Orbital Improvement by Overall and Local Scaling A Simple Example Marten J. ten Hwr Dr. Aletta Jacobsscholengemeenschap, Hoogezand. Nederland Orbitals are the gist of quantum chemistry. They are the solutions of the SchrBdinger equations of one-electron atomic or molecular systems. Such one-electron functions form the basis for Hartree-Fock type calculations on more com~ l e atoms x or molecules. Methods that go - bevond . the Hart m - ~ o c klevel, such as the configuration interaction method, are usually described in terms of orbitals. Unfortunately, these orbitals cannot, in general, be given in a closed form. They are either calculated numerically, or approximated by one (or more) simple variational function(~).In the latter case, one may choose either some function of suitable mathematical form that contains a number of variational parameters or a linear combination of members of a suitable c o m ~ l e t eset of functions. Orbital improvement may be accomplished either by ins the function of troducing more variational ~ a r a m e t e r into chosen form, or by takingmore members of the chosen complete set in the linear combination. Examples hereof may be found in recent work by Smith and co-workers ( I ) , who investigated the usefulness of nearly 100 such approximate orbitals in describing the Hartree-Fock ground state of the helium atom. Overall and Local Scallng I t is well known (2) that any approximate atomic orbital, d r ) , can beimproved by overall scalingwithacomtant scale factor, k. Thus, +(r) is replaced by k3fi+(kr), and the optimum value of k is found from the condition aE/ak = 0,in which E denotes the total energy of the system under consideration. Less well known seems to be the fact that any +(r) can also be improved by local scaling (3).This is done by replacing the variable r by a suitably chosen function, f = f(r). T o avoid mathematical complications, i t is desirable to choose f(r) in such a waythat the inverse function, r = r(fl, is simple (4). However. one can restrict oneself t o choosine a suitable transformation, r(fl, which should increase monotonically with increasine f (rendinn to infinitv. ".i f .f tends to infinitv). and which sh&idsatisfyihe condition
~.
Instead of +(r), one now takes the function
x
( )
f df =;
'(2
om = xv,
as a transformed orbital, because this form enables the use of f as a new (integration) variable. Moreover, x will be normalized, if is normalized. Obviously, the best result will he obtained, if local scaling is complemented with overall scaling. The first application of the local scaling method, in the way outlined above, seems to be due to Hall (4). H e improved the simple exponential, using the transformation
+
in which u is a variational parameter. Taking the transformed orbital as an approximation t o the 1s orbital, Hall treated the Hartree-Fock ground states of the first six members of the helium isoelectronic sequence. Unfortunately, owing t o some errors, Hall's total energies turned out to be too low. But even the higher, correct values (5) are cousiderably lower than those obtained with the untransformed function. For example, for helium the correct energy obtained with the Hall-transformed orbital is -2.86091 233 au, which is much closer to the Hartree-Fock value of -2.86168 000 au (6)than to the value of -2.84765 625 au found with the ulitransformed simple exponential (7). I t thus seems to be worthwhile t o investigate the method of local scaling further. In this work we consider transformation other than the one of eq 3. T o he able to compare our results with those obtained with Hall's transformation, we choose to improve the normalized (but unsealed) function
~
and, like Hall, we treat the Hartree-Fock ground states of atomic two-electron systems. :
Volume 66 Number 8
August 1989
633
Application of a Slmple Transformation For sufficiently small a , the transformation
Using
is not much different from the one of eq 3. However, when eq 5 is used, the expressions of the potential energy integrals are simpler than those found with eq 3. Combining eqs 2,4, and 5, we find
and
The mathematics used to find these results is discussed in detail in ref 4. Hall's transformation leads to more complex expressions for u,, u,, and t, all of which contain the function F1, buteach withadifferent argument (4,5).This function is defined hy FI(x) = exp ( x )El(%),in which%(%) denotes the exponential integral (8)
$[F,(%)I
1 a(1 + a )
- ?pl
=-
aZ
(T)
the derivative oft with respect to a, calculated from eq 8, can be written as dt -= dn
- 8 + 44a + 84aa+ 72a3+ 28a4 + 7m5 6a5(1+ a)'
When we now try to find (for any Z)the optimum value of afrom eq 11,we run into aprohlem that we also encountered in the direct determination of the minimum of E(a): if t is calculated from eq 8 as the difference between two nearly equal positive terms, a number of significant figures are lost. This problem is more serious for larger values of Z: if a = 0.04 (Z .2: 2), four significant figures are lost, hut, if a = 0.01 (2-7). already six such figures are lost. A similar problem is encountered in the calculation of dtlda. In an attempt to circumvent this new problem, we substitute eqs 6-8 and 12-14 into eq 11. After some rearrangements we end up with
Now that the basic formulas have been obtained, the remaining part of the calculation is standard procedure. Ensuring overall scaling with the scale factor k, the kinetic energy T and the total potential energy Vcan be written as T = k2t and V = -2Zku, ku,. After substitution of these expressions into E = T V, the optimum value of k can he found from dElak = 0 to he
+ +
J
The total energy may now be calculated from
Problems wlth Optlmlzatlon The optimum value of a is usually found by the "method of trial and success" (9),i.e., E(a) is calculated from eq 10 for different values of a and the value for whichE(a) reaches its minimum is the optimum one. Unfortunately, in the neighborhood of its minimum, E(a) varies very slowly with a. To find the optimum value of a very accurately, these calculations have to he done in double precision. But, if the calculations have to be carried out on a programmable (pocket) calculator, whichmay give results with an accuracy of 9 to 10 figures, then a high degree of accuracy cannot he obtained. This problem may he circumvented by determining the optimum value of a directly from the condition dElda = 0. Using eqs 9 and 10, this condition leads to the equation
Values of the function FIare best calculated by numerical integration, which can easily be done with a programmable calculator, such as the HP 41C. For a selected value of Z, we now calculate the left-hand side and the right-hand side of eq 15, by substituting suitable values of a. It turns out that a serious loss of significant figures occurs on both sides of eq 15 when a is close to its optimum value. Thus, if Z = 1, we cannot find amore accuratevalue than a = 0.1380, and in the case Z = 2, our best result is a .2; 0.044. Despite agood deal of extra work, we do not seem to be better off with eq 15 than we were with eq 10. Such is not the case, however, as will be shown in the next section. But first we should mention that accurate values of a can be found on the basis of eq 11. To prevent the loss of significant figures in the determination of t and dtlda, we only have to calculate the original integrals, t = ( A - 4 ~ )and dtlda = d(A-4x)ldx, directly by numerical integration. Our optimum results, obtained in this way, are presented in Table 1. Asymptotic Expansions For sufficiently large x , the function Fl(x) can be approximated sufficiently accurately by the asymptotic expansion (10)
-1
"="I
F,(x)
The derivatives of u, and u, with respect to a are readily found from eqs 6 and 7, respectively, to be
-due =-
da
834
+
+
256(32 25a 5a2) (2 + d 3 ( 4+ a)'
Journal of Chemical Education
(13)
,
-I"+,]
(16)
+
Taking x = (2 2a)Ia and substituting eq 16 into eq 15, this last equation (many pages of simple algebra later) becomes
+
and
"
(-) n.r
n=o
+ + +
+
16Za(l+ (1112)a (423/16)m2 (401/16)a3 (68535/256)a4 - (6 27565/512)a5 (415 74825/4096)a6- .) = 1 + 15a + (289/4)a2 (4065/16)a3+ (1311/4)a4 (52683/32)a6- (4 07697/64)a6+ (34 57059/64)a7- . .. (17)
+
..
For sufficiently large Z (or small a ) this relation gives an excellent result for the optimum value of a. But even for Z = 2, the prediction a = 0.04393 13 is not bad a t all, since the optimum value happens to be equal to 0.04393 1985 in this case. The reason why eq 17 performs so much better than eq 15 is, of course, the fact that the large terms that cancel out ineq 15, are not present ineq 17. More important than this is the use we can make of eq 17 to solve the problem of optimization in a general way: we can find an expression for the ootimum value of a in terms of Z onlv. T o do this. we first divide the right-hand sideofeq 17 by
