《基础化学 Fundamentals in Chemistry》课程教学课件（分子形状与结构 The Shapes and Structures of Molecules）分子形状与结构原子分子结构（英文讲义）

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 struc￾ture. 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 momen￾tum. 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

Theideathatwecanonlytalkabouttheprobabilityoffindingtheelectronatagivenpositionisin contrast to classical mechanics in which the position of an object can be specified preciselyTheBorn interpretation imposes certainrestrictions on just whatis acceptablefor a wavefunction: must be single-valued, that is at any given value of x there can only be one value of sincetherecanonlybeoneprobabilityoffindinganelectronatanyonepoint.Wa3 values of electron begin at x= aNotallowed! must not diverge; the total area under ? for all values of x must be finite since it is theprobabilityoffindingtheelectron anywhere.y2yr2shaded area =probability ofinfinite area - no good!electronbeingfound in all spacemustbe finiteEachwavefunctionhasagivenenergyassociatedwithit,forexamplethewavefunctionfortheIs orbital has a different energy from the 2s etc.The way to calculate wavefunctions and theenergies associated with them is to use Schrodinger's equation.The SchrodingerEquationIngeneral,theSchrodingerEquationmaybewritten:constant(theenergyassociatedwithHY=EYthe wave function)same wavefunctionwavefunctionoperator6

The idea that we can only talk about the probability of finding the electron at a given position is in contrast to classical mechanics in which the position of an object can be specified precisely. The Born interpretation imposes certain restrictions on just what is acceptable for a wave￾function: • ψ must be single-valued, that is at any given value of x there can only be one value of ψ since there can only be one probability of finding an electron at any one point. ψ x x = a 3 values of electron begin at x = a Not allowed! • ψ must not diverge; the total area under ψ2 for all values of x must be finite since it is the probability of finding the electron anywhere. ψ x 2 ψ x 2 shaded area = probability of electron being found in all space infinite area - no good! must be finite Each wavefunction has a given energy associated with it, for example the wavefunction for the 1s orbital has a different energy from the 2s etc. The way to calculate wavefunctions and the energies associated with them is to use Schr¨odinger’s equation. The Schrodinger Equation ¨ In general, the Schr¨odinger Equation may be written: H Ψ = E Ψ operator wavefunction same wavefunction constant (the energy associated with the wave function) 6

Tounderstandthisequation, weneedtobeclearonthedistinctionbetweenanoperator and afunction:AFUNCTIONisadevicewhichconverts aNUMBERtoanothernumber.eg the function 'sine';sin:converts numbernumber+1AnOPERATORisadevicewhichconvertsaFUNCTIONintoanotherfunctiondconvertsfunctionsinx→functioncosxegdxd sinx=COSxdxThe Hamiltonian operator, H, is constructed so as to give us the energy associated with thewavefunction“Solving'the Schrodinger equation means finding a suitablefunction, ,that when we op-erate on it using the operator H we get the same function, ,multiplied by a constant E.Theconstant E is the energy associated with the particular wavefunction, .TheenergyEis composedofbothpotential energy(i.e.‘storedup'energyforexamplebyinteractionwithan electricfield)andkineticenergy(i.e.duetoitsmovement).TheHamiltonianoperatorcontainspartstoworkoutboththesecomponents.For thehydrogen atom,theHamiltonian operatoris:e2=2me4元80rthis part gives the ki-this part gives the ponetic energy associatedtential energy associ-with the electronatedwiththeelectron2? (pronounced 'del-squared') =So the Schrodinger equation for the a hydrogen atom could be written:e?w（E山2max20v20-24元801whereh=Planck'sconstant,h,dividedby2nme=mass of theelectron80 = vacuum permittivitye = elementary charger=distancebetweennucleusandelectron=x?+y?+z?7

To understand this equation, we need to be clear on the distinction between an operator and a function: A FUNCTION is a device which converts a NUMBER to another number. eg the function ‘sine’; sin π 2 converts number π 2 −→ number +1 An OPERATOR is a device which converts a FUNCTION into another function. eg d dx converts function sinx −→ function cosx d sinx dx = cosx The Hamiltonian operator, Hˆ, is constructed so as to give us the energy associated with the wavefunction ψ. ‘Solving’ the Schr¨odinger equation means finding a suitable function, ψ, that when we op￾erate on it using the operator Hˆ we get the same function, ψ , multiplied by a constant E. The constant E is the energy associated with the particular wavefunction, ψ. The energy E is composed of both potential energy (i.e. ‘stored up’ energy for example by interaction with an electric field) and kinetic energy (i.e. due to its movement). The Hamiltonian operator contains parts to work out both these components. For the hydrogen atom, the Hamiltonian operator is: Hˆ = − 2 2me ∇2 − e2 4πε0r this part gives the ki￾netic energy associated with the electron this part gives the po￾tential energy associ￾ated with the electron ∇2 (pronounced ‘del-squared’) = ∂2 ∂x2 + ∂2 ∂y2 + ∂2 ∂z2 So the Schrodinger equation for the a hydrogen atom could be written: ¨ − 2 2me ∂2ψ ∂x2 + ∂2ψ ∂y2 + ∂2ψ ∂z2 − e2ψ 4πε0r = Eψ where = Planck’s constant, h, divided by 2π me = mass of the electron ε0 = vacuum permittivity e = elementary charge r = distance between nucleus and electron =  x2 + y2 + z2. 7

SolvingtheSchrodingerEquationfortheHydrogenAtomWe can solvethe Schrodinger equation exactlyforthehydrogen atom.There is not one uniquesolution but a whole series of solutions defined bythe quantum numbers we have already met.Each solution has adifferent mathematical form (these are listedfor reference in theappendix).In general, these solutions to the Schrodinger equation for a one-electron system may bedenoted:Wn, I, m, (x, y, z)quantum numbersand its energy, En, is given bynuclearcharge2mee4=1forHatomEnn28 h2aconstantfortheprincipalquantumnumberparticular orbitaldefined by n onlyThisismuchmoreconvenientlywrittenaswhereRuistheRydbergconstantEn = - RH ×h2whosevalue depends ontheunits usedThefirst pointtonoticeis that the energy of an orbital depends on n only.This means that the2sand2porbitals have the same energy (that is,theyaredegenerate)and the3s3pand3d orbitalsarealso degenerate.Wehave already seenthat this isnotthecaseformulti-electronatoms wherethe2sorbitalislowerinenergythanthe2p,forexampleThe second point to notice is that the predicted energies are negative. As n gets larger, E,tends towards zero. Zero energy corresponds to the electron being separated completely fromthenucleus.Theenergyneeded topromote the electronfromthelowest energylevel tozeroenergyistheionization energyforthe atom.The energylevelsfor thehydrogen atom,drawn toscale, can be represented:0separateelectronandionn=n=2E, = -Rμ/ 9n=2E2 = -R/ 4energylonizationenergyEi = -RHn三8

Solving the Schrodinger Equation for the Hydrogen Atom ¨ We can solve the Schr¨odinger equation exactly for the hydrogen atom. There is not one unique solution but a whole series of solutions defined by the quantum numbers we have already met. Each solution has a different mathematical form (these are listed for reference in the appendix). In general, these solutions to the Schr¨odinger equation for a one-electron system may be denoted: ψn, l, ml (x, y, z) quantum numbers and its energy, En, is given by: En = - z2 n2 me e4 8 ε0 h 2 2 nuclear charge =1 for H atom a constant for the principal quantum number particular orbital defined by n only This is much more conveniently written as En = - z2 n2 where RH is the Rydberg constant whose value depends on the units used RH The first point to notice is that the energy of an orbital depends on n only. This means that the 2s and 2p orbitals have the same energy (that is, they are degenerate) and the 3s, 3p and 3d orbitals are also degenerate. We have already seen that this is not the case for multi-electron atoms where the 2s orbital is lower in energy than the 2p, for example. The second point to notice is that the predicted energies are negative. As n gets larger, En tends towards zero. Zero energy corresponds to the electron being separated completely from the nucleus. The energy needed to promote the electron from the lowest energy level to zero energy is the ionization energy for the atom. The energy levels for the hydrogen atom, drawn to scale, can be represented: energy separate electron and ion Ionization energy 0 n = 1 E1 = -RH E2 = -RH / 4 E3 = -RH / 9 n = 2 n = 2 n = 8 8

Representingthe HydrogenOrbitalsWhilst it is possible to solve the Schrodinger equation exactly for a hydrogen atom,the actualmathematicalformforthethree-dimensionalwavefunctionsrapidlybecomescomplicated.Inthis course, we will use a variety of graphical means to represent the solutions.In order to dothis,itis much more convenient to use polar coordinates ratherthan Cartesian coordinates tospecifywheretheelectron is relativeto thenucleus.22polarcartesianelectron0≤0≤元0≤Φ≤2元nucleus1Yn, I, m, (r, e, Φ)n, l, m, (x, y, z)One of the advantages in converting the wavefunction to polarcoordinates is that can then bewrittenastheproductoftwofunctions,eachofwhichcanberepresented separatelyRn, I(r)Yi, m, (e, Φ)Yn, I, m,(r, e, 0) = IXradial partofwavefunctionangularpartofwavefunctiondefined by n and Idefined by andmfunction of r onlyfunction of eandβonlyIs orbitalShown below is a graph of the radial part of the ls wavefunction as a function of the distancefromthenucleus,r.r/Bohr radi0A341sα e-rone(r≥0)1e-r-erd050100150200250r/pm9

Representing the Hydrogen Orbitals Whilst it is possible to solve the Schr¨odinger equation exactly for a hydrogen atom, the actual mathematical form for the three-dimensional wavefunctions rapidly becomes complicated. In this course, we will use a variety of graphical means to represent the solutions. In order to do this, it is much more convenient to use polar coordinates rather than Cartesian coordinates to specify where the electron is relative to the nucleus. z x y z x y ψn, l, ml (x, y, z) ψn, l, ml (r, θ, φ) y x z r θ φ 0 0 ) e-r er 1 = 9

A3-Dplotshowshowthewavefunctionvarieswithanglesand@.Foransorbital,thewavefunction isindependentof and@and onlydepends ontheradius fromthenucleus.Thismeansthat all s orbitals are spherical.surface withthesurfaceshowsa smalleraparticularvaluevalue of yofthewavefunctionsurface withalarger value of yOne way to try and show how the value of the wavefunction varies at different positions is todraw a picture of a slice through the orbital in a given plane.A contourplot joins togetherregions of the same value of the wavefunction.decreasingvalues ofy13f21213145equalvalues ofynucleusalong contoursXAnotherwayofrepresentingthewavefunctionisadensityplot.Here,themoredarklyshadedaregion is, the greater the value of .y2αprobability offinding electron.The darker the region,thegreatertheprobabilityFinally,a further common way to represent theorbitals is to draw a graph showingthe electrondensity at a set distance,r,from the nucleus, summed over all angles, and o.This functionknownastheRadialDistributionFunction.RDF,maybethoughtofasthesumoftheelectrondensity ina thin shell at a radius rfrom thenucleus.The volume of this shell goes up as rincreases since the surface area ofthe shell increases.RDF=[R(r)×4元r?i.e.[R(r)]?xsurfaceareaofsphere10

A 3-D plot shows how the wavefunction varies with angles θ and φ. For an s orbital, the wave￾function is independent of θ and φ and only depends on the radius from the nucleus. This means that all s orbitals are spherical. y z x y y z x the surface shows a particular value of the wavefunction surface with a larger value of ψ surface with a smaller value of ψ One way to try and show how the value of the wavefunction varies at different positions is to draw a picture of a slice through the orbital in a given plane. A contour plot joins together regions of the same value of the wavefunction. y x r 1 r 1 r 2 r 2 r 3 r 3 r 1 r 2 r 3 r 1 r 2 r 3 wavefunction nucleus ψ decreasing values of ψ equal values of ψ along contours Another way of representing the wavefunction is a density plot. Here, the more darkly shaded a region is, the greater the value of ψ. ψ2 probability of finding electron. The darker the region, the greater the probability 8 Finally, a further common way to represent the orbitals is to draw a graph showing the electron density at a set distance, r, from the nucleus, summed over all angles, θ and φ. This function known as the Radial Distribution Function, RDF, may be thought of as the sum of the electron density in a thin shell at a radius r from the nucleus. The volume of this shell goes up as r increases since the surface area of the shell increases. RDF = [R(r)]2 × 4πr2 i.e. [R(r)]2× surface area of sphere 10