THE STORY OF QUANTUM CHEMISTRY.PART 1: THE OLD QUANTUM THEORYKc-Uh J. ialdJLo.fiVc.paAfine.tvt o{, Chem-uViyUniveA&JAy c& OttawaOttawa OH KM 6W5

(This is the first in a series of three articles onquantum chemistry. Look for the next instalment inFebruary.)

At the beginning of the present century most physicistsand chemists were satisfied that their basic problemswere solved, all that remained being the filling in ofsome details. The principles of mechanics had been wellworked out, heat was understood to be a mode of motion,

and light to have wave properties. There was nosuspicion that within a few years two revolutionarytheories would require a complete revision of allscientific concepts. These were the quantum theory,born in 1900, and the theory of relativity, born in 1905-Einstein had much to do with both of these theories, andof the two the quantum theory has much more profoundimplications for chemistry. Even in elementary teachingwe are bound to make use, even if only implicitly, ofquantum concepts. In a sense the expression 'quantumchemistry' is just a synonym for 'modern chemistry1.

Until the quantum theory was introduced it had beentaken for granted that energy, including radiation, iscontinuous; in other words, one could think of an atomor molecule as having any amount of energy. The essenceof the quantum theory is that this is not the case;instead energy comes in small packets, or quanta (fromthe Latin quantum, how much?). The quanta are small onan atomic scale, and on a macroscale they are negligible;in driving one's car one does not have to worry aboutquantum transitions. In view of this it is amusing tofind that journalists, politicians and others are nowtalking about 'quantum leaps', which they seem to thinkare very large.

The quantum theory originated in 1900, and littleattention was paid to it for a number of years. Thereason was that it first emerged as an ad hoc explanationfor the behaviour of black-body radiation, which hasnever been a problem of outstanding interest amongscientists. X-rays had been discovered in 1895,radioactivity in I896, the electron in 1897, and radiumin I898, and it was to such matters that the best mindsin physics and chemistry were directing their attention.Chemists in particular did not begin to think in termsof quanta until 1913 when Niels Bohr applied the quantum

theory to the problems of atomic structure, whichobviously had chemical implications. It is significantthat Planck did not receive his Nobel Prize until 1918,after Bohr had applied the quantum theory to atomicstructure.

Quanta of Energy

Max Planck (1858-19^7). who in 1900 was professor ofphysics at the University of Berlin, put forward histreatment to explain results that had just been obtainedin Berlin on the variation with frequency of the energyemitted by a 'black body', which is one that absorbs allthe radiation incident upon it. In October, 1900. someresults obtained by Heinrich Rubens (1865-1922) andF. Kurlbaun were communicated privately to Planck.These new results showed for the first time that the

energy emitted, when plotted against the frequency,passes through a maximum. At a meeting of the Berlin

Physical Society held on 25 October, 1900, Plancksuggested an empirical formula that would explain thisbehaviour. The very next day Rubens, who had workedthrough the night, told Planck that his formula fittedall the data within the experimental error.

Much encouraged, Planck worked hard during the next fewweeks to find a theoretical basis for his formula, andhis famous paper on this appeared in Oecember, 1900.Planck's main interest had always been in thermodynamics,and he based his treatment on the statistical

interpretation of entropy that had been developed byLudwig Boltzmann (18M-1906). As a mathematicalconvenience Boltzmann had often treated energy as comingin small packets, but at the end of his derivations healways allowed the size of the packets to go to zero.Planck used a similar treatment and considered packets,

or quanta, of energy of magnitude hv, where v is thefrequency and h is a constant now known as the Planckconstant. At the end, however, he failed to make h goto zero; on the contrary, he obtained a value for it bycomparing his formula with the experimental results.

For several years what little attention was paid toPlanck's treatment was largely unfavourable. JamesHopwood Jeans (1877_19^6), later famous as a cosmologistand a writer on science, published in 1905 a particularlysevere criticism of Planck's treatment, his main pointbeing that it was simply wrong to fail to make the energypackets equal to zero. Jeans was unable, however, tosuggest any better explanation for the behaviour ofblack-body radiation.

Memorial stone at Max Planck's grave in the citycemetery in Gottingen, West Germany.

January 1990/CHEM 13 NEWS 7

Quanta of Radiation

;nt of great importance was the publication in 1905oi—y paper by Albert Einstein (1879~1955). That yearwas an annus mirabilis for Einstein, since in the springof that year he published three papers each one of whichwas worthy of a Nobel Prize. The first was on thequantization of radiation, which (and not relativity)did actually get him his Nobel Prize in 1921; the secondwas on Brownian movement, and the third on the specialtheory of relativity. Einstein was then, and remainedfor four more years, a junior official in the patentoffice in Berne, Switzerland, having become a Swisscitizen in 1901.

Modern textbooks usually refer to Einstein's 1905 paperon quanta of radiation as his paper on the photoelectriceffect, and say that it was based on Planck's 1900paper. A perusal of the paper shows that this is notthe case. Einstein made only a passing reference toPlanck's work, and did not base his arguments on it atall. The photoelectric effect is mentioned in only oneof eight sections, which deal with other matters such asfluorescence and photoionization in gases.

Planck's idea had been that the energy in materialoscillators is quantized, and he did not consider thatradiation is quantized. Einstein, on the other hand,argued that radiation itself consists of particles whichhe called quanta and which the American chemist GilbertN. Lewis (1875-19^6) later called 'photons'. At firstEinstein did not believe that oscillator energy isquantized, but he soon changed his position and became astrong supporter of that idea. In fact, he was muchm enthusiastic about it than Planck, who wasc rvative by nature and had great difficulty indiscarding the principles of mechanics — now calledclassical mechanics — on which he had been brought up.

Specific Heats of Solids

In 1907 Einstein made another great contribution toquantum theory, with a paper entitled "Planck's theoryof radiation and the theory of specific heats." Theimportance of this paper was that up to that time thequantum theory seemed to be concerned only withradiation, but now it was seen to have broaderimp 1i cat ions.

In 1819 the French physicists Pierre Louis Dulong (1785-I838) and Alexis-Therese Petit (1791-1823) had statedtheir famous law: "Les atomes de tous les corps simplesont exactement la meme capacite pour la chaleur." Afterthe kinetic theory was developed it appeared that thespecific heat of a solid element should be 3 R —approximately 6 cal mo!-1 or 25 J mol"1. Measurementsat room temperature showed that this was usually thecase, but when measurements could be made at liquid-airtemperatures the values were much lower.

Einstein showed that the decrease in specific heats withdecreasing temperature is explained on the basis of aslight adaptation of Planck's formula for the energy ofa black body. The physical idea behind the explanationis that at lower temperatures there is not enough energyavailable for an oscillator to be able to change fromone quantum state to a higher state. Einstein did not/ with the specific heats of gases, but they can bev-^.ined in the same way (see CHEM 13 NEWS, March 1988,pages 12-13).

8 CHEM 13 NEWS/January 1990

Memorial stone for Walther Nernst in the city cemeteryin Gottingen, West Germany. "After Nernst's death[18 November 1941] the urn with his ashes had beentaken to Berlin, as had been his wish. He hadobjected to being buried in Gottingen because thatuniversity had become a citadel to Nazism. However,after [his wife, Emma's] death [4 May 1949] thedaughters decided to bring both their parents' ashesto Gottingen. Times had changed and access to Berlinhad become difficult. Nernst's ashes rest next to

those of Max Planck and Max von Laue, three names thatmark a unique period in the history of science."(Quoted from "The World of Walther Nernst", byK. Mendelssohn, 1973, The Macmillan Company, London.)

Photochemical Equivalence

At about the same time (1907) Johannes Stark (187^-1957),then at the University of Hannover, also becameenthusiastic about the quantum theory and began to applyit to a variety of problems. Stark was an excellentexperimentalist but was weak on theory, and some of hissuggestions involved serious errors in physics. Oneuseful idea that he put forward in 1907 was that thereis a one-to-one correspondence between a quantum ofabsorbed radiation and the formation of an excited

molecule that may become involved in a chemical reaction.This relationship was obtained in a more satisfactoryway by Einstein in 1912 in his paper entitled

"Thermodynamic basis of the law of photochemicalequivalence." The law provides a fundamental basis forall photochemical studies involving ordinary radiation.

A later important contribution to photochemistry was thesuggestion by Walther Nernst (188*4-19^1) in 1918 of amechanism for the photochemical reaction between hydrogenand chlorine. In this mechanism a single photon, denotedas /iv, brings about the dissociation of a chlorinemolecule into atoms:

hv 2 CI

The resulting atoms can then participate in reactionssuch as

CI + H2 •+ HC1 + H

H+CI2-HCI+CI

The law of photochemical equivalence applies to reactionsinduced by ordinary radiation, but when reactions areinduced by laser radiation, where there is a much higherintensity, more than one photon may be absorbed by amolecule.

Nfejgw

Stark's initial enthusiasm for the quantum theory wasshort-lived; by 1913 he had become an intemperateopponent of the theory. As he grew older he becameIncreasingly paranoid, and seemed to find it necessaryto oppose any view that was becoming generally accepted.He quarrelled violently with all his colleagues, andwhen the Nazis came into power he adopted a viciousanti-Semitism in which he also attacked non-Jewish

scientists — "white Jews" as he called them —who adopteda reasonable position. Planck, von Laue and Sommerfeldall came under his attack.

Bohr's Theory of the Atom

In 1913 there was published a paper of great importance,the treatment by Niels Bohr (1885-1962) of the structureof the hydrogen atom. Bohr introduced the boldassumption that an electron can remain in an orbit,circular or elliptical, without radiating, and thatradiation is absorbed or emitted only when an electronjumps from one orbit to another. Modern textbooks oftenstate that Bohr applied his quantum condition to theangular momentum of an orbiting electron, but in fact hefirst applied it to the energy and only later showedthat the treatment involving angular momentum ismathematically equivalent.

Probably the most impressive aspect of Bohr's paper washis interpretation of atomic spectra and it comes as asurprise to learn that this was an afterthought. Bohrhad done the first part of his work in Rutherford'slaboratory at the University of Manchester; he hadoriginally gone to work with J.J. Thomson at Cambridge,but Thomson did not appreciate Bohr's well-meantcriticism of his 'plum-cake' theory of the atom and hadencouraged him to go elsewhere. Bohr returned fromManchester to Copenhagen with his theory of energy levelsin the hydrogen atom, and there he was visited by thespectroscopist H.M. Hansen who asked him if he hadinterpreted the spectral lines. Bohr replied that hehad decided not to deal with them, as they would beimpossibly complicated. Hansen then brought Bohr'sattention to some spectral regularities of which Bohrsurprisingly had been unaware. These regularities canbe expressed by the general formula

1 2

which relates the wavenumber v, which is the reciprocalof the wavelength \, to two whole numbers Hi and H2; Ris known as the Rydberg constant. Bohr's theory had ledto the conclusion that the energy of an electron in itsorbit should be inversely proportional to the square ofa quantum number n. He at once realized that by takingthe difference between two energies he obtained anexpression of the form of this equation. The fact thathis theory led to a reasonable value of the Rydbergconstant gave it strong support. Bohr received the 1922Nobel Prize in physics for this work.

At about the same time that Bohr put forward his theory,Stark made a discovery of great significance. In hislaboratory at the Technische Hochschule in Aachen hefound that spectral lines, such as the lines in theBalmer series of hydrogen, are split by an electricfield into a number of component lines. His paperdealing with this appeared in November, 1913, just a fewonths after Bohr's paper was published. Thespectroscopist Emi1 Warburg (1846-1931) and Bohr himselfsoon showed that Bohr's theory could explain the mainfeatures of this splitting, although some of the detailsremained puzzling. Stark received the 1919 Nobel Prize

for this work, but he perversely used the results asthe basis of an attack on Bohr's theory and on thequantum theory in general!

It soon became clear that although Bohr's theory was agreat advance on any atomic theories that had gonebefore, it could by no means explain all the featuresof atomic structure and spectra. Besides not completelyexplaining the Stark effect, it could not explain at allthe Zeeman effect, which is the splitting of spectrallines by a magnetic field; this had been discovered in1896 by the Dutch physicist Pieter Zeeman (1865-19^3).Also, Bohr's theory was quite deficient in explainingthe spectra of atoms containing more than one electron.

Some improvement was achieved by the German physicistArnold Sommerfeld (1868-1951) who had succeededBoltzmann as professor at Munich. Sommerfeld suggesteda second quantum number, the azimuthal quantum number,£, associated with the eccentricity of an ellipticalorbit, and also a third quantum number, the magneticquantum number, m, which gave a partial explanation forthe Zeeman effect. However, there was more Zeemansplitting than could be accounted for in this way andthis anomalous Zeeman effect had to be explained interms of a spin quantum number, 6, relating to the spinof the electron on its own axis. The Austrian physicistWolfgang Pauli (1900-1958) made a great contribution toatomic theory by his exclusion principle, according towhich no two electrons in an atom can have the same setof quantum numbers n, t, m, and 4.

Attempts to provide a complete theory of atomic structureby modifying Bohr's original theory did not in the endprove satisfactory, and there was need for a radicallynew approach. Bohr himself was one of the leaders inthis endeavour, which led in the 1920s to the new quantummechanics, the subject of Part 2 of this article.

Suggested Reading

The old quantum theory is treated in detail in M. Jammer,The Conceptual Development of Quantum Mechanics, McGraw-Hi 11, New York, 1966. Other excellent sources areG. Gamow, Thirty Years that Shook Physics, Doubleday,New York, 1966, Dover Publications, New York, 1985; andsome articles by Martin J. Klein: "Max Planck and thebeginnings of the quantum theory", Archive for Historyof Exact Sciences, J_, 1»59-|»79 (1961); "Planck, entropyand quanta", in The~~Natural Philosopher, Blaisdell, NewYork, Volume 1, pages 83-IO8 (1963); and "Einstein'sfirst paper on quanta", ibid, Volume 2, pages 59-86(1963).

For discussions of Bohr's theory of the atom seeM. Jammer, loc. cit.. Chapter 2 (pages 62-88) and BiancaL. Haendler, "Presenting the Bohr atom", Journal ofChemical Education, 59, 372-376 (1982).

ELEMENT NAMES/SYMBOLS

Some time ago we noted that the name, astatine, iscomposed entirely of the symbols of other elements -AsTaTiNe. We asked for other examples, and George Grossof Union High School, Union NJ 07083, submitted these.

Carbon - CaRbON

Copper - CoPPEr

- COPPEr

Arsenic - ArSeNiC

- ArSeNIC

Tin TiN

Bismuth - BiSmUTh

January 1990/CHEM 13 NEWS 9

THE STORY OF QUANTUM CHEMISTRY.PART 2: THE NEW QUANTUM MECHANICSKeJXh J. LaldleAVzpaAtmo.nt o£ Chml&tAyUnive.>uity ol OttawaOttawa ON KIN 6N5

(This is the second of three articles on quantumchemistry* The first instalment appeared in our Januaryissue, pages 7-9; part 3 will appear in March.)

The inability of the Bohr theory to deal with complexatoms and with molecules was the main reason for theemergence of quantum mechanics. The first paper on thissubject, by Max Born, appeared in 1924, but it was in1925 that there was a tremendous surge of activity, byHeisenberg, Dirac and Schrodinger. It was Heisenberg'spaper, completed in July, 1925, which inspired thealternative treatments of Dirac and Schrodinger whichwere formulated later in that same year. Eventually allof the treatments, although very different in theirapproach, were shown by Schrodinger to be mathematicallyequivalent. Today that of Schrodinger is usuallypreferred, as providing the greatest physical insight.

Born and Heisenberg

In 1921 Max Born (1882-1970) became professor of physicsat the University of Gottingen, and in 1924 he publishedan important paper entitled "Uber Quantenmechanik",which was the first occasion that the expression"quantum mechanics" was employed. In this paper Bornproposed modifications to the equations of classicalmechanics that would lead to quantum mechanics.

In 1925 one of Born's research assistants at Gottingenwas the 24-year-old Werner Heisenberg (1901-1976), whohad obtained his doctorate two years earlier. Like someothers who later had careers of great distinction healmost failed to obtain his degree, as he had neglectedhis laboratory work and was unable to answer any of thequestions put to him in his oral examination by WilhelmWien (1864-1928), the professor of experimental physics.Wien wanted to fail him, but was persuaded by Sommerfeldto agree to giving him third-class standing in his oralexamination. It is amusing to note that one of thetopics that caused Heisenberg trouble was the resolvingpower of a microscope, and he still got this wrong whenhe proposed his uncertainty principle in 1927.

In the spring of 1925 Heisenberg was suffering from sucha severe attack of hay fever that he obtained leave tostay for some time on Heligoland, a rocky island freefrom vegetation. While there, he developed amodification of Born's treatment in which physicalquantities were represented by complex numbers whichunderwent a non-commutative multiplication rule; thatis to say, the order of multiplication is important, sothat pq is not the same as qp. Heisenberg sent hismanuscript to Born, who at once saw that it was importantand submitted it for publication in Heisenberg's nameonly.

Born did not at once appreciate the significance ofHeisenberg's complex numbers but "after eight days ofconcentrated thinking and testing" saw that they hadthe properties of matrices. At the time matrices wereused by some mathematicians but were hardly ever usedby physicists; Born recalled them from his student days20 years earlier. Heisenberg later admitted that heknew nothing of matrices; in developing his matrix

mechanics he had re-invented them without knowing thathe was doing so.

Born later developed Heisenberg's treatment in terms ofmatrices, partly in collaboration with the Frenchphysicist Pasqual Jordan. This association came aboutin an interesting way. In the summer of 1925 Borntravelled by train from Gottingen to Hannover anddiscussed with a colleague his struggles with matrixalgebra. On arrival at Hannover another occupant ofthe compartment introduced himself as Jordan, admittedto overhearing the conversation, mentioned that he hadmade a special study of matrices, and offered his help.

Another contribution of Heisenberg, made in 1927, was ofvery great importance. This was his uncertaintyprinciple, or principle of indeterminacy. If q is theposition of a particle, p its momentum and Aq and Ap arethe uncertainties with which they can be measured, theuncertainty principle states that

A?Ap -{L

where h is the Planck constant. In other words, if bysome means one property is determined accurately,another can only be measured with a high degree ofuncertainty. Heisenberg arrived at the principle byconsidering an imaginary experiment in which one uses abeam of light to determine the position and momentum ofan electron. Because the light disturbs the electron(Compton scattering) there is a limitation on theaccuracies. The resolving power of a microscope isinvolved in the argument and Heisenberg's treatment isincorrect; Bohr had to straighten the matter out for him.

Heisenberg was awarded the 1933 Nobel prize in physics.It was always a matter of regret to him that Born hadno share in this honour. The award of the prize jointlyto Born and Heisenberg would indeed have been moreappropriate. Not only had Born initiated the work onquantum mechanics: it was he who clarified Heisenberg'sobscure mathematics by relating it to matrix mechanics.A less generous man would have delayed the submissionof Heisenberg's paper until a joint paper of broadersignificance could b.e prepared. Born did receive aNobel prize much later, in 1954, for his work on thequantum mechanics of collision processes.

Heisenberg had an open and attractive personality, withnone of the eccentricities displayed by Dirac andSchrodinger, and had a wide popularity. During the Naziera he remained in Germany and directed the atomic energyprogram, which proved unsuccessful. He by no meansapproved of Nazi policies and for.this he suffered fromthe vicious attacks of Stark and Lenard who were powerfuland fanatical Nazis. Born had to leave Germany, andbecame professor at the University of Edinburgh in 1936;he returned to Germany after his retirement in 1954.

Dirac

Shortly after the appearance of Heisenberg's first paperon quantum mechanics, another important contribution wasmade by Paul Adrian Maurice Dirac (1902-1984). Dirac

February 1990/CHEM 13 NEWS 7

was trained as an electrical engineer at the Universityof Bristol, and being unable to obtain employment in1923 secured a research scholarship to work for his Ph.D.degree at Cambridge. In July, 1925, Heisenberg lecturedaj Cambridge and Dirac was present at the lecture but

said that he had paid little attention to it. In•V^ .mhor ne saw the proofs of Heisenberg's firstpaper, and initially saw nothing useful in it. He soonchanged his mind and quickly produced an alternativeformulation, his first paper on quantum mechanicsappearing late in 1925. The essential feature of hisquantum mechanics, which is equivalent to Heisenberg'sformulation but more comprehensible, is that it is basedon Hamilton's classical mechanics, the basic equationsof which are transformed into quantum mechanics by theuse of a number of postulates.

In a later paper Dirac combined his system with thetheory of relativity, with far-reaching results. Hisrclativistic quantum mechanics was able to deduce fromtheory alone that there are two possible quantum numbersfor electron spin, and that there should exist anelementary particle having the mass of the electron buta positive charge. This particle, the positron, wasdiscovered in 1932 by the American physicist Carl DavidAnderson (b. 1905).

Dirac, who shared the 1933 Nobel Prize in physics withSchrodinger, had many amusing eccentricities. Althoughbilingual in English and French (his father was Swiss),he made little use of either language in oralcommunication, although he wrote fluently. On oneoccasion he shared a cabin on a boat with a Frenchman

who knew little English, but what communication they hadwas in English; when the Frenchman later discoveredDirac's knowledge of French and indignantly asked whyDirac had not revealed it, the reply was "You didn'tas1' me." Dirac's favourite word was "Yes" but he

ionally explained that this did not indicatea^w-jment, merely that one should go on talking.

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Victor de Broglie

Before Heisenberg and Dirac made their pioneeringcontributions, the idea had been put forward that, justas radiation has particle properties, particles mighthave wave properties. This idea was first clearlyexpressed by Louis Victor, Prince" de Broglie (1892-1987]in a series of papers that appeared in 1923 and in hisdoctoral dissertation presented in 1924 to theUniversite de Paris. Victor de Broglie had firstintended to become a civil servant and in 1910 obtaineda licenciate in history. Later he became interested inphysics and philosophy, and in 1913 obtained hislicenciate in the physical sciences. He then workedtowards his doctorate, specializing in fundamentalproblems of space, time and atomic structure.

He was particularly impressed by the demonstration in1923 by the American physicist Arthur Holly Compton(1892-1962) that X-rays can be scattered by electrons.This further experimental confirmation of Einstein'shypothesis that radiation can exhibit particle propertiessuggested to de Broglie the converse hypothesis, thatelectrons can have wave properties. Using an argumentbased on Einstein's special theory of relativity hededuced that a particle of mass m and velocity u wi11have a wavelength of

x-i.mv

where h is the Planck constant. He further pointed outthat Bohr's atomic theory can be visualized in terms ofwave patterns fitted into the circumference of anelectronic orbi t.

The reaction of de Broglie's doctoral examiners to theseideas was by no means favourable. One of them wasJean Perrin (1870-1942), another Paul Langevin (1872-1946), and both were skeptical. However, at Langevin'ssuggestion, de Broglie sent his thesis to Einstein, whoreturned an enthusiastic endorsement which included the

phrase, "He has lifted a corner of the great veil." Asa result of this intervention de Broglie did pass, butthe examiners remained doubtful. Present at the oral

examination was the American physicist Robertvan de Graaff (1901-1967), then a student and laterinventor of the electrostatic generator, and he remarked,"Never has so much gone over the heads of so many."

Most other physicists, including Born, Heisenberg andDebye, were also initially skeptical of de Broglie'ssuggestion. Soon, however, confirmation came from theelectron-diffraction experiments of Davisson and Germerat the American Telephone and Telegraph Companylaboratories, and by G.P. Thomson and Reid at theUniversity of Aberdeen. This work was published in 1927,and ten years later Davisson and Thomson shared the Nobelprize in physics. It has been commented that J.J.Thomson won the 1906 Nobel prize for showing that theelectron is a particle, while his son gained a 1937 Nobelprize for showing that it is a wave.

Schrodinger

The quantum mechanics of Erwin Schrodinger (1887-1961)was directly inspired by de Broglie's contribution.

' 'Prince' was a courtesy title conferred upon ade Broglie ancestor by Francis I of the Holy Roman Empireand to which all descendants were entitled. After hiselder brother Maurice died in I960, Victor inherited theFrench title of Due, which took precedence over the titleof Prince.