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Home , Excited state, Ground state, Singlet state, Wave function

o Schrödinger equation for

o Two- .

o Multi-electron atoms.

o -like atoms.

o Singlet and triplet states.

o Exchange .

o See Chapter 9 of Eisberg & Resnick

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o See http://en.wikipedia.org/wiki/Electron_configuration

PY3P05 o What is of simultaneously finding a particle 1 at (x1,y1,z1), particle 2 at (x2,y2,z2), etc. => need joint .

o N-particle system is therefore a 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 )) 2m 2m % 1 ( % 2 ( 2 2e i =1,2 o The electron-nucleus potential for helium is Vi = " 4#$0ri !

o The 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 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 . 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 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 , n(1) = 1, n(2) = 2 => E =-68 eV. -110 ! Neglecting electron-electron interaction

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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-, 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 " 1 2 N =#1 1 #2 2 !#N N E = " E 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 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 . 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 and !nl is the shielding constant. This gives rise to the shell model for multi-electron !atoms.

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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 . ! o The “+” sign applies if the particles are . These are said to be symmetric with respect to particle exchange. The “-” sign applies to , which are antisymmetric with respect to particle exchange. o As electrons are fermions ( 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.

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o State of atom is specified by configuration of two electrons. In , 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 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, 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 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

! !

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o Physical interpretation of singlet and triplet states can be obtained by evaluating the total spin angular (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 () +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 : 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 (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?

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o 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 of quantum numbers”.

1 o Consider the 1 S0 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

1 o 1 S0 is therefore allowed by Pauli principle as ms quantum numbers differ.

3 o Now consider the 1 S1 state: L = 0, S = 1, J = 1

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

3 o 1 S1 is therefore disallowed by Pauli principle as ms quantum numbers are the same.

PY3P05 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 3 o Produces P3,2,1 o Therefore have, n1 = 1, l1 = 0, s1 = 1/2 No violation of Pauli principle and n = 2, l = 1, s = 1/2 3 2 2 1 => P3,2,1 are allowed terms o For L = 1, S = 0 => J = 1 1 o Term is therefore P1 o Allowed from consideration of Pauli principle

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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 Spin angular momentum: s1 = s2= 1/2 = > S =1, 0 and multiplicity is 3 or 1

L S J Term 1 0 0 0 S0 1 1 0 1 P1 1 2 0 2 D2 *3 0 1 1 S1 3 1 1 2, 1, 0 P2,1,0 *3 2 1 3, 2, 1 D3,2,1 o *Violate Pauli Exclusion Principle (See Eisberg & Resnick, Appendix P)

PY3P05 o Singlet states result when S = 0. o Parahelium. o Triplet states result when S = 1 o Orthohelium.

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o Need to explain why triplet states are lower in energy that singlet states. Consider

& 2 2 ) & 2 2 ) 2 ˆ "! 2 Ze "! 2 Ze e H = ( #1 " + + ( # 2 " + + ' 2m 4$%0r1 * ' 2m 4$%0r2 * 4$%0r12 Hˆ Hˆ Hˆ = 1 + 2 + 12 * ˆ 3 ˆ 3 ˆ o The expectation value of the Hamiltonian is E = ## "spatial H" spatial d r1 d r2 o The energy can be! split into three parts, E != E1 + E2 + E12 where * ˆ 3 ˆ 3 ˆ E i = ## "spatial H i"spatial d r1 d r2 * ˆ 3 ˆ 3 ˆ E12 = ## "spatial H12 "spatial d r1 d r2 o The expectation value of the first two terms of the Hamiltonian is just E = E + E ! 1 2 # 1 1 & = "4E R % 2 + 2 ( $ n1 n2 ' where ER = 13.6 eV is called the Rydberg energy.

PY3P05 ! o The third term is the electron-electron Coulomb repulsion energy:

* ˆ 3 ˆ 3 ˆ E12 = ## "spatial H12 "spatial d r1 d r2 e2 * 3 ˆ 3 ˆ = ## "spatial "spatial d r1 d r2 4$%0r12 1 o Using " = ( " ( rˆ ) " ( rˆ ) ± " ( rˆ ) " ( r ˆ )) this integral gives spatial 2 # 1 $ 2 $ 1 # 2 ! 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:

e2 1 * ˆ * ˆ ˆ ˆ 3 ˆ 3 ˆ D"# = '' &" (r1 )&# (r2 ) &" (r1 )&# (r2 )d r1 d r2 4$%0 r12 e2 1 * ˆ * ˆ ˆ ˆ 3 ˆ 3 ˆ J"# = '' &" (r1 )&# (r2 ) &# (r1 )&" (r2 )d r1 d r2 4$%0 r12 o The resulting energy is E12 ~ 2.5 ER. Note that in the exchange integral, we integrate the expectation value of 1/r with each electron in a different shell. See McMurry, Chapter 13. ! 12 PY3P05

# & o The total energy is therefore 1 1 E = "4ER % 2 + 2 ( + D)* ± J)* $ n1 n2 ' where the + sign applies to singlet states (S = 0) and the -sign to triplets (S = 1).

o Energies of the singlet and! triplet states differ by 2J"#. Splitting of spin states is direct consequence of exchange symmetry.

o We now have, E1 + E2 = -8ER and E12 = 2.5ER => E = -5.5ER = -74.8 eV 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.

ˆ ˆ o Exchange energy is sometimes written in the form "E exchange = #2J$% S1 & S 2 which shows explicitly that the change of energy is related to the relative alignment of the electron spins. If aligned = > energy goes up.

! PY3P05 o Orthohelium states are lower in energy than the parahelium states. Explanation for this is:

1. Parallel spins make the spin part of the wavefunction symmetric.

2. Total wavefunction for electrons must be antisymmetric since electrons are fermions.

3. This forces space part of wavefunction to be antisymmetric.

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.

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