Irwin Cohen and Thomas ~ustard'
Youngstown University Youngstown, Ohio
I
I
Atomic Orbitals Limitations a n d variations
Atomic orbitals are increasingly being used in beginning classes in chemistry, yet the limitations of the atomic orbital concept and the differences among the several types of atomic orbitals are rarely discussed a t a level suitable for teachers and students not familiar with quantum mechanics. Recent introductory treatments of quantum chemistry ( 1 ) have emphasized valence bonds and molecular orbitals rather than the concept of the atomic orbital, although the paperback by N. N. Greenwood (3) offers some excellent discussion in this direction. A description of the atomic orbital concept itself is the subject of this account. The three most widely used methods of arriving a t a set of atomic orbitals afford respectively hydrogen-like orbitals, self-consistent field orbitals, and various analytical approximations such as the Slater or Morse orbitals, all of which may d i e r greatly in shape and size from each other. The following discussion concerns the origins, basic features, and limitations of these different kinds of orbitals. It is intended to show that the orbital concept is not fundamentally correct for atoms with more than one electron, but rather that it is a useful approximation, and that hydrogen-like orbitals with a simple modification represent the best introductory picture.
form theset of H-like wave functions, or orbitals. These solutions are usually obtained as products of three factors, R, 8, and which are functions of the spherical coordinates r, 8, and 4, respectively, and which are characterized by the three quantum numbers n, I, and m:
*,
+ d r , 49)
=
(3)
RnL~).ed8).~.(+)
The radial function R includes the factor, e-Zr'n, which describes the concentration of the wave function at the nucleus with an exponential decay toward zero in all directions outward. All three functions, R, 8, and 9 also include complicating factors which may introduce nodes into the simple distribution. These nodes, together with the exponential radial decay, determine the shapes of the orbitals. Since the radial function R depends only on r, the distance from the nucleus, the radial nodes are spherical surfaces. The number of radial nodes is given by the expression, n - 1 - 1. The radii of some of these nodes are given in Table 1 in terms of Zr; that is, the figures in the table are to be divided by Z to give the distances in Bohr radii.
H-Like Orbitals
The Schrodinger equation for an atom may be written in the form v V 2(E - V ) $ = 0 (1) where V2 is a differential operator whose exact form is not needed here, is the wave function, E is the energy of the system in atomic units (one unit = 627.7 kcal/ mole), and V is the potential energy in atomic units. For a hydrogen-like atom (an atom containing only one electron, such as H, He+, or Li++) the potential energy of the electron of charge -e a t a distance r from the nucleus of charge +Ze is -ZeZ/r, or in atomic units, V = -Z/r (2) where r is expressed in atomic units (Bohr radii, 0.529 A). Thus, the potential energy of an electron one Bohr radius from a proton is - 1atomic unit. For simplicity, atomic units of energy, charge, and distance will be used throughout the remainder without explicit comment. If eqn. (2) is put into (I),then the solutions to (1)
+
+
Presented in part before the Division of Chemicd Education a t the 149th meeting of the American Chemicd Society, Detroit, A~ril.1965. -'~iesent address: Chemistry Department, University of Cincinnati, Cincinnati, Ohio.
e
Figure I. Kinds of angular nodes. (01 Node in functim~is double verticolronc obout the r oxir l b l Onodereducesto single horirontol plane o t 0 = 90'. k1 Node in P, function is vertical plone posing through the r oxis, a t dihedral angle from the r x plane.
+
There are a number of diierent possible sets of 8 and Here we shall use only one set: the usual real, spherical harmonics. In this set, the nodes developed by the 8 function depend only on 8, the angle of the radius vector from the +z axis, in such a way that they are always described by the pair of angles 8 and (180" - 8 ) together. Therefore, these nodes are either
* functions.
Volume 43, Number 4, April 1966
/
187
Table 1. Orbital 1s
%
4f Orbital
Radii of the Spherical Nodes, H-like Orbitals (in units of Z X r.)
Nodes none none none none Nodes
Orbital
Nodes
Orbital
Nodes
2s 3P 4d
2.00 6.00 12.00 20.00 Nodes
39 4~ 5d
1.90, 7.10 5.53, 14.47 10.89, 24.11 18.00, 36.00
Sf Orbital
two vertical cones, which are symmetrical about the z axis, or the xy plane, which is perpendicular to the z axis (Fig. l a , b ) . The nodes introduced by the function depend only on the angle of rotation, $, about the z axis; these nodes, therefore, are planes passing through the z axis (Fig. lc). The total number of angular nodes for any orbital is given by 1, so that s orbitals have none, p one, d two, and so on. The po orbital (1 = 1,m = 0) has as its node the horizontal plane (0 = 90') and the p , ~orbitals ; (1 = 1, iml = 1) have one vertical node, at 4 = 0' or a t 6 = 90". The po is often called p, and the two p l l orhitals are usually called p, and p,. Figure2 shows the angular nodes for some of the d and f orhitals. Note that the number of $ nodes (vertical planes) is given by Irnl and that the number of 8 nodes (counting the double cone as two) is 1 - lm(,so that the total number is always I . One can readily extend this pattern to the higher orbitals, starting with two concentric double cones for the go orbital, two double cones and a horizontal plane for the ha, and so on. I n this way, one sees the system of orbitals as a natural and logical development rather than as an arbitrary selection of unusual patterns.
Sf Orbital
nodes. An analogy to vibrating strings or drumheads where the number of nodes determines the frequency of vibration, together with the relation E = hu (energy is proportional to frequency) makes the qualitative explanation of orbitals in terms of nodes particularly satisfying for beginners."ccording to this concept, the energy of all orhitals of the same principal quantum number should be identical, and this is in fact very nearly the case for H-like atoms; the energy depends almost exactly on the number of nodes and not on the kinds of nodes. The spatial distribution of anorbitalmay hevisualized from Figure 2 by imagining the orbital to be concentrated toward the center but confined to the regions b e tween the nodes, approaching zero as a node is approached and changing sign on crossing the node. E. A. Ogryzlo and G. B. Porter (3) have given excellent twoand three-dimensional figures and a mathematical treatment of several H - l i e orbitals and hybrids. The probability density ($2) has essentially the same distribution as $ (4) since the nodes are perfectly unchanged in going from $to fi2. However, is everywhere positive (instead of alternating in sign on opposite sides of the nodes) and approaches zero more rapidly close to the nodes. There are available some excellent sets of figures (6) based on graphs of the angular functions 0 and @, which give a useful diierent perspective on d and f orbitals. These should be compared with Figure 2, although care must be taken that such angular fuuction graphs are not confused with orbital contours (6). Hybrid orbitals are linear combinations of the basic orbitals. For example, (sp) = (sp2)=
(d%pa)= Figure 2. The angulor nodes of the d and f orbitolr The total number of ongulor nodes is given by the azimuthal quantum number, I,ond the number of vertical (6nodes) is given by he absolute value of the orbital mognetis quantum number, Iml. Volves of imj ore given or rubscript,. Rotation aboutthe. axis b y 90°/1ml generoter the otherd and f orbitals, one for each non-zero value of lm/.
An increase in the principle quantum number, n, produces an increase in the number of radial nodes (Table 1) with no change in the angular nodes. For example, the 3d orbital has only the nodes shown in Figure 2 whereas the 4d orbital has one radial node in addition to these angular nodes, the 5d orbital has two additional radial nodes, and so on. Since the number of radial nodes in any orbital is n - 1 - 1, and since the number of angular nodes is I, the total number of nodes is n - 1. Thus, there is a simple relation between energy level and number of 188 1 Journal of Chemical Education
Nodes
a + ($2.
$2,)
(haf
.\/Z$sr)
4% (@$sa
+
$6.
f
dh)
Figure 3 shows contours for the H-like s p (digonal) hybrid and for the H-like d 4 p 3 (octahedral) hybrid. Unfortunately, the set of H-like orbitals can serve as only a rough picture of the electron distribution in atoms containing more than one electron. In such atoms, the Schrodinger equation (1) still holds but $ is then the 2 For those who care to go a bit deeper, it may also be noted that although the set of orbitals shown here may be pictured in terms of well-defined standing waves and nodes, these orbitals do not have a well-defined orientation of angular momentum. Roughly speaking, position is defined but momentum is not. If we oboose a set of €and I @ functions to yield orbitals with welldefined orientation of angular momentum (for example, m = f1 or rn = -1 instead of lml = 1) then position becomes undefined; we lose the 6 nodes and obtain orbitals describable in terms of running waves, circulating about the z-axis.
Figure 3. H-like hybrids. la) sp orbital with Z = 4, for a finCrow atom. (b) d%paorbital with Z = 10, for a firsCrow honrition meld. Distance "nib are Bohr rodii. Contours ore drawn a t probability densities of 0.1 (hatched line) and 0.01 (plain line) electronr/cubic Bohr rodiu., or JI volve of 0.316 and 0.1 atomic unlb, respectively. Broken liner indicate noder Rotation of the figurer .bout their horirontol a x e s generates the threedimensional contours.
wave function for all the electrons simultaneously. Furthermore, the potential energy equation (2) has to be replaced by a more complicated expression, for all of the electrons instead of just one. For example, in a twoelectron atom, (2) becomes V
=
+ (llrd
-(Zlr,) - ( Z h )
(4)
where n and rz are the distances of the two electrons from the nucleus and rlz is the distance between the electrons. (The r,, term is positive since it represents the repulsion between the two electrons.) For an atom containing several electrons, we obtain
where r, is the distance of the ith electron from the nucleus and rc, is the distance between electrons i and j. The substitution of (4) or (5) into (1) leads to an equation that cannot be solved by any direct analytical method. Several approximation procedures have been developed, some of them very accurate, others less accurate but more convenient. The differences are not only in accuracy and convenience but also in fund* mental viewpoint. The simplest approximation method for obtaining a solution for the many- electron atom consists in setting all the l / r , terms in (5) arbitrarily equal to zero. Given this crude assumption, it turns out that the wave function of (1) can be expressed as a product of one-electron wave functions, the H-like orbitals: where the $i's for a ground state are chosen by the aujbau prin~iple.~For example, the products $,,$I,, h,b., Ashp, etc., represent some possible wave functions (in this approximation) for a two-electron atom, with the first of these representing the ground state. I n particular, the (normalized) ground state for helium would he
But since setting the llr,, terms equal to zero amounts to ignoring completely the effects of the electrons on each other, we cannot expect such representations to be very good, and indeed they are not. Even if the many-electron Schrodinger equation could be directly solved, the result would not be a set of orbitals to he combined as in (6), hut rather it would be (for the ground state) a single many-electron wave function which would describe all the electrons at once. To illustrate this and to provide a background for understanding what is involved in the orbital concept and its approximations for many-electron atoms, we first consider the most accurate approximation method now available, the Hylleraas functions. The Orbital Concept
In the 1930's E. A. Hylleraas proposed a number of wave functions for helium, such as (7) @ = e-1.827,
e-1.82**
j ( ~r,, ,
(8)
yil)
where $ is expressed in terms of its maximum value (i.e., $ = 1 a t r1 = rz = 0) rather than in absolute units, rl, rz, and r,z have the same significance as in (4), and j(71,rl. TIP)
=
1
- 0.1008(r1+ r.)
+ 0.1285(~-
+ 0.0331(h+ r# + 0.354 r~ - 0.0318
TI?
(9)
Others, following Hylleraas, have obtained functions like (8) in which (9) is replaced by a more extensive power series. The results have given extremely close agreement with the most precise measurements, better than with any other method (8). But what is the meaning of these functions? The exponential factors in (8) may be considered as H-like 1s functions with a nuclear charge Z = 1.82 instead of Z = 2 as in equation (7). This change in apparent or effective nuclear charge may he attributed to a partial shielding of the nucleus from each electron by the other electron. This shielding effect will be discussed further on. But the major reason for the success of (8) lies in the power series (9), whichintroduceselectron correlalion. The effectof electron correlation is to make the wave function depend explicitly on the distance between the electrons, in such a way as to decrease the value of the wave function when the electrons are close together and to increase the value when the electrons are relatively far from each other. To illustrate this, let n = n = r a n d let r12 = 0,so that we have the wave function for the two electrons coincident at a distance r from the nucleus. Then (8) becomes
$&C,
"Ahough we are concerned here with atomic orbitals rather than atomic struclure, it may he noted that in order to represent an atomic structure we must aetuallv use a combination of further, see any quantum mechanics treatment of atomic structure nnder "Slater determinants'' or "antisymmetrized wave functions."
Now compare this with the wave function for the two electrons equidistant from the nucleus but exactly opposite instead of coincident. Then rl = rz = r and r12 = 2r, so that The quotient of (11) and (10) now represents the ratio of the wave functions for finding the electrons opposite or coincident at a distance r:
from which it is evident that a t small r the wave function is greater when the electrons are opposite each other Volume 43, Number 4, April 7 966
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189
than when the electrons are together. (This relation breaks down a t large r, but then $ is very small. The formula gives the correct relation when r is small enough for $ to he significantly large. A more accurate function than (9) would give the correct relation to larger distances.) The fundamental significanceof these results is simply that the probability distribution of one electron can not be completely stated independently of the other. In other words, the concept of the electron orbital, a wave function which describes one electron independently of any others, is in principle incorrect if there is more than one electron. These results are pictured in Figure 4. Here the solid line shows a cross section of a contour surface for the wave function of one of the two electrons when the other is found a t the point A. Where A is 0.5 Bohr radii from the nucleus, the contour is a cross section of the spheroidal surface on which the wave function is 0.2 of the maximum. If we find the first electron a t the point B instead, then the 0.2 contour for the other is as shown by the dashed line. These contours were computed from the 39-term function by T. Kinoshita (8).
Figure 4. Contours for He-like wave function with one electron Rxed. Solid line: Rxed electron ot point A. Dashed line, Rxed electron a t point 8. Contours ore drown from the Kinoshita function la).
A complete representation of the two-electron atom would show a set of contours for every possible position of one of the electrons and would also indicate the relative probabilities of the diierent contours. Now, we may simplify this by drawing a single contour which is an average of all the possible one-electron contours, weighted according to the relative probabilities. Such a diagram omits the detailed stmcture due to electron correlation but it is the best possible representation of the atom in terms of independent one-electron wave functions, or orbitals. One procedure for finding such a representation is calledtheself-consistent field method, which obtains the distribution of an electron in the average potential of all the other electrons. A simpler procedure is to devise some variation of the H-like functions which will closely approximate this result. Clearly, a one-electron function can, a t least in priuciple, be a completely accurate average; what, then, is wrong with it? The error is in its implications for the distribution of two or more electrons, for if we assign the two helium electrons, for example, to any specific orbital, we are postulating that the distribution of each electron is independent of the other, that the ratio in equation (12) is identically one, or that the two electrons are as likely to be found close together as far apart. An orbital, then, is a useful and accurate concept, hut only (a) if we ohtain it in some way which averages the effects 190 / Journol o f Chemical Education
of other electrons, and (b) if we remember that it shows the time-averaged distribution of an electron and does not a t all indicate electron correlation (9). Although Hylleraas functions are probably the most accurate yet devised, their use is limited because they are difficultto obtain even for helium and they have not yet been prepared for atoms with more than three electrons. We therefore need other methods in order to deal with the electron stmctures of other atoms. Self-consistent Field Orbitals
The self-consistent field (SCF) method for many-electron atoms was originated by D. R. Hartree (10) and improved by V. Fock (11). The basic features of the Hartree-Fock SCF approach are as follows. (1) We assume that the orbital concept is valid. (2) We arhitrarily select a set of reasonable orbitals. The H-like orbitals may be used for this purpose, or a set of analytical functions such as those described below. For a ground state, electrons are assigned by the aufbau principle. (3) We calculate the effecton one of the electrons of all the others, and we correct its orbital for this effect. We do this in turn for each orbital, thus ohtaining a set of first-improved orbitals. The calculations are then repeated with the first-improved orbitals to ohtain a set of second-improved orbitals. The process is continued until no further significant improvement is obtained; the orbitals are then self-consistent. Hartree-Fock SCF orbitals are probably the best that have been obtained up till now, within the limitations of the orbital concept. They will correctly predict the energy of the system to within one or two percent of the observed value. But the result of this SCF method is not a set of equations like (3) but rather a set of tabulated data, so that the orbit& are somewhat difficult to use.
+
Figure 5. Grophs of v r x for (01 2 p , (bl 2s. and (cl 2spz oorbitdr for carbon. Solid liner, SCF orbitalsi dashed lines, H-like orbitals. Abvcissas are morked in units of 0.5 Bohr rodii, ordinates in units of = 0.5 atomic units SCF curves are cdculoted from dato b y A. Jucyr 112).
+
Figure 5 compares H - l i e and SCF wave functions for the 29, 2p, and 2sp hybrid orbitals of carbon, and
Figure 6 compares H-like and SCF contours for the carbon sp, sp2,and spahybrids.' These diagrams make clear that although the H-like orbitals underestimate the size of the best orbital approximation, they correctly approximate the shapes of the orbitals. This is because the SCF method does not change the angular nodes but only changes the radial part of the wave function. Therefore, the H-like orbitals and Figure 2 may always be correctly used in a qualitative presentation.
Figure 6. Contour liner for sorbon (a)H-like and (b) SCF hybrids, ot probability density of 0.01 eiectrons/subic Bohr radius. Solid line, rp; dmrhed fine, rpz; dotted linc, sp3 Scoie in Bohrradii. Rotation aboutthe horizonto1 oris generates the full three-dimenriond flgure. SCF contours are calculoted from doto by A. Jvcyr 112).
The differeuce in size between the SCF and H-like orbitals may be interpreted in terms of a screening effect, as follows? In the SCF method, equation (1) becomes a one-electron Schrijdinger equation where qt is the orbital for the ith electron and, instead of as in equation (2) or ( 5 ) , the potential energy may be expressed as
where S, represents inner shielding, the effect on net charge at r, of all the other electrons within r,, and So represents outer shielding, the effect of all the other electrons farther out than r,. The distinction between inner and outer shielding is this. Consider an electron a t a distance r from a nucleus which is shielded or screened by a uniformly charged negative sphere (SI) a; a distance less than r and another such sphere (So) at a distance greater than r. To calculate the potential of the electron (equation 13) we must consider both spheres, but the electrostatic force is independent of the outer sphere, for the potential within such a sphere depends on its charge, whereas the force is zero regardless of its charge. Therefore, we may write z, = z - S, - So and
z,
=
z - ST
where Z, is the net or effective nuclear charge which may be used to calculate potential energy (V = - Z,/r) and Z f is the effective nuclear charge for force (F = Z,/r2). Therefore, the shape of the orbital, which depends on the motion of the electron and so on the effective force, is determined by Z f whereas the energy is Contours artre drawn at the probability density of one electron per 100 robic Bohr radii. These contours generally show best the overlap of atomic orbitals to form bonds. T h e following is a much simplified treatment of the discussion by Shter (IS).
determined by 2., At r = 0, Z, = Z , = Z, and as T approaches intinity, Z, = Z, = 1, hut a t intermediate distances, especially in the region near the maxima and nodes which determine the shape of the orbital, Z. d i e m appreciably from Z,. Figure 7 shows Z, and Z, for the sodium 1s electron (14). We therefore conclude that an SCF orbital is larger than an H-like orbital because in the former we take account of the screening of the nucleus from each of the electrons by the others. This in effect reduces the nuclear charge so that the electrons are not so tightly held. We also find here a simple explanation for the observation that in atoms containing more than one electron the energy is not the same for all electrons of a given n but rather follows the pattern E, < E, < Ed < E , < . . . . This is because screening is less efficient a t the center of the atom (Fig. 7), so that an electron a t or near the center is subject to a greater Z, and is more stable than one far removed from the center. Now, an s electron has no nodes a t the center, a p has one, a d two, and so on (Fig. 2), so that the sequence s, p, d, f, . corresponds to shifting the orbital to regions of lower Z, and consequently lower stability. The spliti ting of an energy level into the sub-levels can thus he readily explained on the basis of nodes and distance variable screening.
..
Figure 7. Effective nuclear charge vr. distance for the sodium laelectron ( 1 4). Distance in Bohr radii ond Z in atomic units.
Analytical Functions
We now turn to the possibility of describing orbitals in terms of analytical expressions which may be more tractable than SCF orbitals. The simplest method of this sort is the screening constant device. What we do here is to replace the shielding effects described above, which are complex functions of distance, with a simple numerical constant, the screening constant, S: Z.JJ = Z
-S
(14)
We assume that the orbital concept is valid, that equation (6) holds, and that the orbitals are H-like functions (equation 3) with the actual nuclear charge Z replaced by an effective nuclear charge, ,Z ,,, defined by (14). The effect of such a screening constant on the geometry of an orbital is to change the size without changing the shape; reducing the effective nuclear charge makes the orbital everywhere larger. The best value of S may be found by the variation Volume 43, Number 4, April 1966
/
191
method, a mathematical procedure for finding the value of a parameter corresponding to the lowest energy of the system (within the limitation of the equation used). A screening constant orbital is clearly a significant improvement over an H-like orbital, and just as clearly it cannot be as good as an SCF orbital with its distancevariable screening. The advantage of these orbitals is that they give a t least roughly quantitative results while still retaining a high degree of mathematical simplicity. We have constructed some screening constant orbitals for carbon (Fig. 8) which show a good fit with the SCF orbitals, in contrast to the poor fit of the Hlike orbitals (Fig. 6). Screening constants were selected togive the best fit as judged by comparison of many computer-run trials. The best orbitals were obtained with Z,,! = 4.0 for the carbon 2s orbital and 3.1 for the 2p orbital. The 2s orbital thus shows less screening, as averaged out by the screening constant, than the 2s and this is presumably because its average distance from the nucleus is less, because there are no nodes a t the center. I n fact, it is a general result of the screening constant method that in any given energy level, the effective Z's are in the order Z, > Z, > Zd> . . . , because, as noted above, the s orbitd is closest to the nucleus and least screened, the p next, and so on. This order of effective Z's again results in splitting the energy level iuto the s, p, d, f, . . . sequence. Since screening constants are easily interpreted-each electron screens the others from the nucleus-such orbitals may be included readily in fairly elementary presentations.
Figure 8. SCF (dashed line) ond screening constant (full line) orbitals for carbon. lo1 rp hybrid. ( b l spa hybrid. Contours d r a w n o t probobilivy density of 0.01 electrons/cubic Bohr radius. Di3tonce units are Bohr radii.
A more precise technique has been used by P. M. Morse (15) and others to obtain analytical expressions for is, 2s, and 2p orbitals in the first row of the periodic table. They used H-like 1s and 2p orbitals, with Z,, and Zz, as effective nuclear charges determined by the variation method, and instead of the H-like 2s orbital, (2s) = ( Z 5 / 8 r ) " a ( e - " ~ / 2 - Z v / ? . e - Z ? / 2 )
(15)
they used
with two different effective nuclear charges, Z2$ and 225, and with constants A and B inserted to maintain normalization and orthogonality. (However, their notation is different.) Thus, the three orbitals were defined in terms of a set of four effectivenuclear charges. By use of a computer, A. Tuhis (16) was able to obtain values for these which gave excellent predictions of the energy for several first-row atoms and ions; the results were 192
/
Journal o f Chemical Education
almost as good as those of the SCF method. For carbon Tubis found the effective nuclear charges, 21,= 5.69, ZZ, = 3.12, Zzl = 3.31 and ZA = 10.24. That these values are considerably different from the values we found (Fig. 8) of Z Z=~ 3.1 and Za = 4.0 is due to the difference between our concern here with showing the shape of the orbital, in particular matching the outer parts of the wave function with the SCF values, and Tubis' concern with finding the best energy of the system. The difference between Z,and Z, is also relevant. This emphasizes that the screening constant is an essentially mathematical device which may be given exact physical interpretation only with caution.
Figure
9. The 2s orbitel as a
rum of t w o parts.
One mathematical interpretation of (16) is very important and suggestive. It may be considered as the sum of two curves (Fig. 9) wbere the inner behaves according to the Z;,exponent and the outer according to the Z, exponent. The inner curve especially determines the hehavior of the electron a t low r and the outer curve is especially important at high r, so that the inner portion should show less shielding (higher effective Z) than the outer. Now, then, it is conceivable that one may derive an even more exact representation by expandiug not only (15) but also any other orbital iuto the form, where the b's determine the effective nuclear charges for different mathematical portions of the orbital, the x's select the mathematical f o r m (as in the inner, x = 0, and the outer, x = 1, for equation (16), and the a's are coefficients to be selected, subject to normahation and orthogonality. Presumably, an infinite series in the form of (17) could give perfectly accurate results, but since the orbital concept itself is only of limited validity, a relatively small number of t e r m for each orbital may give results as good as possible (17). R. E. Watson (18) has used four to 10 terms for each orbital to obtain results for the iron series atoms and ions which apparently are just as good as those of the Hartree-Fock method. Watson's method consists of selecting a set of x's and b's and solving for the a's by the variation method. The b's are then varied to see if the new a's give better results (lower energy) than the first set, and the process is continued until no further improvement can be made. The orbitals are then self-consistent, so that this method may be
called the analytical SCF method. Although it is not an almost automatic algorithm, as is the HartreeFock method, and in fact is said to require a great deal of wisdom in selecting the starting functions, it has the advantage of yielding its results in analytical form rather than rn tabulated data. A description of orbitals in terms of functions like (16) or (17) is a t least very nearly accurate, and it certainly is better than one based on simple screening. Yet, the diierence qualitatively is small, so that there appears to be little reason for using such descriptions in an introductory picture. H-like orbitals, with simple screeninp: a very satisfactory - if desired,. uaint . . qualitative picture. One final variation must be mentioned. One of the earliest and most influential approaches to the orhital was the suggestion by Slater (19) that if the radial nodes are replaced (mathematically) by a point node a t the center, then the outer parts of an orbital can be described by a radial function of the form, dn-1)
e-Zeiir/n
(18)
where n is the principal quantum number. Slater also gave a set of rules, now widely quoted and used, for finding the Z,,'s for these functions by counting the numbers of electrons in each sub-level. Functions in the form of (18) are called Slater orbitals and are much used. One application is in calculations of overlap integrals, which depend primarily on the outer parts of the orbitals; such calculations are particularly accessible in terms of Slater orbitals and, for the light atoms, are good first approximations. A significant improvement is possible if combinations of Slater orbitals, in a form like that of (17), are made, with the a's chosen so that the combinations form an orthogonal, normalized set. These orthogonalized Slater orhitals are not so good as the Hartree-Fock or analytical SCF functions but they are better than H-like orbitals with screening constants. However, since they are rather complicated, there does not seem to be any advantage to using them in a qualitative introductory picture. Furthermore, it must be strongly emphasized that the simple Slater orbitals in the form of (18) are not, and were not intended to be, qualitative representations of the orbitals. They are a highly useful mathematical device for making certain calculations, and they gain this usefulness a t the expense of a correct description of the radial portions of the wave functions. Summary
An orbital is a one-electron wave function. It is not a perfectly valid concept for atoms or molecules with more than one electron, since it assumes that each electron is independent of each other electron. However, if the averaged-out effect of electronic interactions is taken into consideration, we obtain a selfconsistent field orbital which constitutes a reasonably close approximation to the actual system and which is the most accurate picture possible in terms of the orbital concept. H-like orbitals are valid as qualita-
tive representations, and H-like orbitals with simple screening constants are also roughly quantitative. Functions due to Morse and to Watson (among others) are much better, but are much more complicated. Slater orbitals, on the other hand, should not be used as qualitative pictures. For any correct system of orbital representation, the angular nodes of Figure 2 areuseful; they not only fit all types of orbitals but they also enable a satisfactory introduction to the shapes of the orbitals, the energy levels, and the s-p-d-f energy splitting. Acknowledgment
Janet Del Bene calculated the d?sp3 orhital and prepared all the drawings. Literature Cited (1) GRAY,H. B., ''Electro~~~ and Chemical Bonding," W. A. Benjamin, Inc., New York, 1964; SERER*,D. K., "Elec-
tronic Structure and Chemical Bonding," Blaisdell Pub. R. M., "Behavior Ca., New York, 1964; HOCHSTRASSER, of Electrons in Atoms," W. A. Benjamin, Inc., New York, 1964. inter a l k . (2) GREENWOOD, N. N., "Principles of Atomic Orbitals," Royal Institute of Chemistry, London, 1964. (3) OGRYZW, E. A., AND PORTER,G. B., J. CHEM.EDUC.,40. 256 (1963). (4) COHEN,I., J. CHEM.EDUC.,38, 20 (1961). (5) FRIEDMAN, H. G., JR., CHOPPIN, G. R., AND FEUERRAGHER, D. G., J. CUEM.EDUC.,41, 354 (1964); BECKER,C., J. CHEM.EDUC.,41,358 (1964); PEARSON, R. G., Chem. and Eno. News. 37. No. 26. 72 (19591. also reminted in MANCH. \----,-
(6) OGRYZLO, E. A,, J. CHEM.EDUC.,42, 150 (1965); FRIEDMAN,H. G., el al., J. CHEM.EDUC.,42,151 (1965); COHEN. I., J. CHEM.EDUC.,42, 397 (1965). (7) HYLLERAAS, E. A,, Z. Physik, 54, 347 (1939), in SLATER, J. C., "Quantum Theory of Atomic Structure," McGrawHill Book Co., Inc., New York, 1960, vol. 2, p. 36. (8) KINOSHITA, T., Phys. Rev., 105, 1490 (1957); P E ~ R I ~ , C. L., Phys. Reu., 112, 1649 (1958); see S L A ~ RIOC. , cit., for additional references. (9) SINANOGLU, O., Proe. Natl. Acad. Sei., 47,1217 (1961); also reprinted in PUR, R. G., "The Quantum Theory of Molecular Electronic Structure," W. A. Benjamin, Ine., New York, 1963, p. 492. (10) HARTREE,D. R., Proe. Cambridge Phil. Soe., 24, 89, 111 (1927), in HARTREE, D. R.,"The Calculation of Atomic Structures," John Wiley h Sons, Inc., New York, 1957, p. 18.
(11) FOCK,V., Z. Physik, 61, 126 (1930), 62, 795 (1930), in HARTREE, D. R., op. tit., p. 55; also see SLATER,J. C., "Quantum Theory of Atomic Structure," vol.2, pp. 1-30. (12) J u c ~ s A,, , PTOC. Roy. SOC. (London) A173, 59 (1939). (13) SLATER, J. C., "QuantumTheory of Atomic Structure," vol. 1, pp. 227-9; also see HARTREE, D. R., "Calculation of Atomic Structures." , DD. 77-9. FOCK,V., AND PETRASHEN, M., P h y ~ i k2. . Sowjetunzon, 6, 368 (1934), in SLATER, J. C., o p . eit., "01. 1, pp. 203, 228.
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MORSE,P. M., YOUNC,L. A,, AND HAURWITZ, E. S., P h y ~ . Rev., 48, 948 (1935). TUBIS,A,, Phys. Reo., 102, 1049 (1956). L ~ W D I P.-O., N, Phys. Rev., 90, 120 (1952). WATSON,R. E., Phys. Re"., 119, 1934 (1960); also see RICHARDSON. J. W.. NIEUWPOORT. W. C.. POWELL. R. H.. AND EDGELL; W. F:, J. Chem. P&., 36, i057 (196i). (19) SLATER, J. C., Phys. Reu., 36, 57 (1930).
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