3.1 Basis of singleelectron wavefunctions ( orbitals )


 Kory Richards
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1 3 Atomc physcs multelectron atoms 3. Bass of sngleelectron wavefunctons ( orbtals ) Manypartcle quantum mechancs Multelectron atoms Manybody perturbaton theory (frstorder) HartreeFock selfconsstent feld Atomc physcs multelectron atoms Important to note: ths s not a general overvew to atomc physcs. There are many extremely mportant topcs I am not dscussng at all, e.g., spnorbt effect, theory of angular momentum addton and couplng etc., whch are crucal for understandng quantum mechancs of atoms. However, those are essentally regular quantum mechancs, whch you have seen n other courses, and do not requre a computer. Here, I focus on the aspects of manybody atomc physcs for whch computatonal calculatons are essental. Some excellent sources to pursue these other topcs n more detal are books by Sakura, Johnson, Sobelman 3, and Bethe and Salpeter 4, whch are avalable n the lbrary. 3. Bass of sngleelectron wavefunctons ( orbtals ) Here we brefly revew another method for solvng the radal Schrödnger equaton that s partcularly useful for multelectron atoms. As we shall see below, the total wavefuncton for a multelectron atom s formed from combnatons of snglepartcle wavefunctons (whch we often refer to as orbtals, though that term s used dfferently n many sources). 3.. Algebrac soluton to radal equaton Use some (fnte) set of bass functons {b (r)}, and wrte: P (r) = c b (r). () In general, ths s approxmate, snce the bass set s fnte. As such, a good choce of bass s essental; ths wll be dscussed below. Remember that here P (r) s defned va the radal decomposton for the snglepartcle wavefunctons and s a soluton to the radal Schrödnger equaton: ψ nlm (r) = P nl(r) Y lm (θ, φ), () r HP = εp. (3) Solvng ths equaton s cast to determnng the expanson coeffcents {c }. Note: n general, the bass states b j are orthogonal or normalsed, and they are not egenstates of the Hamltonan. Usng Drac notaton wth = b, ths gves: H b c = ε b j c j (4) j b j H b c = ε b k b c, (5) J. J. Sakura, Modern Quantum Mechancs (0) [n partcular Chapters 3, 5, 7] W. R. Johnson, Atomc Structure Theory (007) 3 I. I. Sobelman, Atomc Spectra and Radatve Transtons (99) 4 H. A. Bethe and E. E. Salpeter, Quantum Mechancs of Oneand TwoElectron Atoms (977)
2 where we multpled on the left by b j (and ntegrated). The result s just a matrx equaton: H j c = ε B j c (6) = Hc = εbc. (7) Ths s a generalsed egenvalue matrx equaton, whch can be solved to yeld a set of egenvalues ε, and egen vectors c. Each egenvector s the set of c expanson coeffcents. Note that f the bass set f orthonormal, the S matrx B j = b b j s just the dentty; however, ths s not true n general. The Fortran routne, DSYGV, wll solve the generalsed egenvalue problem for real, symmetrc matrces. (DSYGV s smlar to DSYEV, but takes n also second matrx, B.) Here, H and B are N b N b square matrces, wth elements: H j = Ĥ j = b (r)ĥb j(r) dr, B j = j = b (r)b j (r) dr, (8) wth N b beng the number of bass states (Bsplnes) used the the expanson () not the number of radal grdponts used for ntegrals. If the bass was orthonormal, S would just be the dentty matrx; n general t s not. For the general case of H = + V (r), we have r H j = b (r)b j (r) dr + b (r)v (r)b j (r) dr (9) = + b (r)b j(r) dr + b (r)v (r)b j (r) dr (0) (ntegraton by parts). In theory, these ntegrals can be found wth extremely hgh accuracy usng Gaussan quadrature. For our purposes, any reasonable ntegraton scheme wll do just fne. Example 3.: DSYGV Parameters. Documentaton: extern "C" nt dsygv_ ( nt * ITYPE, // = for problems of type Av = ebv char * JOBZ, // = V means calculate egenvectors char * UPLO, // U : upper trangle of matrx s stored, L : lower nt * N, // dmenson of matrx A double * A, // c style array for matrx A ( ptr to array, ponter to a [ 0]) // On output, A contans matrx of egenvectors nt *LDA, // For us, LDA =N double * B, // c style array for matrx B [ Av = ebv ] nt *LDB, // For us, LDB =N double * W, // Array of dmenson N  wll hold egenvalues double * WORK, // workspace : array of dmenson LWORK nt * LWORK, // dmenson of workspace : ~ 6* N works well nt * INFO // error code : 0= worked. ); Note: FORTRAN (language LAPACK s wrtten n) uses columnmajor orderng to access D arrays, wle c and c++ use rowmajor. Ths means m[][j] n c++ s m[j][] n FORTRAN.. so we often need to transpose the matrx before sendng to LAPACK Our matrx s symmetrc, so ths doesn t matter, except for uplo
3 uplo : U means upper trangle n FORTRAN s stored so lower n c++ [you may just fll entre matrx] For other LAPACK functons, you can often just tell them the matrx s a transpose, so we don t need to waste tme transposng t ourselves Don t forget to declare the dsygv functon wth extern C, and use the llapack lnker (comple) flag (you may also need the lblas flag) 3.. Bsplne bass functons There are many optons for whch set of bass functons to use n Eq. (). One opton that works very well for atomc physcs s to use Bsplnes. b (r) = r/a B Fgure : Every 3rd splne from a set of 30 Bsplnes of order k = 7 defned on r [0, 50] a.u. The frst nonzero knot was placed at r 0 = 0 3 a.u., and the knots where dstrbuted unformly over [r 0, R] (though t s typcal to nstead dstrbute them logarthmcally). Bsplnes are are a set of N pecewse polynomals of order k, defned over a subdoman r [0, R]. Each splne functon, b, s nonzero for only a small subregon of the doman. Techncal detals (you don t need to know): b (r) n nonzero only for t r < t +n, where {t 0,..., t N } are a seres of N + knots. Frst k knots are placed at r = 0; the fnal k knots are at r = R; the remanng knots are dstrbuted between some nonzero r 0 and R. All splnes go to zero at r = 0, except the 0th splne (ndex 0); the 0th splne (ndex 0) s only nonzero for r < r 0 The kth splne and above [ndex k] are nonzero only for r > r 0 All splnes go to zero at r = R, except for the fnal one (N )th splne 3
4 For any r [0, R], b (r) = Form a complete bass for any polynomal of order k on nterval r [0, R] Typcally, we choose r a.u., R a.u., N = 30 00, and k = Enforce boundary condtons There are several ways to enforce the boundary condtons, ncludng addng extra fcttous nfnte potentals to the Hamltonan at r = 0 and r = R. We wll take a smpler approach, and enforce the boundary condtons by dscardng some of the splnes. Dscard b 0 ths s the only splne nonzero at r = 0, forcng wavefunctons to be P (0) = 0 Dscard fnal splne b N ths s the only splne nonzero at r = R, forcng wavefunctons to be P (R) = 0 ( hard boundary condton) Dscard b We can further take advantage of lowr behavour of the wavefunctons; P (r) r l+. Due to form of the splnes, we can force ths behavour by dscardng b for s and p states. So, wth N total splnes, we actually use only N 3 n the expanson Eq. (). 3. Manypartcle quantum mechancs 3.. Exchange symmetry, product states Quck revew; see Chapter 7 of Sakura 5 for a more complete dscusson (or any other QM textbook). Consder system of two nonnteractng dentcal partcles (they do not nteract wth each other, but may each nteract wth external potental, V ). The total Hamltonan s the sum of those for each partcle (same as n classcal mechancs): H = h = ( ) p m + V (r ). () If ψ a (r) s a soluton to hψ a = ε a ψ a (smlarly for ψ b ), then t s clear that Ψ ab (r, r ) = ψ a (r )ψ b (r ) () s also a soluton to H, wth egenvalue (ε a + ε b ). So s the nterchanged soluton ψ b (r )ψ a (r ) (swapped quantum numbers). In fact, snce two partcles are dentcal, such two solutons are completely ndstngushable. Snce the soluton must of course return to ts orgnal state after two nterchanges of partcles a and b, a sensble general soluton can be wrtten, Ψ ab (r, r ) = [ψ a (r )ψ b (r ) ± ψ b (r )ψ a (r )]. (3) Wthout gong nto the detals, t turns out that for Fermons (/nteger spn partcles), the total wavefuncton must be antsymmetrc under the exchange of any two partcles: Ψ ab (r, r ) Fermon = [ψ a (r )ψ b (r ) ψ b (r )ψ a (r )], (4) 5 J. J. Sakura, Modern Quantum Mechancs (0) 4
5 whle for Bosons (nteger spn partcles) the total wavefuncton must be symmetrc under the exchange Ψ ab (r, r ) Boson = [ψ a (r )ψ b (r ) + ψ b (r )ψ a (r )]. (5) Ths s to ensure Fermons obey the FermDrac statstcs, and Bosons obey the BoseEnsten statstcs. Note that the antsymmetry of Fermon wavefunctons encodes the Paul excluson prncple (spnstatstcs theorem). The same general form apples for systems of many partcles. So long as the system s nonnteractng, the total wavefuncton can be bult of products of snglepartcle wavefunctons, called product states (or Fock states). 3.. Slater determnant For manyfermon systems (assumng them to be nonnteractng), we can form wavefunctons usng product states, but we must ensure those wavefunctons are properly antsymmetrc under exchange. A nce notatonal trck to do ths s to wrte the wavefunctons as determnant, called Slater determnants, of the form: ψ Ψ ab...n (r, r,..., r N ) = a (r ) ψ a (r )... ψ a (r N ) ψ b (r ) ψ b (r )... ψ b (r N ) N!......, (6) ψ n (r ) ψ n (r )... ψ n (r N ) whch s an egenstate of the total Hamltonan H = N h(r ). To avod notaton confuson, note that each r s a dfferent coordnate varable (not a specfc pont on the coordnate grd). It s easest to see that ths works n the case of N = Fermons, but works n general Matrx elements of multfermon wavefunctons Let F and G be general one and twopartcle operators ˆF = ˆf(r ), Ĝ = <j ĝ(r, r j ), (7) where f(r ) acts on the th partcle; g(r, r j ) acts on the par of partcle {, j}. The sum extends over each of the N electrons (or each par of electrons, j; the < j ensures pars are not doublecounted). In regular quantum mechancs, ths s all that s needed; there are no threebody operators. It s farly straghtforward, though a lttle cumbersome, to derve the rules for calculatng matrx elements between manybody Slaterdetermnant wavefunctons. If you do, you wll start to see patterns, whch wll turn nto general rules. I wll not prove the rules here, but just state them; they are easy to verfy for twopartcle wavefunctons. For dagonal matrx elements (expectaton values; ntal state = fnal state): Ψ F Ψ = Ψ G Ψ = <j f ( j g j j g j ). Dagrams representng these are shown n Fg.. Here, the Drac notaton refers to the snglepartcle wavefunctons: a = ψ a, ab = ψ a (r )ψ b (r ), meanng a h b = ψ a h ψ b dv (9) ab h cd = ψ a(r ) ψ b (r ) h ψ c (r )ψ d (r ) dv dv. (8) 5
6 j j j j Fgure : Dagrams representng nteracton of snglepartcle operator wth Fermon (left), the drect part of a twobody operator (mddle), and exchange (rght). Dashed lne represents source of nteracton. For nondagonal matrx elements (transtons), the rules are smlar. Usng a shorthand notaton 6, we have for matrx elements between wavefunctons that dffer by a sngle state: Ψ n F Ψ = n f Ψ n G Ψ = j ( nj g j jn g j ), (0) and for matrx elements between wavefunctons that dffer by two states: Ψ nm j F Ψ = 0 G Ψ = nm g j mn g j. Ψ nm j () 3.3 Multelectron atoms 3.3. Meanfeld (ndependent partcle) model We can use the above mechansms to study the quantum mechancs of manyelectron atoms. H = N ( p Z ) + r <j r r j. () The last term s the electronelectron repulson term, whch ncludes a sum over every par of electrons (the < j ensures pars are not doublecounted). Clearly, ths s not a nonnteractng system. So how can we use the above formalsm? Frst, rewrte the Hamltonan as H = N ( p Z ) r + umf (r ) + δv, (3) where δv = <j r r j u MF (r ). (4) Here, u MF s an (asofyet undetermned) potental, whch s taken to be roughly the average electronelectron nteracton term called the mean feld. If t s chosen wdely, then δv should be small, and can therefore be treated perturbatvely. The physcal justfcaton for ths approxmaton s 6 Here, I use a shorthand notaton Ψ n a = a na a Ψ, (a and a are creaton and annhlaton operators) meanng that one partcle n state a s replaced by one n state n. e.g., f Ψ s a 3partcle wavefuncton wth partcles n states a, b, c, then Ψ n a has partcles n states n, b, c. By state I mean a gven full set of quantum numbers (e.g., n, l, m). 6
7 clear  for a system of large number of nteractng partcles, the ndevdual nteractons between all pars of partcles wll average out, and the average nteracton potental seen by each partcle wll be essentally the same. We therefore frst solve the mean feld Hamltonan wth H H MF H MF = N ( p Z ) r + umf (r ), (5) whch can now be treated as a nonnteractng system. Then, we can deal wth the remanng δv term later, usng perturbaton theory: H = H MF + δv. Ths s called the meanfeld approxmaton, or the ndependent partcle pcture. Ths means we form total (approxmate) manyelectron wavefuncton, Ψ, whch satsfes H MF Ψ = EΨ, (6) from Slater determnants made from sngleelectron solutons to the snglepartcle equaton: hψ = εψ, (7) h = p Z r + umf. (8) In practcle calculatons, we just calculate and store the set of snglepartcle solutons, ψ, and use the rules from Sec to perform manybody calculatons. Importantly, as can be deduced from these rules, the snglepartcle energes ε, correspond to the bndng energes (negatve of the onsaton energes) for the ndvdual electrons n the atom Intal choce for meanfeld There are many ways to choose a good startng approxmaton for u MF ; here we wll dscuss a very smple approach. The atomc potental seen by an electron at very small r s essentally unscreened by other electrons. So, n ths regon, we have V (r) Z/r. Smlarly, at very large r there s total screenng by the (Z ) other electrons, so V (r) /r; see Fg. 3. As a very rough frst guess for the meanfeld potental, we can chose a smooth potental that connects these two regons. A smple choce s to use a parametrc potental; one that works reasonably well s the Green potental (Z ) h ( e r/d ) V Gr (r) = r + h (e r/d ), (9) where h and d are parameters, whch may be tuned to gve reasonable results. Note that V nuc (r) + V Gr (r) behaves lke Z/r for small r, and /r for large r. Before modern computers ths was essentally the best one could do. Now, we can do much better. However, we wll use ths Green s potental as a startng pont, and mprove upon t usng perturbaton theory and the HartreeFock routne. 3.4 Manybody perturbaton theory (frstorder) 3.4. Snglevalence systems We wll frstly focus on atoms that have a sngle valence electron above closedshell core; these are the smplest of the manyelectron atoms. All the electrons, besdes one, are n full shells (each l and m occuped for each n) ths s called the core. The sngle valence electron les n an n shell above 7
8 Fgure 3: Total atomc potental seen by electron at very small r s essentally unscreened by other electrons, V (r) Z/r, whle at very large r there s total screenng by the (Z ) other electrons, V (r) /r. the core. Due to the energy scalng /n, the energy requred to excte a electron from the core s much much larger than that requred to excte the valence electron. Therefore, for snglevalence systems, the core confguraton typcally remans always the same, whle all atomc dynamcs are due to the sngle valence electron. Also, snce the valence electron wavefuncton s mostly located at larger radus, the snglepartcle core wavefunctons depends only very lttle on the nfluence of the valence electron. From the rules presented n Sec. 3..3, all these greatly smplfy the calculatons. The snglepartcle energy of the valence electron can be dentfed as (approxmately) the bndng energy of ths electron. Therefore, dfferences n the snglepartcle energes of dfferent valence states correspond to the transton energes between atoms n those valence states; see Sec Frstorder perturbaton correcton to energes From before, the perturbaton to the total Hamltonan we must consder s: δv = <j r r j u MF (r ), (30) V ee U. (3) (second lne just defnes notaton V ee and U, whch nclude the summatons). quantum mechancs, the frstorder correcton to the total atomc energy s thus Just as n regular δe = Ψ V ee Ψ Ψ U Ψ. (3) To evaluate each of these terms, we must use the rules from Sec The U part s smple, snce these are snglepartcle operators (depend on sngle electron coordnate only): Ψ U Ψ = u MF, (33) where the sum runs over atomc electrons. Note the sum extends over all electrons, and therefore ncludes summaton over all n, l, m and spn quantum numbers. For snglevalence atoms, we can break ths nto the core and valence parts: core Ψ U Ψ = c u MF c + v u MF v, (34) c 8
9 where v denotes the state of the valence electron. Notce that the frst part, the sum over the core electrons, s the same for any valence state (t s just the correcton to core energy). Snce all we can observe n experments s the energy for transtons, and for snglevalence atoms the core essentally remans the same, we can drop ths part (t wll cancel n transtons).e., t does not contrbute to transton energes between valence states, or therefore to the onsaton energy of valence electron. The V ee part s more complcated, snce these are twopartcle operators. From Sec. 3..3, we arrve at [ j r j j r j ] (35) where Ψ V ee Ψ = <j r r r. The frst term s known as the drect contrbuton, and the second s called exchange. Agan, we may separate ths nto terms nvolvng only core states, and those wth core and valence states: core Ψ V ee Ψ = a<c [ ca r ca ca r ac ] core + c [ cv r cv cv r vc ]. (36) Agan, the core part s the same for all valence states, and wll thus cancel n transton and onsaton energes for valence states. Therefore, the frstorder perturbaton theory correcton to the energy for valence state v s Evaluaton of Coulomb ntegrals core [ ] δε v = cv r cv cv r vc v u MF v. (37) c Perturbaton theory requres us to evaluate ntegrals of the form ab r cd, whch correspond to the Coulomb nteracton between electrons. As a remnder, wrtng the ntegral explctly, ths means: ab r cd = ψ a(r )ψ b (r ) r r ψ c(r )ψ d (r ) d 3 r d 3 r. (38) To evaluate ths ntegral, we use the Laplace expanson 7, whch you have lkely come across before n your electrodynamcs course: r r = k=0 r< k r> k+ P k (cos γ), (39) where P s a Legendre polynomal, r < mn( r, r ), r > max( r, r ), and γ s the angle between r and r. For the sphercally symmetrc problems of atom physcs, t becomes much easer to rewrte ths n terms of the sphercal harmoncs (just usng ther defnton): r r = k=0 4π k + r< k r> k+ q ( ) q Y k, q (θ, φ ) Y k,q (θ, φ ), (40) q= k thus separatng the radal and angular parts n a way that makes calculatng ntegrals wth wavefunctons n the form of Eq. () smple. We get expressons n the form: ab r cd = kq C k,q angular Rk abcd, (4) 7 J. D. Jackson, Classcal Electrodynamcs (00) 9
10 where C angular s the angular ntegral, and Rabcd k s the radal ntegral: Rabcd k P a (r )P b (r ) rk < P r> k+ c (r )P d (r ) dr dr, (4) whch s convenent to wrte as R k abcd = 0 P a (r) y k bd(r) P c (r) dr, (43) where y k bd s called a Hartree screenng functon y k bd(r) = 0 r 0 P b (r ) rk < P r> k+ d (r ) dr (44) P b (r ) (r ) k r k+ P d(r ) dr + r P b (r ) r k (r ) k+ P d(r ) dr. (45) The angular part of the ntegral bols down to ntegrals of sphercal harmoncs; note that t depends only on the angular quantum numbers l and m (and multpolarty k and q), and not the specfc form of the wavefuncton. The angular factor can be calculated geometrcally by dong the analytc ntegrals over angular coordnates. In realty, t s much easer to do these ntegrals algebracally, by makng use of the orthogonalty propertes of the sphercal harmoncs and the quantum theory of angular momentum. Smple n concept, such calculatons are often very complex, and we wll not deal wth them at all here. For more nformaton, see the textbooks by Johnson 8, or (for the partcularly brave) Varshalovch HartreeFock selfconsstent feld 3.5. HartreeFock potental Motvated by the above, we defne a potental, whch we wll call v HF (HartreeFock potental), such that a v HF a = [ ] a r a a r a. (46) }{{}}{{} a Drect Exchange Note that ths s the expectaton value of r for the state a.e., the average value of the electronelectron repulson. Therefore, we may use ths potental nstead of u MF when solvng the Schrödnger equaton! I wll justfy ths below, but frst, lets work out explct formulas for v HF. The drect part s easy; we just don t ntegrate over coordnates for ψ a ; vhf drect (r ) = a ψ (r ) r d 3 r. (47) The exchange part s more complex, and cannot be wrtten as a local potental. Instead, we have: vhf exch. ψ a (r ) = ( ) ψ (r )ψ a (r ) d 3 r ψ (r ). (48) r a It s farly easy to check that these formulas agree wth Eq. (46) (v HF = vhf drect for valence states, the summatons smply extend over all core electrons. + v exch. ). Notce that, HF 8 W. R. Johnson, Atomc Structure Theory (007) 9 D. A. Varshalovch, A. N. Moskalev, and V. K. Khersonsk, Quantum Theory of Angular Momentum (988) 0
11 After performng the angular ntegrals, and after summng over magnetc quantum numbers, we can present the effectve drect potental that enters the radal Schrödnger equaton. v drect P v (r) = core n cl c (l c + ) y 0 cc(r) P v (r), (49) where the sum extends over just the prncple and orbtal angular momentum quantum numbers for the electrons n the core, and y 0 s gven by Eq. (44). Only the k = 0 term from the Laplace expanson survves n the drect part. The angular part of the exchange term s a lttle more complcated, and requres some theory we have not yet covered. There s a second way to derve the drect part of the potental, whch further justfes ts use as a potental n the Schrödnger equaton. Consder the electrostatc potental seen by an electron, a, due to the other Z other electrons n an atom. Ths s due to the electrc charge densty of the electron cloud, call t ρ(r). Now, we know that ths s just the same as the electron probablty densty, tmes the electron charge: ρ(r) = e ψ (r). a We can use Gauss law to work out what the electrostatc potental due to ths charge densty s; t wll not be a surprse that we get exactly back the drect part of the potental, Eq. (47) Selfconsstent feld method In the prevous secton we derved the HartreeFock potental, whch we hope to use n place of the meanfeld potental u MF when solvng the Schrödnger equaton to determne the wavefunctons. However, the HartreeFock potental tself depends on the wavefunctons! Therefore, we must start wth an ntal guess for the wavefunctons, and teratvely mprove our approxmaton. Use ntal approxmaton for u MF (e.g., Green potental), solve Schrödnger equaton to generate set of sngleelectron wavefunctons Use these wavefunctons to form v HF, whch s a better potental than the ntal approxmaton Use ths better potental to generate a set of better wavefunctons Now that we have a better set of wavefunctons, can form a betteryet potental, and soon We contnue ths procedure teratvely, untl the solutons converge. The physcal justfcaton for ths procedure s smple; the nterelectronc potental depends on the dstrbuton of atomc electrons. By contnuously mprovng the model for the electron wavefunctons, we wll mprove our model for the nterelectronc potental, whch wll mprove our resultng wavefunctons and so on. That ths method s an accurate approxmaton can be proven by consderng the frstorder perturbaton theory correcton to the energy, when v HF was used as the potental n the Schrödnger equaton. In fact, due to the very defnton of the HartreeFock potental, t s a smple matter to see that ths s exactly zero! Therefore, there are no frstorder correctons to energes (or, ndeed, to the wavefunctons) when the HartreeFock potental s used. In order to mprove the accuracy further, we must consder hgherorder perturbaton theory correctons, whch become much more complcated, and s an actve area of current research (honours/phd projects avalable!).
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