Many Electron Spin Eigenfunctions An arbitrary Slater determinant for N electrons can be written as ÂΦ,,,N )χ M,,,N ) Where Φ,,,N functions and χ M ) = a)b ) c N ) is a product of N orthonormal spatial,,,n ) is a product of N α α spin functions and N β β spin functions and therefore has an eigenvalue of z =. Note that Ŝ equal to M Nα Nβ) we do not consider double occupied spatial orbital when investigating the spin characteristics of a Slater Determinant. To determine the effect of Ŝ on the Slater,,,N ). Accordingly if we want a determinant we need only investigate Ŝ χ M Slater determinant to be an eigenfunction of need only to write ÂΦ,,,N )χ SM,,,N ) Ŝ with an eigenvalue ) S S+ we Where Ŝ χ SM,,,N ) = S S +)χ SM,,,N ). The problem is how to find χ SM,,,N ) and there are several general approaches. Consider for example a three- electron system. We may form = 8 spin functions and we list them below labeled by the M quantum number. We assume the electron order is,,, so αβα = α ) β ) α ) Number Function M ααα / ααβ / αβα / 4 βαα / 5 ββα -/ βαβ -/ 7 αββ -/ 8 βββ -/ Note that ααα & βββ must be eigenfunctions of Ŝ with S=/ and M =±. Functions, and 4 must span the space containing one quartet S=/) and two doublets S=/) all with M=/ while functions 5,, and 7 lie in the M = subspace of the eigenfunctions of Ŝ. So for the M=/ subspace we may write J. F. Harrison /5/07
ϕ = S = /,M = / = aααβ + a αβα + a βαα ϕ = S = /,M = / = bααβ + bαβα + b βαα ϕ = S = /,M = / = cααβ + c αβα + c βαα Or in matrix form ϕ a a a ααβ ϕ = b b b αβα ϕ c c c βαα We may invert the matrix and write Where ααβ A A A ϕ αβα = B B B ϕ βαα C C C ϕ A A A a a a B B B = b b b C C C c c c The spin function ααβ is a linear combination of the quartet, ϕ and the two doublets, ϕ& ϕ ααβ = Aϕ + Aϕ + Aϕ To determine ϕ we will operate on ααβ with a projection operator that will annihilate ϕ& ϕ ) ) ) ) ) Ŝ + ααβ = A + + ϕ = A ϕ And since And + z z ) + 4) ˆ ˆ ˆ ˆ ˆ S ααβ = S S S + S ααβ = S S ααβ ) SS ˆ ˆ ααβ = Sˆ βαβ + αββ = ααβ + βαα + αβα + + J. F. Harrison /5/07
So we have Ŝ ααβ = 74ααβ + βαα + αβα And therefore Ŝ ) 4ααβ = ααβ + βαα + αβα = Aϕ After normalization we have ϕ = S =,M = = ααβ + βαα + αβα) To find the doublets we annihilate the quartet. ) ) ) ) ) ) Ŝ + ααβ = + + Aϕ + Aϕ Note that since ϕ& ϕ are both doublets any linear combination is a doublet so )) Ŝ + ααβ = ααβ + βαα + αβα is a doublet. The other linearly independent doublet is found by projecting the quartet out of one of the remaining functions, either αβα or βαα. )) ) Ŝ + αβα = αβα + βαα + ααβ ) Ŝ + ααβ = ααβ + βαα + αβα We then have three normalized doublet spin functions ααβ + βαα + αβα ) αβα + βαα + ααβ ) βαα + αβα + ααβ ) Note that these three doublets can not be linearly independent because there are only two in the space. We can form two orthonormal functions by taking ααβ + βαα + αβα ) J. F. Harrison /5/07
as one doublet and forming the second by subtracting αβα + βαα + ααβ ) from βαα + αβα + ααβ ) We have, after normalization the second doublet. αβα βαα ). We can form the remaining M = functions by the same projection operator technique but it instructive to use the raising and lowering operators. Recall that for any angular momentum eigenfunction J,M operating with Ĵ ± keeps J the same and changes M to M ± Ĵ± J,M = AJ,M ) J,M ± Given S =,M = we may form S =,M = by operating with Ŝ so ) ) S =,M = Sˆ ααβ αβα βαα αββ βαβ ββα + + = + + and after normalization we have S =,M = = αββ + βαβ + ββα ) Exercise Show that the effect of the lowering operator Ŝ on ααα is the same function S =,M = obtained by using the projection operator on ααβ. Branching Diagram A very convenient way of keeping track of the number of spin eigenfunctions of a given multiplicity for an N electron system is the branching diagram shown below J. F. Harrison /5/07 4
The diagram is constructed as follows. A two-electron system is constructed from a one electron system by coupling the additional spin and the original spin into either a singlet down) or a triplet up). To form the spin eigenfunctions for a three electron system we couple the two electron triplet into a quartet up) and a doublet down). Additionally the singlet may be coupled up) into a doublet. We then have quartet and two doublets. To form the 4 electron system we couple the quartet into a quintet up) and into a triplet down). The two doublets couple up to form two additional triplets and down to form two singlets. The four electron S = 0 for a system then can form quintet S = ), triplets S = ) and singlets ) 4 total of states. This is of course the required. Note the rapid increase in the number of low spin eiegenfunctions as the number of unpaired electrons increases. For N=0 we have 4 singlets, 90 triplets, 75 quintets, 5 septets, 9 nonets and bigtet. We gather below a few explicit spin eigenfunction for N= to 5 and specifically for the case M = S. Lower values of M for a particular S may be generated using the lowering operator Ŝ. These are from the Sanibel notes of Frank E Harris. Note that the functions have been factored where possible to highlight the physical couplings. For example the N=4 singlet, αβ βα ) αβ βα ) / obtains from the singlet coupling of electron pairs & and &4 while the N=5 doublet, αβ βα αβ βα α / also obtains from the singlet coupling of electron pairs ) ) J. F. Harrison /5/07 5
& and &4 with the doublet spin angular momentum being carried by the α spin of electron 5. J. F. Harrison /5/07
Orthonormal Spin Functions for N spins with total spin S through N=5 N S χ SS / α αβ + βα ) / 0 αβ βα ) / / ααα αβ βα α / / ) / βαα + αβα ααβ ) 4 αααα αβ βα αα / / 4 ) 4 βαα + αβα ααβ ) α / 4 βααα + αβαα + ααβα αααβ ) / 4 0 αβ βα ) αβ βα ) / 4 0 ααββ + ββαα αβαβ βαβα αββα βααβ ) / 5 5/ ααααα αβ βα ααα / 5 / ) 5 / βαα + αβα ααβ ) αα / 5 / βααα + αβαα + ααβα αααβ ) α / 5 / βαααα + αβααα + ααβαα + αααβα ααααβ ) 4 / 0 5 / αβ βα ) αβ βα ) α / 5 / + ) 5 / αβ βα ) βαα + αβα ααβ ) / ααββ ββαα αβαβ βαβα αββα βααβ α / 5 / αααββ + ααβαβ + ββααα αβααβ βαααβ ααββα ) / 5 / ααβαβ αααββ + αβαβα + βααβα αββαα βαβαα ) / J. F. Harrison /5/07 7