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 wavefunction: • 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 operate 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 kinetic energy associated with the electron this part gives the potential energy associated 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 < θ < π 0 < φ < 2π nucleus electron cartesian polar One of the advantages in converting the wavefunction to polar coordinates is that ψ can then be written as the product of two functions, each of which can be represented separately. ψn, l, ml (r, θ, φ) Yl, ml = Rn, l (r) (θ, φ) radial part of wavefunction defined by n and l function of r only angular part of wavefunction defined by l and ml function of θ and φ only 1s orbital Shown below is a graph of the radial part of the 1s wavefunction as a function of the distance from the nucleus, r. 0 50 012345 0 100 150 200 250 wavefunction r / pm r / Bohr radii ψ1s e-r 8 ( r > 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 wavefunction 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