A Lesser Known One-Parameter Wave Function
for the Helium Sequence and the Virial Theorem Wai-Kee Li The Chinese University of Hong Kong. Shatin, N.T., Hong Kong In the variational treatment of the helium atom and heliumlike ions, the simplest trial function is often cited to be
*, =
e-"!'l+'z'
For function $1, when a = Z - (5/16), it assumes its minimum energy
(1)
where the parameter a may he physically interpreted as the effective nuclear charge felt by each of the electrons. In this note, another one-parameter trial function, having the form is introduced. Here the parameter a may be taken as a measure of the (angular) correlation effect in the two-electron system. I t would be of interest to determine whether or $2 is a better trial function. The result of such a determination would also indicate the relative importance of the screening and correlation effects. In atomic units, the Hamiltonian operator for the helium sequence is
and the ionization potential (IP) for the helium sequence is
Even though it is less well known, the energy expression for $2 may be easily derived from results found in the literat u ~ e . ' .In ~ order to facilitate the discussion concerning the virial theorem, it is more convenient to break down this energy expression into three parts, namely, nucleus-electron attraction (NE), electron-electron repulsion (EE), and kinetic energy (KE):
'
Bethe, H. A,; Salpeter, E. E. Quantum Mechanics of One- and Two-Electron Atoms; Academic: New York. 1957. Li, W.-K. J. Chem. Educ. 1987, 64, 128.
Volume 65 Number 11 November 1988
983
Results 01 the Two Simple Trial Functions tor the Helium Sequence
HHe Lit Be2+ B3+
C4+ NSt
On+ F"
If'e,
fpmcs
lPrslcb
0.02758 0.9033 2.7789 5.6538 9.5290 14.4051 20.2827 27.1617 35.0434
-0.0273 0.8477 2.7227 5.5977 9.4727 14.3477 20.2227 27.0977 34.9727 43.8477
-0.0110 0.8771 2.7556 5.6321 9.5078 14.3833 20.2585 27.1338 35.0089 43.8840
-
Nest
NEE -1.6296 -7.1341 -16.6555 -30.1853 -47.7193 -69.2555 -94.7930 -124.3314 -157.8703 -195.4097
EEd
PE'
KE'
PE/KE
a
0.4247 1.0072 1.6200 2.2399 2.6622 3.4656 4.1096 4.7339 5.3584 5.9830
-1.2450 -6.1269 -15.0355 -27.9454 -44.8571 -65.7698 -90.6634 -119.5975 -152.5119 -189.4267
0.7159 3.2498 7.7799 14.3134 22.8493 33.3866 45.9248 60.4637 77.0030 95.5427
-1.6831 -1.8653 -1.9326 -1.9524 -1.9632 -1.9699 -1.9746 -1.9780 -1.9806 -1.9826
0.9682 0.5572 0.4803 0.4477 0.4297 0.4162 0.4103 0.4046 0.4001 0.3967
+
+
0.9427, which would lead to an IP of -[X2(KE) X(PE)] $ 2 2 = 0.8879 au; this IP is closer to the experimental value (0.9033 au) than the one (0.8771 au) obtained with $2. If the two parameters in G3 are allowed to vary concomitantly, the optimalvalues fork, c, and IPare2 1.8497,0.3658, and 0.8911 au, respectively. Thus, for He, both +30
-zur =e
+r
z)(I+ ahrIZ)
with and Furthermore, the sum of N E and EE gives the potential energy (PE). The IPSobtained with $1 and $2, along with theexperimentalvalues as well as the NE, EE, and K E using 4%for the first 10 members of the helium sequence are summarized in the table. Examining the tabulatedresults, i t can be seen that, for all the species studied, J 2 is a better trial function than $1. Phvsically speaking, for these two-electron systems, the w r relation e f f k is 'ore important than the screening effect. Remarkably, the IP's obtained with $2 are better than those with &, bv a nearlv constant value. about 0.03 au. Also, both func&nsfail to &edict a stable hidride ion. Moreover, it can be seen that, upon partitioning the total energy obtained with $2, the quantum mechanical virial theorem for such svstems (PEIKE = -2) is not satisifed. [It when a is a t the optimum, the is well known that, for virial theorem is obeyed. Indeed, this is often used as a mathematical check.] The failure of $2 to fulfill the virial requirement indicates that $2 can he improved by a simple scaling of rl and r2: +3
=e
-a~,,+h)
(1
+ airl,) = e-klr
tr
( 1
+c
,
(10)
which is a combination of $1 and $2. While the results yielded by $3 have been discussed in detail in a recent note2, it is worth pointing out that, with the tabulated data, an approximate value for the optimal X can he obtained. Take He as an example. For the giuen a, the best X value is -PEI2KE =
964
Journal of Chemlcal Education
lead to P E and K E values that satisfy the virial theorem, even though h b yields a better energy value (the optimum) than +30 This brings about an interesting feature concerning the virial theorem: PEIKE = -2 does not guarantee the total energy (PE KE) is a t the minimum. This point does not seem to have attracted the attention of many textbook writers. Finallv. .. it is seen that. as Z increases in the helium sequence, parameter0 ingzhecreases steadily. Physically, this means that El? becomes less and less important as a fr~ctorin PE, or e-z('1+'2' becomes increasingly better in describing the electronic behavior of the three-body system. As a result, the ratio PEIKE approaches -2, as predicted by the virial theorem. The data tabulated support this argument. In conclusion, since $2 is a better trial function than $1 for the helium sequence, i t should be included in the variational treatment of these svstems in auantum chemistrv texts. Such an inclusion aGords a bet'ter understanding of the relative imnortance of the correlation and screening - effects in simple atomic systems. In addition, $2 is an excellent example to illustrate a couple of interesting points that are seldom mentioned in textbooks: (1)a trial function, which after optimization does not give results satisfying the virial theorem, can be improved by scaling, and (2) fulfillment of the virial theorem does not necessarily mean that the total energy is at the minimum.
+
