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System of non-interacting particles Lectures 10-11: Multi-electron atoms 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: !(x1,y1,z1; x2,y2,z2; … xN,yN,zN) o Must solve o First consider two particles which do not interact with one another, but move ! in potentials V1 and V2. The Hamiltonian is (2) Hˆ = Hˆ 1 + Hˆ 2 $ h2 2 ' $ h2 2 ' = &" #1 + V1 ( rˆ1 )) + &" # 2 + V2 ( rˆ2 )) % 2m1 ( % 2m2 ( o Schrödinger equation for o Two-electron atoms. o Multi-electron atoms. o Helium-like atoms. o Singlet and triplet states. o Exchange energy. Hˆ "( rˆ1, rˆ2 ,..., rˆN ) = E"( rˆ1, rˆ2 ,..., rˆN ) (1) 2e 2 4 #$0 ri r1 x i = 1,2 o The eigenfunctions of H1 and H2 can be written as the product: ! ! ! e1 y The electron-nucleus potential for helium is Vi = " e2 r2 o PY3004 z "( rˆ1, rˆ2 ) = #1 ( rˆ1 )# 2 ( rˆ2 ) PY3004 ! System of non-interacting particles o Using this and Eqns. 1 and 2, o That is, Hˆ "( rˆ1, rˆ2 ) = E"(rˆ1, rˆ2 ) where E = E1+E2. o Application to helium o Hˆ "( rˆ1, rˆ2 ) = ( Hˆ 1 + Hˆ 2 )#1 ( rˆ1 )# 2 ( rˆ2 ) = (E1 + E 2 )#1 ( rˆ1 )# 2 ( rˆ2 ) Assuming each electron in helium is non-interacting, can assume each can be treated independently with hydrogenic energy levels: 13.6Z 2 n2 Total energy of two-electron system in ground state (n(1) = n(2) = 1)is therefore ! 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, o Eigenvalues of each particle’s Hamiltonian determine possible energies. Total energy is thus o ! "( rˆ1, rˆ2 ,...rˆN ) = #1 ( rˆ1 )# 2 ( rˆ2 )L# N ( rˆN ) N ! E = " Ei i=1 o Observed En = " Can be used as a first approximation to two interacting particles. Can then use perturbation theory to include interaction. ! o E = E1 (1) + E1 (2) # 1 1 & = "13.6Z 2 % + ( $ n(1) 2 n(2) 2 ' #1 1& = "13.6(2) 2 % 2 + 2 ( $1 1 ' = "109 eV -50 -60 Energy (eV) For first excited state, n(1) = 1, n(2) = 2 => E =-68 eV. -70 -80 -90 -100 -110 ! Neglecting electron-electron interaction PY3004 PY3004 System of interacting particles o For He-like atoms can extend to include electron-electron interaction: & "h 2 2 Ze 2 ) & "h 2 2 Ze 2 ) e2 Hˆ = ( #1 " #2 " ++( ++ 4 $%0 r1 * ' 2m 4 $%0 r2 * 4 $%0 r12 ' 2m o o z e2 The final term represents electron-electron repulsion at a distance r12. e1 r12 r2 ! o Wave function for system of interacting particles The solutions to the equation can again be written in the form " i ( rˆi ,# i , $ i ) = Rn i li ( rˆi )%li m i (# i , $ i ) r1 o d 2 Rn i li ! are solutions to The radial wave functions y Or for N-electrons, the Hamiltonian is: ' #h 2 Ze 2 e2 * Hˆ i" i ( rˆi ) = )) $ 2i # + ," i ( rˆi ) 4 %&0 ri 4 %&0 rij ,+ ( 2m = E i" i ( rˆi ) dri 2 ! + 2 dRn i li 2µ $ e2 ' + 2 &E + )Rn l = 0 r dri h % 4 "#0 ri ( i i and therefore have the same analytical form as for the hydrogenic one-electron atom. x N Hˆ = " Hˆ i o i=1 ' #h 2 2 Ze 2 * N e 2 = ") $i # ,+" 2m 4 %&0 ri + i> j 4 %&0 rij i=1 ( ! for Allowable solutions again only exist N 2 En = " where Zeff = Z - !nl. and the corresponding Schrödinger equation is again of the form Hˆ "( rˆ1, rˆ2 ,..., rˆN ) = E"( rˆ1, rˆ2 ,..., rˆN ) ! N where "( rˆ1, rˆ2 ,..., rˆN ) = #1 ( rˆ1 )# 2 ( rˆ2 )L# N ( rˆN ) and E = " E i o Z eff µe 4 (4 #$0 ) 2 2h 2 n 2 Zeff is the effective nuclear charge and !nl is the shielding constant. This gives rise to the shell model for multi-electron ! atoms. i=1 ! PY3004 ! PY3004 ! Shell model o Shell model Electrons in orbitals with large principal quantum numbers (n) will be shielded from the nucleus by innershell electrons. Zeff = Z - !nl. o !nl increases with n => Zeff decreases with n. o !nl increases with l => Zeff decreases with l. o In hydrogenic one-electron model, the energy levels of a given n are degenerate in l: En = " o Z 2µe 4 (4 #$0 ) 2 2h 2 n 2 3s 3p 3d 3s 3p 3d Not the case in multi-electron atoms. Orbitals with the same ! n quantum number have different energies for differing values of l. 2 En = " n=1 n=2 n=3 n=4 o PY3004 Z eff µe 4 (4 #$0 ) 2 2h 2 n 2 As Zeff = Z - !nl is a function of n and l, the l degeneracy is broken by modified potential. ! PY3004 Shell model Shell model o Wave functions of electrons with different l are found to have different amount of penetration into the region occupied by the 1s electrons. o This penetration of the shielding 1s electrons exposes them to more of the influence of the nucleus and causes them to be more tightly bound, lowering their associated energy states. o In the case of Li, the 2s electron shows more penetration inside the first Bohr radius and is therefore lower than the 2p. o In the case of Na with two filled shells, the 3s electron penetrates the inner shielding shells more than the 3p and is significantly lower in energy. PY3004 PY3004 Atoms with two valence electrons Helium wave functions 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: o He atom consists of a nucleus with Z = 2 and two electrons. o Must now include electron spins. Two-electron wave function is therefore written as a product spatial and a spin wave functions: | "( rˆ1, rˆ2 ,..., rˆK , rˆL ,..., rˆN ) |2 =| "(rˆ1, rˆ2 ,..., rˆL , rˆK ,..., rˆN ) |2 which will be satisfied if z r2 " = # spatial ( rˆ1, rˆ2 )# spin "( rˆ1, rˆ2 ,..., rˆK , rˆL ,..., rˆN ) = ±"( rˆ1, rˆ2 ,..., rˆL , rˆK ,..., rˆN ) o ! 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 anti-symmetric with respect to particle exchange. o As electrons are fermions (spin 1/2), the wavefunction of a multi-electron atom must be antisymmetric with respect to particle exchange. ! PY3004 As electrons are indistinguishable => ! must be antisymmetric. See table for allowed symmetries of spatial and spin wave ! functions. r12 e2 y e1 r1 Z=2 x PY3004 Helium wave functions: "spatial Helium wave functions: "spin 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 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 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 $. o There are four possible ways to combine the spins of the two electrons so that the total wave function has exchange symmetry. o Total wave function for a excited atom can therefore be written: " = #$ ( rˆ1 )# % ( rˆ2 ) o Only one possible anitsymmetric spin eigenfunction: o But, this does not take into account that electrons are indistinguishable. The following is therefore equally valid: " = # $ ( rˆ1 )#% ( rˆ2 ) ! o Because both these are solutions of Schrödinger equation, linear combination also a solutions: ! where ! 1/ 2 is 1 "S = (#$ ( rˆ1 )# % ( rˆ2 ) + # % ( rˆ1 )#$ ( rˆ2 )) 2 1 "A = (#$ ( rˆ1 )# % ( rˆ2 ) & # % ( rˆ1 )#$ ( rˆ2 )) 2 1 [(+1/2,"1/2) " ("1/2,+1/2)] 2 o There are three possible symmetric spin eigenfunctions: ! o o (+1/2,+1/2) Symmetric 1 [(+1/2,"1/2) + ("1/2,+1/2)] 2 ("1/2,"1/2) Asymmetric triplet a normalisation factor. ! PY3004 ! Helium wave functions: "spin o singlet PY3004 Helium wave functions Table gives spin wave functions for a twoelectron system. The arrows indicate whether the spin of the individual electrons is up or down (i.e. +1/2 or -1/2). 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. 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. The Sz value is indicated by the quantum number for ms, which is obtained by adding the ms values of the two electrons together. S ms %spin Singlet 0 0 1/ 2("1#2 $#1"2 ) Triplet 1 +1 0 ! -1 1/ 2("1#2 + #1"2 ) #1#2 ! ! PY3004 "1"2 %spatial 1 ("# ( rˆ1 )" $ ( rˆ2 ) + " $ ( rˆ1 )"# ( rˆ2 )) 2 1 ("# ( rˆ1 )" $ ( rˆ2 ) % " $ ( rˆ1 )"# ( rˆ2 )) 2 ! PY3004 Singles and triplet states o Helium terms Physical interpretation of singlet and triplet states can be obtained by evaluating the total spin angular momentum (S), where Sˆ = Sˆ1 + Sˆ 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)h ! Sz = m sh where ms = -s, … +s and s = 0, 1. o o If s1 = +1/2 and s2 = -1/2 => s = 0. o Therefore ms = 0 (singlet state) If s1 = +1/2 and s2 = +1/2 => s = 1. o Therefore ms = -1, 0, +1 (triplet states) singlet state s1=1/2 ! s1=1/2 s2=1/2 s2=-1/2 ms +1 0 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. o Therefore there are two states: 11S0 and 13S1 z triplet state s=1 -1 o (also using n = 1) But are they both allowed quantum mechanically? s = 0, ms = 0 PY3004 PY3004 Helium terms o o Helium terms Must consider Pauli Exclusion principle: “In a multi-electron atom, there can never be more that one electron in the same”; or equivalently, “No two electrons can have the same set of quantum numbers”. o First excited state of He: 1s12p1 o Orbital angular momentum: l1= 0, l2 = 1 => L = 1 o Gives rise to an P term. o Spin angular momentum: s1 = s2 = 1/2 => S = 0 or 1 o Multiplicity (2S+1) is therefore 2(0) + 1 = 1 or 2(1) + 1 = 3 o For L = 1, S = 1 => J = L + S, …, |L-S| => J = 2, 1, 0 o Produces 3P3,2,1 Consider the 11S0 state: L = 0, S = 0, J = 0 o n1 = 1, l1 = 0, ml1 = 0, s1 = 1/2, ms1 = +1/2 o n2 = 1, l2 = 0, ml2 = 0, s2 = 1/2, ms2 = -1/2 o 11S0 is therefore allowed by Pauli principle as ms quantum numbers differ. o Now consider the 11S1 state: L = 0, S = 1, J = 1 o o n1 = 1, l1 = 0, ml1 = 0, s1 = 1/2, ms1 = +1/2 o n2 = 1, l2 = 0, ml2 = 0, s2 = 1/2, ms2 = +1/2 Therefore have, n1 = 1, l1 = 0, s1 = 1/2 and n2 = 2, l2 = 1, s1 = 1/2 o For L = 1, S = 0 => J = 1 o Term is therefore 1P1 o Allowed from consideration of Pauli principle o 11S1 is therefore disallowed by Pauli principle as ms quantum numbers are the same. PY3004 No violation of Pauli principle => 3P3,2,1 are allowed terms PY3004 Helium terms Helium Grötrian diagram o Now consider excitation of both electrons from ground state to first excited state: gives a 2p2 configuration. o Orbital angular momentum: l1 = l2=1 => L = 2, 1, 0 o Produces S, P and D terms o o o Singlet states result when S = 0. o Parahelium. o Triplet states result when S = 1 o Orthohelium. Spin angular momentum: s1 = s2= 1/2 = > S =1, 0 and multiplicity is 3 or 1 *Violate L S J Term 0 0 0 1S 0 1 0 1 1P 1 2 0 2 1D 0 1 1 *3S 1 1 2, 1, 0 3P 2 1 3, 2, 1 *3D 2 1 2,1,0 3,2,1 Pauli Exclusion Principle (See Eisberg & Resnick, Appendix P) PY3004 PY3004 Exchange energy o Exchange energy o Need to explain why triplet states are lower in energy that singlet states. Consider The third term is the electron-electron Coulomb repulsion energy: & "h 2 2 Ze 2 ) & "h 2 2 Ze 2 ) e2 Hˆ = ( #1 " #2 " ++( ++ 4 $%0 r1 * ' 2m 4 $%0 r2 * 4 $%0 r12 ' 2m = Hˆ + Hˆ + Hˆ 1 o o 2 The expectation value of the Hamiltonian is E = ## " * spatial Hˆ " spatial d 3 rˆ1d 3 rˆ2 o ! The energy can be split into three parts, E! = E1 + E2 + E3 where * E i = ## " spatial Hˆ i" spatial d 3 rˆ1d 3 rˆ2 ## " * spatial Hˆ 12" spatial d 3 rˆ1d 3 rˆ2 E = E1 + E 2 #1 1& = "4 E R % 2 + 2 ( $ n1 n 2 ' o where ER = 13.6 eV is called the Rydberg energy. ! * spatial = ## " * spatial Evaluating this integral gives Hˆ 12" spatial d 3 rˆ1d 3 rˆ2 e2 " spatial d 3 rˆ1d 3 rˆ2 4 $%0 r12 E12 = D#$ ± J#$ ! where the + sign is for singlets and the - sign for triplets and D#$ is the direct Coulomb energy and J#$ is the exchange Coulomb energy: The expectation value of the first two terms of the Hamiltonian is just ! ## " 12 E12 = o E12 = PY3004 D"# = e2 4 $%0 '' & J"# = e2 4 $%0 '' & ( rˆ1 )& #* ( rˆ2 ) 1 &" ( rˆ1 )& # ( rˆ2 )d 3 rˆ1d 3 rˆ2 r12 ( rˆ1 )& #* ( rˆ2 ) 1 & # ( rˆ1 )&" ( rˆ2 )d 3 rˆ1d 3 rˆ2 r12 * " * " The resulting energy is E12 ~ 2.5 ER. Note that in the exchange integral, we integrate the expectation value!of 1/r12 with each electron in a different shell. See McMurry, Chapter 13. PY3004 Exchange energy o The total energy is therefore Helium terms #1 1& E = "4 E R % 2 + 2 ( + D)* ± J)* $ n1 n 2 ' o Orthohelium states are lower in energy than the parahelium states. Explanation for this is: where the + sign applies to singlet states (S = 0) and the -sign to triplets (S = 1). 1. Parallel spins make the spin part of the wavefunction symmetric. o Energies of the singlet and!triplet states differ by 2J#$. Splitting of spin states is direct consequence of exchange symmetry. 2. Total wavefunction for electrons must be antisymmetric since electrons are fermions. o We now have, 3. This forces space part of wavefunction to be antisymmetric. o Compares to measure value of ground state energy, 78.98 eV. o Note: o Exchange splitting is part of gross structure of He - not a small effect. The value of 2J#$ is ~0.8 eV. E1 + E2 = -8ER and E12 = 2.5ER => E = -5.5ER = -74.8 eV o Exchange energy is sometimes written in the form "E exchange = #2J$% Sˆ1 & Sˆ 2 which shows explicitly that the change of energy is related to the relative alignment of the electron spins. If aligned = > energy goes up. ! PY3004 4. Antisymmetric space wavefunction implies a larger average distance between electrons than a symmetric function. Results as square of antisymmetric function must go to zero at the origin => probability for small separations of the two electrons is smaller than for a symmetric space wavefunction. 5. If electrons are on the average further apart, then there will be less shielding of the nucleus by the ground state electron, and the excited state electron will therefore be more exposed to the nucleus. This implies that it will be more tightly bound and of lower energy. PY3004