An Investigation of the Quality of Approximate Wave Functions Lynne H. Reed' and Arthur R. Murphy2 Fairleigh Dickinson University, Rutherford, NJ 07070 Most chemistry students are introduced to the Variation Principle when thev takean undereraduate ohvsicalchcmis-
where E(a) is the trial energy, Eo is the true ground state energy of the system, and +(a) is the trial wave function satisfying the boundary conditions of the problem. In general, the trial function could depend on several variational parameters, but the trial wave functions considered in this article will have a single parameter designated as o. A perusal of many physical chemistry books and quantum chemistry texts indicates that few of these books contain caveats with regard to the quality of the wave functions obtained from the Variation Principle. Unfortunately, students may get the impression that the values of the parameters that yield the lowest trial energy also give the "best" overall trial wave function. Such a view is not correct. Indeed, Lesk's physical chemistry text (2) contains the following warning: The wave function that carresponds to the best approximation to the energy may not be the best approximation to the wave function. It need not give the best approximation to properties other than the ground state energy. Pilar's quantum chemistry text (3) also contains comments about the quality of the wave functions obtained from the Variation Principle. He states that the weakness of the Variation Principle is due to the fact "that the energy is an insensitive criterion with respect to a'best' wave function for other physical properties". A pseudo helium model system will be used to illustrate some of the limitations of the wave functions obtained from the Variation Principle. This model system was chosen because of its intrinsic interest and tractability. Students can verify the results because all of the integrals may be done in closed form. No elaborate computer programming is required. Also, since the exact solution of the model system is known, the expectation values computed from trial wave functions obtained from the Variation Principle and a maximum overlap method may be compared with the exact averagevalues. The model system is not intended to shed light on specific expectation values of a real helium atom. Rather, it is used to illustrate a general feature of the Variation Principle. The major point to be illustrated has been documented in the quantum chemistry literature for syatems other than that used in the present paper. For example, Bishop and Macias (4)have shown that there is no correlation between the improvement of a wave function for a diatomic molecule, using energy as a criterion, and the corresponding harmonic force constant. Mukherji and Karplus (5) have also noted
' Present address: Princeton University, Princeton, NJ 08544. Author to whom correspondence should be sent.
that there need not be any correlation between the accuracy of the energy computed from various trial wave functions for hydrogen fluoride and the accuracy of such properties as the molecular dipole moment, diamagnetic susceptibility, etc. Thus the main result to be presented in this paper is not an artifact of the model system being employed. Similar results manifest themselves in more computationally complicated realistic systems. The Model System In a recent paper Crandall et al. ( 6 ) considered a system consisting of two "electrons" interacting with one another via an inverse cube force law. Each "electron" is also attracted to a single fixed nucleus via a Hooke's law force. For a spherically symmetric state, the dimensionless time-independent Schrodinger equation may be written as fi$ = E$ where
and The first two terms of the Hamiltonian account for the kinetic energies of particles 1and 2, while the last two terms arise from the nuclear-particle and electron-electron potentials, respectively. Note that the parameter o determines the strength of the nuclear-particle potential, while A controls the strength of the electron-electron repulsion. The exact ground state solution of this equation is given by
where
and the normalization constant, which contains the gamma function, is given by
Since the Hamiltonian being used does not contain spin variables, the total antisymmetric wave function will consist of a space part of proper symmetry multiplied by a spin part of proper symmetry. Because the normalized spin portion of the wave function yields unity when average values of spinfree operators are computed, the spin part of the total wave function has been suppressed. Variation and Maximum Overlap Calculations When the normalized Gaussian wave function is inserted into eq 1 using the Hamiltonian of eq 3, several Volume 63 Number 9
September 1986
757
integrals must he evaluated. The kinetic energy contribution to the total energy is given by
Average Values ot Operators for Various Wave Functionss Alpha
(r:)
Energy
I
r
(r:)
whereas the integral from the nuclear-particle potential is given by
Details for handling integrals involving llr:, can he found in the appendix. The interparticle potential energy contrihution is
12, for which the results are
Summing eqs 9-11 yields Additional average values can he ohtained from the general formula (A) =
The condition
(+,AJ.)/(w
(21)
For example, (r:) ohtained from the exact wave function is given by
results in a value of ar giving the minimum value of the energy
By evaluating the overlap integral between the true wave function of eq 5 and the trial wave function of eq 8, another expression for a may he ohtained. Using method 1 of the appendix, the integral is given by
whereas the average value of ( r ; ) ohtained from the Gaussian trial wave function is
By arbitrarily choosing w = 1 and X = 0.5, the numerical values appearing in the tahle are ohtained. Discussion of Results
The condition -= 0 dS do
results in a value of a which gives the maximum overlap
Computation of Average Values
For a Hamiltonian depending on a parameter 0, the Hellmann-Feynman Theorem (7) states that
This theorem, which greatly simplifies the calculation of certain average values, holds for the exact wave function as well as for trial wave functions which obey Hurley's condiLetting 0 = wand inserting the Hamiltonian of eq 3 tion (8). and Eo from eq 4 into the H-F theorem yields the average value
Note that (r:)LIZ may be taken as a measure of the "size" of the atom. For 0 = X the following average value is ohtained
What conclusions can he drawn from the averages comhe considered as puted in the previous section? Can $(n,i.) the best trial function? What significance can he attached to the word "best" in this context? If by "best trial wave function" is meant a single trial function for which all computed averaees - are closest to the true averaees. - . then a maior . .orohlem is encountered. In general, no single trial wave function with these ~ r o ~ e r t iexists. es This ~ o i nbecomes t more transparent!~ foc&ing on several fea&res of the tahle. which gives a t ~ i aenergy l closest to the exact It is $(a,;,) energy of the system, but it is $(a,,,), and not $(amin), which provides a value for th_e"size" of th_eatom closest to the true result. Indeed, both $(rum,,) and $0 yield exactly the same formula for the "size" of the atom: the square root of the formula of eq 17. This is a rather unexpected result. When ( ~ / r : ~is) considered, i t is $(amin) which gives an average value closer to the true result. The tahle also contains results for an arhitrarily chosen value of a, nargely a = 0 . 4 5 . Note that the value of (rf) ohtained from $(%= 0.45) yLelds a result closer to the exact value than either $(amin) or $(a,,,). Thus it is seen that there is no guarantee that a given trial wave function which yields the best average value for a given operator will also yield the best average value for another operator. Clearly, choosing a "best trial wave function" from among a set of trial functions on the basis of an average value computation using a single operator is not correct. Such an approach does not provide a criterion for judging the overall quality of the trial wave function. Literature Cited
i11 ~ ~ ~ t ~ i ~ , s . ' r . ~ ~ ~ h ~ v ~ ~ i ~ t i ~ ~ ~ ~
197kChao1.
Similar calculations can he done using the trial energy of eq 758
Journal of Chemical Education
(21 1.esk. A. M:'"lnlrductiun to Physical Chemistry"; Prentice-Hall: Englewood Cliffs. N.I. 1882; p 375. (31 Pilar. F. L. "ElementaryQuantum Chemistry": Mffiraw~Hill:New York, 1968: p 238. i4l Hishop, 0. M.: Maciar,A. J. C h e m Phyr 1971.15,647.
A.; Karplur. M. J. C h e m Phys 1963,38,44. 181 Crandall, R.; Whitnell, R.; B e l t e ~ aR. , Arner.i. Ph>r I984,S2,4:38. (71 Levine, I. R. "Quantum Chemistry": 3rd ed.: Allyn & Bacon: Bustun. 1983; p 404. IRI ~ p s t e i n S. . T. l n he h r e e Ccmeeur in chemis~ry";~ e hB, M., ~ d .van : ~ostrsnd Reinhnld: Now York, 1981; p 15. 191 Calais..l.: Lowdin,P.J. Mol. Specl. 1962,8,203. 110) Omellas, F.R. J. Chsm. Educ. 1985.62. 378. i l 1) Braewell. R."The Fourier Transform and its Applicatians": 2nd ed.: McGraw-Hill: New York, 1978: pp7.2hI-253.422. ( 5 ) Mukherji,
(- I I -). $,
= (2a.h) 16r2A
r
,' 0
e-2"i"+"1r,3
0
1"
[g] dr,dr2
The transformation x =
(ipr,+
r2)
Appendix The first method that will be considered for handling integrals containing functions of rls uses a formula for spherically symmetric eases from Calais and Ldwdin (9).
r
allows the last integral to be written as
,'
+
( 2 a / ~ ) ~ 8 ~ ~e-2"'z2ty21(~Z h y2)(lnx - 1ny)dxdy (1
0
After writing this integral as the sum of four integrals, changing the order of integration of two of the integrals, relabeling integration variables, and regrouping terms, the desired result is obtained
where dvi is the volume element for electron i. The integral of eq 11 will now be evaluated using this method. Using
);(
f(r) = g(r) =
Za
312
exp(-Zar2)
and performing the integration over rlr results in
..
Other methods exist for handline.. inteerals involvine functions of In pnrtirulnr, three-dimensional Fmrirr rran-t.rm wchnquer ~ l f ~ . lhl w r e pnwen t t r be v r q pawerful. Readrri are also urged tu ~ t u d )thetranrfurmatiun uitd by ('randall rt al (ti, in their papwon the pseudo helium atom model system, r,:.
Volume 63
Number 9
September 1986
759
