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The Variety of Structures Which Interest Chemists
Chapter 1 The Variety of Structures Which Interest Chemists S. H . BAUER
I. Introduction
1
II. The Dynamic Aspects of Molecular Structures References
6 15
I. Introduction In the following "baker's dozen" of chapters on molecular properties attention is focused on those observations of the properties of matter which have been interpreted in terms of features ascribed to individual molecules. In these discussions the effects of neighboring molecules are treated by assuming that the latter produce relatively small, but not necessarily negligible, perturbations. T h e unifying concept is that the properties of molecules are determined by their individual structures. For over a century it has been recognized by chemists that to fully characterize a molecule, specification of its static structure was as essential as the statement of its molecular formula. Initially, the term was reserved for a description of the connectivity between the atoms. T h e significance of spatial relations was recognized by Pasteur in 1860, but his comments were overlooked by chemists for over a decade, until the need for a three-dimensional structure was reemphasized by Wislicemus (1873), LeBell (1874), and van't Hoff (1875). Their proposals were bitterly, but unsuccessfully attacked by Kolbe (1877). From this qualitative beginning the concept of molecular structure evolved during the past half century into a quantitative description based on a dynamic model, so that at present, average interatomic distances and root-mean-square amplitudes of vibration can be quoted in some cases to a precision of 0.002 Â. T h e 1
2
S. H. Bauer
current trend is to characterize intramolecular motions in detail by means of specific distribution and correlation functions. Also, there is ample recognition by chemists that while for each combination of atoms there is a three-dimensional arrangement of nuclei for which the total energy is a minimum, there often exist closely similar arrangements at adjacent energy minima which are separated by relatively low barriers. Prior to considering the variety of physical techniques now in use by chemists for molecular-structure investigations it will prove instructive to note their interrelationship and to indicate which questions are answered by each method. For purposes of identification of any molecular species it is necessary to specify which atom in the molecule is bonded by "valence forces'' to what other atoms. This was the major problem of Kekulé, Couper, Butlerov, and their contemporaries (see Kazansky and Bykov (1961)). T h e answers are of a topological nature. For compounds which consist primarily of hydrogen and carbon, with occasional N, O, S, and Ρ atoms, organic chemists have developed ingenious techniques for the determination of connectivity. One of the fundamental points was the early recognition of the existence of isomers for all but the simplest molecular species. * A prerequisite of this approach is the acceptance of "rules of valence" and the principle of minimum structural and configurational change induced by carefully selected structure-probing reactions. T h e method of classical structure analysis may be applied with reliability to organic compounds because of the covalent nature of their bonding; its extension to compounds which include the elements in the upper righthand corner of the periodic table and to the metalloorganics proved quite successful. However, because of the prevalence of ionic bonding in the very large group of materials identified as metallo-inorganics, this approach to structure determination generally breaks down. For carefully written analyses of the postulates inherent in the classical method for structure assignment refer to Wheland (1960) and to Senior (1935, 1938). They make the interesting observation that chemists are not perturbed by the fact that compounds have not been prepared for all the diagrams which may be written according to the formal rules of valence. T h e converse problem, that for some (classically) unique connectivities more than one compound has been isolated at first proved disconcerting; in due time the existence of geometric and stereo isomer* The term unimer has been proposed (Senior, 1951) for compounds with molecular formulas which permit no structural isomers.
1. The Variety of Structures Which Interest Chemists
3
ism was recognized. Methods for distinguishing between different conformations (in a sense, a more subtle form of isomerism) on the basis of chemical properties were brought to the fore by Barton (1950, 1951), after the existence of distinguishable conformers was demonstrated by physical techniques.* In most discussions of chemical properties little notice is taken of the fact that any macroscopic system, no matter how purified, consists of molecules which in some respects are unlike, in that every system, to be properly described, must be represented by an ensemble over a distribution in energies. It follows that whatever the property which is measured for a macroscopic system, it is an average for a distribution over a very large number of states. In some instances these states include differing connectivities between the atoms (tautometers, dimers, and Timers) which are in rapid equilibrium; in others only slightly different, but nevertheless distinct atomic configurations are involved; all these comprise the accessible conformations. Clearly, the observed magnitudes depend on the particular structural probes used in the measurements, since each procedure invokes its characteristic weighting function over the ensemble distribution. The physical methods now available for supplementing structural conclusions derived from classical chemical procedures may be arranged in a hierarchy of types. Consider first the topological statements which can be made on the basis of "additivity parameters." These were discovered by empirical correlation of bond types with tabulated heats of formation (Janz, 1967; Benson, 1965), molar refractivities (Bauer et aL, 1959), parachors (Sugden, 1930), and other physical properties, such as molar volumes, boiling points, diamagnetic susceptibilities, and melting points (Weissberger, 1959). Several of these empirical relations have been so refined that at present one may use them not only to check proposed atom connectivity, but also to test for the presence of functional groups. With regard to the latter, identification is greatly facilitated by the many empirical regularities which have been discovered in the ultraviolet, infrared, raman, and nuclear magnetic resonance spectral patterns (this volume, 1969, Chapters 3, 4, 9 ; Braude and Nachod, 1955; Nachod and Phillips, 1962). An interesting current application to the mapping * Whether different configurations of a molecule are classified as distinguishable isomers or merely as conformers appears to be an accident of the fact that general operating temperatures are in the vicinity of 300°K. When the activation energy for interconversion is of the order of 2 5 i ? T o p c r a tgi nor higher the rate of interconversion is sufficiently slow to permit their individual characterization, and then they are called isomers.
4
S. H. Bauer
of protein structures is the use of "explorer-indicator" molecules, the fluorescence spectra of which change in characteristic ways depending on the structural features of the protein site at which they are adsorbed (Stryer, 1968). T h e possibility of detecting and identifying free radicals produced in electrical discharges or by photolysis, when trapped in rare-gas matrices at liquid-helium temperatures, has led to the study of many molecular fragments and their characterization on the basis of their vibrational frequencies (Chapter 5 ) . Finally, great strides are currently being made in the use of empirically established mass spectral fragmentation patterns to identify the presence of specific groups of atoms in large molecules (McLafferty, 1966). The second catagory of methods concerns tests for various types of symmetry ; the many aspects of chemistry which are affected by molecular symmetry were summarized by Jaffe and Orchin (1965). For example, the nonequivalence of mirror images (chirality) is observable through the rotation of the plane of polarized light on passage through the specimen (Mislow, 1965; Moskowitz, 1962). Current developments in this field concern empirically discovered regularities for optical rotatory dispersion and circular dichroism, which permit the identification of chiral groups and provide information as to their relative dispositions (Ripperger, 1966; Gillard, 1966; Buckingham and Stephens, 1966). These are particularly powerful tools for conformation analysis. T h e noncoincidence of the centers of density of the positive and negative charge distributions in a molecule is discernable as a permanent electric moment (Debye, 1929; McClellan, 1963). As do nearly all other molecular properties, the magnitudes of the moments vary with time. However, even when the fluctuations are so large as to change the sign of the moment relative to some initial orientation of the molecular axis, if they occur with a period which is longer than the characteristic rotational period of the molecule, the average moment is measurable by conventional techniques. The value observed also depends on the ratio of the frequency of the exploring electric field to the fluctuation frequency in the molecule (dielectric dispersion). More precisely than ensemble averages deduced from dielectric constant data, magnitudes of dipole moments for molecules in specified vibrational states may be derived from microwave (Chapter 2 ; Wollrob, 1967) and radio frequency spectra (Chapter 13; Kusch and Hughes, 1959). T h e partial moments associated with specific bonds in molecules may be approximated from correctly measured intensities of infrared bands assigned to vibrations primarily localized in those bonds or in adjacent bond angles (Gribov and Smirov, 1962). T h e presence of
1. The Variety of Structures Which Interest Chemists
5
unpaired electrons, and, in infrequent cases, the presence of residual orbital electronic moments are detectable at the macroscopic level as paramagnetism. Langevin's analysis (1905a,b) relating the magnitude of the elementary magnetons to the macroscopic magnetic induction was adapted by Debye to polar molecules. Paramagnetic resonance spectra are now being used for probing the unpaired electron spin density distributions in a wide range of molecules (Chapter 10; Nachod and Phillips, 1962; Carrington and Luckhurst, 1968). T h e number of equivalent aspects into which a molecule may be placed by rotation, either overall or internal, i.e., the rotational symmetry number, may be verified by comparing the entropy calculated by statistical mechanics for the known molecular configuration using measured vibrational frequencies, with the value obtained from heat capacity data, on the basis of the third law (Aston, 1955). T h e first estimates of internal rotational barriers were obtained in this way during the mid 1930*8. Besides the current extensive use of nuclear magnetic resonance spectra for group identification, their study as a function of sample temperature provides information on the type and number of spin nuclei (H, F , B , 13 1 7 C, 0 ) which are present in equivalent molecular environments, as observed over a time characteristic of the measuring device, which is of the order of a millisecond. Such data are currently being exploited for conformational analysis. Similar structural information may be derived in favorable cases from Mössbauer spectra (Chapter 12), paramagnetic resonance spectra, and nuclear quadrupole spectroscopy (Chapter 11). Finally, since the point group symmetries of molecules determine selection rules for permissible transitions in the infrared and for Raman scattering (Chapters 3, 4 ; Wilson et al.y 1955), often a unique assignment of point group symmetry can be made from the observed number and coincidence of the fundamental and combination bands. However, care must be taken to apply the appropriate group-theoretic analysis to molecules which undergo rapid internal conversions (Longuet-Higgins, 1963; Hougen, 1965). For triatomic and tetratomic molecules distinctive features of their spectral bands, such as the occurrence of alternating intensities for the rotational lines, sometimes permit characterization of their structures as linear vs bent, planar vs nonplanar. Of course, the most significant structural information derived from the analysis of infrared and Raman spectra are the force constants for the normal-mode vibrations and their root-mean-square amplitudes. T h e third catagory consists of two techniques which lead to quantitative values for average interatomic distances and bond angles in molecules.
6
S. H. Bauer
One is the analysis of diffraction patterns produced by X-rays, electron or neutron waves (Chapter 14), and the second, the analysis of rotational band structures recorded with high resolution over one of several spectroscopic regions (Chapters 2, 4, 13). T h e possibility of estimating interatomic distances in relatively simple molecules from N M R spectra in nematic liquid crystal solvents is currently under development (Meiboom and Snyder, 1968). A qualitative comparison of the specific information which may be derived from these techniques is given in the following section. T h e determination of both static and dynamic structural parameters for molecules (or radicals) which have very short lifetimes is currently limited to spectroscopic methods (Herzberg, 1966). In the last catagory of physical methods for structure analysis one may group those experiments which provide information on electron charge distributions, both in the ground and in excited molecular states. This is a rapidly expanding area, but it is not an exclusive one, since information on the relative probability distributions of electrons within the molecular framework may be derived from some of the techniques included in the first three groups, as well as from novel experiments now being developed. For example, (1) the hyperfine splittings of spectral lines observed in microwave spectra (Chapter 2 ) , in electron spin resonance spectroscopy (Chapter 10), and in nuclear quadrupole spectra (Chapter 11) permit the allocation of electron spin density to specific nuclei; (2) the magnitudes of electric dipole and quadrupole moments may be interpreted as measures of local excesses or deficiencies in charge density (Chapter 8 ) ; and (3) electron-density plots obtained by superposing X ray and neutron diffraction Fourier patterns show (somewhat coarsely) localized accumulations of charge due to bonding (Chapter 14). T h e most recently introduced technique for the study of the inner electronic structure of atoms and molecules is photoelectron spectroscopy (Turner, 1966, 1968), which supplements and extends the range of conventional ultraviolet spectral analysis. For example, differences in the carbon core electron energies, such as between methane and ethane, have been measured by velocity analysis of the electrons ejected from these molecular species when the gases were exposed to soft X-rays ( H a m r i n ^ ö / . , 1968).
Π. T h e D y n a m i c Aspects o f M o l e c u l a r Structures In a classical paper Born and Oppenheimer (1927) showed that by expanding the complete wave function for a molecule in a power series
1. The Variety of Structures Which Interest Chemists
7
1/4
in ( m / M ) , where m is the mass of the electron and M is an average nuclear mass, the dominant term is a product of (1) an electronic wave function solved for clamped positions of the nuclei and (2) a nuclear wave function for which the potential energy term is the electronic energy for any specified nuclear configuration. That is, because of the large disparity between the electronic and nuclear masses, the wave function for the molecules is factorable, to a good approximation, into one which describes the electronic states for any specified set of nuclear positions, and another which describes the motion of the nuclear masses in a net potential field produced by the nuclei and electron clouds. Thus, for any w-atomic molecule there is a surface V(q1... # 3 n_ 6 ) which relates the total energy of the nuclei plus electrons to (3n — 6 ) independent nuclear position variables (3n — 5 for a linear molecule). Exceptions to the Born-Oppenheimer approximation are well known. They occur: (1) when the potential function changes very rapidly with the nuclear position variables, and (2) when there is a near-crossing of potential energy surfaces for two electronic states. T h e objective of a complete structure determination may now be stated as follows: (1) for each set of electronic quantum numbers (and particularly for the lowest energy set) locate precisely the minima in this surface (the position variables for the lowest minimum are of primary interest); (2) measure the curvatures of the surface at this minimum; and, if possible, (3) obtain estimates of the barrier heights which separate adjacent minima. It is evident that were structural measurements made even at the lowest absolute temperature, the deduced magnitudes of the position variables would be averages over the zero-point vibrations, in the lowest energy conformation. At any finite temperature the experimentally observed values are averages over all rotational and vibrational states and for all low-energy conformations, weighted by the corresponding Boltzmann distribution factors. Obviously, the numerical values depend on the weighting function characteristic of the particular probe and on the sample temperature. Hence, the most extensive exploration of the molecular potential-energy surface for a given species is achieved when it is studied both by diffraction and spectroscopic techniques, over as wide a range of temperatures as is possible. T h e unavoidable averaging can be formulated in general terms. In diffraction techniques, the coherently scattered intensity (Chapter 14) due to randomly oriented pairs of scattering centers (Î, j) separated by r{j is proportional to the characteristic weighting function W(r^) = (sin sr^jsr^, where s = (4π/Α) sin θ/2. Since λ is the wavelength of the radiation used,
8
S. H. Bauer
s has the dimensions of a reciprocal length; θ is the angle between the directions of the primary and scattered rays. T h e recorded intensity is a superposition of contributions from all such pairs /(*; r\r · · ) cc
Σ Jdj J all pairs
dri}.
P(ry -
(2.1 )
0J
By applying a Fourier transformation to the measured I(s; r%) function, r r n t ne it is possible to obtain a curve which is ΣΛιι pairs P( ij — %)- ^ above equation r\ is the separation of i and j at the minimum in the potential energy surface for the nuclei; J { and Jj are intensity factors which depend on the radiation used; and Ρ(τ^ — r%) is the probability density function which characterizes the distribution of distances r{j e about r ij. T o deduce the r?/s and to interpret the Ρ function, note that it is a product of two factors. Because all macroscopic samples consist of molecules distributed among the accessible states (in thermal equilibrium this is given by the Maxwell-Boltzmann relation), Ρ(Τ« - r'v) = ( 1 / 0 ) Σ βη[πρ(-βΕη)]ρη(τϋ
1
- τ »);
β = 1/kT.
(2.2)
η
In Eq. (2.2), η stands for a collection of quantum numbers which characterize rotational and vibrational states (for diffraction studies only the ground electronic state is sufficiently populated), grj is the degeneracy of the state specified by the energy En above the zero-point level, and Q is the partition function; ρη(τ^ — r\j) is the intrinsic probability distribution of in state η. T o evaluate ρη, it is necessary to know Ψη(£Η), the wave function which is the solution of the dynamic problem for the motion of n nuclei constrained by the potential energy surface V(qx). T h e separation of vibrational and rotational motions is not always clear cut (Chapters 6, 7 ) , particularly when there are possibilities for internal rotations, inversions, and pseudorotations. There is no unique set of qx coordinates; for instance, in addition to the angle variables which characterize the overall molecular rotation (Θ, 99, χ), one may introduce (3n — 6 ) normal coordinates if the potential function is harmonic. However, for the following discussion it is convenient to invoke the linear transformation properties of such coordinates, and to use 3« Cartesian displacement coordinates (of course, for such a set the secular equation for the vibrations has six Γ vanishing roots). Then one may replace Ψη(ς„) by Ψη(ζι.. . ξ η ΐ Λ · - *n)> e where rn locates the nth atom at the potential minimum with respect to an arbitrary coordinate system. T h e normalized probability that rl is in
1. The Variety of Structures Which Interest Chemists
9
e
the interval between ( r / + ξ χ ) and (rx + ξ ι ) + d\x is given by ^
|ΐ^Ι ^···^η-Φ,(ξ,; 2
!Ζ"'
e
e
similarly, for r 2 to be in the interval (r2 + ξ 2 ) ; (r2 + £ 2 ) + ^ζ: ^
iZ
''' J
r 1 2 = r 2« -
1
^
|
2
^ ^
3
• ''
+ ξ2 - ξ ι;
D
LN Ξ
Φ (
^
Ξ ΐ2
Γ Ι
*'''
Γ
Λ
^
ξ 2 = r 1 2 - rf 2 + ξ ι ; lall orientations
·
3 )( 2
(2.4) (2.5)
Since we are concerned only with the magnitude of r 1 2 , it is necessary to integrate over all combinations of d\x and d%2 which produce a specified I r 1 2 1 , and the integral must be averaged over all orientations of r 1 2 . The above formulation is general, but the evaluation of Eq. (2.5) is not trivial even for simple harmonic motion, when the dependence of the potential on displacements is quadratic and the distribution functions ρη and Ρ are Gaussian (Chapter 14). * It is worth noting that the consequences of anharmonicity of the potential function assume greater importance as the precision of measurement is increased. On the one hand, typical root-mean-square amplitudes of vibration are 0.050 Â or greater, even in the zero-point vibrational state, while, on the other hand, diffraction techniques are being refined so that the minimum in the potential energy surface may be located to within 0.001 Â for the (3n — 6 ) geometric parameters. This inevitably requires that the correction for anharmonicity be accurate to that level. Equations (2.3), (2.5), and, hence, (2.1) can be made explicit only when e the wave functions Ψη(ξ\ t ) for the rotational, vibrational (anharmonic), and coupling parameters are known. These are available for many diatomic molecules, and various quantitative correlations between rotational and vibrational spectroscopic constants in diatoms have been proposed, recently by Calder and Ruedenberg (1968). For polyatomic molecules only extensions by analogy are available (Chapter 14). * In terms of a mental construct, even for coupled simple harmonic oscillators, were it possible to observe the instantaneous separation between a pair of nuclei directly, that magnitude would not appear to fluctuate in a sinusoidal manner. In any state η, whatever the excitation, the vibrational motions are superposed on the zero-point motions for all the normal modes, but these occur with random phases; the net effect is a complex irregular sequence of internuclear separations, but the net result is a Gaussian distribution function.
10
S. H. Bauer
In the determination of molecular structures from spectroscopic data, whether they are radio-frequency, microwave, or high-resolution Raman, infrared, or optical band systems, one must measure spectral line fre,, quencies. While the intensities are determined by "transition moments, their positions are given by differences between the total energies of the corresponding states. T h e expectation value for the energy increment in the transition between a pair of well-defined states (η <- η) may be expressed in terms of mean molecular rotational constants (designated Αη, Βη, Cn) for the upper and the lower states. These constants are given by the reciprocals of the principal moments of inertia for the molecule in the states involved. T h e moments of inertia for the hypothetical molecule were the nuclei held rigidly at the potential minimum must be deduced from the observed rotational constants by extrapolation. For example, the rotational-term values for a slightly asymmetrical prolate rotator, are given, to a first approximation, by (Chapter 2) EJhc
= F„(J, K) = i(B + C)J{J
+ 1) + [Αη -
HB +
C\]K\ (2.6)
where J and Κ are quantum numbers which specify the angular momentum around the principal axes of rotation:
A„ = {%lAnc)(\lI y ; A
, = ( Σ M \X\
n
iQ
(2.7)
in which the averages are taken over the state 77, and the sum over all atoms i in the molecule, as located with respect to the major axis A; similarly for Βη and Cv. It appears that, to a good approximation, when η refers to vibrational quantum numbers only, 3w-6
A„ = A
t
-
Σ α^(η+
ig,),
(2.8)
where the sum is evaluated over all normal modes, and the gx are the corresponding degeneracies. Then Ae can be estimated provided a sequence of transitions in η have been observed. The following aspects of molecular spectroscopy should be noted. (1) When a complete rotational analysis can be made, the species involved is fully characterized with respect to composition, geometry, and vibrational motions. When more than one conformation is present, detailed analysis of microwave or far-infrared line splittings provide information on barrier heights for internal rotations, inversions, or ring puckering
1. The Variety of Structures Which Interest Chemists
11
motions (Chapter 2 ; Ueda and Shimanouchi, 1967, 1968; Laane and Lord, 1967; Scharpen and Laurie, 1968). However, there are limitations. (2) Selection rules for transitions between states which lead to absorption or emission of radiant energy in the radio-frequency, microwave, or infrared require that the corresponding dipole-moment matrix elements be greater than zero; for Raman spectra it is necessary that the polarizability matrix elements be greater than zero. Hence, spectroscopic techniques are not universally applicable. (3) Since at most only three principal moments of inertia characterize any molecule, for polyatomics measurements must be made on isotopically substituted species to obtain a sufficient number of independent data for a complete structure determination. (4) In order to deduce the geometric parameters at the minimum energy from the observed rotational constants, corrections must be made for anharmonicity and for vibrational-rotational interaction ; otherwise, a consistent set of position coordinates cannot be derived (Herschbach and Laurie, 1962). A measure of the magnitude of the anharmonicity present in relatively massive diatoms is provided by data on the alkali halides. For these, some of the most precisely determined internuclear distances were obT A B L E P» INTERNUCLEAR
Molecule
B
DISTANCES IN DIATOMIC ALKALI
Molecule
U (A) 1.563892 ± 0.00005
39
Li Cl 7 79 Li Br
2.02067
±
0.00006
39J£127J
2.17042
±
0.00004
7
L ii 2 7 I
2.39191
±
0.00004
NaF 35 Na Cl 79 Na Br 127 Ha I 3 9 KF
1.92593
±
0.00006
*L;F 7
35
39
K3
5
C1
2.3606
±
0.0001
2.50201
±
0.00004
2.71143
±
0.00004
2.17144
±
0.00005
2.6666
±
0.0001
85
K7
9
Br
RbF 86 35 Rb Cl 85 79 Rb Br 85 127 Rb I CsF 35 Cs Cl 79 Cs Br 127 Cs I
HALIDES
re (A) 2.82075 ± 0.00005 3.04781 ±
0.00005
2.26554 ± 0.00005 2.78670 ±
0.00006
2.94471 ± 0.00005 3.17684 ±
0.00005
2.3453
±0.0001
2.9062
±0.0001
3.07221 ± 0.00005 3.31515 ±
0.00006
° Kusch and Hughes (1959), Honig et al (1954). Inspection shows that even to within an error of ±0.005 A, these internuclear distances cannot be represented by sums of assigned radii. Honig et al. (1954) proposed 3 the empirical relation (valid to ±0.005 A ) : rc = r M ± r x — 0.410[(α Μ ± a x ) / r ] + 0.175[(χρ — xCs) — ( χ χ — XM)]> where or^ and a x are the atomic polarizabilities and the #'s are the corresponding electronegativities. b
12
S. H. Bauer
tained by the molecular-beam-radio-frequency technique. T h e r / s are listed in Table I. T h e dependence of the measured rotational constants on the vibrational quantum number for lithium bromide and for sodium fluoride are cited in Table I I . Typically, as the vibrational quantum TABLE II ROTATIONAL CONSTANTS
0
(Molecular-Beam-Radio-Frequency Spectra) Constant (MHz)
e
79
e
Li Br
19,090.296 ±0.006 18,882.800 ±0.009 18,677.242±0.055
Bo B1 B2
81
Li Br
19,057.005 ±0.006 18,850.050 ±0.020 18,645.035 ±0.217
2 3
N
ia 9
F
13,029.811 ±0.002 12,894.543 ±0.003 12,760.674±0.004
Hebert et al. (1964), Hollowell et al. (1964).
number increases, the magnitude of the rotational constant decreases by \ to 1% per quantal increment, indicating an increasing value for the mean of the reciprocal of the square of the internuclear separation. Table I I I illustrates the effect of increasing vibrational quantum number TABLE III DIPOLE MOMENTS FOR DIATOMIC ALKALI HALIDES
0
(Molecular-Beam-Radio-Frequency Spectra) Molecule e
L ii 9 F
7
19
Li F e 35 Li Cl 3 9 K3 5 23
C1
37
Na Cl 23 79 Na Br 23 127 Na I 85 85
R
1b 9
F
35
Rb Cl
1 3 3 C 1s 7 F 133
35
Cs Cl
ν = 0
ν = 1
ν = 2
6.32736(20) 6.3248(10) 7.1289(10) 10.2688(10) 9.0017(7) 9.1183(6) 9.2357(30) 8.5465(5) 10.510(5) 7.8839(9) 10.387(4)
6.41472(20) 6.4072(10) 7.2168(10) 10.3288(15) 9.0610(7) 9.1715(6) 9.2865(30) 8.6134(7) 10.564(5) 7.9546(10) 10.445(4)
6.50317(20) 6.4905(10) 7.3059(10) 10.3822(22)
° Hebert et al. (1968). Parentheses enclose error limits.
—
9.2246(6) 9.3368(30) 8.6809(9) 10.618(5) 8.0257(10) 10.503(4)
1. The Variety of Structures Which Interest Chemists
13
on the mean dipole moments for selected alkali halides. Their magnitudes also increase by approximately 1% per quantal increment. In general, the anharmonic correction is larger for molecules which consist of lighter atoms. I f the proposal of Snyder and Meiboom (1967) proves practical, such that precise geometric parameters may be derived from well-resolved N M R spectra of solutes in nematic liquid crystal solvents, a third weighting function will be available. From extensive spectral patterns they evaluate by trial and error the magnetic dipole-dipole interaction constants for pairs of nuclei z, j: Dij = Λ μ ^ φ
-
2
3 cos e y ) r ^ > a v.
(2.9)
T h e angle which the vector r{j makes with the direction of the applied field is θφ μί and μ} are the corresponding nuclear magnetic moments, and Λ is a collection of fundamental constants. By defining the task of a 3w-atomic molecular-structure determination to be the exploration of its multidimensional potential energy surface o ne ΡΧίι- · -^η-β)» limits consideration to molecules in field-free space, unperturbed by their surroundings and unconcerned with packing, stacking and space-filling problems, which are in the province of crystallographers. It is therefore essential to assess the influence of intermolecular forces on intramolecule geometry and dynamics. As expected, materials in which adjacent groups of atoms cannot be uniquely identified as molecular units (ionic structures, extended chains, sheets, or three-dimensional networks) show large changes in adjacent-atom separations upon sublimation, consistent with the near equality of their heats of sublimation and the associated bond-breaking energies. Three examples are given in Table I V . For molecular crystals, heats of sublimation are of the order 1/5 to 1/15 of the covalent bond dissociation energies, and comparable to heats of dissociation of Η bonds. In these, one may anticipate that vaporization or sublimation would, at most, induce changes in bond lengths and valence angles of a magnitude observable for the dimermonomer interconversions of formic acid (Table V ) . When only van der Waals forces are operative, changes of phase have lesser consequences, particularly on relatively rigid structures such as benzene rings (Chapter 14). In contrast to the approximate constancy of bond lengths, one should anticipate that on sublimation readily discernable changes should occur in the conformational aspects of molecular structures. There are numerous
14
S. H. Bauer TABLE
IV
STRUCTURAL CHANGES ON SUBLIMATION (IONIC OR INFINITELY EXTENDED SYSTEMS)
System
Bond length (A)
a
Lithium Chloride (Li—Cl) Crystal Dimer Monomer
2.570 2.23 2.0207
a
Lithium Bromide (Li—Br) Crystal Dimer Monomer
2.751 2.35 2.1704
Lithium Iodide (Li—1)° Crystal Dimer Monomer
3.00 2.54 2.3919
Elementary carbon (C—C) Diamond Graphite C3
c
6
1.5442 1.4210» c 1.277 d 1.3117
2 ft
Phosphorous pentachloride (P—Cl) Crystal: PC1 4+ [Td] PCI«" [O h] Vapor: P C 1 5 [D 3 h] P C 1 3 [C 3 V] [ P C l 4 + . P C l e - ] c r y s tl a a b c d
1.98 2.07 (av) 2.04±0.06 (eq) 2.19 ±0.02 (ax) 2.043 ±0.003 P C 1 5 + PCI3 +
Cl 2
Bauer et al. (1960). Bowen et al. (1958). Herzberg (1966). Herzberg (1950).
systems in which the energies of adjacent minima in the V(qH) surface differ by only a few kilocalories, and the barriers which separate them are of the order of lOkcal/mole. Since crystal forces are of comparable magnitude (i.e., favorable packing arrangement may provide energy lowering greater than the increases due to conformational distortions), the most stable free molecule structures may have significantly different
1. The Variety of Structures Which Interest Chemists TABLE V
15
e
STRUCTURAL CHANGES IN FORMIC ACID
^O—Η
Bond H—C (A) C = 0 (A) C—Ο (A) Ο—H (A)
Χ)—Η··· er
Monomer 1.103 1.216 1.343 0.973 125.5 107.6
± ± ± ± ± ±
0.001 0.001 0.001 0.001 0.7 0.6
Dimer 6
—
1.216 1.338 0.978 121.2 114.0 2.764
± ± ± ± ± ±
0.0015 0.001 0.001 1 0.4 0.001
° Bonham and Su (1968). Least squares standard deviations in matching the experimental and theoretical e Σ P(r{j — r {j) functions; reasonable error limits are 3-4 times larger. b
appearances from those in the condensed phase. Large, relatively flexible molecules undergo comparable changes when dissolved in different types of solvents, with magnitudes which depend on the concentrations. Laser beat-frequency spectroscopy is a new tool for the investigation of such effects (French et al., 1969). Finally, the more sensitive aspects of the V{qx) surface, the curvatures at the potential minimum, are definitely altered by close encounters with other molecules. Rotational motions are transformed into rotatory oscillations, and normal mode frequencies are changed by magnitudes which range from 50 to several hundred reciprocal centimeters upon condensation (Chapter 6, Section I V , B ; Chapter 7; Bellamy, 1958).
GENERAL REFERENCES
ASTON, J . G. (1955). In "Determination of Organic Structures by Physical Methods" (E.A. Braude and F . C. Nachod, eds.), Vol. 1, Chapter 13, pp. 525-564. Academic Press, New York. BAUER, N., FAJANS, K., and LEWIN, S. Ζ. (1959). In "Techniques of Organic Chemistry"
(A. Weissberger, ed.), 3rd ed., Vol. II, Pt. 2, p. 1139. Wiley (Interscience), New York. BELLAMY, L . J . (1958). "Infrared Spectra of Complex Molecules," Pt. V. Methuen, London. BENSON, S. W. (1965). J. Chem. Educ. 42, 502.
16
S. H. Bauer
BOWEN, H. J . M., et al. (1958). "Tables of Interatomic Distances and Configurations in Molecules and Ions," Spec. Publ. No. 11. Chem. Soc, London. BRAUDE, Ε. Α., and NACHOD, F. C , eds. (1955). "Determination of Structures by Physical Methods," Vol. I. Academic Press, New York. DEBYE, P. (1929). "Polar Molecules." Chem. Catalog Co., New York. HERZBERG, G. (1950). "Molecular Spectra and Molecular Structure," 2nd ed., Vol. I. Van Nostrand, Princeton, New Jersey. HERZBERG, G. (1966). "Molecular Spectra and Molecular Structure," Vol. III. Van Nostrand, Princeton, New Jersey. JAFFE, H. H., and ORCHIN, M. (1965). "Symmetry in Chemistry." Wiley, New York. JANZ, G. J . (1967). "Thermodynamic Properties of Organic Compounds." Academic Press, New York. KAZANSKY, Β . Α., and BYKOV, G. V., eds. (1961). "Centenary of the Theory of Chemical Structure." USSR Acad, of Sei., Moscow. KUSCH, P., and HUGHES, V. W. (1959). "Handbuch der Physik" (S. Flügge, ed.), Vol. 37 [1]. Springer, Berlin. MCCLELLAN, A. L. (1963). "Tables of Experimental Dipole Moments." Freeman, San Francisco, California. MCLAFFERTY, F. W. (1966). "Interpretation of Mass Spectra." Benjamin, New York. MEIBOOM, S., and SNYDER, L. C. (1968). Science 162, 1337. MISLOW, K . (1965). "Introduction to Stereochemistry." Benjamin, New York. NACHOD, F. C , and PHILLIPS, N. D. (1962). "Determination of Organic Structures by Physical Methods." Vol. II. Academic Press, New York. TURNER, D. W. (1968). In "Physical Methods in Advanced Inorganic Chemistry" (M. A. O. Hill and P. Day, eds.), Chapter 15. Wiley (Interscience), New York. WHELAND, G. N. (1960). "Advanced Organic Chemistry," 3rd ed. Wiley, New York. WILSON, Jr., Ε. B., DECIUS, J . C , and CROSS, P. C. (1955). "Molecular Vibrations." McGraw-Hill, New York. WOLLRAB, J. E. (1967). "Rotational Spectra and Molecular Structure." Academic Press, New York. SPECIAL
REFERENCES
BARTON, D. H. R. (1950). Experientia 6, 316. BARTON, D. H. R. ( 1 9 5 1 ) . / . Chem. Soc. p. 1048. BAUER, S. H., INO, T., and PORTER, R. F. ( 1 9 6 0 ) . / . Chem. Phys. 33, 685. BONHAM, R. Α., and Su, L. S. (1968). Symp. Gas Phase Mol. Structures, 2nd, Univ. of Texas, Austin, February 1968, Abstract M-l. BORN, M., and OPPENHEIMER, J . R. (1927). Ann. Phys. 84, 457. BUCKINGHAM, A. D., and STEPHENS, P. J . (1966). Ann. Rev. Phys. Chem. 17, 399. CALDER, G. V., and RUEDENBERG, K . (1968). / . Chem. Phys. 49, 5399. CARRINGTON, Α., and LUCKHURST, G. R. (1968). Ann. Rev. Phys. Chem. 19, 31. FRENCH, M. J . , ANGUS, J . C , and WALTON, A. G. (1969). Science 163,
345.
GILLARD, R. D. (1966). Progr. Inorg. Chem. 7, 215. GRIBOV, L. Α., and SMIROV, U. N. (1962). Soviet Phys. Usp. (English Transi.) 4, 919. HAMRIN, K . , JOHANSSON, G., GELIUS, U., FAHLMAN, Α., NORDLING, C , and SIEGBAHN,
Κ. (1968). Chem. Phys. Letters 1, 613. HEBERT, A. J . , BREIVOGEL, F. W., and STREET, Jr., K . (1964). J . Chem. Phys. 41, 2368.
1. The Variety of Structures Which Interest Chemists HEBERT, A. J . , LOVAS, F. J . , MELENDRES, C. Α., HOLLOWELL, C. D.,
STORY, Jr., T.
17 L.,
and STREET, Jr., K. (1968). J. Chem. Phys. 48, 2824. HERSCHBACH, D. R., and LAURIE, V. W. (1962). J. Chem. Phys. 37, 1668, 1687. HOLLOWELL, C. D., HEBERT, A. J., and STREET, Jr., K. ( 1 9 6 4 ) . / . Chem. Phys. 41, 3540. HONIG, Α., MANDEL, M . , STITCH, M . L., and TOWNES, C. H . (1954). Phys. Rev. 96, 629. HOUGEN, T. J . (1965). Pure Appl. Chem. 11, 481. KOLBE, H . (1877). / . Prakt. Chem. [2] 15, 473. LAANE, J . , and LORD, R. C. (1967). / . Chem. Phys. 47, 4941. LANGEVIN, P. (1905a). J. Phys. 4, 678. LANGEVIN, P. (1905b). Ann. Chim. Phys. 5 (8), 70. LEBELL, J . A. (1874). Bull. Soc. Chim. [2] 22, 337. LONGUETT-HIGGINS, H . C. (1963). Mol. Phys. 6, 445. MOSKOWITZ, A. (1962). Advan. Chem. Phys. 4, 67. RIPPERGER, H . (1966). Z. Chem. 6, 161. SCHARPEN, L. H., and LAURIE, V. W. ( 1 9 6 8 ) . / . Chem. Phys. 49, 221. SENIOR, J . K . (1935). J. Chem. Educ. 12, 409, 465. SENIOR, J . K . (1938). J. Chem. Educ. 15, 464. SENIOR, J . K . ( 1 9 5 1 ) . / . Chem. Phys. 19, 865. SNYDER, L. C , and MEIBOOM, S. (1967). Chem. Phys. 47, 1480. STRYER, L. (1968). Science 162, 526. SUGDEN, S. (1930). "The Parachor and Valency." Routledge, London. TURNER, D. W. (1966). Advan. Phys. Org. Chem. 4, 31. UEDA, T., and SHIMANOUCHI, T. ( 1 9 6 7 ) . / . Chem. Phys. 47, 5018. UEDA, T., and SHIMANOUCHI, T. (1968). / . Chem. Phys. 49, 470. VAN'T HOFF, J . H . (1875). Bull. Soc. Chim. [2] 23, 295. WEISSBERGER, Α., ed. (1959). "Techniques of Organic Chemistry," 3rd ed., Vol. II. Wiley (Interscience), New York. WISLICEMUS, J . (1873). Ann.
167,
302.
I. Introduction
1
II. The Dynamic Aspects of Molecular Structures References
6 15
I. Introduction In the following "baker's dozen" of chapters on molecular properties attention is focused on those observations of the properties of matter which have been interpreted in terms of features ascribed to individual molecules. In these discussions the effects of neighboring molecules are treated by assuming that the latter produce relatively small, but not necessarily negligible, perturbations. T h e unifying concept is that the properties of molecules are determined by their individual structures. For over a century it has been recognized by chemists that to fully characterize a molecule, specification of its static structure was as essential as the statement of its molecular formula. Initially, the term was reserved for a description of the connectivity between the atoms. T h e significance of spatial relations was recognized by Pasteur in 1860, but his comments were overlooked by chemists for over a decade, until the need for a three-dimensional structure was reemphasized by Wislicemus (1873), LeBell (1874), and van't Hoff (1875). Their proposals were bitterly, but unsuccessfully attacked by Kolbe (1877). From this qualitative beginning the concept of molecular structure evolved during the past half century into a quantitative description based on a dynamic model, so that at present, average interatomic distances and root-mean-square amplitudes of vibration can be quoted in some cases to a precision of 0.002 Â. T h e 1
2
S. H. Bauer
current trend is to characterize intramolecular motions in detail by means of specific distribution and correlation functions. Also, there is ample recognition by chemists that while for each combination of atoms there is a three-dimensional arrangement of nuclei for which the total energy is a minimum, there often exist closely similar arrangements at adjacent energy minima which are separated by relatively low barriers. Prior to considering the variety of physical techniques now in use by chemists for molecular-structure investigations it will prove instructive to note their interrelationship and to indicate which questions are answered by each method. For purposes of identification of any molecular species it is necessary to specify which atom in the molecule is bonded by "valence forces'' to what other atoms. This was the major problem of Kekulé, Couper, Butlerov, and their contemporaries (see Kazansky and Bykov (1961)). T h e answers are of a topological nature. For compounds which consist primarily of hydrogen and carbon, with occasional N, O, S, and Ρ atoms, organic chemists have developed ingenious techniques for the determination of connectivity. One of the fundamental points was the early recognition of the existence of isomers for all but the simplest molecular species. * A prerequisite of this approach is the acceptance of "rules of valence" and the principle of minimum structural and configurational change induced by carefully selected structure-probing reactions. T h e method of classical structure analysis may be applied with reliability to organic compounds because of the covalent nature of their bonding; its extension to compounds which include the elements in the upper righthand corner of the periodic table and to the metalloorganics proved quite successful. However, because of the prevalence of ionic bonding in the very large group of materials identified as metallo-inorganics, this approach to structure determination generally breaks down. For carefully written analyses of the postulates inherent in the classical method for structure assignment refer to Wheland (1960) and to Senior (1935, 1938). They make the interesting observation that chemists are not perturbed by the fact that compounds have not been prepared for all the diagrams which may be written according to the formal rules of valence. T h e converse problem, that for some (classically) unique connectivities more than one compound has been isolated at first proved disconcerting; in due time the existence of geometric and stereo isomer* The term unimer has been proposed (Senior, 1951) for compounds with molecular formulas which permit no structural isomers.
1. The Variety of Structures Which Interest Chemists
3
ism was recognized. Methods for distinguishing between different conformations (in a sense, a more subtle form of isomerism) on the basis of chemical properties were brought to the fore by Barton (1950, 1951), after the existence of distinguishable conformers was demonstrated by physical techniques.* In most discussions of chemical properties little notice is taken of the fact that any macroscopic system, no matter how purified, consists of molecules which in some respects are unlike, in that every system, to be properly described, must be represented by an ensemble over a distribution in energies. It follows that whatever the property which is measured for a macroscopic system, it is an average for a distribution over a very large number of states. In some instances these states include differing connectivities between the atoms (tautometers, dimers, and Timers) which are in rapid equilibrium; in others only slightly different, but nevertheless distinct atomic configurations are involved; all these comprise the accessible conformations. Clearly, the observed magnitudes depend on the particular structural probes used in the measurements, since each procedure invokes its characteristic weighting function over the ensemble distribution. The physical methods now available for supplementing structural conclusions derived from classical chemical procedures may be arranged in a hierarchy of types. Consider first the topological statements which can be made on the basis of "additivity parameters." These were discovered by empirical correlation of bond types with tabulated heats of formation (Janz, 1967; Benson, 1965), molar refractivities (Bauer et aL, 1959), parachors (Sugden, 1930), and other physical properties, such as molar volumes, boiling points, diamagnetic susceptibilities, and melting points (Weissberger, 1959). Several of these empirical relations have been so refined that at present one may use them not only to check proposed atom connectivity, but also to test for the presence of functional groups. With regard to the latter, identification is greatly facilitated by the many empirical regularities which have been discovered in the ultraviolet, infrared, raman, and nuclear magnetic resonance spectral patterns (this volume, 1969, Chapters 3, 4, 9 ; Braude and Nachod, 1955; Nachod and Phillips, 1962). An interesting current application to the mapping * Whether different configurations of a molecule are classified as distinguishable isomers or merely as conformers appears to be an accident of the fact that general operating temperatures are in the vicinity of 300°K. When the activation energy for interconversion is of the order of 2 5 i ? T o p c r a tgi nor higher the rate of interconversion is sufficiently slow to permit their individual characterization, and then they are called isomers.
4
S. H. Bauer
of protein structures is the use of "explorer-indicator" molecules, the fluorescence spectra of which change in characteristic ways depending on the structural features of the protein site at which they are adsorbed (Stryer, 1968). T h e possibility of detecting and identifying free radicals produced in electrical discharges or by photolysis, when trapped in rare-gas matrices at liquid-helium temperatures, has led to the study of many molecular fragments and their characterization on the basis of their vibrational frequencies (Chapter 5 ) . Finally, great strides are currently being made in the use of empirically established mass spectral fragmentation patterns to identify the presence of specific groups of atoms in large molecules (McLafferty, 1966). The second catagory of methods concerns tests for various types of symmetry ; the many aspects of chemistry which are affected by molecular symmetry were summarized by Jaffe and Orchin (1965). For example, the nonequivalence of mirror images (chirality) is observable through the rotation of the plane of polarized light on passage through the specimen (Mislow, 1965; Moskowitz, 1962). Current developments in this field concern empirically discovered regularities for optical rotatory dispersion and circular dichroism, which permit the identification of chiral groups and provide information as to their relative dispositions (Ripperger, 1966; Gillard, 1966; Buckingham and Stephens, 1966). These are particularly powerful tools for conformation analysis. T h e noncoincidence of the centers of density of the positive and negative charge distributions in a molecule is discernable as a permanent electric moment (Debye, 1929; McClellan, 1963). As do nearly all other molecular properties, the magnitudes of the moments vary with time. However, even when the fluctuations are so large as to change the sign of the moment relative to some initial orientation of the molecular axis, if they occur with a period which is longer than the characteristic rotational period of the molecule, the average moment is measurable by conventional techniques. The value observed also depends on the ratio of the frequency of the exploring electric field to the fluctuation frequency in the molecule (dielectric dispersion). More precisely than ensemble averages deduced from dielectric constant data, magnitudes of dipole moments for molecules in specified vibrational states may be derived from microwave (Chapter 2 ; Wollrob, 1967) and radio frequency spectra (Chapter 13; Kusch and Hughes, 1959). T h e partial moments associated with specific bonds in molecules may be approximated from correctly measured intensities of infrared bands assigned to vibrations primarily localized in those bonds or in adjacent bond angles (Gribov and Smirov, 1962). T h e presence of
1. The Variety of Structures Which Interest Chemists
5
unpaired electrons, and, in infrequent cases, the presence of residual orbital electronic moments are detectable at the macroscopic level as paramagnetism. Langevin's analysis (1905a,b) relating the magnitude of the elementary magnetons to the macroscopic magnetic induction was adapted by Debye to polar molecules. Paramagnetic resonance spectra are now being used for probing the unpaired electron spin density distributions in a wide range of molecules (Chapter 10; Nachod and Phillips, 1962; Carrington and Luckhurst, 1968). T h e number of equivalent aspects into which a molecule may be placed by rotation, either overall or internal, i.e., the rotational symmetry number, may be verified by comparing the entropy calculated by statistical mechanics for the known molecular configuration using measured vibrational frequencies, with the value obtained from heat capacity data, on the basis of the third law (Aston, 1955). T h e first estimates of internal rotational barriers were obtained in this way during the mid 1930*8. Besides the current extensive use of nuclear magnetic resonance spectra for group identification, their study as a function of sample temperature provides information on the type and number of spin nuclei (H, F , B , 13 1 7 C, 0 ) which are present in equivalent molecular environments, as observed over a time characteristic of the measuring device, which is of the order of a millisecond. Such data are currently being exploited for conformational analysis. Similar structural information may be derived in favorable cases from Mössbauer spectra (Chapter 12), paramagnetic resonance spectra, and nuclear quadrupole spectroscopy (Chapter 11). Finally, since the point group symmetries of molecules determine selection rules for permissible transitions in the infrared and for Raman scattering (Chapters 3, 4 ; Wilson et al.y 1955), often a unique assignment of point group symmetry can be made from the observed number and coincidence of the fundamental and combination bands. However, care must be taken to apply the appropriate group-theoretic analysis to molecules which undergo rapid internal conversions (Longuet-Higgins, 1963; Hougen, 1965). For triatomic and tetratomic molecules distinctive features of their spectral bands, such as the occurrence of alternating intensities for the rotational lines, sometimes permit characterization of their structures as linear vs bent, planar vs nonplanar. Of course, the most significant structural information derived from the analysis of infrared and Raman spectra are the force constants for the normal-mode vibrations and their root-mean-square amplitudes. T h e third catagory consists of two techniques which lead to quantitative values for average interatomic distances and bond angles in molecules.
6
S. H. Bauer
One is the analysis of diffraction patterns produced by X-rays, electron or neutron waves (Chapter 14), and the second, the analysis of rotational band structures recorded with high resolution over one of several spectroscopic regions (Chapters 2, 4, 13). T h e possibility of estimating interatomic distances in relatively simple molecules from N M R spectra in nematic liquid crystal solvents is currently under development (Meiboom and Snyder, 1968). A qualitative comparison of the specific information which may be derived from these techniques is given in the following section. T h e determination of both static and dynamic structural parameters for molecules (or radicals) which have very short lifetimes is currently limited to spectroscopic methods (Herzberg, 1966). In the last catagory of physical methods for structure analysis one may group those experiments which provide information on electron charge distributions, both in the ground and in excited molecular states. This is a rapidly expanding area, but it is not an exclusive one, since information on the relative probability distributions of electrons within the molecular framework may be derived from some of the techniques included in the first three groups, as well as from novel experiments now being developed. For example, (1) the hyperfine splittings of spectral lines observed in microwave spectra (Chapter 2 ) , in electron spin resonance spectroscopy (Chapter 10), and in nuclear quadrupole spectra (Chapter 11) permit the allocation of electron spin density to specific nuclei; (2) the magnitudes of electric dipole and quadrupole moments may be interpreted as measures of local excesses or deficiencies in charge density (Chapter 8 ) ; and (3) electron-density plots obtained by superposing X ray and neutron diffraction Fourier patterns show (somewhat coarsely) localized accumulations of charge due to bonding (Chapter 14). T h e most recently introduced technique for the study of the inner electronic structure of atoms and molecules is photoelectron spectroscopy (Turner, 1966, 1968), which supplements and extends the range of conventional ultraviolet spectral analysis. For example, differences in the carbon core electron energies, such as between methane and ethane, have been measured by velocity analysis of the electrons ejected from these molecular species when the gases were exposed to soft X-rays ( H a m r i n ^ ö / . , 1968).
Π. T h e D y n a m i c Aspects o f M o l e c u l a r Structures In a classical paper Born and Oppenheimer (1927) showed that by expanding the complete wave function for a molecule in a power series
1. The Variety of Structures Which Interest Chemists
7
1/4
in ( m / M ) , where m is the mass of the electron and M is an average nuclear mass, the dominant term is a product of (1) an electronic wave function solved for clamped positions of the nuclei and (2) a nuclear wave function for which the potential energy term is the electronic energy for any specified nuclear configuration. That is, because of the large disparity between the electronic and nuclear masses, the wave function for the molecules is factorable, to a good approximation, into one which describes the electronic states for any specified set of nuclear positions, and another which describes the motion of the nuclear masses in a net potential field produced by the nuclei and electron clouds. Thus, for any w-atomic molecule there is a surface V(q1... # 3 n_ 6 ) which relates the total energy of the nuclei plus electrons to (3n — 6 ) independent nuclear position variables (3n — 5 for a linear molecule). Exceptions to the Born-Oppenheimer approximation are well known. They occur: (1) when the potential function changes very rapidly with the nuclear position variables, and (2) when there is a near-crossing of potential energy surfaces for two electronic states. T h e objective of a complete structure determination may now be stated as follows: (1) for each set of electronic quantum numbers (and particularly for the lowest energy set) locate precisely the minima in this surface (the position variables for the lowest minimum are of primary interest); (2) measure the curvatures of the surface at this minimum; and, if possible, (3) obtain estimates of the barrier heights which separate adjacent minima. It is evident that were structural measurements made even at the lowest absolute temperature, the deduced magnitudes of the position variables would be averages over the zero-point vibrations, in the lowest energy conformation. At any finite temperature the experimentally observed values are averages over all rotational and vibrational states and for all low-energy conformations, weighted by the corresponding Boltzmann distribution factors. Obviously, the numerical values depend on the weighting function characteristic of the particular probe and on the sample temperature. Hence, the most extensive exploration of the molecular potential-energy surface for a given species is achieved when it is studied both by diffraction and spectroscopic techniques, over as wide a range of temperatures as is possible. T h e unavoidable averaging can be formulated in general terms. In diffraction techniques, the coherently scattered intensity (Chapter 14) due to randomly oriented pairs of scattering centers (Î, j) separated by r{j is proportional to the characteristic weighting function W(r^) = (sin sr^jsr^, where s = (4π/Α) sin θ/2. Since λ is the wavelength of the radiation used,
8
S. H. Bauer
s has the dimensions of a reciprocal length; θ is the angle between the directions of the primary and scattered rays. T h e recorded intensity is a superposition of contributions from all such pairs /(*; r\r · · ) cc
Σ Jdj J all pairs
dri}.
P(ry -
(2.1 )
0J
By applying a Fourier transformation to the measured I(s; r%) function, r r n t ne it is possible to obtain a curve which is ΣΛιι pairs P( ij — %)- ^ above equation r\ is the separation of i and j at the minimum in the potential energy surface for the nuclei; J { and Jj are intensity factors which depend on the radiation used; and Ρ(τ^ — r%) is the probability density function which characterizes the distribution of distances r{j e about r ij. T o deduce the r?/s and to interpret the Ρ function, note that it is a product of two factors. Because all macroscopic samples consist of molecules distributed among the accessible states (in thermal equilibrium this is given by the Maxwell-Boltzmann relation), Ρ(Τ« - r'v) = ( 1 / 0 ) Σ βη[πρ(-βΕη)]ρη(τϋ
1
- τ »);
β = 1/kT.
(2.2)
η
In Eq. (2.2), η stands for a collection of quantum numbers which characterize rotational and vibrational states (for diffraction studies only the ground electronic state is sufficiently populated), grj is the degeneracy of the state specified by the energy En above the zero-point level, and Q is the partition function; ρη(τ^ — r\j) is the intrinsic probability distribution of in state η. T o evaluate ρη, it is necessary to know Ψη(£Η), the wave function which is the solution of the dynamic problem for the motion of n nuclei constrained by the potential energy surface V(qx). T h e separation of vibrational and rotational motions is not always clear cut (Chapters 6, 7 ) , particularly when there are possibilities for internal rotations, inversions, and pseudorotations. There is no unique set of qx coordinates; for instance, in addition to the angle variables which characterize the overall molecular rotation (Θ, 99, χ), one may introduce (3n — 6 ) normal coordinates if the potential function is harmonic. However, for the following discussion it is convenient to invoke the linear transformation properties of such coordinates, and to use 3« Cartesian displacement coordinates (of course, for such a set the secular equation for the vibrations has six Γ vanishing roots). Then one may replace Ψη(ς„) by Ψη(ζι.. . ξ η ΐ Λ · - *n)> e where rn locates the nth atom at the potential minimum with respect to an arbitrary coordinate system. T h e normalized probability that rl is in
1. The Variety of Structures Which Interest Chemists
9
e
the interval between ( r / + ξ χ ) and (rx + ξ ι ) + d\x is given by ^
|ΐ^Ι ^···^η-Φ,(ξ,; 2
!Ζ"'
e
e
similarly, for r 2 to be in the interval (r2 + ξ 2 ) ; (r2 + £ 2 ) + ^ζ: ^
iZ
''' J
r 1 2 = r 2« -
1
^
|
2
^ ^
3
• ''
+ ξ2 - ξ ι;
D
LN Ξ
Φ (
^
Ξ ΐ2
Γ Ι
*'''
Γ
Λ
^
ξ 2 = r 1 2 - rf 2 + ξ ι ; lall orientations
·
3 )( 2
(2.4) (2.5)
Since we are concerned only with the magnitude of r 1 2 , it is necessary to integrate over all combinations of d\x and d%2 which produce a specified I r 1 2 1 , and the integral must be averaged over all orientations of r 1 2 . The above formulation is general, but the evaluation of Eq. (2.5) is not trivial even for simple harmonic motion, when the dependence of the potential on displacements is quadratic and the distribution functions ρη and Ρ are Gaussian (Chapter 14). * It is worth noting that the consequences of anharmonicity of the potential function assume greater importance as the precision of measurement is increased. On the one hand, typical root-mean-square amplitudes of vibration are 0.050 Â or greater, even in the zero-point vibrational state, while, on the other hand, diffraction techniques are being refined so that the minimum in the potential energy surface may be located to within 0.001 Â for the (3n — 6 ) geometric parameters. This inevitably requires that the correction for anharmonicity be accurate to that level. Equations (2.3), (2.5), and, hence, (2.1) can be made explicit only when e the wave functions Ψη(ξ\ t ) for the rotational, vibrational (anharmonic), and coupling parameters are known. These are available for many diatomic molecules, and various quantitative correlations between rotational and vibrational spectroscopic constants in diatoms have been proposed, recently by Calder and Ruedenberg (1968). For polyatomic molecules only extensions by analogy are available (Chapter 14). * In terms of a mental construct, even for coupled simple harmonic oscillators, were it possible to observe the instantaneous separation between a pair of nuclei directly, that magnitude would not appear to fluctuate in a sinusoidal manner. In any state η, whatever the excitation, the vibrational motions are superposed on the zero-point motions for all the normal modes, but these occur with random phases; the net effect is a complex irregular sequence of internuclear separations, but the net result is a Gaussian distribution function.
10
S. H. Bauer
In the determination of molecular structures from spectroscopic data, whether they are radio-frequency, microwave, or high-resolution Raman, infrared, or optical band systems, one must measure spectral line fre,, quencies. While the intensities are determined by "transition moments, their positions are given by differences between the total energies of the corresponding states. T h e expectation value for the energy increment in the transition between a pair of well-defined states (η <- η) may be expressed in terms of mean molecular rotational constants (designated Αη, Βη, Cn) for the upper and the lower states. These constants are given by the reciprocals of the principal moments of inertia for the molecule in the states involved. T h e moments of inertia for the hypothetical molecule were the nuclei held rigidly at the potential minimum must be deduced from the observed rotational constants by extrapolation. For example, the rotational-term values for a slightly asymmetrical prolate rotator, are given, to a first approximation, by (Chapter 2) EJhc
= F„(J, K) = i(B + C)J{J
+ 1) + [Αη -
HB +
C\]K\ (2.6)
where J and Κ are quantum numbers which specify the angular momentum around the principal axes of rotation:
A„ = {%lAnc)(\lI y ; A
n
iQ
(2.7)
in which the averages are taken over the state 77, and the sum over all atoms i in the molecule, as located with respect to the major axis A; similarly for Βη and Cv. It appears that, to a good approximation, when η refers to vibrational quantum numbers only, 3w-6
A„ = A
t
-
Σ α^(η+
ig,),
(2.8)
where the sum is evaluated over all normal modes, and the gx are the corresponding degeneracies. Then Ae can be estimated provided a sequence of transitions in η have been observed. The following aspects of molecular spectroscopy should be noted. (1) When a complete rotational analysis can be made, the species involved is fully characterized with respect to composition, geometry, and vibrational motions. When more than one conformation is present, detailed analysis of microwave or far-infrared line splittings provide information on barrier heights for internal rotations, inversions, or ring puckering
1. The Variety of Structures Which Interest Chemists
11
motions (Chapter 2 ; Ueda and Shimanouchi, 1967, 1968; Laane and Lord, 1967; Scharpen and Laurie, 1968). However, there are limitations. (2) Selection rules for transitions between states which lead to absorption or emission of radiant energy in the radio-frequency, microwave, or infrared require that the corresponding dipole-moment matrix elements be greater than zero; for Raman spectra it is necessary that the polarizability matrix elements be greater than zero. Hence, spectroscopic techniques are not universally applicable. (3) Since at most only three principal moments of inertia characterize any molecule, for polyatomics measurements must be made on isotopically substituted species to obtain a sufficient number of independent data for a complete structure determination. (4) In order to deduce the geometric parameters at the minimum energy from the observed rotational constants, corrections must be made for anharmonicity and for vibrational-rotational interaction ; otherwise, a consistent set of position coordinates cannot be derived (Herschbach and Laurie, 1962). A measure of the magnitude of the anharmonicity present in relatively massive diatoms is provided by data on the alkali halides. For these, some of the most precisely determined internuclear distances were obT A B L E P» INTERNUCLEAR
Molecule
B
DISTANCES IN DIATOMIC ALKALI
Molecule
U (A) 1.563892 ± 0.00005
39
Li Cl 7 79 Li Br
2.02067
±
0.00006
39J£127J
2.17042
±
0.00004
7
L ii 2 7 I
2.39191
±
0.00004
NaF 35 Na Cl 79 Na Br 127 Ha I 3 9 KF
1.92593
±
0.00006
*L;F 7
35
39
K3
5
C1
2.3606
±
0.0001
2.50201
±
0.00004
2.71143
±
0.00004
2.17144
±
0.00005
2.6666
±
0.0001
85
K7
9
Br
RbF 86 35 Rb Cl 85 79 Rb Br 85 127 Rb I CsF 35 Cs Cl 79 Cs Br 127 Cs I
HALIDES
re (A) 2.82075 ± 0.00005 3.04781 ±
0.00005
2.26554 ± 0.00005 2.78670 ±
0.00006
2.94471 ± 0.00005 3.17684 ±
0.00005
2.3453
±0.0001
2.9062
±0.0001
3.07221 ± 0.00005 3.31515 ±
0.00006
° Kusch and Hughes (1959), Honig et al (1954). Inspection shows that even to within an error of ±0.005 A, these internuclear distances cannot be represented by sums of assigned radii. Honig et al. (1954) proposed 3 the empirical relation (valid to ±0.005 A ) : rc = r M ± r x — 0.410[(α Μ ± a x ) / r ] + 0.175[(χρ — xCs) — ( χ χ — XM)]> where or^ and a x are the atomic polarizabilities and the #'s are the corresponding electronegativities. b
12
S. H. Bauer
tained by the molecular-beam-radio-frequency technique. T h e r / s are listed in Table I. T h e dependence of the measured rotational constants on the vibrational quantum number for lithium bromide and for sodium fluoride are cited in Table I I . Typically, as the vibrational quantum TABLE II ROTATIONAL CONSTANTS
0
(Molecular-Beam-Radio-Frequency Spectra) Constant (MHz)
e
79
e
Li Br
19,090.296 ±0.006 18,882.800 ±0.009 18,677.242±0.055
Bo B1 B2
81
Li Br
19,057.005 ±0.006 18,850.050 ±0.020 18,645.035 ±0.217
2 3
N
ia 9
F
13,029.811 ±0.002 12,894.543 ±0.003 12,760.674±0.004
Hebert et al. (1964), Hollowell et al. (1964).
number increases, the magnitude of the rotational constant decreases by \ to 1% per quantal increment, indicating an increasing value for the mean of the reciprocal of the square of the internuclear separation. Table I I I illustrates the effect of increasing vibrational quantum number TABLE III DIPOLE MOMENTS FOR DIATOMIC ALKALI HALIDES
0
(Molecular-Beam-Radio-Frequency Spectra) Molecule e
L ii 9 F
7
19
Li F e 35 Li Cl 3 9 K3 5 23
C1
37
Na Cl 23 79 Na Br 23 127 Na I 85 85
R
1b 9
F
35
Rb Cl
1 3 3 C 1s 7 F 133
35
Cs Cl
ν = 0
ν = 1
ν = 2
6.32736(20) 6.3248(10) 7.1289(10) 10.2688(10) 9.0017(7) 9.1183(6) 9.2357(30) 8.5465(5) 10.510(5) 7.8839(9) 10.387(4)
6.41472(20) 6.4072(10) 7.2168(10) 10.3288(15) 9.0610(7) 9.1715(6) 9.2865(30) 8.6134(7) 10.564(5) 7.9546(10) 10.445(4)
6.50317(20) 6.4905(10) 7.3059(10) 10.3822(22)
° Hebert et al. (1968). Parentheses enclose error limits.
—
9.2246(6) 9.3368(30) 8.6809(9) 10.618(5) 8.0257(10) 10.503(4)
1. The Variety of Structures Which Interest Chemists
13
on the mean dipole moments for selected alkali halides. Their magnitudes also increase by approximately 1% per quantal increment. In general, the anharmonic correction is larger for molecules which consist of lighter atoms. I f the proposal of Snyder and Meiboom (1967) proves practical, such that precise geometric parameters may be derived from well-resolved N M R spectra of solutes in nematic liquid crystal solvents, a third weighting function will be available. From extensive spectral patterns they evaluate by trial and error the magnetic dipole-dipole interaction constants for pairs of nuclei z, j: Dij = Λ μ ^ φ
-
2
3 cos e y ) r ^ > a v.
(2.9)
T h e angle which the vector r{j makes with the direction of the applied field is θφ μί and μ} are the corresponding nuclear magnetic moments, and Λ is a collection of fundamental constants. By defining the task of a 3w-atomic molecular-structure determination to be the exploration of its multidimensional potential energy surface o ne ΡΧίι- · -^η-β)» limits consideration to molecules in field-free space, unperturbed by their surroundings and unconcerned with packing, stacking and space-filling problems, which are in the province of crystallographers. It is therefore essential to assess the influence of intermolecular forces on intramolecule geometry and dynamics. As expected, materials in which adjacent groups of atoms cannot be uniquely identified as molecular units (ionic structures, extended chains, sheets, or three-dimensional networks) show large changes in adjacent-atom separations upon sublimation, consistent with the near equality of their heats of sublimation and the associated bond-breaking energies. Three examples are given in Table I V . For molecular crystals, heats of sublimation are of the order 1/5 to 1/15 of the covalent bond dissociation energies, and comparable to heats of dissociation of Η bonds. In these, one may anticipate that vaporization or sublimation would, at most, induce changes in bond lengths and valence angles of a magnitude observable for the dimermonomer interconversions of formic acid (Table V ) . When only van der Waals forces are operative, changes of phase have lesser consequences, particularly on relatively rigid structures such as benzene rings (Chapter 14). In contrast to the approximate constancy of bond lengths, one should anticipate that on sublimation readily discernable changes should occur in the conformational aspects of molecular structures. There are numerous
14
S. H. Bauer TABLE
IV
STRUCTURAL CHANGES ON SUBLIMATION (IONIC OR INFINITELY EXTENDED SYSTEMS)
System
Bond length (A)
a
Lithium Chloride (Li—Cl) Crystal Dimer Monomer
2.570 2.23 2.0207
a
Lithium Bromide (Li—Br) Crystal Dimer Monomer
2.751 2.35 2.1704
Lithium Iodide (Li—1)° Crystal Dimer Monomer
3.00 2.54 2.3919
Elementary carbon (C—C) Diamond Graphite C3
c
6
1.5442 1.4210» c 1.277 d 1.3117
2 ft
Phosphorous pentachloride (P—Cl) Crystal: PC1 4+ [Td] PCI«" [O h] Vapor: P C 1 5 [D 3 h] P C 1 3 [C 3 V] [ P C l 4 + . P C l e - ] c r y s tl a a b c d
1.98 2.07 (av) 2.04±0.06 (eq) 2.19 ±0.02 (ax) 2.043 ±0.003 P C 1 5 + PCI3 +
Cl 2
Bauer et al. (1960). Bowen et al. (1958). Herzberg (1966). Herzberg (1950).
systems in which the energies of adjacent minima in the V(qH) surface differ by only a few kilocalories, and the barriers which separate them are of the order of lOkcal/mole. Since crystal forces are of comparable magnitude (i.e., favorable packing arrangement may provide energy lowering greater than the increases due to conformational distortions), the most stable free molecule structures may have significantly different
1. The Variety of Structures Which Interest Chemists TABLE V
15
e
STRUCTURAL CHANGES IN FORMIC ACID
^O—Η
Bond H—C (A) C = 0 (A) C—Ο (A) Ο—H (A)
Χ)—Η··· er
Monomer 1.103 1.216 1.343 0.973 125.5 107.6
± ± ± ± ± ±
0.001 0.001 0.001 0.001 0.7 0.6
Dimer 6
—
1.216 1.338 0.978 121.2 114.0 2.764
± ± ± ± ± ±
0.0015 0.001 0.001 1 0.4 0.001
° Bonham and Su (1968). Least squares standard deviations in matching the experimental and theoretical e Σ P(r{j — r {j) functions; reasonable error limits are 3-4 times larger. b
appearances from those in the condensed phase. Large, relatively flexible molecules undergo comparable changes when dissolved in different types of solvents, with magnitudes which depend on the concentrations. Laser beat-frequency spectroscopy is a new tool for the investigation of such effects (French et al., 1969). Finally, the more sensitive aspects of the V{qx) surface, the curvatures at the potential minimum, are definitely altered by close encounters with other molecules. Rotational motions are transformed into rotatory oscillations, and normal mode frequencies are changed by magnitudes which range from 50 to several hundred reciprocal centimeters upon condensation (Chapter 6, Section I V , B ; Chapter 7; Bellamy, 1958).
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S. H. Bauer
BOWEN, H. J . M., et al. (1958). "Tables of Interatomic Distances and Configurations in Molecules and Ions," Spec. Publ. No. 11. Chem. Soc, London. BRAUDE, Ε. Α., and NACHOD, F. C , eds. (1955). "Determination of Structures by Physical Methods," Vol. I. Academic Press, New York. DEBYE, P. (1929). "Polar Molecules." Chem. Catalog Co., New York. HERZBERG, G. (1950). "Molecular Spectra and Molecular Structure," 2nd ed., Vol. I. Van Nostrand, Princeton, New Jersey. HERZBERG, G. (1966). "Molecular Spectra and Molecular Structure," Vol. III. Van Nostrand, Princeton, New Jersey. JAFFE, H. H., and ORCHIN, M. (1965). "Symmetry in Chemistry." Wiley, New York. JANZ, G. J . (1967). "Thermodynamic Properties of Organic Compounds." Academic Press, New York. KAZANSKY, Β . Α., and BYKOV, G. V., eds. (1961). "Centenary of the Theory of Chemical Structure." USSR Acad, of Sei., Moscow. KUSCH, P., and HUGHES, V. W. (1959). "Handbuch der Physik" (S. Flügge, ed.), Vol. 37 [1]. Springer, Berlin. MCCLELLAN, A. L. (1963). "Tables of Experimental Dipole Moments." Freeman, San Francisco, California. MCLAFFERTY, F. W. (1966). "Interpretation of Mass Spectra." Benjamin, New York. MEIBOOM, S., and SNYDER, L. C. (1968). Science 162, 1337. MISLOW, K . (1965). "Introduction to Stereochemistry." Benjamin, New York. NACHOD, F. C , and PHILLIPS, N. D. (1962). "Determination of Organic Structures by Physical Methods." Vol. II. Academic Press, New York. TURNER, D. W. (1968). In "Physical Methods in Advanced Inorganic Chemistry" (M. A. O. Hill and P. Day, eds.), Chapter 15. Wiley (Interscience), New York. WHELAND, G. N. (1960). "Advanced Organic Chemistry," 3rd ed. Wiley, New York. WILSON, Jr., Ε. B., DECIUS, J . C , and CROSS, P. C. (1955). "Molecular Vibrations." McGraw-Hill, New York. WOLLRAB, J. E. (1967). "Rotational Spectra and Molecular Structure." Academic Press, New York. SPECIAL
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