o Schrödinger equation for o Two-electron atoms. o Multi-electron atoms. o Helium-like atoms. o Singlet and triplet states. o Exchange energy. o See Chapter 9 of Eisberg & Resnick PY3P05 o See http://en.wikipedia.org/wiki/Electron_configuration PY3P05 o What is probability of simultaneously finding a particle 1 at (x1,y1,z1), particle 2 at (x2,y2,z2), etc. => need joint probability distribution. o N-particle system is therefore a function of 3N coordinates: z !(x1,y1,z1; x2,y2,z2; … xN,yN,zN) ˆ ˆ ˆ ˆ ˆ ˆ ˆ o Must solve H " (r1 ,r2 ,..., rN ) = E"(r1 ,r2 ,...,rN ) (1) e2 e1 o First consider two particles which do not interact with one r 2 r1 another, but move !in potentials V1 and V2. The Hamiltonian is Hˆ = Hˆ + Hˆ (2) y 1 2 x 2 2 $ ! 2 ' $ ! 2 ' = & " #1 + V1(rˆ1 )) + & " # 2 + V2(rˆ2 )) % 2m1 ( % 2m2 ( 2 2e i =1,2 o The electron-nucleus potential for helium is Vi = " 4#$0ri ! o The eigenfunctions of H1 and H2 can be written as the product: "(rˆ1 ,rˆ2 ) =#1(rˆ1 )#2 (rˆ2 ) ! ! PY3P05 ! o Using this and Eqns. 1 and 2, ˆ ˆ ˆ H" (rˆ1 ,rˆ2 ) = (H1 + H 2 )#1(rˆ1 )#2(rˆ2 ) = (E1 + E 2 )#1(rˆ1 )#2(rˆ2 ) ˆ o That is, H" (rˆ1 ,rˆ2 ) = E " ( rˆ 1 , r ˆ 2 ) where E = E1+E2. ! o The product wavefunction is an eigenfunction of the complete Hamiltonian H, corresponding !to an eigenvalue E which is the sum of the energy eigenvalues of the two separate particles. o For N-particles, ˆ ˆ ˆ ˆ ˆ ˆ " (r1 ,r2 ,...rN ) =#1(r1 )#2 (r2 )!#N (rN ) o Eigenvalues of each particle’s Hamiltonian determine possible energies. Total energy is thus N ! E = " E i i=1 o Can be used as a first approximation to two interacting particles. Can then use perturbation theory to include interaction. ! PY3P05 o Assuming each electron in helium is non-interacting, can assume each can be treated independently with hydrogenic energy levels: 13.6Z 2 E = " Observed n n 2 o Total energy of two-electron system in ground -50 state (n(1) = n(2) = 1) is therefore -60 ! E = E1(1) + E1(2) Energy -70 # & 2 1 1 (eV) = "13.6Z % 2 + 2 ( $ n(1) n(2) ' -80 # 1 1 & = "13.6(2)2% + ( $ 12 12 ' -90 = "109 eV -100 o For first excited state, n(1) = 1, n(2) = 2 => E =-68 eV. -110 ! Neglecting electron-electron interaction PY3P05 o For He-like atoms can extend to include electron-electron interaction: z & 2 2 ) & 2 2 ) 2 ˆ "! 2 Ze "! 2 Ze e H = ( #1 " + + ( # 2 " + + 2m 4$% r 2m 4$% r 4$% r e r e ' 0 1 * ' 0 2 * 0 12 2 12 1 o The final term represents electron-electron repulsion at a r 2 r1 distance r12. ! y o Or for N-electrons, the Hamiltonian is: x N ˆ ˆ H = "H i i=1 N 2 2 N 2 ' #! 2 Ze * e = ) $ i # , + " 2m 4%& r " 4%& r i=1 ( 0 i + i> j 0 ij and the corresponding Schrödinger equation is again of the form Hˆ " (rˆ ,rˆ ,...,rˆ ) = E"(rˆ ,rˆ ,...,rˆ ) ! 1 2 N 1 2 N where N "(rˆ ,rˆ ,...,rˆ ) =# (rˆ )# (rˆ )!# (rˆ ) and E = E 1 2 N 1 1 2 2 N N " i i=1 ! PY3P05 ! ! ' 2 Ze2 e2 * ˆ ˆ #! 2 ˆ o The solutions to the equation H i"i (ri ) = ) $ i # + ," i (ri ) ( 2m 4%&0ri 4%&0rij + ˆ = E i"i (ri ) can again be written in the form (rˆ , , ) R (rˆ ) ( , ) "i i #i $i = ni li i %li mi #i $i 2 2 ! d Rn l 2 dRn l 2µ $ e ' o The radial wave functions are solutions to i i i i E R 0 2 + + 2 & + ) ni li = dr r dri ! % 4"#0ri ( ! i and therefore have the same analytical form as for the hydrogenic one-electron atom. o Allowable solutions again only exist! for Z 2µe4 E = " eff n (4#$ )22!2n 2 0 where Zeff = Z - !nl. o Zeff is the effective nuclear charge and !nl is the shielding constant. This gives rise to the shell model for multi-electron !atoms. PY3P05 o Includes He and Group II elements (e.g., Be, Mg, Ca, etc.). Valence electrons are indistinguishable, i.e., not physically possible to assign unique positions simultaneously. o This means that multi-electron wave functions must have exchange symmetry: 2 2 | "(rˆ1 ,rˆ2 ,...,rˆK ,rˆL ,...,rˆN ) | =| "(rˆ1 ,rˆ2 ,...,rˆL ,rˆK ,...,rˆN ) | which will be satisfied if "(rˆ1 ,rˆ2 ,...,rˆK ,rˆL ,...,rˆN ) = ±"(rˆ1 ,rˆ2 ,...,rˆL ,rˆK ,...,rˆN ) ! o That is, exchanging labels of pair of electrons has no effect on wave function. ! o The “+” sign applies if the particles are bosons. These are said to be symmetric with respect to particle exchange. The “-” sign applies to fermions, which are antisymmetric with respect to particle exchange. o As electrons are fermions (spin 1/2), the wavefunction of a multi-electron atom must be anti- symmetric with respect to particle exchange. PY3P05 o He atom consists of a nucleus with Z = 2 and two electrons. z o Must now include electron spins. Two-electron wave function is therefore written as a product spatial and a spin e2 r12 e1 wave functions: r " =#spatial (rˆ1 ,rˆ2 )#spin 2 r1 o As electrons are indistinguishable => ! must be anti- y Z=2 x symmetric. See table for allowed symmetries of spatial and spin wave! functions. PY3P05 o State of atom is specified by configuration of two electrons. In ground state, both electrons are is 1s shell, so we have a 1s2 configuration. o In excited state, one or both electrons will be in higher shell (e.g., 1s12s1). Configuration must therefore be written in terms of particle #1 in a state defined by four quantum numbers (called "). State of particle #2 called #. o Total wave function for a excited atom can therefore be written: " =#$ (rˆ1 )#% (rˆ2 ) o But, this does not take into account that electrons are indistinguishable. The following is therefore equally valid: ˆ ˆ " =#$ (r1 )#% (r2 ) ! o Because both these are solutions of Schrödinger equation, linear combination also a solutions: 1 ! " = (# (rˆ )# (rˆ ) +# (rˆ )# (rˆ )) Symmetric S 2 $ 1 % 2 % 1 $ 2 1 " = (# (rˆ )# (rˆ ) &# (rˆ )# (rˆ )) Antisymmetric A 2 $ 1 % 2 % 1 $ 2 where 1 / 2 is a normalisation factor. ! PY3P05 ! o There are two electrons => S = s1+ s2 = 0 or 1. S = 0 states are called singlets because they only have one ms value. S = 1 states are called triplets as ms = +1, 0, -1. o There are four possible ways to combine the spins of the two electrons so that the total wave function has exchange symmetry. o Only one possible anitsymmetric spin eigenfunction: 1 [(+1/2,"1/2) " ("1/2,+1/2)] singlet 2 o There are three possible symmetric spin eigenfunctions: ! (+1/2,+1/2) 1 [(+1/2,"1/2) + ("1/2,+1/2)] triplet 2 ("1/2,"1/2) ! PY3P05 o Table gives spin wave functions for a two- electron system. The arrows indicate whether the spin of the individual electrons is up or down (i.e. +1/2 or -1/2). o The + sign in the symmetry column applies if the wave function is symmetric with respect to particle exchange, while the - sign indicates that the wave function is anti-symmetric. o The Sz value is indicated by the quantum number for ms, which is obtained by adding the ms values of the two electrons together. PY3P05 o Singlet and triplet states therefore have different spatial wave functions. o Surprising as spin and spatial wavefunctions are basically independent of each other. o This has a strong effect on the energies of the allowed states. S ms $spin $spatial 1 Singlet 0 0 1/ 2("# $#" ) ("# (rˆ1 )"$ (rˆ2 ) +"$ (rˆ1 )"# (rˆ2 )) 1 2 1 2 2 +1 "1"2 1 1 0 1/ 2("# +#" ) (" (rˆ )" (rˆ ) %" (rˆ )" (rˆ )) Triplet ! 1 2 1 2 # 1 $ 2 $ 1 # 2 ! 2 -1 #1#2 ! ! PY3P05 o Physical interpretation of singlet and triplet states can be obtained by evaluating the total spin angular momentum (S), where ˆ ˆ ˆ S = S1 + S 2 is the sum of the spin angular momenta of the two electrons. o The magnutude of the total spin and its z-component are quantised: S = s(s +1)! ! S = m ! z s where ms = -s, … +s and s = 0, 1. z o If s1 = +1/2 and s2 = -1/2 => s = 0. m ! s triplet o Therefore ms = 0 (singlet state) +1 state s1=1/2 o If s1 = +1/2 and s2 = +1/2 => s = 1. s2=1/2 s = 1 o Therefore ms = -1, 0, +1 (triplet states) 0 -1 singlet s1=1/2 s2=-1/2 state s = 0, ms = 0 PY3P05 o Angular momenta of electrons are described by l1, l2, s1, s2. o As Z<30 for He, use LS or Russel Saunders coupling. o Consider ground state configuration of He: 1s2 o Orbital angular momentum: l1=l2 = 0 => L = l1 + l2 = 0 o Gives rise to an S term. o Spin angular momentum: s1 = s2 = 1/2 => S = 0 or 1 o Multiplicity (2S+1) is therefore 2(0) + 1 = 1 (singlet) or 2(1) + 1 = 3 (triplet) o J = L + S, …, |L-S| => J = 1, 0. 1 3 o Therefore there are two states: 1 S0 and 1 S1 (also using n = 1) o But are they both allowed quantum mechanically? PY3P05 o Must consider Pauli Exclusion principle: “In a multi-electron atom, there can never be more that one electron in the same quantum state”; or equivalently, “No two electrons can have the same set of quantum numbers”.