ThegraphsbelowshowthewavefunctionandtheRDFforthelsorbitalr/Bohr radir/ Bohr radi23234product of y2 x r2Hence reaches amaximumandthenfalls dueto e-rtermB501001502002505010015020025000r/pmr/pmTheradial distributionfunction and?tellus different things.TheRDFtells us theprobabilityoffindinganelectronatagivenradius,summedoverall angleswhereas?tellsustheprobabilityof finding an electron at a given set of coordinates (either r, and or x, y, and z).RDFgivesprobabilityof finding electron inathin shell of8thickness Srat radiusrfromthenucleusy?givestheprobabilityoffinding electron in a tinyvolumeelementOvatpositionx,y,zfromthenucleus (oratposition r, 0, Φ)Note:? for the ls is greatest at the nucleus so this is the position (of tiny volume element)whereiselectronismostlikelytobefound.The Bohr radius, ao, is the radius (tiny thin shell) at which the electron is most likely tobe found..The radial distribution function is zero at the nucleus because of the r?term.(Essentiallywearelookingfortheelectroninashellofzerovolume.11
The graphs below show the wavefunction and the RDF for the 1s orbital. 0 50 012345 0 100 150 200 250 wavefunction r / pm r / Bohr radii 0 50 12345 0 100 150 200 250 RDF r / pm r / Bohr radii 1s r 2 product of ψ2 x r2 Hence reaches a maximum and then falls due to e-r term The radial distribution function and ψ2 tell us different things. The RDF tells us the probability of finding an electron at a given radius, summed over all angles whereas ψ2 tells us the probability of finding an electron at a given set of coordinates (either r, θ and φ or x, y, and z). x y z RDF gives probability of finding electron in a thin shell of thickness δr at radius r from the nucleus ψ2 gives the probability of finding electron in a tiny volume element δv at position x, y, z from the nucleus (or at position r, θ, φ) δr Note: • ψ2 for the 1s is greatest at the nucleus so this is the position (of tiny volume element) where is electron is most likely to be found. • The Bohr radius, a0, is the radius (tiny thin shell) at which the electron is most likely to be found. • The radial distribution function is zero at the nucleus because of the r2 term. (Essentially we are looking for the electron in a shell of zero volume.) 11
2sorbitalNoneof thewavefunctionsforanys orbital containanyor @terms;thevalueof dependsonlyonthedistancefromthenucleus,r.Consequently,all sorbitalshavethesame shape-theyareallspherical.However,the wavefunction for each s orbital is unique and each has a very different appear-ance.Thewavefunction and theRDFforthe2s orbital are shownbelow.maximumprobabilityof finding electron atSthis radius告radial nodeo1202468101246810r/Bohrradiir/BohrradiiNoticethatthe sign(orphase)of thewavefunctioncanbepositiveornegative.We shall seehowthisisimportant later,butthis does notmatterwhen wecometoconsiderwhere theelectron issincetheprobability offindingthe electron isproportional tothe square ofthewavefunction.However, where the wavefunction changes sign from positive to negative, as it crosses the x-axis,there is a point at which =O.Such aposition is called a node.This is clearly visible inthe density plot, shown below.radial nodemaximumprobabilityoffindingelectronatthisradiusbut at this positionwhere density isgreatest (atthe nucleus)Notethatthenodeisjustamathematical surfaceandthereforehasnovolume.Itismeaninglesstoask about the chance of finding anelectron in any spacewith no volume.As soon as wespecifyavolumeby sayingitisashell witha tinythickness,thereisthena small chanceoffindingtheelectroninthatshell.]12
2s orbital None of the wavefunctions for any s orbital contain any θ or φ terms; the value of ψ depends only on the distance from the nucleus, r. Consequently, all s orbitals have the same shape – they are all spherical. However, the wavefunction for each s orbital is unique and each has a very different appearance. The wavefunction and the RDF for the 2s orbital are shown below. wavefunction r / Bohr radii r / Bohr radii RDF 0 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 2s 2s radial node maximum probability of finding electron at this radius Notice that the sign (or phase) of the wavefunction can be positive or negative. We shall see how this is important later, but this does not matter when we come to consider where the electron is since the probability of finding the electron is proportional to the square of the wavefunction. However, where the wavefunction changes sign from positive to negative, as it crosses the xaxis, there is a point at which ψ = 0. Such a position is called a node. This is clearly visible in the density plot, shown below. radial node maximum probability of finding electron at this radius but at this position where density is greatest (at the nucleus) [Note that the node is just a mathematical surface and therefore has no volume. It is meaningless to ask about the chance of finding an electron in any space with no volume. As soon as we specify a volume by saying it is a shell with a tiny thickness, there is then a small chance of finding the electron in that shell.] 12
2porbitalsThe radial part of the 2p orbitals does not depend on the value of my and so the radial parts forthe Px, Py, and pz are all the same.noradial nodesVRDFR(r)1268104r/BohrradiiTheangularpartsofporbitalsdodepend onthevalueof mi,whichdeterminesthewaytheorbitalisoriented.Thisisbestillustratedinthe3-Dplots.allporbitalshaveoneangularnode2px2py2pzΦ= 0°0=900angular node for Φ = 9o°ieyznodalplanexzplanexyplaneΦ from x-axisθfromz-axisangularnode13
2p orbitals The radial part of the 2p orbitals does not depend on the value of ml and so the radial parts for the px, py, and pz are all the same. r / Bohr radii 0 0 2 4 6 8 10 12 R(r) RDF no radial nodes The angular parts of p orbitals do depend on the value of ml, which determines the way the orbital is oriented. This is best illustrated in the 3-D plots. x y z 2px x y z 2py x y z 2pz all p orbitals have one angular node angular node for φ = 90o ie yz nodal plane φ = 0o xz plane φ from x-axis θ from z-axis θ = 90o xy plane angular node 13
3sorbitalTheradialpartofthe3sorbitalhastworadialnodesR(r)ROF10121416182022242628308r/Bohrradiradial nodes3porbitalsIn addition to the angular node common to all porbitals,the3p orbital also has aradial node.oneradial nodeRDFR(r)2628302081O12161822244r/BohrradiiTheradialand angularnodesof the3porbitalaremostclearlyshowninthedensityand3-Dsurfaceplotsangularnode-radial node14
3s orbital The radial part of the 3s orbital has two radial nodes. R(r) RDF r / Bohr radii 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 radial nodes 3p orbitals In addition to the angular node common to all p orbitals, the 3p orbital also has a radial node. R(r) RDF r / Bohr radii 0 0 4 6 8 10 12 14 16 18 20 22 24 26 28 30 2 one radial node The radial and angular nodes of the 3p orbital are most clearly shown in the density and 3-D surface plots. angular node radial node 14
3dorbitalsTheradial partsofallfive3dorbitals arethesame,buttheangularpartsdepend onthevaluesofmy which determines the orientation of the orbital.no radial nodesRDFR(r)6242628304810121416182022r/ Bohr radiiangularnodes(nodalplanes)3dxy3dxz3dyznodal coneHwhen 0 = 54.702ndnodalconewhenθ=125.303dx2-y23dz2Eachdorbital has two angularnodes.The angular nodes defineplanes forall theorbitals exceptthed,zforwhichtheangularnodesarecones.15
3d orbitals The radial parts of all five 3d orbitals are the same, but the angular parts depend on the values of ml which determines the orientation of the orbital. R(r) RDF r / Bohr radii 0 4 6 8 10 12 14 16 18 20 22 24 26 28 30 2 no radial nodes xx x yy y zz z 3dxy 3dxz 3dyz angular nodes (nodal planes) xy y x z z 3dx2-y2 3dz2 nodal cone when θ = 54.7o 2nd nodal cone when θ = 125.3o θ Each d orbital has two angular nodes. The angular nodes define planes for all the orbitals except the dz2 for which the angular nodes are cones. 15