AN ESR STUDY OF LINEWIDTHS IN METAL'AMINE SOLUTIONS AND AN AB INITIO INVESTIGATION OF THE LITHIUM AMMONIA INTERACTION Thesis for the Degree of Ph. D‘ MICHIGAN STATE UNIVERSITY VINCENT A. NICELY 1969 Mk. ‘ . LIBRA {Y M ichigan S‘ .catc $4.4? I”! i f5.- §}"y ;- f‘ I " ~~‘-"¢-'-' ‘M ~1u§-‘:p'-|nu at» nib This is to certify that the thesis entitled AN ESR STUDY OF LINEWIDTHS IN METAL-AMINE SOLUTIONS AND AN AB INITIO INVESTIGATION OF THE LITHIUM-AMMONIA INTERACTION presented by VINCENT A. NICELY has been accepted towards fulfillment of the requirements for PH 0 D degree in Chemis t! y Major professé/ (films/07% Date fifi‘b“. ///(/1 /96? 0469 amomc av 5"? NOAH & SONS' BOOK BINDERY INC. LlIMHY BINDERS IIT. IlCIlfll IA; \\_________. ”ammu— ' ABSTRACT AN ESR STUDY OF LINEWIDTHS IN METAL-AMINE SOLUTIONS AND AN A§_INITIO INVESTIGATION OF THE LITHIUM-AMMONIA INTERACTION BY Vincent A. Nicely An electron Spin resonance (ESR) study of the mI dependent linewidth of the hyperfine pattern from alkali- metal-amine solutions and an 22 initio self-consistent field molecular orbital study of the electronic structure of com- plexes of lithium atoms and one or two ammonia molecules are reported in this thesis. The ESR spectrUm of the monomer in dilute solutions has Lorentzian-shaped lines. A two-state equilibrium model is sufficient to describe the linewidth dependence upon mI of an isothermal spectrum. Due to persistent problems with solution decomposition, however, data could not be obtained over a sufficient range of temperature and solvent composition to permit a distinction among several possible models. A systematic study was made of the solution decomposi- . tion at room temperature. Solutions could be prepared which were stable for periods of weeks to months in the presence of excess metal after some initial decomposition. However, Vincent A. Nicely for freshly-prepared dilute solutions without excess metal the decomposition was much faster and depended on the surface to volume ratio of the container. Zero order decay was observed for solutions in a 2 mm diameter tube. The spectrum of sodium in frozen ethylenediamine below 0°C was reinvestigated. Solutions made with carefully purified solvent gave a Spectrum which was very similar to that of highly dispersed sodium metal. Solutions to which ammonia had been added or which were made with solvent which had been allowed to react with metal prior to distil— lation gave the two line pattern previously reported (1). The narrow line observed from solutions with impure solvent is probably due to liquid solution dispersed throughout the solid. An §b_initio molecular orbital study was made of the stability and electronic structure of lithium-ammonia com— plexes with a basis of gaussian lobe functions. A complex of lithium with one ammonia molecule is predicted to be stable by 20 kcal/mole relative to a free lithium atom and a free ammonia molecule. The bonding in the complex is primarily through sharing of the lone pair of electrons on ammonia between the ammonia molecule and the lithium atom. The unpaired electron orbital is rearranged upon formation of the complex so that the unpaired electron has very low density in the region of the ammonia with the exception of Vincent A. Nicely a sharp peak in density near the nitrogen nucleus. The com- plex with two ammonia molecules is nearly isoenergetic with the one ammonia complex plus an ammonia molecule. The ad- dition of the second ammonia causes an increase in the preferred lithium-nitrogen distance and an increase of the occupancy of the nitrogen orbitals by the unpaired electron. The ammonia appears repulsive to the unpaired electron in the diammonia complex, also. The implications of the results for metal ammonia solutions are discussed. REFERENCE (1) Vos, K. D., Ph. D. Thesis, Michigan State University, East Lansing, Michigan, 1965. AN ESR STUDY OF LINEWIDTHS IN METAL-AMINE SOLUTIONS AND AN fig INITIO INVESTIGATION OF THE LITHIUM-AMMONIA INTERACTION BY Vincent A. Nicely A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1969 (W757 6527/20 To My Parents ii ACKNOWLEDGEMENTS Several peOple have been helpful during the course of the research reported in this thesis. Professor James L. Dye suggested the problems and advised the author throughout the investigation. Professor Jerry L. Whitten provided helpful suggestions as well as the programs which were used to do the molecular orbital integration and self-consistent field calculations. Professor James F. Harrison gave help- ful advice about the molecular calculations on several occasions. Dr. John Bartelt was very helpful with the numerical analysis. The author wishes to exPress to these people his special gratitude for their generous assistance. The National Science Foundation provided a traineeship for the author from 1965 through 1969. The Atomic Energy Commission provided equipment and materials through a con— tract with Professor J. L. Dye. The Michigan State University Computer Center provided a generous grant of time on the CDC 3600 computer. iii TABLE OF CONTENTS CHAPTER I . INTRODUQION O O O O O O O O O O O O O O O O O Ammonia Solutions. . . . . . . . . . . . . Non-ammonia Solutions. . . . . . . . . . . Models . . . . . . . . . . . . . . . . . . The Problem. . . . . . . . . . . . . . . . II. ESR LINEWIDTHS OF THE MONOMER . . . . . . . . The Model. . . . . . . . . . . . . . . . . Transition Energy . . . . . . . . . . . Line Shape and Width. . . . . . . . . . Characterization of Line Shape . . . The Formalism. . . . . . . . . . . . Chemical-Physical Models . . . . . . . . . The Experiment . . . . . . . . . . . . . . Measurement of the Spectrum . . . . . . The ESR. . . . . . . . . . . . . . . Temperature Control. . . . . . . . . Field Measurement. . . . . . . . . . Klystron Frequency Measurement . . . Use of the CAT . . . . . . . . . . . Preparation and Use of the Samples. . . The Rubidium Solutions . . . . . . . The Cesium Solutions . . . . . . . . Sodium in Ethylenediamine. . . . . . Date Analysis. . . . . . . . . . . . . . . Parameter Estimation. . . . . . . . . . Reduction of Raw Data to Line Positions and Widths . . . . . . . . . . . . . Rb Data. . . . . . . . . . . . . . . Cs Data. . . . . . . . . . . . . . . Reduction to Spectrosc0pic Quantities . Results. . . . . . . . . . . . . . . . . . Qualitative Results . . . . . . . . . . The Type of Spectrum for Several Systems . . . . . . . . . . . . . Systematic Investigation of the Decomposition . . . . . . . . . . iv Page «mm» H 10 11 12 15 15 16 19 25 25 25 26 26 27 27 28 28 51 55 34 35 45 44 45 47 49 SO SO 53 m!‘ 14 A. ’Avv' Lu: TABLE OF CONTENTS--Continued CHAPTER The Flow System . . . . . . . . Quantitative Results . . . . . . . Solutions in Ethylamine-Ammonia Mixtures . . . . . . . . . . Spectra in Solid Ethylenediamine Solutions. . . . . . . . . . Discussion. . . . . . . . . . . . . . The m Dependent Linewidth . . . . Solution Stability . . . . . . . . Solid Solutions. . . . . . . . . . III. é§_INITIO STUDY OF LITHIUM AMMONIA INTER- ACTIONS C O C O O C O O O O O O O O 0 Introduction. . . . . . . . . . . . . The Ab_Initio M. 0. Technique. . . The Application. . . . . . . . . . The Methods . . . . . . . . . . . . . Construction of the Wave Function. The Basis Functions . . . . . . . Geometry of the Nuclear Framework. Analysis of the Wave Function. . . Results . . . . . . . . . . . . . . . The SCF and POpulation Analysis. . Charge Density Distributions . . . Discussion. . . . . . . . . . . . . . Energy . . . . . . . . . . . . . . POpulation Analysis. . . . . . . . Charge Densities . . . . . . . . . Density of Unpaired Electron at .Atom . . . . . . . . . . . . Charge Densities in General . . Influence of the Basis on the Charge Distribution. . . . . Implications for Monomer Models. . LITERATURE CITED . . . . . . . . . . . . . . . . APPENDIX I--A Differentially Pumped, Greaseless High Vacuum Valve and End Joint . . . . . APPENDIX II--List of Programs. . . . . . . . . . Page 57 59 59 61 78 78 80 82 85 85 85 85 88 9O 92 95 96 96 97 129 129 152 155 154 156 159 159 142 146 149 TABLE II. II. III. IV. V. LIST OF TABLES CHAPTER II Residual Width in Metal-amine Solutions. . . . A and g in Some Cs Solutions in a Mixed Ethyl— amine Ammonia Solvent. . . . . . . . . . . . . CHAPTER III Some Results of the SCF Calculations . . . . . Some Reaction Energies . . . . . . . . . . . . Total Overlap Populations Between Atom Pairs . Gross POpulation in an Orbital on an Atom. . . Total Gross POpulation on an Atom. . . . . . . vi Page 64 64 99 101 102 105 104 FIGURE 10. LIST OF FIGURES Page CHAPTER II Schematic diagram for the flow system. . . . . 65 A representative Spectral line recorded in the C—1024 . . . . . . . . . . . . . . . . . . . . 66 Cesium in ethylamine-ammonia . . . . . . . . . 67 A representative line from the hyperfine pat- tern of a Cs-ethylamine-ammonia solution . . . 69 ~An experimental and theoretical Cs-ethylamine- ammonia hyperfine pattern. . . . . . . . . . . 71 Hyperfine constant and g for two solutions of e Rb in ethylamine-ammonia mixture . . . . . . 72 The results of the linewidth analysis. . . . . 75 Linewidth of the single line spectra in sodium- ethylenediamine (en) solutions . . . . . . . . 74 Linewidth of the narrow line in sodium-ethyl- enediamine (en) "solid" solution above -90 C . 75 The upper graph shows the first order decay in the flow system with flow stopped. The lower graph shows the zero order decay of absorbance for a sample carefully prepared under high vacuum conditions in a well-degassed system. . 77 CHAPTER III The total charge density of ammonia. . . . . . 106 The total charge density of the lithium atom . 108 The total charge density of the lithium ammonia complex. . . . . . . . . . . . . . . . 110 vii LIST OF FIGURES--Continued FIGURE 4. 5. 10. 11. 12. The total charge density of the lithium di- ammonia complex. . . . . . . . . . . . . . . . The charge density difference is for the lithium ammonia complex total charge density . The charge density difference is for the total charge density of the lithium diammonia com- plex . . . . . . . . . . . . . . . . . . . . . The unpaired electron charge density of the lithium atom . . . . . . . . . . . . . . . . . The unpaired electron charge density of the lithium ammonia complex. . . . . . . . . . . . The unpaired electron charge density of the lithium diammonia complex. . . . . . . . . . . The charge density difference is for the un- paired electron density of the lithium ammonia complex. . . . . . . . . . . . . . . . . . . . The charge density difference is for the un- paired electron density of the lithium di— ammonia complex. . . . . . . . . . . . . . . . This is a comparison of the unpaired electron distribution in the lithium, the lithium am- monia complex in a minimum basis, and the lithium ammonia complex in the extended basis. APPENDIX I Construction of a greaseless high vacuum valve A high vacuum end joint. . . . . . . . . . . . viii Page 112 114 116 .118 120 122 124 125 128 147 148 1.. . t 1.4 i , I. I I e a s .3 a. s L .nh PEI. .TA R». an. a s x». a: v A . 4 .a u x: S». CHAPTER I INTRODUCTI ON Ammonia Solutions Since the first report that alkali metals dissolve in liquid ammonia (68),the nature of the solute has been an active area of investigation. The solutions have properties which range from electrolytic to metallic as the concentra- tion of dissolved metal is increased. Only solutions which are dilute enough to be non—metallic are considered here. Numerous references to the original literature can be found in several recent articles which review the nature of the physical prOpertieS of metal ammonia solutions (54,56,61,65). The solute species in infinitely dilute solutions are generally considered to be solvated electrons and metal cations. In order to account for the concentration dependent pr0perties of the solutions, several ways of combining these species to form new Species have been proposed (see 54 for references). Upon reviewing the data available in the literature, Dye (54) has concluded that the concentration dependence of the conductance, transference number, and activity coefficient data is well represented by simple ion— pairing phenomena. This is in conflict with the usual interpretations which require a substantial fraction of the total solute to be present as species of stoichiometry M- or M2 (where M represents the presence of a metal nucleus in the species). The concentration dependence of the mag- netic susceptibility and enthalpy of solution, however, require that some exothermic reaction(sL forming at least one diamagnetic species,be considered. On the basis of these results, Dye has concluded that if the experimental data are accurate, the processes which lead to the concentration dependence of the susceptibility and enthalpy must have only small effects on the electrochemical prOperties. Furthermore, he has suggested a type of interaction which might cause such results. The electron spin resonance Spectra of europium in ammonia (16) Show signals from both the solvated electron and the Eu+2 cation, even in the concentration region where extensive ion pairing is expected. Catterall and Symons have concluded that since cation-electron interactions influ- ence the solvated electron linewidth monomeric species must 2 and the be formed. However,since the spectra of both Eu+ solvated electron were observed, they concluded that the monomer must not be centro-symmetric. This led to the con- clusion that ion-pairing is the predominant mode of catione electron interaction. All of the theoretical studies of the cation-electron interaction (6,45,46,54) assume a centro-symmetric unit n. :1 5i (I) I ‘1‘ with stoichiometry M. These calculations indicate a density of unpaired electron at the metal nucleus which agrees approximately with the NMR shift data for ammonia solutions. In summary, the nature of the solute in metal ammonia solutions is still an unsolved problem, since the models which have been proposed cannot be made to agree with all of the available data. Furthermore, the mechanism of the cation electron interactions which are inferred from experimental data has not been clearly established. Non-ammonia Solutions Although metal—ammonia solutions have been studied more extensively, the alkali metals also dissolve in a variety of other solvents including polyethers, amines (15) and hexamethylphosphoramide (see 10 for references). Some mix- tures of these solvents with inert diluents also dissolve the metals. As with ammonia solutions, these solutions are electrical conductors, blue in color, and distill into metal and solvent fractions. In the details of their behavior, however, they are significantly different from metal-ammonia solutions. The limited conductance data (50) Show some of the differences particularly for sodium solutions. The opti— cal Spectra (for references see 10) in some cases are very. different from ammonia solutions. Although three bands are commonly observed for some of the metals, one of these bands has been attributed to sodium formed by reaction with Pyrex (45). Another of the absorption bands (as well as the sodium "impurity" band) has recently been attributed to a species of stoichiometry M- (51) with unknown structure. The third band is thought to be the solvated electron and electrostatic aggregates of it (25,29). Just as for am- monia solutions (54), the Optical spectra do not appear to give specific information about the nature of the species. Electron spin resonance measurements on some non-ammonia solutions give a hyperfine pattern in addition to or in place of the single line absorptflon which is observed in ammonia solutions (2,5,4,17,18,19,20,24,55,65,66). The hyperfine pattern is attributed to a contact interaction of the un- paired electron with a metal nucleus, except for lithium solutions in which it is attributed to a contact interaction with four magnetically equivalent nitrogen-14 nuclei. Because of the detailed structural information inherent in the hyperfine structure constant, the electron spin resonance Spectra of solutions with measurable concentrations of this species have been intensively investigated. The reported results for a number of different solvent-metal combinations which have the hyperfine splitting from a single metal nucleus can be summarized as follows: The A value increases and the 9 value decreases with increasing temperature at constant solvent composition. Addition of ammonia at con- stant temperature to the n-aliphatic amine solutions de- creases the A value. The linewidths for the hyperfine lines depend upon solvent, temperature, excess cation concentration, and perhaps other unknown factors. The linewidths are dependent upon the line, a result referred to as an mI de- pendent linewidth. Spectra from a few solid solutions have been reported to give a single line (66) or a hyperfine pattern (24,55). The nature of the Species in the solid has not been studied extensively, however. Models Tuttle and co—workers (5,4) postulate that the para- magnetic species with the hyperfine interaction is a monomer similar to that proposed for ammonia by Becker, Lindquist and Alder (5). The solventedependent A value is attributed to the medium-dependent volume of the unpaired electron orbital on the monomer. They assume that solvent molecules are bound in the monomer structure with a binding energy which depends on the molecule and the "solvation shell" and that the volume of the unpaired electron is related to the volume of bound solvent. They explain that the solvent dependent prOperties result from different binding energies for different types of molecules and that the marked tempera- ture dependence occurs because the binding energy is close to kT. For rubidium and cesium solutions they advocate the same model and assume that the m dependent linewidth I demonstrates the asymmetry of the Species; a result which they claim confirms their earlier conjectures about solvent binding. By an extension of earlier approximate calculations for the ammonia monomer (54), O'Reilly and Tsang (55) inter— preted the temperature dependent A value of potassium in ethylamine. The variation in A was attributed to the change in cavity radius of the monomer species. Dye and co-workers (24) have concluded that neither explanation is quantitatively applicable to their more com— plete data for potassium in ethylamine-ammonia mixtures. They interpret the variations of A value with temperature and solvent composition as the result of an equilibrium among distinct species, each of which has different magnetic properties. They were able to fit their data by assuming that a solvated metal atom, a monomer with all ethylamine in the first solvation shell and a monomer with one ammonia molecule in the first solvation shell were in equilibrium. Later, they (55) attempted to apply the same model to the m dependent linewidth observed for solutions of cesium in I ethylamine. Although the m dependence could be fit, the I electron spin resonance spectra they obtained did not have Lorentzian—shaped lines as required by their model, and their linewidth expression had the wrong temperature dependence. Two models have been proposed to explain the hyperfine pattern observed in lithium solutions. Tuttle and co-workers (5) have suggested that the species is a monomer as with the other alkali metals. The differences presumably occur because this monomer has a more expanded unpaired electron orbital and the first solvation shell exchanges with the bulk solvent more slowly. Dye (55) and Catterall gt _l. (21) have prOposed that the Species is an ion-pair between a solvated electron and a tetra-coordinated lithium cation. The magnetic equivalence of the nitrogens is obtained by the rotational motion of the lithium cation and its solva- tion shell according to this proposal. The Problem The preceding sections indicate that a number of un- solved problems still remain before a complete physical description of the solute can be given. The study of cation- electron interactions in the amine solutions which have hyperfine splitting should provide information about the nature of the solute as aggregation of cations and solvated electrons occurs. In spite of the many different effects observed, however, no model which is quantitatively applicable has been prOposed. Furthermore, the theoretical investigation of the experimental observations has not been very complete. This thesis reports an investigation of the cation- electron interactions. Both experimental and theoretical methods were used for the study. The mI dependent relaxation has been examined in order to find additional information which could be used to ascer— tain the correct model for the cation-electron interaction. The method of application of a general relaxation formalism with unspecified chemical models is outlined. The data obtained, although limited in quantity because of experimental difficulties, indicate that the relaxation formalism is applicable. Solution instability was the primary limitation on the collection of usable experimental data. A systematic investigation of the decomposition re— vealed that dilute solutions containing only small amounts of decomposition products undergo a decomposition reaction which is catalyzed by the container surface. This is a serious problem because the influence of decomposition products would probably interfere with interpretation of the data. Although the data indicated that the general relaxation formalism is applicable, they could not be used to distinguish among the various models for the cation-electron interaction. A limited investigation of solid solutions of sodium in ethylenediamine was made to determine whether the monomer could be observed. The two patterns observed by V08 (66) were reproduced. The sharp line above -900C is concluded to be due to a liquid dispersed throughout the solid due to impurity. Pure ethyl— enediamine with sodium shows a resonance characteristic of Na metal, but small quantities of ammonia change the resonance of the solid below -90°C. No explanation is given for this observation. The electronic structure of some complexes of lithium and ammonia were examined by use of 22 initio molecular orbital calculations. The resulting wave functions were analyzed by population analysis and by contour mapping. Although the calculations leave out all long range inter- actions, the results are of some significance for solution chemistry. The calculations indicate a stable complex of the stoichiometry LiNHs could be formed. Analysis of the charge distribution shows that a net charge transfer from ammonia §Q_lithium occurs and that the unpaired electron avoids the region of space near the ammonia. A substantial contact interaction at nitrogen is predicted, but this represents a peak in the unpaired electron density at nitrogen rather than a large charge transfer into the nitrogen orbitals. Calculations for Li(NH3)2 were made and analyzed. The results are compared to the results for LiNH3 and found to be very similar. CHAPTER II ESR LINEWIDTHS OF THE MONOMER In this chapter the characteristics of some electron Spin resonance (ESR) patterns from solutions of alkali metals dissolved in amines are examined. The study was undertaken to obtain a better understanding of the unusual effects of the environmental parameters, solvent and temperature, on the Spectroscopic parameters, hyperfine Splitting constant, g-factor, and linewidth. On the basis of previous work, other investigators (5,4,17,18,19,20,24,55,55) had suggested explanations for the observed effects of varying the environmental parameters. However, no one has given a description of the phenomena which fits all of the results. The multi-state model (19, 20.24.55) (a special case of a distribution model) is the most promising model proposed to date for quantitative purposes. The author intended at the start to collect quantitative data under conditions which would permit further testing of the prOposed models. If the proposed models proved unsatiSfactory,an improved model was to be sought. .The conditions required to obtain data appropriate for testing the models proved to be extremely difficult to meet. 10 11 In particular, decomposition was found to be rapid for dilute metal-amine solutions placed in containers small enough to use in the ESR spectrometer cavity. Since the study of dilute solutions was required to prevent complications caused by solute-solute interactions, only a few solutions provided information which could be used to test models. One dilute solution of rubidium and one of cesium in a mixed solvent of ethylamine and ammonia were examined in detail over a limited temperature range. The data indicate that the lines are of Lorentzian-Shape, contrary to a previous report in the liter- ature (55); thus the relaxation can probably be described by use of the relaxation formalism as adapted for ESR spectra by Freed and Fraenkel (57). This relaxation formulation of the two state model is capable of fitting an isothermal Cs solution Spectrum. Insufficient experimental data were available to adequately test this model. Sodium in solid ethylenediamine was examined in some detail because two different Spectral lines had been reported for this system (66). The results indicate that one of the lines probably results from the presence of a multicomponent liquid solution dispersed throughout the solid giving rise to one of the observed patterns. The Model Chemical information is derived from spectroscopic observations by means of an interpretative model. An electron 12 Spin resonance (ESR) experiment is usually interpreted by using a mathematical model based upon the formalism of the spin Hamiltonian (59). The spin Hamiltonian operator for a system contains the spin variables explicitly and all non— Spin variables as parameters which are chosen either to give the best comparison of the experimental observations to the formal predictions or to satisfy some other, more funda- mental experiment. A time independent spin Hamiltonian leads to stationary energy states for the Spin system which, with the transition selection rules, yields a theoretical pre- diction of the spectral line positions. A more detailed knowledge of the processes which govern the transition probabilities is needed to predict the line shapes. The time dependent processes usually responsible for the transi- tions are treated by time dependent perturbation theory. Formal solutions of both the time dependent and independent Hamiltonians have been presented by other investigators. These formal solutions along with their inherent assumptions will be quoted from their work and used. Transition Energy The transition energy is constructed from a spin Hamiltonian which describes the coupling of the external field to the magnetic moments of the system and the inter4 moment magnetic coupling in the system. Only two types of magnetic interactionsare important in the metal amine solu- tions as observed in the ESR transition energies. 15 The first type of resonance shows no resolvable structure (17,18,65,66) thus requiring coupling of the electron mag- netic moment to the external field only. The second type of resonance is a multiplet (2,5,4,17,18,19,20,24,55,65, 66) which can be interpreted as resulting from hyperfine interaction of the unpaired electron with one or more nuclei, thus requiring coupling of the electron moment to one or more nuclear moments. The Single line is characterized by the Spin Hamiltonian (1) H = -g a H-E where: g is the electronic 9 factor for the system, B is the Bohr magneton, S'is the electron spin operator, and H is the external magnetic field. This Hamiltonian for a spin of one-half leads to the transition energy given by (2) AE =hy= 9 8H where: AB is the transition energy, 7 is the frequency of the radiation for the transition, and h is Planck‘s constant. The Hamiltonian for the multiplet spectrum is written (5) H = -gBH-S +3.2 gNflNl-I I. J+j§1 AjIj S all nuclei all nuclei where: is the nuclear 9 factor, 6N is the nuclear magneton, gN Ij is the spin operator for the j-th nucleus, Aj is the hyper- fine coupling constant for the j-th nucleus, and all other 14 symbols are as before. ‘Both Aj.and g are treated as scalar constants rather than the more general tensor constants because in the amine solutions only the isotrOpic (scalar) part can be observed. The energy levels described by the Hamiltonian 5 with one unpaired electron (S = i) and one nucleus (arbitrary I) are given by (9) _ _A_W___ EI A_w (4)3(F'mF) — -2(21+1) I Hm1?i 2 { v1 + __2mFx + X2 } (I +.5) with: + for F = I + i, - for F = I - fi'. AW = A(21 + 0/2. x = HO(gB + gNBN) /A(21 +1). in which Fl= I'+ S'is the total angular momentum, mF — mI + mS is the Z projection of the total angular momentum, “I is the nuclear moment, and all other symbols are as before. The selection rule for absorption is AmF = 1. Then, the expression for the ESR transition energy is P D 2m X 'r — — _ .1. éfl. __E_.. 2 ‘2 (5) AE(AITlF - 1) _ I HO + 2 ([1 +(I+05) + X] L 2% X 1 2 15 + [1 +(I+.5) + X 1 I where: m? refers to the upper state, m: to the lower state, and the other symbols are as before. The unknown quantities in equation 5 are A and the electronic 9 value. Even though measurement of the transi- tion energy and field of any two transitions is sufficient to permit a solution of the expression for A and 9/ better 15 values are obtained by measuring more transitions and fit- ting equation 5 to all of the measurements using a least- squares method. The Sign of A is not given by 5 (the equation is written for a + IA I) since changes in Sign on various terms accompany a change in sign on IA Ito -IA I so that the transition energy expression is the same. Note that equation 5 is non-linear in A and 9 so an iterative procedure must be used to solve for them. A function which satisfies equation 5 for each eigen- value is needed for the relaxation problem to be considered later. These functions are the eigenvectors of the matrix constructed using equation 5 and a suitable complete basis. Line ghape and Width The shape and width of spectral lines are often de- termined by the dynamic interactions of the energy absorbers (or emitters) being observed. Following definitions of shape and width, the formal details of investigating a class of dynamic mechanisms using line shape information will be given. Characterization of Line Shape (58). A compact descrip- tion of the distinguishing features of a spectral line represents the desired characterization of it. This descrip- tion might be a table of constants such as the moments of the line or it might be the Specification of a mathematical form (called a shape function) and the values of certain constants in this form. Because of its compactness, the 16 latter method is preferred if it is possible. The linewidth is the value of a particular constant in a shape function. Two of the most frequently occurring shape functions for symmetrical lines are the Gaussian and the Lorentzian functions. The Gaussian function is (6) g(H-Ho) = Cnexp {-°695[(H'Ho)/%AH;j2} where: C is a normalization constant, H0 is the field value at the line center, H is the field value of observation, and Aflfi is the full width at one-half maximum height. The Lorentzian function is (7) £(H-Ho) = C/(l. + ((H—HO)/§AH%)2) with the same symbol definitions as before. Most ESR spectra are recorded as the first derivative of the absorption so that the first derivative with respect to H of the given forms should be compared to the recorded Spectra. The Formalism (1,37). The equations presented are the result of a second order, time dependent perturbation treat- ment of a random stationary process with the restriction that [132<<1° where: H; is the perturbation magnitude expressed as a fre- quency, TC is a characteristic time for the interaction, and < > symbolizes a statistical average. The working equa- tions are summarized as follows: 17 00 (a) IN) = ‘73; of G(t)cos(wt) dt where: I(w) is the normalized frequency distribution of power absorption and, (9) G(t) = tr[ sx(t)-sx(o)] where: tr is the trace Operator, Sx(t) is the x component of Spin magnetization at time, t, SX(O) is the x component of Spin magnetization at time 0. (10) -Sx(t) = exp{iHot} S:(t)eXp{-iHOt] 1 where: H0 is the time independent Hamiltonian, Sx(t) iS the x component of magnetization in the interaction representation. (11) %E'< A Is:(t)IB > = C2 < c Is:(t)ID > I R D ABCD where C and D are states chosen to satisfy (12) E - E = E - E where E is the energy of a A B C D’ state, (15) RABCD = 2 JACBD(WAC) ’5: (éBDqAFCF(WFC) + éACJFBFD(WbF)) All states where A, B, etc., symbolize the states of the system. Finally, (I) . (2L4) JABCDW) = 5-ch < A IH1(t)I B >< CI H1(t+T)l D) e lWTdT 18 where H1(t) is the time dependent portion of the Hamiltonian for the system. To use these formulas one must know both the form of the time dependent Hamiltonian and the zero order states. This information is used in the function given by 14. By substitution in the various equations, I(w) is obtained. Freed and Fraenkel (57) indicated that this formalism predicts a Lorentzian Shape for each transition and that non-Lorentzian Shapes are due to SUperposition of lines from (almost) degenerate transitions with either different widths or different centers. It is important, however, to note that this theory predicts Lorentzian lines unless two or more transitions in the spectrum are degenerate. Recently, work by Freed (58) indicates how unsymmetrical lines can occur due to terms from a higher order time- dependent theory. The more detailed theory is not needed for this work. The special case of hyperfine splitting from a single nucleus is especially important for the metal—amine solu- tionS. .For hyperfine Splitting much larger than the line- width, equation 11 becomes (due to 12) d 1: _ 1 (15) 3—1:- < A ISx(t)IB > - RABAB< A ISx(t)IB >. For this case, Lorentzian shaped lines centered at EA-EB with half-width at half-height of HA are predicted. BAB 19 Chemical-Physical Models Use of the formalism requires knowledge of the chemical or physical behavior causing the time dependence of the spectroscopic system. Some possible mechanisms are reviewed in this section. Formulas are presented which aid in testing some of the suggestions for the nuclear magnetic quantum number (mI) dependent width of the ESR absorption. For solu— tions of cesium and rudibium in ethylamine Tuttle and co- workers (4) have suggested that the observed linewidth de— pendence on m is due to the low symmetry of the species I causing anisotropy of the A and g tensors. The anisotropic Species is assumed to tumble in solution which results in the observed mI dependence by the mechanism of McConnell (48, 52, 71). No quantitative comparison of the model to the eXperi— mental results was made by them to confirm their hypothesis. This model will be discussed later even though no quantita- tive test of it has been made in this work. To interpret the observed variations of A values as a function of temperature and solvent composition for potassium in a binary solvent of ethylamine and ammonia, Dye and co- workers (24) considered an equilibrium model with two or more Species in (rapid) equilibrium causing a complete averaging of the A value. They were able to fit the A value variations. The potassium system exhibits very little mI dependence of the linewidth; therefore, it does not permit a test of the linewidth predictions of this mechanism. 20 Cesium in ethylamine, however, does exhibit sufficient mI dependence of the linewidth to permit comparison with the relaxation predictions of the model. They applied the model to some cesium-ethylamine solutions (55) but did not achieve complete agreement with all of the observations. Other authors (19,20) have suggested this model but made no quanti— tative comparisons of the model with experiment. There iS the possibility that both the anisotropic rotor and the isotropic modulation of the equilibrium model are present. Other mechanisms also exist which could give the mI dependence or contribute to it. The necessary at- tribute of all such mechanisms is time dependence of the A and/or 9 tensors. One further example of such a process is weak exchange coupling (47). -At this point the method of application of the two state model as a Special case will be presented using the previous formalism. It is necessary to assume that the processes which govern the residual contributions to the linewidth have Hamiltonians which commute with the Hamiltonians of the mI dependent process. If this assumption iS not applic— able, then coupling terms for the processes must be con- sidered. Since nothing is known of the residual linewidth mechanisms, this assumption will be used. The time dependent Hamiltonian and the zero order states of the system are needed to calculate the m dependent line- I width. The states of the system are the eigenvectors of the 21 Breit-Rabi Hamiltonian. The time dependent Hamiltonian is assumed to be (16) H(t) = g(t)f3 RE + A(t) 5-5 where: g(t) is the instantaneous deviation of the 9 value from its time averaged value, and A(t) iS the similar quantity for A. By definition, then, these quantities average over time to zero. The first step in evaluating the linewidth is to con- struct the integral, J, of equation 14 using 16 and the zero order states of the system. This expression is _ _ _ GD _. (17) JABCDW) = 62 «IR-'81 D>CI g(t)9(t+T)e ”Tar T + s { < AIH°§ I13 > foog(t)A(t+T)e-iw dT ‘ O T + < A If~§ I 13> foo A(t)g(t+T)e-iw d'r )7 O + < A I'f-‘S’ IB> fOOA(t)A(t+T)e—iWTdT. o with < > indicating expectations, and the bar in the time integral indicating an ensemble average. One must now define more Specifically the nature of the time variation of A and 9 so that the correlation functions of these quantities can be calculated. The four correlation integrals are solved (following Abragam (1), pp. 270-1) in this case by assuming that the correlation is of two random stationary quantities, fa(Y1) and fb(Yg), which are functions \ 22 of time through their dependence on Y1(t) and Y2(t) which are random functions of time. Then one calculates the time correlation of the two functions of interest as (18) GAb(’U) = fA(t)fb(t+T)'-' ff P(Y1)P(Y1'Y2'T)fA(Yl)fb*(Y2)dY1dY2 where: P(Y1) is the probability that the value Y occurs, I P(Y1,Y2,T) iS the probability that the value Y2 occurs at T after the value Y; is known to occur, fA and fB are two func— tions of Y for which the correlation iS desired? For the two state problem, the functions of the random variable are 6A and Ag, the change of A and g from the average values. (The two state model Simply assumes that the paramagnetic Species 'can exist in two distinct forms which have different para- magnetic observables. There is assumed to be an equilibrium between the states which follows the laws of thermodynamics.) For this model Freed and Fraenkel (57) have Shown that (19a) A(t) A(t+T) P PII(AI—A )2 exp{ -T/TC,A ) I II By analogous arguments (19b) g(t) g(t+T) = P )2 exp{-T/TC } IPII(91'9II ,g and (19c) g(t) A(t+1) A(t)g(t+T) = P PII(AI-A I II)(gI_gII) eXp[_T/TC'gA} One should also note that using this formalism, th§_mean value of a derived distribution would be given by f = f P(Y1)f (Y1)dY1. This is in fact the result that Dye gt _l. Y1 (24,53) have used to interpret the A value variations in the solutions of potassium in ethylamine-ammonia mixtures. 25 where PI and P are the fractional populations of states I II and II, AI and AII are the A values of states I and II, gI and gII are the g values of states I and II, and TC S is a correlation time for the process S. By the nature of this model, one expects both A and g to change simultaneously so their correlation times will be the same. The first two functions are positive but the third may be negative if the large value of A goes with the small value of g and gigg- versa. The third term can be considered as (19d) <69A> =-¢kfiA2> x (correlation coefficient) Using these results, equation 17 becomes (20) JABCD = [82 <<5gé> + B [ + ] ° <6Ag> + <6A2>] 1/1C _ I (ma) 2 + w2 i W 5] (1/1'C)2 + w2 ’ For each occurrence of a J in equation 15, the form given by equation 20 can be inserted. The spin integrals are easily evaluated with the known zero order functions, but the fluctuation terms (e.g., <6A2>;) and the correlation times are not known at this time. If no independent determination 24 of these quantities is available, one can adjust them as parameters to give a least squares fit of the Spectrum. Anticipating some later results, the last multiplicative term will be discussed here because it will need to be Simplified. The imaginary term contributes a line shift which is small; therefore, it will be neglected. The value of the real term depends on the value of w. For the J's used in the expansion of (equation 15), there are RABAB two types of terms. In one type of term, the w magnitude is comparable to 1/TC, but these terms contribute little because the Spin matrix elements multiplying it are small. In the second type of term w is zero; therefore, the term Simplifies to Tc' The terms with non-zero w are the key to an inde- pendent evaluation of TC. Otherwise, only products of Tc and the fluctuation quantities occur. The temperature dependence of the product of one of the fluctuation terms times a correlation time is given by (use A for example) (21) (TC 6A2 > = constant exp[ (-2AF + EB)/RT} [1 - exp{-AF/RT H3 where the symbols have been'previously defined. The thermo- dynamic parameters would be the same as those used to fit the A and 9 value temperature dependence. 25 The Experiment Since the test of the model is based on linewidths and line shapes, the experiment must allow accurate measurement of these. The solutions must be dilute enough in metal to avoid observable concentration dependent solute-solute inter- actions and effects due to changes or inhomogenities in the environment of the sample must be negligible. Due to the relative instability of the solutions, the low relative concentrations of paramagnetic material, and the sensitivity of the observables to environmental parameters, good experi- mental data for the system are difficult to obtain. This section presents the methods used in this investigation. Measurement of the Spectrum The ESR. A Varian V-4500 Spectrometer with 100KHz field modulation, a Varian 12 inch magnet and a TE—102 rectangular cavity operating at 9.2 to 9.5 GHz were used. Field sweep was controlled by a V-4280A precision scanning unit or by a Helmholtz coil of 4 inches diameter which was driven by a Wavetek model 116 multipurpose VCG wave—form generator. The field produced by the Helmholtz sweep coil was checked for inhomogenity and non-linear sweep using a TCNE radical anion solution with linewidths of about 150 milli— gauSS. Over the sweep range used for the amine Spectra, no undesirable effects were observed. After the large magnet 26 was allowed to warm to operating temperature (at least one hour), it was observed to drift about 5 milligauss per minute. Temperature Control. A special apparatus, Similar to the standard equipment but with some differences, was used to control the temperature of the metal-amine solutions. This apparatus has 5 parts: a heat exchanger, a transfer tube, and a sample holder. The heat exchanger is a flat coil of one-quarter inch c0pper tubing with inlet and outlet through vacuum insulated glass tubes oriented with their axes perpendicular to the plane of the coil; an arrangement which permits a constant heat exchange length regardless of coolant level. The vacuum insulated transfer tube holds the heater-sensor of a V-454O Varian temperature controller, and the sample holder iS a vacuum insulated fused quartz Dewar of 11 mm .od; which passes through the cavity. When the flow rate of dry nitrogen was maintained constant through this controller, a thermocouple at the sample indicated that the temperature was constant to within i.O.1OC. Field Measurement. The magnetic field was measured with a proton gauss meter described by Buss and Bogart (15,49). The proton magnetic resonance (PMR) was detected by using a 400 Hz field modulation and phase sensitive detector. The PMR signal was displayed on an oscillosc0pe and used to mark- the ESR Spectrum as it was recorded in a first derivative mode. The difference of field inside and outside of the 27 cavity was measured to be about 0.15 gauss. The frequency of the PMR was measured using a Hewlet-Packard Model 524C frequency counter with a model 525A frequency converter to permit the approximately 15 MHz Signal to be measured. Klystron Frequency Measurement. The klystron frequency was measured with a calibrated wavemeter or by comparison to a harmonic from a Micronow calibrater as described by Kuska (49). This frequency measurement Should be accurate to 0.1 MHz. Use of the CAT. Some Spectra were recorded in a Varian computer of averaged transients (CAT), C-1024. When the CAT was used the field was swept by the Helmholtz coil (see above). The C-1024 Sweep was triggered by a pulse from the Wavetek wave form generator so that the sweep of the field and of the CAT were always started synchronously. The ESR spectrum was stored in the first half and the magnetic field markers (proton resonance signals recorded directly) were stored in the second half of the CAT memory. The forward and reverse scan (increasing and decreasing of field with time) of the Spectra were recorded so that differences in electronic delay time for the ESR and proton Signals could be measured and the field measurement corrected. The spectra were recorded in a time period Short enough so that the drift of the large external field was a negligible perturbation on the spectra (see above). After the Spectra were recorded in the CAT, a graphic permanent display was produced by 28 recording the contents of the CAT memory on an X—Y recorder. In this recording the ESR and field markers were displayed with their scales overlayed by using the overlay feature of the C-1024 (see Figure 2). The ESR spectrum was then punched on 80 column cards for further processing by a digital computer (CDC 5600). Preparation and Use 9; the Samples The Rubidium Solutions. These solutions were prepared in a Pyrex vacuum system with all Teflon valves and joints. The sample tube and the associated tube from which the rubidium (Rb) metal was distilled were made of fused quartz. The glassware was rinsed with a cleaning solution of 2.5% HF, 1% acid soluble detergent, 55% nitric acid, and the remainder distilled water. Rinsing with the cleaning solu- tion was followed by rinsing with distilled water Six to ten times. If the glassware would later be in contact with a metal solution, it was further treated with hot aqua regia, again rinsed with distilled water Six to ten times, and then, rinsed three times more with triply distilled water. The vacuum line was evacuated using a Consolidated Vacuum Corporation MCF-60 three stage oil diffusion pump with a Cenco "High Vac 7" used as a backpump. The pumps were separated from the working section of the vacuum line by a liquid nitrogen cooled glass trap. A Dow Corning 29 Silicone oil was used in the pump. An ion gauge was at- tached near the cold trap on the manifold side of it. A two centimeter diameter tube acted as the manifold to which the various equipment for the solution preparation was at— tached by means of 5 mm Teflon endjoints made by Fisher- Porter. The solutions were to be composed of a binary solvent of ethylamine and ammonia. In order to make a known solvent composition, the following procedure was employed. A solvent reservoir containing a pure liquid solvent and a glass bulb with a valve were attached to the vacuum manifold. The glass bulb was immersed in a constant temperature bath. To prepare a measure of solvent, the solvent reservoir was brought to a known, reproducible temperature by using a Slush of a pure liquid in equilibrium with its solid. The amine or ammonia vapors were then allowed to equilibrate with the constant temperature bulb. The bulb was closed after the system reached equilibrium, trapping a repro- ducible amount of solvent in the bulb as vapor. The glass bulbs used for different solvents were calibrated by con- densing the vapors in a small glass tube and weighing them. The ethylamine bulb gave 0.01027 moles and the ammonia bulb gave 0.00101 moles so that a sample with one measure from each would be 0.91 mole fraction ethylamine and 0.09 mole, fraction ammonia. The standard error from replicate measures of these numbers was less than 0.005 mole fraction. 50 The ethylamine reservoir was equilibrated by using a water Slush and the ammonia reservoir by using a 1,2—dichloroethane slush. The gas bulbs were kept near 250C with an absolute control of.i 0.050C. Before the metal solution was prepared, the container for it was rinsed with ammonia as follows. The vacuum system was evacuated to at least 5 x 10"6 torr. The vacuum line had attached to it an extra vessel which contained a solution of an alkali metal in ammonia. With the pump closed from the line, some of the ammonia was distilled into the sample tube using a dry ice-propanol bath. Then the ammonia was removed to its original container by distilla- tion. The metal ammonia solution was frozen, pumped on high vacuum, and thawed. This procedure was repeated ten times. After the rinsing with ammonia, the alkali metal to be used to make the solution was distilled to form a thin film on the tube above the region in which the solution was to be contained. Then a portion of ethylamine was measured in the gas bulb and transferred to the sample tube by cooling the sample tube. The metal was introduced into the solution at this time by condensing some of the ethylamine on the metal and letting a few drops of solution run down into the con- tainer. Then a portion of ammonia was measured and trans- ferred into the container by condensing it into the container. The solution was frozen and the system was opened to the high vacuum pump. The sample was sealed by fusion of the 51 container above the solution and removed from the vacuum line. The tubes were sealed so that at room temperature only about 10-20% of the volume of the tube was not filled with solution. This procedure gave solutions which were not saturated with metal. (Most of the solutions were not very stable but two were stable enough to provide some ESR Spectra. The original ethylamine (Eastman anhydrous) was probably not ammonia-free because it was dried over sodium-potassium alloy before use. No VPC analysis of this solvent was made. The ammonia (Matheson anhydrous) was dried over alkali metal before use. The samples were stored in liquid nitrogen until used. The ESR Spectra were recorded as described above, by using-the C-1024 technique. During the recording of the spectra care was exercised to avoid overmodulation, saturation, and effects of field drift and sweep Speed distortion. The Cesium Solutions. No fresh solutions of Cesium (Cs) which were both dilute enough for use and stable enough to measure could be prepared in containers small enough to fit into the ESR cavity. Because of this instability of dilute solutions in small containers, a flow system was built to provide a steady source of the CS solution. Only one success- ful run was made with: this flow system, which was intended 52 to take advantage of the fact that at low temperatures and in large vessels, solutions could be made which were stable for long periods of time (months at least). The flow system had a large reservoir in which the solution could be prepared and kept. This reservoir was connected to a transfer tube which went through the cavity to another storage reservoir. The first reservoir which contained the solution was kept at ~780C in a dry ice- alcohol bath. ‘An overpressure of about two atmospheres of He was kept on the solution to cause it to flow. The solu- tion transfer tube waS enclosed in a Dewar into which temperature controlled gas (obtained as described previously) was blown. The transfer tube held the solution long enough for it to reach thermal equilibrium with the gas. A triple walled Dewar which was of fused quartz and evacuated between the outer two walls passed through the cavity. The solution flowed through the inner tube and gas which had previously reached thermal equilibrium with the solution flowed between the inner two walls. A thermocouple could be placed in the intermediate region to measure the temperature of the gas. Matheson ethylamine (stated purity, 98.5%) was freeze purified prior to use. _After freeze purification, no am- monia or methylamine could be detected by VPC using a column of 15% tetrahydroxyethylethylenediamine and 5% tetraethylene— pentamine on chromosorb W 60/80 mesh (64). Before preparation 55 of the final sample, however, the solvent had been allowed to stand over Rb metal for a few days and then over a CS metal for a few days at room temperature. No blue solution could be obtained over the Rb metal, but with the Cs metal a blue solution formed very slowly (over a period of days). After the blue CS solution formed, a sample of the vapor over the solution revealed ammonia (the exact amount was not determined but the VPC peak heights were in a ratio of about 1:25, NHagEtNHg). An attempt was made to prepare a Cs solu— tion at -780C from this solvent. The solution could not be made concentrated enough to use. In order to increase the solubility of the metal, about 5 volume percent at -780C of 99.99% ammonia (J. T. Baker Co.) was added to the ethyl— amine. After the ammonia was added a solution of sufficient concentration of metal to study could be obtained. About 1.5 liters of solution were in the bulk vessel.. An optical sample was taken from this bulk vessel and then the flow system was assembled and evacuated. The ESR Spectra were taken with the solution flowing in the flow system. These Spectra were recorded using an X-Y recorder. AS for the Rb Spectra, the spectrometer was Operated to avoid saturation, overmodulation, and sweep distortion. The Spectra were recorded by scanning without markers. Then one scan was field marked. Then the mI = -1.5 line was expanded and field marked to give a measure of the absolute width. Sodium in Ethylenediamine. Spectra of sodium (Na) in ethylenediamine (en) (27,40) were recorded for solutions 54 below 0°C. The en is a solid at these temperatures. Two preparations of en were used. The first had been freeze purified and then had been allowed to stand over potassium metal for a few months. There had been significant decomposition during storage as judged by the precipitate in the solvent vessel. Before use the solvent was degassed by repeated freeze-pump-thaw cycles. The second preparation of en was also freeze purified. However, in this case the en was not allowed to stand for an extended period of time with metal. The liquid was distilled over potassium metal and the resulting solution allowed to remain overnight. Then the solvent was removed from the potassium by distillation to a clean storage vessel. The en was degassed and then kept in the storage vessel until it was used a few days later. The ammonia used in the mixed solvents was 99.99% anhydrous ammonia from J. T. Baker Co. Data Analysis The data analysis to be discussed in this section is the numerical analysis used to find the parameters which give the best fit of the zero order Hamiltonian (equation 5) solutions (equation 5) to the observed transition energies and fields of the Spectrum, and to find the parameters which give the best linewidth constants (equation 15) to the ob- served linewidths of a Spectrum. The important questions, 55 how to obtain the best estimate of the parameters and how to obtain an estimate of the uncertainty in the knowledge of these values will be briefly reviewed. After a brief eXplanation of the methods used to obtain parameter esti— mates and uncertainties, the procedure used to analyze the data from the hyperfine splitting pattern will be explained. Parameter Estimation The estimation of parameters in equations which are presumed to describe observations can be done in several ways. The method chosen for this work is to apply a least squares criterion, i.e., to take as "best" estimates of a set of parameters those which minimize a functional, ¢, of the form (22) ¢ = Z wi(Res(i))2 where W1 is the weight at the i-th observation, ReS(i) is the residual at the i-th observation. Two important questions which arise are: 1) How does one Specify the weight? 2) How does one formulate the residual (Res(i))? These questions have been considered by Wentworth (67) for non-linear parameter estimation (also applicable for problems linear in the parameters) using a method due to Deming (28). The summary given here will parallel the work of Wentworth. For data from samples described by independent gaussian distributions of errors with individual variances, a maximum likelihood parameter +4 I 5 d z. o. si (I) 56 estimation leads to minimizing 22 with wi defined as prOpor- tional to the inverse of the variance for the distribution from which the i-th point was taken. Since it is not pos- sible in general to consider all error to reside in a given one of the observables, however, one minimizes the functional 2 Z wi. Rig observations j = variables 3 J (23) “I 1 where wij is the weight of the i-th observation j-th variable Rij is the difference between the measured and calcu— lated value of the j-th variable using data from the i-th observation, with the constraint that the function (F) which relates the parameters sought and the observed variables is zero at each point, i.e., F (parameters, observables) = 0. This formulation treats each of the variables symmetrically so that it is not necessary to assume one or more of the variables to be known exactly. This form gives a maximum likelihood estimate if each of the Rij is from an independent distribution with variance o‘.2 = 1/wi Wentworth, using ij j° the method of Deming, with the above assumptions Shows how to derive the normal equations used to obtain corrections to a given set of parameter estimates. The result is (24) 55:5 where = observations (25) ij= 2 235 351: . 1 K=1 ou. bu L 57 0’ 2 variables OF X K (26) LK = 2 (5—5) 7%.“:— m=1 Kn O __ observations FK OFK (27) (c)i = 2 L— Ou K=1 K i FK is the value Of the constitutive equation with the present parameter estimates at the K-th observation XE is the m—th observable ui is the i-th parameter LK is the inverse Of the total weight at the K-th observation as given by 26 2 cng is the variance of the xm-th variable at the K-th observation 662 is the variance of unit weight (U)i is the negative of the i-th parameter correction. An Operational procedure to make use of these equations is as follows: Using initial estimates of the parameters, solve for the corrections to the parameters using equation 24. Then form the corrected parameters using the elements of U and iterate the above procedure until convergence is Obtained. Before considering the question of convergence, a more powerful method to Obtain a non-divergent solution to 24 will be considered. A divergent solution is one which on successive iterations of parameter corrections gives a 58 sequence of successively larger sums of squares of residuals rather than the desired decreasing sequence. The sum of squares of residuals being considered is (67) Observations (28)‘ s = i=1 Fi/LK The Simple cycling procedure Often diverges when the initial estimates of the parameters are very different from the best estimates or when the sum of squares of residuals is very non-linear in the parameters. Wentworth suggests a method for overcoming this convergence difficulty but in this work another method has been found much more efficient. A damped least squares method which has been discussed and compared to other methods by Pitha and Jones (57) was used. It is a modification of a procedure which is designed to Simul- taneously minimize the step length, 53 and the functional S. The iteration step of the method is u w I w E. o (29) {I where: A is a matrix which has diagonal elements equal to those of E and zero Off diagonal elements, P is an adjustable constant which is chosen to cause S to be a minimum (Optimized at each iteration). A quadratic search procedure on P has been found to be very efficient. Two basic questions about convergence must now be con- sidered. The first is how to tell when a minimum of S has been found and the second is how to establish that it is the 59 lowest minimum Of S. The first question is the simpler of the two and will be considered first. The most direct test for a minimum is to examine the first derivative of S with respect to each Of the parameters to see whether it is zero. (Actually a second derivative test is necessary to establish a minimum but this will not be examined.) Equation 27 is prOportional to this derivative if it is assumed that the weighting function, L is independent Of the parameters. KI Actually, LK is not independent of the parameters, but its dependence is much less than that of the F so that equation K 27 is a good approximation to the quantity of interest. Consider the relation of the parameter corrections to the elements of the vector given by 27. Equation 24 shows that the parameter corrections are given by a matrix product of E and of the inverse of E. This means that if the elements Of C are zero then the parameter corrections will also be zero. However, it must be pointed out that the parameter corrections can be small from other causes such as large elements in 5. Nevertheless, one way to test approximately for convergence is to examine the magnitude of the parameter corrections. If they are below some predetermined accept- able value, then the iteration is terminated. The point that this is an approximate minimum test iS important and implies that the predetermined level for termination should be Significantly smaller than the required accuracy of the parameters. This test is used because it is efficient since 40 the quantities needed have already been calculated, but some more refined techniques Should be used until experience has been gained with any given type of problem. Another technique which can be used in conjunction with that just discussed is to start with different sets of initial guesses for the parameters to see whether the same parameter values within the needed tolerance are given at convergence. If the different initial parameter estimates give the same result, it is likely that a minimum has been found. When a minimum has been found, one must ask whether it is the lowest minimum in the functional being minimized. There is no general method known to the author to be sure that a minimum is the lowest or "global" minimum. However, certain considerations can help to solve the problem in many cases Of practical importance. An investigator usually knows approximate ranges within which the parameters must lie. If unusual values of the parameters are Obtained one might suspect that the minimum of interest has not been found. In other cases, the investigator knows approximately what final value of S is expected. If the value obtained is larger one might suSpect the possibility Of convergence to a "local" minimum. In such cases a different minimum may be found if significantly different initial guesses for the parameters are employed. For many problems, however, one can never Show that convergence to the lowest minimum has been Obtained. 41 When "best" parameters have been obtained, there are some Significant points which must be considered about the meaning of the parameters. It is important to know how error in the data effects the values of the parameters. It is also important to know whether the range of the data is sufficient to Specify the parameters to within the desired accuracy, and if so, how well the data determined the para- meters. Finally, it is important to know how an error in one parameter estimate is likely to effect the estimated values of the other parameters. These important points have been studied and are understood for least squares estimates using mathematical models which are linear in the para- meters. For models which are non-linear in the parameters the situation has been studied but the answers are much more restricted. Briefly, the answers may be summarized as follows. For the non—linear estimation problem the inverse of the matrix 5 (equation 24) is proportional to the approxi— mate variance-covariance matrix, just as the inverse of the coefficient matrix Of the normal equations is prOportional to the variance—covariance matrix for the parameters of the linear estimation problem. In order to find the approxi- mate variance-covariance matrix the proportionality constant which multiplies 3—1 must be found. The prOper value of this constant is the magnitude of the variance of unit weight. Usually this constant is unknown, but Deming points out that an approximation to it is given by the sum of Squares Of the 42 residuals divided by the number of points minus the number of parameters. (This divisor is Often called the degrees of freedom of the problem.) The normalized inverse of E is only approximately the variance—covariance matrix since it is based on equations linearized in the parameter corrections and these equations only approximately describe the functional, 8, and the normalization constant is also only approximate. AS the name suggests, the variance-covariance matrix for the parameters contains the desired information about the distributions on the parameters. The diagonal elements are the variances for the corresponding parameters, and the off diagonal elements contain the covariance or the corre- lation coefficients for the parameters. Other statistical quantities related to these may be calculated from the variance-covariance matrix elements but the derived quanti— ties will not be discussed here. The above information answers most of the questions raised except the point alluded to when questioning how errors in the data affect the para— meters. The Obvious influence of uncertain data is that the variance of the parameters goes up as the data become more uncertain, but there iS another more subtle influence possible. In a non-linear problem there is no certainty that the parameter estimates are unbiased, i.e., they may not converge to the true value as the number Of data points iS increased, holding the variance of the data constant. This problem can be overcome by having better data since 45 this results in a distribution with smaller variance for the parameters which tends to make any bias less important. The above techniques have been used in the analysis of the data presented later. The variances on the parameters indicated were derived as indicated above. A program to carry out the manipulations and print the results was coded in FORTRAN for a CDC 5600 computer. The CDC 5600 carries a 48 bit word (the equivalent of 10—11 decimal digits in floating point form) to represent each number. In summary, the constants in the equations which are presumed to describe the data have been adjusted using the method Of generalized, weighted least squares analysis. The outputs of the analysis are estimates for the parameters, estimates for the variances (or standard error) on the para— meters, and estimates of the correlations among the para- meters. Reduction Of Raw Data to Line Positions and Widths The data which are obtained from the ESR are not suit— able for direct application of the mathematical models of the Spectroscopic phenomena. Since the data for the Rb solu- tionS and for the CS solutions were in somewhat different physical forms, they needed different treatments to yield data which are suitable for studying the model. Therefore, the reduction to widths and positions of the different data will be discussed separately. 44 Rb Data. The Rb data were Obtained using the rapid scan method outlined in the experimental section. Therefore, the data were recorded in the CAT and then from it recorded in two forms: on chart paper (see Figure 1) with the field markers overlaying the ESR lines, and on punched cards. A spectrum, using this technique, is obtained region by region where a region means a portion of the spectrum with- out Significant parts which are not of interest. The Rb spectra were recorded one line at a time with a scan width sufficient to Obtain the outer portions of the line. In the cases where Rb85 and Rb87 lines overlap,both lines were recorded Simultaneously. The widths and positions of the lines are such that the lines from a given isotOpe are well separated in these solutions. Thus, a Single line or a pair of lines is represented in each set of punched cards from CAT. From the plotted recording, the fraction Of a full scan corresponding to a particular magnetic field could be determined. Usually about five or more markers were used for each line. The recording was arranged so that both the forward and reverse scan were recorded. The field values obtained from the recording were punched on cards and given to the CDC 5600 computer along with the CAT punched, numerically coded Spectrum. The computer fitted one or more Lorentzian lines to the segment of the Spectrum. (No Sig- nificant deviations from Lorentzian shape could be found.) 45 The fact that a forward and reverse scan were used is important Since the field marker was amplified by one channel Of electronics and the ESR signal by another channel of electronics, and the different electronic pathways have dif- ferent delay times. What this means is that two Signals which were input to the amplifiers at the same time may arrive at the recorder at different times. This was noted to happen when recording the Rb data. However, the fact of the different delays does not affect the results reported Since when both upfield and downfield scans are used, the relative delays are equal but in Opposite field directions SO that an average of the two scans gives cancellation of the delay. This procedure was used in Obtaining the posi- tions of the lines for the Rb solutions. Each line was represented by 100 to 200 points in the data analyzed by the computer. The results given by the computer analysis of the spectra include the width and posi- tions Of the lineS and estimates of the standard error of these quantities. CS Data. The data for the Cs solution were recorded directly on an X-Y recorder. Recording the data was done in several steps as described in the experimental section. The positions of the lines were Obtained directly from the chart by using the field marking and linear interpolation. 'Phe variance on the position cannot be estimated directly when IJsing this technique, but it is very likely to be equal for 46 all Of the lines. This equality Of the variance is an important input for the program to calculate A and g values. The next quantity of interest is the relative widths of the lines Since the relaxation information lies in this Observation. .Examination of the lines reveals that the lines are very nearly Lorentzian in shape (see Figure 4). The Lorentzian shape means that the area is given by (50) A = constant °I°AH2 where: A is the area Of the integrated line, I is the peak- to—peak intensity of the first derivative presentation, and AH is the peak-to-peak width of the first derivative presen- tation. Since the radical under Observation has hyperfine splitting from one nucleus, the lines should all have equal area under the conditions of the eXperiment. (In cases which have been examined carefully, the lines do, in fact, have equal areas within experimental error.) Using the equal area hypothesis along with equation 50, one can Obtain the most sensitive measure of the relative widths Of the lines by measuring I and calculating AH from it. Then, (51) AH = constant/Iit where the constant in 51 is independent of ml, the nuclear magnetic quantum number. The relative variance in the widths can be calculated from the formulas for the propagation of error and the 47 'assumption that I can be measured to the same absolute accuracy for each line of the spectrum. This assumption iS plausible since the electronically generated noise on the Signal is the main limiting factor in the determination. Under this assumption the relative variance on the width is given by (32) o’ 2 (8%?” 0‘12 AH constant 6&2 IS The one remaining quantity needed is the absolute width of the lines. This was Obtained by measuring the absolute width of one line in the pattern. The measurement was ob- tained by displaying one line with field markers on a 15 inch recording so that the features were clearly Shown. This permitted measurement Of the peak—to-peak width as well as the Shape of the line being examined. For Lorentzian lines all other widths can be calculated using equation 50 since thiscnualine allows the constant to be determined. Reduction to SpectrOSCOpic Quantities After the data had been treated aS listed above, they all had the same form, i.e., the quantities line position, relative variance on line position, line width, relative variance on line width, and transition energy were determined. {The A and g values could be calculated directly with equa- t:ion 5 and the parameter estimation scheme outlined before. 48 Preliminary analysis indicated that all of the quantities in the relaxation equations (using 15) could not be determin- able from the data. In particular, the correlation time and the mean squared width of the distributions are separable only if the nuclear relaxation is a significant factor in the line widths of the Observed Spectrum. Since this was not eXpected, a method of analysis was adOpted to seek Optimum information about this separation. First the analysis used only the terms with w = 0 in equation 15. This reduced to solving for the products Of the correlation time and the second moment of the distribution, e.g., Tc <6A2> plus the residual width. Then the full analysis was performed with- all terms to see if the correlation time could be found. In no case was there sufficient information to find this constant. It is important to realize that with Spectra from sys- tems having large hyperfine coupling constant, each line is Observed in a different magnitude of external field. This implies that the energy level diagram for the system changes from line to line in both position of levels and in their characteristic dependence on parameters such aS-A and g, for example. This means that in order to Obtain the correct Use of the relaxation formula,one must obtain the complete «analysis Of the energy levels at each transition Since at 1relatively low fields the mixing of the high field states (ful, ms) iS a function Of the field. Results The original purpose of this experimental work was to test the prOposed mechanisms for the linewidth dependence on mI; therefore further testing the model for the environ— mental dependence Of the A and g values. As was pointed out in the previous chapter, a sizable amount Of qualitative and semi-quantitative information is available concerning the dependence of A and g on various external parameters of the system. At least the qualitative trends are believed to be well established. However, no quantitative explana- tion of these trends which fits all of the observations has been demonstrated to be completely satisfactory. The previous sections have indicated the type of experimental results which are necessary to test the linewidth mechanism. Previous work from various laboratories indicated that the linewidths are sensitive to some unknown parameters (18), with concentration of metal probably being one of them. Therefore,it was thought necessary to make measurements at low enough metal concentrations so that all concentration effects were absent; thus showing the intrinsic m dependence. I Although the preparation of stable solutions has been re— ported (24), the early attempts to make such solutions 49 50 proved unsuccessful and so a more thorough examination of this problem was made. The first part of this section will concentrate on the preliminary examination of the suitability of various systems of solvent and metal, and on the subsequent study of the stability of the solutions under various conditions. The mI dependence of some samples was studied quanti- tatively. The analysis Of these measurements will be Shown after the preliminary data are presented. Then, the results of a study of solid solutions of sodium in ethylene— diamine will be given. Qualitative Results The experimental ESR Spectrum and the solutions must have certain well-defined characteristics to be used for the study of the m dependent linewidth. These character- I istics include stability, a relative concentration of the m species to total metal SO that m is observable at low metal concentrations, an easily measurable m 'dependence, I and a line separation sufficiently larger than the width so that the lines do not overlap significantly. The Type of Spectrum for Several Systems. .A survey was made to assess the prOperties of several systems as possibly usable for studying the m dependence. Studies by I ESR for a relatively large number of alkali metal-amine systems have been reported (see for example (26)). 51 These studies have either reported only if paramagnetism is Observed or have reported the spectra and/or the A and g values as a function Of temperature. In many cases the reported data are not sufficient to Show whether the system would be useful to studies of the mI dependence of linewidth. In others, the reported data suggest that the systems may be useful. Therefore, ESR Spectra for a number of alkali metal solutions in various amines were recorded. In most cases, the solvent had not been carefully purified but had been dried by reaction with excess alkali metal (Similar to the procedure used for most work reported in the literature). The results are Of qualitative value in representing the expected spectral characteristics since no features are ex— pected to exhibit large changes because Of a small amount Of impurity. The ethylamine and propylamine samples used dissolved the alkali metals very slowly, to form very pale blue solu- tions. The ethylamine samples, when checked by VPC (vapor phase chromatography) analysis, always showed some ammonia (by retention times) after a blue solution had formed, even if none had been detected before solution preparation. The propylamine sample was not checked by VPC. The solutions in ethylamine (with ammonia) Showed a hyperfine pattern from one metal nucleus and a Single line near the middle of the pattern. Because of the greater number of lines in the 52 cesium. and rubidium spectra, they were studied most exten- sively. The result was that both the Signal to noise ratio and the stability were poor. The n-propylamine sample (of Cs) was blue and gave an ESR signal. The stability and signal to noise ratio of these systems were not good enough to permit them to be used in the linewidth study. Some Of the diamines solutions do not give an observable ESR hyperfine splitting pattern but do give a Single line resonance. In some cases the hyperfine split resonance is observed, however, but in all cases examined two problems occur. All the alkali metals are quite soluble in these lower diamines (ethylenediamine; 1,2—prOpanediamine; 1,5- propanediamine) and thus very deep blue solutions can be formed, but the monomer (m Species) is not favored by these solvents so that its relative concentration is small. The second problem is that, if Observed at all, the lines are relatively broader in these solvents. Taken together, these results mean that when the monomer is Observed, its ESR Spectrum is not useful for linewidth studies Since the lines overlap and the single line has become concentration broad- ened, implying that some concentration dependent fine broad- ening mechanisms have become Operative. For the spectra which were recorded, the A and g were not measured Since these measurements were not necessary for the immediate purpose Of the investigation. 55 For the solutions in the diamines, a relatively larger concentration Of metal could be achieved and consequently the decomposition was not such a severe limitation. Some Spectra Of diamine solutions were recorded at temperatures as high as +800C. Even though the solutions were decomposing, eventually turning colorless (pale yellow if enough metal were present) after short periods at high temperatures, they did remain blue long enough for the Spectra to be recorded. It Should be noted that at higher temperatures the monomer can be seen more easily at constant metal concentration. Some other systems surveyed included metals in methoxy- ethylamine and in mixed solvents of dioxane and the di— amines. These solvents Offered no advantage over the others, and the stability was noticeably less, SO only a few solu- tions were prepared. After the survey of solvents, a choice of a mixed sol— vent Of ethylamine and ammonia was made. This mixed solvent (gives relatively narrow lines, and the monomer can usually be Observed. Instability does not seem to be more of a problem than with the other systems. Systematic Investigation of the Decomposition. In Spite of the fact that stable solutions have been reported in the literature, the preliminary search for a suitable system to study was hindered by the persistent tendency Of the solutions to decompose, leading to a loss of blue color and paramagnetism. It became apparent that if 54 the linewidth study were to be continued, the decomposition had to be either controlled or eliminated. Therefore, in order to better understand the factors involved in this reaction, a systematic investigation was made of the con- ditions which influence solution stability. The first question to be considered is whether some impurity is re- sponsible or whether the reaction occurs with solvent alone. NO direct method is available to answer this question, so some experiments were performed which gave indirect evidence. First, the solvent was kept over metal for a long time (months) to remove any residual reactive materials which were present in it. In this case the source Of reducible impuri- ties must be either the vacuum system or the metal. The metal was distilled several times prior to use after its last exposure to the atmosphere, so that one would eXpect any easily reduced materials in the metal to have been re- duced by the metal vapor during distillation. Two sources of impurities can arise from the vacuum system: 1) residual materials on the walls of the vacuum system, and 2) leaks in the system (including outgaSSing the parts or diffusion Of gas by or through parts). The vacuum system was constructed such that all parts not made Of Pyrex were made of Teflon (e.g., valves, joints). The pumping system has been described previously, and Appendix I gives the description of the valves and joints invented to give a vacuum tight system. When these parts were used,the 55 system Showed no pressure change when the joints were rotated or when the valves were moved with the pressure at 2 x 1077 as measured with an ion gauge. Therefore, the‘ leaks were considered removed. What about impurities on the walls Of the apparatus used? Three approaches were used to examine this question. The system was repeatedly flamed until the sodium D line caused the flame to be yellow. The system was repeatedly exposed to the vapors of dried ammonia or ethylamine and evacuated cyclically, and the solution containing part of the apparatus was rinsed (poured in, left to stand for up to 5 days, and poured out) with a solution of alkali metal in ammonia. With any Of these procedures, the results were about the same; the blue color faded in about 10-20 minutes for solutions stored in a 2 mm i.d. tube but Slower in larger diameter tubes. This was clearly Shown in one experiment in which the solution was contained in a tube which had two sections Of different diameter. The solution faded in the small tube much more rapidly than in the larger diameter tube. Subsequent tests showed that very pale blue solutions could be made and kept in 2 cm tubes for a few hours before the color faded. The temperature for these experiments was near room temperature and the total metal was low enough SO that the color was pale blue (easily transparent) in a two centimeter tube. The decay of one sample was Observed 56 spectrophotometrically and the results are recorded in Figure 10. Note the fact that the decay is very nearly zero order. One further eXperiment was performed which is important. It has been reported that heating an alkali metal in the Pyrex vessel which is to contain the solutions causes greater stability for ammonia solutions (44). To test the applica- bility Of this result to the amine system, a similar test was made using rubidium metal. The metal was refluxed in the sample tube until the tube became lightly amber in color (about 15 minutes). Then all of the metal was distilled into a side arm of the vessel to cause one part of the vessel to have no metal in it. Then solvent was distilled into the metal-free section of the vessel and frozen in liquid nitrogen. The vessel was removed from the vacuum line by fusion of the connection. ~A solution was made by carefully pouring the solvent onto the metal and then removing the light blue solution which formed into a 2 mm i.d. part of the cell. The solution behaved as in previous prepara- tions, decomposing after a few minutes. Then a new solution was made and the experiment was repeated. After a few of these cycles the solutiOn stability had increased markedly, to give a half life of a few hours. This same solution has now been kept at -200C for 12 months and is blue,showing no signs of decomposition, i.e., no precipitate is formed and the small amount of metal present is shiny and does 57 not appear reacted. At the normally Observed reaction rate one would eXpect 0.5 cc of metal to have reacted, but in fact, 1 percent Of this amount could have been detected and has not reacted. The Flow System. The results of the study of the de- composition indicated that a solution could be made stable for longer periods of time if the volume of the sample were large compared to the surface area. These results could be utilized in a flow system in which the solution is kept in a large container until just before the measurement and then allowed to flow into the measurement cavity. Because of the nature of the ESR Spectrometer, only a small volume of solu- tion can be placed in the resonant cavity. A flow system was built to make use Of these facts, and was used for a number of runs. However, certain difficulties in the design and construction of the flow system were never overcome. One would like to have the flow system designed in such a manner that only a negligible fraction of the solution de- composes before the measurement is completed. In a large vessel (e.g., a 10 cm diameter cylinder) the decomposition rate of the solutions is negligible (months) for solutions Which are no more than 10-5 molar in total metal and at -780C. Therefore, the only concern is decomposition in the tube which transfers the solution to and through the cavity. From the decomposition standpoint then, fast flow is desirable. 58 However, from a practical standpoint, Since linewidth and shape measurements require a long time, a slow flow, which conserves solution, is better. From the known reaction rates (approximate) and the geometry of the flow system, the rate for this flow system which gave less than 5% decomposition at the cavity was 1 ml/min. This rate was also reasonable from the viewpoint Of temperature control, since during the transfer stage the temperature of the liquid must be changed from the value in the storage con- tainer to that desired in the cavity. The storage container must be kept cold (-78OC proved sufficient) to prevent dif- ferential volatization Of the components of the ethylamine— ammonia mixture . The control Of the rate of flow proved to be the most difficult problem. The valve for this job must be: 1) free flnmn influence on the solution; 2) Operable over a wide temperature range; and 5) relatively insensitive to small temperature changes. NO commercial valve with these prop- ertiescould be found. Plastic components (e.g., Teflon) ‘k3INDt satisfy criteria 2 and 5. The author designed and buili: a number of glass valves, none of which worked com- Pletfiely successfully even though the runs were made using thes€3 valves. Breakage of these valves or loss of flow con- trOlwere the primary difficulties encountered in using the flow system . 59 If a volatile solvent is used, the pressure Obtained as the temperature is increased gives problems. The large vessels used in this eXperiment will not withstand large pressures so that the range of temperature is limited. A final difficulty to be mentioned is lack of stability in the transfer tube. In every case, the stability in the transfer tube was less than exPected based on the previous decomposition studies. Figure 10 shows a quantitative decomposition study using the flow system. The decomposition of both the single and the multiplet pattern is first order in contrast to the zero order decay Shown in Figure 10. The difficulty probably arises from poor evacuation Of the transfer tubes. Since about 4 feet of 2 mm i.d. tubing is involved, it is not surprising that the system might be con— taminated with impurities due to the large surface area involved. This stability is much less than in the bulk, implying impurity contamination during transfer. The first order decay is also consistent with this assumption. Quantitative Results Solutions in Ethylamine-Ammonia Mixtures. Of the many samples examined, quantitative linewidth data were col- lected on two solutions and line position data on a third. These solutions were thought to be dilute in metal, and did not decompose noticeably prior to use so that very little decomposition product could have been present in the solu- tions. 60 Figure 5 gives an Optical spectrum Of the solution used for the CS run taken in a 1.0 cm Pyrex cell. Figure 4 shows the line of the hyperfine pattern for the CS solution corresponding to mI = 1.5. Also in Figure 4 one can see the superimposed line shape of a Lorentzian and Gaussian Shape function. The line is very nearly Lorentzian in shape as required for application of the relaxation formulas given before. Other solutions which were examined also showed the Lorentzian shape when they were dilute in total metal. Figure 5 Shows a representative Spectrum Of the Cs solution and the calculated Spectrum using the line widths given by the best fit of the relaxation formula. Note that the center line is missing from the simulated Spectrum, since it comes from another species. Table II Shows the A and g values Obtained from some Cs Spectra. For the Rb solution no Optical Spectrum was recorded but the sOlution was pale blue. Figure 2 shows a line from the ESR spectrum. The lines correspond to m = 0.5 for I 87Rb, and m = 1.5 for 85Rb. Figure 6 Shows the A and g I value for 8SRb as a function Of temperature for two samples. NO difference in the g values of 85Rb and 87Rb could be detected within the rather large experimental uncertainty. The A values were in the ratio expected from the nuclear moments. Figure 7 summarizes the relaxation analysis as a func- tion of temperature. AS is indicated by Figure 5, the two 61 state model of fluctuation of the isotrOpic A and g values is capable of describing a given isothermal spectrum within the rather large eXperimental uncertainties. The question. Of the applicability Of the two state model rests on the ability to correlate the temperature dependence of the con— stants obtained from the analysis. The figure shows that the Rb solutions have not yielded any definite information. The scatter in the data is large for the A fluctuations, and the analysis found no Significant g fluctuations. The CS data are somewhat better. Both the A fluctuations and g fluctuations Show an increase Of the quantity on the ordinate with increasing temperature. It is important to note that the fluctuation quantities, <1C592> and (TC-5A2), both have the same Sign of Slope on the figure as is re- quired by equation 21. The one further quantity to be noted is that the fluc- tuations of A and g were found to be anti-correlated in every case for CS. This means that a fluctuation which tends to increase the A value, tends to decrease the 9 value and vice versa. Spectra in Solid Ethylenediamine Solutions. Solutions of alkali metals in en have been reported to give two different ESR absorptions depending on the temperature of the sample. In preliminary studies of Spectra for this work a Similar phenomenon was observed in potassium in 1,2-propanediamine. .Whereas the en gave an abrupt 62 transition between Spectra at about -900C, the propanedi— amine gave two patterns which appeared to be in equilibrium over a temperature range. This suggested that more than one type of species may be present in the solid. The ques- tion is: can this be the m unit? In order to examine these properties further, solution of Na in ethylenediamine were reexamined. The first experiment was done to verify the reproduci- bility of the reported phenomena in en. Samples were prepared as indicated earlier using solvent that had been stored over metal for a long time. The result was that two patterns were Observed and as indicated by the previous work, the abrupt transition from one type of pattern to the other occurred at -90°C. Since the solvent had been dried over metal and was known to have decomposed Significantly during drying, the possibility Of impurities in the solvent causing the result was recognized. The most probable impurity is ammonia, but others are certainly possible. The second experiment, then, required the use Of care- fully purified solvent which had not been allowed to undergo extensive reaction with metal prior to its use. With this solvent, the effect of added ammonia could be tested. The experiment was performed as described earlier. The results were that the "pure" solvent gave only a broad line up to 0°C and no transition to a sharper line at -900C, but solu- tions with ammonia added behaved as in the previous work by 65 exhibiting two patterns. The results are summarized in Figures 8 and 9. The measured quantity is the peak-to-peak linewidth at each temperature. The Significant point to observe is that the sample of the solvent thought to be most pure does not give the abrupt change, but that the samples of solvent which have undergone prior decomposition with metal or have had ammonia added do give the abrupt transition. Furthermore, the samples with ammonia or the ones having solvent with prior decomposition give similar variations of linewidth with temperature, and they are dif— ferent from the solvent thought to be most pure. Since the two impure EH1 samples had widely different metal concentra- tions (one was dark blue and the other very pale blue) the metal concentration is apparently not an important factor. 64 TABLE I RESIDUAL WIDTH IN METAL-AMINE SOLUTIONS T(OC) Metal fi-AHé_(Gauss) -42 Cs 2.25 -55 CS 2025’ 2054: 2050 -62 Cs 1.55 -64 CS 1.57, 1.65 -54 Rb 1.04, 1.08 -65 Rb 1.19 -72 Rb 0.98 TABLE II A AND 9 IN SOME CS SOLUTIONS IN A MIXED ETHYLAMINE-AMMONIA SOLVENT T(OC) A(MHz) g -41 122.7 _+_ 0.5 1.9979 :1: 0,0001 -54 115.5 i_0.2 1.9984.i 0.0001 -65 104.5 i.1.6 1.9991.i 0.0004 65 .Emummm 30am wSu Mom Emummso Uflumfimsum .a shaman coausaom own: How mmmuoum.IWx umHuDO mmm Houucou muzumummfima .IIY.l~ HHO>MW wwuv 5.280 mmm IV. r I cowusHom Mom \\\\\\ Houucoo muzumsmmamu ocm Hmmmcmna ummcmsoxw umwm.|¢Y mmm cmmouuw: mun v, r :3 Hmaaouucoo musumummfima.l1._ _ mmm Honucou musumnmmamu mom nmmmcmuu Hm3ma cowusaom.llv 3 / ummHHmQ mm A Nd 66 .Hmcmwm mCHxHME pamflm wsu.m« Boawm .mmcaa HOEOSOE EsanQDH mo Mama mcwmmmaum>o on no snow mmum>mn ocm UHMBHOM m ma m>nsu Hmmms one .emoalo map cw omouoomu mafia Hmnuowmm m>HumucwmmnmmH 4 (11+ I A ucflom mcwcusa .N muomflm 67 .mocmnuomnm ucm>aom mom pom snow may mcfluzo mmomo How omuuwnuoo cwmn mm: Esnuommm one .Hamo Xmuwm EU a .MAcOEEmnmcwEmamnum cw Enammu .m musmflh Aiev numcmam>m3 ooaa oooa com com -00» com 4 A _ a _ a _ J . _ q . .Il one. on J o as... _ _ b p _ _ _ _ F _ _ LI N.o «.0 w.o m.o o.d eoueqxosqv 68 .coHuucsm mmmnm cmflmmsmw m mucwmwummu flu .cofluocsm mmmnm cmfluucmuoq m mucmmwummu O .cowusaom MHGOEEM ImcflEmenumlmU m mo cumuumm wchHmmhn mnu Eoum OGHH m>wuchmmmHmwu d .w musmwm 69 e musmwm 70 Figure 5. An experimental and theoretical Cs-ethylamine- ammonia hyperfine pattern. The calculated Spectrum uses Lorentzian Shaped lines with widths given by the two state model. F iiiiii .musuxfle MHGOEEMImcwEmaanum cm as QMmm mo mcofiusaom OBu How m can unsumcou mcfimummmm .w musmflm Soc B Soc 9 ON) owl owl owl , 0 ON) Ce) om) om) 72 - . a A 4 E. q - - n. U v G fiv .. on - 1 1 oaoo.m (ZHW) E}_, L maoo.m 75 l N.O m.o $.H 33$ 6 +0T X J 91' .AUPNmov Esammo loos a Odl Oml Owl OBI .O.N a. AV mwv .o.oa -o.aa — — F . T_Oes 9_OI x VJ m can .mwmwamcm nuoHBOGHH map mo muasmmu one .5 musmflm oa- Esflofinsm loos s om- om- Owl A e a .N.o m.o O.H .a.a T.oes 9_01; x VJ 74 .cowusaom ommomaoowo Bonn :0 AV cm :93? + mmz "D . G 50 :UHDQ: " G .mcowusaom Acmv mosamflomcwamzumlasfloom as muuommm mafia mamcwm may no nu0w3mcwq .m musmflm AUOV B owl owl om) Oman Oman T _ u 4 G o.¢ o.m dd HV (ssnes) 75 .owwwflnzm maasmmumo sewn om: QOHgB cm Ou venom mmz BV.mu musumummEmu EOOH as $5.3 Mnmo 58> .coquHOm UOmOQEOomo EOHM cmq musumnmmfimu EOOH um mfian mamm hnm> .soflusaom owmomeoomo Eosm cw O .0 0m: w>onm cofiusaom =owH0m= Acmv mcwEMHomcmahnumleswoom cw mmfla 3ounmc may mo £u©a3mcflq .m wusmflm Ob) 1 Od.o om.o HV dd (ssnes) Figure 10. 76 The upper graph shows the first order decay in the flow system with flow stOpped. 0 represents the area under'the hyperfine lines EJrepresents the area under the Single line. The lower graph Shows the zero order decay of absorbance for a sample carefully prepared under high vacuum conditions in a well- degassed system. xv. n _ ... _ _ _ _ no 0. 0.8.7654 5 2 1 00000 O Amnmspwnusv cowumnucmocou Time (min) mocwnuomnd 0.5 600 700 800 0 500 40 200 500 100 Time (sec) 78 Discussion The original goal Of this investigation was to examine the proposals for the m dependent linewidth. Persistent I decomposition of the required dilute solutions necessitated a systematic investigation of the decomposition. This section contains a discussion of the meaning of and rela- tions among the results Obtained. The mI Dependent Linewidth The interpretation Of the only previous systematic investigation of the m dependent linewidth (55) did not I account for all of the experimental evidence. These investi— gators found the line to be nearly gaussian in shape with a residual width Of about 5 gauss. However, this author found the lines to be very nearly Lorentzian in shape with a residual width of about 2 gauss (Table I) under Similar conditions. Catterall _p__l. (18) report that excess cation broadens the hyperfine lines in potassium-ethylamine solu- tions. This author's decomposition study makes it plausible to assume that Dye §£_§l, had some excess cation in their solutions due to decomposition. If the cations in cesium 79 solutions act like those in potassium solutions, it is plaus- ible to expect the lines Observed by Dye e£_§l, to be broader than those Observed by the author. It is also likely that the cation exchange would cause the line Shape to become non-Lorentzian, Since with small excess cation concentrations the process would be in the near Slow exchange region. The Lorentzian line shape is important because it indicates the process(es) which cause the m dependence is I rapid on the ESR time scale which iS necessary for the appli- cation Of the relaxation formalism used. The two-state model is capable of fitting a given spectrum but this does not mean that there are actually two Species in equilibrium. It does mean that the relaxation equations which were used are sufficient. Therefore, rapid modulation of the isotrOpic A and g constants in an anti- correlated manner can describe the data at one temperature. In order to know more Specifically the nature of the modula- tion, the effect of solvent, metal, temperature and pressure on the correlation functions and means of the distributions must be studied. A model Should not be rejected if it describes all of the variations Observed. It is likely, however, that data for any small range of the variables can be described by more than one model. The data Obtained in this study are for an insufficient range of temperature to distinguish among the possible mechanisms for the m dependent linewidth. The data do I 80 indicate, however, that a distribution model with a rapid equilibration among states could describe the data since this study Shows the lines to be of the proper shape and a previous study (24) Shows that A can be described by a distribution including at least three states. The remain- ing unsolved problems are the cause and distribution of the states of the system. A much more extensive set Of data which is collected under carefully controlled conditions will be necessary for the solution of these problems. The mI dependence may be due to anisotrOpy of the Species as has been suggested (4). Since the isotropic A value is strongly influenced by the environment, the aniso- trOpic components of the A tensor are likely to be similarly influenced. In order to test an anisotropic rotor model, the relations of the anisotrOpy to the environmental con- ditions must be specified. Therefore, this proposal cannot be tested in the same manner as for solutions Of inorganic complexes which have nearly constant magnetic parameters. Whereas the distribution model explains not only the mI dependent linewidth but also the A and 9 value Shifts, the anisotropic Species model explains only the m dependent I linewidths. Solution Stability The systematic investigation of the factors influencing the solution stability indicated that the surface-to-volume 81 ratio of the container was important because solutions which had a relatively larger surface area at constant volume decomposed more rapidly. These decay studies gave qualitatively reproducible results. The one quantitatively measured decay was zero order. The Observations are in agreement with a surface catalyzed reaction mechanism; a result which is Similar to that for ammonia solutions (51, 44). Deactivating the surface by reacting metal vapor with it was not completely successful but the reaction did be- come slower after some initial reaction had taken place. These results are Significant because they indicate that either quartz or Pyrex catalyzes the decomposition reaction. Therefore, it is very difficult to use tube dimensions which permit study of the sample in the 9.2 GHz ESR spectrometer, yet do not cause solution decay. The decay observed under some circumstances is quali- tatively different. One solution in the flow system Showed a first order decay of the paramagnetism when the flow was stopped. The rate of this reaction was much more rapid than expected for a well cleaned system at a comparable temperature. The result is probably due to impurities introduced by the flow system reacing with the solute. In summary, the decomposition in pure solutions appears to be catalyzed by the container surface. NO method was. found to inhibit the process and avoid decomposition product simultaneously. The ESR experiment requires that one must 82 use small containers and dilute solutions while avoiding decomposition products if the linewidths are to be reliable. Therefore, the decomposition is a severe limitation to the ,possible eXperimentS. Solid Solutions The reinvestigation Of the solid solutions Of sodium .in.ethylenediamine Shows that the presence of the Sharp line aabove -90°C is due to impurities in the solid. Ammonia, xnhen introduced deliberately, gives the same results, whereas, (:arefully purified ehtylenediamine does not give the sharp ssingle line, but rather a resonance characteristic of dis- E>ersed Na metal (50). The most likely interpretation is tzhat the impurity, which probably is ammonia introduced by Ciecomposition Of the solutions in the usual drying procedure, forms a liquid mixture with ethylenediamine which has a estatectic at -900. Under this hypothesis, the sharp line is <211aracteristic of a liquid solution of sodium in the mixture. CHAPTER III Ag INITIO STUDY OF LITHIUM AMMONIA INTERACTIONS Introduction The Ab Initio M. 0. Technique The ap_initio calculation Of an all electron wave func- tion for polyatomic non-linear molecules,using a basis Of gaussian functions, has become practical Since the develOp— ment of the electronic digital computer. BoyS (8) has Shown that all of the required integrals can be readily evaluated in terms of tabulated functions if the gaussian function is used as the basic radial function. Two modifications of this procedure are now commonly used. The nuclear centered gaussian function uses a gaussian radial function but main- tains the Spherical harmonic function to give the angular dependence. Therefore, this technique is the same as that using Slater type orbitals except that the eXponential function has been replaced by a gaussian type radial function. A number of recent calculations have successfully used this type Of basis for the description Of rather large polyatomic molecules (22,25). A Second approach uses only primitive' spherical gaussian functions, but combines these into fixed linear combinations which represent other types of functions. 85 84 In the gaussian lobe modification, as this technique is called, angular dependence is Obtained by using linear combi- nations Of functions which have different centers. This technique has also been used to describe the electronic structures of some large molecular systems (11,12,70). The major problem in past ap_initio calculation of electronic structures Of polyatomic molecules has been the evaluation Of the "multicenter" integrals, that is, integrals \vhich contain functions centered on more than two nuclei. {The gaussian technology has removed this problem. Other lproblems arise, however, for "larger" molecules. The number <>f integrals needed for the calculation of an electronic sstructure increases as the fourth power of the number Of 1>asis functions. The number Of basis functions increases vvith both the number and complexity Of the atoms used. flDhe expense of doing an ab_initio calculation is, therefore, 51 rapidly increasing function Of the Size Of the molecular saystem and is the primary limitation on the Size of the system which can be investigated. In a recent review article (25) Clementi has discussed tine various levels of approximation used in calculations of electronic structure. The method used in the work discussed lit) this chapter is the single determinant self consistent field molecular orbital scheme in the Born-Oppenheimer approxi- Inaition which is an adaptation of the atomic Hartree-Fock me thod to the molecular problem. The best solution in this 85 approximation is known to be incomplete. Furthermore, if a finite basis set is used, there are errors introduced because the solution Obtained iS not the best single determinant. The effects of these errors on the prOperties calculated will be considered when the results are discussed. Recent publications by many investigators have examined the application of §p_initio methods to the solution of prob— lems involving physical and chemical prOpertieS of isolated molecules. For example, the energy as a function Of nuclear position has been used to calculate force constants (7); the wave functions for near Hartree-Fock solutions have been used to calculate charge densities and to see how bonding affects them for diatomic molecules (14); reaction energies and rotational barriers have been examined (60) and even the energy surface for the reaction Of NH3 and HCl has been (studied (22). The method is able to explain (or predict) many prOperties. The accuracy for Specific properties will be discussed in a later section. The Application The experimental data relating to monomers in metal— amine solutions potentially contain significant structural information about the solute Species present. The lithium monomer is a particularly interesting case because the Splitting constant is Obtained for some of the solvent as well as an upper limit for Splitting by the metal ion. The most puzzling feature Of these data is that the unpaired electron 86 density at the lithium nucleus is small while the unpaired electron density at the solvent nitrogen is significantly larger. The qualitative eXplanations for the Observations are reviewed in Chapter I. The quantitative models for the monomer (6,45,46,54,55) are extensions of the solvated electron models to include a cation. The effects of the radium and metal inner Shell electrons have been treated in several ways. For the ammonia monomer of sodium, Jortner treated the solvent and metal inner Shell electrons as a continuous dielectric medium. O'Reilly has calculated wave functions for ammonia monomers of all the alkali metals using a continuous dielectric medium with a spherical cavity which contained a point positive charge to represent the cation and point multipoles to represent the first solvent layer. The unpaired electron function was made orthogonal to the metal inner Shell orbitals and in some cases, also, the orbitals of the first solvent shell. All molecular orbitals except for the unpaired electron were those for isolated molecule calculations. He later extended the model to the ethylamine monomer of potassium. These calculations assume that the long range forces are most important in the formation Of the Species. Any Short range interactions of the metal cation, unpaired electrOn and solvent molecule are specifically left out in these calculations. If the monomer is formed by partial covalent bonding of the solvent and metal or if Significant charge is 87 transferred from the metal atom to Specific solvent mole— cules, the continuous dielectric model would be incapable of predicting it. Using the gaussian integration technology, the short- range interactions can be examined in detail. With present techniques, examination of conditions representative of the liquid state is not possible because Of the amount Of labor required. Model systems which can help conceptualize the nature of the interactions in the liquid can be examined, however. This chapter presents the calculations and analyses of the interaction Of a lithium atom with one or two ammonia molecule(S). The gaussian lobe function basis is used to construct the single determinant solution to the Born- Oppenheimer Hamiltonian. Analysis Of the results using the energy, charge density, and various pOpulations leads to the conclusion that: 1) a lithium atom--one ammonia molecule complex is stable relative to the separate gas phase mole- cules, 2) significant hyperfine contact density can occur at the nitrogen nucleus without appreciable charge transfer to the ammonia molecule, 5) ammonia is bound to the lithium atom primarily by a covalent bond utilizing the lithium 2p orbital and the ammonia lone pair Of electrons, and 4) the unpaired electron avoids the Spatial region containing the ammonia. Although the calculation does not include possible long range interactions Of the unpaired electron with the solvent, 88 some of the results provide information about short range interactions which might take place in the solutions. The calculations used ammonia molecules rather than amine molecules because of the time difference involved in the different calculations. It is unlikely, however, that the results for amine molecules will be Significantly different, in their qualitative features. The Methods The method used to construct the wave function for the molecular species can be described as a self consistent field linear combination of group orbitals (SCF-LCGO). The resulting wave function has been analyzed by using a popula— tion analysis scheme due to Mulliken (55) and by contour maps of the charge density and changes in charge density. Although the methods are rather common, they will be outlined in some detail here SO that the assumptions made in the cal- culation can be stated explicitly. Construction of the Wave Function The starting point for this calculation is the electronic Hamiltonian in the Born-Oppenheimer approximation. The Hamiltonian is n m Zi n i (l) H = Z [-§VE+ (-Z 7— )] + Z T i= j= ij k>i ki where: 89 - é-V: is the kinetic energy Operator of the i-th q electron - -—- is the potential energy Operator for the in i-th electron in the field of the j—th nucleus. 'l— is the electron-electron electrostatic Tki interaction (repulsion). A solution to 1 must Obey the Pauli principle for the electrons as a minimum condition. The self consistent field solution is the result Of minimizing the expectation value Of the Operator 1 with a prOperly anti-symmetric trial solution constructed using a basis Of independent particle solutions. The solution to the resultant equations from the minimization are non-linear in the trial function coefficients so that the solution must be achieved in an iterative manner. The particular method used for the examination Of the interaction of lithium and ammonia is based upon a descrip— tion of the basic independent particle (one electron) func— tions in terms of linear combinations of gaussian type functions. (This can be viewed as a LCAO basis in which the atomic orbitals (A.O.) have been replaced by group orbitals (G.O.)). A Single determinant trial function is formed from the one electron functions and the total energy of the system (eXpectation value of Operator 1) iS minimized with respect to the coefficients of the groups of gaussian functions in the eXpansion of the one electron basis functions. 90 The important point is that the Hamiltonian given by 1 is assumed to apply to the system, a basis Of gaussian functions is assumed for the system and then the solution to 1 is achieved without further approximation. The programs to generate the integrals and to use them to calculate the SCF solution were generously donated by Professor J. L. Whitten. Two SCF programs were used: one calculated singlet wave functions and the other calculated doublet wave functions. The one used depended on the number of electrons in the system. The Basis Functions The molecular orbitals are expanded as a linear combi- nation of basis functions. The basis functions for lithium and nitrogen were taken from the table given by Whitten (70). The hydrogen function contained five terms which had been Optimized for use in calculations with C-H bonds (70). The lithium basis was augmented by addition of a two-term gaussian lobe representation Of a p function which was scaled to an effective charge of about 0.65 on the lithium (this function was not actually Optimized to represent a Slater type orbital). The collection of functions outlined above will be referred to as the minimal basis set. TO test the adequacy of the minimal basis, an extended basis was used for some calculations. For the hydrogen function, the five-term representation was split into a L_w. 91 short range group with three terms and a long range group of two terms. The relative coefficients within a group were maintained, and the exponents were not changed. The nitrogen p function was Similarly split into a Short range function of three terms and a long range function Of two terms. In order to allow the possibility Of charge transfer, the nitrogen basis was augmented by including a gaussian representation Of a Slater three S function with an effective change of 0.75 (overlap with Slater 5 S of 0.9996). The lithium basis was extended to include two 2 p gaussian representations(overlap with Slater 2 p of 0.9984) with effective charges of 0.95 and 1.50 and two 58 representations with effective charges of 1.0 and 0.5. The gaussian repre- sentations were constructed by maximizing the overlap of a Slater function Of the desired type and the gaussian representation of it. The constants were chosen so that either the neutral or positively charged lithium could be represented. For nitro- gen, the 5 S function was chosen to represent a neutral nitrogen atom. This basis is believed to be sufficient to represent the various possible hybridization schemes for the lithium atom and/Or to allow the possibility Of a transfer of the unpaired electron to the ammonia. 92 Geometry of the Nuclear Framework Because these calculations are time consuming on a digital computer, only a limited search of geometry was made. First, the lithium atom was assumed not to strongly perturb the electronic structure Of the ammonia molecule. Therefore, it seemed reasonable to assume an equilibrium ammonia geometry for the NH3 framework. However, the ques- tion of the position of the lithium atom relative to the NH3 molecule arose. One might suSpect that placing the lithium on the C3 axis Of the ammonia would be favored, but should it be placed on the same Side of the ammonia as the hydrogens or on the Opposite Side? NO Obvious means (except calculation) of answering this question was found. Therefore, a number of configurations with the lithium atom on each Side Of the NH3 part were tried. All of these calculations used Csv symmetry. There is also the question of whether such a complex would be stable to autoionization to a free electron and a positively charged species. If a positively charged Species could form what would be its prOpertieS? To obtain the answers to these questions, a series of calculations were performed for the positively charged complex. After the calculations for one lithium and one ammonia predicted a bound Species, a series of calculations were performed for one lithium atom interacting with two ammonia molecules. All of the two-ammonia calculations were for species with D3H symmetry. 95 The ammonia molecule was assumed to be of C3v symmetry with r(N-H) = 1.918 a.u. and an angle between the C3 axis and the NH axis Of 68.15 degrees. This is very close to the experimental geometry given in the literature (42). NO search Of the ammonia geometry was made. Apalysis Of the Wave Function Since it will not be possible in the immediate future to perform §b_initio calculations on Species likely to be found in solution, understanding the mechanisms of the electronic interaction in model Species is important so that they may form a conceptual basis for thinking about the Species in solution. In order to better understand the way in which ammonia and lithium interact, the wave functions have been analyzed according to the population analysis scheme of Mulliken (55). Results of only a few Of the pOpu- lation breakdown equations will be tabulated. One quantity which is informative is the net overlap between two atoms. This is given by All orbitals All orbitals All m.o.'s of atom K of atom E . (2) n(k,1) = f z I 2n(1) cir cisfl s 1 K FKSE with the definitions: n(k,l) net overlap between atoms k,l 94 n(i) number of electrons in m.o. i Cir 'CiS coeff1c1ent of baSis function rk or Sl k l in m.o. i S overlap between basis functions r and S rKSl K 1 Two other quantities which are Of particular interest are the gross population in a given molecular orbital on a particular atom and the total gross pOpulation on a given atom. All basis All atoms functions except K (5) N(i7k) = z [n(i) c. (c. + 2 c. s )] r HK HK 1 f K 181 rKS1 and All m.o.'s (4) N(k) = 2 N(i;k) i They are defined by equations 5 and 4 in which the symbols have the same meaning as above. The method of pOpulation analysis is not a rigorous quantum mechanical formalism which relates to observables as the expectation value of physical observables does, but rather, it is a means of forming a conceptual connection between chemical concepts of bonding and the results of quantum mechanical calculations. One must remember that the results of the population analysis are to be used only when they agree conceptually with the results given by expectations of observables (when such a comparison is possible). 95 The total charge density is a physical observable. One can also measure the density of unpaired electron in the region of certain nuclei because the isotrOpic hyperfine Splitting is directly related to this density. Therefore, these quantities have been calculated for the different nuclear configurations. The Fermi contact density is calcu- lated as the square of the value of the molecular orbital containing the unpaired electron at the position Of the ‘nucleus. The total charge density at a point in space is 'the sum Of the squares of the magnitudes of the occupied Inolecular orbitals times their occupation number. In order to illustrate the changes in charge density ‘Mhich occur on formation of the interacting aggregate Species. 'two types of contour maps have been prepared. Total charge (density maps have been plotted. Since the changes in mmm.e +mmzmq mmooo.a .. u- .. memm¢.mo mamma.a +mmzma 00000.d I) In In Nmmme.m0 memmm.m +mmqu maoo.a u: u: I: eaome.mm memmm.m +mm2wq aeaoo.a u- I- In mmmme.mm msmms.m +mmzfiq e¢H00.d muoaxm>.m 0000.0 mma.0 mommm.m0 mumm0.¢ mmZHA emaoo.a muoaxmm.s maeo.o mma.o mmooo.mm mamma.a mmzmq mmdoo.a mloaxmm.> mee0.0 NNH.0 smaow.m0 msmmm.m mmzmq 0m00.d muoaxeo.m m0¢0.0 0ma.0 mdm00.mm mewmm.m mszA mm000.« muoaxam.m emeo.0 mad.0 sma00.mm msmm>.m oflqu In In In mmaefi.mm II 032 mmemm.» nu +«q 0000.6 In In mmd.0 madm¢.> In HA nmms qumH> m z . an om\mwu mucmumao mmaowmm >\MNI uumucoo AEHmm LA “vs; wmamcm ZIHA mZOHBSDQHAwU mom NEH. m0 mBADmmm NEOm H mqm¢9 wwmmm.o 000000.fi mfiooo.a 100 m>ooo.a m-oaxa.e mIdem.m >¢0.0 mm0.0 OOH.O mm0.0 mmddm.mm ommdm.m0 omdm¢.> nbmwd.wm mhmma mwmmm .d .m mmzmq mmmmm omocmuxm «mama mmmmm omsemuxm . sq mommm cmpcmuxm mmz mwmmm omeemuxm mooo.a Ndoo.d $HOO.H mdoo.d ommmm.mdd bmhmm.mdd Hmmmm.mdd mammm.mdd mbmmd mummm mhmmm mhmmw .¢ .m .m .m N m a +A mzv.fl mlmmzvma ”Amazoma «Amazons + >NOO.d mNoo.H omoo.d mmoo.d vIOHx>.fi v.0984 euoaxom.a «noaxso.m mmmm>.mdd mmNN>.mHa ¢>NN>.de hmfimh.mdd mhmmm 050mm mhmmd mhmmm .d .0 .d .m mxmmzoaq «impasse «Amazons «Amazons mHOO.d >N00.H vuoaxom.n vuoaxaa.m meo.o hmmo.0 >mmmm.mm momma.m0 .fi 101 TABLE II SOME REACTION ENERGIES* A . Reaction AE(products- reactants) kcal/mole Li + NH3-9’L1NH3 (minimum basis -18.2 for each) Li + NHs-fi'LiNHg (extended basis -20.2 for each) .+ . + L; + NH3—> (LlNHa) -45.0 (LiNHE)+ +e-—’ LiNHg -91.57 Li + 2NH3 -+»Li(NH3)2 —4.93 LiNHa + NH3 —*L1(NH3)2 +15.5 Li+ + 21m3 —> (L1(NH3)2)+ -128.8 (L1(NH3)2)+ + e- —"Li(NH3)2 ‘35.? *- Calculated using the results of Table I. 102 TABLE III TOTAL OVERLAP POPULATIONS BETWEEN ATOM PAIRS Atom Pair Species Li,N Li,H N,H H,H NHS (Minimum -- -- 0.699 -0.029 Basis) LiNHs (Minimum +0.074 -0.016 +0.696 -0.024 Basis) LiNHa (Extended -O.225 -0.012 +0.695 -0.026 Basis) Li (NH3) 2 (Minimum -O.115 -0.011 0.698 -0.026 Basis) (Atoms in (Atoms in the same the same NH3Imoiety)mOiety) 105 TABLE IV GROSS POPULATION IN AN ORBITAL ON AN ATOM Quantity . . * Gross Population in the (M.O.number) Orbital on the Atom Species (Total number of m.o.'s) Li N H Li 2/2 1 -— -- NH3 5/5 -- 1.950 0.017 LiNHa 6/7 0.088 1.848 0.021 minimum basis 7/7 0.994 0.006 0.000 r ._ = 5.85... (unpaired L1 N electron) LiNHg 6/7 0.096 1.840 0.021 extended basis 7/7 1.084 -0.085 0.000 r . = 5.85... (unpaired L1 N electron) LiLNHg); 10/12 0.098 0.921 0.010 rL. = 4.25... 11/12 0.097 0.924 0.009 1-N 12/12 0.944 0.026 0.001 (unpaired electron) *- Based on order of increasing orbital energy. 104 TABLE V TOTAL GROSS POPULATION ON AN ATOM Atom Species Li N H Li 5.000 --- --- NH3 --- 7.907 0.697 LiNH3 5.067 7.955 0.667 (Minimum basis) LiNH3 5.160 7.866 0.658 (Extended basis) Li(NH3)2 3.123 7.895 0.682 (Minimum basis) Figure 1. 105 The total charge density of ammonia. The contours in the upper graph are for values of 10.0, 2.8, 0.77, 0.41, 0.22, 0.11, 0.06. 0.05, 0.02, 0.0088, 0.0046, 0.0025, 0.0015. 0.0007, 0.0004 (ac-3). The upper graph is for a reflection plane and the lower graph is along the C3 axis of the molecule. ’3 3 c .2 z: 8 O. o 3:; l0 8 o. q I0 := m c 0’ .2 0 l0 0 is” .c 3 0 no 106 4 Position Figure 1 (Go) Figure 2. 107 The total charge density Of the lithium atom. The contours in the upper graph are for values of 10.0, 2.8, 0.77, 0.41, 0.22, 0.11, 0.06, 0.05, 0.02, 0.0088, 0.0046, 0.0025, 0.0015. 0.0007, 0.0004 (ao's). The upper graph is for a plane containing the lithium nucleus and the lower graph is for a line containing the lithium nucleus. 108 Ac 3 Seas... l0° ** 9.3 .m 22.8 385 Figure 2 Figure 5. 109 The total charge density Of the lithium ammonia complex. The contours in the upper graph are for values of 10.0, 2.8, 0.77, 0.41, 0.22, 0.11, 0.06, 0.05, 0.02, 0.0088, 0.0046, 0.0025, 0.0013, 0.0007, 0.0004 (ac-3). The upper graph is for a reflection plane and the lower graph is along a C3 axis Of the molecule. 110 3 8:8... .m €60 W m .0 3800 ".5905 Figure 5 111 Figure 4. The total charge density of the lithium diammonia complex. The contours in the upper graph are for values of 10.0, 2.8, 0.77, 0.41, 0.22, 0.11, 0.06, 0.05, 0.02, 0.0088, 0.0046. 0.0025, 0.0013, 0.0007, 0.0004 (ac-3). The upper graph is for a reflection plane and the lower graph along a C3 axis of the molecule. Only one—half Of the molecule is shown. Position (Go) 6") Charge Density (0 Position F iiiiii (ac) Figure 5. 115 The charge density difference is for the lithium ammonia complex total charge density. The values Of the contours in the upper graph are 0.01, 0.008, 0.006, 0.004, 0.002, 0.001, -0.01, -0.008, -0.006, -0.004, -0.002, and -0.001 (ao's). The upper graph is for a re- flection plane of the molecule and the lower graph is along a C3 axis Of the molecule. The two maxima which are not shown in the lower graph have values of 0.052 and 0.056. A positive difference Shows charge density decrease in the complex relative to the sum Of the lithium atom and ammonia molecule. (0.) Position Chcge Difference (03’) .0 )0 .005 ‘ 114 2 ‘ 21 ' 6 Position (Go) Figure 5. Figure 6. The charge density difference is for the total charge density of the lithium diammonia complex. The values of the contours in the upper graph are 0.01, 0.008, 0.006, 0.004, 0.002, 0.001. -0.01, -0.008, -0.006, -0.004, -0.002, and -0.001 (ac-3). The upper graph is for a re- flection plane of the molecule and the lower graph is along a C3 axis of the molecule. The minimum and maxima for which the values are not shown in the lower graph are: -0.19, +0.055, +0.050. .A positive difference shows charge density decrease in the complex relative to the sum of charge densities of the lithium atom and two ammonia molecules. (0o) Position Chage Difference (05’) 116 2 4 Position Figure 6 (0e) 8 117 Figure 7. The unpaired electron charge density of the lithium atom. The contours in the upper graph are for 0.1156, 0.0599, 0.0516, 0.0167, 0.0088, 0.0046, 0.0024, 0.0013 and 0.0007 (ac-3). The upper graph is in a plane containing the lithium nucleus and the lower graph is along a line containing it. 118 Figure 7 Figure 8. 119 The unpaired electron charge density Of the lithium'ammonia complex. The contours in the upper graph are for 0.1156, 0.0599, 0.0516, 0.0167, 0.0088, 0.0046, 0.0024, 0.0013 and 0.0007 (ac-3). The upper graph is for a reflection plane and the lower graph is along a C3 axis of the molecule. The lithium is on the left. be) Position goo-8) 'Chage Density t0° )0 I0" I0" . 120 0 ‘ 2 4 Position Figure 8 (ac) e 121 Figure 9. The unpaired electron charge density of the lithium diammonia complex. The contours in the upper graph are for 0.1156, 0.0599, 0.0516. 0.0167, 0.0088, 0.0046, 0.0024, 0.0015 and 0.0007 (ao's). The upper graph is for a reflection plane of the molecule and the lower graph is along a C3 axis of the molecule. Only one half of the complex is Shown. The lithium is on the left edge of the graph. (0.) (Co-3) PCBIIIOU Charge Density 5 2 122 (0o) Figure 10. 125 The charge density difference is for the unpaired electron density of the lithium ammonia complex. The values of the contours in the upper graph are 0.01, 0.008, 0.006, 0.004, 0.002, 0.001, -0.01, -0.008, -0.006. -0.004, -0.002, and -0.001 (ac-3). The upper graph is for a reflection plane Of the mole- cule and the lower graph is along the C3 axis Of the molecule. For the lower graph the maximum has a value of 0.055 and the minimum has a value of 0.05. -A positive dif- ference Shows a decrease in the charge density of the complex relative to the sum of the lithium atom and ammonia molecule densities. 124 as cofimod .OIO m. m m is 858:0 326 O O a. . Position (Go) Figure 10 Figure 11. 125 The charge density difference is for the unpaired electron density of the lithium diammonia complex. The values of the contours in the upper graph are 0.01, 0.008, 0.006, 0.004, 0.002. 0.001, -0.01, -0.008, -0.006, -0.004, -0.002, and -0.001 (ao'a). The upper graph is for a reflection plane and the lower graph is along a C3 axis of the molecule. The maximum not Shown in the lower graph has a value of 0.017 and the minima have values of -0.052 and -0.22. A positive difference shows a decrease in charge density in the complex relative to the sum of charge densities of the lithium atom and ammonia molecules. 126 u.)(. 11‘. 13 .® 2 I seen. m. m m Figure 11 127 .Aoaosc comouuflc can annuwa on» nmsonnu mcwmmmm mafia o co coxmu ma huwmcmo mmumno 0:8 .¢IIIV manna ooosmuxo on» Ca memEOo MAGOEEM Emanuwa mnu 0cm .AIIIV manna EDEwcHE m cw xmameoo owcoaem Esanufla may .1...V sous asaeusa may as soeusnanumap couuooao oouwmmcs on» no consummeoo m we mane .NH musmwm 128 I2 POSITION is.) Figure 12 (.Ib) Ausuao asuvno 129 Discussion The meaning of the various quantities which have been calculated and their probable relation to physical systems are related in this section. First, the reliability Of the quantities will be examined and then their chemical Significance will be considered. Energy The energy shown in Table I and certain energy differ- ences Shown in Table II are the result of the Single determinant, small basis representation of the electronic wave function for the syStem of nuclei and electrons. The differences Of this method from experiment can be listed as: 1) errors due to the limited basis, 2) errors due to the inability Of a single determinant to describe the exact solu— tion, and 5) errors due to neglected effects. ,Ritchie and King (59) have Obtained near Hartree-Fock (best Single deter- minant) solutions for ammonia and other molecules. From their results, they have estimated the correlation energy (contribution not represented by a best single determinant) in ammonia to be -0.55 a.u. (207 kcal/mole). Thus, one can see that if an energy difference is obtained (as in Table II), it will only be meaningful if the correlation energy changes are small compared to the total correlation energy. Ritchie and King also estimated'the Hartree-Fock energy of ammonia to be -56.201 a.u. which means that the minimum basis 150 calculation of ammonia listed in Table I is high by 0.059 a.u. (57 kcal/mole); a substantial amount of energy on a chemical scale. Therefore, the errors due to basis limitations must cancel if energy differences for reactions are to be meaningful. Recently, Snyder and Basch (62) have investigated the calculation of reaction energies by the SCF method. For the systems investigated, they found a root mean square deviation C”? the calculated reaction energy from experi- ment Of 6.5 kcal/mole. .However, several points must be made concerning the differences between this work and the work Of Snyder and Basch. First, they were using comparable (but slightly better) basis sets (their NH3 energy is -56.1714). Most Of the reactions they investigated involved systems with multiple bonding going to systems with differ— ent bonding (in a number of cases having no multiple bond— ing). Thus, the changes in bonding in their calculation are much greater than those in these calculations (see later). Most of the systems they investigated were substantially larger than the Li-NH3 calculations reported here. .Therefore, equal fractional changes in correlation energy, or changes due to basis incompleteness wouldlxzmore pronounced in their calculations. The one further difference between this work and the other is that the present calculations have an Open Shell involved.in some cases. Since the unpaired electron is reasonably localized in the lithium orbitals and because 151 the calculation is on the same level as Snyder and Basch's calculation, one can probably conclude that the presence of the Open Shell is not significant. Taking all of these arguments into account, one can probably conclude that the error shown by the calculations of Snyder and Basch are expected to be outer limits for the energy calculation of L1NH3, Li, and NH3. Based upon the calculated energy one can conclude the following; although the absolute values of thezenergies calculated are significantly in error (i.e., they have relatively large known errors), the differences are probably accurate enough to permit the conclusion that the complex of lithium and ammonia would be stable with respect to dissoci- ation into the (gas phase) atom plus an ammonia molecule by about 20 kcal/mole. (As is to be expected, it is also very stable with respect to ionization. The process of adding a second ammonia molecule (in a D3h configuratiOn) does not appear favored but two things must be considered. The Cgv configuration has not been considered. This geometric arrangement might be more suitable, but the increased repul- sion of the ammonia molecules would make it seem unlikely. Secondly, the energy differences are smaller in this case and the overall energy is larger, leading to greater uncer- tainty in the result. 152 POpulation Analysis At the outset it was decided not to alter the NH3 internal geometry since it was thought that the bonding in this structure would not be significantly perturbed by the presence of the lithium atom. The pOpulation analysis provides the first Opportunity to examine this assumption. A comparison of the N—H parameters given in Tables III to V Shows, using the same basis, they are changed very little by the presence of lithium. In Table III the net overlap between atoms is given. The interpretation of this quantity is that two atoms are bonded, non-bonded or anti-bonded to each other depending on whether this quantity is positive, zero, or negative.' The minimum basis calculation shows a positive overlap between lithium and nitrogen orbitals but the extended calculation Shows the Opposite. This result cannot be explained but is not to be accepted Since it is in disagreement with the observable, i.e., the eXpectation value Of the binding energy. Table IV contains the gross pOpulationS Of the orbitals on an atom. From this quantity one desires to learn how the charge is rearranged on forming a molecule. One can, in M O. theory, assign electron pairs to orbitals and this pOpulation breakdown is supposed to inform about which elec- trons are in which regions of the molecule. Only the valence Shells are included since the lower lying orbitals are primarily confined as they are in the constituents. 155 The result Of primary interest here is the location of the unpaired electron, but one also Obtains insight into the mechanism of the bonding of the lithium to the ammonia. Using the data in Table IV we can conclude that the unpaired electron remains with 99.4% of its pOpulation, in the lithium orbitals. Notice that the sixth orbital in the LiNH3 has an 8.8% pOpulation in the lithium orbitals, also. This orbital corresponds essentially to the lone pair of the ammonia molecule and populates the lithium orbitals by overlap with the P2 orbital of lithium. It is primarily through this mechanism that the lithium gains the excess population of 0.067 electrons which is Shown in Table V for the LiNH3. The extended basis calculations Show the same basic effects. The negative pOpulation on nitrogen given in Table VI is Similar to those attributed by Mulliken to imperfections in the definitions of the orbital pOpulations. Table IV also Shows that for the complex with two ammonias, each nitrogen gains about 2.6% of the unpaired electron, or equivalently the unpaired electron has a popula- tion of 94.4% in the lithium orbitals. Notice also in Table V that the lithium again Obtains a Significant increase in gross charge. Change Densities Charge density as a function Of position has been calcu- lated for some of those Species for which the wave function was Obtained. Two quantities have been calculated; the 154 unpaired electron density at a nucleus (related to the hyperfine splitting) and both the total and unpaired electron density in planes Of symmetry. They will be discussed separately. .Density of Unpaired Electron at an Atom. Before con— sidering the results Obtained, a discussion of the general accuracy of the method in predicting the Observed value is needed. Here, as with the energy calculation, one needs to know how errors are likely to enter and what their magnitude iS likely to be. This method has two predominant sources of error. First, the Hartree-Fock method may not give the exact solution and second, the functions which have been used as a basis for the expansion may not behave "prOperly" at the nucleus. Using lithium atoms as an example Of the accuracy available, one finds that the Hartree‘Fock solution yields a value of 2.09 (atomic units) (41) with an experi— mental value Of 2.906 (atomic units) (41) for the value of the square of the wave function (we) at the nucleus. Therefore, it can be concluded that the HartreeeFock method itself is deficient with respect to this expectation value. Using Whitten's gaussian eXpanSion of the HartreehFock solu— tion (70) gives a value of we at the nucleus of 1.55 (Table I). This happens because the gaussian function differs from the natural solution in the region of the nucleus be- cause it approaches the origin with a lepe tending toward zero whereas the natural solutions approach the origin with 155 a finite lepe. Addition of more functions to the expansion can improve this to the point of the HF solution. Thus, the small size of the value calculated from Whitten's func- tion is a result of the limited basis. These inaccuracies will also be present in other numbers calculated with a Similar basis. .Even though the absolute magnitude of the result is eXpected to be in error, the trends observed due to the formation of a complex are probably correct. The results are listed in Table I. First, the LiNH3 results will be considered. The most Obvious trend iS that the formation of the complex Significantly lowers the contact at the lithium nucleus. It is also to be noticed that the value of the unpaired density at lithium is a rather sensi- tive function of the position of the ammonia. Note that a Significant decrease in contact density occurs even though the population analysis indicates small decrease in the population of the lithium orbitals. .Notice also that the unpaired Spin density at the nitrogen is large, even though the unpaired electron transfer to the ammonia orbitals is small (Table IV). This is not a contradiction and will be commented on further when the contour maps of density are presented. Examination of the values for the diammonia complex Shows different results. 'The value of the density at lithium is smaller than for the atom but larger than for the one ammonia complex. .Note, however, that the value of 156 the Spin density at nitrogen has increased about an order of magnitude over the value for the one ammonia complex. This increase at nitrogen is consistent with the increase in unpaired electron density in the nitrogen orbitals as Shown by the population analysis (Table IV). The value at hydrogen has also increased by about the same factor even though it is still very small. Charge Densities in General. Although the value of the contact density is not well-represented by the basis used, one would eXpect the charge density in other regions to be much better represented. In Whitten's earlier work (69,70) he has Shown how well the gaussian lobe expansion represents the atomic orbitals. One might expect Similar representation of the molecular charge density; eSpecially in this case where little rearrangement is seen. The figures have been ordered so that Figures 1 through 4 Show the total charge density for the constituents and for the complexes at the geometry Of their lowest energy. I However, because the shifts in charge density on forming the complex are small, a plot of the shifts has been created by forming a difference density of the reactants minus the product. This iS shown in Figure 5 for LiNH3 and Figure 6 for Li(NH3)2. ~A word of caution is due here: the changes in charge density are a small fraction of the total charge density. .Note the magnitude Of the changes are less than 0.5 ao'3 while in some regions the total density may be as high as 180.0 ao*3. 157 Figures 5 and 6 Show that in both the one-to-One and the one-to-two complexes of lithium and ammonia, the total charge in the region of the lithium nucleus increases. In both cases, one can also see that near the nitrogen nucleus the density changes oscillate in Sign along a line connecting the two nuclei. The upper half Of the figures Shows that what is happening is a decrease of charge density in the region Of the lone pair of electrons on the ammonia and an increase in the plane perpendicular to the axis of the lone pair. This charge Shift probably results from two causes. First, the lithium atom could act repulsively toward these electrons and second, some of the charge could be transferred into the orbitals on the lithium. The trans— fer to lithium is consistent with the pOpulation analysis, and also provides an explanation of the increase of charge behind the lithium atom (relative to the ammonia position). Figures 7 through 11 are similar plots except that the density of the unpaired electron is shown. The changes in this density are a much larger fractional change than with the total charge density since the unpaired electron density is initially much smaller. The line plots of density versus position along the C3 axis Show clearly that the presence of the ammonia molecule changes the number and lo— cation of nodes in the unpaired electron wave function.) Comparison of Figures 7 and 8 also shows that the unpaired electron density peaks Sharply in the region of the nitrogen 158 nucleus, even as the bulk of the orbital tends to avoid the region of the ammonia molecule. This results from the requirement that the unpaired electron orbital remain orthogonal to the other occupied orbitals and that the region close to the nitrogen has a large density of electrons. The peak arises from the mixing Of a small amount of nitrogen orbitals to achieve this orthogonal con- dition but the total electron transfer to nitrogen is very small as is shown by the population analysis (Table IV). One can see by comparison Of Figures 8 and 9 that the frac- tion of unpaired electron in the region Of the ammonia is greater in the two—ammonia complex than in the one-ammonia complex. The peak in the region of the nitrogen is also higher in the two-ammonia complex. Figure 9 Shows that the unpaired electron avoids the ammonia when only one is present by obtaining a higher density "behind" the lithium. Since this is not possible in the two-ammonia complex, one should not be surprised to find increased occupancy of the nitrogen orbitals by this electron. -Figures 10 and 11 are difference density maps of the unpaired electron density. The changes in density are about the same magnitude as the changes in total charge density (Figures 5 and 6) but a comparison shows that the changes in a given region Of space are of Opposite Sign to the changes in total density; that is, where the total density increases, the unpaired density increases and vice versa, twith the magnitudes of the changes being Similar. 159 Influence Of the Basis on the Charge Distribution. AS mentioned earlier, there was concern about the need for a larger basis set to describe the gross features of the problem. If the unpaired electron shows any tendency to transfer to the ammonia molecule, it would probably transfer into what would be essentially a 53 orbital of nitrogen. On the other hand, if hybridization arguments are used, one might expect the lithium to form an sp hybrid resulting in promotion of the unpaired electron to a 5s of lithium, followed by donation of the lone pair of ammonia into the Sp hybrid to form the bond. Either of these schemes would change the charge distribution from that given by the minimum basis calculations because the 5S orbitals for either atomic system are poorly represented in this basis. Figure 12 represents the charge distribution calculated using different basis sets. we can see once again how the ammonia perturbs the unpaired electron distribution by com- parison with the lithium atom, but we can also see that the effects of extending the basis are rather minor. From this we conclude that the 5S orbitals of nitrogen and lithium are rather unimportant in the description of this complex. Implications for Monomer Models These calculations Show that a lithium atom transfers little charge to ammonia molecules which are near it and that ammonia bonds to a lithium atom by a coordinate covalent bond. Because alkyl groups are electron donating groups, 140 the aliphatic amines would be expected to have a charge density at the NH3 group at least as large as in ammonia. Therefore, the charge transfer to ammonia is expected to be at least as large as in the amines. .The predominant effect Of the neighbor molecules is repulsion of the metal electrons; particularly the unpaired electron. These calculations indicate that any model of the monomer which treats it as an expanded atom should consider the repulsion Of the individual solvent molecules for the unpaired electron. The charge density maps from this calcu- lation of the unpaired electron orbital Show it to have a pronounced angular dependence. Therefore, if the short range interactions Of the ammonia molecules with the unpaired electron are correctly described by a model which treats the ammonias as an array Of electrostatic point dipoles, the angular dependence Of the unpaired electron orbital which is shown in the 32 initio calculations implies that the electro— static potential terms with angular dependence are important. If a lithium atom were inserted into the solvent lattice, these calculations indicate that the value of some properties such as the hyperfine contact density at various nuclei would be sensitive to the number and position of neighboring sol- vent molecules. Furthermore, the values Of the observables might be Significantly different from the free atom values. The experimental hyperfine contact in lithium-ethylamine solutions is very much smaller than the free atom and that at the nitrogen is larger than at the lithium. It has been 141 suggested (5) that this occurs because of a spherically symmetric redistribution of the unpaired electron from lithium onto nearby solvent molecules. These calculations do not Show any mechanisms by which such behavior can be explained. In particular, if the long range solvent inter- actions are predominant in the reduction of contact at the atom, a Similar reduction in contact at the nitrogen should be expected. LITERATURE CITED (1) (2) (5) (4) (5) (6) (7) (8) (9) (10) (11) (12) (15) (14) LITERATURE CITED Abragam, A. (1961), "The Principles of Nuclear Magnetism." Amen House, London, England, Oxford University Press. Bar-Eli, K. and Tuttle, T. R., Jr. (1965), Bull. Amer. Phys. Soc. 8, 552. Bar-Eli, K. and Tuttle, T. R., Jr. (1964), J. Chem. Phys. 42. 2508 . Bar-Eli, K. and Tuttle, T. R., Jr. (1966), J. Chem. 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Thesis, Michigan State University, East Lansing, Michigan, 1965. Wentworth, W. E. (1965), J. Chem. Educ. 42, 96 and 165. Weyl, W. (1865), Pogg. Ann. 121, 601. Whitten, J. L. (1965), J. Chem. Phys. 88, 549. Whitten, J. L. (1966), J. Chem. Phys. 44, 359. Wilson, R. and Kivelson, D. (1966), J. Chem. Phys. 44, 154. APPENDICES APPENDIX I A DIFFERENTIALLY PUMPED, GREASELESS HIGH VACUUM VALVE AND END JOINT Greaseless high vacuum valves and joints Often leak small quantities of gas, especially when they are Opened or closed or parts are moved relative to one another. In this appendix a modification of commercially available valves and joints which stOpS this leakage is described. Figure 1 Shows how to construct the valve from com- mercially available valves. A new Teflon plug must be machined to fit the longer barrel and Should have "0" rings above and below the intermediate pumping chamber. When the valve is used, the bottom parts are used as with a normal valve and the top part is pumped by an auxiliary system. Both 4 and 10 mm valves have been constructed using Delmer- Urry valves. Figure 2 shows the arrangement Of parts used to con- struct a rotatable joint. The outer chamber formed when two Of these parts are joined is pumped by the same auxiliary pumping system as the valves. When constructing this joint, the two individual joints must be made COplanar and con- centric. Although 5mm and 15mm joints have been used,. 5mm and 20mm joints would be better. 146 147 Figure 1 148 Figure 2 APPENDIX II LIST OF PROGRAMS This appendix lists several computer programs for data handling and analysis which were develOped during the course Of this work. The names and functions Of these programs will be listed here. All of these programs were written in FORTRAN for the CDC-5600 computer. A G VALUE--The program calculates the best least square value of the A and g constants for an isotrOpic ESR Spectrum with hyperfine Splitting from a Single nucleus. LINSEP--This program separates overlapping first derivative ESR lines by least squares fitting a set of shape functions to the magnitude versus field data. The data from CAT is used as described in Chapter II. LINWID--This program calculates linewidth as a func— tion of mI using the formalism described in Chapter II. It fits by the criterion of least squares the fluctuation constants. The version used assumed a two—state model, but could be easily modified for more general cases. CONPLT--This is a subroutine which calculates and plots the contour maps ShOwn in Chapter III. The input data give the magnitude of the function to be plotted at a rectangular array of points in the plane of interest. 149 iii 3 03168 9023 HI YI ”III I" III E” V" NHIII I I I 3 129 IIIIIIIIII!)