《无机化学 Chemical Theory》课程教学资源（讲稿）Part I Basic concepts of Quantum mechanics Chapter 6 Many-electron atoms

6.Many-electronatomsTheheliumatomTheHamiltonianfor He (with clamped nucleus)ishHHeV2mnV2meZe2(1)4元9斤Ze24元0/224元0:/12Because oftheterm in1/r1-this can't be separated into H(1)+H(2), sothe wavefunctioncan'tbewrittenas=(1)(2)

6. Many-electron atoms The helium atom The Hamiltonian for He (with clamped nucleus) is Because of the term in 1/𝑟12this can't be separated into 𝐻(1) + 𝐻(2), so the wavefunction can't be written as Ψ = 𝜓(1)𝜓(2). 2 2 He 1 2 2 2 2 0 1 2 0 2 2 0 12 2 2 4 4 4 e e H m m Ze r Ze r e r            ò ò ò (1)

6.1 Central field approximationHoweverwecan writeapproximately(2)HHe ~ H(1)+H(2)withn_?+V(r)H:(3)2mwhere Vincludes a spherical average of therepulsion from theother electron.This is the central field approximation, and it allows us to treat the electrons as if theymoveindependentlyofeachotherThat is, = (1)(2), with H = EWriting out the kinetic energy term gives (in atomic units)121a22aH:or +2r2 +V(r)2r? Or

However we can write approximately (2) with (3) where V includes a spherical average of the repulsion from the other electron. This is the central field approximation, and it allows us to treat the electrons as if they move independently of each other. That is, Ψ = 𝜓(1)𝜓(2), with 𝐻𝜓 = 𝐸𝜓. Writing out the kinetic energy term gives (in atomic units) 6.1 Central field approximation H H H He   1 2      2 2 2 e H V r m       2 2 2 2 ˆ 1 2 2 H r V r r r r r         l

The important feature is that V still depends only on r, not on and BecauseofthiswecanstillwriteV/nlm = R. (r) Ym(0, p)where Yim(, ) is a spherical harmonic, just as before, but Rn(r) now satisfies a differentradial equation:1(1+1)1 0,20RmR., + V(r) R., = E.m,R.l2r22r2 Orar

The important feature is that V still depends only on r, not on 𝜃 and 𝜑. Because of this we can still write where 𝑌𝑙𝑚(𝜃, 𝜑) is a spherical harmonic, just as before, but 𝑅𝑛𝑙(𝑟) now satisfies a different radial equation:   ( , )    nlm nl lm  R r Y    2 2 2 1 1 2 2 nl nl nl nl nl R l l r R V r R E R r r r r         

TheSelf-ConsistentFieldmethodTheradialequation is1(1+1)1aaR,!m +V(r)Rm=EmRlR2r.22r2 arOrVis an average of the interactions with the other electron (or electrons, in general), so wecan't calculate it until we know where the electrons are. We have to start by guessing theform oftheorbitals.andthen(i) use the orbitals to evaluate V(r),(ii) solve the eq. to get new orbitals.andrepeattheprocessuntiltheneworbitalsagreewiththeoldonesThisiscalledtheSelf-ConsistentFieldorSCFmethod

The Self-Consistent Field method The radial equation is V is an average of the interactions with the other electron (or electrons, in general), so we can't calculate it until we know where the electrons are. We have to start by guessing the form of the orbitals, and then (i) use the orbitals to evaluate 𝑉(𝑟) , (ii) solve the eq. to get new orbitals, and repeat the process until the new orbitals agree with the old ones This is called the Self-Consistent Field or SCF method.     2 2 2 1 1 2 2 nl nl nl nl nl R l l r R V r R E R r r r r         

Thepotential actingontheelectron is sphericalEach electron in an atom moves independently in a central potential due to the CoulombattractionofthenucleusandtheaverageeffectoftheotherelectronsintheatomHartree'sprocedureisasfollowss, = f(r)Yl(0,g)do = s(ri,0.9)s2(r,02.Φ)...s,(r,0n.n)eelectron 1Q.TP2P, = -e|s, P2dy. :dy4元804元80Y121Vi2 +Vi, +...+V.dyryi=2Ze12V(ri,0,g)=V(r)ririj=2n2V? +V(r) |t;(1) = s,t(1)2me

Each electron in an atom moves independently in a central potential due to the Coulomb attraction of the nucleus and the average effect of the other electrons in the atom. The potential acting on the electron is spherical electron 1 ( ) ( , ) l i m s f r Y    Hartree’s procedure is as follows 0 1 1 1 1 2 2 2 2 ( , , ) ( , , ) ( , , ) n n n n         s r s r s r 1 2 12 2 0 12 4 Q V dv r     2 2 2 2 12 | | ' s e dv r   0 ' 4 e e   2 2 2   e s| | 2 2 12 13 1 2 | | ' n j n j j ij s V V V e dv  r      2 2 2 1 1 1 1 1 1 2 1 1 | | ' ( , , ) ' ( ) n j j j j s Ze V r e dr V r r r         2 2 1 1 1 1 1 ( ) (1) (1) 2 e V r t t m           

Orbitalenergiesinmany-electronatomsBecausewe nowhavea potential v(r)instead of z/r,theenergies change,andthey nowdepend on I as well as n. We can see why by comparing the radial functions for the 2s and 2porbitals in hydrogen.Remember that in the hydrogen atom, the 2s and 2p orbitals have the same energy.Thepotential energy contribution is(V)n, =[y nm (r)'V(r)nim (r)r2 sin drdodp= J(R.Ym) V(r)R.Ymr2 sinOdrdedp=[ RV(r)r’dr= [V(r) P. (r)drWhen V = -1/r this is the same for both 2s and 2p:Forexample(V)2,=J R2,V(r)r?dr = - J°re'dr =-3 / 24 =-1 /4

Orbital energies in many-electron atoms Because we now have a potential 𝑉(𝑟) instead of 𝑍/𝑟, the energies change, and they now depend on l as well as n. We can see why by comparing the radial functions for the 2s and 2p orbitals in hydrogen. Remember that in the hydrogen atom, the 2s and 2p orbitals have the same energy. The potential energy contribution is When 𝑉෠ = −1/𝑟 this is the same for both 𝜓2𝑠 and 𝜓2𝑝. For example                 * 2 * 2 2 2 ˆ sin d d d ˆ sin d d d ˆ ˆ d d nl nlm nlm nl lm nl lm nl nl V V r r r R Y V r R Y r r R V r r r V r P r r                 r r   2 2 3 2 2 0 1 ˆ d e d 3!/ 24 1/ 4 24 r V R V r r r r r p p           

Radial probability densities in theH atom0.2HP(r)2pRadial probabilitydensities2s0.1P(r)= r?R. (r)for 2s and 2p hydrogen atom orbitals0.0Schematic potential energy curves for the2468O10hydrogen atom (solid line) and for a many-electron atom (dashed line).V(r)

Radial probability densities for 2s and 2p hydrogen atom orbitals. Schematic potential energy curves for the hydrogen atom (solid line) and for a many￾electron atom (dashed line). Radial probability densities in the H atom     2 2 P r r R r  nl

Orbital energies in many-electron atomsForthemany-electronatom,theoperator0.2V = -1/r is replaced by the function shown by theP(r)2p2sclashed line in the figure, which behaves like -1/r for0.1large r, but like -z/r near the nucleus.Theorbitalsinamany-electronatomarenotthesameas inthehydrogen atom, butthe 2s wave-function110.04826010always has a radial node, and consequently a peak in1the probability density near the nucleus, while the 2pradial density has no node.v(r)Consequently a 2s electron in a many-electron atom1feels the attraction of the nucleus more strongly than a2p electron does, so it has a lower energy

Orbital energies in many-electron atoms For the many-electron atom, the operator 𝑉෠ = −1/𝑟 is replaced by the function shown by the clashed line in the figure, which behaves like −1/𝑟 for large r, but like −𝑍/𝑟 near the nucleus. The orbitals in a many-electron atom are not the same as in the hydrogen atom, but the 2s wave-function always has a radial node, and consequently a peak in the probability density near the nucleus, while the 2p radial density has no node. Consequently a 2s electron in a many-electron atom feels the attraction of the nucleus more strongly than a 2p electron does, so it has a lower energy

A2pelectronismoreeffectivelyscreenedfromthenucleusbytheotherelectronsthana2selectron, or equivalentlythe2s electronpenetrates through the otherelectrons tothenucleus more effectively than the 2p.Either way, the 2s orbitals have lower energy than the 2p, and the difference in energyincreases as we go across the period from Li to Ne.The same thing happens in each row of the periodic table, so the orbital energy level patternbecomessomethinglikethefollowing(omittingthelslevel):E=04f4d4p3d4s3p3s2p2s

A 2p electron is more effectively screened from the nucleus by the other electrons than a 2s electron, or equivalently the 2s electron penetrates through the other electrons to the nucleus more effectively than the 2p. Either way, the 2s orbitals have lower energy than the 2p, and the difference in energy increases as we go across the period from Li to Ne. The same thing happens in each row of the periodic table, so the orbital energy level pattern becomes something like the following (omitting the 1s level):

In the hydrogen-like atom the orbital energies are -Z2/2n2. In many-electron atoms, thenuclear charge is screened by the other electrons and the energy does not increase inmagnitude so quickly with increasing Z.However it does increase, so the electrons withgiven principal quantum number become more strongly bound as Z increases. The orbitalenergies can be expressed very roughly as -Zeff? /2n?2, where Zefr depends on both n and l,and isgivenby Slater'srules:Zefr = Z - Snl, where for electrons with principal quantum number n > 1,Sm =0.35(N, -1)+0.85Nn- +NcoreHere N, is the number of electrons with principal quantum number n, and Ncore is thenumberwithprincipal quantumnumber less thann-1.Sis =0.3 isa special case

In the hydrogen-like atom the orbital energies are −𝑍 2/2𝑛2 . In many-electron atoms, the nuclear charge is screened by the other electrons and the energy does not increase in magnitude so quickly with increasing Z. However it does increase, so the electrons with given principal quantum number become more strongly bound as Z increases. The orbital energies can be expressed very roughly as −𝑍eff 2/2𝑛2 , where 𝑍eff depends on both n and l, and is given by Slater's rules: 𝑍eff = 𝑍 − 𝑠𝑛𝑙 , where for electrons with principal quantum number 𝑛 > 1, Here Nn is the number of electrons with principal quantum number n, and Ncore is the number with principal quantum number less than 𝑛 − 1. 𝑠1𝑠 = 0.3 is a special case. 1 core 0.35( 1) 0.85 nl n n s N N N     