Resource Papers-V Prep.&
under the aponrorrhip of
The Advisory Council on College Chemistry
R. Stephen Berry1 The University of Chicago Chicago, Illinois
Atomic Orbitals
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. I. Atomic Orders of Magnitude and the Bohr Alom 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.
Quantity
Expressions for Characteristic Properties of Circular Bohr Orbits
Expression Zs
Energy, E Radius, r Velocity, V Period
2h'
n'
tLZ nz e2m X ez
t- X
Z
2rha
n8 X Za
Value 109,687 cm-I or 13.58 ev or 2.178 X 10-LLerg n2 0.529 X X em
z
2 188 X lo8 X l* S X
Z
;cm/sec
n3 X Z, see
tL = Planck's constant h divided by 2r, e = charge on the e l e c tron, m = mass of the electron, and Ze = nuclear charge.
effective charges and effective quantum numbers further along.) The Bohr-Sommerfeld model is admittedly 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
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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 r u c t u r e . ~ With it, we can say, for example, that phenomena occurring 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 affectthese 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 examining the representation of waves and the basis of the wave theory of matter. The historical background of wave-particle dualitythe 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 sa familiar and rather romantic subject. The entire period of its invention and development was so short that many, perhaps most, of its major figures contributed 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~?&igentlyand 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 preM., Phy8 Rev. Letters, liminary form quite recently [GRYZINSKI,
14,1059 (1965)l. 284
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Journd of Chemical Education
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), corresponds 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)
pA = h
(2)
The deBroglie relation 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 descrihing a wave function fi is a relationship between the
time and space derivatives of J.. What is that relationship? I n the function J.(x,t), the quantity u is a frequency, and must be associated with the time t to give a diiensionless 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 corresponds 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
- 2 r u t ) + i sin ( k z - 2 r v 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
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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 X J . 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 i n 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
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 O n e 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
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 continuous 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 I n essence, the imposition of houndary conditions a t both ends of the range, plus the conditions that the houndstate wave have no kinks or discontinuities and he quadratically integrable, eliminates all possible functions except those having an integral number of half-
*
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 g e h no further from x = 0,y = 0 than 1, the lengthof the rope-i.e., ylendl 1.
5
(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 rbound t 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. I n 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 & w r o i d d curve; ib) and (4 phygically impossible situations, corresponding to no true V is only o dying exponential on one states; the wove function with E ride, and grows exponentidly on the other. Arrows mark points where E = V a n d cvrvotvrechonger sign.
oiwrl I,? diwct suhslitution for r UNI u > n r u [ A l l h ~ itt i? wwtl~whikI 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
E = - - mu2 2
and (c) the quantum condition that the aetimz he quantized, and from (A4) or (A5), Here n is any positive integer and h is Planck's constant, to be
Volume 43, Number 6, June 1966
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