Atomic Orbitals and Valence W . F . LUDER Northeastern University, Boston, Massachusetts INTRODUCTION
0
NE of the most powerful tools the chemist has is the electronic theory of valence. Yet it still is not being used generally to full advantage. The two prinapal factors responsible for this seem to be: reluctance to abandon the "rule of eight" and failure to make use of the concept of atomic orbitals. Ever since the beginning of the theory as suggested by G. N. Lewis in 1916, i t has been obvious that there were numerous compounds which did not "obey" the rule of eight. However, several followers of Lewis have so strongly emphasized the rule of eight that it is still generally accepted by those unfamiliar with the recent literature in the field. The idea of atomic orbitals from elementary quantum theory is now often presented as an aid to the understanding of atomic structure [even in one freshman textbook ( I ) ] but is still rarely applied to the problem of valence. I t is the purpose of this paper to show in an elementary way how abandoning the rule of eight and applying the theory of atomic orbitals can lead to a fuller understanding of valence. In order to do this, sohe of the details, such as are involved in hybridization of orbits for example, either will be omitted or modified for the sake of simplicity. Some of the modifications explicitly proposed or implicit in the discussion may give rise to objection on the ground that they do not give the complete picture. But after all, we do not yet understand all the phenomena of valence by any means. THE RULE OP EIGHT
The familiar dictum that in forming chemical bonds atoms gain, lose, or share electrons to make the outermost shell of each atom contain eight electrons probably gained its iirm adherents because the majority of known chemical compounds are made up primarily of elements in the first two rows of the periodic table. With but few exceptions these elements do obey the rule of eight. The presence of carbon with its thousands of compounds seems to lend added support to the rule. Actually, of course, the sheer number of componnds of any one element has no bearing whatever on the rule of eight. The carbon atom counts merely as one atom in the ten. But even in these ten, two of the eight elements which do not already have complete valence shells do not always obey the rule of eight-namely, beryllium and boron. There is little doubt that boron trichloride is covalent and, therefore, that the boron atom has only six electrons in its valence shell:
Beryllium chloride also seems to be covalent, as shown by its high solubility in benzene. This means that the beryllium atom has four outer electrons in beryllium chloride:
When the third period is considered, three out of seven atoms show "exceptions" to the rule of eight: aluminum, phosphorus, and sulfur. Aluminum chloride, phosphorus pentafluoride; and sulfur hexafluoride are typical covalent c o m p o ~ d s . They must, therefore, contain six, ten, and twelve bonding electrons, respectively. Proceeding further in the periodic table many other such examples of covalent compounds containing six, ten, and twelve electrons in the valence shell may be found. Most of these are in Groups 111, V, and VI of the representative elements (Figure 2). Several occur in Group VII, but will be discussed later. The existence of iodine heptafluoride shows that fourteen electrons are possible in the valence shells of some atoms. Other violations occur also in the related metals (see Figure 2). Tungsten, in addition to the di- and tetrachlorides, forms covalent penta- and hexachlorides, WCls and WCle. . WBrs and WBra are also known, while Wehas the low boiling point of 19.5'C.* In addition to the di-, tri-, and tetrachlorides, molybdenum forms covalent compounds having the formulas MoC4 and M0F6. Osmium forms a covalent hexafluoride (b. p. 205'C.). but even more embarrassing for the adherents of the rule of eight, i t forms OsFs, a stable, typically covalent compound having a boiling point of 47.3"C. Here is one compound a t least having twice the number of bonding electrons permitted by the rule of eight! The examples given above have included only those compounds in which the bonding is of the simple covalent type. Coordination compounds and metal carbonyls offer many more "violations" of the rule of eirrht. Other cases in which some electrons are not * In view of the rarity of pentacovalency, doubt may arise concern in^ its exirtence in W C l a and WBro. However, the low
boiling points together with the fact that the boiling point of the I,rornidc is the hicher - - - - - ~ - -~"~ of the two mav be taken as sufficient evidence for covalency until crystal structure measurements are made. If the compounds were ionic the reverse order of boiling points would be expected. ~~~~
~~~~~
~
-
used as bonding electrons (analogous to the lone pair in ammonia) will be discussed subsequently. Turning to a consideration of the valences of the related metals (Figure 2) when acting as positive ions, we find that not one of the many positive ions formed conforms to the rule of eight. In none of them does the loss of electrons leave a completed shell of eight electrons. This fact is largely obscured in the older type of eightcolumn periodic charts, but is very obvious in Figure 2. Most of the evidence illustrated by the cases just discussed has been ignored by the more ardent advocates of the rule of eight. A few compounds have been considered by them, e. g., phosphorus pentafluoride and sulfur hexafluoride, and various devices have been adopted in the attempt to force compliance with the rule. One was the hypothesis of single electron bonds in such compounds as phosphorus pentafluoride and sulfur hexafluoride. It is no doubt true that oneelectron bonds do exist in a very few cases. The hydrogen-molecule ion is evidently held together by one electron, but the bond is much weaker than the electronpair-ven in this, the most favorable rase. If oneelectron bonds exist in the boron hydrides they are still bility of these compounds.
To maintain eight valence electrons for the sulfur atom in the hexafluoride would require four one-electron bonds, which (becauseof the large size of the sulfur atom compared to hydrogen or boron) would be far more unstable than those in boron. Yet sulfur hexafluoride is one of the most stable compounds known (5). In spite of this conclusive evidence it has been claimed ( 6 ) that parachor measurements support the presence of one-electron bonds in such compounds. But as pointed out by R. Samuel the parachor may lend itself equally well to different schemes of valence depending upon the original assumptions (7). Additional exceptions to the rule of eight will be considered in more detail below. Abandoning the rule of eight, it would seem then, is the first step in acquiring the fuller understanding of valence made possible by the progress in our knowledge of atomic structure. ATOMIC ORBITALS
According to the quantum theory, the periodic system can be regarded as built u'p from hydrogen by adding one proton a t a time to the nucleus (plus the required number of neutrons) and one electron a t a time to the atom outside the nucleus. From the periodic table and spectroscopic data it is deduced that the electrons arrange themselves in main levels and sublevels in a regular way as shown in Figure 1. Figure 1 is a plot of approximate, relative energy levels (8). Each circle represents one atomic orbital. On paper each circle may be regarded as a box which can be occupied either by one electron or by two electrons spinning in opposite directions. The numbers in Figure 1 are values of the total quantum number n which represents the distance from the nucleus. Each main level is designated by one value of the total quantum number. The letters s, p, d, f correspond to the values 0, 1, 2, 3 for the azimuthal quantum number 1, which determines the "eccentricity" of the orbital. The difference in eccentricity leads to sublevels in the main levels. Both types of numbers must be used in the equations for the energy of the orbitals. The azimuthal quantum number has a series of values for each value of n: For example, if n = 3, 1 has the values 0, 1, 2 which results in three sublevels in the third main level. These are designated as 3s, 3p, 3d. From the periodic table and from spectroscopic data obtained when the atoms are placed in a magnetic field it is found that s sublevels always contain one orbital, p-sublevels always contain three orbitals, d subleve:s contain five orbitals, and f sublevels contain seven orbitals. It is obvious from Figure 1 that in the upper energy levels there is considerable overlapping of main levels. Furthermore, some of the closer sublevels apparently interchange in some cases. This is indicated by vertical lines joining the sublevels ifivolved. In general, the electrons fdin the lowest energy levels first. One electron goes into each orbital on a given
I s 1
I
P
d
FIGURE2.-ATO.UrC STRUCTURE CHARTOF THE ELEMENTS line until all the orbitals a t that level are occupied by one electron. When every orbital in a given sublevel contains one electron, then and only then, they begin filling up with two electrons in each orbital. Two electrons in an orbital are said to be coupled or paired. Supposedly the opposite spins of coupled electrons resnlt in electromagnetic attraction more than sufficient to overcome the repulsion of like charges. The way in which electrons are added to the orbitals can be made clearer by taking some examples from Figure 2. From the chart the electron-confi uration of $8 ,om 8~ V Jobdg M aluminum is seen to be: 1 9 2Q,
from the chart and written as follows (writing the 2 p orbitals together since there is no change in them) : 123: 1s' 2 9 2pa3s' 3 39
@&.
/r g, a, L
phosphorus
aPr
f l3P da
DP: Iss 2s22p' 3s' 3p 39 3p
Sulfur
,.s:
IS=2s22p83sZ3pa 3p 3 p
Chlorine
,dl: ls"2sP 2p 3sP 39
1rC1: Is22r22Pa3r43pa 3p23p
where in this case the superscripts represent the numher of electrons in each kind of orbital. If we wish to use the superscripts to represent the number of electrons in each orbital the three 2p orbitals must be separated as follows:
Note that phosphorus has three unpaired p electrons, but that since there can he only three P orbitals sulfur has only two unpaired electrons and chlorine only one. We shall now make a slight modification in the usual way of writing electronic symbols. The five atoms just discussed may be written.= follows:
,&I: 1 9 2sz2pz 29' 2pp3s' 3p
Remembering that the orbitals take electrons one a t a time until each orbital has one electron in i t before coupling begins, the electron-configurations are taken
:A].
:si.
:P.
:S:
:Cl:
Written this way the symbols indicate the paired and
of the representative elements (Figure 2) are concerned, they will not be discussed here. Neither will the rare earths be considered. Readers interested in the rare earths may consult an article by Pearce and Selwood (10).
One fact should be called to attention before proceeding with a brief discussion of the related metals (Figure 2). There are practically no monatomic negative ions having a charge greater than minus two. There are practically no monatomic positive ions having a charge greater than plus three. For example, we are accustomed to think of "stannic ion" as having a valence of plus four, but actually, there is no such thing as a stannic ion in pure stannic chloride, a typical covalent compound. Even in water the stannic ion has no existence as a monatomic ion with four positive charges on it. Apparently insufficient energy is involved in ordinary chemical reactions to build up charges greater than minus two or plus three. According to Coulomb's Law, the size of the atoms involved must also have an important bearing upon the formation of ions so that beryllium, a small atom, does not show muchkendency to f o m an ion with a valence of plus two. 'IYle rule of two centers attention on those atoms in a molecule which are completing their orbitals in the combination, so i t does not add much to the treatment of the related metals as positive ions. However, it might be well to point out that the related metals do not violate the rule of two as they do the rule of eight. As mentioned previously, none of the many positive ions of these metals conforms to the rule of eight, but the rule of two does not specify the nature of the positive ion. The number of electrons removable to complete the orbitals of some other atom is determined by size and charge as well as by the numbers of unpaired electrons. For example, the electronic configuration of iron is:
and the electronic symbol showing electrons from both incomplete shells as valence electrons is:
factor. We shall do so by first applying the rule of two to each compound chosen as an example assuming i t to be formed from neutral atoms whose electron configurations are given in Figure 2. Considering the representative elements first, we find that the maximum number of valence electrons for elements in the second period acwrding to the Panli principle is eight.$ Beryllium and boron chlorides have already been discussed sufficiently to illustrate compounds in which the maximum of eight is not reached. The properties of boron trichloride as a strong acid (2) make i t apparent that the one empty orbital of the boron atom can still be filled by two paired electrons from an atom in a basic molecule. The remaining four active elements in the second period require no further discussion, since for them the rule of two becomes equivalent to the rule of eight. In the third period (considering only the representative elements in that period for the time being) the number of orbitals is nine. The tendency of. magnesium to fill up its orbitals seems to be the reason for the activity of Grignard reagents. The tendency of aluminum chloride to fill up an empty orbital in the aluminum atom results in its familiar activity as a strong acid catalyst (4). The SiF8c2ionapparently involves the filling of two orbitals of silicon in silicon tetrafluoride by paired electrons from two fluoride ions to give a total of twelve electrons.$ Carbon in the preceding period cannot do tbis because in its compounds all its orbitals are filled by eight electrons. The phosphorus atom has two paired and three single electrons in its valence shell so phosphorus trifluoride is written (using crosses for phosphorus electrons):
Phosphorus pentafluoride must be formed by uncoupling the lone pair of 3s eleetroas. This might be formally represented as follows:
It is easy to remove two electrons (suppose we say two of the unpai1M38 electrons) to give the ferrous ion, a little more difficult to remove a third one, and impossible because of the large electrical field to remove the fourth electron by ordinary chemical means. Most of the ions of these metals are left with unpaired electrons, as is shown by their color and paramagnetism.
-
..
:F: x.
__
:P:F:..
.x. :F:
..
+ 2 . F..:
-:'8 ..
''p" '.p.; . c +. .... P fP:
..
f
-.
.?..
where the small arrow indicates that the two paired
COVAL.ENCE
-
f R . Samuel (7) mentions the possihle existence of nitrogen
When atoms comolete one or more orbitals to form pentatluoride. Such a compound if it does exist would involve covalent bonds, thd number of pairs shared between promotion from = to = 3. S The question which arises at this point-amely, what them depends upon various factors including: (1) the determines in detail the number of atoms grouped about the number of unpaired electrons in each atom; (2) spatial central atom in such cases?-is beyond the scope of this article. it to say here that completion of orbitals may be opposed considerations; and (3) bond energies. Bearing num- Suffice by the repulsion of built-up charges or hindered by lack of space bers (2) and (3) in mind, we shall emphasize the first for enough atoms to gather round the central atom.
electrons are uncoupled during the reaction. Actually the two fluorine atoms are combined in a molecule and the paired electrons between them must alm be uncoupled, but they are written separately for the sake of simplicity. This uncoupling of paired electrons requires energy, but not more than the reaction can amply supply, since the compound is formed and is fairly stable. The extreme instability of nitrogen pentafluoride if i t exists a t all (7) must be due to the fact that the nitrogen atom has no more orbitals available for which n = 2 and promotion of electrons from n = 2 to n = 3 would require more energy than the reaction can furnish." The series of sulfur compounds &Fa SF%, SF,, S2Fla,SF6(5) offers an interesting application of the rule of two. The sulfur atom has two pairs and two single electrons:
both molecules the .sulfur atoms have twelve valence electrons:
The interhalogens of the types AB3,ABs, and AB? are also interesting examples of the rule of two. TCls must involve uncoupling of two of the paired electrons in the iodine atom of iodine monochloride :
This gives ten electrons in the valence shell of iodine. I n iodine pentachloride there should be twelve, one more pair being uncoupled.
SFz must be formed by direct pairing of the two unpaired electrons in the sulfur atom:
The formation of SF, involves the uncoupling of one pair, leading to ten electrons in the sulfur valence shell. Both pairs are uncoupled in SF6giving a total of twelve:
Two dierent electronic formulas have been suggested for &Fz (13):
The weight of the experimental evidence seems to favor the second of the two. I n the second formula two of the paired electrons on the upper sulfur atom have been uncoupled to pair off with the two unpaired electrons of the lower sulfur atom. This gives the upper sulfur atom ten electrons in its valence shell. S2Roresembles SF6to some extent in its chemical properties, but is more reactive (5). The electronic formula is similar in that in
11 The pentachloride (11) and pentabromide (12) of phosphorus have been the subject of crystal structure investigations which have shown them to be ionic in the solid state. The ions are PCl.+1 and PCls-' for the chloride and PBrdtl and Br-' for the bromide. This does not mean that they are not covalent in the gaseous state nor dws i t imply that I'F. i i not ~wntacwalmtin all three s t a i r s . The boiling puint f o r the fluoride i i far hclow the Imiline noinls of the chluridr and brum>dr I 1 thc fluoritle were ionicijs boiling point would he higher
Iodine heptafluoride should be written with fourteen bonding electrons, all seven of the original iodine electrons being shared with the seven fluorine atoms:
From these examples chosen from the representative elements it is apparent that (in many more cases than for the rule of eight) the rule of two automatically gives the correct result when molecules are considered as if formed in the same way as the periodic system is built up. A few more examples taken from the related metals will be considered next. The older eight-column periodic charts obscured the similarities observed in the chemical properties of the related metals. The type of chart illustrated by Figure 2 clears up this source of confusion. All these atoms have either one or two electrons in the outermost shell and therefore behave primarily as metals. Most of them have two electrons in the last shell and from the chemist's point of view most of the remainder could be considered as having two also. Their similarities when acting as cations have already been considered. I n spite of their primary characteristics as metals some of them in one respect do behave as nonmetals. Chromium and manganese, for example, form negative ions with oxygen which correspond to sulfate and perchlorate ions. Yet neither chromium nor manganese forms a negative "chromide" or "manganide" ion as do sulfur and chlorine. The answer to these apparent con-
and the 0 s atom in OsFs must have sixteen electrons:
The failure of the related metal carljonyls and nitrosyls to obey the rule of eight was recognized from the beginning. [Details may be found by consulting reference (5).] So it is not surprising that our present concepts of their valence relationships already are so close to the ideas presented here that no further discussion of them is necessary. I t would he interesting to speculate about other covalent compounds of the related metals, but in just these cases it would be merely speculation since insufficient experimental data are available. However, the above examples of covalent compounds, together with those considered in the previous section, should be sufficient to illustrate the application of the rule of two to the problem of valence. It is evident that proper use of the rule of two demands: (1) application of our knowledge of atomic orbitals, (2) modification of our present method of writing electronic symbols so as to represent two partly filled energy levels of practically the same energy as one valence shell. SUMMARY
Certain modifications in the electronic theory of valence as customarily presented are proposed. These involve:. 1. Abandoning the rule of eight. The majority of
the ninety-two elements show exceptions to this rule. 2. Substituting the rule of two. This implies that the formation of molecules takes place in the same manner as the periodic system is built up. 3. Writing electronic symbols to show in one valence shell those electrons which belong to two partially complete energy levels of practically the same energy. These modifications seem to lead to a more satisfactory theory of valence since the theory now applies much more widely than hitherto. It is interesting t o note that in some cases the new electronic formulas are equivalent to the older formulas in use before the rule of eight was proposed. LITJ3RATURE CITED
(1) (2) (3)
..
(4) (5)
FOSTER. W., AND H. N. ALYEA, "An Introduction to Elrmentary Chemishy," D. Van Nostrand Company New York, 1941. LUDER, W. F., Chem. Revs., 27, 547 (1940). LUDER.W. F.. AND S. ZWPANTI. I. Am. Chem. Soc.. 66. 524 (1944).
'
LUDER, W. F.. AND S. ZWPANTI, Chem. Revs., 34,345 (1944). EMELOUS, H. J., AND J. S. ANDERSON, "Modern Aspects of Inoreanic Chemistm." . . . . D. Van Nostrand Comoanv. ~ e w - ~ o r 1939. k, SUGDEN, S., "The Parachor and Valemy," George Routledge &Sons. Ltd.. London. 1930. SAMUEL, R., J., Chcm. Phys., 12, 180 (1944). PAULING, L., "Nature of the Chemical Bond," Corndl University Press. Ithaca, New York, 1939. LONDON, F., 2. Physik.. 46. 455 (1928). PEARCE, D. W., AND P. W. SELWOOD, J. CREM.EDUC., 13, 224 (1936).
POWELL, H. M., D. CLARK, AND A. F. WELLS, I. Ch~m. Soc., 642 (1942). POWELL, H. M., AND D.
CLARK, Nature, 145,971 SAMUEL, R., I. Chm. Phys., 12, 380 (1944).
I t is said t h a t o sampling of diets has shown t h a t half thepeople of thin country still suffer f r o m a deficiency of v i t a m i n Bn,or riboflavin. Says t h e Deportment of Agriculture in Food and Home Notes, "Riboflavin m y n o t be t h e magic sought b y P o m e d e Leon, who dreamed of drinking a t t h e fountain of youth, b u t it i s essentialfor growth and general good health and for preserving y o u t h and uigor." T h e Department of Agriculture further states: "As w i t h almost all uitamin deficiencies, people who d o n o t get enough riboflavin tend t o be nervous and irritable and feel tired and r u n down. Eyes i t c h and b u r n and look tired and bloodshot. Vision m y be blurred in d i m light, and bright light hurts. Air Corpsfliers have oecasionolly reported eye disturbances after flights in bright weather. Some scientists t h i n k t h i s m o y be d u e t o t h e riboflosin of t h e eyes being erposed t o the destructive effects of light. "Characteristicsigns of a diet deficient in riboflauinfor a considerable length of t i m e show u p in t h e tissues of t h e eyes, m o u t h , and skin. S y m p t o m s include cracked and scaly places a t t h e corners of t h e m o u t h , and shining red sores around nose, eyes, and eyelids. T h e tongue becomes inflamed-deep red and gloss-nd swallowing m a y be diffceult. Comparatively few people show these signs of r i b o f i t i n deficiency t o a n acute degree, b u t too m a n y i n t h e country are n o t u p to par became t h e y do n o t get t h e a m o u n t t h e y need for m a r i m u m health. "Milk is probably t h e m o s t important source of riboflauin, since i t furnishes almost half t h e supply of this vitamin in our notional diet. Milk i s n o t t h e m o s t concentrated Nutrisource. Liver, kidney, cheese, eggs, and some greens like kale are richer. tionists estimate t h a t bread a n d f l o u r enrichment has raised t h e riboflamin content of t h e nwrage American diet about one seventh."
.. .
(1940).
