1966_Berry_J Chem Educ_V - Atomic Orbitals

Resource Papers-V Prep.& under the aponrorrhip of

The Advisory Council on College Chemistry

Today the teaching of chemistry probably leans more heavily on the theory of atomic structure than on any other single pillar. We use the concepts of shell structure to develop the periodic table and all the characteristic physical and chemical properties associated with chemical periodicity. The Bohr model of atomic shells gives us a universal tool to estimate and exhibit the magnitudes characteristic of virtually all atomic and molecular properties. Orbitals are a t the base of our interpretations of molecular structure, and we almost always express these orbitals in terms of, or a t least in relation to, the orbitals of the constituent atoms. We even can begin to use orbital concepts to interpret many reaction mechanisms. The subject is, in fact, so integrated into our whole approach to chemistry that we are astonished when a freshman comes to us from a high school chemistry course that did not interpret chemistry in terms of orbitals.

R. Stephen Berry1 The University of Chicago

Chicago, Illinois

I. Atomic Orders of Magnitude and the Bohr Alom

Atomic Orbitals

The Bohr-Sommerfeld theory of atomic planetary orbits was the first quantitative statement of atomic shell structure, and is still the source of much of our

intuition about atoms. Even more important, this model was the statement of the postulate of stationary states, a statement that simply defied the laws of classical physics: an electron in a "Bohr atom" remains in orbit forever, and does not spiral in toward the nucleus. Classically, the angular acceleration of such a bound electron would force it to radiate its energy away slowly and eventually fall into its attracting nucleus. With the postulate of stationary states and the postulate of the quantization of action, the theory of circular Bohr orbits can be developed in afew lines of algebra see references (21 -34; also see Appendix). From it, we develop a powerful little table (see Table 1) of numerical expressions for the radius, velocity, energy, and period of the circular orbit characterized by two numbers: 2, the number (or effective number) of positive unit charges attracting the electron of interest, and n, the quantum number (or effective quantum number) characterizing that particular stable orbit. (We shall say more about

Table 1. Expressions for Characteristic Properties of Circular Bohr Orbits

Quantity Expression Value

Zs 109,687 cm-I Energy, E 2h' n' or 13.58 ev

or 2.178 X 10-LLerg tLZ nz n2

Radius, r - e2m X 0.529 X X z em

ez Z Z Velocity, V t- X 2 188 X lo8 X ; cm/sec

2rha n8 n3 Period X Za l * S X X Z, see

tL = Planck's constant h divided by 2r , e = charge on the e lec tron, m = mass of the electron, and Ze = nuclear charge.

effective charges and effective quantum numbers further along.) The Bohr-Sommerfeld model is admit- tedly inconsistent with classical mechanics and it gives some results that do not agree well with experiment, or that simulv are w r o n ~ . ~ Nevertheless its

Alfred P. Sloan Fellow. ' One example is the nonzero angular momentum of the ground

state of hydrogen, and the properties such as magnetic moments associated with angular momentum. The theory also prohibits the electron from entering the nucleus; electrons actually can penetrate nuclei. The ability of electrons to approach and penetrate nuclei to varying degrees is the reason that proton magnetic resonance lines occur a t s. varietu of energies in a given field. Without this property, nuclear magnetic resonance would not he anything like the powerful analytical tool that i t actually is.

Volume 43, Number 6, June 7 966 / 283

utility as a tool for estimating orders of magnitude is universally recognized, and it is surely the source of much of our visceral intuition about atomic ~t ructure .~ With it, we can say, for example, that phenomena oc- curring in times longer than about 10-l5 sec can best be described in terms of the time-averaged distribution of an electron and not by a moving point-charge, simply because the classical electron would go through many orbits during the event. Phenomena requiring less than lo-" see, say, are not well described by average distributions of charge in atoms. By the same token phenomena involving energies much greater than 1&20 ev are necessarily quite disruptive to the outer shells of atoms, but energies a hundredfold smaller do not affect these shells very much.

Naturally, the classical estimates we have just made have their parallels in quantum mechanics, in terms of wave structure and the uncertainty principle. We shall examine these; hut first let us spend a moment examin- ing the representation of waves and the basis of the wave theory of matter.

The historical background of wave-particle duality- the particle-like properties of light exhibited (for example) in the photoelectric effect, their parallel in the wave character of particles as suggested by deBroglie and demonstrated by diffraction of electrons, and finally the flowering of quantitative quantum m e c h a n i c ~ i s a familiar and rather romantic subject. The entire period of its invention and development was so short that many, perhaps most, of its major figures con- tributed to its very origins and were still aliveand active when it had become a mature and universally accepted cornerstone of physical science. Moreover the logic of its growth and the relationships between different theories and between experiment and theory is exceedingly clear. And fortunately it is very well documented; consequently we shall make no attempt to discuss this background here. A few of the author's own favorite references are given in the bibliography.

II. Matter Waves

Let us examine some of the relationships associated with matter waves and with waves in general. For, after all, atomic orbitals are nothing more than the forms msumed by the standing waves characteristic of single electrons hound in an atomic potential field. We can discuss them i~?&igently and correctly only if we are ready to recognize and use their wave-like properties.

To begin, let us distinguish standing waves from running waves. The latter are worth a little of our attention here for two reasons. First, running waves are less cumbersome in a development of the relationship between the quantum conditions and the wave equation. Second, the transition between running and standing waves gives us a natural and mathematically simple example of the principle of superposition, in its most precise way-in terms of the interaction of two readily visualized waves.

8 It is worth noting that occasionally, respectable attempts are made to find classical models that fit better with quantum mechanics than the Bohr atom. One in fact appeared in pre- liminary form quite recently [GRYZINSKI, M., Phy8 Rev. Letters, 14,1059 (1965)l.

Traveling waves are appropriate for describing free particles; running waves, for describing bound particles. Any mathematical function of two or more variables-the time, t, and the spatial coordinate, x, for example--describes a running wave if the independent variable or variables can he written in one particular form. If a function f(x,t) can be written as f(z) where z = kx - wt-that is, if x and t are always related so that the real independent variation off always can be given in terms of such a a-then the function f is a traveling wave. It need not be a periodic function like siu[A(kx - at)], hut it may be. The crucial property is this: a given point on a plot of f(z), say f(ao), cor- responds to an infinite number of pairs of values of x and t , corresponding to the solutions of kx - at = a. with k and w fixed by f. Since t moves inexorably forward, the value of x that keeps the quantity kx - ot, called the phase, equal to zo must also move inexorably forward, a t just the velocity w/lc, the phase velocity. Note that we have not introduced any explicit concept of wavelength or frequency because we have not yet discussed periodic waves.

Periodic waves, and sinusoidal waves in particular, play a unique role in wave mechanics. Their mathematical simplicity, and the fact that combinations of sine and cosine waves can be used to represent any smooth function to an arbitrary degree of accuracy, make sinusoidal waves attractive. However, the property that makes them so important for atomic physics and chemistry is the fact that they give the exact representation of the simplest stationary states that occur in quantum mechanics, the states of a free particle. Let us see how these waves are a consequence of the Einstein relation between energy E and frequency v in cycles/sec [ (2~) - ' X o radians/sec],

E = hv, (1)

The deBroglie relation pA = h (2)

between momentum p and wavelength A, and the classical expression for the total energy of a free particle, is simply the kinetic energy:

E = pn/2m (3)

From relations (I), (Z), and (3) we obtain the connection between the frequency and wavelength of matter waves. tbthdispersion relation

often written in terms of 1c = 2s/A instead of A, so that

where 15 = h/2a. This is quite a diierent thing from the relation for light waves

Free matter waves have a frequency that varies inversely as the square of the wavelength, not as the first power--or alternatively, the phase velocity of matter waves in free space varies inversely with wavelength. If matter waves can be described by a wave equation, then we can infer that equation from their dispersion relation, eqns. (4a) and (4b). A wave equation descrih- ing a wave function fi is a relationship between the

284 / Journd of Chemical Education

time and space derivatives of J.. What is that relationship?

In the function J.(x,t), the quantity u is a frequency, and must be associated with the time t to give a diien- sionless variable of the form vt. Similarly X must be associated with x to give a dimensionless variable x/X, or lcx. Since v appears to the first power, the wave equation must involve only the first derivative of J. with respect to time. The wavelength X appears as C2, SO that it must he the second derivative of J.(x,t) with respect to x which is related to the first time derivative:

More specifically, using eqn. (4a), we have

What about the numerical constant of proportionality? If it were m1, then J. would be a real exponential of the form e x p [ i (2svt + kx)]. This is a formal solution to the equation but is not quite acceptable in polite physical company because of its annoying property of becoming infinite when the exponent becomes large. A physically allowable free-state solution is one whose amplitude is at least bounded everywhere; this is readily achieved if our constant is i = 47. We have

which has the solutions + ( z , t ) = A e W - 2 4 (6)

(We choose +i and not -i so that positive k corre- sponds to a wave moving to + m as t increases.) The form (6) for J. allows us to identify three differentiating operations with the evaluation of physical quantities:

so that the time rate of change of J. for any x is a constant proportional to the energy of the state: next, the slope of the function J. is a constant proportional to the momentum:

giving us the momentum p, and finally, we recognize that the identity

is just equivalent to the dispersion relation, and amounts to writing

E = p2/2m

I n the general case, we identify ikb/bt with the total E-the sum of T, the kinetic, and V, the potential, contributions.

Finally we conclude this little exposition of running waves by writing the running wave, eqn. (6), in its equivalent forms

+(z , t ) = A [cos ( k z - 2rut ) + i sin ( k z - 2rv l ) l (9a)

It is the last of these which is most important for our understanding of standing waves and atomic orbitals.

Standing waves are waves that oscillate in time but whose crests and troughs remain fixed in space. For example $(x,t) = f(x) sin(2s vt) is such a function; at any point xo, $(x,t) oscillates between f(xa) and -f(xo). The function (9) is not a standing wave, but a superposition of them. We can rewrite (9) as the sum of

+ ( z , t ) = A [cos Zsvt cos k z + i eos 2 n d sin kz + sin Zrvt sin k z - i sin 2sut cos k z ]

four separate standing waves having just the right phases to give one real and one imaginary running wave moving together, as (9a) displays them.

The mathematics that make a superposition of standing sine and cosine waves into a traveling wave are quite clearly exhibited in eqns. (9) and (10). At this point one could discuss the more philosophical aspects of superposition. One might, for example, ask about the probability of finding the electron in a sin kx distribution, if we know that the wave function is J.(x,t) of eqns. (9) and (10). We shall not pursue this point here [cf. Reference (5) 1.

The point we must make now is this: the proportionality of v and E, and therefore the proportionality of the first time derivative and E, require that all stationary (constant E ) solutions of the quantum mechanical wave equation for matter waves have a factor e ~ l r i u ~ . I n other words, the functions representing all the stationary states of an electron (or of a complex system) contain a complex oscillating factor containing the time. If the system is free, then the spatial part may be complex also, and the function J. can be a running wave.

The foregoing aside on running and standing waves has served two functions. It has developed in a sort of painless way a primitive example of the mathematical statement of the superposition principle. This principle is the very basic quantum mechanical concept that there are always alternative and equivalent descriptions of an electron wave. No one description necessarily tells us explicitly all the properties of the electron that we might want to know, or makes apparent all the useful ways of interpreting the physical properties of a wave function.

The concept of superposition will become a very important one in our discussion of alternative representations of orbitals; of hybridization; and of valence bond, molecular orbital, and mixed representations. The basic concept to be grasped now is the existence of equivalent descriptions, any one of which can be obtained from any other by a re-expression or transformation no more subtle or complicated in principle than the transformation that gives us eqn. (9) as an alternative to eqn. (6).

The second main reason we have dwelt on the concept of a wave is to develop the time dependence of an electron wave in its simplest example. Now, as we proceed into a discussion of bound states and of atomic orbitals, we will be using a form and a physical picture that will let us drop the explicit time dependence of our wave function. Nevertheless, all along, it is important to remember that every wave has a time dependence, that electron waves describing states of constant energy have factors and therefore have real and imaginary

Volume 43, Number 6, June 1966 / 285

parts whose amplitudes oscillate sinusoidally in time about mean values of zero.

We conclude this general discussion by amplifying briefly the physical and mathematical notions associated with the concept of a stationary state, and with the idea that a wave function should have a factor e""'. Let us generalize eqns. (5) and (7) a bit for our later use by supposing that E is not necessarily p2/2m hut may also contain some potential energy V(x) that may or may not vary with distance hut does not vary with time. Then, instead of eqn. ( 5 ) , the general statement of (7) is

or, by way of defining the Hamiltonian X,

If the Hamiltonian X does not contain time explicitly, and ours clearly does not, then X acts only on the space variable of $(x,t) and not on t. But the partial time derivative acts only on the time part. These two conditions can be satisfied only if itia$/at and XJ . are one and the same constant multiple of J. itself. The multiplicative factor is obviously just E, from our previous discussion. But this implies that the energy E is constant in time, i.e., that the energy is stationary, or that the system is in a statiaar?~ state. "Stationary" in this usage does not imply that the wave function is constant, but only that the energy is constant and the wave function is a periodic traveling or standing wave. The second implication follows from the foregoing because the form of eqn. (11) implies that J.(x,t) can he written as a product +(x)t(t), and that

and

The function t(t) is simply e-""'; +(x) is just a function of x independent of time, so that J.(x,t) is necessarily a wave with period Elti.

Where does the idea of discrete quantized states enter our physics? So far, all we have done applies to con- tinuous distributions of states. The discrete quantiz* tion is, in essence, a result only of the introduction of finite boundary conditions. So long as we make no restriction on how the wave function behaves as it goes off to * m, there is no quantization. (We require only that the free functions remain bounded by some upper and lower limit.) However as soon as we introduce "finity" boundary conditions-like saying that +(x) must vanish a t the walls of a box located a t *a, or that +(x) must correspond to a function on a ring and +(x) = +(x+211), or that +(x) must go to zero exponentially as x approaches hoth * m -any such conditions immediately remove the possibility of a continuum of E values and of a corresponding continuum of states. [A detailed discussion of this is given in Ref. (,$).I In essence, the imposition of houndary conditions a t both ends of the range, plus the conditions that the hound- state wave have no kinks or discontinuities and he quadratically integrable, eliminates all possible functions except those having an integral number of half-

waves within some region (not arbitrary) appropriate to each individual problem. All other formal solutions would in some way fail to satisfy the various conditions.

Ill. Discrete Stationary States for Single Particles: One-Electron Systems

Orbitols in One Electron Atoms

Under what circumstances do we find discrete quantized states for a single particle? These circumstances are the kind that lead to the quantized states of the particle in a square box, the harmonic oscillator, the rigid rotor, and the one-electron atom or molecule. The circumstances require that the potential energy V have a dip or well of some sort, so that V(m), its value at infinity, is higher than its value somewhere in the well. If there are any states of the particle whose energy E is less than V(m), then these states must be discretely quantized. For example, the simple harmonic oscillator with V = kx2/2 has infinite V(m), so that all its states are quantized. The hydrogen atom has V(m) = 0, conventionally, so that any state of negative energy, E < 0, must be discretely quantized.

Graphically, the discrete quantization is a generalization of the discrete quantization of the oscillatory states of a rope with hoth ends fastened, or of a particle in a box. For the rope or particle-in-box, a continuum of oscillations is possible if only one end is held or if one end of the box is open. However, if both ends are fastened or closed, respectively, the only oscillations are those that give constant displacements of zero at hoth ends (see Fig. 1). If the rope's length is L, these have the spatial form A sin (nrrx/L) (or some combination of these) where n is any positive integer. The generalization

nodal point

(b )

Figure 1. I.) Rope with a free end; the dirplocement of the end of the rope con hove any volue. The oniy conditions are that ot one end (x = 01, the dirplasement y(O1 = 0, and the1 the other end geh no further from x = 0, y = 0 than 1, the lengthof the rope-i.e., ylendl 5 1.

(bl Rope with both ends swashed, the conditions y101 = 0 and yIL1 = 0 allow the rope oniy a discrete (but infinitel number of vibrational stoles, nomely thwe with 0, 1.2. ... nodes between the ends.

comes when we replace the two rigid fastenings with exponential decreases to zero a t both ends, + m , for all three Cartesian coordinates-or tie the two ends of the rope together.

It is worth noting that a central potential may be attractive and still not be able to s u ~ n o r t bound states. In the region where E < V, the wave function is curving away from the axis, so that i t must be leaving the axis as it enters the potential well. In the region where E > V, and the wave function is curving back toward zero amplitude, its curvature is always proportional to the depth of the well below its energy value. If the well is both shallow and narrow, the wave may be unable to bend back to re-enter the forbidden region with its slope inclining toward the axis, the slope required to make the function die exponentially as it penetrates the region where E < V . If no wave can re-enter properly, then no wave can correspond to a bound state. Figure 2 illustrates this behavior; we can picture it in terms of trying to fasten a very stiff rope to two hooks in a tight space.

Bound States in a Potential Figure 2. Three types of behavior; (a) bound, quantized rtote with curvature just suitable far matching both decaying curves with the &wroidd curve; ib) and (4 phygically impossible situations, corresponding to no true states; the wove function with E < V is only o dying exponential on one ride, and grows exponentidly on the other. Arrows mark points where E = Vand cvrvotvrechonger sign.

The exponential decrease is a very simple consequence of the mathematics whenever the total energy E is less than the potential V. We refer again to Ref. ( 2 ) , pp. 51-58, for a particularly clear exposition of this topic. It is worth noting one point now that will become especially important in the last section of this paper, dealing with electron correlation. Whenever E < V, the kinetic energy is necessarily negative, and the momentum, being p2/2m, becomes imaginary. This seems strange and formal, but it is just as strange, formal, and above all nonclassical to have E < Vat all. The existence of the wave function in nonclassical regions is one of the most important physical differences between classical and quantum mechanic^.^

With our conclusions from the previous section and the paragraphs just preceding, we have a basis for a

clear but still qualitative picture of a stationary state of a singleelectron atom. The electron is described by a complex wave in three dimensions; the wave's real and imaginary parts oscillate sinusoidally in time, 90' out of phase, with a frequency v = E/h. The energy E of the electron is negative, and can only assume certain discrete values, and the wave itself goes to zero exponentially as r, the electron's position vector, goes to infinity. Such a function is the simplest example of an atomic orbital.

Quantum Numbers and Constants of the Motion

So far we have characterized an atomic orbital by one number only, the energy. We all know perfectly well that there are other characteristic numbers or quantum numbers associated with orbitals. Let us inquire into their origin.

That E is a good quantum number is a consequence of the conservation of energy. This is a rather trivial statement when it is put this way, but we can say it slightly differently: if X, the Hamiltonian of a system, does not change explicitly with time, then the energy of that system will be a constant or a good quantum number. That means that if the potential and the parameters such as mass, and the universal constants do not depend on time, so that X a t t , is identical with X a t ts then E will be a good quantum number.

The other familiar quantum numbers of atomic orbitals represent other physical constants of the motion associated with other invariances of a Hamilton- ian, i.e., of the physical description of a system. The quantum number 1 of a particle is a good quantum number when the total orbital angular momentum of the particle is a constant. This comes about if, and only if, the Hamiltonian of the system is spherically symmetric. Since the kinetic energy depends only on momentum and makes no reference to any spatial coordinate, i t is as symmetrical in space as it can be; we need only examine the potential energy for its symmetry. If it depends only on r, the distance of the particle from the origin, and not on any angle, then naturally the potential energy is spherically symmetric.

Suppose we are studying an electron bound by a potential V. The electron has some instantaneous angular momentum. We now seize the apparatus that produces V, and rotate the apparatus to a new angular orientation without translating it a t all. If the potential were not spherically symmetric, such a motion would clearly disturb the electron, in general, but if V were spherically symmetric, rotating the apparatus would leave the electron entirely unaffected. Specifically, if V is not spherically symmetric, the rotation would in general introduce a torque on the electron and change its orbital angular momentum. If V is spherically symmetric, then the electron's orbital angular momentum is unchanged by the operation applied to V. In other words the orbital angular momentum is a constant of the motion.

For the sphere, or for a spherically symmetric system,

Q mmacroseopie example of the penetration of a. quantum mechanical wave through a classiedly forbidden barrier zone i3 the Josephson effert. In this phenomenon current emtrrien in semi- and super-conductors are able to penetrate layers of insnla- tor between two semiconditcting or mpereondneting bodies.


re-orientation into any angle is a symmetry operation: re-orientation leaves the system in a condition in- distinguishable from its initial condition. Symmetry operations never come singly (except those for the most unsymmetrical things we can think of, things that can only be left alone) and do not come in arbitrary combinations; in general one operation followed by another is equivalent to a third operation. The set of all operations associated with a particular symmetry type is the grmp of operations for that symmetry. The set of all rotations constitutes the simplest full symmetry group for a spherical system. The rotations about the figure axis constitute a symmetry group for a cylinder or a helix.

Just as the total orbital angular momentum is a constant if V is spherically symmetrical, the angular momentum component about the figure axis of a cylindrical system is a good quantum number. The in- variance of the cylindrical system with respect to any rotation about its axis implies the existence of a constant of its motion, the corresponding component of orbital angular momentum.

A special case of a system with cylindrical symmetry is the sphere--and, indeed, a spherical system has a constant orbital angular momentum component along an arbitrarily chosen axis, as well as a constant total orbital angular momentum. This quantum number is m,, of course. (We shall sometimes use m for m,.)

Why can we not h d the angular momentum com- ponents along three axes of a sphere, instead of just one? At the risk of being too brief, we can answer this just by saying that such knowledge would localize the particle's orientation too much, to the point of violating the uncertainty principle.

In general, in a spherically symmetric potential, the energy of a hound electron depends on its angular momentum-i.e., on 1, as well as on the principal quantum number n, but never on m. (The exception is the Coulomb potential, for which all states of the same n are of equal energy, regardless of I.) There are 21 + 1 values form, for any 1, so that there are 21 + 1 different degenerate orbitals having the same n but different m

"-1

[n2 or C (21 + I), for the Coulomb or hydrogen-like 1-0

case]. These 21 + 1 different states transform into mixtures of each other if we redefine the orientation of the coordinate axes of our spherical system. But no matter how we re-orient the coordinates, a given set of 21 + 1 wave functions transform only into each other, and never into any other functions. This is exactly analogous to the way sin no and cos n8 can be transformed by addition and subtraction into en' and e-"O or mixtures of these, but never into anything containing e-"n+l)e. The set of 21 + 1 functions is called a basis for a representation of the rotation group.

I n fields of lower symmetry than spherical, the orbital angular momentum is no longer quantized. However, the symmetry invariances may remain in part. Sometimes we can suppose that 1 is a good, constant quantum number but m is not (Russell-Saunders coupling: there, the same thing happens to spin; S is a constant but M, is spoiled as a quantum number). In other cases, even 1 is not constant. It is very often useful to start with a set of free atom functions, either orbitals or many-electron state functions, with all the

degeneracy appropriate to a spherical potential; then one asks what happens to this particular set of orbitals if the symmetry of the potential is lowered to something less than spherical, like octahedral or simple threefold for example. One uses the symmetry properties of the wave functions to determine their behavior in a t best a semiquantitative way. This can be an exceedingly powerful way of elucidating chemical problems. The most famous application of this sort is of course crystal and ligand field theory in its phenomenological form. We shall return to this topic for some examples, but we shall not try to develop the entire theory of group representations in atomic physics and chemistry. Several references a t various levels are given in the bibliography. The pertinent conclusion for us now is this: if the symmetry of the potential V is lowered from spherical to some new form, then some but not all the degeneracies of the spherical case may remain. The rather straightforward algebra of group representations lets us determine very easily exactly how any given basis set of 21 + 1 degenerate functions from the spherical case will split into smaller sets in the new and lower symmetry. The answers are expressed in terms depend- ing only on 1 and on a smaller number (sometimes only one) of angle-independent quantities that can be treated as empirical parameters or can be evaluated from atomic wave functions. We shall return briefly to this subject later.

The Forms of One-Electron W a v e Functions

The mathematics of a spherically symmetric one- electron problem lead directly to a set of standing waves that we can understand and describe very easily, a t least in part. Putting together the pieces of the foregoing discussion, we can tell just what to expect. In general, the characteristic stationary waves must describe constant-energy states whose total orbital angular momentum is characterized by 1, and these waves must come in degenerate sets of 21 + 1. If we make use of all the spatial constants of motion, then these functions correspond to the states of definite mi. We are free to describe a system of energy E in terms of any combination of m, eigenfunctions. As we have indicated previously, it is sometimes useful to suppose that the appropriate wave function is not an eigenfunction of any one component of angular momentum, but has some other property, like directionality.

The solution of the spherically symmetric Schro- dinger equation appears almost as soon as we convert the Hamiltonian into a form that can be written as a sum. In the sum, one set of terms contaiosall the radial dependence including the entire potential, and the other, all the angular part, which is only kinetic. (We let 3, = rZT, and 3 0 , ~ = r2 X l(1 + 1)RZ/2mr2, or rZ X Tn,* rotational energy.)

r'X = [3, + +V(T)] f %.p (14) r-dependent angledependent

only only

We say X is separable when it can he so expressed, as a sum of independent terms. Using the same reasoning that gave us equations (12) and (13) from (l l) , we obtain a product form for fi(r, 8, q) :

288 / Journal of Chemicol Education

and one equation for each of the three factors R(r), O(8) and %(q). The function R(r) must depend on the specific problem. It is known, naturally, for the Coulomb potential V(r) = -e2/r, and for the general Ar-" potential as well. I n the next section we shall see how such a problem arises and is treated in many- electron atoms. The Coulomb solutions are given by Laguerre functions; some of the lower ones are shown in elementary texts.

Basically, all the radial solutions R&) for bound states have certain properties in common. They all go exponentially to zero as r goes to infinity; the function of lowest energy is nodeless and each successive higher function with the same 1 has one more node than the one before it, and the functions are orthogonal and can always be normalized in the sense that

6 R,:*(r) R,,i(r) r2d~ =

S,.,' (i.e., 0 if n f n', and 1 if n = n')

These conditions have the usual consequence that each function reaches its maximum amplitude in its outermost lobe, and that each successively higher function has its outer lobe a t larger r than the one before. This is all clear and obvious in the case of the hydrogenic functions, but is is worth noting that these character- istics are quite general. I n the next section we show some radial functions for atoms.

The angular functions are the very well known spherical harmonics, the forms taken by standing waves in any spherical problem. For example the standing waves on a flooded planet exhibit exactly the same angular behavior as do atomic orbitals hut occur only on a single spherical surface, so they are perhaps easier to visualize than the wave function of an atomic electron. This example is developed in Ref. (3).

Table 2. Some Lower Spherical Harmonics Y d 8 . d

s-function: 1 = 0

pfunctions: 1 = 1

Y P = & eos 8

d-functions: 1 = 2

~ ~ - 1 = 415 sin 0 cos 0 ec's = 8s

4z sin 20 e'v -

yZs = $3 ,in2 6 = !+5 (1 - eels 28) e2tq 3% 2 32%

Table 2 represents some of the analytical expressions for the spherical harmonics Ytm(8, q) = O(O)%(q). Certain general properties are worth explicit mention.

First, we see that each Yim(8,u) is a polynon~ial of degree 1 in sin 8 and cos 8. These can be expressed, alter- nately, as trigonometric functions of multiples of 8. Either form lets us visualize the 8-dependence explicitly. We could combine functions like YI1 and Y1-I to get real (i.e., not complex) comhinationsvarying as sin 8 cos a (i.e., as x) and as sin 0 sin q (i.e., as y).

Another point to recognize about the spherical harmonics is a symmetry property with respect to the origin. The functions with even 1 all contain only even powers of sin 8 and cos 8, and the odd 1 functions, only odd power. If we change s to -x, y to - y, and z to - z (i.e., invert the coordinate system through the origin), then Y's of even 1 go into themselves (even or gerade functions) while Y's of odd 1 go into their nega- tives (odd or ungerade functions).

Third, as the number of angular nodes increases with I, so each lobe becomes more directional or pointed. Very high 1 functions, with large numbers of angular nodes, describe electrons that are nearly classical in their orbital rotations. For, according to the Corre- spondence Principle, when the wave length of an electron wave is small compared with the dimensions of the volume in which the electron moves, the particle ceases to show its wave character and behaves like a particle.

By contrast, the radial parts of the one-electron bound state wave functions retain quantum character even for high quantum numbers because the outermost lobes are always both the largest and the most spread out. The inner parts of highly excited states, the parts at low r values, do have rapid oscillations. Consequently, in states of high n, electrons behave classically in their radial coordinate when they come near the nucleus but as waves when they move to large r and their local momentum is small.

IV. Atomic Orbitals in Many-Electron Atoms

The Many-Elecfron Problem

Up to this stage, we have examined the wave functions for a single electron moving in a potential field, especially a spherical potential. But most atoms do not consist of a single electron moving in a potential. Let us now explore the way one can develop well-defined wave functions for single electrons in many-electron systems. Then we can look at some of the properties of these orhitals and see how they are used in some representa- tive chemical problems. In the next and final section we can look at the limitations of the orbital method, and see how its limitations affect our interpretations of physical problems and how we can try to overcome these limitations.

Suppose we consider first the two-electron case, the helium atom. The lowest state of He must have both electrons in 1s orbitals, we say. But the Hamiltonian of the system contains potential terms -ez/r,, -eZ/rz, and e2/rlz--that is, not only is the nuclear attraction for each electron part of the potential; the electron-electron repulsion is part also. Obviously a t any instant neither electron moves in a spherically-symmetric potential. Then how can we possibly refer to a 1s orbital, much less assign both electrons to it?

The rationale for assigning the electrons in a qualitative building-up model of the atom and the method by

Volume 43, Numher 6, June 1966 / 289

which we calculate the shapes of orbitals depend on the notion that we can find some effective potential V(r,) for each electron. Each V(rj) must have spherical symmetry and approximate the true potential felt by electron j . Could such an effective potential be found, and could one solve the resulting equations to find orbitals for a many-electron atom?

Both questions have been answered with an almost positive "yes." The proper effective potential was developed by Hartree, Fock, and Slater in the early 1930's for atoms with closed shells or with one electron or one hole in a closed shell. Their original method left some ambiguities about the best way of defining V(rJ for arbitrary open-shell atoms. Then, the method was exteuded by several people so that potentials could be defined for open shells. (These can probably never be made to approximate real potentials as well as the Hartree-Fock potentials for closed-shell atoms. This is inherent in the slipperiness of an open shell system-the electrons tend to move simultaneously from one m, state to another within a given open shell of lixed 1. The mathematics of open shells tells us this when it requires that a single stationary state be a superposition of several specific assignments of electrons to mi- states, i.e., to be a multideterminant function, if we may use a term to be defined later.)

The Hartree-Fock method consists of using a Hamiltonian for each electron that contains V(r,) defined by the Hartree-Fock prescription, which we shall describe shortly, and solving all the equations together for all the electrons of an atom. The original solutions for HartreeFocB orbitals were obtained by numerical integration and were therefore tabulated functions with no analytical expressions. Numerical solutions are still obtained and, at their most refined level, are probably still the most accurate. Analytical approximations for Hartree-Fock orbitals can be obtained with high-speed computers; they are extremely useful and often are very close approximations to the best functions we have.

Note that when the method reaches the stage of computation, the Hartree-Fock equations are ordinary, not partial differential equations. This is because the Hartree-Fock potential is chosen to be spherical, so that the orbitals must he spherical harmonics multiplied by radial functions. These radial functions are the solutions of the differential equations. The symmetry of the problem allows it to be reduced this way by letting us use our general knowledge about spherical systems to go most of the way.

The seemingly mysterious potentials of the Hartree- Foclr equations are defined this way: the potential for each electron is the mean potential due to the nucleus and all the other electrons-more specifically, it was shown to be the root-mean-square potential. The original form used by Hartree utilized a simple Coulomb field based on Born's probabilistic interpretation of the wave function. If $(r) is the amplitude of the wave function at r, then '$(r) or $*(r) $(r) is the inten- sity of the wave there, a quantity everywhere positive and whose space integral is normalized to unity. These properties led to the identification of $(r)i2 as the density of probability, or, in our minds replacing a time average with a space average, led to the identification of l$(r)i2 as the mean charge density at r. Hence each

electron must feel the Coulomb field of the spherical average or the spherically symmetric sum of the charge densities defined by all the i$(r),2's for all the other electrons.

But, we ask, how can we find one orbitd unless we know all the others? The answer to this question is the crux of the success of the method. We can start with a guess for all the Vsof the system, with a most outlandish collection of orbitals if we wish. We use all but one of these to determine the potential for the last electron, and then solve the differential equation for this last electron's orbital, $(N). Then we use the new orbital with all the originals but one, say $(N-,), to determine a refined version of the deleted $(N-I) from the original set. The two new orbitals $(N) and $(N-l) plus the old ones give us a third new one, $(N. 2), and we continue until we've determined an orbital for each electron. Then we start through the process again, refining our first $(,,, $(,-I), . . . , $cl) revisionsinto second revisions, and then go through again and again until we find that the orbitals are not changed by further recycling. The potential field defined by this series of operations is now consistent with the orbitals that it determines and that determine it. It is called a self-consistent field or SCF, in this case a Hartree SCF.

Exclusion Principle and Many-Electron Wave Functions

We must interject a brief acknowledgment of the existence of electron spin and a review of its role in many-electron problems. (It could have been part of our discussion of symmetry; cf. References (36) and (27).) And we must introduce the Pauli Exclusion Principle, too. The former adds the quantum number 112, to our set n, 1 and m, for a oneelectron function; strictly, it adds s as well, but since all the electrons we know haves = '/s, we don't bother carrying it explicitly. We do have to pay attention to m, (= the component of s along one chosen axis, in any many-electron or magnetic field problem. Them, quantum number's values do contribute very much to magnetic properties; more germane for this context, spin adds one extra degree of freedom to each orbital.

A spatial function $(r) with an assigned m, of +'/2

is said to be a spin orbital with or spin, and if it is neces- sary to designate the spin state explicitly, is usually written either as $(r) a or as $(r) alone. If m, is assigned as one commonly writes $(r)B or &(r). Sometimes one need only say that $ should stand for the entire spin orbital.

The Pauli Exclusion Principle is the statement that no two electrons may be assigned to the same spin orbital. This can be restated many other ways, perhaps less concisely: no more than two electrons can be in the same orbital, and if two are in the same one-electron state in coordinate space, they must have different spin states; or, no two electrons in a system can have the same values for all their quantum numbers. If two electrons with the same spin are forced together in space, they must go into different energy states. One may remain in a low-energy state, but the other must go to at least the next higher energy orbital. Such a restriction is equivalent to the requirement that the two electrons stay apart in momentum space if their wave functions are close together in coordinate space. The measure of this requirement is closely related to the

290 / Journal of Chemical Education

uncertainty relation ApAq 2 R/2 for any single coordinate of one particle. The requirement, stated in terms of the six-dimensional space of three spatial coordinates and three momentum coordinates is simply that each electron needs a volume AT = AxAyAz Ap,Ap,Ap, = h3.

Electrons are identical particles, so that no physical property can be affected if we rename or renumber them. For example (and not an accidental example), if we deal with a wave function q(rl, . . . , rN) for N electrons, then -?(r,, . . . , rN)= must be unaffected if we interchange two electrons; moreover the wave function *(rl, . . . , rN) must go back into itself if we interchange the same pair of electrons twice. These two properties imply that interchanging two electrons must either leave Y unchanged ( q is symmetric) or, at most, change it into its negative ( q is antisymmetric). Now \E must always he zero if two electrons are assigned to the same spin orbital, according to the Pauli Principle. This is just what the situation would be if Y were to change into - q whenever a pair of electrons was interchanged. If electrons 1 and 2 were arbitrarily assigned to the same spin orbital, then the identity of electrons and their spin orbitals would imply that -?(r,, r,, . . , r = ( r r , . . ., rw) but also our guessed permutation property would require that q(r,, r2, . . ., rN) = -V(r%, rl, . . ., rN), so that this Y would necessarily be zero. We have never seen a nonzero Y in nature with two electrons in the same spin orbital; we can infer, therefore, that the antisymmetric choice correctly describes the behavior of real many-electron functions. This argument is not meant as a rigorous proof of the antisymmetric property of many-electron functions but only as a demonstration of the relationship between the Exclusion Principle and the property of antisymmetry.

The Form of Many-Electron Orbital Wave Functions

Now we can go on to consider the form of many- electron wave functions and the relationship of this form to atomic orbitals, and then we shall return to the Hartree-Fock problem.

If we assume that we can find appropriate self-consistent potential fields V(rJ for each electron of an atom, then the Hamiltonian of the atom can be written as a sum of Hamiltonians, each one containing the kinetic energy T, and potential energy V(r,) for only one electron. That is,

x = C xi sll electrons

j

Once again, as withequations (12) and (13) and with (14) and (15), we have a separable Hamiltonian. This time it is separable into its one-electron terms. As before, whenever the Hamiltonian is separable, the wave function can be expressed as a product of functions, each of which is the solution of its own equation:

XAr;) = v&;) (17)

That is, we may write

*(r,,. . . r ~ ) = +(11(11). . . I L I N ~ I N ) (18)

where we let each indicate the J t h spin orbital. Equation (18) is a very useful form and tells us very

explicitly something about the kind of atomic wave function we get if we start with the orbital concept. This equation says that in this picture, the electrons have probability amplitudes, and therefore probability distributions, that are independent of the position of any other electron. The only effect elertron 1 has on electron 2 is through mutual effect of their average potentials of interaction with each other and to a lesser degree, with the other electrons.

Expression (18) cannot present the oomplete picture even in terms of orbitals because it does not have the property of being antisymmetric with respect to exchange of any pair of electrons. If we exchange electrons 1 and 2, and require that the function q be repaired so that it changes sign with this exchange, then we can do it this way: replace (18) with

Now we can make any other pair interchange and sub- tract the corresponding new functions from each of the two terms in (18a) to get to a more repaired state. Eventually we'll reach some of the rearrangements again; in fact we can construct all the permutations of the N electrons among the N spin orbitals &,). If we continue to change the sign when we make a pair exchange, we will eventually construct the totally antisymmetric function that can be constructed with the spin orbitals $ill, . . . , fiIwl. The factor 1 / 4 2 in (18a) was added to keep the function normalized. When we have all N permutations, the normalizing factor is I/.\/#!!, instead of 1 / 4 2 .

I t was pointed out by J. C. Slater that the construction of a completely antisymmetric function from a set of products, adding and subtracting as we have just described, is exactly equivalent to making a determinant out of the J.ln(rk) spin orbitals. One lets the number (J) of the spin orbital be the column index and the electron number k be the row index, or vice versa.

In this way we have

This expression is the Slater determinant, or the determinantal function based on the spin orbitals $(I,, . . . , \I.(N). Because Slater determinants are so commonly used, and because it is really redundant to write more than the principal diagonal or the N spin orbitals themselves, once we know that a determinant is meant, we frequently use a shorthand:

Even the normalizing factor is left implicit. Another common notation applicable even if \Ir is not a product function uses the antisymmetrizing operator (2 (which is usually defined to do the normalizing also). We say simply that if (2 acts on the function (IS), it generates the function (19). Although this seems like an arbitrary and rather useless formal definition, it is possible to write out explicit prescriptions for (2, and to use this


shorthand as a powerful tool for dealing with many- electron problems.

It turns out that a single determinant function like (19) is a very suitable form for a Hartree-Fock wave function whenever all the spin orbitals of a given n and 1 are either full or empty-i.e., in the closed shell cases like He, Be, Ne, or Zn+2. The form is also fine when there is one electron outside a closed shell structure or one hole in a closed shell system. In other cases, if we try to use a single Slater determinant to define the wave function of N electrons, we find we are in trouble.

Except for special cases, we cannot obtain functions with quantized total orbital angular momentum L or total spin S with single determinant functions. Physi- cally, in the proper functions with quantized L and 8, the inl and in, values of the individual electrons in the open shells are no longer good quantum numbers. The correct function based on orbitals must be written as a sum of two or more determinants. Here are two examples. The lowest triplet state of He has orbitals 1s and 2s each singly occupied. In the notation of (19a), we have

The extra 1 / 4 2 is another normalizing factor. The two assignments in (20) differ only in assignments of m,. In effect, the electrons flip each other's spins. A more drastic reorganization takes place in the 8P ground term of the carbon atom:

Similarly the 2p2, 'D, and '8 levels of carbon also involve 1 and s reorganization (we omit 1s and 2s electrons in writing).

P(p:'D,Mfi = 0, MI = 0) = 1 - 1212~0%l + 12p+%I - l%+2~- I I, and

*(p,= '8, Mr. = 0, M. = 0) = 1 -- 43 [ PP+% I - 1 2 x 2 ~ - l - 12~02x1 I

In this example, the 2p electrons change both their m. values and their m:s. In all cases of this sort, the changes are not random but occur in concert. Here, by the way, we have our first explicit inkling that there might be a limitation to the orbital picture, just because in the second of these examples, we cannot assign a specific set of four quantum numbers to the electrons. We can specify only n and I for each electron.

The set of states defined by assigning n and I for each orbital is called a configuration. A real state is based on a single configuration just so far as each one-electron nand 1 are true constants of the system.

If, in addition to all then's and l's, we specify L and S, then we are said to specify a term, like ls22s22p2 to give aP of carbon. From any single configuration we can always find one or more terms.

What physical effects result because we must use the antisymmetrized function (19) instead of the simple product (18)? We have discussed how the antisymmetry property was associated with the Exclusion Principle and with the fact that the electrons can't all

condense in the 1s orbital of an atom. The antisymmetry property exhibits its effects on the

forms of atomic orbitals in a very explicit way. This is through the famous exchange interaction which Fock added to Hartree's simple average Coulomb interaction in the self-consistent field. In Hartree's original expressions, the potential energy of electron j in an atom had the form

Fock showed that an antisymmetric function like (19) introduced a similar-looking but physically different term, due to single pair permutations:

Because of the additional pair exchange, these always appear in closed shell cases as the combination FJK - GJK. Also the G's are zero if J.c,l and $yK1 have different m,, while the F's are not. The effect of GJK is to reduce the effect of Coulomb repulsion between electrons of the same spin in $qJ, and J . ( K ) . This in turn permits the orbitals to be closer to the atomic nucleus and therefore to be more strongly bound. The im- provements in energies with the exchange interaction included are quite dramatic; the effects on wave functions are sometimes subtle. Figure 3 shows a compari- son of typical Hartree and Hartree-Fock radial functions.

Hartree-Fock orbitals have been computed numeri- cally to high accuracy for some atoms, and approximately for the entire periodic table. They have also been computed in analytic form to rather high accuracy for a number of atoms. The very first analytic form for the radial wave function, historically, was the simple exponential, introduced in 1930 by Frenkel. The most popular extension of this form is a monomial multiply- ing an exponential:

The parameter 3 is fixed by normalization, so that only a is a free parameter. These are the Slater type orbitals; Slater's original set of orbitals was based on a simple set of rules for choosing optimum ru's [J. C. SLATER, Phys. Rev. 36,507 (1930)l. The most popular of the modern accurate analytical forms is a sum of functions like (24), often with several terms of the same n. The other popular form in current use is based on a sum of Gaussians, which replace ar or (24) with j3rBr2. These have the advantage that they lead to simpler integrals than do exponentials in r, but more of them are required to reach a given level of accuracy. Refer- ences to some of the more extensive tables and sources are given in the bibliography (4f-44).

The Hartree-Fock orbitals, as we said, represent the best ae+lectron functions based on the mean field of all the electrons, in the sense of giving the best one-electron energies in this mean field. Because of this energy

292 / Journol of Chemical Education

property, Hartree-Fock orbital energies give very good estimates of ionization potentials (and electron affini- ties, if we start with Z + 1 electrons around a nucleus of charge +Ze). The orbital energy of the highest filled orbital is approximately equal to the ionization energy. I t was also shown that Hartree-Fock orbitals give good estimates of all other atomic properties that involve interactions of a potential field without inducing transi- tions. By "good," we mean accurate to about the same percentage accuracy as the total energy. Typical properties that are evaluated well by Hartree-Fock fnnc- tions are charge distributions, electric field gradients a t nuclei (the source of nuclear quadrupole coupling in chlorine compounds), and in certain cases like the alkali atom principal series, spectral line frequencies and in- tensities. Other properties that are likely to be less accurately described by Hartree-Fock functions are properties like spin-spin and spin-orbit coupling, that depend on electron-electron interactions, and spectral frequencies and transition probabilities that involved more than a simple s-to-p transition, like the forbidden atmospheric line of oxygen.

One property for which there is a significant number of calculated and experimental values is the ordinary electric dipole polarizability, the induced dipole per unit of applied electric field. Table 3 contains some of the calculated (Hartree-Fock) and experimental polar- izahilities for several atoms and ions. It appears that the lower the polarizability, the lower the relative error of the calculation, which is probably because the species with high polarizabilities have many low-lying excited states including doubly-excited states or states that are not included in the Hartree-Fock calculation.

Table 3. Experimental and Theoretical (Hartree-Fock) Dipole Polarizabilities

Atom or Experjmentd~ Ion Theoretical (A3) (Aa)

COHEN, H. D., J . Chem. Phys. 43, 3558 (1965). L:ANGHOFF, P. W., AND HURST, R. P., P h ~ s . Rev. 139, A1415

(1965). Experimental values compiled by A. Dn~GanNo, Advan. Phys.

11, 281 (1962).

Transformations of Orbitals: Hybrids and Ligand Field Orbitals

One of the most striking properties of molecules is the relative rigidity of their structures. If we had no information about their structures, we might well attribute much looser structures to them, structures more like liquid drops. It has been one aim of theoretical chemistry to understand the directional character of chemical bonds, and to do this in terms of atomic orbitals, if possible.

One of the most attractive ways to interpret the directional character of bonds is based on using directional atomic orbitals. These are the hybrid orbitals

in a free atom. I n a molecule, one can use the same hybrids, or the closely related equivalent orbitals, or any of several other similar localized orbitals; they vary a little in their definitions and manner of construction but prove to be quite similar in the long run. We shall here restrict ourselves to atoms, sidestepping a lengthy discussion of these various localized molecular orbitals. Let us first examine the mathematics relating the Hartree-Fock orbitals to the localized orbitals. This way we can see how the best approximations to one- electron states of constant energy are related to the best approximations to localized orbitals for electron pairs. Then we shall compare the two types of orbitals.

The approach is based on a transformation like one to which we alluded in our discussion of spherical harmonics. The localized orbitals appear when we super- pose the standing waves that are Hartree-Fock orbitals. We quote and then use a theorem about our determinantal functions like (19): we can make any unitary transformation we like of the filled orbitals and leave the value of the determinantal function (19) unaffected. Recall that a unitary transformation is like a pure rotation of a coordinate system: it preserves all the lengths (normalization) and angles (orthogonality) when it transforms the old coordinates (functions) into the new ones. This means that if we mix the individual factors +(rJ with each other by adding and subtracting them in any size pieces, so long as we maintain the orthogonality and normalization of the new $'s, the determinant (19) formed from the new +'s will be equal to the old determinant, for any fixed choice of electron coordinates. We can mix functions with 1 = 1 and mr's of +1 and - 1 to get functions that look like sin 0 cos p and sin 0 sin p in their angular dependence; that is we can go from

R(n)R(n) sin Ole+ sin B d w - R(r,)R(rl) sin A e - i ~ , sin B&

to R(?,)R(m) sin 8, eos o, sin 82 sin m -

R(r,)R(m) sin 8, sin q: sin & cos oz

by letting

= .@ [Rfr) sin B sin ql 4a

We have transformed from complex orbitals with quantized mr into directional orbitals looking like cos 0 but with respect to the z and y axes.

The orbitals can be made still more directional if we are willing to spoil not only mr but also 1 itself as a good one-electron quantum number. We can mix a spherically-symmetric s-function with a cosine-like p-function; if we mix them by adding equal parts of each, then on the positive z-axis, where cos 0 has its maximum, the p- wave will add its maximum to the outer (positive) lobe of the s-function and reinforce the total wave amplitude. On the negative z-axis, the p-wave will tend to


cancel the s-function, or, in optical terms, there will be maximum destructive interference of the waves. If the radial parts of the s and p-waves are very similar, as they are if n = 2 for both waves, then the reinforcement and cancellation is very significant. In fact, for this case, the p-wave slightly more than cancels the s-wave on the negative z-axis. Written out, we have (using the normalizing factors from Table 2),

With equal parts of the normalized s and p-functions, we have constructed an sp hybrid. If we had subtracted the p-function from the s-function, we would have simply changed z for -z and put the reinforcement onto the negative z lobe, and the cancellation on the positive side.

In this example, we constructed two equivalent sp hybrids. We could do the same to construct three equivalent sp2 hybrids from two p's and an s. Let us do this example with p, and p,-orbitals, keeping to the x, y plane, and use the fact that p-orbitals are likevec- tors in their angular behavior. First, one may be chosen as 2/3 pU:

Now, the other two functions must contain A; they must contain equal parts of J.nz with opposite signs and these parts must contain all of J.,, (i.e., the sums of the squares of the coefficients of each component orbital must add to 1); and they must contain all the rest of J.vn divided evenly and with the same sign, so that both of them point toward the negative y direction. This fixes the other two as

and

For convenience in illustration, one sometimes im- plicitly supposes that Rz0(r), the 2s radial function, is identical with the 2p radial function Ral(r), so that the angular dependence of the hybrid orbital can be plotted unambiguously in one or more planes. In truth, Rzl(r) and Rza(r) are not generally identical, as Figure 3 shows.

What relationship exists between the localized hybrid orbitals or localized orbitals in general, and the Hartree-Fock orbitals that we examined previously? First of all, the two sets give the same total energy of any atom, by virtue of our quoted theorem. Second, they can be obtained from similar but by no means identical sets of equations. The Hartree-Fock orbitals are in one sense the best orbitals we know how to find- the energetic sense. The Hartree-Fock equations and all the fundamental equations we have for deriving orbitals ab initio (not just by transforming them into each other) are variational equations. They say that the one-electron energy of each orbital is the lowest possible

energy that we can find for it, so long as each orbital is orthogonal to the other orbitals of the same atom that have still lower energies. The only condition the Hartree-Fock orbitals satisfy, other than the minimum energy and orthogonality conditions, is the normalization condition, to conserve our scale of probability. It was shown only recently that localized orbitals, even the most localized orbitals in the sense of maximum intra-orbital interaction of electron pairs, can also be solutions of well-defined and rigorous variational equations. This is an important point historically and didactically. It seemed for quite some time that since there were physically-based and mathematically sound equations for Hartree-Fock orbitals, they rested on a firm basis while the localized orbitals were in a more vulnerable situation, without any rigorous physical basis. They were always widely used, hut with some trepidation about their physical interpretation. In any event, the variational equations for localized orbitals necessarily carry other conditions than the energy condition of the Hartree-Fock equations. There is avariety of conditions one can use, like the one requiring that the electron-electron mean interaction within an orbital be maximized (and therefore the inter- orbital mean interaction becomes minimized).

r(Bohr radii)

Figure 3. Radio1 functions RIr) for neutral carbon in in ground state. The solid curves ore Hortree-Fosk functionr UUCYS. A., Proc. Roy. Soc. (London1 A173, 5 9 (1939)l. m d the dotted curve is a Hortree 2p function [TORRANCE, C. C., Phyr. Rev. 46, 3 8 4 1193411. The Hortree Is and 2s functions are very rimiior to the Hortree-Fock functions .how" here. Note the rimilority of the outer parts of the 2s and 2 p functions. (The functions plotted are mluolly R = r +, ro that J P R'dr = 1 .I

The equations for localized orbitals are somewhat less tractable than those for Hartree-Fock orbitals. If we are only interested in visualizing and describing localized orbitals qualitatively, it is relatively easy and satis- factory to use existing Hartree-Fock orbitals to make hybrids within a specific shell. Symmetry alone fixes all the mixing coefficients for us and we can derive them with a little algebra, just as we did in the previous section. However if we want the localized orbitals that are best by some criterion like the one we just cited, then we are forced to do a moderate amount of computation.

294 / Journal of Chemicol Education

The variational equations for localized orbitals are more cumbersome than the Hartree-Fock equations, so that i t may be best to find the Hartree-Fock orbitals first and scramble them to meet the extra conditions.

Naturally, since localized orbitals must meet extra conditions, they cannot be the best orbitals in the orbital energy sense. Nor can they satisfy the symmetry conditions, the conditions that lead to the spherical harmonic form for each Hartree-Fock orbital. Nor can they have quantized angular momentum. (It may be that for open shell atoms, the energetically-best Hartree-Foclc orbitals don't have quantized angular momenta either. This question is currently unsolved.) Moreover the fact that localized orbitals do not give the best one-electron energies means that they do not give good values for ionization potentials, a t least from one- electron energies. Of course if we compare the total energy of atom and corresponding positive ion to get the ionization potential, it makes absolutely no difference whether we use the Hartree-Fock functions or some local set obtained by scrambling the Hartree-Fock functions. However, the Hartree-Fock orbital energies are generally better approximations to ionization potentials than the total energy differences.

It may turn out that localized orbitals are more u s e ful than Hartree-Fock orbitals in other ways. One hope is that they will offer a straightforward way to evaluate correlation energies. Correlation energy in an atom or molecule is the difference between the actual total (usually with relativist,ic contributions subtracted) and the Hartree-Fock total energy. It represents the extra energy of the system coming from the fact that electrons really interact with real electrons, not with mean fields of other electrons. It would be helpful if it turns out that we can calculate correlation effects by looking a t electrons just two at a time. If that is the case, then localized orbitals are likely to be extremely useful for this purpose. As yet, we cannot say for sure.

V. Beyond Orbitals-The Correlation Problem

As we just said, the Hartree-Fock energy differs from the true energy of an atom or molecule by a relativistic contribution that need not concern us here: and a correlation contribution. The origin of the correlation energy is quite clear and we can even attribute to it a precise potential field, the Jluctuation potential. This total field is the difference between the mean field and the exact field felt by an electron. An example is shown in Figure 4. In a two-electron atom one can visualize this field clearly. It acts between two electrons but depends somewhat on the distances between electrons and nuclei. It is highly repulsive and very short-ranged, stronger when the two electrons are far from the nucleus than when they are close. When one electron comes sufficiently close to the other, the electrons' potential energy becomes greater than their potential and, correspondingly, their kinetic energy

'This is not to say that relativistic effects are negligible. In terms of the total energy, relativistic effects become important even in the second-row elements. However, these contributions occur primarily in the inner shells, part,icols;rlp because of the high kinetic energies associated with the inner shells. As a result, the relativistic effects have little direct effect on the chemical behavior of atoms.

must become negative and their momenta, imaginary. This is a two-particle, three-dimensional example of the phenomenon we discussed in connection with the existence of quantized bound states of a single particle.

Figure 4. The fluctuation potentiol for I s electrons in beryllium (after ref. 26). The fluctuation potentiol is the difference between the exact and mean potentidr. It is o function of n 0% well as of rs; the value r~ = 0.27 a,"., the most proboble Is rodiur, w m chosen. The figure described po- tentiolr dong the linecontaining the nucleus and electron l .

5

We should interject a comment on the quantitative importance of the correlation problem. These energies in atoms are just the size of chemically important energy differences-roughly 1 or 2 ev per pair of opposite-spin electrons in the valence shell and higher for shells nearer the nucleus. Moreover correlation is the primary source of London forces, and therefore of the cohesive energy of molecular crystals. I n other words, even though we may get qualitative, graphic, and clear notions about electronic wave functions from orbital descriptions, and even though we may find that orbital calculations are useful for correlating chemical phenomena, we must be exceedingly cautious about using any wave function for quantitatiu~ purposes unless it contains correlation effects or unless we can show that the correlation effects drop out in our particular problem.

The tendency of one electron to repel another and to force the wave functions of the two electrons to have low amplitudes when the electrons are near gives rise to the concept of the correlation hole. This hole takes the form of a region around each electron, the region where the fluctuation potential is very large, where no other electron is likely to be. The Pauli Exclusion Principle establishes this hole moderately well for electrons of the same spin, but it has no effect on the spatial distribution of electrons with opposite spins, so does not help to introduce any correlation effect. I n this case the correlation hole must be a pure Coulomb hole and can only be introduced into a wave function by inclusion of specific terms above and beyond the Hartree-Fock function. I n other words we can improve on the Hartree- Fock function by superposing additional terms, which are the result of the fluctuation potential, to account for electron correlation.

What does the fluctuation potential do to our orbital concept? One thing it cannot do. I t cannot spoil


4- ex~cttw-elsctmn Wtentiol e2/r,, when electron 1 is of

- 3-

averope potentiol seen by 2-

-3 -2 -I t

7 7 I

6 -

-

I I I I

I I

Bcryilium 1s electrons - - fluct~ation potential - *en by electron 2

the overall symmetry or angular momentum of our total wave function. It can, however, spoil virtually all the other bases of our orbital picture. First of all, it obviously spoils any quantization of the angular momentum and energy of individual electrons. This means, in turn, that individual electrons cannot he described by specified quantum numbers n and 1. This is turn means that we cannot, strictly, specify the configuration of an atom or molecule. The entire structure of our atomic physics seems momentarily to be crashing down.

I n fact, the situation is not disastrous. To a reason- able degree of approximation, we can specify atomic configurations and the n and 1 quantum numbers of individual electrons. (We generally cannot specify individual m(s and m,'s though; the electrons' interactions do spoil their orientations.) At least we can specify the most likely or most important atomic configuration, or one-electron n,l and orbital energy.

This brings us directly to the problem of how one can actually go beyond the orbital picture, and how one can get a better wave function than the Hartree-Fock function. There are two principal approaches. The first historically and the most extensively explored is the method known as configuration interaction. The second has several variations, but all are based essentially on the concept of cluster expansions.

Configuration interaction is, as its name suggests, the addition to the Hartree-Fock N-electron function of one or more similar functions having different assignments of the individual n's and 1's of the electrons. These new functions correspond then to configurations differing from the Hartree-Foclc configuration. For example the beryllium Hartree-Fock configuration is obviously 1~2292. We could imagine adding to this some lsZ2s2p and ls22s3s. However neither of these will do: the first is necessarily an overall P state (total L = 1) while our Hartree-Fock and true functions are both S states, and in the long run, we can only build the latter with S configurations. The second, ls22s3s, does not affect construction of the lowest state provided we have really used Hartree-Fock functions. Strictly, the ls22s3s can enter to a very slight degree, but for a first and very good approximation, we can neglect it. I n fact we are justified in saying that neither it, nor any other configuration that differs from the one of interest in the quantum numbers of a single electron, will contribute to configuration mixing with the configuration of interest. Inotherwords the ls22s2configuration beryllium mixes with ls22p2 and ls23s2 but not with 1sZ2s2p in the overall zero-angular momentum IS state of the atom.

What physical effect is associated with this configuration interaction? The basic idea is relatively easy to see. Let us neglect the two 1s electrons. The 2s radial function of Be looks roughly like that of Figure 3 and the angular function is of course constant. In the 2sZ configuration, the radial and angular coordinates of one electron are entirely unrelated to the radial and angular coordinates of the other. Now, if we add some 2p2 character to the total wave function, we introduce some correlation between the two sets of angular coordinates. The 2p2 part has the effect of accounting partially for the polarization of one electron by the field of the other. This occurs because the two electrons' p functions reinforce their s functions in different angular regions.

When one electron's probability amplitude is large in a particular direction as a result of constructive interference of its s and p parts, the other electron's proh- ability amplitude is largest about 90' away, due to its own simultaneous interferences. The admixture of 3sZ has a similar effect on the radial variables. In effect, mixing 3sZ with 2s2 introduces some in-out correlation (cf. References 52-55).

Configuration interaction is a formally exact method that must converge to the exact wave function of the Hamiltonian of choice. I t is computationally straightforward and we are learning to interpret the physical significance of its various sorts of terms. In general, it is a very slowly converging proce~s.~ The beryllium atom was treated with 37 conf?gurations, for one of the most accurate calculations available on any atom more complex than helium. Configuration interaction is probably the method of choice at present if one simply wants a very accurate wave function of a moderately complex atom and cares very little about computational efficiency.

The other general approach to correlation has its origins in the fact that one can treat the interaction of a single pair of electrons. One can get a much more accurate representation of helium-like a t o m than one gets from the orbital model alone. This better rellre- sentation comes from the addition of terms in the two- electron wave function that contain the interelectronic distance explicitly. One of the simplest examples is a dying exponential factor e- ""'?' that can be included as a factor only for a small range of the inter-electronic distance r12 around the value rlz = 0. A much more elaborate one is the wave function of He originally introduced by Hylleraas, which contains a number of terms in r12 is its most highly developed form.

The general approach to correlation through pair interactions can be paraphrased somewhat generally this way. I n atoms, it may be valid to suppose that the order in which one should calculate interactions is:

(1) interaction of each electron with the mean field of all the others (HartreeFock);

(2) deviations from (1) due to simple pair interactions (binary encounters in two electrons in the HartreeFock field of the rest);

(3) deviations from (1) and (2) due to three-body encounters; etc.

If binary encounters are much more important than three-body effects, then we should be able to treat part (2) by using methods similar to those developed for the two-electron problem. This approach n-as developed by Szasz, Tsang, and Sinanoglu, and is perhaps most graphically described in an article by Sinanoglu (51).

The results indicate that for atoms up to beryllium, the cluster expansion approach is very efficient and physically clear. It is too early to tell whether it will be useful for calculation of the wave functions of more com-

-

6 I t is possible to choose a. set of orbitals that maximizes the rate at which a. configuration interaction series converges to the exact wave functions. These orbitals are the natural spin orbitals introduced by LBwdin znd developed by him, Shull, Gilbert, Wahl, and others. I t seems at present that the natural spin orbital expansion might lead to quantitatively useful wave functions with about three configurations per electron pair. This is the most optimistic estimate of the efficiency of the configuration interaction expansion.

296 / Journal o f Chemical Education

plex systems. I t may be that tho three-, four-, and n- body terms will be as important as the pair terms just because of their numbers, even though they may be small individually. There is some reason to fear that this could happen simply because the Coulonlb correlation hole can be comparable to the size of an atom. This means that the electrons might be best thought of as though they are constantly in collision with all the rest, at least those with the same principal quantum number. On the other hand estimates of the pair correlation energies of a number of atoms suggest that higher contributions are probably small.

In fairness to the many people actively working on the problem of electron correlation, it should he said that most of these workers recognize that both configuration interaction and cluster expansions may have their own domains of utility. Just where these domains extend and overlap, and whether they cover all the problems of chemical interest, is still a very open question. Bibliography

Within each section, the order is approximately from the most fundamental and/or diacult to the most elementary. The parenthetic numbers 1 4 and G indicate in a rough way the level at which the author of this article has found the book useful or would guess it to be useful. The sections are ordered from the most general to the most specific.

We should refer the reader to Val. 2 of Reference (9) for a particularly extensive bibliography, especially of atomic theory. Quontum Mechonicol Foundations (1) Dmnc, P. A. M., "The Principles of Quantum Mechanics,"

Oxford University press, New York, 1947. The elassio treatise, its first two chapters often lend clarity to the broadest fundament& of the quantum mechanical viewpoint, even at the intermediate level. (3,4,G)

(2) KRAMERS, H. A,, "Quantum Mechanics," IntersciencePub- lishers, (division of John Wiley & Sons, Inc.), New York, 1957.

Extremely clear basic treatise with particularly elegant and efficient mathematics. Treatment of matter waves is especially useful at a relatively early level (2, for the mathematically prepared). Otherwise (3,4,G).

(3) KAUZMANN, W., 'iQuantum Chemistry," Academic Press, Inc., New York, 1957.

Particularly useful because of its extensive development of the mathematics of waves in a way that gives a lot of physical insight. In all, a good reference far self-teaching as well as a good text. (2-G)

(4) FEYNMAN, R. P., LEIGHTON, R. B., AND SANDS, M., '*The Feynman Lectures on Physics. Quantum Mechanics," Addison-Wesley Publishing Co., Reading, Mass., 1965.

The book speaks for itself. A pleasure to read, and a. fimt-class book; to be used at thelowest possible level, before bad habits set in. (1-G)

(5) S I ~ R ~ I N , C. W., 'Tntraduction to Quantum Mechanics," Henry Holt and Co., New York, 1959.

This is a straightforward and very clear elementary text, with many illustrations m d diagrams.

(6) HEISENBERG, W., "The Physical Principles of Quantum Theow." Dover Publications. Inc.. 1930.

page Appendix. (1 or 2-G). ( 7 ) HINSHELWOOD, C. N., "The Struclure of Physical Chemis-

try," Oxford University Press, 1951. Parts I11 and IV present a qualitative and well orran- ieed storv that ca i be undekood bv a well-educated freshman. An excellent overview. (1-3)

Atoms: General Monographs and Texts

(8) CONDON, E. U., AND SHOILTLEY, G. H., "Theory of Atomic Spectra," Cambridge Univemity Press, 1953. [First printing, 1935.1

Long the single most important work in the field, i t is fundamental, comprehensive (for its time), and an excellent source for study if you are willing to work at it. (4,G)

(9) SLATER, JOHN C., "Quantum Theory of Atomic Structure," McGraw-Hill Book Co., 1960, 2 velum%?.

The only competition for Candon and Shortley in scope and level. Shter's style is less ten-e than Condon and Shortley's. This book places more emphasis on the mathematics and computation and less on the physics than Condon and Shortley. It has a large number of useful tables and probably the world's most extensive selected bibliography of (primarily) theoretical treat- ments of atoms. (3,4,G)

(10) BETHE, H. A,, AND SALPETER, E. E., "Quantum Mechanic8 of One- and Two-Electron Atoms." Academic Pms. New York, 1957.

Concise and broad survey of the area a t a relatively high level, intended primarily a? a definitive reference. (4 C,) ~.,.,

(11) Reviews of Modern Physics, 35, No. 3, July (1963). Proeeed- ings of the Internetional Symposium on Atomic and Molecular Quantum Mechanics, Sanibel Island, Florida, January, 1963.

A very good representation of the state of the science with a large variety of clearly presented material of current interest. Contains extensive discussion following the papers and two "soft" articles, a memoir by Egil Hylleraas and a review by J. C. Slater. Most of the other articles are not elementary. (Some articles, I-G; a few, 3 or 4,G; most, G.)

(12) J . C h m . Phys., Special Issue in Honor of R. S. Mulliken, November 15, 1965.

Somewhat like the preceding reference, this too is based on the proceedings of a Srtnibel Island meeting. It has same review and survey articles, one or two almost historical in their approach, and many other articles that represent the state of the art fairly well. (Like the preceding reference, mostly G, with same I-G)

(13) LOWDIN, P.-O., AND PULLMAN, B., editors, r'M~lecular Orbitals in Chemistry, Physics and Biology," Academic Press, New York, 1964.

Much the same as the two preceding references but with considerably more emphasis on molecules and molecular orbitals and correspondingly less on atoms.

(14) Fmo, U., AND FANO, L., "Basic Physics of Atoms and Molecules," John Wiley & Sons, Inc., New York, 1959.

A clear, fundamental treatment, unique in its ability to combine physical and mathematical ideas into a coherent and comprehensive teaching book. This has proved itself at all levels, inclnding freshman honors. Highly recommended. (14 ,G)

(15) SINANOGLU, O., AND TUAN, D. F.-T., "Quantum Theory of Atoms and Molecules," Ann. Rev. Phys. Chem., 15, 451 il!464\. -,

Particularly useful as an introduction to the literature. 12 4 r.1 \-,-,-,

(16) BALLHAUSEN, C. J., "Introduct~on to Ligand Field Theory," MeGraw-Hill Book Co., New York, 1962.

Chapter2 offers agentler introduction than Condon and Shortley but leads you to almost the same point, a t least with regard to manipulation. Chapter 3 is a dear and useful but rather brief treatment of symmetry properties. (3,4,G)

(17) K~NDRATYEV, V., "The Structure of Atoms and Molecules," translated from the Russian by G. YANHOVSBY, P. Noordhoff N. V. Groningen, The Netherlands.

Contains good descriptions of Bohr-Sommerfeld model and considerable material about molecules.

(18) L I N N E ~ J. W., "Wave Mechanics and Valemy," John Wiley & Sons, Inc., New York, 1960.

A brief and elementary-to-intermediate discussion ori- ented toward interpretation of structure, with considerable emphasis on orbital concepts. (2-G)


(19) HERZBERO, G., "Atomic Spectra and Atomic Structure," Prentice-Hall, Inc., New York, 1937.

An old but classic treatment that deals more with atomic spectra than with orbitals.

(20) J@RGENSEN. CHR. KLIXBULL, "Orbitals in Atom and Male- cules," Academic Press, 1962.

A short but densely packed intermediate-level book. Emphasis is on symmetry and on coordination chemistry applicatiom. ( 2 4 )

(21) GREENWOOD, N. N., "Principles of Atomic Orbitals," Royal Institute of Chemistry Monographs for Teachers, No. 8, The Royal Institute of Chemistry, London, 1964.

Thorough elementary discussion. Possibly bard to oblrtin locally.

(22) GRAY, HARRY G., "Electrons and Chemical Bonding," W. A. Benjamin, Inc., 1964.

Elementary, but quite ready to use simple mrtthe- mstics; the emphasis in the purely atomic part is on manipulations. (1,2)

(23) RYSCHKEWITSCH, G. E., "Chemical Bonding and the Geome try of Molecules," Reinhold Pub. Corp., 1962.

An elementary survev, about half of which is devoted to conventiond presentation of atomic structure. (1)

(24) HOCHSTRASSER, ROBIN M., "Behavior of Electrons in Atoms," W. A. Banjamin, Inc., 1964.

A very elementary and expository text with essentially no mathemah, it is probably most useful for people who want to acquaint themselves with the general ideas, concll~4ions, and vocabulary of electronic structure of atoms, but have little or no intention of making use of it. (1,2) See also reference (3).

(25) THOMSON, G., "The Atom," Oxford University Press, New York, 1962.

A small readable qualitative discussion.

Symmetry (26) WIGNER, E., ''Group Theory and Its Application to the

Quantum Mechanics of Atomic Spectra," Academic Press, New York, 1959.

The standard and classic reference; a revised transla- tion of the old edition in German. (3 or 4,G)

(27) HAMERMESH, M., "Group Theory and Its Application to Phvsicd Problems." Addison-Weslev Publishine Co.. - Reading, Mass., 1962.

One of a small number of very good books for self- teaching as well as for course texts in this subject. Considerably broader in scope than books such as the following reference. (3,4,G)

(28) COTTON, F. A,, "Chemical Applications of Group Theory," Interscience Publishers (division of John Wiley & Sons), New York, 1963.

A popular and clear exposition of haw to use methods of group theory in problem like ligand field theory. (2-G) See also references (9), (14), and (16).

Correbtion (29) LOWIN, P. O., "Correlation in Quantum Mechanics. I.

Review," Ad". in Chemical Physies II, p. 207 (1959). An extensive review of the field. Some significant advances have been made since this was written. (4,G)

(30) Yosarzu~r, A,, "Correlation in Quantum Mechanics. 11. Bibliography," Ado. in Chemical Physies 11, p. 323 (1959).

An annotated bibliography of atomic theory accom- panying the previous reference.

(31) SINANOGLU, O., "Many-Electron Theory of Atoms and Molecules," Pmc. Nat. Acad. Sci., 47, 1217 (1961).

A rather qualitative description of the physics and formalism of the correlation problem, with emphasis an cluster method. (3,4G).

(32) LENNARD-JONES, J. E., AND POPLE, J. A,, "The Spatial Correlation of Electrons in Atoms and Molecules. I. Helium and Similar Two-Electron Systems in Their Ground States," Phil. Mag. 43, 581 (1962).

(33) LENNARDJONES, J. E., "The Spatial Correlation of Elec- trons in Atoms and Molecules. 11." PTOC. Nat. Amd.

A very useful and extremely clear discussion of the physical phenomenon, with much discussion of very simple models. Definitely recommended for anyone trying to discuss physical origins of localization. (2-G) See also Refs. (9), ( l l ) , (ld), and (14).

Localized and Direcfionol Orbitols

(35) LMNETT, J. W., 'The Electronic Structure of Molecules, A New Approach," John Wiley & Sons, Inc., New York, 1964. ? 3 - ~ )

(36) LINNETT, J. W., Valency and the Chemical Bond," Ameri- can Scientist, 53, 459 (1964). (I-G)

These t,wo references resent an ad hoe but somewhat

with some correlation effects. (37) BENT, H. A,, "An Appraisal of Valence-Bond Structures and

Hybridization in Compounds of the First-Row Ele- ments." Chem. Revs., 611 275 (1961).

An empirical and semiquantitative review; develops structure bv usine orbital ideas as the oreanizine frame- " - work without delving into quantum mechanics of the orbitals themselves. Bent has also written less comprehensive and more descriptive articles for J. CHEM. EDUC., e.g., 40, 446, 523 (1963); 42, 302, 348 (1965). (2 or 3,4)

(38) OORYZLO, E. A., AND PORTER, G. B., "Contour Surfaces for Atomic and Molecular Orbitale," J. CHEM. EDnc., 40,256 (1963). (I-G)

(39) ADAMSON, A. W., "Domain Representations of Orbitals," J. &EM. EDUC., 42, 141 (1965).

A device for visualizing orbitals, similar to those used in solid state work.

(40) JOHNSON, R. C., AND RETTEW, R. R., "Shapes of Atoms," J . CEEM. EDTIC., 42, 145 (1965).

These three are primarily useful as teaching aids for their presentations of visualization devices. They contribute essentially no physical ideas, other than those already inherent in the orbital concepts. (Ideas me useful for 1,2,3).

(41) COHEN, I., AND BUSTARD, T., "Atomic Orbitals: L i m i b tions and Variations," J. CHEM. EDUC., 43, 187 (1966).

Tables

(42) MOORE, CHARLOTPI: E., "Atomic Energy Levels," US. National Bureau of Standards Circular 457, Vol. 1, 1949; Vol. 2, 1952; Vol. 3, 1958.

The comprehensive tabulation of known and predicted atomic energies. A must for research, and very useful for examples.

(43) HERMAN, F., AND SKILLMAN, S., "Atomic Structure Cslcu- lations," Prentice-Hall, Inc., Englewood Cliffs, N. J., ,062 A=-".

Approximate orbital energies, orbitals, and total potentials for the element^. A convenient source.

Analytic Hartwe-Foek Functions. CLEMENTI, E., ROOTAAAN, C. C. J., AND YOSHIMINE, M.,

Phys. Rev. 127, 1618 (1962). Firsbrow atoms.

CLEMENTI, E., J . Chem. Phys., 38, 996 (1962). Ground and excited states of isoelectronic series with 2 to 10 electrons.

CLEMENTI, E., AND MCLEAN, A. D., Phys. Rev., 133, A419 (1964).

Li-, B-, C-, N; 0-, F-. CLEMENTI, E., J. Chem. Phys., 38, 1001 (1964).

11 to 18 electrons. CLEMENTI, E., ET AL., P h g ~ Rev., 133, A1274 (1964).

Ns-, Al-, Si-, P-, S-, C1-. CLEMENTI, E.. J. Chem. Phvs.. 41. 295 (1964). . . .

19 to 30 electrons. CLEMENTI, E., J. Chem. Phys., 41, 303 (1964).

31 to 36 electrons. See also Reference (a), Vol. 1, Appendix 16.

Sei., 38.496 (1952). General discussions (3-G). (34) DICKENS, P. G., AND LINNETT, J . W., "Electron Correlation Hisfory Presentofions for

and Chemical Consequence%" Quart. Rars., 11, 291 (45) GAMOW, G., "Mr. Tcmpkins in Wonderland," Maemillan (1957). Co., New York, 1940.

298 / lournol of Chemical Education

(46) GAMOW, G., "Mr. Tompkins Explores the Atom," Macmil- Ian Co., New Yark, 1945.

(47) HOFFMAN, B., "Strange Story of the Quantum," Dover, New York, 2nd Rev Ed., 1959.

148) JAFFE. B.. "Crucibleg." Fawcett Publications. New York. . . rev. ed.. 1960. , -

(49) SHAMOS, M. H., ed~tor, "Great Experiments in Physics," Henry Holt and Co., New York, 1960. See also Ref. (25).

Appendix This represents the briefest derivation of the circular orbit

Bohr atom known to this writer. We write three equations, in terms of the mass m of the electron (strictly the reduced mass, but we neglect this distinction), its charge -e, the charge +Ze of the nucleus, the velocity v and the radius r.

(a) the definition of the total energy

of an electron in a Coulomb field; (b) thearbit,rary condition that the orbit bestable-centrifugal

force equaling centripetal force,

and (c) the quantum condition that the aetimz he quantized,

Here n is any positive integer and h is Planck's constant, to be

ZeP ZeZ y = - = -

rnvr nfi

and r:

The period r is

Tllc clergy Level.; call be dercn>oiwrl I,? diwct suhslitution for r UNI u > n r u [ A l l h ~ t it i? wwtl~whik I U u e (W 18, I d 1 w a purely c l a 4 n l rdarion that sirnpllfivc ( k l ) , itarnvl\, t h : vlrial relnricm for a l/r2 force:

rn"' = Ze'/r

or the potential energy is -2 X the kinetic energy, so that

mu2 E = - - 2

and from (A4) or (A5),


1966_Berry_J Chem Educ_V - Atomic Orbitals

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