Pauli Principle and Permutation Symmetry Heinz Kleindienst, Arne Liichow, and Rene Barrois institut f u r Physikalische Chemie und Elektrochemie, Heinrich-Heine-Universitat Dusseldorf, D-40225 Dusseldorf, Germany

In the quantum-mechanical treatment of the stationary states of atomic or molecular systems, the corresponding Hamiltonian (1,2), using the Born-Oppenheimer approxi- mation and neglecting relativistic a s well a s spin-orbit in- teractions, in atomic units is

where N and M are the numbers of electrons and nuclei; 2, is the charge of nucleus a; and ria, r,, Re* are the distances between electron i and nucleus a, the electrons i and j, and the nuclei a and b.

The stationary states are characterized by eigenvalues Efs of the operator H and by square integrable wave func- tions Yi's, depending on spatial and spin coordinates. Fur- thermore, the admissible wave functions must fulfill the Pauli principle: They must be antisymmetric with respect to the interchange of the coordinates (spatial and spin) in each pair of electrons. Simultaneously, Yi mus t he a n eigenfunction of s Z , where s2 is the square of the operator of the spin angular momentum. s2 is given by (3)

N N

sZ = C C ( 4 ~ ~ 0 ) + s ~ ( ~ ) s ~ O + sZ(Wi) 1 (2) &l j=1

where s,(i), sy(i), s,(i) are the components of the angular momentum (vector) operator a i ) of the ith electron.

From these quantum-mechanical facts, a serious prob- lem arises: The eigenfunctions of the operator H are func- tions of the spatial coordinates only. How can the Pauli principle be fulfilled with spin-free eigenfunctions? The answer is found using arguments based on group theory (5, 6 ) and is one of the main subjects in Matsen's spin-free quantum mechanics (7).

Permutation Symmetry of Hand .S2 Obviously the Hamiltonian H is invariant with respect

to the permutations of the coordinates of the electrons. These permutations form a group of order N!, called the symmetric group SN (8-10). A fundamental statement in quantum mechanics is the irreducibility postulate (4): The eigenfunctions of H transform a s a n irreducible repre- sentation (IR) of the symmetric group SN. The IR's of this group can be illustrated by so-called Young frames (9-13). Every Young frame is characterized by a partition of the number N. A partition of N is defined by N nonnegative integers Xi's with hi 2 Az t ... t AN whose sum is N, that is,

Ashort abbreviation for a partition is P.112 ... AN] with the convention that I, = 0 i s omitted.

Usually, the corresponding Young frame (pattern) is given by a diagram containing 2.1 boxes in the first row, hz boxes in the second row, and so on. We demonstrate these patterns for the two- and three-electron systems: helium or Hz and lithium. We get the following partitions.

For N = 2,

[21,1111 = [l21

F o r N = 3,

The following are the corresponding Young frames,

For a n N-electron system, the spin eigenfunctions of s2 transform also as the IR's of the symmetric group SN be- cause s Z , a s is immediately seen from eq 2, i s invariant with respect to the permutations P E SN. But i t can he shown (11) that Young frames are not admissible if they contain more than two rows; they must be of the type [AiAzl.

With a theorem based on group theory (5) we can show how to characterize the admissible spatial eigenfunctions. This theorem states that the Young frames for the spatial eisenfunctions have a t most two columns. More nreciselv. - , . to an admissible spatial rigenfunction of I1 rorrcspund.; an awrciatcd Younr frame 11 of the suin functims whcrc the columns are intkchanged by rows. For example, the fol- lowing spatial and spin Young frames of the symmetric group SQ are associated.

On the left of each pair of diagrams stands the IR for the spatial eigenfunction, and on the right is the associated Young frame for the corresponding spin eigenfunction.

Because the Young frame

for the spin eigenfunctions is forbidden, the corresponding Young frame

on for the spatial eigenfunctions is not allowed.

Examples of Atomic Systems We consider in detail the situation for two atomic sys-

tems: the helium atom and the lithium atom.

Helium Atom

In the case of the helium atom, the permutation group of H i s the symmetric group Sz that is isomorphic to the point group Cz. The IR's of Sz are represented by the Young frames

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that correspond to the symmetric representation Al and the antisymmetric representationA* of the point group Cz. Spatial functions that transform as the IR

m are symmetric functions. Antisymmetric spatial functions belong to the IR

0 Both frames are allowed because the associated Young

frames for the spin functions, that is,

1 and m

have at most two rows. The spin functions transforming as

1 and D

yield the singlet system and the triplet system. Quantum- mechanical calculations and spectroscopic measurements show that the ground state of the helium atom belongs to the singlet system; the corresponding spatial eigenfunc- tion y ~ , is a symmetric square integrable function, that is,

where Lf2, denotes the set of symmetric and square inte- grable functions.

For the triplet system, the corresponding spatial wave function is antisymmetric,

The final result for the spatial eieenfunctions can be stated as follows: The space of squareintegrable functions L' is divided into two symmetry-adapted subspaces

For the calculation of the eigenvalues of H, for example, using the Rayleigh-Ritz variational principle, spatial test functions chosen from L??, vield t he sinelet svstem. ~-. " - whereas test functions from L?& vield the triplet svstem. ~- , " From high-precision auantum-mechanical calculations (14) usin; correlated wave functions, we get the following for the lowest eigenvalues of the singlet and the triplet sys- - tem in atomic units.

Lithium Atom

We now consider the lithium atom. The corresponding resolution of the spatial space L' with resped to the IR's of the symmetric group S3 is given by the symmetry-adapted subspaces

Lt3], Lf2,j, and ~ 6 ~ 1

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Spatial test functions for the Rayleigh-Ritz variational method selected from the subspaces

yield the doublet and the quartet system. In order to con- struct symmetry-adapted spatial test functions, the so- calledYoung operators can be used (15). If the test function is chosen from Lf3,, then an eigenvalue E, of H is approxi- mated, corresponding to a spatial eigenfunction

- However, E, and Go do not represent a real physical state

because the associated Young frame

1 for a s ~ i n function is not allowed. Comnarison of the eicen- . values fur the lowest etgenstates in tho corrc.ipmding suh- .ip:rcrs'quantum-mechanical ~alcu1:itim; shwvs I l(i -18

The result is surprising. Specifically, the lowest eigen- value of the atomic three-electron Schrodinger operator H, that is, for the lithium atom, is not realized physically. The ground-state e n e r q for the lithium atom is given by the lowest eigenvalue E, of the doublet system. This agrees completely with the spectroscopic result: The term for the ground state of lithium is a term.

The General Solution Now we have a complete answer to the problem: Pauli

principle and permutation symmetry In a spin-free quan- tum mechanics the admissible eigenstates ofH are charac- terized by eigenvalues Ei and corresponding spatial eigen- functions ~IL'S where the y12s must transform as those IR's of the svmmetric erouD SN whose Young frames do not - .~ have more than two columns. Only in this case do admissi-

2 ble spin eieenfunctions ri's of S exist-the corresponding - ."~ - associated Young frame has only two rows-and the auti- symmetrizer d applied to the product yIi(r)~i(a) yields an antisymmetric wave function Yi.

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