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Studies in History and Philosophy of Modern Physics 41 (2010) 178–182

Contents lists available at ScienceDirect

Studies in History and Philosophyof Modern Physics

1355-21

doi:10.1

E-m1 Re

journal homepage: www.elsevier.com/locate/shpsb

On the alleged non-existence of orbitals

Peter Mulder 1

Kardinaal Mercierplein 2, P.O. Box 3200, B-3000 Leuven, Belgium

a r t i c l e i n f o

Article history:

Received 26 January 2009

Received in revised form

15 February 2010

Accepted 23 February 2010

Keywords:

Philosophy of chemistry

Quantum mechanics

Orbitals

Electron correlation

Pauli principle

98/$ - see front matter & 2010 Elsevier Ltd. A

016/j.shpsb.2010.02.003

ail address: peter.mulder@hiw.kuleuven.be

search Foundation—Flanders.

a b s t r a c t

I argue that the claim made by Scerri that in many-electron atoms, orbitals do not exist according to

quantum mechanics, is incorrect, for it relies on the view that orbitals are entities. Orbitals are states,

not entities, and their use in describing many-electron atoms should be seen as an approximation. The

writings by Scerri and others on the issue of realism that are based on the claim therefore lead astray. I

furthermore disentangle two issues that Scerri discusses in arguing for his claim: that of electron

correlation and that of the Pauli principle. Finally, I point out that more generally there is a

misconception in chemistry of what quantum states are.

& 2010 Elsevier Ltd. All rights reserved.

When citing this paper, please use the full journal title Studies in History and Philosophy of Modern Physics

1. The electron configuration model

The use of orbitals in chemistry is based on the electron-configuration model, in which an atom’s electrons are taken tohave either of the eigenfunctions of the electron in the hydrogenatom as their states. Let us spell this out in some detail.

The Hamiltonian of the hydrogen atom is given by

H ¼�1

rð1Þ

in which r denotes the distance between the nucleus and theelectron. Its eigenfunctions are the electron’s possible time-independent, or stationary, states. These eigenfunctions c of (1)are called hydrogenic orbitals. The term ‘orbital’ applies to one-electron wave functions in general, of which hydrogenic orbitalsare merely a subset, but because of the preponderance ofhydrogenic orbitals in chemistry, they are often simply referredto as ‘orbitals’. In this section I will only be concerned withhydrogenic orbitals, and in the next section extend the discussionto one-electron functions in general.

Hydrogenic orbitals differ by their quantum numbers, whichstand for conserved quantities:

�

the principal quantum number n (1,2,3,y), which correspondswith the orbital’s energy;

ll rights reserved.

�

the azimuthal quantum number l (0,1,y,n�1), correspondingwith orbital momentum; � the magnetic quantum number ml (� l,y,l), representing

the projection of the orbital momentum along a certainaxis.

Rather than by enumerating their quantum numbers, hydrogenicorbitals are named as follows. The principal quantum numbern is represented by its numerical value. The azimuthalquantum number l is denoted by a letter. For the purposes ofthis paper it suffices to mention s, which corresponds withl¼0, and p, corresponding with l¼1. Thus one has thehydrogenic orbitals 1s, 2s, 2p, 3s, y . For l¼1, there are threepossible values for ml, which are represented by subscriptsaccording to the axes they correspond with, giving 2px, 2py

and 2pz.A fourth quantum number, the spin quantum number ms (�1/

2,1/2), which stands for spin, or intrinsic orbital momentum, isnot included in the orbital concept, although spin plays animportant role in the explanation of electron configuration, towhich I turn now.

In chemistry, the electron configuration of an atom gives the‘arrangement’ of its electrons. This is to be understood as follows:each electron in the atom is taken to have a certain hydrogenicorbital as its quantum state, and exactly which of these are‘occupied’ by electrons in this sense is given by the electronconfiguration. For example, the electron configuration of lithium

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P. Mulder / Studies in History and Philosophy of Modern Physics 41 (2010) 178–182 179

is 1s22s1, meaning that two electrons occupy the 1s orbital andone the 2s orbital.2

In conceiving hydrogenic orbitals as possible electron states inmany-electron atoms, an approximation is made. The Hamilto-nian of an atom with nuclear charge Z and Z electrons is given by

H ¼�XZ

i ¼ 1

Z

riþXZ�1

i ¼ 1

XZ

j ¼ iþ1

1

jri�rjjð2Þ

in which the first term signifies the Coulomb attractions betweenthe nucleus and the electrons and the second term the Coulombrepulsions among the electrons. In the electron configurationmodel, the second term is neglected. In the next section I willdiscuss this in more detail. First consider how the electronconfigurations of specific atoms are determined. This is done byprogressively ‘filling up’ orbitals with electrons, hereby observingthe following three principles:

�

stat

elec

(n¼

the Pauli exclusion principle: no two electrons can have thesame four quantum numbers, or, put differently, only twoelectrons can occupy the same orbital, and if they do, theyhave opposite spin values;

� the Aufbau principle: orbitals with lower n+ l, and for given

n+ l, those with lower n are filled first3;

� Hund’s rule: for given n, every orbital is singly occupied with

one electron before any one orbital is doubly occupied, and allelectrons in singly occupied orbitals have the same spin value.

Let us illustrate this with an example. Carbon has six electrons.First the hydrogenic orbital with lowest n+ l (n¼1, l¼0), 1s, isfilled with two electrons, which have opposite spins. Thehydrogenic orbital with n+ l¼2 (n¼2, l¼0), 2s, is filled second,again with two electrons with opposite spins. There are fourhydrogenic orbitals with n+ l¼3, namely 3s (n¼3, l¼0), 2px, 2py

and 2pz (which have n¼2, l¼1 and different ml). The 2p orbitalshave lower n than the former and are therefore filled first. The tworemaining electrons each occupy a different 2p orbital, and theyhave parallel spins, in accord with Hund’s rule. This electronconfiguration is written as 1s22s22p2.

2. The limitations of the electron configuration model

The electron configuration model neglects interelectronicinteractions and is therefore merely approximate. In many-electron atoms, electrons do not have hydrogenic orbitals as theirstates. But what are their states? Consider again the Hamiltonianof a many-electron atom:

H ¼�XZ

i ¼ 1

Z

riþXZ�1

i ¼ 1

XZ

j ¼ iþ1

1

rijð3Þ

in which rij ¼ jri�rjj, the distance between electrons i and j. For thesimplest many-electron atom, helium, it reads

H ¼�2

r1�

2

r2þ

1

r12: ð4Þ

Its eigenfunctions are two-electron wave functions that cannot bedecomposed into two one-electron wave functions. Hence theelectrons are not themselves in individual stationary states. Byextension, the eigenfunctions of an n-electron atom are functions

2 To be more precise, this is the electron configuration of lithium in its ground

e. It is different in excited states. Throughout this paper when I speak of ‘the’

tron configuration of an atom it refers to the atom’s ground state.3 There are some exceptions to this rule. For instance, in copper, the 3d orbital

3, l¼2) is full while the 4s orbital (n¼4, l¼0) contains only one electron.

of the coordinates of all n electrons and cannot be decomposedinto n one-electron functions. Although the Hamiltonian ofhelium cannot be analytically solved, let alone those of largeratoms, this can be seen by considering the operators correspond-ing with quantum numbers of individual electrons. As Scerri(2001, p. S80) has pointed out, they do not commute with theatom’s Hamiltonian, and therefore the electrons are not them-selves in stationary states.

It should be emphasized that the impossibility of writing thestate of a many-electron atom as a product of one-electronfunctions is not merely a practical one; it is impossible inprinciple. Essentially, this is because the electrons are correlated

because of their interactions, meaning that the probabilitydistributions of their variable quantities (observables) such astheir positions are not independent. To give an example, if, in thehelium atom, P(x,y) is the joint probability distribution of findingthe one electron at x and the other at y, and P(x) and P(y) are theirmarginal probability distributions, then Pðx,yÞaPðxÞPðyÞ.

To give an example of a many-electron wave function, considerhelium again. As its Hamiltonian cannot be analytically solved,one needs to approximate it numerically. One such way is throughso-called Hylleraas functions

c¼ e�aðr1þ r2Þf ðr1,r2,r12Þ ð5Þ

in which f is a power series of r1, r2 and r12.So far when considering orbitals I have been concerned with

hydrogenic orbitals specifically. Wave functions that are productsof these orbitals are the result of completely neglecting theinterelectronic repulsion term in the Hamiltonian (3). The aboveargument, however, applies to one-electron functions—orbital-s—in general. States that are products of one-electron functionsdifferent from hydrogenic orbitals feature for instance in theHartree–Fock method in quantum chemistry. In the Hartree–Fockmethod, the interelectronic interactions are dealt with in anaverage way by replacing the individual interactions by the meanfield of all electrons. While the resulting wave functions are inbetter agreement with experiment than products of hydrogenicorbitals, they still fail to account for electron correlation, whichresults from the instantaneous rather than the average interactionbetween electrons.4 I will therefore henceforth consider orbitalsin general rather than just hydrogenic orbitals.

Before turning to the philosophical consequences of the factthat the wave function of a many-electron system cannot bewritten as a product of orbitals, consider a theorem of quantummechanics which makes for another shortcoming of the electronconfiguration model: the Pauli principle. In Section 2 I introducedthe Pauli exclusion principle, which is often considered aderivative of the more general Pauli principle. It has two readings:no two electrons can have the same four quantum numbers, andonly two electrons can occupy the same orbital, and if they do,they have opposite spin values. Before turning to the more generalPauli principle, consider what the Pauli exclusion principle resultsin for the simplest many-electron atom, helium. On the electronconfiguration model, its ground state is

1sð1Þ1sð2Þ: ð6Þ

To be more precise, the above state is only the spatial part ofhelium’s state. The total quantum state of a system of electronsalso contains a spin part. The spin value of an electron(ms¼{1/2,�1/2}) corresponds with a certain spin state (either m,corresponding with ms¼1/2, or k, corresponding with ms¼�1/2).

Now the Pauli exclusion principle states that no two electronscan have the same four quantum numbers, or, alternatively, that

4 In quantum chemistry, there are several post-Hartree–Fock methods to deal

with electron correlation, such as configuration interaction.

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6 For the ground state of helium, Hylleraas functions are symmetric like (6),

P. Mulder / Studies in History and Philosophy of Modern Physics 41 (2010) 178–182180

only two electrons can occupy the same orbital, and if they do,they have opposite spin values. Hence, according to this principle,in the current example one electron has spin state m and the otherk, resulting in the state

1sð1Þ1sð2Þmð1Þkð2Þ: ð7Þ

To recapitulate, when we ‘translate’ the electron configuration ofhelium to a quantum state, we obtain the above state. Theproblem is that this state is not physically possible. This isbecause the Pauli principle, of which the Pauli exclusion principleis merely a derivative, dictates that the wave function of a systemof two or more electrons be antisymmetric, meaning that itchanges sign under exchange of the electrons. For helium, thisimplies that its state is given by

12

ffiffiffi2p

1sð1Þ1sð2Þ½mð1Þkð2Þ�kð1Þmð2Þ�: ð8Þ

This is a so-called entangled state, in which one cannot say thatelectron 1 has spin ‘up’ and electron 2 spin ‘down’ or vice versa.The electrons do not have individual spin states, only the whole ofthe two electrons does. As they furthermore share the samespatial state, there is no way to physically distinguish betweenthem and hence they are called indistinguishable.

Above I gave two definitions of the Pauli exclusion principle,the first being ‘no two electrons can have the same four quantumnumbers’ and the second ‘only two electrons can occupy the sameorbital, and if they do, they have opposite spin values’. Bothdefinitions are frequently found in textbooks. I also mentionedthat the Pauli exclusion principle is a derivative of the moregeneral Pauli principle. For the first definition of the exclusionprinciple, this is indeed the case. In an antisymmetric state,electrons do not have the same four quantum numbers as theylack a well-defined value of the spin quantum number ms.However, while at first sight the second definition of the exclusionprinciple seems simply a consequence of the first, the example ofhelium shows that it is false. The inference from the first to thesecond definition relies on the assumption that electrons do haveindividual quantum numbers in many-electron systems, which assaid is not true of ms.

The implications of the Pauli principle are even more strikingwhen the antisymmetry requirement cannot be met by merely anappropriate spin part of the wave function and the spatial part isentangled as well, which is the case for all atoms containing threeor more electrons and for some states of helium. The simplestexample is the first excited state of helium, in which, in electronconfiguration terms, one electron occupies the 1s orbital and theother the 2s orbital.5 This state is given by

12

ffiffiffi2p½1sð1Þ2sð2Þ�1sð2Þ2sð1Þ�: ð9Þ

The orbitals being entangled means that it is meaningless to saythat electron 1 is in 1s and electron 2 in 2s or vice versa. Thepredicate ‘is in orbital x’ does not apply to electrons in anentangled state. The Pauli principle thus makes for a second wayin which the electron configuration model is incorrect.

It should be emphasized that the two issues discussed sofar—electrons not having individual states due to electroncorrelation and their states being antisymmetric because of thePauli principle—are independent, although Scerri suggests other-wise when he writes that

since [individual quantum numbers] can no longer be placedon individual electrons they must be regarded as indistin-guishable. The Pauli Exclusion Principle remains valid, butmust be re-expressed in such a way as to uphold the non-individuality of electrons and this is done by stating that the

5 From this point on, I will leave spin out of the discussion again.

atomic wave function is antisymmetric with respect to theinterchange of any two electrons. (Scerri, 2001, p. S79)

Both issues arise in many-electron systems, but both their originsand their consequences are different. The Pauli principle is atheorem about permissible quantum states and has nothing to dowith interelectronic interactions. The issue of electron correlation,on the other hand, is a result of interactions between electronsand leads to states like the Hylleraas functions (5). These statesare not antisymmetric, and need not be if it were not for the Pauliprinciple. Furthermore, electron correlation only concerns thespatial part of the system’s wave function. The electrons’ spin isunaffected by it. Therefore—again, if it were not for the Pauliprinciple—electrons are allowed to have individual spins andhence spin quantum numbers, which makes for a way tophysically distinguish between them.6

3. The reality of orbitals

On the basis of the fact that the wave function of a many-electron system cannot be written as a product of orbitals, Scerriargues:

Individual electrons in a many-electron atom are not ofthemselves in stationary states whereas the atom as a wholedoes possess stationary states. This [y] shows definitively thatthe orbital model is an approximation in many-electronsystems. It also requires that the scientific term ‘orbital’ isstrictly non-referring with the exception of when it applies tothe hydrogen atom or other one-electron systems. (Scerri,2001, p. S79)

Let it be noted that Scerri’s speaking of orbitals not referring ismeant in an ontological sense: in many-electron atoms, orbitalsdo not exist:

Quantum mechanics tells us that orbitals and configurationsdo not strictly exist, that is to say, they [do not] refer to realentities in the natural world. (Scerri, 2000b, p. 420)

Before discussing Scerri’s argument, it should be mentioned thatmany physicists and philosophers have an antirealist view ofquantum mechanics altogether. In their view, wave functions aremere mathematical constructs that have no physical meaning.However, for the discussion at hand to make sense a realism vis- �a-

vis wave functions is presupposed. Scerri’s assertion aboutorbitals not referring is based not on an instrumentalism withrespect to quantum mechanics, but rather on the specifics ofmany-electron atoms.

Consider, then, his argument. Scerri assumes that orbitals areentities. This is a category mistake. Orbitals are states, not entities.In physics, the state of a system can be described as its conditionat a certain point in time. In classical mechanics, it is made upfrom the basic variable properties—positions and momenta—ofthe particles comprising the system. In quantum mechanics, therelation between a system’s state and variable properties of itsconstituents is not so straightforward; the state merely gives theprobability density of values of observables upon measurement.Nevertheless, the essence of the concept of a physical state—thecondition of a system at a certain time—remains. Specifically,orbitals are—purported—states of electrons. Note the last twowords: of electrons. Orbitals are predicated of electrons and hence

and antisymmetry can be enforced by an antisymmetric spin part of the total

(spatial plus spin) state. For excited states, the spatial part must be antisymme-

trized as well.

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Fig. 1. Graphical representation of the probability density regions corresponding with the 2p orbitals.

P. Mulder / Studies in History and Philosophy of Modern Physics 41 (2010) 178–182 181

do not have an independent existence like entities do. When oneuses orbitals to describe a many-electron atom, one thereforedoes not posit the existence of certain entities.

Because orbitals are not entities, it simply makes no sense tospeak of them as either existing or not existing, or as eitherreferring or not, just like in a classical-mechanical context thestatement ‘a velocity v exists’ is meaningless. Rather, whendescribing an atom by a wave function that is a product oforbitals, one predicates a state of the system that is not the exactstate but instead a simplified one. This realization has importantrepercussions for the issue of realism which Scerri and othershave taken up in the light of the claim that orbitals do not exist.Consider the following dilemma posed by Crasnow:

Given this state of affairs [orbitals not existing], we are facedwith several options, none of which seems particularlyattractive. One would be to acknowledge that the orbitalmodel is strictly false. But this leaves something of a puzzleabout its success and the centrality of the atomic orbital modelin both the theory and practice of chemistry. Since one of theprimary arguments that realists have traditionally used is thesuccess of science, the puzzle deepens. The alternatives may beeven worse. We could claim that both quantum physics andchemistry should be treated realistically but that there is aradical disunity of these areas of science (metaphysicaldisunity). If we take the other leg of the traditional approachand move to antirealism, we give up truth of the theories andthere is no conflict, but then we are left with other questions.Why is there so much convergence between chemistry andphysics if they are not both describing one world? (Crasnow,2000, p. 129)

When one drops the premise that orbitals do not exist, thedilemma dissolves. There is no radical disunity between chem-istry and physics. Similarly, Scerri’s attempts to defend a realisticstance towards orbitals despite their purported non-existenceaccording to physics—by taking up an ‘intermediary position’Scerri (2000a)—are uncalled for.

Conceiving orbitals as featuring in wave functions that stand infor more complex wave functions also allows the question howchemistry can be successful if it is based on a model that is strictlyfalse to be readily answered. While these wave functions are notexact, they are approximations to the true wave function of thesystem. While they fail to account for electron correlation, they doquite well in other respects, which explains their widespreaduse.7

7 For a discussion of the merits and shortcomings of orbitals in describing

many-electron atoms, see Cohen and Bustard (1966).

That orbital wave functions are to be seen as approximationshas already been defended by Jenkins (2003) and by Ostrovsky(2005), although neither has pointed out that viewing orbitals asnon-referring entities is meaningless. Thus Ostrovsky writes that

the orbitals in a multi-electron atom are approximations. Inthis regard, the term ‘orbitals’ is [y] ‘strictly non-referring’,although that terminology is hardly appropriate, becauseit underestimates the physically justified approximation.(Ostrovsky, 2005, p. 110)

I stress again that the ‘strictly non-referring’ is not just hardlyappropriate, but simply meaningless. The use of orbitals is anapproximation, and no deeper philosophical issues arise.

4. The conception of orbitals in chemistry

As I argued above, orbitals are not entities. Consider, however,the graphical representation of the 2p orbitals in Fig. 1.

Of these representations Scerri writes:

[y] orbitals are regularly discussed and pictured in gloriouscolor diagrams by chemists as though they were real andconcrete entities (2000a, p. 52).

Indeed such pictures suggest that orbitals are real entities.However, while Scerri charges the ‘real’, the falsehood of thesuggestion lies elsewhere. What is represented in Fig. 1 is not ‘the2p orbitals’, but rather the following. Among other things, anorbital gives a probability density for the electron to be foundupon measurement, which is given by the square of the wavefunction: r¼ jcj2. This probability density—called electrondensity—in turn gives a region where the electron is very likelyto be found,8 and it is this region that is graphically shown.According to the Oxford Dictionary of Chemistry, both the wavefunction and the region of high electron density correspondingwith it can be called ‘orbital’ (Daintith, 2004). This twofold use ofthe term ‘orbitals’, however, is highly misleading, for it conveysthe wrong idea of what quantum states are. Orbitals are wavefunctions and emphatically not a kind of balloon-like entities assuggested by pictures like Fig. 1. This is not to say that suchpictures are not useful, but one should be careful in interpretingthem.

There is a more general misconception by chemists and indeedmany others of the relation between electrons and their quantumstates, regardless of whether the latter are conceived as entities ornot. The misconception is that electrons are taken to havepositions within orbitals. Consider for instance what Nelson

8 Typically, ‘very likely’ means a probability of 90–95%.

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P. Mulder / Studies in History and Philosophy of Modern Physics 41 (2010) 178–182182

(1990) writes about an electron in state n with correspondingprobability density r:

The latter represents the distribution that would be obtained ifa large number of measurements were made of the position ofthe particle in state n, the particle being restored to this stateafter each measurement. This suggests that, if a series ofmeasurements could be made of x without disturbing themotion of the particle, the resulting distribution would be r.The latter would then reflect the motion of the particle in thesame way in which the density of the image on a long-exposure photograph reflects the motion of a macroscopicobject. (Nelson, 1990, p. 643, emphasis original)

Nelson tacitly assumes that electrons move—have positions thatvary through time. But there is nothing in the formalism ofquantum mechanics that suggests this. The recourse to thehypothetical scenario in which measurements do not lead tostate collapse is misleading, for state collapse is an essentialaxiom of quantum mechanics and hence this scenario wouldcome down to one in which quantum mechanics is notempirically adequate. The answer to the question that appearsin the title of Nelson’s paper—‘how do electrons get acrossnodes?’.9—is simply: they don’t, for they do not move throughspace.

9 A node is a region of an orbital where the probability of finding an electron is

zero—for instance the origin in the 2p orbitals drawn in Fig. 1.

The real issue for chemists, then, is how to come to terms withthe nature of quantum states in general and their relation to theelectrons they are predicated of. This is a notoriously difficult task,and I do not have a ready solution to it. What I do hope to havemade clear, however, is that the use of a specific type of quantumstates, viz. orbitals, is not philosophically problematic.

References

Cohen, I., & Bustard, T. (1966). Atomic orbitals. Journal of Chemical Education, 43(4),289–299.

Crasnow, S. (2000). How natural can ontology be?. Philosophy of Science 67(1),114–132.

Daintith, J. (2004). Oxford dictionary of chemistry. New York: Oxford UniversityPress.

Jenkins, Z. (2003). Do you need to believe in orbitals to use them?: Realism and theautonomy of chemistry. Philosophy of Science, 70(5), 1052–1062.

Nelson, P. (1990). How do electrons get across nodes?. Journal of ChemicalEducation 67(8), 643–647.

Ostrovsky, V. N. (2005). Towards a philosophy of approximations in the ‘exact’sciences. Hyle, 11(2), 101–126.

Scerri, E. (2000a). Naive realism, reduction and the intermediate position. In N.Bhushan, & S. Rosenfeld (Eds.), Of minds and molecules (pp. 51–72). New York:Oxford University Press; 2000.

Scerri, E. (2000b). The failure of reduction and how to resist the disunity of sciencein chemical education. Science and Education, 9(5), 405–425.

Scerri, E. (2001). The recently claimed observation of atomic orbitals and somerelated philosophical issues. Philosophy of Science, 68(3), S76–S88.