Shown below are the density plots for the 3d orbitals, taken through the specific planes.:dx2-y2 through xy planedxy or dxz or dyzdz2throughxzpryzplanedz2 through xyNodesThetotalnumber of nodes inthehydrogen orbitalsdepends onlyon theprincipal quantumnumber, n.totalnumberofnodes=angular+radial=n-1Numberofangularnodes=Ii.e.all sorbitalshave0porbitalshave1d orbitals have 2etc:.Numberofradialnodes=n-1-l16
Shown below are the density plots for the 3d orbitals, taken through the specific planes. dx2-y2 through xy plane dz2 through xz pr yz plane dz2 through xy dxy or dxz or dyz Nodes The total number of nodes in the hydrogen orbitals depends only on the principal quantum number, n. total number of nodes = angular + radial = n − 1 Number of angular nodes = l i.e. all s orbitals have 0 p orbitals have 1 d orbitals have 2 etc ∴ Number of radial nodes = n − 1 − l 16
Building up multi-electron atoms using hydrogenorbitalsWhilst it is possible to solve the Schrodinger equation exactly for single electron systems this isnot the case formulti-electron systems.For such systems,in addition to the attractionbetweenthenucleusandelectronsthereisalsorepulsionbetweenelectrons.Itisthisaddedcomplication of electron-electron repulsion that makes it impossibleto solvethe Schrodinger equationfor multi-electron systems. However, reasonably good approximations for the energies of theelectronsinatoms canbemadeusingtheorbital approximation.This assumesthat,fromthepoint of view of one particular electron, the effect of all the other electrons can be averaged outtogivea modified potential, centred on thenucleus.As an example,takeanatomof lithiumwhichhasthreeelectrons.When working outtheenergyofthe electron l,weaverage out theeffects of the other two electrons.focus onthiselectrone,0Othink of as&Oaverage effects of"hydrogen-like" system withthetwoelectronsmodified charge at nucleusThe orbital approximation assumes that the modified potential is spherically symmetric andcentredonthenucleus.Thismeansthatthewavefunctionsforeachelectronhavethesameformas thosein hydrogen.In the above example,electrons 2 and 3 are said to screen the effects of the nucleus fromelectron 1.Exactly how well electrons screen one anotherdepends on which orbital they are in.Forexample,thelselectronsinlithiumscreenoffthenuclearchargefeltbytheelectroninthe2sorbital rather well.In contrast,since the electron in the2s orbital is on average further awayfrom the nucleus than the ls electrons, it has little effect on the nuclear charge experienced bythe Is electrons. This means the nuclear charge each electron experiences, the effective nuclearcharge,Zeff,isdifferentforthedifferentelectrons.1selectronsshieldeachotherbyabout3o%ofoneproton'schargebutarenotreallyshieldedbythe2sInstead of feeling a charge of +3, each 1s experiences a charge of about 2.7.Thetwo1selectronsshieldthe2selectronmorefully;Zefr(2s)isaround1.317
Building up multi-electron atoms using hydrogen orbitals Whilst it is possible to solve the Schr¨odinger equation exactly for single electron systems this is not the case for multi-electron systems. For such systems, in addition to the attraction between the nucleus and electrons there is also repulsion between electrons. It is this added complication of electron-electron repulsion that makes it impossible to solve the Schr¨odinger equation for multi-electron systems. However, reasonably good approximations for the energies of the electrons in atoms can be made using the orbital approximation. This assumes that, from the point of view of one particular electron, the effect of all the other electrons can be averaged out to give a modified potential, centred on the nucleus. As an example, take an atom of lithium which has three electrons. When working out the energy of the electron 1, we average out the effects of the other two electrons. 3+ e1 - e2 - e3 - x+ e1 - focus on this electron average effects of the two electrons "hydrogen-like" system with modified charge at nucleus think of as The orbital approximation assumes that the modified potential is spherically symmetric and centred on the nucleus. This means that the wavefunctions for each electron have the same form as those in hydrogen. In the above example, electrons 2 and 3 are said to screen the effects of the nucleus from electron 1. Exactly how well electrons screen one another depends on which orbital they are in. For example, the 1s electrons in lithium screen off the nuclear charge felt by the electron in the 2s orbital rather well. In contrast, since the electron in the 2s orbital is on average further away from the nucleus than the 1s electrons, it has little effect on the nuclear charge experienced by the 1s electrons. This means the nuclear charge each electron experiences, the effective nuclear charge, Zeff, is different for the different electrons. 1s electrons shield each other by about 30% of one proton’s charge but are not really shielded by the 2s. Instead of feeling a charge of +3, each 1s experiences a charge of about 2.7. The two 1s electrons shield the 2s electron more fully; Zeff(2s) is around 1.3. 17
In hydrogen, the 2s and 2p orbitals have the same energy (as do the 3s, 3p and 3d etc). This isnot obvious from looking at the radial distribution functions, but is perhaps not too surprising.RDFhydrogen1sclearlyclosertothe1snucleus onaverage2p2s0.40.50.60.70.80.00.10.20.30.9In multi-electron systems, such as lithium, orbitals which share the same principal quantumnumber but which differ in their orbital angular momentum quantum number are no longerdegenerate.Inspection of the radial distribution functions for lithium 1s, 2s and 2p orbitalsenables us to understand why the degeneracy is lost.RDFlithiumNotethecontraction of theorbitals (especiallythe 1s) due to the increased nuclear charge.1sItishardtosaywheretheaveragedistancefromthenucleus is, but an electron in the 2s orbitalspendsasignificantamountoftimeclosertothenucleusthan anelectron inthe2p orbital.2s2p0.2 0.30.40.50.60.70.80.9 nm0.0greaterpenetrationofthe1sbythe2sorbitalThe 2s orbital is said to penetrate the ls orbital to a greater extent than does the 2p orbital.Thismeans itexperiencesmoreofthenuclearchargethanthe2porbital whichinturnmeansthatanelectron inthe2sorbitalhasa slightlylowerenergythantheoneinthe2porbital.(Rememberthe greater thenuclear charge, the lower the energy:En=-RZ-/n?.Also the wavefunctionsfors orbitals areall non-zero at thenucleus,in contrastto otherorbitals.)18
In hydrogen, the 2s and 2p orbitals have the same energy (as do the 3s, 3p and 3d etc). This is not obvious from looking at the radial distribution functions, but is perhaps not too surprising. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 RDF hydrogen 1s 2p 2s 1s clearly closer to the nucleus on average In multi-electron systems, such as lithium, orbitals which share the same principal quantum number but which differ in their orbital angular momentum quantum number are no longer degenerate. Inspection of the radial distribution functions for lithium 1s, 2s and 2p orbitals enables us to understand why the degeneracy is lost. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 nm RDF lithium Note the contraction of the orbitals (especially the 1s) due to the increased nuclear charge. It is hard to say where the average distance from the nucleus is, but an electron in the 2s orbital spends a significant amount of time closer to the nucleus than an electron in the 2p orbital. 1s 2s 2p greater penetration of the 1s by the 2s orbital The 2s orbital is said to penetrate the 1s orbital to a greater extent than does the 2p orbital. This means it experiences more of the nuclear charge than the 2p orbital which in turn means that an electron in the 2s orbital has a slightly lower energy than the one in the 2p orbital. (Remember the greater the nuclear charge, the lower the energy: En = −RHZ2/n2. Also the wavefunctions for s orbitals are all non-zero at the nucleus, in contrast to other orbitals.) 18
Theeffectsoforbitalpenetrationmaybecomesopronouncedthatitbecomesdifficulttopredicttheenergyorderfororbitalsinlargeratoms.Forexample,inpotassium and calcium,the4sorbitalisfilledbeforethe3dorbital.Inspectionof theradial distributionfunctionsforthehydrogen 3d and 4s orbitals clearly indicates that if the electron is in the 4s orbital it is, on average,furtherfromthenucleushydrogenRDF3d4s1.01.50.00.52.0However,in atoms likepotassium,thegreater nuclear chargecontractstheorbitals and thepen-etrationof the lowerorbitals by the 4s orbital is sufficient tomeanthat this orbital is lower inenergy than the 3d orbitals.RDFpotassiumgreaterpenetrationby 4s0.51.52.00.01.0The exact ordering of energy levels in multi-electron atoms rapidly becomes complicated andimpossibletopredict:For a hydrogen atom, the ordering of energy levels is:1s<2s=2p<3s=3p=3d<4s=4p=4d=4fetc.ie depends on n onlyFor lithium:1s<<2s<2p<<3s<3p<3d<4s<4p<4d<4f<5setc.degeneracy nowlost. Order still depends mainly on n but also on l.For sodium:1s<<2s<2p<<3s<3p<4s<3d<4p<5setc.4s lower than 3d (the same order as when fill up periodic table)Forpotassium:1s<<2s<2p<<3s<3p<4s<4p<5s<3d etc4s,4pand5sall lowerthan3dFor Cat:1s<<2s<2p<<3s<3p<4s<3d<4p<5s etc.For Sc2+:1s<<2s<2p<<3s<3p<3d<4s<4p<5setc.K, Cat and Sc2+ all isoelectronic but could not predict the ordering!19
The effects of orbital penetration may become so pronounced that it becomes difficult to predict the energy order for orbitals in larger atoms. For example, in potassium and calcium, the 4s orbital is filled before the 3d orbital. Inspection of the radial distribution functions for the hydrogen 3d and 4s orbitals clearly indicates that if the electron is in the 4s orbital it is, on average, further from the nucleus. RDF 0.0 0.5 1.0 1.5 2.0 hydrogen 3d 4s However, in atoms like potassium, the greater nuclear charge contracts the orbitals and the penetration of the lower orbitals by the 4s orbital is sufficient to mean that this orbital is lower in energy than the 3d orbitals. RDF 0.0 0.5 1.0 1.5 2.0 greater potassium penetration by 4s The exact ordering of energy levels in multi-electron atoms rapidly becomes complicated and impossible to predict: For a hydrogen atom, the ordering of energy levels is: 1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4 f etc. ie depends on n only For lithium: 1s << 2s < 2p << 3s < 3p < 3d < 4s < 4p < 4d < 4 f < 5s etc. degeneracy now lost. Order still depends mainly on n but also on l. For sodium: 1s << 2s < 2p << 3s < 3p < 4s < 3d < 4p < 5s etc. 4s lower than 3d (the same order as when fill up periodic table) For potassium: 1s << 2s < 2p << 3s < 3p < 4s < 4p < 5s < 3d etc. 4s, 4p and 5s all lower than 3d For Ca+: 1s << 2s < 2p << 3s < 3p < 4s < 3d < 4p < 5s etc. For Sc2+: 1s << 2s < 2p << 3s < 3p < 3d < 4s < 4p < 5s etc. K, Ca+ and Sc2+ all isoelectronic but could not predict the ordering! 19
Theenergyof anatomwithaparticularelectronconfigurationdependsontheenergiesof allthe electrons it contains.Removing or addingan extra electron changes the energy of all theelectrons present.Because of this, it is meaningless to saythat‘the 4s orbital is lower in energythan the 3d orbital';whilst itmight be trueforoneparticular atom or ion, itmaywell change ontheadditionorsubtractionofanextraprotonorelectron.Orbital energies across thePeriodic TableSince electrons in the same shell do not shield each other very well (typically only by about30-35%),theeffectivenuclearchargeof thevalenceelectrons increases as anextraprotonandelectronareaddedaswemoveacrosstheperiodictablecNLiBeB0FNeelement2.53.94.55.81.31.93.25.2ZeffofvalenceelectronItisthis increaseinZefftogetherwiththedifferentdegrees of penetrationand shielding oftheelectrons present that explains the trends in orbital energies as wemove across the periodic table.Agraphoforbital energiesvs.atomicnumber00valence e-LiKCaScTiV.CrMnFeCoNiCuZnaNatAlGaXMMgosiXBeBAA0CO.S-10--10aCONIxOKroArQ04pe2020ONeJHe3080core1s4s40AOD2s50503d3s2p33p-60-60AtomicNumber. orbital energy decreases across a period due to increased ZefCore electrons.i.e.electronsinshellslowerthan valence electrons.have verylowenergyand take littlepart in reactions.Thetrends intheorbital energiesarereflected inthe ionization energies and electronegativitiesoftheatoms(whichincreasewithincreasingZef)andalsointheiratomicradii(whichdecreasewithincreasingZef).ThesetrendswillbeexploredmorefullyintheChemistryoftheElementscourseintheEasterterm.Weshallnow seehowthesetrendsaffectthechemicalbondingbetweenatoms.20
The energy of an atom with a particular electron configuration depends on the energies of all the electrons it contains. Removing or adding an extra electron changes the energy of all the electrons present. Because of this, it is meaningless to say that ‘the 4s orbital is lower in energy than the 3d orbital’; whilst it might be true for one particular atom or ion, it may well change on the addition or subtraction of an extra proton or electron. Orbital energies across the Periodic Table Since electrons in the same shell do not shield each other very well (typically only by about 30-35%), the effective nuclear charge of the valence electrons increases as an extra proton and electron are added as we move across the periodic table. element Li Be B C N O F Ne Zeff of valence electron 1.3 1.9 2.5 3.2 3.9 4.5 5.2 5.8 It is this increase in Zeff together with the different degrees of penetration and shielding of the electrons present that explains the trends in orbital energies as we move across the periodic table. –60 –50 –40 –30 –20 –10 0 –60 –50 –40 –30 –20 –10 0 H He Li NaMgAl Si P S Cl Ar K CaScTi V CrMnFeCoNiCuZn GaGeAsSeBr Kr BeB C N O F Ne 1s 4s 3d 2p 3s 3p 4p 2s Orbital Energy / eV Atomic Number A graph of orbital energies vs. atomic number valence e - core e - • orbital energy decreases across a period due to increased Zeff • Core electrons, i.e. electrons in shells lower than valence electrons, have very low energy and take little part in reactions. The trends in the orbital energies are reflected in the ionization energies and electronegativities of the atoms (which increase with increasing Zeff) and also in their atomic radii (which decrease with increasing Zeff). These trends will be explored more fully in the Chemistry of the Elements course in the Easter term. We shall now see how these trends affect the chemical bonding between atoms. 20