Richard 1. Snow and J a m e s 1. Bills Brigham Young University
Provo, Utah 84602
I
I
Textbook Error, 777
The Pauli Principle and Electronic Repulsion in Helium
Several recent textbooks in quantum chemistry use a discussion of the excited states of the helium atom to demonstrate the importance of the Pauli principle in determining electronic repulsions.' The argument goes as follows. 1) Zeroth order wave functions for excited states of helium are
where
and S+ is one of the symmetric functions a(l)a(2), P(1)0(2), and (1#/\/2) (a(1)0(2) + 0(l)n(2)); a and 0 are spin functions; and is, 2s, and 2 p are hydrogen-like atomic orbitals with screening parameters ((1s) = ((2s) = ( ( 2 ~ )= 2. 2) These wave functions are used with the electronic repulsion, l/rlz (in atomic units), lo calculate the fxstorder correction (El) to the zeroth-order energies.
If the optimum value of the screening parameter I is used instead of the usual f = 2, the result is the same: Triplets have lower energies than corresponding singlets, apparently because of decreased repulsion in the triplets. - Although accurate wave functions have been known for these excited states for many years (1, 2), i t appears that only recently have the total energies been analyzed and found to contradict the common interpretation given above (3-5). A comparison of calculations using selected wave functions is shown in Tables 1and 2. Examination of the tables shows that for helium a wave function with only two parameters gives results which are very different from those shown for simple perturbation theory or the optimized one-parameter wave function. The total energies and the individual terms in the total energies for the two-parameter wave function are remarkably similar to those calculated using the accurate wave functions. These accurate calculations give electronic repulsion in the triplet state which is greater than the repulsion in the corresponding singlet state, rather than less as implied by the usual argument. The triplet energy is lower Table 1. Parameters and Expectation Values for P States
..
. exact
WP)
axad
is'de6Nd in esn. (1).WP) and +(W,together ~6ththe expectation exact exact values, are given in C. L. Pekeris, et el., Phys. Reu., 140, A1104 (1965): A4, 516 (1971). c
Table 2. Parameters and Emectation Values for S States 3) It is concluded that the triplets have lower energy than corresponding singlets because the electronic repulsion (El) is less for the triplets. The decrease in repulsion of triplet relative to singlet is explained by stating that "Pauli forces" require that electrons with parallel spins he farther apart on the average than electrons with antiparallel spins.
..
WS) $I(*)
*1('8) $>(*S)
W'S)
+a(W *('S) exact
$(as) exact
Suggestions of material suitable for this column and guest columns suitable for publication directly should be sent with as many dr~ailsas possible, end particularly with reference to mod. em textbooks, to W. H. Ebrrhardr. School of Chemistry. GeorKia InstituIeofTechnoloev. Atlanta. Georgia 3 0 3 2 . 1 Since the purpos-of this column is to prevent the spread and continuation of errors and not the evaluation of individual texts, the sources of error discussed will not he cited. In order to he presented, an error must occur in at least two independent recent standard books.
.
2.000 2.000 1.815 1,850 1.990
2.000 2.000 1.815 1.850 1.058
2.009
1.389
... ...
. ...
3.131 3,400 3.385 5.318 4.474 5.270
0.464 0.376 0.421 0.348 0.249 0.272 0.250
4.541
2.124 2.058 2.138 2.143 2.172 2.146
4.448
0.268
4.619
2.175
3085
5.m 5.0W 4.536 4.624
4.529 4.616
2.036
+
1.0000 c +I is d e h d in eqn (1). +@S) = 0.1207 ls'(1) ls'(2) 18(lS).The coeffiiciente were determined by the linear variation method. The screening parameter of the 1s' orbital is 1.6875. Even with this Parameter fixed, the virial theoremissatidad within 17,. In both the triplet and singlet states lahas the same form as lyl except that the 28 orbital in l a is b2i where li and 23 are Slaternot a hydrogenic orbit& 10 +I, zs = d i tylls orbitas with orbital exponent equal to 1(2d/2. T h e eonstant. o and b were chosen Bo that the 2~ orbital is orthogonal to the 1s orbital with sereenins parameter {(IS). ibex-% is found in H. 0.~ritchardand A. Wallis, J. C h m . Phys.,42,3548,1965. The expectstion valvesfor are those in D. Xahl, J. Chem, ~ h y ~56, . , 4236 (1972),except that the values for (n,)are fmm C. L. Pekeris as cited in Table I.
+
Volume 51, Number 9, September 1974
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585
due to greater nuclear attraction, which more than compensates for the greater repulsion (3). The average interelectronic distances also confirm that electrons are closer in the triplet state than in the corresponding singlet state. The unex~ectedf a d that the electronic re~ulsionis sometimes greater in the state of higher multiplicity has been given an interpretation by Katriel (6). whose approach is used in the following interpretation of the 2'P and Z3Pstates of helium. The change in the wave function from +lPP) to + 1 ( 3 P ) is assumed to take place through an intermediate function +1(3P), which is + 1 ( 3 P ) written with 'the screening parameters optimized for + l P P ) . The total energy of +1(3P) is only slightly higher than that of + I C P ) , but the kinetic and potential energies of +I'(~P)do not satisfy the virial theorem. The virial theorem requires that 2 T = -V, where T is the kinetic energy and V is the potential energy
Since T and the first two terms of V all have the same values for +l'fP) as for +,CP), the (nearly correct) difference in total energy between W P ) and +l'fP) shows u p entirely in the (incorrect) value of ( r 1 2 - I ) for +1'(3P), which is less than the (correct) value of ( r l 2 - l ) for + 1 C P ) . Although the virial theorem is satisfied by ILlCP), the decrease in (r12-l) for +l'CP) makes -V too large (or T too small) for the virial theorem to he satisfied by +I'(~P). Hence this intermediate function is not a suitable wave function, but it can serve to indicate what to expect of the (correct) function + l C P ) . Since the total energy of +ICP) is lower than that of +lCP), then T of +lfP) must be larger than T of + I C P ) , and hence larger than T of k ' f P ) . On going from +I'CP) to ILlfP), the increase in T comes about through an increase in f(2p), partially offset by the small decrease in {(ls). However, the increase in t(2p) affects (rI2-I) even more than i t does T. The effect on ( r 1 2 - I ) can be interpreted by realizing that the 1s or-
= This point-change model is less satisfactor$ for the more penetrating 2s orbital, as expected.
586
/ Journal of Chemical Education
bital is scarcely penetrated by the 2p orbital, and hence the potential of the 1s electron is nearly the same as that of a point charge a t the nucleus. Using this point-charge model, we find
and we estimate the difference in ( r 1 2 - l ) between ILI'CP) and hfP) to be 0.032, compared to 0.030 obtained by complete c a l c ~ l a t i o n . ~By contrast, the corresponding change in T is only 0.008 (-0.024 in T(ls)and +0.032 in T(2p)), and the change in (Zlr) is 0.039 (-0.024 in (Z/ r)lS and +0.063 in (Z/r)z,, all in atomic units). Hence, for ( r ~ z - l ) ,the decrease on going from +lPP) to +1'fP) is more than offset by the increase on going from ILl'fP) to W P ) , and the overall result is a higher value of (r12-l) for +lfP) than for + I C P ) . The total energy of +,PP) is lower not hecause of lower electronic repulsion, but because the greater electronic repulsion is more than offset by the greater nuclear attraction. We conclude that calculations on the helium atom do not show that "Pauli forces" tend to keep electrons of parallel spins separated in space. Although this discussion of helium can not be applied iqdiscriminately to other atomic and molecular systems (7), Katriel (8) and Colpa and Islip (5) have shown that helium is not unique with respect to the reversal of electronic repulsion. The results for helium are important since helium is so frequently used as an example. This simple calculation clearly demonstrates the inadequacies of approximate solutions by showing that very subtle changes may alter substantially our qualitativeinterpretation of theoretical results. Literature Cited
15) (6) 11) 18) 19) (10) IIII
Colpa, j. P., and la lip,'^. F. J., MOI. Phys, 25,101 (1973). Kafrie1.J.. Theorel. Chim. Act0.. 26,103(1972). Me~srner,R.P..sndBims, F. W., J. Phvr. Chem.. 73,2085 11969). Kafriel. J., Theorel. Chim. Acto.. 23.309 11972). Pekoria, C . L . , e t sl.,Phys. RPU.,I40,AllM 11965); A4.516(1971). Pritchard.H. O.,and Wallis, A,, J. C h m . Phys.. 42.354811965). Kohl. D..J Chem. Phrr., 56,4236 119121.
