or the equal-spacing rules) are entirely equivalent to those derived from the trace invariance properties of the Hamiltonian. However, Whitman (107) first used the rules in a form which is more convenient for computer programming and somewhat more obvious physically. Examination of the schematic energy-level diagram.in Fig. 3 I2 .... 2 l --' 0 I]. _ -2 Fig. 3. Schefiic energy-level diagram for a general four? spin system, illustrating equal-spacing conditions. shows that the two transitions A and B with a common origin differ in energy by the spacing between theenergy levels which form their termi- ~“nations. Similarly, the transitions D and C with a common terminal state differ by this same spacing. Thus, after assigning two spectral lines to the transitions A and B, we must seek two lirwswi Lh identical Spacing to assign an to the transitions L;and.C, Similarly,~the lines assigned to the tran- sitions F and E must have this same spacing. Such equalities of spacing occur throughout the energy-level diagram, and impose severe restrictions upon the possible assignments of spectral lines to transitions between energy levels. Because of the experimental error it is again futile to require that the correct assignment of the observed spectral lines obey 63 the equal-spacing rules exactly. Therefore it is necessary to introduce some validity limit on equal line spacings in such a way that two energy spacings are taken as equal if they differ by no more than this limit. The magnitude chosen for this quantity will depend upon the accuracy of ‘the experimental spectrum. The number of experimental spectral lines is usually much smaller than the number of possible transitions. For this reason an actual transition diagram will contain many incomplete transi- tion "loops", and many of the equal-spacing rules will not be applicable. IV. B-c. Description of the Computer ProgramJ As mentioned before, a correct assignment of the experimental spec- tral lines to the permitted transitions within the energy-level diagram must be consistent with the intensity-sum rules and the line-spacing rules. So we use these two sets of rules as the criteria for the computer“ assignment program. Because of the considerable experimental errors in» volved in the measurements of the intensities and the positions of spectral lines, the proper choice of the permitted limits of validity of the line— * spacing and intensity-sum rules is of considerable importance. If these ‘limits are chosen so small as to be less than the experimental errors, then - 'even the correct assignment will be excluded as unsatisfactory. On the other hand, if these limits are too large a great number of assignments may be found which are apparently equally satisfactory. In addition, if the validity checks are not sharp,the pragram running time-may-be~consi- derably lengthened. The optimal technique is to choose the validity limits to be about equal to a liberal estimate of the experimental errors and to remember that the line positions are normally known with far greater acc- ~- uracy than are the line intensities,so the validity limit on line spacings 63 the equal-spacing rules exactly. Therefore it is necessary to introduce some validity limit on equal line spacings in such a way that two energy spacings are taken as equal if they differ by no more than this limit. The magnitude chosen for this quantity will depend upon the accuracy of the experimental spectrum. The number of experimental spectral lines is usually much smaller than the number of possible transitions. For this reason an actual transition diagram will contain many incomplete transi- tion "loops", and many of the equal-spacing rules will not be applicable. IV. B-c. Description of the Computer Progzam As mentioned before, a correct assignment of the experimental spec- tral lines to the permitted transitions within the energy-level diagram must be consistent with the intensity-sum rules and the line-spacing rules. So we use these two sets of rules as the criteria for the computer“ assignment program. Because of the considerable experimental errors int volved in the measurements of the intensities and the positions of spectral lines, the proper choice of the permitted limits of validity of the line- ' spacing and intensity-sum rules is of considerable importance. If these ~ ‘limits are chosen so small as to be less than the experimental errors, then ‘even the correct assignment will be excluded as unsatisfactory. On the other hand, if these limits are too large a great number of assignments may be found which are apparently equally satisfactory. In addition, if the validity checks are not sharp,the program running time may be=consi- derably lengthened. The optimal technique is to choose the validity limits to be about equal to a liberal estimate of the experimental errors and to remember that the line positions are normally known with far greater acc- - uracy than are the line intensities,so the validity limit on line spacings 6h will be chosen relatively much smaller than the validity limit on inten- sity sums. If no satisfactory assignment is obtained, the limits are increased and the assignment procedure is repeated. Or, if a number of different satisfactory assignments are obtained, the assignment program ‘is rerun with the reduced validity limits until only one satisfactory 'assignment is obtained, or at most several. If a single basic spectral line assignment is distinctly better than any other, this assignment is used to determine the best set (or sets) of the chemical shifts and spin‘ coupling constants by the method described in section IV. A. However, if several almost equally satisfactory assignments are Obtained, it is desirable to use each of these assignments to calculate a set of champ cal shifts and spin coupling constants. A priori estimates of some para- meters can then be used to immediately exclude some of these sets from further consideration. The input data for the computer pragram are the number of experi- - mental (observed) spectral lines, the number of possible transitions, the‘number of nuclei in the system, the number of energy levels, the permissible error in intensities, the permissible error in positions, the positions of observed spectral lines in order of decreasing inten- sities and the normalized intensities of lines in decreasing order. After the data have been read into the computer, the binomial coeffi- cients of the number of nuclei in the system, the lowest numbered tran- sition from each energy level, the total number of possible transitions from each energy level, the normalized intensity factor by which inten- ‘sities of transitions £222.“ level exceed those 32Dthe level, the energy 'level numbers from which transitions occur [e.g., K(29) I 6 means the 65 29th transition is from the energy level 6] and the energy level DUMP bers §g_which transitions occur (e.g., L(2h) . 7 means the zhth tran-' sition is £g_the energy level 7] are calculated, then the size of the energy-level diagram is determined. The observed spectral lines are numbered in order of decreasing intensity and the program steps systematically through all of the possible transitions, attempting to assign to each transition the most intense spectral line available. The most intense line available - is assigned-into the first possible transition. The intensity sum for "“the level involved is constructed; the line is either accepted or reJected by this test. If it is rejected, the next most intense line is tried and the testing begins anew, and so on until finally a zero- intensity (or unobserved) line is.eeed. If some line' should pass the 'first test,g§hi program then constructs the line spacing test when applicable (or possible). If it is not possible to construct this test, as is the case for transitions 1, 2 and 3 for the two-spin system, and l, 2, 3,‘h, 5, 6 and 13 for the three-spin system, or if some number ”of this test has been assigned a zero intensity transition, the tents- tive-assignment will be accepted and the program will proceed to the assignment of the next level. If a zero-intensity (or unobserved) line is used, find it still fails both the intensity-sum test and the line- spacing test, then the program.backs up and reassigns to the preceding transition an unused, less intense line. A proper assignment has been obtained when every spectral line has been assigned to some transition in such a way that all of the spacing and intensity-sum rules are obeyed. Computer time can be saved by taking advantage of the equivalent 66 nuclei and (or) the molecular symmetry. For example, a plane of sym- metry exists in the ABB'CC' system. Choosing the basic functions which are either symetric or antisymmetric enables the secular determinant to be factored into noncombining symmetric and anti symmetric portions. The symmetric functions lead to secular subdeterminants of order 1, 3, 6, 6, 3 and 1 corresponding to the values hi. , £3. , +31. , -_;. , -3. and 2 -2 for 1,, respectively. The antisymetfic functions yield secular 2 3 E- O * 3.. . - g. and--' .3. '. The schematic energy-level diagram is shown in 2 . “subdeterminants of orders 2, h, h and 2 corresponding to Iz - + Fig. 3:. Symetric Anti-symmetric E‘s. —- 331 "" 332 "'""" E33 —- 321-— 3:2 ‘- 3:1 -—E‘12 4:3 ‘4?» 435 4816 -- -‘ Eii 4:2 “—4.13 4.1.1. "" 38-1 14112411 34115—3: 54:15“ 1"3:1 14:1 24:1 3—821 6"" 3:31 _..' E132:— 313 3— E13 1-- 3:32.... 3:5 -- Fig. lb. The schematic energy-level diagram of AA'BB'C system As mentioned“ before, we identify the experimental energy eigenvalues by a subscript indicating twice the corresponding Iz value and a super- script s or a indicating symetric or antisymetric. The schematic energy-level diagram consists of two sets of levels between which no transitions occur: an antisymmetric set of two levels for Iz -; ; four levels for Iz -.;_ ; four levels for Iz - -% and two levels for I2 I “2:. , and the remaining set of symmetric levels. 67 There are a total of 32 possible transitions within the anti- symmetric levels and l78 possible transitions within’the symmetric levels. The assignment procedure can be done in two segments. In this case, if we have to observed spectral lines, the 32 antisymmetfic transitions were first assigned from among the total of ho observed lines and then the remaining observed lines were assigned to symme-, tric transitions. When we assign the antisymmetric portion we con- sider that the 178 transitions within the symmetric levels are for- bidden.-Ia this case, besides the data mentioned previously, two other - sets of data should be read in. The extra two sets of data are the number of unassignable (or~forbidden) transitions and the set of values ( -l or 0) assigned to the transitions, e.g., KD(h) I -1 means the transition h is forbidden or unassignable; KD(5) = 0 means the tran- sition 5 is available to be assigned. So we give values of--l to the transitions which are forbidden or unassignable, while we give values of 0 to the transitions which are available to be assigned before the computer starts to make the assignments. This set of values (either -l or 0) is read into the computer. When the computer starts to make theassignments, it will skip the transitions which have values of -l and only try to assign the observed spectral lines to the transitions which have values of 0. For the symmetric portion, a similar proce- dure can be applied. A lot of computer time can be saved in this way. 68 IV. C. Examples The technique described above is now applied to the study of the spectrum of cyclopropyl cyanide, a five-spin ABB'CC' system. The spectra of 2-bromo-S—chlorothiophene (two-spin AB system), styrene oxide (three-spin ABC system) and o-dichlorobenzene (four- spin AA'BB' system) are also treated here for sake of completeness and to illustrate the procedure. IV. C-a. Two-spin System Z-Bromo-S chlorothiophene has been studied by the pertubation method (48). It belongs to the two-spin AB system. In this case, the number of observed spectral lines is four (N-h), the number of possible transitions is four (NA-4), the number of proton is two (NB-2), and the number of energy levels is four (NWHZNB-Zz-é). The permissible error of intensities is chosen to be lfq,about 102 of the intensity of the strongest line. The permissible error in position is selected as 5 which usually is about twice the observed fluctuation in position of the center of gravity of corresponding lines in the symmetrical spectrum. In this case, we set both the number of unassignable transitions (ND) and the number of the last initially assigned transition (NT) equal to zero for the first run. The experimental spectrum is taken from Anderson's paper (48). For convenience of computer calculation the experimental values of relative intensities are initially normalized to 1000-NB-2(NB-1)- 1000o2'2-4000, and the experimental values of the positions are converted to fixed point values and referenced to the center of 69 gravity of the spectrum as the origin. This corresponds to setting “A + “B I O in the analytical form of the matrix elements. After these manipulations, we arrange the positions of the experimental lines in order of their decreasing intensities and arrange the normalized intensities of the experimental lines in decreasing order. That is, RA (1, 2, -°°° N + 1) are -11, 11, -50 and 0, while KB (1, 2, .... N + 1) are 1680.1680,320.320,and 0. All the values mentioned above are the input data. We then put these input data into Program."Assign" and obtained two possible assignments as follows: The first assignment gives the following values of the energy levels: E1 I 0 cps, I -5.0 cps, I 1.1 cps and E__1 I 0 cps. E01 E02 These values of the energy levels are substituted into the following equations which have been derived for AB systems (108): J I -(E01 + (IV-18) £02) 2 6 I -4 EDIEOZ (IV-19) These give J I 3.9 cps and 6 I 4.7 cps which agree with Anderson's results. From the second assignment, we have E1 I 0 cps, E01 I 5.0 cps, I 1.1 cps, and E_1 I 0 cps. We substitute these values E02 into (IV-18) and (IV-19) and get J I -6.6 cps and 6 I 4.71cps. The second solution is excluded since it is physically impossible. 70 IV. C-b. Three-Spin sttem Reilly and Swalen have analyzed the 40-MHz proton.resonance spectrum of styrene oxide by using the iterative method (50). In the computer assignment technique, we have the following input data for this spectrum: N I 12, NA I 15, NB I 3, NW I ZNBI23I8, NPEI I 160, NPEP I 190, KA(1, 2, --~ N + l) I -655, -1423, -l729, ~1061, -2254, 2508, 2652, 2902, -95, -2047, -496, -2306 and O; and KB (1, 2, .... N + l) I 1521, 1456, 1324, 1248, 1141, 1088, 956, 812, 696 648, 564, 536 and 0. The best assignment we obtained is: These lead to the energy levels E3 I 0 cps, E11 I -14.25 cps, E12 I -95 cps, E1 I 29.02 cps, E I ~21.38 cps, E I 11.69 cps, E I 24.06 cps, 3 -11 -12 -13 and E_3 I 0 cps. These values are then substituted into the following equations (108): J + J + J I - (E11 + E AB AC BC + E13) I - (E_ + E + E 11 -12 -13) (IV'ZO) 12 71 JAB(5B’250) + JAc(5c'263) + Jsc(53 + 5c) ' 3(1'311E12 + E11E13 + E12E13 + E12313 ‘ E-llE-lz ' E-llE-lB ’ E-12E-13) (IV‘ZI) 2 2 2/3 GB + 2/3 6C - 2/3 686C - 3/2 (JABJAC + JABJBC + JACJBC) . '(E11E12 + 311213 + E12E13 + E_11E_12 + E_11E_12 + E_11E_13 + E_12E_13) (IV-22) J (a -25 )2 + J (5 -25 )2 + J (5 + a )2 AB B C AC C B BC B C ' 9 “31113121311+ E—11E-12E-13) (IV'23) 2(53- 25C) (6C ; 253) (5B + cc) - -27(E11E12E13 - E_11E_12E_13) (IV-24) and the above equations solved numerically to obtain J I 5.63 cps. AB J I 2.47 cps, I 4.08 cps, GB I 11.95 cps, and 6C I 43.87 cps AC JBc which are in good agreement with Reilly and Swhlen's results. IV. C-c. Four-Spin System The 40 MHz proton resonance spectrum of o—dichlorobenzene has been analyzed by Pople et a1”(122) using the conventional methods and has been analyzed by Whitman (107) using the computer assignment method. It is included here for the sake of illustration and to compare different versions of the assignment program. Since the molecule of owdichlorobenzene contains a plane of symmetry the spin-energy-level diagram consists of two sets of levels between which no transitions occur-- an antisymmetric set of two levels each for m I +1, 0, and -1, and the remaining set of symmetric levels. There are a total of eight permitted transitions within the 72 antisymmetric levels and 20 transitions within the symmetric levels. In order to save computer time, the assignment procedure can be done in two segments. First the eight antisymmetric transitions are assigned from among the total of 24 observed lines, and then the remaining observed lines are assigned to symmetric transitions. The input data for the first segment of the assignment are NI24, NAI56, NBI4, NWI16, NPEI-zso, urns-28, NDI48, NTIO, KA(1,2, ---N + 1) - 353, -353, 416, -416, 680, -680, 843, -843, 967, -967, 1394, -l394, 1340, —1340, 1006, —1006, 41, -41, 72, -72, 1645, -1645, 1738, -1738 and 0. KB(1.2.1~matN +11) I 3651, 3651, 3081, 3081, 1896, 1896, 1341, 1341, 1288, 1288, 1145, 1145, 959, 959, 919, 919, 855, 855, 669, 669, 104, 104, 90, 90 and 0. To save computer time, another set of data KD(1,2, --- NA) should be read in, i.e. KD(1,2, ... NA) - f -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -l, -l, -l, -l, -1, -l, -l, -l, —1, 0, 0, -1, -1, -1, -l, 0, 0, -1, -l, -l, -1, -1, -1, -l, -1, -l, -1, -l, -1, -l, -l, -1, -1, -1, -1, 0, 0, -1, -l, 0, 0, -1, -1, -1, -1. From the above input, we obtain the best assignment for the antisymmetric portion as follows: 18 16 For the second segment of the assignment, NI16, NDI36, KA(1,2,'°°N + l) I 353, -353, 416, -416, 843, -843, 967, -967, 1394, -l394, 1340, -l340, 72, -72, 1738, ~1738, o, KB(1,2, ... N + 1) - 3651, 3651, 3081, 3081, 73 1341, 1341, 1288, 1288, 1145, 1145, 959, 959, 919, 919, 669, 669, 90, 90, o. KD(1,2, NA) - o, o, -1, -1, o, o, o, o, -1, -1, 019,0, 0, o, -1, -1, -1, -l, -1, —1, -l, -1, -l, —1, -1, -l, -l,-l, 0, 0, -1,-l, 0, 0,-1,-1, -l, 0, 0, -1, -l, 0, 0, -l, -1, -l, -l, -1, -1, -1, -l, -1, -l, 0, 0.-1, -1. All the other input data are the same as the first segments of the assignment. The best output for the symmetric portion is as follows: The lines are now renumbered to the order of the input KA (l,2°--N + l) of the first segment assignment, which leads to the following assignment: K 3 12 14 2 7 20 13 11 74 The experimental values of the energy levels obtained from the experimental spectrum using this assignment are E2S I 0, E+lls I S S S 4.16, E+12 I -l3.94, E0s I 7.69, E02 I -4.27, E -l3.22, S S S 0 04 I -27.34, E_11 I 4.16, E_12 —13.94, E_2 E_na ..+ 5.24, E_12a I 5.24. The transition energies are measured 8 03 E - o, a - 11.62, E+11 relative to the average of the energies of the transitions 1, 2, 3, and 4, and the energy of the state m I +2 is chosen as zero. Since the antisymmetric levels are not joined by transitions to the symmetric levels an additional zero of energy is needed for the antisymmetric levels. This has been chosen such that the sums of the antisymmetric energies for a given m value is zero. The values of the energy levels are then substituted in the following equations (107) and solved, 8 S N I -(E11 + E12 ) I JAB + JAB' (IV-25) S S S ' S S S KI3(E11 + E12 )-(E01 + E02 + E03 + E04 ) JBB' + JAA' (IV-26) 1/2 - a. a_ a - S S) - J ' -J ' M (E-11 E-12 E11aE12 )’( 4E11 E12 BB AA (IV-27) 2_ a a_ a a 2 S S _ a. _ , 2 L [(E-11 E~12 E11 E12 ) ’4E11 E12 ] 4Eo1anz (JAB JAB ) (IV-28) 2 S S a a a a 5 ' ‘4E11 E12 4EOianz ' 2(“11:35:12 + E~11 E-12 ) (Iv-29) This leads to dIvB-uA = 15.23 cps, JAB - 8.17 cps, JAB' I 1.01 cps, JBB' I 7.44 cps, and JAA' I 0.36 cps, which are in excellent agreement with Whitman's results. 74 The experimental values of the energy levels obtained from the S S I O, E+11 - . -13e22, experimental spectrum using this assignment are E S 4.16, E+12 S - -27.34, E S 2 S S S I -13.94, EOS I 7.69, E02 I -4.27, E03 5 o 04 -11 ’ -12 -2 ' 0' E+11 a E_11a I + 5.24, E_12 I 5.24. The transition energies are measured E - 4.16 E = -13.94, E a - 11.62, relative to the average of the energies of the transitions 1, 2, 3, and 4, and the energy of the state m I +2 is chosen as zero. Since the antisymmetric levels are not joined by transitions to the symmetric levels an additional zero of energy is needed for the antisymmetric levels. This has been chosen such that the sums of the antisymmetric energies for a given m value is zero. The values of the energy levels are then substituted in the following equations (107) and solved, s s N - -(E11 E12 ) - JAB + JAB. (IV-25) s s s - s s s K"30311 + E12 )'(Eo1 + E02 + E03 +’E04 ) ' an' + JAA' (IV-26) 1/2 .. a_SS) IJ,-J, M (E-11aE-ua'E1laE12 )/( 4311 E12 BB AA (IV-27) 2 a a a a 2 S 812 L '[(E-11 E-12 ‘E11 E12 ) (4311 E12] Wanoz M(JAB AB ) (IV-28) 2 a a 5 ' '4E11% 12 S'4E01anz ‘ 2“311%12‘1" E-11aE-12 ) (Iv-29) This leads to dIv -v I 15.23 cps, J - 8.17 cps, J I 1.01 cps, B A AB AB' J I 7.44 cps, and JAA' I 0.36 cps, which are in excellent agreement 33' with Whitman's results. 75 IV. C-d. Five-Spin System The proton resonance spectrum of cyclopropyl cyanide belongs to the five-spin ABB'CC' system. As mentioned before, the ABB'CC' is one of the most general system of five-spin spectra since the A283 , AZBZX , --- etc. systems can be considered as special cases of ABB'CC'. The experimental 60 MHz proton spectrum was taken by Dr. Myra Gordon with a Varian A—60 spectrometer. Since a plane of symmetry exists in these systems the spin-energy-level diagram also consists of two sets of levels between which no transitions occur. The assignment procedure can also be done in two segments. First, the 32 antisymmetric transitions are assigned from among the total of 40 observed lines, and then the remaining observed lines are assigned to the 78 symmetric transitions, or vice versa. However, in this case, the best assignment can be obtained by assigning 40 observed spectral lines among 110 possible transitions, since from the spinrenergy-level diagram we know there are 100 unassignable or forbidden transitions in this system. The input data for this spectrum are: NI40, NAI210, NBIS, NWI32, NPEII900, NPEPI35, NDI100, NTIO, KA(1,2, ... N + 1) I 356, 349, 346, 342, 276, 340, 293, 286, 520, 469, 296, 281, 576, 302, 594, 384, 991, 310, 246, 507, 313, 377, 392, 539, 652, 649, 600, 657, 534, 548, 620, 743, 644, 0, 612, 694, 712, 721, 640, 727, o; KB(1,2, ... N + 1) - 11693, 9517, 8346, 5496, 5479, 4733, 3800, 3200, 3147, 2681, 2600, 2215, 1982, 1632, 1426, 1372, 1201, 1066, 933, 816, 799, 779, 610, 583, 579, 535, 439, 401, 350, 233, 233, 233, 148, 116, 116, 116, 93, 93, 85, 47, 0, KD(l,2, °°° N + 1) I 0, O, 0, -1, -1, 0, O, O, O, 0, 0, -l, -1, -1, -l, 0, 0, O, 75 IV. C-d. Five-Spin System The proton resonance spectrum of cyclopropyl cyanide belongs to the five-spin ABB'CC' system. As mentioned before, the ABB'CC' is one of the most general system of five-spin spectra since the A283 , AZBZX , --- etc. systems can be considered as special cases of ABB'CC'. The experimental 60 MHz proton spectrum was taken by Dr. Myra Gordon with a Varian A—60 spectrometer. Since a plane of symmetry exists in these systems the spin-energy-level diagram also consists of two sets of levels between which no transitions occur. The assignment procedure can also be done in two segments. First, the 32 antisymmetric transitions are assigned from among the total of 40 observed lines, and then the remaining observed lines are assigned to the 78 symmetric transitions, or vice versa. However, in this case, the best assignment can be obtained by assigning 40 observed spectral lines among 110 possible transitions, since from the spinrenergy-level diagram we know there are 100 unassignable or forbidden transitions in this system. The input data for this spectrum are: NI40, NAI210, NBIS, NWI32, NPEII900, NPEPI35, NDI100, NTI0, KA(1,2, ... N + 1) I 356, 349, 346, 342, 276, 340, 293, 286, 520, 469, 296, 281, 576, 302, 594, 384, 991, 310, 246, 507, 313, 377, 392, 539, 652, 649, 600, 657, 534, 548, 620, 743, 644, 0, 612, 694, 712, 721, 640, 727, 0; KB(1,2, ... N + 1) I 11693, 9517, 8346, 5496, 5479, 4733, 3800, 3200, 3147, 2681, 2600, 2215, 1982, 1632, 1426, 1372, 1201, 1066, 933, 816, 799, 779, 610, 583, 579, 535, 439, 401, 350, 233, 233, 233, 148, 116, 116, 116, 93, 93, 85, 47, 0, KD(1,2, °°° N + 1) I 0, 0, 0, -1, -1, O, 0, O, 0, 0, 0, -1, -1, -l, -1, O, 0, 0, O, 0, 0, 76 '12 ’19 ’19 '19 09 09 02 09 O: 09 ’1: ‘1: '19 '19 '1: ’19 -1, -l, -1, -1, 0, O, 0, 0, -1, -1, -1, -l, -l, -l, 0, 0, 0, 0, O, 0, 0, 0, 0, 0, -1, -l, -1, -1, 0, 0, O, O, 0, 0, -1, -1, -1, -l, O, 09 09 09 0, 09 ‘19 ’19 '12 '19 0: Os 09 0: 09 0, '19 '19 '19 09 O: 09 0, O, 0, ’19 '19 ’19 ‘19 0’ 09 09 02 09 09 ’1: '1: '19 '1: '19 '19 -l, -l, -l, -l, 0, O, 0, 0, -1, -1, -l, -l, -l, -1, O, 0, O, 0, -l, .1: ’19 ‘19 “19 “19 09 09 09 09 "1: ‘19 “19 “19 "19 '12 08 09 09 0, 0’ 09 0: '19 '19 0, 09 09 '19 '19 09 09 09 '19 '19 0: 09 09 '19 "19 09 09 09 '19 '12 0: 09 09 '19 '19 '19 '1: ‘19 09 O: '19 '1: '19 09 O: -1, -1, -1, o, o, -1, -1, -1, o, o, o, o, o, -1, -1. The output of the assignment 4,28, 41, -l, -1, 3, 19, 34, 41, 41, 41, -1, -1, -l, -l, 41, '1: '12 ‘1’ '19 16, 41, 41, -1, 10, 15, 41, 41, -1, 18, 41, 41, 20, 41, 41, 7, 27, 4o, -1, -1, -1, -1, 8, 41, 41, 24, 41, 41, -1, -1, -1, -1, -1, -1, 9, 23, 35, 41, —1, -1, -1, -1, 41, 33, 1, 14, 36, 41, 41, 41, -1, -1, -1, -1, 41, 41, -1, -1, -1, -1, 37, 25, 41, 41, 17, 38, -1, -1, 41, 41, 22, 41, -1, -1, -1, -1, 41, 41, 41, 41, 41, -1, 41, 41, 41, 41, 41, 41, -1, -1, -1, -1, -1, -1, 5, 12, 41, 41, -1, -1, -1, -1, -1, -1, 41, 41, 11, 71, -1, -1, 41, 41, 41, 13, -1, -1, -1, -1, -1, -1, 2, 21, 3o, -1, -1, 41, 41, 41, -1, -1, 26, 31, 41, 32, -1, -1, 41, 41, 41, -1, -1, 41, 39, 41, -1, -1, 41, -l, -1, -1, 41, 41, -1, -1, -1, 41, 41, -1, -1, -1, 6, 41, 41, 41, 41, 41, -1, -1. Before we evaluate all the values of the energy levels, we rewrite all the positions of the observed spectral levels relative to the center of gravity of the spectrum as the origin. 50 we have KA(1.2. N + 1) . ~130, ~137, -14o, -144, 77 -210, -146, -193, -200, 34, -17, —190, -205, 90, -184, 108, -102, 505, -176, -240, 21, -173, -109, -94, 53, 166, 163, 114, 171, 48, 62, 134, 257, 158, -486, 126, 208, 226, 235, 154, 241, and 0. The energy of state m I +5 is chosen as zero and for the antisymmetric energy levels the sum of the antisymmetric energies for m I 3 is chosen as an additional zero as in the four-spin case. From this assignment, we obtain the following "experimental" values of energy levels. E5 0 cps, E31 14.4 cps, E32 17.1 cps, E33 a a S S 3.1 cps, E31 4.1 cps, E32 -4.1 cps, E11 I -28.4 cps, E12 I S S S S 38.4 cps, E13 I 53.0 cps, E14 I 2.2 cps, I 14.5 cps, E16 I E15 a a a a 32.5 cps, E11 7.5 cps, E12 5.3 cps, E13 I 16.7 cps, E14 I S -13 a s “ u a I -24.3 cps, E_14 24.8 cps, I-41.4 cps, E S I -18.4 cps, E I -7.6 cps, ~12 S I 0.9 cps, E S F=11 I -27.8 cps, E 15.8 cps, s -14 -13.5 cps, E_12 _138 S S E_31 I 17.7 cps, E_32 29.1 cps, E_33 _34 S a cps, E_5 I 0, E_31 I 7.3 cps, and E_32 I -7.3 cps. Substituting these values into equ: tions (IV-4) to (IV-l6) gives GB I 48.0cps, -15 a I -l3.0 cps, E S I 17.4 cps, E S I 18.7 6C I 42.2 cps, J12 I 7.6 cps, J13 I 4.1 cps, J14 I 7.6 cps, J1 7.1 I 4.1 cps, J23 I-5.6, J24 I 10.4, J I 7.1, J 5 25 ' ' J34 35 ' 10.1, J I-5.6. 45 IV. C-e. Conclusions The computer assignment method described here has considerable applicability. If the high-resolution spectrum to be analized is well resolved, the present method always gives a clean-cut solution without guessing initially any parameters. For spectra for which 78 we cannot estimate a set of good parameters for the iterative methods, the present method thus has advantages. The major limitation of the computer assignment method is the resolution of the experimental spectrum, as mentioned in a previous section. .Also, this method usually needs more computer time than other ‘methods do, especially if we cannot choose sharp validity limits on the line-spacing rules and on the intensity-sum rules; the computer time required may then be considerably lengthened. 5 cps 5 cps 79 J I‘ll 10 cps ‘ f . l‘ (5.) V ioonene b h fi ‘- .-oro an." E ‘ ...-J-J_L.. chlorobenzene -..”qu k H‘- 1; IV- 0. \/ /\ — -- .- 15. v \.s.. - \J/ 00 am of Cyclopropyl Cyanide , ,\p C ...M , -.F x). . II a /.\..l l.\ . [K A .. a; n ..v IV! 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IO 2:: 26 :7 28 3O 31 32 33 41 45 46 51 600 601 92 94 95 93 96 97 521 522 523 52 APPENDIX I PROGRAM 'MATRtLE' PROGRAM MATRELE DIMENSION ID(323405)9CJ(595)9N(32)9CD(3204) PCAD lOeIeNN FORMAT(215) READ lSoJ FORMAT(I2) JL=J+I-1 READ 209(N(K)9K=J0JL) FOPMAT<3212) PRINT 200(N(K)0K=J0JL) DO 27 K=JOJL NK=N(K) DO 27 NCD=IoNK READ 269CD(KONCD) FORMAT(FIOe7) PRINT 280C0‘K0NCD) FOPMAT(IH eFlOe7) DO 32 JK=J0JL NK=N(JK) DO 32 KN=10NK '\ READ 310(ID(JKCKNCLN)9LN=ICNN) FURMAT(DI1) PRINT 33o (ID(JKCKNCLN)0LN=10NN) FORMAT(IH 0511) JF=J-l JF=JF+1 IF(JF-JL)519510350 JI=J DO 601 K=10NN DO 601 L=19NN CJ(KeL)=O KF=0 NJF=N(JF) KF=KF+1 -” KI=O IF(KF-NJF)961969321 NJI=N(JI) KI=KI+1 IF(KI-NJI)1119111995 PRINT 5220 JFeJI FOPMAT(215) PRINT 5279 ((CJ(K0L)0 K=19NN)9 L=10NN) FOPMAT¢12FIOe6//) JI=JI+I 1F}Ji-JL)600.600.4S 89 111 121 122 131 141 161 161 161 162 184 201 202 £11 221 222 231 241 261 271 281 291 301 311 321 350 NSUM=O NABS=O MD=0 L1=O L1=L1+1 . IF(L1‘NN)13191310201 ND=ID(JFOKFOLI )’ID(JI‘OKI 9L1) NSUM=NSUM+ND ‘ NAB=XABSF(ND) NABS=NABS+NAB 1F(NAS1151.121.181 STOP MD=MD+1 IF(MD-2)182eld4e121 L5=L1 GO TO 121 LT=LI GO TO 121 1F(NAas-3)202.97.97 Icho-11211.97.211 1F221.231.97 1F0 COTozoI ”\1 (0 ISI IND: Tour, KF,L I)-lD(JI,KI.LI)I—J 91 Iv NW . 1x, II, 12, 92 APPENDIX II. PROGRAM "ASSIGN" VARIABLES, NAMES AND ARRAYS Number of observed spectral lines (input). Number of possible transitions (input). Number of nuclei with spin % (input). Number of unassignable (forbidden) transitions (input). Number of times zero-intensity (unobserved) spectral line has been used. Number of last initially assigned transition (input). Next available location in.KH array, initially IP - l. Permissible error of intensities (input). Permissible error of positions (input). Next available network number, initially IQ = 5. Next available index of I and J arrays. Number of energy level from which current transition occurs. Index to avoid meaningless permutations of equivalent energy levels. If 1383 = 0, try the first spectral line in the list for the next transition. If 1338 = 1, try the next spectral line in the list for the next transition. Number of energy level to which current transition occurs. Number of spectral line being considered. Number of transition currefifily to be assigned. Number of energy levels, 2 (input). IYI and 122 are dummy variables. KA(1,2,...,N+1) Positionsrof lines in order of decreasing_inten- LAB(l,2,... sities. KA(N*1) e 0 (input). ,NW) Normalized intensity factor by which intensities of transitions from a level exceed those to the" KB(1,2,...,N+1) Normalized intensities of lines in decreasifiE’order. KB(N*1) a 0 (input). IBc(l,2,...,NB+1) BinOmial coefficients of NB. KC(1,2,...,N) Transition assignments of lines. 0 - unassigned, KD(1,2,...,N1) Numbers of lines assigned to transitions; e.g., KD(h)=7 means line 7 has been assigned to transi- tion h, KD(5)=-l means transition 5 is forbidden. KE(1,2,...,NU) Energies of energy levels, relative to KE(1) - O. KF(1,2,...,NW’ Network assignments of energy levels. 0 - unassigned, l - network 1, etc. , KG(1,2,...,NA) Number of energy levels assigned to networks by each transition. KG(12)-3 means three levels were assigned to a new network by the 12th transition. KH(1,2,...) Numbers of energy levels assigned to networks by each transition, stored consecutively. KG array keeps track of how many are due to each transition. I(l,2,...) Network numbers of levels before IV transition assignment , indexed by IR. J(1,2,.... ) K(1,2,...,NA) L(1’2’0 00,“) 11(1,2,...,NW) " M1(1,2,...NW) MB(1,2,.0.NW) m(1,2’00 CNN) 93 Amount added to network to bring it into correlation with another network, through transition V. Energy level numbers from which transitions occur. e.g, K(29)=6 means the 29th transition is from level 6. Binary level numbers to which transitions occur. Lowest numbered transition from each energy level. Total number of possible transitions from each energy level. Eccunmlated intensities of transitions to each energy level. Accumulated intensities of transitions from each energy level. NOTES All data are introduced as integers and all calculations are fixed point . Intensities: are initially normalized to 1000 NB° 2(NB-l). Networks identify the possibility of non-interacting sets of levels, e.g., symmetric and mtisymmetric. 94 PROGRAM ASSIGN DIMENSION KACSO)oKB(bO)9KCC50).KD(3003)0Ki(128)9KF(128)QKGC300310 1KH(400)01(20019J(200)0K(300310L(3003)0M(128)0NA(128)9MB€12810 2MC(128)9LAB(128)915C(9) 100 READ 1019 NQNAONBONWQNPEIQNPCPQNDQNT 101 FORMAT(8151 , NEE=N+1 READ 104. (KACIX). 1X=19NEE) 104 FORMAT(1316) RfiAD 1049 (Kb(JX)9JX=1oNEE) . READ 21000(KD(JX)9JXB19NA) 2100 FORMAT(4012) 12:1 NbV=NB+1 DO 102 1x=19N8V IUC(1X)=IZ IZ=IZ*(NB-1X+11/1X 102 PRINT 1030 (IBC(IX)) 103 FORMAT (14) IV=1 1YY=0 122:1 DO 106 JX=10NB IdCE=IBC(JX) DO 116 1Y=1QIBCE MX=IY+1YY M(MX)=IV JXA=JX+1 MA(MX)=IBC(JXA) LAB(MX)=1000*(NB+2-2*JX) IdCV=IBC(JXA) DO 116 IZ=IQIBCV K(IV)=1Y+1YY L‘IV)=IZ+IZZ PRINT 80 (K(1V)9 L(1V)) 8 FORMAT(215) 116 Iv=1v+1 IYY=1YY+IBC(JX) KX=JX+1 106 IZZ=IZZ+IBC(KX) 200 DO 201 1X=10N 201 KC(1X)=0 DO 202 Jx=10NW KE(JX)=0 KF(JX)=0 MB(JX)80 202 MC(JX)=O 9S 1p=1 10=5 19:1 KF(1)=1 1V=1 NN=0 , KAN=0 KBN=O KCN=0 KON=0 KEN=O KFN=O KGN=0 KHN=O K1N=O KJN=O KKN=0 KLN=O KMN=O KNN=Q KON=0 KPN=0 PRINT 2049 N9NA9NB9NW9NPEI9NPEP9ND9NT 204 FORMAT(8151 . NE=N+1 PRINT 2059(KA(1X)91X=19NE) 205 FORMAT(1316) PRINT 2069 (KB(JX)9JX319NE) 206 FOPMAT(1316) IF(NT-1130099999999 999 DO 2201 LX=19NT 1F(KD(LX))300930092202 2201 CONTINUE 2202 DO 2222 NX=19NT IY=KD(NX) KC(1Y)=NX IS=KCNX1 1T=L(NX) KE(1T)=KE(IS)+KA(1Y) KF(1T)=KF(IS) KG(NX)=1 KH(1P1=1T IP=1P+1 MC(1518MC(1$)+KB(1Y) MB(1T)8MB(1T)+KB(1Y) 2222 1V=NT+1 300 301 400 500 511 501 502 503 504 700 701 702 703 704 705 706 707 708 1100 800 880 811 881 812 882 813 883 814 884 815 96 IF(KD(1V))30194009400 1v=1v+1 ‘ GO TO 300 ISSS=O IU=1 . IF(1U-(N+111511950191800 IF(KC(IU))60005019600 IS=K(1V) 1T=L(1V) IF((MC(IS)+K8(IU))-(MB(IS)+LA8(IS)+NPEI)150395039502 1F(IU-(N+1)16009180091800 1X=M(IS)+MA(IS)‘1V 1F(ISSS-1)50497009504 IF(1X-1)70197009701 IF((MC(IS)+1X*K8(IU))‘(Mb(IS)+LA8(161-NPE1))180097019701 IF(1U-(N+l))70497020702 1F(NN-(NA-N-ND)17039180091800 NN=NN+1 KG(1V)=O GO TO 800 KFS=KF(IS) KFT=KF(1T) 1F(KFS-KFT)90097059900 1F(KFS)7O6910009706 IF(KE(IT)-(KE(IS)+KA(IU)+NPEP)1707.7079600 IFKKE(IT)-(KE(IS)+KA(IU)fNPtP)160097089708 KG(1V1=O KC(IU)=1V M8(1T)=M8(1T)+KU(IU) MC(IS)=MC(IS)+K8(IU) KD(1V)=IU IF(1V-26)140098109880 KAN=KAN+1 1F(KAN-5)801980191400 1F(1V’31)140008119881 KBN=K8N+1 1F(KBN-5)801980191400 IF(1V-45)140098129882 KCN=KCN+1 1F(KCN-5)801980191400 1F(1V-55)140098139883 KDN=KDN+1 IF(KDN-5)801980191400 1F(1V-58)140008140884 KEN=KEN+1 IFCKEN-518019801914OO 1F(1V-61)140098159885 KFN=KFN+1 885 816 886 817 887 818 888 819 889 820 890 821 891 822 892 823 893 824 894 825 801 1151 1400 1401 1402 1600 1601 1602 1603 1604 1605 1700 1F(KFN-5)801980191400 1F(1V-68)140098169886 KGN=KGN+1 1F(KGN-5)801980191400 1F(1V‘71)140098179887 KHN=KHN+1 1F(KHN-5)801980191400 1F(1V“78)140008189888 K1N=K1N+1 1F(K1N-5)801980191400 1F(1V-81)14009819o889 KJN=KJN+1 1F(KJN-5)801980101400 1F(1V-105)140098209890 KKN=KKN+1 IF‘KKN-51801980191400 1F(1V-145)140008219891 KLN=KLN+1 1F(KLN-5)801980191400 1F‘1V-1631140098229892 KMN=KMN+1 1F1KMN-51801980191400 1F(1V'1801140098239893 KNN=KNN+1‘ 1F(KNN-5)801980191400 1F(1V-190)140098249894 KON=KON+1 1F(KON-5)801980191400 1F (1V-200)_l40098259825 KPN=KPN+1 1F(KPN-5)801980191400 PRINT 11519(KD(1X19 1X=19NA) FORMATC3014) 1F(1V-NA)14019115091150 1V=1V+1 1F(KD(1V1)14009140291402 1F(I$-K(IV))400916009400 1T=L11V1 1F(KF(1T))400916019400 1F(1S“1116029170091602 1F(IS'2)16039170091603 IFCIS-(N8+21116049170091604 1F(1$$S‘1’16059170091605 155531 1U=1 GO TO 500 1SSS=1 97 600 1701 1000 1200 900 901 902 903 904 907 908 1300 1304 1306 1307 1308 1800 98 1F(1U-(N+1))60095009500 1U=IU+1 GO TO 500 KF(IS)=10 IO=IO+1 KG(IV)=2 KHIIP)=IS IPV=IP+1 KHtIPV)=1T 1(1R)=0 KE(IS)=O J(IR)=0 IP=IP+2 IR=IR+1 KF(1T)=KF(IS) KE(1T)=KE(IS)+KA(IU) GO TO 1100 1F(KFT)90299019902 KG(IV)=1 KH(IP)=1T 1P=1P+1 GO TO 1200 1F(KFS)9O4.9039904 KG(1V)=1 KH 1300... networks' Sheet II 800.. Assign this line to transition IV lhOO..|Last transition? \ V i No I) No,‘ 102 FLGN SHEET III Yes Consider next transition Print assignment ———>1500...(Sheet IV) 1’ Is this to be . assigned? Yes Same initial level _ as previous trans- Yes Is final level ition?< 1600.. in any network? l Yes No Not _ - * h00...(Sheet I) Is initial level ' first of a row? Yes No 1700... ’ 1888 = 1 , Yes Is ISSS-l? \L A fl'Non Is this zero- ISSS - 1 intensity line? No - Start with Yes first line v? 600...(Sheet I) 5m000(5h%t I) 103 FLOW SHEET IV Back up JV Consider previous -—§ 1800.. transition Jz - 9 Already at first Y Double permissible transition? as error of intensity - 1N0 ' I ‘Was this transition 200...(Sheet I) 1500.. to be assigned /N/O I i Yes ..Remove line originally«assigned. \1/ I I ‘Was zero-intensity' No Restore previous. line assigned? \‘transition inte , , I .sities \V Yes I W ‘1’ Reduce number of ”were any levels zero-intensity assigned to a line used network by'this transition? Restore level to Yes zero-network [level assigned? - V“ 1 No 'Was final level . _ assigned? Restore all levels ' '* to previous networks Yes 1700...(Sheet III) i ISSS - 0 600...(Sheet I)