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Lecture 20 Helium and heavier atoms (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsoring agencies. Helium and heavier atoms We use the exact solutions of hydrogenic Schrödinger equation or orbitals to construct an approximate wave function of a manyelectron atom, the helium and heavier atoms. Unlike the hydrogenic atom, the discussion here is approximate and some rules introduced can have exceptions. Spins and antisymmetry of fermion wave functions start to play a critical role. Helium and heavier atoms The Schrödinger equation for hydrogenic atoms can be solved exactly, analytically. Those for many-electron atoms and molecules cannot be solved analytically. The wave function is a coupled function of many variables: ( Y r1,r2 ,… ,rn ) Coordinates of electron 1 The orbital approximation We introduce the following approximation (the orbital approximation): ( ) ( ) ( ) Y r1,r2 ,… ,rn » j1 r1 j 2 r2 Hydrogenic orbital j n (rn ) For the helium atom, this amounts to r1, r2 1 r1 2 r2 The orbital approximation The approximation is equivalent to neglecting interaction between electrons 1 and 2, Hˆ = - 2 2m Ñ 2 1 Ze 2 4pe 0r1 - 2 2m Ñ 2 2 Ze 2 4pe 0r2 + e 2 4pe 0r12 Hydrogenic electron 1 Hydrogenic electron 2 Interaction … so that, Hˆ » - 2 2m Ñ 2 1 Hˆ 1 Z - 2 4pe 0r1 2m Ñ 2 2 Hˆ 2 Z 4pe 0 r2 = Hˆ 1 + Hˆ 2 The orbital approximation H »- 2 2m Ñ 2 1 Z 4pe 0 r1 - 2 Ñ 2 2 2m Z 4pe 0 r2 = H1 + H 2 ( ) ( ) exact hydrogenic H j ( r ) = E j ( r ) problem ( H + H )j ( r )j (r ) = ( H j ( r ) ) j ( r ) + ( H j ( r ) )j ( r ) = ( E + E )j (r )j (r ) Eigenfunction H1j1 r1 = E1j1 r1 2 2 1 1 1 2 2 1 2 1 2 1 2 2 1 2 1 1 2 2 2 2 2 2 2 2 1 1 The orbital approximation We construct a helium wave function as the product of hydrogenic orbitals with Z = 2. r1, r2 1 r1 2 r2 Issue #1: an electron is fermion and fermions’ wave function must be antisymmetric with respect to interchange (the above isn’t): Y ( ,rn , ,rm , ) = -Y ( ,rm , ,rn , ) Issue #2: each electron must be either spin α or β (the above neglects spins). Spins Let us first append spin factors r1 , r2 1 r1 2 r2 (1) (2) r1 , r2 1 r1 2 r2 (1) (2) r1 , r2 1 r1 2 r2 (1) (2) r1 , r2 1 r1 2 r2 (1) (2) None of these is antisymmetric yet Y ( ,rn , ,rm , ) = -Y ( ,rm , ,rn , ) (Anti)symmetrization Symmetrization: Sym. g+ ( x, y ) = f ( x, y ) + f ( y, x ) g+ ( y, x ) = f ( y, x ) + f ( x, y ) = g+ ( x, y ) Antisymmetrization: Antisym. g- ( x, y ) = f ( x, y ) - f ( y, x ) g- ( y, x ) = f ( y, x ) - f ( x, y ) = -g- ( x, y ) Antisymmetric function Antisym. Sym. Antisym. h- ( x, y ) = g+ ( x, y ) g- ( x, y ) { } h- ( y, x ) = g+ ( y, x ) g- ( y, x ) = g+ ( x, y ) -g- ( x, y ) = -h- ( x, y ) Helium wave functions Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 )a (1)a (2) Already symmetric and cannot Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 ) b (1)b (2) be made antisymmetric Neither sym. or antisym. Antisym. Sym. { } Y ( r ,r ) µ {j ( r )j ( r ) - j ( r )j ( r )} b (1)b (2) Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 ) - j1 ( r2 )j 2 ( r1 ) a (1)a (2) 1 2 1 { Antisym. { Sym. 1 2 2 1 2 2 1 } Sym. } Antisym. Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 ) - j1 ( r2 )j 2 ( r1 ) {a (1)b (2) + b (1)a (2)} Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 )a (1)b (2) Neither sym. Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 ) b (1)a (2) or antisym. Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 ) + j1 ( r2 )j 2 ( r1 ) {a (1)b (2) - b (1)a (2)} Triplet states These three have the same spatial shape – the same probability density and energy – triply degenerate (triplet states) Antisym. Sym. { } Y ( r ,r ) µ {j ( r )j ( r ) - j ( r )j ( r )} b (1)b (2) Y ( r ,r ) µ {j ( r )j ( r ) - j ( r )j ( r )} {a (1)b (2) + b (1)a (2)} Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 ) - j1 ( r2 )j 2 ( r1 ) a (1)a (2) 1 2 1 1 2 2 1 2 2 1 1 2 1 1 2 2 1 2 2 1 φ1 and φ2 cannot have the same spatial form (otherwise this part becomes zero). Electrons 1 and 2 cannot be in the same orbital or same spatial position in triplet states (cf. Pauli exclusion principle) Singlet state There is another state which is nondegenerate (singlet state): { Sym. } Antisym. Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 ) + j1 ( r2 )j 2 ( r1 ) {a (1)b (2) - b (1)a (2)} Opposite spins φ1 and φ2 can have the same spatial form because the anti-symmetry is ensured by the spin part. Electrons 1 and 2 can be found at the same spatial position. Energy ordering For the helium atom, depicting α- and β-spin electrons by upward and downward arrows, we can specify its electron configurations. 2s 2s 2s 1s 1s 1s Triplet states Singlet state A Singlet state B The orbital approximation Ze Ze e 2 H =Ñ Ñ2 + 2m 4pe 0 r1 2m 4pe 0 r2 4pe 0 r12 2 2 2 2 2 2 1 Hydrogenic electron 1 Hydrogenic electron 2 Interaction 2s 2s 1s 1s 2s Triplet states 159856 cm-1 Singlet A 166277 cm-1 1s 0 cm-1 Singlet B Beyond helium … A many-electron atom’s groundstate configuration can be obtained by filling two electrons (α and β spin) in each of the corresponding hydrogenic orbitals from below. When a shell (K, L, M, etc.) is completely filled, the atom becomes a closed shell – a chemically inert species like rare gas species. Electrons partially filling the outermost shell are chemically active valence electrons. Shielding In a hydrogenic atom (with only one electron), s, p, d orbitals in the same shell are degenerate. However, for more than one electrons, this will no longer be true. Nuclear charge is partially shielded by other electrons making the outer orbitals energies higher. Shielding Electrons in outer, more diffuse orbitals experience Coulomb potential of nuclear charge less than Z because inner electrons shield it. Zeff Z Effective nuclear charge Shielding The s functions have greater probability density near the nucleus than p or d in the same shell and experience less shielding. Consequently, the energy ordering in a shell is Lower 40 energy s< p<d < f < 3p 30 3d 20 3s 10 0 5 10 15 20 Aufbau principle This explains the well-known building-up (aufbau) principle of atomic configuration based on the order (exceptions exist). 1s < 2s < 2 p < 3s < 3p < 4s < 3d < 4 p < 5s < 4d < 5p < 6s < 6s 6p 6d 6f 6g 6g 5s 5p 5d 5f 5g 4s 4p 4d 4f 3s 3p 3d 2s 2p 1s Hund’s rule An atom in its ground state adopts a configuration with the greatest number of unpaired electrons (exceptions exist) – why? 2p 2s 1s Oxygen Hund’s rule Spin correlation or Pauli exclusion rule explains Hund’s rule. { } Y ( r1 ,r2 ) µ j 2 p ( r1 )j 2 p ( r2 ) - j 2 p ( r2 )j 2 p ( r1 ) a (1)a (2) y z y z 2p Y ( r1 ,r2 ) µ j 2 p ( r1 )j 2 p ( r2 ) {a (1)b (2) - b (1)a (2)} y y Two electrons can be in the same spatial orbitals and the same position Spatial part is antisymmetric and the two electron cannot occupy the same spatial orbitals or the same position – energetically more favorable Summary We have learned the orbital approximation, an approximate wave function of a manyelectron atom that is an antisymmetric product of hydrogenic orbitals. We have learned how the (anti)symmetry of spin part affects the spatial part and hence energies and the singlet & triplet helium atom and explains Hund’s rule. Shielding explains the aufbau principle.