TheelectronicstructureofatomsThe chemical reactivity of any species be it atom,molecule or ion,is due to its electronic struc-ture.Forexample,fluorineisincrediblyreactiveformingcompoundswithallbutthreeoftheelementswhereasthefluorideion,withoneextraelectronmakingitisoelectronicwithneon,ismuch less reactive and has a completely different chemistryOur goal inthispartof the courseistogain an understanding of theelectronic structure ofmolecules and to seehowthishelps in predictingtheshapes ofmolecules and the outcome ofchemical reactions. The simplest way to predict the energy levels that are present in a moleculeis to combine the energy levels of the constituent atoms.Hence we first look at the electronicstructureoftheatom.Energy levels and photoelectron spectroscopyInformation about theelectronic structureof atomscanbegainedfroma techniquecalledphotoelectron spectroscopy.In this technique, the sample is bombarded with high energyX-rayphotons ofknownenergy whichknockoutelectronsfromthesample.Byanalyzingtheenergies of the emitted electrons,it is possibleto deduce theenergy they possessed in the atom.Shownbelowarethephotoelectronspectraofhelium,neonandargon.He15040302010Nehardto2s15tens ofevremove2p1I5040208808708603010Ar3p2p2stens ofev1shundreds3s3200eVofev33034032026025024040302010ionization energy/eVmostenergetic'valence'electrons-allprettysimilarinenergy1
The electronic structure of atoms The chemical reactivity of any species be it atom, molecule or ion, is due to its electronic structure. For example, fluorine is incredibly reactive, forming compounds with all but three of the elements whereas the fluoride ion, with one extra electron making it isoelectronic with neon, is much less reactive and has a completely different chemistry. Our goal in this part of the course is to gain an understanding of the electronic structure of molecules and to see how this helps in predicting the shapes of molecules and the outcome of chemical reactions. The simplest way to predict the energy levels that are present in a molecule is to combine the energy levels of the constituent atoms. Hence we first look at the electronic structure of the atom. Energy levels and photoelectron spectroscopy Information about the electronic structure of atoms can be gained from a technique called photoelectron spectroscopy. In this technique, the sample is bombarded with high energy X-ray photons of known energy which knock out electrons from the sample. By analyzing the energies of the emitted electrons, it is possible to deduce the energy they possessed in the atom. Shown below are the photoelectron spectra of helium, neon and argon. 50 40 30 20 10 50 40 30 20 10 340 330 320 260 250 240 40 30 20 10 880 870 860 ionization energy / eV He Ne Ar 1s 2p 2s tens of eV most energetic 'valence' electrons - all pretty similar in energy tens of eV hundreds of eV hard to remove 1s 2s 2p 3s 3p 1s 3200 eV 1
When helium atoms are bombarded with high energy X-rayphotons,only electrons of oneenergy are emitted. This tells us that both the electrons in helium have the same energy. In thespectrumofneon,therearethreemainpeaks-onearound870eVwhichcorrespondstothefirstshell electrons being removed and twopeaks in the 20-50eV region.This tells us that there areelectrons withtwodifferent energies inthe2nd shell.In theargon spectrum,thereis onepeakaround3200eV(notshownontheabovespectrum),twomainpeaksinthehundredof eVs,and two inthetens ofeV.Thephotoelectron spectra showthatthere arefurther subdivisions ofenergy levels within the main energy shells.You will probably already be familiar with thesetheseenergylevelsandhowtheyaredenoted.Thedifferentpeaksinthephotoelectron spectracorrespondtoremovingelectronsfrom thesedifferent shells.egforArdenotetheelectronicstructureas2s21s22p63s23p6small differenceslargedifferences in energyWhilst there is a huge difference in energy between different energy shells (i.e. between the ls2s and 3s electrons), there is a much smaller energy difference between the subdivisions in eachshell (i.ebetween the2s and 2porbetweenthe3s and3pelectrons.Orbitals and Quantum NumbersThe electrons in atoms are said to occupydifferent orbitals.We shall see exactlywhat an orbitalis a little later, but for the moment we can think of it as an energy level.Any one orbitalcan accommodatetwoelectrons.An atomhas one1s orbital and this can accommodatetwoelectrons. Similarly, the 2s orbital can occupy two electrons. There are three 2p orbitals, all ofequal energy and each can hold two electrons.Orbitals that have thesame energy aredescribedas degenerate. The d orbitals have a five-fold degeneracy so, for example, there are five 3dorbitalsofthesameenergy.In order to distinguishbetween atomic orbitals,we need to describethree things:which‘shell'it is in, whether it is an s, p, d, or f orbital and then which one of the three p orbitals orthe five d orbitals or the seven f orbitals it is.Each of these things is expressed by a quantumnumber.What shell we are referring to is denoted by the principal quantum number, n.ntakesintegralvalues1,2,3,4,andsoonFor a one electron stsyem, the value of n alone determines the energy of the electron.Whether we are referring to an s, p, d, or f (or g, h, etc.) is denoted by the angular momentumquantum number, I (sometimes called the azimuthal quantum number).I can take integer values O, 1,2,...but the particular value it takes depends on the value of n.I takes integral values from 0 up to (n - 1)2
When helium atoms are bombarded with high energy X-ray photons, only electrons of one energy are emitted. This tells us that both the electrons in helium have the same energy. In the spectrum of neon, there are three main peaks – one around 870 eV which corresponds to the first shell electrons being removed and two peaks in the 20-50 eV region. This tells us that there are electrons with two different energies in the 2nd shell. In the argon spectrum, there is one peak around 3200 eV (not shown on the above spectrum), two main peaks in the hundred of eVs, and two in the tens of eV. The photoelectron spectra show that there are further subdivisions of energy levels within the main energy shells. You will probably already be familiar with these these energy levels and how they are denoted. The different peaks in the photoelectron spectra correspond to removing electrons from these different shells. small differences eg for Ar denote the electronic structure as large differences in energy 1s2 2s2 2p6 3s2 3p6 Whilst there is a huge difference in energy between different energy shells (i.e. between the 1s, 2s and 3s electrons), there is a much smaller energy difference between the subdivisions in each shell (i.e between the 2s and 2p or between the 3s and 3p electrons. Orbitals and Quantum Numbers The electrons in atoms are said to occupy different orbitals. We shall see exactly what an orbital is a little later, but for the moment we can think of it as an energy level. Any one orbital can accommodate two electrons. An atom has one 1s orbital and this can accommodate two electrons. Similarly, the 2s orbital can occupy two electrons. There are three 2p orbitals, all of equal energy and each can hold two electrons. Orbitals that have the same energy are described as degenerate. The d orbitals have a five-fold degeneracy so, for example, there are five 3d orbitals of the same energy. In order to distinguish between atomic orbitals, we need to describe three things: which ‘shell’ it is in, whether it is an s, p, d, or f orbital and then which one of the three p orbitals or the five d orbitals or the seven f orbitals it is. Each of these things is expressed by a quantum number. What shell we are referring to is denoted by the principal quantum number, n. n takes integral values 1,2,3,4, and so on For a one electron stsyem, the value of n alone determines the energy of the electron. Whether we are referring to an s, p, d, or f (or g, h, etc.) is denoted by the angular momentum quantum number, l (sometimes called the azimuthal quantum number). l can take integer values 0, 1, 2, . but the particular value it takes depends on the value of n. l takes integral values from 0 up to (n − 1) 2
Thevalueofldeterminestheorbital angularmomentumoftheelectron:angular momentum = Vl(I + 1)whereh=Planck'sconstant/2元Wecanthinkofthisangularmomentumasbeingthemomentumof theelectronasitmovesaroundthenucleus.Weshall seelaterthatorbitalswithdifferentvaluesofIalsohavedifferentshapesEachvalueof I has adifferentletterassociated withit:4,1=O23.5.1dfSpghWhen n =1, the only value for I =O. This is the Is orbital.When n = 2, I can be O, or 1 which correspond to the 2s and 2p orbitals.When n =3, I =0, 1, and 2 which correspond to the 3s, 3p and 3d orbitalsThe third quantum number needed to label an orbital is the magnetic quantum number, mi.m, takes integer values from +l to-/ in integer steps.m, tells us something about the orientation of the orbital.(Specifically it tells us the component of the angular momentum on a particular axis.)Foransorbital (l=O):mi=0onlyso just one s orbitalfor eachvalue of nFor a p orbital (l = 1):m = +1,0, -1so three p orbitals with the same value of nFor a d orbital (l = 2):m =+2,+1,0,-1,-2so fivedorbitals with the samevalue of nThethreequantumnumbers,n,I andmydefinetheorbital anelectronoccupiesbut if wearetrying to label an electron, there is one further thing we need to know. We said earlier that theelectron has angular momentum associated with it as it moves in its orbital. It also has its ownintrinsic angular momentum.Whereas the orbital angular momentum might bethoughtof astheangularmomentumthe electron has as it moves about in the orbital, this intrinsic angularmomentum might be thought of as the angular momentum the electron has due to it spinningabout an internal axis (although bear in mind this is just an analogy). spin angularorbital angularmomentummomentum3
The value of l determines the orbital angular momentum of the electron: angular momentum = √ l(l + 1) where = Planck’s constant/2π We can think of this angular momentum as being the momentum of the electron as it moves around the nucleus. We shall see later that orbitals with different values of l also have different shapes. Each value of l has a different letter associated with it: l = 0, 1, 2, 3, 4, 5, . s p d f g h, . . . When n = 1, the only value for l = 0. This is the 1s orbital. When n = 2, l can be 0, or 1 which correspond to the 2s and 2p orbitals. When n = 3, l = 0, 1, and 2 which correspond to the 3s, 3p and 3d orbitals. The third quantum number needed to label an orbital is the magnetic quantum number, ml. ml takes integer values from +l to −l in integer steps. ml tells us something about the orientation of the orbital. (Specifically it tells us the component of the angular momentum on a particular axis.) For an s orbital (l = 0): ml = 0 only so just one s orbital for each value of n For a p orbital (l = 1): ml = +1, 0, −1 so three p orbitals with the same value of n For a d orbital (l = 2): ml = +2, +1, 0, −1, −2 so five d orbitals with the same value of n The three quantum numbers, n, l and ml define the orbital an electron occupies but if we are trying to label an electron, there is one further thing we need to know. We said earlier that the electron has angular momentum associated with it as it moves in its orbital. It also has its own intrinsic angular momentum. Whereas the orbital angular momentum might be thought of as the angular momentum the electron has as it moves about in the orbital, this intrinsic angular momentum might be thought of as the angular momentum the electron has due to it spinning about an internal axis (although bear in mind this is just an analogy). orbital angular momentum spin angular momentum 3
Inananalogousmannertotheorbitalangularmomentumwhichhasitsmagnitudedefinedbyland its orientationdefined bymi,therearetwoquantumnumbers forthespin angularmomen-tum.Themagnitudeofthespinangularmomentumisdefinedbythequantumnumbersanditsorientation is defined by ms.The value of s for an electron is fixed: s = . This means that all electrons possess the sameintrinsicangularmomentum.The values m, can take are integer steps from +s to-s which means the angular momentumcan be oriented in one of two ways: m,=+ and ms=-.For historical reasons, we usuallydenote the spin of the electron by an arrow:↑ for m, =+ I for m, =-[Note: this is exactly analogous to the spin of a nucleus in NMR. There the spin is given thesymbol I. For 'H, I = and this spin can be oriented in two ways, ↑ or I, corresponding to mivalues of + and-↓.For deuterium,H, I = 1, there are 3 ways (21 + 1) oforienting the spincorresponding to m values of +1, 0 and -1.JSummaryTo specify the state of an electron in an atom (to be precise a one-electron atom or ion, see later)weneedto specifyfourquantumnumbers:n (specifies the energy)these3quantumnumbersdefineI (specifies themagnitude of theorbital angularmomentum)the orbital them,(specifiestheorientationoftheorbital angularmomentum)electronisinThis tells usm,(specifiestheorientation of the spinangularmomentum)about the spin ofthe electronNote that there is no need to specify s since it is the same for all electrons (s =)It is a fundamental observation that no orbital (defined by n, l, and mi) may contain more thantwo electrons,and if there are two,then they must haveopposite spin (m,=+ and m,=-)It thereforefollows that any electron in an atom has a unique set of four quantum numbers.ThisisoneformofthePauliPrinciple.4
In an analogous manner to the orbital angular momentum which has its magnitude defined by l and its orientation defined by ml, there are two quantum numbers for the spin angular momentum. The magnitude of the spin angular momentum is defined by the quantum number s and its orientation is defined by ms. The value of s for an electron is fixed: s = 1 2 . This means that all electrons possess the same intrinsic angular momentum. The values ms can take are integer steps from +s to −s which means the angular momentum can be oriented in one of two ways: ms = +1 2 and ms = −1 2 . For historical reasons, we usually denote the spin of the electron by an arrow: ↑ for ms = +1 2 ↓ for ms = −1 2 [Note: this is exactly analogous to the spin of a nucleus in NMR. There the spin is given the symbol I. For 1H, I = 1 2 and this spin can be oriented in two ways, ↑ or ↓, corresponding to mI values of +1 2 and −1 2 . For deuterium, 2H, I = 1, there are 3 ways (2I + 1) of orienting the spin corresponding to mI values of +1, 0 and −1.] Summary To specify the state of an electron in an atom (to be precise a one-electron atom or ion, see later) we need to specify four quantum numbers: l (specifies the magnitude of the orbital angular momentum) ml (specifies the orientation of the orbital angular momentum) ms (specifies the orientation of the spin angular momentum) n (specifies the energy) these 3 quantum numbers define the orbital the electron is in This tells us about the spin of the electron Note that there is no need to specify s since it is the same for all electrons (s = 1 2 ). It is a fundamental observation that no orbital (defined by n, l, and ml) may contain more than two electrons, and if there are two, then they must have opposite spin (ms = +1 2 and ms = −1 2 ). It therefore follows that any electron in an atom has a unique set of four quantum numbers. This is one form of the Pauli Principle. 4
Acloserlookatorbitals-wavefunctionsMany phenomena-such as the swinging of a pendulum or the change in theglobal population-canbedescribedbymathematicalfunctions.Todescribetheripplesonthesurfaceofwater,forexample, we could usea function based ona sine wave; y(x)= sin(x).y(x) = sin x1-+ valuesor+'phase0元/2213元1元values'phaseor-1-A function is a mathematical tool that we can feed a numberinto and it will give out a newnumber.Inthiscase,ifweput inavalueforx,say(/2),thesinefunction returnsthevalue+1.Quantummechanicsprovidesus withthebestunderstandingoftheelectronic structureofatoms and molecules.Theresults from a quantum mechanical analysis reveal that thepropertiesofanelectroncanalsobedescribedbyamathematicalfunctioncalledawavefunction,giventheGreek symbol (psi)isafunction ofcoordinates,forexamplex,y,&zhence we write (x,y,z)We have seen that the properties of an electron depend on which particular orbital it occupiesand thattheorbital isdefined bythequantum numbersn, Iand my.An orbital is a wavefunction(specifically an orbital is a one-electron wavefunction, as we shall see later). A different functionis needed for each orbital and each function is defined by the three quantum numbers n, I andmj.Wecanwriteourwavefunction as:nlm(x,y,z)which says that thewavefunction is a function of position coordinates x, y and z but that thereare different wavefunctions defined by thequantum numbers n, I and mj.Oncethe wavefunction of theelectron is known,it is possibleto calculate useful informationfromitsuchasthepositionormomentumoftheelectron.TheBornInterpretation of theWavefunctionOnephysical interpretation ofthewavefunction,isthat?(or,morepreciselymultipliedbyits complex conjugate,*,since can be a complex number)gives a measure of theprobabilityof finding the electron ata given position.F2for this wavefunction,maximumprobability of electron beingfound in region around x = 0probability of being at x = a(ortechnicallybetweenx=aandx=a+8x)5
A closer look at orbitals – wavefunctions Many phenomena – such as the swinging of a pendulum or the change in the global population – can be described by mathematical functions. To describe the ripples on the surface of water, for example, we could use a function based on a sine wave; y(x) = sin(x). π/ 2 1π 2π 3π 4π -1 0 1 x y(x) = sin x + values or + 'phase' - values or - 'phase' A function is a mathematical tool that we can feed a number into and it will give out a new number. In this case, if we put in a value for x, say (π/2), the sine function returns the value +1. Quantum mechanics provides us with the best understanding of the electronic structure of atoms and molecules. The results from a quantum mechanical analysis reveal that the properties of an electron can also be described by a mathematical function called a wavefunction, given the Greek symbol ψ (psi). ψ is a function of coordinates, for example x, y, & z hence we write ψ(x, y,z) We have seen that the properties of an electron depend on which particular orbital it occupies and that the orbital is defined by the quantum numbers n, l and ml. An orbital is a wavefunction (specifically an orbital is a one-electron wavefunction, as we shall see later). A different function is needed for each orbital and each function is defined by the three quantum numbers n, l and ml. We can write our wavefunction as: ψn,l,ml (x, y,z) which says that the wavefunction is a function of position coordinates x, y and z but that there are different wavefunctions defined by the quantum numbers n, l and ml. Once the wavefunction of the electron is known, it is possible to calculate useful information from it such as the position or momentum of the electron. The Born Interpretation of the Wavefunction One physical interpretation of the wavefunction, ψ , is that ψ2 (or, more precisely ψ multiplied by its complex conjugate, ψ∗, since ψ can be a complex number) gives a measure of the probability of finding the electron at a given position. ψ x a 2 for this wavefunction, maximum probability of electron being found in region around x = 0 probability of being at x = a (or technically between x = a and x = a +δx) 5