H. E. Montgomery, Jr. United States ~ & a ~l c i d e m ~ Annapolis, Maryland 21402
I
I
Helium Revisited An introduction to variational perturbation theory
Bridging the gap between the formal and the computational aspects of quantum theory is always a problem in courses which first introduce quantum theoretical concepts. If the introduction occurs in an undergraduate physical chemistry course, the chemistry student's lack of background in physics and mathematics forces the instructor to concentrate on formalism. Students usually finish a physical chemistry course having a nodding acquaintance with particle in a box problems and with the solution of the Schrodinger equation for the hydrogen atom, hut with no understanding of how quantum theory can he used t o treat chemical problems. T h e undergraduate who wants to learn about computational quantum theory is usually sent to the physics department (where he receives another dose of formalism), or told to "wait until you eet to eraduate school." Since most colleees offer courses in " advanced chemical topics t o their senior chemistry majors, it would amear that a course in com~utationalmethods would he a desirable subject for student; contemplating graduate work in physical or physical organic chemistry. Yet such courses are seldom offered. One of the reasons for this is the scarcity of computational examples which lend themselves to presentation in such a course. The purpose of this paper is to present such an example. The problem considered is a calculation of the ' S ground state of the two-electron atom. The computational technique is a variational-perturbation procedure due to Hylleraas.' Although the calculation lends itself nicely to a simple computer program, it is possible to obtain meaningful results using a four-function calculator.
holds for all values of A. Eqn. (16) is the variational-perturbation equation originally derived and used hy Hylleraas.'
Method
Equations
The equations for Rayleigh-Schrodinger perturbation theory can be found in any introductoryquantum theory texL2 The fundamental principle is that the Hamiltonian, H, for a quantum mechanical system can he partitioned as
The form of the calculation is determined hy the choice of the wavefunction. This calculation uses the one-electron wavefunction
H = H o t AH1
(1)
where H n is called the zero-order Hamiltonian. H,is called the first-brder ~ a m i l t o n i a nand , Xis called the perturbation oarameter. The enerzv ... and the wavefunction describine the ivsttm can then hr ohtained as p w t r ser:es in A. 'I'hii work uses the iorniulntion rivrn hv Scherr and Knieht:' in their work on two-electron atoms of niclear charge Z. ?he following formulas result4
then substitution into eqn. (9) gives a power series in X in which the coefficients of A for powers greater than 0 must equal zero. In particular, the coefficient of XI gives (*ol*,) = 0
(11)
By using the Schrodinger variational principle (&IHI&)-E(oIo) 2 0
(12)
a variational expression for $1 and Ez can be obtained. Substitution of eqns. (1-6) into eqn. (12) gives apower series in A. T h e coefficient of A0 is ($olHol*o) -Eo(+ol+o)
(13)
which vanishes by equ. (7). The coefficient of X1 is 2(+1lHo- Eo(Jio)
+ (+olH~- E ~ l h )
(14)
which can be shown by eqns. (7) and (8)to he zero. The coefficient of X2 is (*~lHo-Enl$o)
+ 2(hlH1-
E~lJlo)-Ex
(15)
For the case of small A, the term involving X2 must necessarily he the dominant term in the series. But the variational principle, eqn. (12), requires that this term be positive or equal to zero. Since there is no X-dependence in eqn. (15), the restriction (hIHo-EollL~) + 2(+11H1- EIIJ.~)5 Ep
(16)
F = 1s + (l1Z)G (17) where 1s is the normalized radial part of the 1s orbital for the hydrogen atom 1s = 2e-'
(18)
and G is a one-electron variational trial function. If 4, the two-electron wavefunction is chosen to be a product of two one-electron wavefunctions, then
Ha = - % V L ~ KvzZ- l / r ~ 111-2
(2)
4 = F(l)F(2) = ls(l)ls(2)t (l/Z)lls(l)G(2) + C(l)ls(2))+ . . . (19)
HI = l l r u
(3)
Thus
A = 1/Z
(4)
E=EoZZ+EIZ+Ep+..,
(5)
+ . ..
$ = J.0 t (1IZ)J.j + (112)2112
&" = ls(l)ls(2)
(20)
Equations (7) and (8) can he used to evaluate Eo and E l
(6)
Eo and El can be evaluated using Eo(J.olJ.o)= (J.olHolh)
(7)
E~(+ol*o) = (v'olH~l$o)
(8)
If $ is normalized so that
(*I*)
=1
(9)
' Hylleraas, E. A.,Z. Physik, 66,209 (1930). 'Pilar, Frank L., "Elementary Quantum Chemistry,'. McCraw-Hill
Book Co., New York, 1968, pp. 249 ff. :'Scherr, C. W., and Knight, R. E., Rcu. Mod. Phys., 35, 431 ~
., .. . . ,
tlnlts nrr w e d thr- Ey (39) The coefficient of a2vanishes by eqn. (71, while the coefficient of n can be shown by eqns. (7) and (8) to be zero. The remaining terms form the variational expression originally used to determine 4,. Thus satisfies both the variational perturbation equation and the normalization requirement. Results Using a 14-term variational trial function, E2 was calculated to he -0.11100317 au. Increasing the number of terms in the wavefunction had no effect on the first seven decimal places of the calculated value of Ep. Substitution of the values of Eo, E l , and E z into eqn. ( 5 ) allows calculation of the energy for two-electron atoms of arbitrary Z. For the helium atom ( Z = 2), a total energy of -2.86100317 au is ohtained. T h s compares favorably with the Hartree-Fock limit of -2.861673 au given by Green, et al.," but is significantly higher than the energy of -2.90372433 au obtained by Scherr and Knight3 in their sixth-order treatment using correlated wavefunctions. The effect of interelectronic repulsion on the one-electron wavefunction can be better seen by considering D(r), the radial probability distribution, where
and eqn. (26) becomes Ez 5 2
D(r) = r2$2
X Z oma,Tmn+ 4 nZ=:1 n,Pn
,=I "=I
(30)
The an's are determined by minimizing Ez with respect to each of the am's. This results in a set of k simultaneous linear equations of the form
These equations may he written in matrix form as (TNo) = -(P) (32) The an's can he determined by matrix inversion or by Gaussian elimination. Substitution of the am'sinto eqn. (30) determines E2 and &. $ r,, and 6, can be converted to the form of eqn. (6) by requiring that the normalization conditions of eqns. (10) and (11) he satisfied. Since & is aprodud of normalized functions, it is itself normalized. Thus eqn. (10) can be satisfied by making *O
$1
(38)
=
mo = 1s(l)ls(2)
(33)
can he ohtained by writing $1
= 61 +
&
where a is a constant determined by inserting $0 and +I eqn. (11)
(34) into
For the unperturbed one-electron function, D(r) has a maximum at r = 0.5 au. Inclusion of interelectron repulsion shifts the maximum in D(r) to 0.594 au for the perturbed oneelectron function. This change in D(r) can be thought of as increasing the average distance between the two electrons thereby making the atom more stable. Thus the sign of E2, the second-order energy, is seen to be in agreement with the change in the electron distribution. If the simple trial function is used, E2 can he evaluated using basic algebra. This two-term trial function gives Ez = -7104 = -0.109375 Thus the two-term ex~ressionaccounts for over 98% of the energy obtnincd using the 14-term expression. 'I'h~si.nlculationdemonstrates the feasibility ofperfurm~nr quantum mechanical calculations giving accurate results without becoming bogged down in mathematical formalism or esoteric programming techniques. As such, it should be useful either as a lecture example or as an exercise for the student who wishes to acquire some familiarity with computational methods. ~
Acknowledgment
($0141+ a$o) = 0
(35)
or=-($oldl)
(36)
Thus
The author would like to thank the United States Naval Academy Academic Computing Center for the use of its Honeywell-635. Gratitude is expressed for very useful discussions with Dr. C. F. Rowell.
and
i l r = 41 - ($obl)$o
(37) T o demonstrate that the normalized, first-order function
"Green, L. C., Mulder, M. M., Lewis, M. N., and Wall, J. W., Jr., Phys. Reu., 93,757 (1954).
Volume 54, Number 12, December 1977 / 749
