J. Phys. Chem. 1993,97, 2457-2460
2457
Wrong Parity States and the Molecular Orbital Description of Doubly-Excited Two-Electron Atoms in D Dimensions M. Dunn' and D. I(. Watson Department of Physics and Astronomy, University of Oklahoma, Norman, Oklahoma 73019-0225 Received: September 29, 1992; In Final Form: December 2, 1992
Rost et al. have examined some of the implications of dimensional scaling for the molecular orbital (MO) description of the "right parity" (u = (-1)'-) doubly-excited states of two-electron atoms. They propose a new and simple classification scheme utilizing interdimensional degeneracies, saddle degeneracies, and A-doubling. The present paper extends this description, in D dimensions, to the "wrong parity" (T = (-l)'-+l) states and provides a justification for the extension of the new classification scheme to include the u = (-l)L+f states. This is achieved by showing that the M O approximation preserves the exact interdimensional degeneracies between u = (-1)'- and u = (-l)L+t states and that there exists approximate interdimensional degeneracies between the u = (-l)L+t MO states. It is also pointed out that the phenomenon of A-doubling is special to three dimensions, although the lifting of this degeneracy is expected to be small for low dimensions. Despite the fact that the energy level spectrum no longer features A-doubling for larger values of D, the internal MO electron center of mass wavefunctions for those states which exhibit A-degeneracy at D = 3 are still identical in any dimension. The lifting of this degeneracy does not affect the classification scheme of Rost et al. since it utilizes A-doubling at D = 3. In a recent paper' J. M. Rost, S.M.Sung, D. R. Herschbach, and J. S. Briggs (hereafter called paper I) explore some of the implications of dimensional scaling2 for the molecular orbital (MO)description of two-electron atom^.^,^ The MO treatment postulates that the rotational and internal degrees of freedom of a two-electron atom are approximately separable. For doubly excited states with A = f l this is known to be a good approximation.s.6 The interelectron distance R is then taken to be an adiabatic coordinate with the result that the two-electron problem parallels the Born-Oppenheimer treatment of the Hz+ molecular ion. The Pdimensional two-center Coulomb problem is separable in prolate spheroidal coordinates7J
where ri is the distance from the nucleus to electron i and wg-2 is a set of D - 2 azimuthal angles about the interelectron vector R = rl - r2. Hence one obtains an approximate separation of the two-electron problem. Although this is not a very good approximation for the ground state of atomic systems? the accuracy of this approximation for doubly-excited states at D = 3 has been explicitly demonstrated by a recent analysis by Rost et aL4 of wavefunctions from an accurate diagonalizationof the full twoelectron problem.lO The D dimensional MO treatment of paper I establishes that approximate interdimensional degeneracies satisfying the dimensional link
+
(D,L) 0 ( D - 2i,L i ) , i = 0, 1'2, 3, ... (1) exist for r = (-l)L states at low D, the interdimensionaldegeneracy
-
(5,U (3,L + 1) being exact in the MO approximation. That this latter exact degeneracy for I = ( - l ) L states is not carried over to the full two-electron system is demonstrated by the work of Dunn and Watson" (paper 11) where they solve for the full two-electron system in D dimensions; Le. it is an approximate interdimensional degeneracy. The authors of paper I use these approximate interdimensional degeneracies and saddle degeneracies of the MO picture and invoke "&doubling" to create a simple classi0022-3654/93/2091-24S7504.00 f 0
fication scheme for two-electron states where the MO potentials of a groupof doubly excited two-electron states in threedimensions are related to a single MO potential of a "generator" IScstate in D 1 3. Thus the generator state characterizes a specific group of doubly excited states at D = 3. The scope of the work in paper I is confined, by the D dimensional angular momentum theory which is used, to u = (-l)L states. In that paper the D dimensional angular momentum theory is derived from the properties of the hyperspherical harmonics, the generalization to D dimensions of the spherical harmonics. The spherical harmonics all have I = ( - l ) L . This is sufficient in three dimensions since the u = (-l)L and u = (-l)L+' states transform under equivalent representations of the rotation groupSO(3). However, in Ddimensionsthis is no longer the case. The two classes of states no longer transform under equivalent representations of the D dimensional rotation group SO(D) (a discussion of this point may be found, for example, in another paper by the authorsI2 hereafter called paper 111). As a result, the D dimensional angular momentum theory one derives from the hyperspherical harmonics is incomplete. These considerationsmay be illustrated by considering the vector and axial vector functions (at D = 3)
respectively, where rjp' is the jith component of the position vector r(a)of the ccth particle. In three dimensions both YVand YA have the same number of independent components (three, they are both P states) but in D dimensions YVhas D components while YAhas D(D - 1)/2 independentcomponents (YAis a scalar in two dimensions). In paper 111 we note that there are no oneparticle functions that transform under an irreducible representation of O(D) (and hence SO(D)I3J5) and have u = (-l)L+l at D = 3, a point which is exemplified by Y, which vanishes identically if constructedfrom the coordinatesof a single particle. The MO description for T = (-1)L" states closely follows the theory developed by Rost et al. for the ?r = (-1)' states in D dimensions in paper I. The D dimensional Hamiltonian H for 0 1993 American Chemical Society
Dunn and Watson
2458 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993
a two-electron species with an infinitely heavy nucleus of charge 2 is
where all the lengths have been scaled by Z,the Hamiltonian has been scaled by 1/Z2, R = rl - rz, r = (II + r2)/2 and the rr are the electron-nucleus distances. The D dimensional Laplacians are
having thesamevalueof (111toensure that $LJw"-r(r,R) transforms correctly under "parity" transformations with determinant = -1 and that the spin-space wavefunction has the correct exchange symmetry. The P - ' / ~ ( , ~ ) ~ ~ , ~ (are ~ ) internal (P,z;R MO ) electron center of mass (ECM) wavefunctions (solutions of the "oneparticle"-like two-center Coulomb problem in the variable r with R held fixed (see ref 7 and paper I)), p is the perpendicular distance of the electron from the internuclear axis, and z is the projection onto that axis measured from the midpoint between the two nuclei. The Jacobian weighted MO ECM wavefunction di,m(D)(p,~;R) satisfies an equation identical to that at D = 3 with the substitution
-
and
11111
m(D) =
+ j1( D - 3 )
(6)
where I,, is the angular momentum quantum number of eq 5 in the body-fixed D - 1 dimensional subspace orthogonal to R where L,12 and 1 0 _ ~ 2 are the sums of the squares of the (IrrZz is a "one-particle"-like operator). At D = 3 1111 = Iml. independent elements of the D dimensional angular momentum Equation 1 shows that the Jacobian weighted MO ECM tensors (Lbl)ijand ID..^)^, respectively. Both (L,I)ijand ( 1 ~ 1 ) ~ ~wavefunction and energy, q,(o)(R),feature, as originally noted are the angular momentum tensors associated with a single vector by Herrick,*Oexact interdimensional degeneracies following the coordinate; i.e. they are both 'one-particle"4ike (quasi-particle) pattern of eq 1. The R(b1)(b3)/4Fi(R) are the expansion operators. These quasi-particle angular momentum tensors have coefficients of the series and allow for interelectron motion. components Following paper I we consider a one channel approximation, as has been pursued with much success by Briggs, Feagin, Rost (3) et al.3 in three dimensions, for which the wavefunction is where eijiiY is the antisymmetrizer 6iidjy - aijdji, and are the elements of the Lie algebra of the D dimensional rotation group SO(D)
+$/#"V,R)
[l(j,lk/l= i h (likaj, + lj$ik - ljkai/ - lipjk) The generalizedangular momentum operator (quadratic Casimir operator I ')
To derive an equation for Fi(R)in the one channel approximation one projects the Schradinger equation onto the complete set of the
=
BLS lMi(lmll(*bl)R I
) ( b 3 ) / 4 ~i
(R)~(~~)/~~~,~(~)(P,
@fdMS**(,;R) r Slmll DLSv" t ~ ~ l m l J *1 )bp ( b 3 ) / z 4 + , m ( o ) ( ~ , z ; ~ )
(4) has the hyperspherical harmonics filml(wrrl) as eigenfunctions with eigenvalues
lblzY/lml(~b= , ) Ibl(lbI + D - ~ ) Y / I , ~ I ( ~ (~5 I) ) where {m)= (1,2,10-3, I M , ..., 11)and ID-, 1 b - 2 1 ... 2 1111. Since Ibl2 is a quasi-particle operator, the eigenvalues of eq 5 span the complete eigenvalue spectrum of I,Iz. The total angular momentum tensor
and drop all off-diagonal couplingsto obtain the one dimensional differential equation
(-d2 + V ( R ) + 1 -E)Fi(R) = 0 dRZ ZR
where V(R) is given by the expectation value
+
Lij = I , Lij is a two particle operator and so L2 (see eq 4 ) has an additional set of eigenvalues and eigenvectors over the quasi-particle operators f 2 and l2 for two particle states which survive down to D = 3 . These new eigenvectors are states with T = (-l)L+l at D = 3.18 Since L&12 commutes with the Hamiltonian and is a scalar operator under the D dimensional rotation group, the eigenvalues of L b 1 2have the same value in space fixed or body fixed coordinates. Expanding the wavefunction +L*lMlJ+*(r,R) in terms of the irreducible representationsDLl~lml( Q,,) of the orthogonal group O(D),I9-I8where Q,, denotes the D(D - 1)/2 generalized Euler angles, one derives
and ei,,(o)(R)is the energy of the two-center Coulomb problem. As noted by Rost et al. in paper I
is a function of i and m(D)only, i.e. it has the same value as the D = 3 result with )mi = m(D). Thus any breaking of the interdimensional degeneracies of eq 1 will come from the term
Rost et al. in paper I find for the
T
= (-1)L states that
Fi(R)P-(,Z)lZdi,m(o)(p,r;R)
The factor Sf"; (abl) rotates the space-fixed coordinate + L(D)(L(D)+ 1) - m2(D)+ ;(D - 3 ) ( D - 5 ) (7) frame into a b & % x e d frame in which R lies along the body fixed axis J D and is a linear combination of the D L , w , m l ( Q ~ l ) where L(D) = L + ( D - 3)/2 and &,(o) is a function of R. The
Doubly-Excited Two-Electron Atoms in D Dimensions
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2459
last term in eq 7 means that most of the exact interdimensional degeneracies between states of parity A = are lifted when two-electron motion in the MO approximation is taken into account. However, there is still an exact interdimensional degeneracy between states in D = 3 and D = 5 in the MO approximation. In addition, Rost et al. in paper I also show that low D states satisfying the dimensional link of eq 1 are approximately degenerate in the MO approximation. The work of Dunn and Watson in paper I1 impliesthat off diagonal couplings lift the exact degeneracy between states of parity T = (-l)L in D = 3 and D = 5 . For the u = (-l)‘+I states one needs to know (@f.IWS.l r,{lmll
,fI
2
LIMS.*)
l@i,ilmll
Following Rost et al. in paper I we write
Since ( ~ o - , ) ~ Ddoes not couple states with the same quantum numbers {m),the last term is zero. It is found in paper I that the expectation values of the quasi-particle operators
-
where Ai,m(o)is the same function of R as the D = 3 result with the replacement Iml m(D). Since (Lo-I)~, = (10-1)ij for i j I D - 1, L&zz = 10-22 and has eigenvalues m2(D) - (D - 3)2/4. In Appendix A it is shown that the eigenvalues of L&l2 for the u = (-1)L+l states are ( L 1)(L + D - 3), where L is the angular momentum quantum number at D = 3. Thus
+
+
Ai,m(D) + L(D)(L(D) 1) - m2(D)+ $D
- l)(D - 3)
+
-
Acknowledgment. The authors express their gratitudeat being able to attend DUDFEST 92 and wish Dudley Herschbach a magnificant seventh decade. The authors also wish to express their gratitude to Jan Michael Rost and Stella Sung for helpful discussions. This work was performed with the support of National Science Foundation Grants PHY-9008721 and PHY-9123199.
(8)
Comparing eqs 7 and 8 one sees that states satisfying the dimensional link of eq 1 fori = 1 (the u = (-1)L+I in D dimensions and the A = (-l)L states in D 2 dimensions) are exactly degenerate; i.e. the MO approximation preserves the interdimensional degeneracies noted by Herrick and Stillinger,20,21 Goodson et and Dunn and Watson (papers I1 and 111). Regarding interdimensional degeneracies between the u = (-1)‘+l states eq 8 would seem to indicate an exact interdimensional degeneracy between states in D = 1 and D = 3 dimensions. This though, is not the case since the u = (-l)L+l states vanish identically at D = 1,23a point exemplifiedby YAof eq 2. However, in the same way that Rost et al. in paper I find approximate interdimensional degeneracies at low D between the A = (-l)L states, the above exact interdimensional degeneracies between the A = (-l)L and A = (-l)L+l states imply similar approximate interdimensionaldegeneracies between the u = (-l)L+l states for low D. These considerations justify the notion of a ‘generator” ISc potential for the u = (-l)L+l states as outlined in paper I. One also notes that eqs 7 and 8 are identical when both are evaluated at D = 3 with the same values of angular momentum. This is the phenomena of A doubling. However, in other dimensions this degeneracy is lifted. This is not surprising since the differential equations involve the quadratic Casimir operators of SO(D) and it is only in three dimensions that the ?r = (-1)‘ and T = (-l)L+lstates transform under equivalent representations of SO(D) which have the same eigenvalue for this Casimir operator. Despite this, the electron center of mass wavefunction +,,m(D)(p,~;R) is the same for both members of the doublet in any dimension. One also expects near degeneracy for low D. In fact, by using the exact interdimensional degeneracies between the ?r = (-1)L and A = states, it is seen that the lifting of the A-doubling in D dimensions for states with a given L(D) and
+
m(D) is exactly the difference between the approximately interdimensionallydegenerate A = (-l)L states of the same L(D) and m(D) in D and D 2 dimensions. Thus from the data in paper I, the splitting of the A-degenerate states will be no more than a few percent of the total energy up to D 10. This lifting of the A-doubling degeneracy will not affect the classification scheme proposed by Rost et al. in paper I which uses approximate and exact interdimensional degeneracies, saddle degeneracies, and A-doubling at D = 3. To summarize: the D dimensional MO description of Rost et al. in paper I for the 7r = states of doubly excited states of two-electron atoms has been extended to A = (-l)L+l states. We have shown that the MO approximation preserves the exact interdimensional degeneracies between the A = (-1)‘- and A = states and that there exists approximate interdimensional degeneracies within the A = (-l)L+l manifold of MO states. This provides the justification for the extension of the new classification scheme proposed by Rost et al. (which uses interdimensional degeneracies, saddle degeneracies, and A-doubling, see paper I) to A = (-l)L+l states. It has also been noted that the phenomenon of A-doubling is specificto three dimensions. This is to be expected since u = (-1)L and A = (-1)L+I states which are equivalent in three dimensions under SO(3) are no longer so in D dimensions under SO(D). Despite this the extent to which the A-doubling is broken will be small for low values of D and the internal MO ECM wavefunctions will still be identical in any dimension for those states which form a A-doublet at D = 3.
Appendix A The total angular momentum tensor Lij is the sum of the two one particle angular momentum tensors 1;) and 1;) of particles 1 and 2, respectively, i.e.
The one particle angular momentum operators have the form of eq 3. Using the fact that up to a multiplicative factor the antisymmetrizer 6 is a projection operator, i.e. e2and 26, and the commutator relations between the coordinate x and the momentump = -i(d/dx) (Le. the Lie algebra of the Heisenberg group) one obtains for the generalized total angular momentum operator (see eq 4)
6,,(D - 3)(r(*).VU)] (Al) In an analogous fashion to the hyperspherical harmonics, the two particle states transforming under irreducible representations of the D dimensional rotation group q D ) , for which T = (-l)‘+I at D = 3, are eigenstates of L2 (the second part of Schur’s Lemma24). A tensor representation, hr(p(l),p(z)),of these basis functions of the irreducible representations may be generated as follows (see papers I1 and 111)
where hr(p(I ) , p ( 2 ) ) ~ ~ ~is, lal rth ~ , ~rank ~ , zhomogeneous l~ polynomial/ tensor in the components of the electron position vectors r(O).The W(l~v”‘zl ,J*lJ”,!J( r(’),r(’)) are tth rank monomials/tensors constructed {JP,
2460 The Journal of Physical Chemistry, Vol. 97, No. IO. 1993
from the p(,)th rank monomials/tensors following fashion
q;;,’l(r(u)) in
the
where
{J,,J is the set of indices of the monomial, being shorthand for ...,j,,(ml, and p ( ~+) p(2) = t . The projection operator rq projects from a rank t tensor space onto the irreducible 1-fold traceless tensor subspace of rank t labeled by r and p. One may write :D;as
j l , j2,
Dunn and Watson
References and Notes (1) Rost, J. M.;Sung,S. M.; Herschbach, D. R.; Briggs, J. S . Phys. Reo. A 1992, 46, 2410. (2) (a) Witten, E. Phys. Today 1980, 33 (7), 38. (b) Witten. E . In Recent Deuelopments in Gauge Theories; NATO Advanced Study Institute Series B, Vol 59; t’Hooft, G.,et al., Us.; Plenum: New York, 1980. (c) Herschbach, D. R. J . Chem. Phys. 1986,81,838. (d) Chatterjee, A. Phys. Rep. 1990, 186, 249. (e) Dimensional Scaling in Chemical Physics; Herschbach, D. R., Avery, J., Goscinski, O., Eds.; Kluwer Academic: Dordrecht, 1992. Related papers are cited therein. (3) (a) Feagin, J.; Briggs, J. S . Phys. Reo. Lett. 1986,57,984. (b) Feagin, J.; Briggs, J. S . Phys. Reo. A 1988, 37, 4599. (c) Rost, J. M.; Briggs, J. S . J. Phys. B 1991, 24, 4293. Related papers are cited therein. (4) Rost, J. M.; Gersbacher, R.; Richter, K.; Briggs, J. S.;Wintgen, D. J . Phys. B 1991, 24, 2455. ( 5 ) (a) Lin, C. D. Ado. A t . Mol. Phys. 1986,22,77. Related papers are cited therein. (b) Alsosee Chen, Z.; Bao, C.-G.;Lin, C. D. J . Phys. B 1992, 25, 61.
where Y: is a Young operator of the group of permutations of t objects, S,, operating on the indices of the monomial W(1~’(21( I ) ( 2 ) The partition r = [ r l , l ] denotes the Young tJu,,jJJB,J(r9r ) diagram associated with YL, where yi is the number of boxes in the ith row and Eni = y I + 1 = 1. The label p denotes which standard Young tableau, drawn from the Young diagram denoted by r, is associated with YL and may be taken to be the number in the box in the second row. The integer quantity gr is the number of standard Young tableaux which may be drawn from a given Young diagram and is equal to 7 1for the r = [rl,l] states. B: projects out the traceless portion of a rth rank tensor space and so annihilates any tensor of the form
.
‘ i F k 1 m . .. The differential operator V,-Vp in L2commutes with YrB:and operates on q~~~,,l(r(1))q~~~!j(r(2)) to give terms of the form
(6) (a) Berry, R. S.;Krause, J. L. Ado. Chem. Phys. 1988,70,35. Related papersarecited therein. (b) Berry, R. S . Contemp. Phys. 1989,30,1. Related papers are cited therein. (7) Frantz, D. D.; Herschbach, D. R. J. Chem. Phys. 1990, 92, 6668. (8) Sung, S. M.;Herschbach, D. R. J. Chem. Phys. 1991, 95, 7437. (9) Hunter, G.; Pritchard, H. 0. J . Chem. Phys. 1967, 46, 2153. (10) Gersbacher, R.; Broad, J. T. J . Phys. B 1990, 23, 365. (1 1) Dunn, M.; Watson, D. K. Continuation of the SchrMinger Equation for Higher Angular Momentum States of Two-Electron Atoms to D Dimensions, Interdimensional Degeneracies and the Large Dimension Limit, in preparation.
(12) Dunn, M.; Watson, D. K. Continuationof the Wavefunction of Higher Angular Momentum States to D Dimensions, in preparation. (1 3) Reference 14, p 164. (14) Weyl, H. The ClassicalGroups; Princeton University Press: Princeton, NJ, 1939. (15) Reference 16, p 397.
(16) Hamermesh, M. Group Theory and its Application to Physical Problems; Dover: New York, 1989. (17) Reference 16, p 317. (18) In even spatial dimensions the irreducible representations of O(D)
for which the Young tableaux are self-associate are reducible under S q D ) to two inequivalent irreducible representations of S q D ) with the same dimension.l3,Is Despite this, the quadratic Casimir operator of S q D ) L?will have the same value for both irreducible representations of SO@), i.e. the original irreducible representations of O(D) with the self-associate Young tableaux are eigenstates of L?. This follows from the facts that these two irreducible representations of S q D ) are connected by “parity” type transformations with determinant = -1 and that L? is invariant under these transformations. These two different irreducible representations of S q D ) will bedifferentiated by higher Casimiroperatorsofoddorder. Theirreducible representations of O(D) with tableaux which are not self-associate remain irreducible when the group is restricted to SO(D).l3.l5 (19) Talman, J. D. Special Functions: A Group Theoretic Approach; W. A. Benjamin: New York, 1968; Section 7-2, discusses the completeness of the
As explained in paper 111, when D = 3 t = L + 1 and so eq A2 may be written
and is the result we are seeking.
as.
(20) Herrick, D. R. J . Marh. Phys. 1975, 16, 281. (21) Herrick, D. R.; Stillinger, F. H. Phys. Rev. A 1975, 11, 42. (22) G d s o n , D. Z.; Watson, D. K.; Loeser, J. G.; Herschbach, D. R. Phys. Rev. A 1991, 44, 97. (23) Reference 14, p 154, Theorem (5.7.A). (24) See, for example, ref 16, p 100, Lemma 11.
