A method for summing strongly divergent perturbation series: The Zeeman effect in hydrogen

Physica 128A (1984) 253-267 North-Holland, Amsterdam

A METHOD FOR SUMMING STRONGLY DIVERGENT PERTURBATION SERIES: THE ZEEMAN EFFECT IN HYDROGEN Gustav0

A. ARTECA,

INIFTA, Divisih

Francisco M. FERNANDEZ, and Eduardo A. CASTRO*

Alejandro

M. MESdN

Qufmica Tehica, Sucursal4, Casilla de Correo 16, La Plats 1900, Argentina

Received 2 January 1984 Received in final form 28 March 1984

A new method for summing strongly divergent power series is presented. It consists of an order-dependent mapping that transforms the divergent series into a convergent sequence. The procedure is applied to the perturbation series of the Zeeman effect in hydrogen obtaining accurate energy values for the states 1s and 2p,t over a wide range of field strengths. Difficulties in handling the 2s state are pointed out.

1. Introduction There has been considerable interest in large-order perturbation theory since the pioneering works of Bender and Wu’) and of Simon*) on the anharmonic oscillator. Recent reviews of the subject are found in ref. 3 and in the issue Int. J. Quantum Chem. 21 (1) (1982). Several models have been very helpful in understanding large-order perturbation theory and this paper is concerned with one of them: the Zeeman effect in hydrogen (ZEH). In addition to its usefulness in astrophysics, atomic and solid-state physics (for a good account of the most relevant literature see refs. 4 and 5 and references therein), the ZEH poses quite an attractive model because it leads to strongly divergent perturbation series6). There were three sound attempts to sum the Rayleigh-Schrodinger perturbation series for the ground-state energy level of the ZEH5,7*8). Though appropriately improved continued fractions have proved to be convergent’), they seem to approach the eigenvalue very slowly for intense field strengths. Better results are obtained by successive applications of a generalized Euler transformation (GET)‘). But the most accurate perturbation calculations are undoubtedly those coming from the order-dependent mapping * To whom correspondence

should be addressed.

0378-4371/84/$03.00 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

254

G.A. ARTECA

et al.

(ODM)‘) (an d re ferences therein). However, to obtain such accurate energy values this method requires knowing the asymptotic behaviour of the eigenvalue and very accurate input data. In fact, Le Guillou and Zinn-Justin had to calculate the first 61 perturbation coefficients with an accuracy of about 20 to 27 significant figures’). On the other hand, it is not necessary to know the analytical properties of the eigenvalues to apply the GEP) and it requires much less computational effort. The purpose of this paper is to develop a new method to sum strongly divergent power series that is simpler than those discussed above597*8).In section 2 we calculate the first perturbation coefficients for the energy of the Is, 2p,, and 2s states of the ZEH. In section 3 we show the resummation method and then, in section 4, we apply it to the aforesaid problem. Further comments are made in section 5 where an improved mapping to deal with the ZEH is discussed.

2. Calculation of the perturbation

coefficients

The part of the Hamiltonian operator describing the ZEH that is relevant to our discussion is (atomic units are used throughoutr) H(Z,A)=H,+AV,

Ho=-;A

where 2 = 1 for the hydrogen satisfies the scaling relation

-%

r

v+x2+y2),

atom. Every eigenvalue

E(Z, A) of H(Z, A)

E(Z, A) = Z*E(l, AZ-4). Therefore, the Zeeman effect in any hydrogenlike atom with nuclear charge Z reduces to that in hydrogen. When applying perturbation theory (PT) to the ZEH, two problems arise. Firstly, the Schrodinger equation is non-separable and secondly the zerothorder energy spectrum is continuous for all positive energies. The latter fact makes a direct application of the standard Rayleigh-Schrodinger PT too cumbersome. There are however several suitable ways of calculating the coefficients Ei in the expansion5X7*9) E&A)=

cE,Ai. i=O

(3)

THE ZEEMAN EFFECT IN HYDROGEN

255

Two of usl’) have recently developed an alternative and simple procedure to carry out such a calculation. Since the calculation of perturbation coefficients is not our present goal and its detailed discussion will appear elsewhere”), we only give here those results that will be useful later on. The pth correction Ep to the energy of the 1s state is given by E

q(P-1) 8 2.2

=

P

7

(44

where the rational numbers I$) (q = 0, 1, . . . ; i = 1,2, . . . ; j = 0,2,4, . . .) obey the following recursion relationship

(4b) Igj==s

PO

c f,.,N

’

=

&(kf

+

1)

-

N(N

+

1))

.

(4c)

Similarly, we have

2

IgL_,,=-

cf,f,I!&J,J

M-l

’

I19’= s po,

M=3,4

,...

+ ; (N2 - 1)1$4J,,_,

;N=l,3

,...

)

(5b)

;p=O,l,...,

(59

for the 2p,, states, and E

P

=

#I&-”

- @,,l)} 9

’

V-4

p-1

(

I$ = - fs, + k c {I?$ - 2Ig}Igq-” q=o

+ I&-f;

)

(6b)

+ pf2-;-l) ) >

(6~)

)

p-1

1::

=

-

;a,,

+

f

(q=o{I$$ - 2I$J;}Igq-” c

G.A. ARTECA

256

M=2,3

,...

;N=0,2

,...

et al.

;p=O,l,...,

for the 2s state. By solving these expressions recursively, we obtain the results shown in table I. The first 20 perturbation corrections for the 1s state agree with those computed previously’) up to the last significant figure. The same accuracy is expected in the coefficients Ei for the 2p,, and 2s states that are not available in the literature.

3. The method We have recently developed a new procedure (called functional method) that has proved to be successful in obtaining accurate results from strongly divergent power series by using a few perturbation terms”-13). This method has been motivated by a sort of semiclassical approach to parameter-dependent systems14’15). Among the models studied, we can mention the anharmonic oscillator”), the funnel-like confining potential13) and the Stark”) and Zeeman”~12) effects in hydrogen. TABLEI energy of the Is, 2p,l and 2s states calculated via the recursion relationships given in eqs. (4)-(6). The numbers between round brackets are powers of 10.

First perturbation correctionsto the n 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

E&s)

E, (2~,1)

E (2s)

-0.5 0.25 -0.2760416666666664 0.1211154513888887 -0.9755405906394653 0.1178630246122380 -0.1959272760583517 0.4274861689521947 -0.1186935282560849 0.4097260186880245 -0.1725156234947565 0.8716665393270882 -0.5210940934011892 0.3640532401232869 -0.2940370393476464 0.2719572430769083 -0.2856379426829852 0.3381522618344236 -0.4482314059482390 -0.6612952596854647 -0.1080080640806817

-l/8 3 -116 16192 -57205264115 624260454415 -0.5246578495235231 0.2719288070866074 -0.1699171093824181 0.1261305580753331 -0.1100183924440788 0.1117401621637270 -0.1310589020557425 0.1761487030881318 -0.2693305161691009 0.4652898048792428 -0.90249112S1888209 0.1953914952529491 -0.4696543856732298 0.1247151762591110 -0.3642185373074384

-l/8

(1) (1) (3) (4) (5) (7) (8) (10) (11) (13) (15) (17) (19) (21) (23) (25) (27) (30)

(12) (15) (18) (21) (24) (27) (30) (33) (36) (39) (42) (46) (49) (53) (56)

712 -47813 20257619 -7447456481135 0.1881650928102710 -0.8204391786512023 0.4389596735728581 -0.2816897679290545 0.2137332357817471 -0.1897900716893362 0.1955823529627524 -0.2321429979067009 0.3151014434719851 -0.4857971635820124 0.8452044545889403 -0.1649430121661903 0.3590184717445867 -0.8670399595821733 0.2312096400478561 -0.6777773205584709

(10) (12) (15) (18) (21) (24) (27) (30) (33) (36) (39) (43) (46) (49) (53) (56)

THE ZEEMAN EFFECT IN HYDROGEN

257

In this section we show an improved version of the abovementioned method that can be easily applied to a large number of problems in quantum and field theory. The viewpoint of the present approach is quite different from the previous one”-‘3). Let E(Z, A) be a real, unknown function of the real variables Z, A (0 s Z, A < 00) that leads to the strongly divergent expansion E(l, A) = i

E$,

(7)

i=O

of which we know the first N + 1 coefficients. Our aim is to rearrange (7) so that we can obtain accurate results for E(l, A). This can be done easily if we know that E(Z, A) = Z”E(1, AZb), where a and b are real numbers and b -C0. Fortunately, encountered in quantum and field theory. We now define J%,

P) = E(KU - B1, P),

(8)

this situation is often

(9)

where K and p are real, positive variables. The scaling relation (8) enables us to write E(l, A) = K-“(1 - ,B)-%(K, /3), where A =

K*p(l-fi)*.

(11)

Since b < 0, eq. (11) maps 0 d A < 03 into 0 S/I < 1 allowing us to expand E(l, A) in powers of the bounded variable /3. To this end we write

l?(K, 6) = i i=O

Ei(K)pi,

(12)

which together with (10) and (11) is supposed to be a better approach to E(l, A) than (7) because of the boundedness of j3. In order to obtain the coefficients Ei(K), we rewrite (10) as J?(K, j3) = K”(l- P)“E(l, A(P)) and expand the right-hand side in powers of /3 by using the mapping (11) and the

2.58

G.A. ARTECA

et al.

expansion (7). The result is:

l?qK)=

$(-l,ii(q’pi)r+“E,,

(13)

where c) = c(c - l)(c - 2). . . (c - i + 1)/i! and 6) = 1. The parameter K has been introduced to attain the largest possible convergence rate for (12) and its proper value is easily obtained as follows. We approach E(l, A) by the sequence

SE, = K-(1 - /3)-“S,(K, /3),

S,(K, /I) =

2 Ei(K)f?‘,

N = 1,2,

.. . ,

i=O (14

where S,(K, p) is supposed to tend to l?(K, p) as N + ~0. Since l?(K, 1) = E(0, 1) is independent of K, then the plot S,(K, 1) vs. K should exhibit a plateau whose extension should increase as N increases. The existence of such a plateau is a suitable convergence criterion and the best results are supposed to occur when K belongs to it. Therefore, it is appropriate to choose K to be the stationary point Ki, @S#K)(K = Ki) = 0, with the smallest absolute value of the second derivative or to be the inflexion point KIN, (a’S@K”)(K = Kk) = 0, with the smallest absolute value of the first derivative. We will go back to this point again in the next section. Obviously, if S,(K,, 1) converges towards E(0, l), then SE,,, will converge to E(l, A) for any positive A value. It follows from the discussion above that K (and thereby the mapping (11)) is order-dependent. In this way, the original power series (7) has been transformed into the sequence (14) (with K = KN). On the other hand, eq. (8) tells us that E(l, A) = A-“lbE(A*‘b, 1) .

(15)

Therefore,

lim AdbE(l, A) = E(0, 1)) A+-

(16)

provided E(0, 1) exists. This last equation suggests that if we know E(0, l), we will be able to improve our results in the large-h regime by fixing KN so that lim A*’ SE, = S,(K, .&+-

1) = E(0, 1) .

(17)

THE ZEEMAN EFFECT IN HYDROGEN

259

In addition to this, it follows from (15) that if E(Z, 1) leads to a formal Taylor expansion about 2 = 0, then E(l, A) will behave like m

E(l,A)=h-“b~e$i’b,

e,=E(O,l),

(18)

i=O

in the large-A regime. If it happens that we do not know any scaling relation like (8) but a formal expansion like (18) we will obtain a and 6 from the latter and build the sequence SE, as in eqs. (11) (13) and (14). Anyway, each term SE, of the sequence can be expanded in powers of either A or Ab. This may be the weak point of our method because sometimes E(Z, 1) does not obey a Taylor expansion about Z = 0 but behaves in a complicated fashion. This failure must also occur when using the GEP), that is a particular case of our mapping (11) when b = -1. We will return to this point later on. It would be very interesting and most helpful to know under which general conditions our sequence converges to the function E(l, A). We cannot do this at present but we are sure that our method applies successfully to some well-known strongly divergent power series. Let us consider briefly two of them here: the simple integral

E’(Z, A) = j exp(-Zx* - Ax4) dx ,

(19)

0

and the eigenvalues of the anharmonic ~*o(z,

A)* = EAo(Z, A)p,

oscillator

HAo(Z, A) = -d*/dx* + ZX* + Ax4 .

(20)

The ODM has proved to sum the A-power series in both case?). In the former we obtain a = -i and b = -2, while c1= i and b = -i hold in the latter. With these values of a and b, our mapping (11) reduces to those used by Seznec and Zinn-Justin16). Though these authors determine the order-dependence in a different wayi6), we have found that both procedures are equivalent”). In fact, the dependence of either K”, or Kk on N is quite similar to that predicted by the saddle-point argument16). There is no doubt that our method and the 0DM16) lead to convergent sequences when applied to the problems just discussed”). Moreover, it can be easily proved that these methods are also equivalent to Caswell’s generalized Wick-ordering18) that has been developed to deal with anharmonic oscillators and two-well potentials. A more detailed discussion about this point is found in refs. 11 and 13.

G.A. ARTECA

260

et al.

The situation is entirely different when considering the ZEH because our mapping has nothing in common with that used by Le Guillou and ZinnJustin’). We discuss this case now. It follows from eqs. (2) (8) (11) and (14) that (a = 2, b = -4)

A = K;P(l-

p)-” )

@lb)

where the coefficients ,!?#C,) are given by (13). SE, is an acceptable approach to E(l, A) because the first N + 1 coefficients of their A-power series are equal. In addition to this, we have lim A-‘” SE, = S&C,, 1) , A--

(22)

the exact result being lim A-“*E(l, A) = E(0, 1) = (M + 1)/2, A-t-

(23)

where M = 0, 1,2, . . . is the Landau quantum number4). By comparing (22) and (23) we check the convergence of our sequence and the accuracy of our results in the very large-h regime. It is worth noting that E(K, p) is an eigenvalue of the Hamiltonian operator

(24) a fact that explains why K must be order-dependent. If it were not so, we would obtain an asymptotic divergent perturbation series in p because the perturbation potential in eq. (24) is as singular at infinity as the original one (i.e. S,(K,, p) is not a mere (x2 + y*)/8). Ob viously, when K is order-dependent, Rayleigh-Schrodinger series for the Hamiltonian (24) and the argument above does not apply. According to what was said above, we expect this problem to be most unfavourable to apply our method because SE, obeys a A-1’4-power series (cf. (18)) in the large-h regime, whereas the binding energy E for the fundamental state, E =

Aln/2-

E,

(25)

THE ZEEMAN EFFECT IN HYDROGEN

261

behaves (2~)~‘~=

- 2

ln(A/8)“2 +

21n -

+ . **,

(26)

where y is the Euler constant, when A +m. We therefore expect that our results for the binding energy will not be as accurate as those obtained via the ODM’) which takes into account the asymptotic expansion (26) explicitly. The advantage of our method is that it is simpler and requires much less computational effort than the ODM. In addition to this, it is not necessary to know the asymptotic form of the binding energy to apply our method. In this sense it resembles the GEP), though the latter appears to be more complicated and effort demanding because of using more adjustable parameters and a reiterative technique.

4. The Zeeman effect in hydrogen

All the attempts to sum the PT series for the ZEH have been devoted to the ground state5’7’8).In this paper we try the 2p,, and 2s states too. If the paramagnetic and spin terms (they are both constants of the motion4)) are included in the Hamiltonian operator (1) the energies of the Is and 2p_, states are found to approach the lowest Landau level as the field strength increases4,20). They are called tight bound states4’m). Under the same conditions the 2s state tends to the third Landau level whereas the 2p+, approaches a higher one. If the paramagnetic and spin terms are taken off (as in the present paper), the eigenvalues of the Hamiltonian operator (1) for the Is, 2p,, and 2s states obey (23) with M = 0, M = 1 and A4 = 0, respectively. We have obtained all the stationary (KL) and inflexion (KIN) points of S,(K, 1) for N = 1,2,. . . , IV,,,, where N,(ls) = 20, N,,,(2p+i) = 15 and N,,,(2s) = 13 according to our computational capabilities. Since every coefficient Ei is a polynomial of degree i in K, the number of stationary and inflexion points increases as N increases. Although the results obtained via the stationary and inflexion points are equivalent, the latter have proved to be more stable and we use them in our analysis below. When applying the ODM to the models in eqs. (19) and (20) Seznec and Zinn-Justin16) concluded that the proper order-dependence is K’, a N and Kr a N, respectively. This suggests that in a general case we should expect Kib a N to give rise to a convergent sequence. In this paper we search the K&N plane for straight lines to deal with the ZEH. The inflexion points for the 1s state are shown in fig. 1. The full lines labelled

262

G.A. ARTECA

et al.

K" 180-

160140120IOO6060-

1

Fig. 1. Inflexion points of S,(K,

1) for the 1s state.

A-E are straight lines KN 0: N. All these paths appear to give rise to convergent sequences that lead to wrong results. To obtain straight lines Ki a N we first separate the K$N plane into two parts, one of them containing no inflexion point, by means of a straight line (labelled a in fig. 1). It passes through the first points on B and C. Then we draw other straight lines parallel to the first one (a) that pass through more than two inflexion points. An example is the dashed line labelled b in fig. 1 that we use in our calculations. When following it, a sequence S,(K, l), N = 4, 8, 12, 16, is obtained which oscillates about 0.5 with an amplitude Due to this oscillation, our results cannot be very accurate regime. However, it must be remembered that the sequence

of about 20.05. in the large-A S,(K, 1) cor-

responds to the most unfavourable case A + 00 (cf. eq. (22)). One cannot expect to be able to sum such a strongly divergent series in the infinite-field limit by adding only the first 16 perturbation terms. To obtain more accurate energies in the large-field regime, we determine KN according to eq. (17). Table II shows the energy of the 1s state obtained by means of the largest inflexion point on the line b (N = 16, E*) and through eqs. (17) and (23) with M = 0 and N = 20 (Et). To check the accuracy of our results we compare them with the very careful and extensive calculation carried out by Wunner and Ruder”) and RSsner et al.“). To this end, we use A”’ in table II instead of A. Though E* contains fewer terms than Et does, the former is more accurate when A”* < 60. The latter approaches the exact eigenvalue2’9”) more closely when Al’*>60 because Et yields the exact leading term when A +a. Both calculation schemes are accurate enough for most purposes. As pointed out before, our method is simpler than the GEP). Besides, our

THE ZEEMAN

263

EFFECT IN HYDROGEN

TABLE II Energy of the 1s state of the spinless ZEH for a wide range of field intensities. A’”

E(1, ,),,,t

21.22)

E(l, A)*

E(l, A)+

1o-2

-0.499975005

-0.499975001

-0.499975005

10-l 0.2 0.6 1 2 3 6 10 20 60 100 200

-0.497526480 -0.490381565 -0.4274595 -0.331114 -0.021305 0.338399 1.54362 3.27316 7.79409 26.427 45.211 92.249

-0.497526480 -0.490381565 -0.4274556 -0.331041 -0.019968 0.343336 1.57283 3.35652 8.06697 27.690 47.572 97.483

-0.497526480 -0.490381565 -0.42746229 -0.3311689 -0.0222139 0.335467 1.531755 3.252204 7.78462 26.795 46.211 95.273

300 2000

139.293 937.97

147.473 997.94

144.645 990.696

* = 2.24699417869. ’ KN from eqs. (17) and (23) with N = 20 and M = 0, Km= 2.756838835.

results are more accurate than Silverman’s’) when 10 i A“’ d 100 in spite of the fact that we use fewer perturbation terms. Our results cannot be as accurate as those obtained by Le Guillou and Zinn-Justin’), especially when considering the binding energy, unless we are able to take into account the asymptotic expansion (26) explicitly. On the other hand, our method is much more straightforward and requires less computational effort than the ODM’). Fig. 2 shows the inflexion points for the 2p,, states. Since this figure is similar to fig. 1, we do not discuss it in detail but point out the main differences. When comparing the coefficients of the perturbation series for the 1s and 2p,, states (table I), it follows that the latter is more strongly divergent. This fact results in a more complex pattern of inflexion points (and also of stationary points) for the 2p,, states as shown in fig. 2. If we compare the areas bounded by N = 1 and N = 15 in figs. 1 and 2, we find that there are more inflexion points in the second case and that they are much larger. Due to this, fig. 2 supports more straight lines KN 0~N than fig. 1 does. We have also been able to draw one more straight line K”, m N (labelled c) in this case than we have been in the previous one. The sequences S,(K,, 1) obtained by following the straight lines b and c approach the exact value E(0, 1) = 1 with an oscillation of

264

G.A.

ARTECA

et al.

A

K4 1500-

lOOO-

500-

)J.; A

c

.____~"_~~~~~~

___.-_&_____1 5

Ol

Fig. 2. Inflexion

about dealing

20.2.

points

This is an acceptable

with the infinite-field

I IO

of &(K,

result

N

I* 15

1) for the 2p+1 states

if one considers

the fact that we are

limit.

Our results for different A values obtained using the largest inflexion point on line c (N = 15) and via eqs. (17) and (23) with N = 15 and A4 = 1 are compared in table III with the very accurate ones reported in refs. 21 and 22. From table III we obtain the same conclusions as those obtained before from

TABLE

Energy

III

of the 2p,l states of the ZEH terms are excluded) for a wide range

AU2

10-2 10-l 0.2 0.6 1 2 3 6 10 20 60 100 200 2000

EC

A)*

(spin and paramagnetic of field intensities. &I, h)Wt21.22)

E(Lh)+

-0.124701 -0.100846 -0.050559 0.223978 0.539055 1.39128 2.29220 5.14423 9.13330 19.5350 63.1870 107.991 221.766 2323.08

-0.124701 -0.100846 -0.050566 0.222923 0.533680 1.35944 2.21587 4.86594 7.89582 17.7721 55.8798 94.5475 192.076 1973.66

-0.124701 -0.100846 -0.050539 0.225376 0.543403 1.400387 2.29645 5.07671 8.87458 18.53453 57.8031 97.3653 196.653 1993.048

* Ki5= 4.39082096726. ’ KN from 4.41293433.

eqs.

(17)

and

(23) with

N = 15 and

M = 1; Kls=

THE

ZEEMAN

EFFECT

IN HYDROGEN

265

table II. Our calculation of the energy of the 2p,, states is however less accurate when Aln < 60, which is due to the fact that the perturbation series is more strongly divergent in this case. Unless a very accurate description of the binding energy is required, our results are useful for most purposes. We deem it appropriate to point out that this is the first perturbation calculation of the energy of the 2p,, states that holds beyond A = 0.01. The 2s state is not so important as the 1s and 2p_, ones due to which it has not received so much attention as the last two have. However, since we have found serious difficulties in applying PT to the 2s state, we deem it worthwhile studying it here briefly. The binding energy for this state is not a continuously decreasing function of the field intensity as in the previous examples but exhibits a maximum value at a finite field strength’l). This particular feature makes our PT calculation a too rough upper bound when A > 1. The pattern of stationary and inflexion points in the K$N plane is similar to those in figs. 1 and 2 but the oscillation of the sequence is so large that it is not possible to estimate E(0, 1). Unfortunately, we have not been able to deal with more than 13 perturbation terms and a much larger order is required to check our method. We have calculated the energy of the 2s state for A 5 1 using eqs. (17) and (23) with N = 13 and M = 0 and the results are compared in table IV with the very accurate ones reported in refs. 21 and 22. Though the accuracy of our computed energy is not exciting, the method appears to be promising if one takes into account that the Rayleigh-Schrodinger series predicts an energy of about lo6 when A”‘= 0.1. Since the ODM’) makes explicit use of the asymptotic expansion for the binding energy, it applies only to those states with 1= m = n - 1 for which such an expansion is known”). On the other hand, our method and the GET apply, in principle, to any state.

Energy

TABLE IV of the 2s state of the spinless the small-field regime.

A’”

E(l,

lo-’ 1o-2 10-l 0.2 0.6 1

-0.12499650 -0.12465157 -0.097734 -0.04375 0.2233 0.5010

A)+

E(1,

ZEH

in

,4)eraer21,22)

-0.12499650 -0.12465157 -0.09808916 -0.04899 0.14723 0.33956

’ KN from eqs. (17) and (23) with N = 13 and M = 0; K,, = 8.0498714.

266

G.A. ARTECA

et al.

5. Further comments

The method just discussed appears to be very useful in dealing with most of the strongly divergent perturbation series usually encountered in quantum and held theory. The procedure is simple and straightforward, requires little computational effort and yields acceptable results even when a few perturbation terms are included. In addition to this, it takes into account part of the analytical properties of E(1, A) via the parameters a and b obtained from either the scaling relation (8) or the asymptotic expansion (18). It must be remembered that the former has been of great importance in understanding the causes of divergence of the Rayleigh-Schrddinger series for the anharmonic oscillate?). The GEP) is obviously unable to take into account this particular feature (i.e. the scaling relation) of the function that gives rise to the power series. When neither a scaling relation nor an asymptotic expansion like (18) are available, the suitable mapping (11) can be obtained by varying b till the largest convergence rate is attained. Such a study would show whether the natural mapping based on the scaling relation is the most efficient procedure. The work of Le Guillou and Zinn-Justin’) has proved that the introduction of the asymptotic expansion (26) improves the convergence rate of the ODM sequence giving very accurate binding energies. We do not attempt to make our sequence obey eqs. (25) and (26) because we want to keep our method as general as possible. However, we will sketch how it could be done. We first notice that the variable u defined by the mapping (A/g)1’2= 4-‘u2 eU+Y,

(27)

obeys exactly the truncated asymptotic expansion given in eq. (26) with U = (2E)Y On the other hand, eqs. (25) and (27) enable us to expand the binding energy in powers of u: & =

go EiUi.

(28)

Then we use our method with alb = -2 and obtain a sequence SF~, SEN =

5 EiUi i=o

(29)

that can be expanded as (cf. eq. (18)) m SeN = ;u’ c Eiuilb, i=O

b< 0.

(30)

THE

ZEEMAN

EFFECT

267

IN HYDROGEN

The parameter b is varied till the largest convergence rate is attained and K is fixed so that E, = 1. In this way our sequence yields the truncated asymptotic expansion (26) exactly. Results obtained by such a procedure will be presented elsewhere in a forthcoming paper.

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62A (1977)