5.61 Physical Chemistry Lecture #23 page 1 MANY ELECTRON ATOMS

Transcription

1 5.6 Physical Chemisty Lectue #3 page MAY ELECTRO ATOMS At this point, we see that quantum mechanics allows us to undestand the helium atom, at least qualitatively. What about atoms with moe than two electons, such as lithium o cabon? Hee, we discuss how the independent paticle model (IPM) is capable of giving a ealistic pictue of atomic stuctue in essentially an analogous fashion to the helium case. To begin with, we set up ou coodinates so that the nucleus is at the oigin and the electons ae at positions,, 3,. In tems of these vaiables, we can quicly wite down the many-electon Hamiltonian Z Ĥ = j + j= j= j j= < j j Kinetic Enegy e-ucleus Attaction e-e Repulsion ote that the numbe of electons () need not be the same as the nuclea chage (Z) we might, fo example, have a cation whee <Z o an anion whee >Z. Thus, the Hamiltonian has the same thee souces of enegy as in the two electon case, but the shee numbe of electons maes the algeba moe complicated. As befoe, we note that we can mae the Hamiltonian sepaable if we neglect the electon-electon epulsion: Z Z Z Ĥ = ˆ IP j = j h j j= j= j j= j j= whee each of the independent Hamiltonians ĥ i descibes a single electon in the field of a nucleus of chage +Z and the subscipt IP eminds us that this Hamiltonian descibes the paticles as if they wee independent of one anothe. Based on ou expeience with sepaable Hamiltonians, we can immediately wite down the eigenstates of this Hamiltonian as poducts with enegies given as sums of the independent electon enegies: Ψ =ψ ()ψ ()ψ (3)...ψ ( ) E = E + E + E E 3 3 Whee () is a shothand fo (,σ ) and i { ni, l i, m i,s i } i {n i,l i,m l,i,m s,i }specifies all the quantum numbes fo a given hydogen atom eigenstate. Of couse, thee is a poblem with these eigenstates: they ae not antisymmetic. Fo the Helium atom, we

2 5.6 Physical Chemisty Lectue #3 page fixed this by maing an explicitly antisymmetic combination of two degeneate poduct states: ψ ( ) sα () ψ ( ) ψ ( sα )ψ () ψ ()ψ () sα = sβ sα sβ ψ sβ () ψ sβ () whee on the ight we have noted that this antisymmetic poduct can also be witten as a deteminant of a x matix. As it tuns out, it is staightfowad to extend this idea to geneate an paticle antisymetic state by computing an x deteminant called a Slate Deteminant: ψ () ψ () ψ 3 () ψ () ψ () ψ () ψ 3 () ψ () Ψ(,,3,... ) = ψ (3) ψ (3) ψ 3 (3) ψ! (3) ψ ( ) ψ ( ) ψ 3 ( ) ψ ( ) As you can imagine, the algeba equied to compute integals involving Slate deteminants is extemely difficult. It is theefoe most impotant that you ealize seveal things about these states so that you can avoid unnecessay algeba: A Slate deteminant coesponds to a single stic diagam. This is easy to see by example: p x sα( ) sβ () sα() p x α() sα sβ sα p α s ( ) ( ) ( ) x ( ) Ψ (,, 3, 4) = sα( 3) sβ (3) sα(3) p x α(3) s sα( 4) sβ (4) sα(4) p x α(4) It should be clea that w..ψ (,,3,4) e can extend this idea to associate a deteminant with an abitay stic diagam. Futhe, ecall that fo the excited states of helium we had a poblem witing cetain stic diagams as a (space)x(spin) poduct and had to mae linea combinations of cetain states to foce things to sepaate. Because of the diect coespondence of stic diagams and Slate deteminants, the same pitfall aises hee: Slate deteminants sometimes may not be epesentable as a (space)x(spin) poduct, in which case a linea combination of Slate deteminants must be used instead. This geneally only happens fo systems with unpaied electons, lie the s s configuation of helium o the p x p y configuation of cabon.

3 5.6 Physical Chemisty Lectue #3 page 3 A Slate deteminant is anitsymmetic upon exchange of any two electons. We ecall that if we tae a matix and intechange two its ows, the deteminant changes sign. Thus, intechanging and above, fo example: Intechange Rows ψ () ψ () ψ () ψ () 3 ψ () ψ () ψ 3 () ψ ( ) Ψ(,,3,... ) = ψ )! (3) ψ (3 ψ 3 (3) ψ (3) ψ ( ) ψ ( ) ψ 3 ( ) ψ ( ) & ψ! ψ () ψ () ψ () ψ () 3 ψ () ψ () ψ 3 () ψ () (3) ψ (3) ψ 3 (3) ψ (3) = Ψ(,,3... ) A ψ ( ) ψ ( ) ψ 3 ( ) ψ ( ) simila agument applies to any pai of indices, and so the Slate deteminant antisymmetic unde any i j intechange. The deteminant is zeo if the same obital appeas twice. We ecall that if we tae a matix and intechange two of its columns, the deteminant also changes sign. Assuming = 3 above: Ψ(,,3,... ) = ψ (3 (3! Intechange C olum ns &3! ψ () ψ ) ψ () ψ () ψ () ψ () ) ψ ) ψ ( 3) ψ (3) ψ ( ) ψ ( ) ψ ( ) ψ ( ) ψ () ψ () ψ () ψ ( ψ () ψ () ψ () ψ ( ) ) 3 ψ ( ) ψ (3) ψ 3 ψ ( ) ψ ( ) ψ ( ) ψ ( ( ) ψ ( 3) = Ψ(,,3... ) ) The only way a numbe can be equal to its opposite is if it is zeo. The same esult holds if we choose any two of the obitals i and j. Thus, in enfocing

4 5.6 Physical Chemisty Lectue #3 page 4 antisymmety the deteminant also enfoces the Pauli exclusion pinciple. Thus, Slate deteminants immediately lead us to the aufbau pinciple we leaned in Feshman chemisty: because we cannot place two electons in obitals with the same quantum numbes, we ae foced to fill up obitals with successively highe enegies as we add moe electons: s, s, s, s,. This filling up is dictated by the fact that the wavefunction is antisymmetic, which, in tun, esults fom the fact that the electon is spin-½. As it tuns out, all half-intege spin paticles (Femions) have antisymmetic wavefunctions and all intege spin paticles (Bosons) have symmetic wavefunctions. Imagine how diffeent life would be if electons wee Bosons instead of Femions! Ψ is nomalized if the obitals ψ i ae othonomal and two deteminants ae othogonal if they diffe in any single obital. These two facts ae elatively tedious to pove, but ae useful in pactice. The non-inteacting enegy of a Slate deteminant is the enegy of the obitals that mae it up. Just as was the case fo helium, ou antisymmetic wavefunction is a linea combination of degeneate non-inteacting eigenstates: Ψ(,,..., ) = (ψ ()ψ ()ψ (3) ψ ( ) ψ ()ψ ()ψ (3) ψ 3 3! ( ) + ψ (3)ψ () ψ ( ) ψ ( 3 ) ψ ( ) ψ 3 ψ ψ...) ( ) 3 ( ) ( ) Thus the deteminant itself is an eigenstate of the non-inteacting Hamiltonian with the same eigenvalue as each state in the sum: E Ψ = E + E + E E 3 We have only completed half of the independent paticle pictue at this point. We have the noninteacting enegy; what emains is the computation of the aveage enegy. To compute this, we would need to do a -dimensional integal involving a Slate Deteminant on the left, the Hamiltonian in the middle (including all inteaction tems) and a Slate Deteminant on the ight. The boo-eeping is quite tedious, but can be woed out in the most geneal case. The esult is that the enegy beas down into tems that we aleady ecognize:

5 5.6 Physical Chemisty Lectue #3 page 5 oninteacting Enegy Aveage Repulsion H ˆ i = = E i + J ij K i < j J ij ψ * ()ψ * () ψ ()ψ ()d d j ij d σ dσ i j i K ij ψ * ψ ψ d dσ dσ ( )ψ * i ) ( ) d j i ( ) j ( This enegy expession has a nice intuitive feel to it: we get contibutions fom the inteaction of each electon obital with the nucleus (fist tem) as well as fom the mutual epulsion of each pai of obitals (second tem). The epulsion has the chaacteistic Coulomb minus Exchange fom because of the antisymmety of the Slate deteminant, which leads to a minus sign evey time we exchange and. Fo example, when we expand the deteminant in tems of poduct functions, we will have a tem lie: X ψ ()ψ ()ψ (?) ψ (? ) 3 we will have a coesponding tem with with a minus sign: Y ψ ( ) ψ ()ψ (?) ψ ( 3? ) When we compute the aveage epulsion we will have: (X Y...) (X Y...) d d... The +XX and +YY tems will give us Coulomb integals and the coss tems -XY and -YX will give us exchange integals. We note that, as defined, the Coulomb and Exchange tems ae both positive, so that the exchange integal always educes the epulsion enegy fo a Slate deteminant. These aguments do not constitute a poof of the enegy expession above they ae meely intended to give you a geneal feeling that this expession is plausible. Deiving the aveage enegy actually taes quite a bit of time and caeful booeeping of diffeent pemutations (e.g. 4 vesus 4 3 ). If you want to delve deepe into these inds of things, we ecommend you tae 5.73, which coves much of the mateial hee in geate depth.

6 5.6 Physical Chemisty Lectue #3 page 6 In any case, most of the time we need not woy about the details of how the aveage enegy expession is deived fo a geneal deteminant. Usually what we want to do is use the fomula to compute something inteesting athe than e-deive it. Towad this end we note that the Coulomb and exchange integals above involve integation ove spin vaiables, which we can be done tivially. Basically, the integation ove σ (σ ) will give unity if the left and ight functions fo electon (electon ) have the same spin, and zeo othewise. Fo the Coulomb integal, the left and ight functions ae the same, so the spin integation always gives and we can get id of the spin integation : J ij = ψ * i ()s * i (σ )ψ * ()s * j (σ ) j ψ i ()s i (σ )ψ j ()s j (σ )d d dσ dσ ψ i ()ψ j ()d d = s* i (σ ) s * j (σ )s i (σ )s j (σ )dσ dσ ψ * i ()ψ * j () = ψ * i ( )ψ * j () ψ i ()ψ j ( )d d J ij whee hee we use the shothand i { n i, m i, l i } to epesent all the spatial quantum numbes fo the i th obital, and similaly fo the j th one. Fo the exchange integals, the left and ight functions ae diffeent, and so the exchange integal is only nonzeo if the i th obital and the j th obital have the same spin pat: i (σ )ψ j ()s j (σ ) ψ i ()s i (σ )ψ j ()s j (σ )d d dσ dσ s (σ ψ * j )dσ dσ i ()ψ * j () ψ i ()ψ j ()d d if s i =s j 0 if s i s j K ij = ψ * i ()s * * * = s* i (σ )s * j (σ )s i (σ ) ψ * i ()ψ * j () ψ ψ d d K if s =s i ( ) j ( ) ij i j = if s i s j 0 In pactice (fo example on a Poblem Set) it is useful to begin with the geneal expession fo the independent paticle enegy and then integate out the spin pat to get a final expession that only involves the spatial integals J and K athe than the space-spin integals J and K. Fo example, say we ae inteested in the s s s configuation of lithium. The enegy is given by

7 5.6 Physical Chemisty Lectue #3 page 7 Ĥ = E + J K i ij i = i < j ij = E + E + E + J K + J K + J K s s s sα;sβ sα;sβ sα;sα sα;sα sβ;sα sβ;sα = E + E + J + J K s s ss ss ss Thus, in going fom space-spin integations to space integations, we acquie a few factos of, but the fomula still loos simila to the expessions we ve seen befoe. The independent paticle enegy expession is extemely poweful: it allows us to mae ough pedictions of things lie atomic ionization enegies and excitation enegies once we have the Coulomb and exchange integals. The independent paticle enegy also gives us a igoous explanation of anothe feshman chemisty effect: shielding. As we now, while s and p ae degeneate fo the hydogen atom they ae not degeneate as fa as the ode of filling up obitals. s comes fist and p comes second. As you now, physically this aises because the s obital sees an effective nuclea chage that is bigge than p because it is shielded less by the s obital. It tuns out that the IPM has a simple means of explaining this esult. If we loo at the s s p x configuation of lithium and go though exactly the same manipulations as we did above fo s s s we get: Ĥ = E + E + J + J K s p x ss s p x s p x taing the diffeence between s s p x and s s s gives: ΔE = E s p E s s = E + E + J + J K s px ss spx sp x ( E + E + J + s s ss J ss K ss) = J J + K K 0 s p x s s s s s p x Thus, while s s p x and s s s give the same enegies if the electons do not inteact, they give diffeent enegies once we include the aveage inteaction. Futhe, it is clea fom the above expession that the elevant inteactions ae between the s and p obitals on the one hand and the s and s obitals on the othe. Using ou physical agument of shielding, we asset that the s and s obitals will epel each othe less than the s and p obitals, leading to a net stabilization of s s s elative to s s p x. We exploe this idea in moe detail on the poblem set.

8 5.6 Physical Chemisty Lectue #3 page 8 While thee ae many things we can pedict using the IPM fo example, we can show that the ionization potential of fluoine is geate than that of oxygen, in line with what we leaned in geneal chemisty this model is not pefect. McQuaie goes to geat lengths to show that one can impove the IPM by using the hydogenic obitals with Z eff <Z. That is to say, one gets bette answes if one uses valence obitals that have been patly shielded by the inne coe. This gives bette answes, but even this model is not always accuate. The most impotant ole that the IPM plays in chemisty is not its ability to give us good numbes but the fact that it gives us a simple famewo to stat taling about the electonic stuctue of complex molecules. We just have to ecognize that thee will still need to be some coections (o pehaps fudge factos) applied to the IPM if we want numeically accuate answes.

9 MIT OpenCouseWae Physical Chemisty Fall 03 Fo infomation about citing these mateials o ou Tems of Use, visit: