Henry A. Bent
University of Minnesota Minneapolis
II I
Tangent-sphere models of molecules, VI
Ion-Packing Models of Covalent Compounds
The purpose of this paper is to suggest that, suitably interpreted, the ionic model of heteropolar compounds may be applied to homopolar compounds and perhaps to metals. Chemical bonding, i t is generally conceded, is an electrostatic phenomenon. The idea is not new. Berzelius pursued it zealously, nearly two centuries ago, following Davy's famous electrochemical experiments, but floundered on the facts of organic chemistry (1).
If compound formation arises from electrical attractions between oppositely charged atoms, why is it that electronegative chlorine atoms can be substituted in compounds for electropositive hydrogen atoms? Worse yet, why can carbon combine with carbon? Or hydrogen with hydrogen? Or any element with itself? All elements should be monatomic? I n 1923 Gilbert N. Lewis stated clearly a solution to this "perplexing quandry" (9). If the properties of substances could not be explained by the mere assumption of charged atoms, might they not he explicable if we should no longer regard the atom as a unit, but rather if we might ascertain where the chal.ge or charges resided within the atom itself?
Continuing, Lewis asserted that [Tlhe suggestion that an understanding of chemical affinity must be sought in U e localization of charges within the atom [italics added] contains the germ of the final successful explanation of the chemical bond.
It seemed to be stretching the laws of physics considerably, however, and merely for the convenience of chemists, to suDDose with ~~~i~ that. in comDonnds. . two electrons-hut, interestingly, never more than two electrons-could share the same region of space. mysterious* was what kept the electrons of Lewis's electron-dot diagrams in space "like Mohamet's coffin." But for that matter one might ask, "What keeps the oppositely charged ions in sodium chloride apart?" The problem of repulsion has been a difficult, if obfor all electrical theories of matter. vious, is that matteroccupies space ~h~ common of other matter. ~~t no general law to the of repulsion is known. It is possible, nonetheless, to describe, if not to explain, the steric effects of repulsion with. a simple model.
.
Presented st the Symposium on "Models for Discussion of Molecular Geometry" sponsored by the Division of Chemical Ed ,,eation at the Society Meeting, Chicago, Ill., September, 1967. hi^ study was supported by a grant from the National Seionee Foundation.
768 / Journal of Chemical Education
In 1883 the chemist William Barlow wrote (3) Some studies pursued by the writer as to the nature of molecules have led him t o believe that in the atom-groups which modern chemistry reveals to Us the several atoms occupy distinct portions of space [italics added]. . . . The object of-the present paper is to show how far this conclusion is in harmony with, and indeed to some extent explains, the symmetrical forms of crystals . .
.
Barlow's paper contains the first detailed drawings of tangent-sphere models of crystals. Thirty years later, with X-rays, the Braggs, father (W. H.) and son (W. L.), verified Barlow's conjecture regarding the structure of sodium chloride. Later they determined the structures of several other important crystal types. Summing up, in 1920, W. L. Bragg (4), who had studied under Pope who had studied under Barlow, observed that the structural relations he and his father had observed could
atoms;~ach snhere is held in d a c e bv touchine several neieh-
Bragg found Barlow's model of tightly-packed spheres extremely useful. He wrote in 1926 (5), I t is very striking, in working out a new structure, to observe how closely the first approximation obtained by packing together an assemblrtge [of spheres] with the dimensions of the atomic domains, corresponds with the final structure attained by a. c a r e ful consideration of the intensities of reflection, One cannot help being convinced that the idea. is useful, and gives a. helpful conception of the reasons for the stability of the structures.
continuing, B~~~~remarks that [I]t must he counted a fortunate result of the nature of atomic fields that the conception of atoms occupying a characteristic amount of space in the crystal structure to the exclusion of others gives results in a fair agreement with ohsenration.
An operational definition of "packing" was given by Bragg and West in 1927. Noting that in all examined silicates neighboring oxygen atoms were about 2.7 A apart, they wrote, ''For brevity this feature will be referred to as 'packing' of oxygen ions, with positive ions between them'' ('). The Barlow-Bragg tangent-sphere crystals was refined by Wasastjerna and Fajans, who pointed out, on the basis of refractivities, that anions generally are larger than cations; by ~ ~ l d ~ ~ who h ~ emphai d t , sized the importance of radius ratios; and by Pauling, who extended previous work and set forth a set of simple rules, which, though neither "rigorous in their der-
ivation nor universal in their application," permit an "intuitive understanding of the stability of crystals in terms of visualizable interionic interactions" (7). At this point, with Lewis's electronic interpretation of the Couper-Keknle-van't Hoff model of homopolar compounds and with the Barlow-Bragg-WasastjernaFajans-Goldschmidt-Pauling model of heteropolar compounds, chemists had Berzelius-like, charged-particle models for many, if not all, types of compounds. Still, there remained the problem of explaining from "fund* mental principles"-i.e., of relating to physics-what it was that kept oppositely charged particles apart. The ancient problem of explaining chemical affinity had become a problem of accounting for physical repulsion. "Hitherto it has been generally assumed," observed Pauling in 1927, "that the repulsive forces between atoms arise from the interaction of quadrupole (and higher) moments of the atoms, that is, from their departure from spherical symmetry" (8). Even if this assumption were useful in interpreting interactions between atoms, it could not account for the internal stability of individual atoms. A clue to a practical solution of the problem of repulsion was given in 1926 by three discoveries: the discovery of the wave properties of electrons; the discovery of the "classically nondescribahle two-valuedness of the electron" (9)(called "spin"); and the discovery of the Exclusion Principle. Taken with the results of modern structural studies, these discoveries suggest (to paraphrase Barlow (10)) that
The structural relations revealed by modem chemistry can be expressed in a simple manner by regarding the electrons in a molecule as an assemblage of spheres packed tightly together. Each sphere is held in place by the attraction of atomic nuclei and by its touching several neighboring spheres. Spheres do not collapse, because the electron cloud must satisfy the Exclusion Principle and the wave equation. It is striking in examining a structure to observe how closely the first approximation obtained by packing together an assemblage of "spherical electrons" corresponds with the final structure attained by careful physical measurements. One cannot help being convinced that the idea is useful, and gives a helpful conception of the reasons for the stability of the structure. It must be counted a fortunate result of the Exclusion Principle and the wave properties of matter that the conception of electrons occupying characteristic amounts of space in molecules to the exclusion of other electrons of the same spin gives results in a very fair agreement with observation. Figure 1 shows tangent-sphere models of ethane,
I n the atom-groupings which modern chemist~yreveals to us the sareml electron-pairs occupy distinct portions of space.
The object of the present series of papers is to examine how far this conclusion is in harmony with, and perhaps to some extent explains, the forms of molecules. Pocking Models of Molecules
The idea that electrons occupy distinct portions of space to the exclusion of other electrons of the same spin may he viewed as a generalization of the hypothesis that molecules and ions (which are composed of electrons) occupy distinct portions of space to the exclusion of other molecules and ions. So familiar is the latter hypothesis, it is not always realized that a different conception was once entertained. Wrote T. W. Richards in 1911 (11), Most physical chemists refer all changes in volume to changes in the extent of the empty space between [gadike] molecules. But are there, after all, any such empty spaces in solidn and liquids? . . . [Tlhe so-called "sphere of influence" of the atom is the actual boundary by which we know the atom and measure its behavior. Why not call this the actual bulk of the atom?
Today, changes in volume of solids and liquids are generally referred, in the conventional interpretation of quantum mechanics, to changes in the average extent of the empty space between gas-like point charges within molecules. But are there, after all, any such "empty spaces" in molecules? The so-called "sphere of influace" of an electron (it's "Fermi hole" (10a)) is the boundary by which we know the electron and measure its behavior in compounds. Why not call this the actual bulk of the electron? One can see that the hypothesis is suggestive. Indeed, to paraphrase Bragg,
Figure 1. Tmgent.sphere models of HsC-CHs, HzC=CHl, and H k C H . Light spheres represent protonated electron pairr of C--H bonds, dark spheres electron pairs of bonds between carbon core,.
ethylene, and acetylene. Models of this type produce, through simple packing considerations, every useful feature of classical structural theory, and modern refinements of that theory (10). I n such models, each sphere represents a negative "particle," a valence-shell electron pair. The smaller positive "particles," the atomic cores, are not shown; they are in the tetrahedral interstices between the larger negative particles. These tangent-sphere models suggest that, perhaps Covalent molecules may he viewed as "ion compounds."
The cations in this analogy are the atomic cores (atomic nuclei plus inner-shell electrons). The anions are the valence-shell electrons. An Instructive Nototion
In developing the analogy between the structures of ionic and covalent compounds, a notation used by Lewis (1%) and Fajans (IS) and described, with disclaimers appropriate to the present discussion, hy Sir William Ramsay in his Presidential Address before the Chemical Society of London in 1908, is useful. Said Ramsay in his address on "The Electron as an Element" (14) I have to bring before you a suggestion which, although not exactly new, admits of definite statement, and affords a mental picture of what may conceivably take place. It is not a "theory"; I do not hope that it may be h e ; it is rather a hypothesis, 8 supposition that I expect to be useful; it may be a "mnkebelieve"; I trust that it will not be s. "mistake." Volume 45, Number 12, December 1968
/ 769
The hypothesis admits of a short statement. I t is: electrons are atoms of the chemical element, electricity; they possess mass; they farm compounds with d h e r elements [atomic cores]; they are known in the free state, that is, as molecules; they serve as t,he "bands of union" between atom and atom. The electron may be assigned the symhol "E."
It is convenient to have a name, appropriate t o the present discussion, for valence-shell electron pairs. We shall call them electride ions, symbol E2, or Ez2-. Protonated electride ions will be called hydride ions, symbol H, or H-. Application of this notation to ethane, ethylene, and hydrogen peroxide is shown in Figure 2.
H
\
/
H
H-C-C-H
/ H
\H
\
/ H
models, Figure 1, suggest a new type of isomorphism. As Goldschmidt noted in 1929 (15), Isomarphism-the great discovery of Mitscherlich that chemically different substances, e.g., phosphates and arsenates, might show a close resemblance in crystalline form, that they are isomorphous-osually demands a close analogy of chemical formulae; that is, the relative quantities of building stones of diiferat kinds must correspond i n the two isomorpkous substances [italics added].
Since a close crystallographic analogy often exists between ionic fluorides and hydrides ( I @ , Figure 4, left
H
/H
C=C
..-0:I
:O
\H
I
H
Figure 4.
Ethane
Ethylene
Hydrogen Peroxide
An o p p i i ~ t i o nof the lroelectronic Principle.
The Ramsay-Lewis-Fajans notation brings out the point, as do the tangent-sphere models, Figure 1, that
arrow, and since the fluoride ion is isoelectronic with the oxide ion, Figure 4, top arrow, which carries the same charge as the electride ion, which is isoelectronic with the hydride ion, Figure 4, bottom arrow, one might expect to find structural analogies (isomorphisms) among oxides and electrides, Figure 4, right arrow. I n fact, therc is for these ions an interesting, if not universally valid, "law of substitution." As Fajans has emphasized (17)
I n coualal compounds the cations C'+, N6+, 0 6 + , and F'' have an electride-ion coordination number of OUT.
Ozide ions onrl eledride ions are, to some eztenl, slructmd equivalents.
Figure 2. Four repremntotions of three familiar molecules. In the molecules' graphic formulas (top line], the symbols H, C. and 0 represent the atomic cores of hydrogen, carbon, and oxygen (charger +I, +4, and +6, respectirelyl. In the molesuler' Rammy-Lewis-Fajam nototions (second linel, (HI represents o protonated, (Ed an unprotonded electride ion.
The notation emphasizes the ionic character of covalent compounds. I n the full Ramsay-Lewis-Fajans notation, for example, H,O, = (H-)(E,z-),@+(E,2-)O(i+(EE2-),(H-) The distribution of charge represented by this instructive, if awkward, notation is shown, schematically, in Figure 3. Isomorphism Revisited
Table 1 illust,rates this point for elements from several Table 1.
0 E Si
=
= =
Oxides and Electrides with Similar Formulas and Similar Structures
Oxide ion (charge -2) Electride ion (charge -2) Silicon core (or kernel, charge +4), etc. Oxide Electride
Tangent-circle drawings, Figure 3, or tangent-sphere
Ethylene Ethane
Hydrogen Peroxide 770
/
Figure 3. Tangent-drcle reprerentotions of the moleculer of Figure 2. Large circler (negative charged) represent electride ions; plus signs in Smdler them represent protons. circler (positively charged) represent atomic core*. For simplicity, the four &ctride ions .bout on atomic core ore shorn in o square-planor arrangement rather than in the electrostatically more favorable tetrahedral arrangement lcf., Figure 1)
Journal o f Chemical Education
groups of the Periodic Table. (In Table 1 and'tables following, the symbol E stands for two electrons.) Replacing the oxide ions in, for example, CllOi= 03C10C103 (one bridging oxide ion and six unshared oxide ions) by electride ions gives E,CIECIE, = C1, (one bridging-i.e., bonding-electride ion and six unshared electride ions). The arrangement of cations or atomic cores (small cations) is infinite chains in SOa (asbestoslike form) and elemental selenium, tetrahedral clusters in gaseous P,Olo and PA,puckered layers in solid PIOj and elemental arsenic, a three-dimensional framework in SiOz and elemental silicon, infinite chains in the complex anion of CaSiOa and the silicon lattice of CaSi,
puckered layers in the complex anion of CaSizOs and the silicon lattice of CaSiz, and a face-centered cubic lattice in CaO and elemental calcium. LLComplex salts" are known, also. Thus XeOl = Xe03E, CLO = ClzOEa,SO2= S O 8 , and PaOa= PtOsEa. Analogy of chemical formula may be extended by representing electride ions, either protonated (H) or unprotonated (E or Ez), by the symbol A (for anion) and atomic cores by the symbol l U (for "metallic" cation). The formulas of ethane and hydrogen peroxide, Figure 1, are then the same, JGAT. Similarly, methane, ammonia, and water have the same formula, MA4. Also, ethylene and formaldehyde have the same formula, M2As. And acetylene, hydrogen cyanide, nitrogen, and carbon monoxide have the same formula, MzA,. Molecules that have the same generalized Ramsay-Lewis-Fajans formula, M,A,, that is, molecules that have the same number of valence-she11 eIectrons, 2y, and the same number of heavy-atom cores, x, are said to be isoelectronic (18). Paraphrasing Goldschmidt, we may say that The great discovery of Lewis (19) and Langmuir (90)that chemically different substancep might show a close resemblance in electronic structure, that they are isolectronic, demands a close analogy of chemical formula; that is, the quantities of building stones of different kinds (electtide ions and atomio cores) must correspond in the two isoelectronic substances.
For molecules that contain many, many building stones-i.e., for crystals-it is the relative, not absolute, quantities of building stones that must correspond if two substances are to be isomorphous or isoelectronic with each other. Possibly the most celebrated example of this principle is Hume-Rothery's discovery that intermetallic compounds with different chemical formulas but the same number of valence electrons per atom have often closely related structures (21). More familiar examples, among electrides, are beryllium oxide, boron nitride, and diamond, or, among oxides (A = 02-,not E P ) , SiOzand AIPOI. The st.ructura1 equivalence of ordinary anions and electride ions may be further illustrated by comparing the two geometrical schemes that have been developed, independently of each other, for describjog the structures of chemical compounds, one about half a century ago, by X-ray crystallographers, for heteropolar compounds, and the other, a full century ago, by organic chemists, for homopolar compounds. In both schemes, markedly different roles are attributed to the positive and negative "ions." In both cases, attention is focused in the first instance upon the arrangement of the larger, negative ions. Many heteropolar compounds, for example, may be described as arrays of close-packed anions with ordinary cations (relatively large atomic cores) in some or all of the tetrahedral and octahedral interstices of the anion lattice (Zlb, 22). Similarly, many homopolar compounds may he described as fragments of arrays of close-packed electride ions with atomic cores (relatively small cations) in the tetrahedral interstices of the electride ion lattice (10a). Briefly stated The bond diagrams of classical structural theory are the anion lattices of coualent compounds.
When the quantities, or relative quantities, of positive and negative ions correspond for two compounds, the
compounds' cation and auion lattices may correspond, also. Such compounds are said to be isoelectronic if covalent (anions = electride ions, cations = small atomic cores), and isomorphous if ionic. Chemical Equivalents
The analogy in formula and structure of oxides and electrides, Table 1, carries over to an analogy in nomenclature and reactivity, Tables 2 and 3. To a remarkaTable 2.
Analogous Terminology
Oxide
Electride
Size of cations: Large (NarO, CaO) Basic Oxideion donor Small (SO8,SiO,) Acidic Oxide-ion acceptor Medium (ALOr, ZnO) Amphoteric Oxide-ion donor and acceptor Large and small (N@SO4) Neither basic or acidic (Neutral) Salt Doubly protonated oxide-ion: HnO
Size of atomic cores: Large (Na, Ca) Reducing Electron donor Small (S, C1) Oziduing Eleotron acceptor Medium (P,Br) Inlermedzate ozidation state Electron donor and acceptor Large and small (NaC1) Neither reducing or oxidieing Salt Doubly protonated electride ion: Hz
Table 3.
Analoaous Reactions
Oxide
Electride
,n ofwater with: Reacti~ Basic oxide HzO = 2 ~ a 0 H NarO (OH- = wnglyprotonsted oxideion) Acidic oxide SOs H20 = H2SOd (H,S04 = proton donor)
Reaction of hydrogen with: Reducing electride H, = 2NeH 2Na (H- = singly protonated electrideion) Oxidizing electride S H2 = H2S (SEs Hb = HISEl = proton donor) Reactmn of reducing electride with oxidizing elect,ride S = NaS 2Na Ca Si = CaSi (CaE SiE1 = CaSiE8)
+
+
+
++
++
2Cs. (2CaE
+
+ Si02 SiO. = ZCaO + Si + = 2CaO f Si&)
ble degree, oxide ions and electride ions are chemical equivalents. It must be counted a fortunate result of the wave properties of electrons and the Exclusion Principle that so simple a model as an ion-packing model corresponds so closely to the central terminology and principal reactions of classical inorganic chemistry, once one introduces as a parameter of importance the sizes of the positive ions. The reactions of water (a doubly protonated oxide ion) with large-core (basic) and small-core (acidic) oxides, for example, are analogous to the reactions of hydrogen (a doubly protonated electride ion) with large-core (reducing) and small-core (oxidizing) electrides (Table 3). Further, just as "the chemical reactioas of sodium [electride] are retained in its hydride [its protonated electride]" (25), so the chemical reactions of sodium oxide are retained in its protonated oxide, NaOH. Similarly, the reaction of a law-core oxide (e.g., CaO) with a small-core oxide (e.g., SiO,) is analogous to the reaction of a large-core electride (e.g., Volume 45, Number 12, December 1968
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771
CaE) with a small-core electride (e.g., SiEa). Both reactions ~ r o d u c ea more or less "neutral" "salt" (Table 3). The reaction of double decomposition, which played an important role in the development of inorganic and nhvsical chemistrv " " (23). , ,, mav " be viewed as an anionexchange reaction, one anion of which may be the electride ion (Table 3). Thus, writes Mendeleeff (23)
may be described by defining the type of atomic arrangement and the numerical data of atomic distances. The effect of chemical substitution [of cesium for sodium, for example] may be shown by alteration of atomic distances and by alteration of atomic arrangement" (15). Noting the transferability of interatomic distances, Goldschmidt remarks, "Such constant or nearly constant interatomic distances lead to the conception that for any particle there is a sphere of action If hydrogen be passed over the compound of oxygen with copper which is practically impenetrable. . . " (15) [italics added]. at red heat, then metallic copper and water are abtainedFollowing Bragg and West, "We speak for convenience CuO Ha = H 2 0 Cu of the 'size' or 'diameter' of the ion although, of This kind of double decomposition [sic] course, this is not a definite physical quantity. I t is a useful expression of the empirical rule that interat,omic [CuO HIE = Ha0 CUE] distance is an additive property to a fair degree of apis called reduction proximation in these inorganic compounds" (6). Reduction of n metal oxide may be viewed as an oxideFrom an extensive study of the structures of inorganic compounds during the years 1923-27, Goldion-electride-ion exchange reaction in which t,he oxide schmidt made a fundamental discovery: " [A] sudden ion is captured by the smaller atomic core. alteration of crystal structure occurs at a certain limit Table 4 is a classification and summary of bond types To explain the meanof the quotient R,.ti..:R.,i,,." Table 4. Bond Tvves ing of such limiting quotients of radii, Goldschmidt introduces a fundamental hypothesis: "We shall, provisionally, set up an hypothesis," he writes, "that the stability of crystal structure of heteropolar compounds requires that anions and cations are in mutal Large and contact" (15). Small Cores Small Cores Large Cores - -Goldschmidt concurs in these matters with the views Covalent bonding Ionic bonding Metallic bonding of W. L. Bragg, whose chief assumptions, in the words of Nonmetal Salt Acidic oxide Neutral oxide RI. L. Huggius, were: "(1) that each atom of a given Oxidizing elecElectride neither element is surrounded by a 'sphere of influence' of the tride oxidizing or reducing same size in different crystals: and (2) that the 'spheres of influence' of adjacent atoms in crystals are tanuent to one another" (24) [italics added]. Actually, as W. L. based upon the sizes of the cations coordinated by an has noted, "[Tlhe contribution by a given ion anion. Open circles represent anions (e.g., 0%-, E22-), Bragg appears to be larger when it is associated with another filled circles, cations. Three limiting cases may be large ion than whenit is near a small ion" (6). recognized: the cations coordinated by an anion may "We now come to a very interesting point as to the be (1) all large, like Na+ ( r = 0.95.A); ( 2 ) all small, position of the molecule in the crystal, " W. H. Bragg like CI7+(r = 0.26 A ) ; or (3) large and small. If the has remarked. ''In one sense it may be said to have anions are electride ions, the substance is said to be (1) disappeared, as in rock-salt, for example. . . . [Tlhe metallic (cations large, as in Na), (2) covalent (cations whole crystal isnow one molecule" ($5) [italics added], small, as in CI,) or (3) ionic (cations large and small, as The supposition that the spheres of influence of in NaC1). Correspondingly, the substance is called a practically impenetrable cations and anions are tangent metal, a nonmetal, or a salt. If the coordinated anion to one another in crystals leads to a set of simple rules is an oxide ion, the substance is said to be basic (cations regarding the coordination numbers of ions in crystals. large, as in NaO), acidic (cations small, as in Cl,O), or An example from plane geometry illustrates this point, neutral (cations large and small, as in NaC104). remakks Goldschmidt (15). Circles of two sizes are to be arranged in such a manner that a smaller circle 11 The Fundamental Law of Crystal Chemistry representing a cation, radius R M , has contact with as The concept of "size" ha8 been used to raticnalize many larger circles A representing anions, radius Rn,as more than the chemical properties of substances. possible, without any intersecting of the larger circles. Table 1 illustrates the importance in structural chemAs can be determined from Figure 5, an arrangement of istry of pure number: substances that have the same cation-to-anion ratio have often similar structures. I t is well-known, however, that sodium chloride and cesium chloride, for example, have different structures, as do elemental oxygen and sulfur, sulfur and selenium, and selenium and polonium. The fundamental problem of crystal chemistry, Goldschmidt has written, is to discover "the causes which determine the type of structure of any given substance; why, for instance, has magnesium fluoridc the structure of rutile [Tioil, strontium fluoride the structure of fluorite ICaFll?" (15). Now, Goldschmidt notes, "Any crystal structure Figure 5. Figurer used by Goldschmidt to illvtrole the radius-ratio effect. A
+
+
772
/
+
+
Journol of Chemkol Fducofion
three anions A around cation IN without rattling of M within the interstice formed by the anions is only possible if the ratio RM : RA is equal to or greater than 0.15. A square-planar arrangement of four anions about the cation demands a radius ratio that equals or exceeds the limit 0.41. These and additional rules, the well-known radius-ratio rules of crystal chemistry, are summarized in Table 5 taken from Goldschmidt's historic paper on crystal structure and chemical constitution (15). Table 5.
The Arrangement of Anions A About a Cation M (Goldschmidt)
Number of surrounding anions A 2 3
4
8
Arrangement of anions A
Lower limit for the ratio
R M : ~
Opposite each other Equilateral-triangle Cube-diaeonals itetmhedhedral) Square Cube-faces, octahedroncorners Cubediagonals
0 0.15 0.22
0.73
To summarize, Goldschmidt writes [Tlhe two essential factors connecting crystal structure and chemical constitution are: (1) The relative proportions of the various kinds of atoms (or ions) in the chemical formula. (2) The relative sizes of the various kinds of particles (atoms or ions) in the crystal.
I have called this chesis the fundamentallaw of crystal chemistry, as it gives a most general statement concerning the factors which determine the structure of crystals a? a function of chemical composition. The Fundamental Law of Structural Chemistry
In molecules, as in crystals, the effect of chemical substitution may be shown by alteration of atomic distances and by alteration of atomic arrangement. Nitrosyl fluoride, for example, has the structure indicated by the graphic formula
Each atomic core (F7+, N5+, 08+) has about i t four electride ions (the Octet Rule). The corresponding sulfur compound has a different structure (N), however. In NSF, the central position is occupied not by the atom with the smallest core charge, nitrogen, but, rather, by the atom wit,h the largest atomic core, sulfur. Interatomic distances, particularly a short. sulfur-nitrogen distance, suggest that the molecule may be represented by the graphic formula
I n this representation of NSF, five electride ions are placed about the sulfur core, as in SFa. In seeking the meaning of this change in structure, we may, after Goldschmidt, set up, provisionally, the hypothesis that the stability of a molecule of a homo-
polar compound requires that the molecule's polar opposites, its atomic cores and electride ions, should be in mutual contact-and that there should be as many such contacts as possible. Briefly, our chief assumptions are: (1) that each component of a molecule, an atomic core or electride ion, is surrounded by a "sphere of influence" of the same size in different molecules (actually, the effective size of an electride ion appears to be somewhat larger when it is associated with large atomic cores than when it is near a small one (1Od)); and (2) that the "spheres of influence" of adjacent polar opposites in a molecule are tangent to one another. For an atomic core to have an electron-pair-coordination-number of six, for example, the radius of the atomic core should be at least 41% of the radius of its valenceshell electrons (Table 5). While this condition is met by second-row elements (e.g., Si, P, S, Cl), which, therefore, can expand their octets, i t is not met by most first-row elements (C, N, 0, F), whose atomic cores would rattle in the octahedral interstices formed by six close-packed valence-shell electride ions. For smallcore elements (rat.,,. < 0.2 A), the maximum eleetron-pair-coordination-number is four. From this geometrical point of view
..,,
The Octet Rule is a statement of the wn-rattling condition for small atomic cores.
Atoms in molecules obey the Octet Rule if their atomic cores are small. They often do not obey the Octet Rule if their atomic cores are large. In summary, the two essential factors connecting molecular structure and chemical composition are: (1) The number of atomic cores and electride ions in a substance (2) The relat,ivesizes of bhe at,omie cores and electride iann
This thesis, stressing the importance in structural chemistry of pure number (27) and relative sizes (lac, d), is applicable, in broad outline, to all packing models. It may be called the Fundamental Law of Structural Chemistry. The Cation-Cation Coulomb Term
We now come, to paraphrase Bragg, to an interesting point as to the position of an atom in a molecule. In one sense it may be said to have disappeared. Molecules are not built of "atoms." R.lolecules are built of atomic cores and electride ions. The whole wzolecule is m e small crystal. Thus, as we have seen, many of the rules of crystal chemistry may be applied to covalent compounds. Pauling's First Rule of Crystal Chemistry, that a coordinated polyhedron of anions is formed about each cat.ion, has its counterpart in the First Rule of Covalent Chemistry In molecules a coordinated polyhedron of electride ions is formed about each atomic core With Lewis (IQ), Sidgwick and Powell (28), and Gillespie and Nyholm (29), we may state, also, a Second Rule of Covalent Chemistry Structurally, lone pairs may ofteu be treated like bonding pairs Volume 45, Number 12, December 1968
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773
The corresponding statement in crystal chemistry is self-evident: The space about the surface of a cation taken up by coordinated anions is about the same whether or not the anions are coordinated by other cations. Anions are shared by cations when a structure is anion deficient, i.e.,,when the number of anions in the structure is insufliclent to complete separate coordination polyhedra about each cation. Shared electride ions are called bonds, more specifically, two-center bonds, when they are shared by only two atomic cores. Electride ions shared by more than two atomic cores are called multicenter bonds. Pauling notes that In a crystal containing different cabions those with large valence [high electric charges]. . . tend not to share polyhedron elements with eaeh other.
"This rule," he says, "follows directly from the fact that cations with high electric charges tend to be as far apart from each other as possible, in order to reduce their contribution to the Coulomb energy of the crystal" (7). A corresponding rule may be stated for covalent compounds. I n a molecule containing different atomic cores those with large eleet~iecharges tend not lo share electride ions with eaeh other.
Highly electronegative atoms tend to be bound in molecules by the fewest possible honds consistent with the Octet Rule (SO). The structure of nitrous oxide, for example, is NNO, not NON, that of nitrosyl fluoride FNO, not NOF or, worse yet, NFO. Similarly, NCO(cyanate) is more stable than CNO- (fulminate) (Sob). If highly-charged atomic cores do share electrons with each other, the resulting molecule, e.g., Fz, OF,, H,Op, is generally reactive. The instability of such systems may be said to stem from high core-charge "st~ain." Drawings of two tetrahedra of anions that share, respectively, one, two, and three of their anions in common are shown in Figure 6. These drawings are taken
This rule applies to all coordinated structures. I t applies, in particular, to the formation of small-ring componnds and multiple bonds. From the viewpoint of the electrideion model of chemical bonds, the "strain energy" of small-ring compounds does not arise from "bond bending" per se. Rather, Baeyer's " s h i n energy" arises from cation-cation w p u l s i m s .
It is to be anticipated, adds Pauling, that [Clation-cation repulsion will operate in some cases to displace the cations from the centers of their coordinated polyhedra.
"This action will be large," he notes, "only in case the radius ratio approaches the lower limit for stability" (7). Examples cited by Pauling are hematite (FezOa) and corundum (A1203). I n these crystds each AlOe octahedron shares a face with another RfOs octahedrot I n hematite (radius ratio 0.45) the iron ions are 2.06 A from the three oxygen ions defining a shared face and 1.99 A from the other three oxygen atoms; in corundum (radius ratio 0.41) the corresponding distances are 1.99 and 1.85 A. Analogous cation displacements appear to occur in covalent componnds. I n ethylene, for example, cationcation repulsions operate t o push the carbon cores from the centers of their coordinated polyhedra of electride ions outward toward the protonated electride ions of the C-H bonds, Figure 7. Two expected consequences
Figure 7. Tangent-circle drawing of a two-dimensional model of ethylene showing the effect of cotion-cation repulsion on the HCH bond angle and bond length. Large, open circler represent electride ions; the C-H small, Rlled circler represent carbon nuclei; plus signs represent protons.
Figure 6. The rhoring of a corner, an edge, ond a face b y a pair of tetrahedra (Fig. 13-21. ref. 14111.
directly from Pauling's classic paper on the structures of complex ionic compounds (7). They serve equally well as drawings of the tangenesphere models of single, double, and triple honds (Figure 1). From the viewpoint of the electride-ion representation of chemical bonds
With regard to ionic componnds, Pauling has observed that The presence of shared edges, and particularly of shared faces, in a coordinated structure decreases its st,ahility.
"This decrease in stability," Pauling adds, "arises from the cation-cation Coulomb term" (7). More generally, A n y structural change whose chief effect i s to bring cations closer togethw tends to decrease stability.
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of this action are that, as observed, the HCH angle in C2Hashould be larger, and the C-H distance smaller, than in C2Hs (fob). I t would appear that, generally, enhanced hond angles opposite and diminished bond lengths adjacent to double honds arise from displace ments of the atomic cores of double bonds away from the shared-edges of their electride-ion polyhedra. Similar displacements help to account for the rule (31) that the s-character of an at,om tends to concentrate in orbitals the atom uses toward electropositive substituents; that is to say, enhanced hond angles opposite and diminished bond lengths adjacent to electronegative groups probably arise, in part, from the displacement of atomic cores away from those corners of their electrideion polyhedra that they share with the highly charged cores of electronegative atoms. This analogy between electride ions and ordinary anions raises, again, an interesting point, which will now be considered, explicitly. A Frame of Reference
The ion-packing model of molecules introduces a new
frame of reference from which to view the structures of homopolar compounds. One cannot speak, for example, of "cation displacements" without having a frame of reference with respect to which the cations can he displaced. For homopolar compounds, this frame of reference is the electride-ion lattice. And, just as the concept of an anion lattice is the key to the simple description of heteropolar compounds (%lb,92), so, too, as the use of graphic formulas illustrates,
"simple network-former transfer," Figure 9 (39). The
The concept of an electrideion latlice is the key to the simple description of homopolar compounds.
The followingexamples further illustrate this point. It is well-known that the lengths of single bonds adjacent to multiple bonds are generally shorter than normal (31). The carbon-carbon single bond in diacetylene (HCEC-CeCH), for example, is almost as short as a normal carbon-carbon double bond, yet the triple bonds in diacetylene are almost the same length as, possibly even a bit shorter than, the triple bond in acetylene. An explanation of these facts is shown diagramatically in Figure 8. For clarity, differences in bond lengths
Figure 9. Simple network-farmer tronlfer [Figure 1 , sequence 2-1, ref. Network-former cations, Si4+,ore not shown. Open circler represent oxide ianr.
(3211.
Figure 8. Interatomic distances ond proposed carbon-core positions in diamond, diacetylene, and acetylene. Valence-stroke$ represent electride ions.
have been exaggerated. Valence strokes in Figure S represent electride ions, ie., each graphic formula in Figure 8 represents an array of tangent spheres. As indicated in the figure, the carbon cores in diamond are in the centers of their coordinated polyhedra of (somewhat "mushy") electride ions. I n acetylene and diacetylene, however, cation-cation repulsions operate to displace the carbon cores away from the shared faces, the triple bonds; toward the shared corners, the single bonds. Consequently, although the centers of the two inner tetrahedral interstices in the electron cloud of diacetylene are probably approximately the same distance apart as are the centers of adjacent tetrahedral interstices in the electron cloud of diamond, the corresponding internuclear distance in diacetylene is less than that in diamond partly because the carbon cores i n diacetykne are not in the centers of their tetrahedral in'erstiees. Although the central bond in diacetylene is, in the model, a single bond, i t is shorter than the single houd in diamond because the location of the carbon cores with respect to the electron cloud is different in the two cases. The concept of an anion lattice provides a frame of reference useful, also, in the description of chemical reactions. The quotation that follows [with our comments in brackets] is a silicate chemist's description of a
mechanism as described is exactly analogous to the organic chemists' S N mechanism ~ for a Walden inversion. As detailed previously for the reaction of a hydroxide ion with methyl chloride ( l o b ) , the first step in the reaction is the formation of a penetration-type complex between a nucleophile and a Lewis acid, followed by the migration of t,he active atomic core in the Lewis acid into the empty tetrahedral interstice of the penetration complex; this st,ep produces a second penetration complex (intimate ion pair) that, by dissociating, completes the react,ion. The two reactions compared are t,he reaction (charges are not shown) 0.Si';O
+ OISiOSiOI = 01Si*OSi03 + OSiO?
and, in the Ramsay-Lewis-Fajans notation, the reaction [to which the commcnts in brackets refer] of OH- with CHsCI, (EddH)O(Ea)
+ (H)aCIEz)CI(E,)a
=
(E,)AH)O(EdC(H)a
+ (Ez)Cl(E,),
Three tetrahedra are involved. A t,etrahedral group [the hydroxide ionl having an nnshared oxygen atom [electride ion] is in apposition to a pair of tetrahedra [the methyl chloride molecule] har ring an oxygen atom [electride ion] in such a way that its unahared oxygen [electride ion] touches thl.ee oxygen atoms [electride ions] of the pair. These three atoms [eleclride ions] do not include the one that is shared between the pair. I t will be seen [Figure 9; or Figure 33, stage 1, ref. ( l o b ) ] that a centrally placed Oa [(E&l triangle may be traced in this grouping ft,hepenetration complex]. The configurational change consists simply in migration of a network-forming atom [atomic core, C'+] from its station on one side of the O3 [(E,),] triangle t o the symmetrical position on the other. The oxygen atom [electride ion1 previously unshared has now become the connect,. ing oxygen [electride ion] shared by two tetrahedra. The oxygen atom [electride ion] previously shared is now onshared. . . . After the transfer of the network-former [C'+] the tetrahedral group with tho unshared oxygen [electride ionl will he free to separate from that part of the configuration to which it had previously been bonded. . . . Such are the moves of silicate chess (82). Volume 45, Number 12, December 1968
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Large Core Electrides
We conclude this discussion of the analogy between heteropolar and homopolar compounds with the suggestion that perhaps metals, also, may be viewed as ' Iion-compounds." . Wrote W. H. Bragg in 1916 (25), It is curious that SO many.. . [orystalsl are based upon the facecentered lattice. I n this respect copper, silver, zinc blende, iron pyrites, fluorspar, and diamond are all alike; in every case the representative points lie on a simple facacentered lattice.
Is i t merely a concidence that the face-centered cubic structure of many metals is the same as the cation lattices of zinc blende, iron pyrites, fluorspar, cuprite, and rock salt? Is it merely a coincidence that the unusual coordination of gallium atoms about gallium atoms in gallium metal (one nearest gallium-atom neighbor and six next-nearest gallium-atom neighbors about each gallium atom) is found, also, in gallium sulfide? These faots would receive a simple explanation if one supposed with Bragg (SS), Goldschmidt (16), and Fajans (84) that in metals electrons play the role of anions. "A crystallized element, a metal for example," wrote Goldschmidt in 1929, "is not an aggregation of single free atoms, but is really a compound, comparable for instance with zinc sulfide" (15). Summary
Reasoning by analogy is always dangerous, but not always wrong. No one would claim that positive inferences could be made.about the electronic structures of homopolar compounds, or of metals, from a knowledge of the structures of heteropolar compounds, or viceversa. Obviously new and important factors arise in passing from one class of compounds to the other. I t is startling, however, how often similarities among heteropolar, homopolar, and metallic compounds emerge once one starts to look for them. Some of these similarities are summarized in Table 6. They support the chief conclusion of the present study, namely that all substances may be viewed as ion-compounds. Differences Table 6.
Heteropolm compounds
Analogous terms
Homopolar compounds
Crystal Ion's sphere of influence Cation' Anion Oxide ion Basic oxide (0" donor) Acidic oxide (0'- acceptor) Isomorphic crystals Local electrical neutrality Coordination polyhedron Coordination nite First coordination shell Second coordination shell
Molecule Electron domain, Fermi hole Atomic care Valenceshell electron pair Electride ion Reducing agent (El- donor) Oxidizing agent (El- acceptor) Isoeleetronic molecules Zero formal charges Sextet, octet, . . . Orbital Valenoe shell Outer darbitals, antibonding orbitals, "pockets" ( l o b ) Cation in tetrahedral interstice Octet Rule obeyed Cation in octahedral interstice Expanded octet Network-former Multivalent atom Brideine ion Bondine air ~ o n : h d ~ i nion ~ Lane Multiply-bridging ion Multicenter bond Cation-cation Coulomb term Baeyer strain energy Anion lattice Bond diagram Anion deficient Electron deficient
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Journal o f Chemical Educafion
in the structures and chemical properties of substances appear to arise from differences in the sizes (and shapes) of their atomic cores and not from differences in the principles that describe the electronic environments of those cores. Ion-packing models have the particular virtue that they focus attention directly upon fundamental physical principles. They emphasize, at every step of the discussion, and not merely in a preliminary mathematical statement, or in a final numerical result, that electrons are de Broglie particles that obey the Exclusion Principle. The Exclusion Principle is represented in the models by the simple device of treating the sphere of influence of a fermion as an impenetrable sphere. The coulombic terms of the electrostatic Hamiltonian are represented by the Doctrine of Coordination and by Pauling's Rules of "crystal" chemistry. The kinetic energy terms are represented by the assumption of ions of finitesize. Ion-packing models are particularly useful for showing the saturation and directional character of valence, multicenter bonding, intermolecular interactions, reaction mechanisms, and the origin of the (approximate) transferability of structural parameters. Not least of all, ion-packing models show why seemingly different models of molecular structure give essentially the same results. The current situation with regard to molecular models is similar to the situation that prevailed in 1861 when Butlerov, one of the fathers of structural theory, wrote (35), [there exists] a mass of different [chemical] theories and different methods of expressing them [and] all these theories, often even those which a t first glance seem entirely different, include much t,hat is similar, and if we separate from them that calortttion which is due to the personality of the authors and discard the prejudice which is caused by the author's vanity, the similarities become apparent and it is easy and simple t o show that there exists a mass of faots that must lead to the same conclusions.
Common to all working chemical theories must be, by design or by accident, through mathematics or through some equivalent procedure, an accommodation to the fundamental principles of electron physics. Most importantly, any electronic interpretation of chemistry must reflect, in one way or another, the operation of the Exclusion Principle. From the viewpoint of the present discussion, van't Hoff's theory of the tetrahedral atom, Barlow's atom-packing theory, Werner's coordination theory, Lewis's electron-pair theory, Pauling's hybrid-orbital theory, Walsh's molecular-orbital theory, Gillespie's valence-shell-electron-pair repulsion theory, Linnett's spin-set theory-all of these theories are, in various guises, a Mechanics of the Exclusion Principle. The importance of the Exclusion Principle to chemistry was stressed by Lennard-,Jones. He wrote, in 1954, that (36) I t does more to determine the shapes and properties of molecules than any other single factor. I t is the exclusion principle which plays the dominant role in chemistry. Its all-pervading influence does not seem hitherto to have been fully realised by chemiqts, hut it is safe to say that ultimately it will be regarded as the most important property to be learned by those concerned with molecular structure.
The introduction of the Exclusion Principle int,o chemistry through the tangent-sphere representat,ion of
electronic structure leads to a set of Universal Structural Principles. I n chemical compounds A polyhedron of anions, which may be localized electrons, is farmed about each cation or atomic core. A cation's coordination number is determined (in part) by the radius ratio. Rattling is bad. Hence, Small cations generally have a coordination number of four. Large cations may have a coordination number greater than four. Polyhedra, may share corners, edges, or faces. Edge- and face-sharing destabilizes a. structure, owing to the cation-cation Coulomb term. Similarly, Small cations with large chargea tend not to s h a x polyhedron elements with each other. When two polyhedra do share an edge or face, core-core repulsions may operate to displace the cations, if small, from the centers of their coordinated polyhedra. Anions may be shared by more than two cations.
Ion-packing models are simple, in the Feynman sense that "a thing is simple if you can describe it fully in several different ways without immediately knowing that you are describing the same thing" (87). Initially, the models were viewed as geometrical expressions of the Exclusion Principle ( I O U ) ; then as an inference based upon the Isoelectronic Principle (18); later as a quantum-mechanical refinement of Lewis's interpretation of the graphic formulas of classical structural theory (88); still later, as a generalization of Werner's coordination theory (89); and now, as an extension of the idea of impenetrability, from whole ions in crystals to individual electrons in molecules. Ion-packing models may be viewed, also, as concrete representations of Berzelius's polar opposites, Earnshaw's detached particles (40),van't Hoff's tetrahedral atom, Pauling's hybrid orbitals ( d l ) ,Fajan's quanticules (IS), Fermi's holes ( I O U ) , Lennard Jones's o-regions (4!2), Gillespie's valence-shell-electron-pair repulsion theory (4.9, Linnett's spin-sets (44), Ruedenberg's localized molecular orbitals (46), and Frost's spherical gaussian orbitals (46). More generally, they are models of the configurations of maximum probability of wave functions that satisfy the Exclusion Principle and that therefore display in some measure the properties of an exclusive orbital representation, of which a tangent-sphere representation is a special case. Ion-packing models are in fact not new. They are, to paraphrase what Schrodinger has said about the wave equation, a completely organic development, one might almosl he tempted to say a more elaborate exposition, of older models (47),Figure 10. Literature Cited (1) BENFEY,0 . T., "From Vital Force to Structural Formulas," A. J., Houghton Mifilin, New York, 1964, p. 12; IHDK, "The Development of Modern Chemistry," Harper and Row, New York, 1964, p. 132. (2) Lriw~s,G. N., "Valence and the Structure of Atom8 and Molecules," The Chemical Catalogue Co., 1923; reprinted by Dover Publicrttions, New York, 1966, p. 74. (3) BARLOW, W., Nature, 29, 186 (1883). (4) BRAGG,W. L., Phil. Mag., 40, 169 (1920). (5) BRAGG,W. L., Phd. Mag., 2, 258 (1926). ROY.SOC. (London), 114, (6) BRAGG,W. L., A N D WEST,J., P~oc. 452 (1927). (7) PAULING, L., J . Am. Chem. Soc., 51, 1010 (1929). L., J . Am. Chem. Sac., 49, 765 (1927). (8) PAULING, (9) PAUL,,W., Sei., 103, 213 (1946). (10) (a) BENT,H. A,, J. CHEM.EDUC.,40,446 (1963); (b) ibid.,
Figwe 10. Models of single, double, ond triple bonds published by Know in 1894 (481. Knorr's models correspond to Crvm Brown's early reprerenAll that ir missing to tation of a carbon-carbon ringle bond or C--C. reduce Knorr'r models to our current represenlotion, Figure 1, is the concept of shoring.
40, 523 (1963); (c) ibid., 42, 302 (1965); (d) ibid., 42, 348 (1965). (11) RICHARDS,T. W., J . Chem. See., 99, 1206 (1911). 112) L ~ w r s .G. N.. J . Am. Chem. Soe.. 38. 762 (1916). . . i13j FAJAG, K., ~ h i m i a 13, , 349 (19i9). ' (14) RADISAY, W., J . Chem. Soe., 93, 774 (1908). V. M., Tram. Fm. Soe., 25, 253 (1929). (15) GOLDSCHMIDT, J. R., (16) MESSER, C. E., MILLER, R. M., AND BARRANTE, Inarg. Chem., 5, 1814 (1966). K., Ceramic Age, 54, 288 (1949). (17) FAJANS, 118) BENT.H. A,. J . CHEM.EDUC..43. 170 11966). i l 9 j LEWIS, G. N., Trans. Far. SO;., 19,454'(1923); J. Fmnklin Tn.rt. 226. 295 . ....., . ., . .II92RI. (20) LANGMUIR, I., J . Am. Chem. Soe., 41,868 (1919). W., AND RAYNOR, G. V., "The Struc(21) (a) Hum-ROTHERY, ture of Metals and Alloys," (3rd ed.), The Inst. of Metals, London, 1954, p. 194; (b) WELLS,A. F., "Stn~ctural Inornanic Chemistry." ed.), Oxford University .. (3rd . ~ r e s i London, , 1962. 122) W. G.. J. CHEM.EDUC..40.54 11963) ~ - -GEHMAN. -, . . (23) MENDELEEPP, D., "Principles of Chemistry," Vol. I., (5th ed.) (lranslator: KAMENRKY, G.), Longmans, Green, and Co., New York, N. Y., 1891. (24) HUGGINS, M. L., Phys. Rev., 28, 1086 (1926). (25) BRAGG,W. H., J. Chem. Soc., 252 (1916). 126) . . KIRCHHOPF.W. H.. AND WILSON,E. B., JR., J. Am. Chem. Soe., 85, i n 6 (1963). 127) . D.. Proo. . , W ~ w n A. . , i n Stereochem... 1.. 1 (1954). H., 31, I'roc. Roy. Sor. 2 9 S l n t i w ~ c ~N. , V., n o l ' o s . ~ ~ . ~ l.on~lon),A176, I i 3 (1!14(1 . R . .I., . t ~ nNI.HOI.\I,R. S., Q m ~ t R . P I . ,11, 339 (29 GII.L~..PI~., 10:7 ,. ,A"-. (30) (a) Cj., LATIMER, W. M., J . Am. Chem. Soe., 51,3185 (1929); S. B., J . Am. Chem. (b) PAULING,L., A N D HENDRICKS, Soe., 48, 641 (1926). (31) BENT,H. A., Chem. Rev., 61, 275 (1961); J. CHEM.EDUC., 37. RIfi - - 119fiOl. ~ , ~ ~ ~ (32) L ~ Y E, . D., cia Cryst., 18, 141 (1965). (33) BRAGG,W. H., "X-rays and Crystal Structure," (4th ed.), G. Bell and Sons, London, England, 1924. (34) FAJANS,K., "Quantieule Approach t o the Metallic State," paper resented a t Gordon Res. Conf. on Metals, New Hampton, N. H., July, 1950; FAJANB, K., AND KREIDL, N. J., J. Am. Ceramic Soe., 31, 105 (1948). \----,-
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(35) BYKUV, G. V. (Translator: LEEESTER,H. M.), Chymia, 10, 199 (1965). (36) LENNARD-JONES, J., Ado. Sei., 136 (1954). (37) FEYNMAN, R. P., Phys. Today, 19, 31 (1966). (38) BENT,H. A,, Chemistry, 3 9 , 8 (December) 1966; Chemistry 40. 1967. .., R.(.Tannnrv) ~ ..--", . -~ . ~ ~ (39) BENT,H . A., J. CHEM.EDUC.,44, 512 (1967). ( 4 0 ) EARNSHAW, S., Tmna. Camb. Phil. Soe., 7 , 97 (1842). (41) PAULING, L., "The Nature of the Chemical Bond," (3rd ed.), Carnell Univ. Press, Ithaca, N. Y., 1960. (42) LENNARD-JONES, J., P ~ c ROV. . SOC.(London), 198, 14 (1949).
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(43) GILLESPIE,R. J., J. CHEM.EDUC.,40, 295 (1963). (44) LINNETT,J. W., "The Electronic Structure of Molecules. A New Approach," John Wiley & Sons, Inc., New York, N. Y., 1964. (45) RUEDENBERG, K., Modern Quantum Chemistry, 1 , 85 11965): EDMINSTON. C.. AND RUEDENBERG. . K... J. Chem. ~ h y s . 4; 3 , 897 (1965). ' (46) FROST,A. A,, PRENTICE, H. B., 111, AND ROUSE,R. A., J. Am. Chem. Soe., 89, 3064 (1967). E., in "Nobel Lectures in Physics," Vol. 2, (47) SCHRODINGGR, Elsevier Puhl. Co., New York, N. Y., 1965, p. 309. (48) KNORR,L., Ann., 279, 203 (1894)
