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Scaling, renormalisation and accuracy of perturbation calculations
CHEhiICAL
Volume 105, number 3
SCALING,
RENORMALLSATION
PHYSICS
AND ACCURACY
16 hlarch 1984
LETTERS
OF PERTURBATION
CALCULATIONS
M. COHEN and S. KAIS Deportment
ofPhysical
Received 2 January
Chemistry,
The
Hebrew
Universiry.
Jerusalem
93904,
Israel
1984
l-or certain eigenvalue problems whrch lead to dkergent Rayleigh-Schrbdinger perturbation series, enegy estimates oi acceptable accuracy may be obtzined casrly m first order by optn-nlsing a vanahonal scale factor. Some simple calculations on the quartic anharmonic osc2lator and on the quadratic Zecman effect shorn errors xxhich are generally quite %mdl over a very wide range of perturbation pammeter valuer
ly, for a given hamiltonian
I_ Intrculuc tion Following the pioneering
work of Bender and Wu
[l]
and of Simon [2] on the anhxmonic oscillator, has been a great deal of recent interest m largeorder perturbation theory, particularly for systems whose Rayleigh-Schriidinger (RS) perturbation series are known to be divergent or asymptotic. A major aim has been to devise successful summation techniques for such series; the work of &Zek and Vrscay [3] and of Silverman [4] provide just two recent examples, and many references. What emerges clearly from these treatments is that different procedures must be used in order to obtain highly accurate results for different systems, and there appears to be no L llversal solution applicable to all problems. Moreove. _Jthough it is now possible to calculate RS energy coefficients accurately up to order 100 (or beyond) for some model there
systems, it remains unlikely that this will become the norm. A generally applicable alternative is clearly desirable. Now, as has been emphasized by Banejee [5] and Killingbeck [6], it is often possible to rescule or tenormaIise a perturbation series. In fact, certain widely used “summation procedures” have their origins in an appropriate change of scale, or equivalently, in a different choice of reference (zero-order) hamiltonian Hb in place of the more natural (physical) operator Ho _ (The celebrated screening approximation of Dalgarno and Stewart [7] is a very good example of this.) Specifical0 009-26 14/84/S 03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
H(J)
= Ho + XH,
(1)
in which h S a ~~ahtrul perturbation parameter, and H,,, H, are mdependent of h, it will often be more convenient to write H(X)=H(X;p=
1).
(2)
where
and Hi, Hi may be (explicitly or imphcltly) h dependent. The formal perturbarion parameter p will be set equal to unity at the end of the calculation. so that ir is sufficient for our purposes if convergence of the pperturbation series can be established only for a rather limited range, 0 < _u< 1; the range of the physical parame ter h is frequently unbounded (0 < X < -)_ Since H&A) may be chosen differently for differenr ranges of h, it becomes possible ro treat in zero order the leading effect of the physical penurbation, AH, _ Moreover, if this X dependence is introduced by means of variational parameters, these may be optimlsedfir my partidar X by appeal to the variation theorem: there is no need to have recourse ro the asymptotic (WKBJ) approximation for large h, although this is a well-documented procedure [1,5] _ In this letter, we reconsider the ground-state energies of two popular “problematic“ hamilronians, the one-dimensional (quartic) anharmonic oscillator and 295
Volume
10s. number
the twodimensional
PllYStCS
CHEMICAL
3
(quadratic)
cffcct. De-
Zccman
Table I
spite
the simplicity of’ both calculations. we obtain energies which arc remarkably accurate over almost tbc
Ground-state
eniite ranges OSboth pcttutbartons.
A -.
2. The qmic
anharmonic
oscillalor
‘The complct e hanultonlan
d2/d.2
H(A) 7
IS
+ r* + xx4
(A > 0)
(4)
and If(A) is anatlytically soluble when A ; 0. hut not for large A. WC make the sln~ple choice II&A) = -.d2/dx2 H;(h).
(I
Where
o(A)
.
+ ,2x2
.0*)x*
+ xx4
.
iS a A&!pCndCnt
transformation
If@)
l
oj-4~:dx2
palamCtC,
SO
thJt , f0,
$9 ”
-
My
Iv”’
F(ya) ‘n
5tdC
= (2n
+
l)o
+ (A/d)x4]
.
(‘I)
- a) + 2,(2n2
+ 2n t
1) A/o2
state (n = 0) and for any A, the perturbation sum E;“)(o) t ICAt) prondes a rtgorous upper bound to the energy. Variation of a now leads to the follawinR equations for oor,, and the corresponding optimiscd CllClgy Eopt : E=$(k-
mum to
CllUI
100,
ol
whereas
k(o*r
) Jars
I)+
l/Za.
(9)
not C.xcecr\ 2 2%
the Q 7 1 wties
is clearly
fat h up
divcr@g. The same proccrlure may be used for excItLY1 states (n 2 1) &hot&t EA”) + EF) now furnishes an upper
296
_-YLT
1.06917 1.2.5x02 1.41606 1.63943 I.80651 1.94377 2.06196 2 50286 3 07s51 4.089jt 5 10458 _
t 04529 174185 I .I39235 t .60754 I .769S9 1.90114 1.01 u34 2 44917 3.009’94 4.UOiY9 4.99942
181
if the variat wn of o 1s suitably if the variatmn
constraints,
It ts easily
whik
for large A and laige
,tsUh
Wa.S obtail,ed
is carried
seen
cons.
th,oup,h
rained. with-
1 when
that a
A = 0,
Ihls latter
n, 0%: (3nX)1/3;
by ktaIlCt)CC [ fi ] 011 the
ptCViOUtiy
basis of a WKBJ
treatment,
hut
3. The quadratic
Zeeman
effect
charge
% placed
z direction.
this
is unncccssacy.
f{(7)
of a one-Clcctron
in a magnetic
the non-rclativistlc
field
atom
of nuclca,
of strength
hamdtonian
H III the
(in atomic
is CffCct ivcly =
- Z/r t g+7?r2 silt30 .
j F2
(10)
(tlerc, 7 IS a dirncnsionlcss constant B/Ho. where Ho = 23505 X 10y G.) ‘Ihts problem is clearly two4ur,Cnsional. and may be therefore expected to hr mo~c dtfficult to treat than the one4imensional oscillator probkin. On the other cally
Table 1 contains mnnerical resulti based on eqs. (7) (9) both for a = I and for aopt,together with accurate values from the work of Biswas et al. 181; the maxi-
.
out
units)
results,
For the ground
l)~3A.
__
(6)
09
~*(a-
oscdta~or
.-
-_- mkJg,r) -
values I’rom ref.
For rn = 0 states
,
= :(211 + I)(l/a
onharmonic
-
1.075 1 375 1.75 2.5 3.25 4 0 4.75 H.S 160 38 5 ‘16.0 -
Nevertheless,
,
and usutg standard E;)(a)
leads to
n
Wli”(X)
(quarlic)
the
.-_
- Eb .-- 1)
0.1 0.5 I.0 2.0 7.0 4.0 5.0 10.0 20.0 50.0 100.0 -
bound only
t0 hC dCtC,-
the well-known propsolutions A simple
x’) ,
1)x2
iIi (A) _ l o[( I /02
of
(5)
x + x/at12 +
rncrey
.-.
a) Accurate
mmed; this allows us to exploit erties of the harmonic oriljator scaling
16 March 1984
LI;‘I-I’ERS
soluble
clearly
hand,
desirable
to choose
correctly
at both
It{:’
cxp[
=N
the hamiltontan
for srnall7
tmrh
limits.
where a : 4-y). fl-
a model ifI
Ihcrcforc.
(ar t ipr2
H(7)
is analyti-
and for large 7, and ,t is
sin%)]
N-y) with the
which
bchnvCs
we choose
(11)
, expectztron
that
at y
-O,o-%andp~Owhileas7-.m,a-rOandP-b~7 Corrcqtonding iq
r -* r/,2)
to this
$‘,o’.
WC have (aftc,
transform-
Volume
105. number 3
H&)-d-iv*
-
Iq(y)-4[p/r-
CHEMICAL
l/r + Gr + $A*)
~r+~zJr2)sin~0]
where we w&e /l=fi/S,
sin%]
PHYSlCS
(12)
for convenience
p=(a-q/a,
v=g
+2/cr4
_
(13)
Here both air) and /I(T) may be varied independently with 7, although it is also possible to vary only a single parameter, either II(~) or p(r) In this case also, all energy integrals may be evaluated analytically and we obtain the results E~qcr,
@) = a’(-
+ + p) ,
[exp(--t)P/(l
+pt)J
dt _
Table 2 contains four sets of results based on eqs. ((Y and j3 fHed, one parameter varied, both parameters varied) together with the most accurate energies available [3,4]. We have taken 2 = 1 for comparison with results of other workers. Even with cr and p tixed (note that our results correct and extend rhose of U-ardt [lo]) there is no divergence, a consequence of the correct behaviour of the trial function at both small and large field strengths. Indeed, only at y = 2 where the correcr energy changes sign, is the error in this simplest calculation unacceptably large. With j3 fured and (Yvaned (these results extend and improve (14)
(14)
(15)
isfactory
where = s
integrals are most easily generated recursively
from
Ground-state Y
results.
We have no reason to beheve that these examples
0 These
integral
those of Rau et al. [ll]) and with Q fned and fl varied, we obtain some improvement for all fLyed strengths although unexpectedly, variation of Q is more effective for large y and of fl for small y_ Independent variation of both (Y and /3 yields generally sat-
E~)(~,~)=d[PC~-~(c~-cC1)-~~(C~-2C2)]/C1,
C, = C,@)
16 March 1984
where X= l/p and I51 (X) is the exponential (see, for example, ref. [9])_
)
)
LETTERS
energy of a hydrogen atom in a magxtic
=)
are in any way exceptional, and conclude that other systems may be treated no less successfully by the same merhcds.
field (in atomic units) Accurarc
Presentcalculations
b)
(1)
(2)
(3)
(5)
(4)
0.1
-0.49579
-0.49659
-0.49743
-0 49747
-0.49
0.2 0.5
-0 48530 -0.43 170
-0.40747 -0.43716
-0.49036 -0.44681
-0.49027 -0.44683
-0.4903 -0.44721
10
-0.30’762
-0.3
-0.3’934
-0.32956
-0.01560
-0.01763
-0.33117 -0.02223
1003
2.0
0.02262
0.01685
3.0 4.0 5.0
0.39239 0.78629 1.1961
0 38885 0.78466 1.1956
10.0 20.0 100
3 3707 7.9771 46.826
3.3678 7.9508 46 536
034785 0.73837 1.1428 3.2939 7.9211 46.789
c)
753
0 3429s 0.72946 1.1324
0.33539
3.2756
3.2564
7.8245 46.361
200 300 1000
96.248 145.89 495.78
95.684 145-10 49297
96 220 145.88 494.77
95.433 144.82 492.72
2000 .____~_
994.12
991.42
994.11
99122
8
0.71913 1.1198 7.7847 46.211 9 5273 144.65 492.36 990.70
__
‘) 7=BjBo where B. = 2 3505 x 10’ G. b’(1)rr=l.~=y/2;(2)~vvaried.~=~~2;(3)cr=1.pvaried;(4)aand~varied. Cl (5) From refs
[3,41_
297
Volume
105. nurnbcr
CliEMlCAL
3
PllYSIcS
Referma 1 I ] CM.
Ucndcr
111 J. ?i7ck
and 1.T. Rev. l&l
121 B. Simon,
Ann
Wu. Yhys. Hcv. I CI~CI~ 21 (1968)
(1969)
Phyr. NY
and ):.H. Vrwxy.
K. Hancrjec.
Phyr
YIoc
Inlrrn.
SOC A368
161 1. Killm~hrxk Rcpl. Prop. Phyr. Al4 (1981) 1005,.
298
5.8 (1970)
J ~uuntum
(:hrm.
498
(19791
PhyL 40(
1977)
I SJ. Y63, J.
21
K. IMta.
V.S. Varma, J. Math lYj
76
Rev. A28 (1981)
Roy.
181 S.N. Birwaq.
1231.
27.
141 J.N. SlIverman. ISI
I6 March
171 A. IJal+?arno and A.I.. (1960) 534.
406,tnyr
i1982)
LErrERS
I IO] I I1 I
M. Ahramowlz
SIcwr1.
I’roc.
R.P. Saxcrw. Phyr
I4 (1973)
and I. !&pm.
malhcmalical luncttons(I)ovcr, W. Ekardt, Solid Slate Commun. A R P. RJU. R.0 All
(lY75)
1865.
Murllcr
edr.
Ray,
SOC.
1984
A257
P.K. Snvzvt;lv;r ;lnd 1190. Iiandb,ok
New York I6 (1975)
and I.. Spruch.
of
1972). 233. Phys
Rev
Volume 105, number 3
SCALING,
RENORMALLSATION
PHYSICS
AND ACCURACY
16 hlarch 1984
LETTERS
OF PERTURBATION
CALCULATIONS
M. COHEN and S. KAIS Deportment
ofPhysical
Received 2 January
Chemistry,
The
Hebrew
Universiry.
Jerusalem
93904,
Israel
1984
l-or certain eigenvalue problems whrch lead to dkergent Rayleigh-Schrbdinger perturbation series, enegy estimates oi acceptable accuracy may be obtzined casrly m first order by optn-nlsing a vanahonal scale factor. Some simple calculations on the quartic anharmonic osc2lator and on the quadratic Zecman effect shorn errors xxhich are generally quite %mdl over a very wide range of perturbation pammeter valuer
ly, for a given hamiltonian
I_ Intrculuc tion Following the pioneering
work of Bender and Wu
[l]
and of Simon [2] on the anhxmonic oscillator, has been a great deal of recent interest m largeorder perturbation theory, particularly for systems whose Rayleigh-Schriidinger (RS) perturbation series are known to be divergent or asymptotic. A major aim has been to devise successful summation techniques for such series; the work of &Zek and Vrscay [3] and of Silverman [4] provide just two recent examples, and many references. What emerges clearly from these treatments is that different procedures must be used in order to obtain highly accurate results for different systems, and there appears to be no L llversal solution applicable to all problems. Moreove. _Jthough it is now possible to calculate RS energy coefficients accurately up to order 100 (or beyond) for some model there
systems, it remains unlikely that this will become the norm. A generally applicable alternative is clearly desirable. Now, as has been emphasized by Banejee [5] and Killingbeck [6], it is often possible to rescule or tenormaIise a perturbation series. In fact, certain widely used “summation procedures” have their origins in an appropriate change of scale, or equivalently, in a different choice of reference (zero-order) hamiltonian Hb in place of the more natural (physical) operator Ho _ (The celebrated screening approximation of Dalgarno and Stewart [7] is a very good example of this.) Specifical0 009-26 14/84/S 03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
H(J)
= Ho + XH,
(1)
in which h S a ~~ahtrul perturbation parameter, and H,,, H, are mdependent of h, it will often be more convenient to write H(X)=H(X;p=
1).
(2)
where
and Hi, Hi may be (explicitly or imphcltly) h dependent. The formal perturbarion parameter p will be set equal to unity at the end of the calculation. so that ir is sufficient for our purposes if convergence of the pperturbation series can be established only for a rather limited range, 0 < _u< 1; the range of the physical parame ter h is frequently unbounded (0 < X < -)_ Since H&A) may be chosen differently for differenr ranges of h, it becomes possible ro treat in zero order the leading effect of the physical penurbation, AH, _ Moreover, if this X dependence is introduced by means of variational parameters, these may be optimlsedfir my partidar X by appeal to the variation theorem: there is no need to have recourse ro the asymptotic (WKBJ) approximation for large h, although this is a well-documented procedure [1,5] _ In this letter, we reconsider the ground-state energies of two popular “problematic“ hamilronians, the one-dimensional (quartic) anharmonic oscillator and 295
Volume
10s. number
the twodimensional
PllYStCS
CHEMICAL
3
(quadratic)
cffcct. De-
Zccman
Table I
spite
the simplicity of’ both calculations. we obtain energies which arc remarkably accurate over almost tbc
Ground-state
eniite ranges OSboth pcttutbartons.
A -.
2. The qmic
anharmonic
oscillalor
‘The complct e hanultonlan
d2/d.2
H(A) 7
IS
+ r* + xx4
(A > 0)
(4)
and If(A) is anatlytically soluble when A ; 0. hut not for large A. WC make the sln~ple choice II&A) = -.d2/dx2 H;(h).
(I
Where
o(A)
.
+ ,2x2
.0*)x*
+ xx4
.
iS a A&!pCndCnt
transformation
If@)
l
oj-4~:dx2
palamCtC,
SO
thJt , f0,
$9 ”
-
My
Iv”’
F(ya) ‘n
5tdC
= (2n
+
l)o
+ (A/d)x4]
.
(‘I)
- a) + 2,(2n2
+ 2n t
1) A/o2
state (n = 0) and for any A, the perturbation sum E;“)(o) t ICAt) prondes a rtgorous upper bound to the energy. Variation of a now leads to the follawinR equations for oor,, and the corresponding optimiscd CllClgy Eopt : E=$(k-
mum to
CllUI
100,
ol
whereas
k(o*r
) Jars
I)+
l/Za.
(9)
not C.xcecr\ 2 2%
the Q 7 1 wties
is clearly
fat h up
divcr@g. The same proccrlure may be used for excItLY1 states (n 2 1) &hot&t EA”) + EF) now furnishes an upper
296
_-YLT
1.06917 1.2.5x02 1.41606 1.63943 I.80651 1.94377 2.06196 2 50286 3 07s51 4.089jt 5 10458 _
t 04529 174185 I .I39235 t .60754 I .769S9 1.90114 1.01 u34 2 44917 3.009’94 4.UOiY9 4.99942
181
if the variat wn of o 1s suitably if the variatmn
constraints,
It ts easily
whik
for large A and laige
,tsUh
Wa.S obtail,ed
is carried
seen
cons.
th,oup,h
rained. with-
1 when
that a
A = 0,
Ihls latter
n, 0%: (3nX)1/3;
by ktaIlCt)CC [ fi ] 011 the
ptCViOUtiy
basis of a WKBJ
treatment,
hut
3. The quadratic
Zeeman
effect
charge
% placed
z direction.
this
is unncccssacy.
f{(7)
of a one-Clcctron
in a magnetic
the non-rclativistlc
field
atom
of nuclca,
of strength
hamdtonian
H III the
(in atomic
is CffCct ivcly =
- Z/r t g+7?r2 silt30 .
j F2
(10)
(tlerc, 7 IS a dirncnsionlcss constant B/Ho. where Ho = 23505 X 10y G.) ‘Ihts problem is clearly two4ur,Cnsional. and may be therefore expected to hr mo~c dtfficult to treat than the one4imensional oscillator probkin. On the other cally
Table 1 contains mnnerical resulti based on eqs. (7) (9) both for a = I and for aopt,together with accurate values from the work of Biswas et al. 181; the maxi-
.
out
units)
results,
For the ground
l)~3A.
__
(6)
09
~*(a-
oscdta~or
.-
-_- mkJg,r) -
values I’rom ref.
For rn = 0 states
,
= :(211 + I)(l/a
onharmonic
-
1.075 1 375 1.75 2.5 3.25 4 0 4.75 H.S 160 38 5 ‘16.0 -
Nevertheless,
,
and usutg standard E;)(a)
leads to
n
Wli”(X)
(quarlic)
the
.-_
- Eb .-- 1)
0.1 0.5 I.0 2.0 7.0 4.0 5.0 10.0 20.0 50.0 100.0 -
bound only
t0 hC dCtC,-
the well-known propsolutions A simple
x’) ,
1)x2
iIi (A) _ l o[( I /02
of
(5)
x + x/at12 +
rncrey
.-.
a) Accurate
mmed; this allows us to exploit erties of the harmonic oriljator scaling
16 March 1984
LI;‘I-I’ERS
soluble
clearly
hand,
desirable
to choose
correctly
at both
It{:’
cxp[
=N
the hamiltontan
for srnall7
tmrh
limits.
where a : 4-y). fl-
a model ifI
Ihcrcforc.
(ar t ipr2
H(7)
is analyti-
and for large 7, and ,t is
sin%)]
N-y) with the
which
bchnvCs
we choose
(11)
, expectztron
that
at y
-O,o-%andp~Owhileas7-.m,a-rOandP-b~7 Corrcqtonding iq
r -* r/,2)
to this
$‘,o’.
WC have (aftc,
transform-
Volume
105. number 3
H&)-d-iv*
-
Iq(y)-4[p/r-
CHEMICAL
l/r + Gr + $A*)
~r+~zJr2)sin~0]
where we w&e /l=fi/S,
sin%]
PHYSlCS
(12)
for convenience
p=(a-q/a,
v=g
+2/cr4
_
(13)
Here both air) and /I(T) may be varied independently with 7, although it is also possible to vary only a single parameter, either II(~) or p(r) In this case also, all energy integrals may be evaluated analytically and we obtain the results E~qcr,
@) = a’(-
+ + p) ,
[exp(--t)P/(l
+pt)J
dt _
Table 2 contains four sets of results based on eqs. ((Y and j3 fHed, one parameter varied, both parameters varied) together with the most accurate energies available [3,4]. We have taken 2 = 1 for comparison with results of other workers. Even with cr and p tixed (note that our results correct and extend rhose of U-ardt [lo]) there is no divergence, a consequence of the correct behaviour of the trial function at both small and large field strengths. Indeed, only at y = 2 where the correcr energy changes sign, is the error in this simplest calculation unacceptably large. With j3 fured and (Yvaned (these results extend and improve (14)
(14)
(15)
isfactory
where = s
integrals are most easily generated recursively
from
Ground-state Y
results.
We have no reason to beheve that these examples
0 These
integral
those of Rau et al. [ll]) and with Q fned and fl varied, we obtain some improvement for all fLyed strengths although unexpectedly, variation of Q is more effective for large y and of fl for small y_ Independent variation of both (Y and /3 yields generally sat-
E~)(~,~)=d[PC~-~(c~-cC1)-~~(C~-2C2)]/C1,
C, = C,@)
16 March 1984
where X= l/p and I51 (X) is the exponential (see, for example, ref. [9])_
)
)
LETTERS
energy of a hydrogen atom in a magxtic
=)
are in any way exceptional, and conclude that other systems may be treated no less successfully by the same merhcds.
field (in atomic units) Accurarc
Presentcalculations
b)
(1)
(2)
(3)
(5)
(4)
0.1
-0.49579
-0.49659
-0.49743
-0 49747
-0.49
0.2 0.5
-0 48530 -0.43 170
-0.40747 -0.43716
-0.49036 -0.44681
-0.49027 -0.44683
-0.4903 -0.44721
10
-0.30’762
-0.3
-0.3’934
-0.32956
-0.01560
-0.01763
-0.33117 -0.02223
1003
2.0
0.02262
0.01685
3.0 4.0 5.0
0.39239 0.78629 1.1961
0 38885 0.78466 1.1956
10.0 20.0 100
3 3707 7.9771 46.826
3.3678 7.9508 46 536
034785 0.73837 1.1428 3.2939 7.9211 46.789
c)
753
0 3429s 0.72946 1.1324
0.33539
3.2756
3.2564
7.8245 46.361
200 300 1000
96.248 145.89 495.78
95.684 145-10 49297
96 220 145.88 494.77
95.433 144.82 492.72
2000 .____~_
994.12
991.42
994.11
99122
8
0.71913 1.1198 7.7847 46.211 9 5273 144.65 492.36 990.70
__
‘) 7=BjBo where B. = 2 3505 x 10’ G. b’(1)rr=l.~=y/2;(2)~vvaried.~=~~2;(3)cr=1.pvaried;(4)aand~varied. Cl (5) From refs
[3,41_
297
Volume
105. nurnbcr
CliEMlCAL
3
PllYSIcS
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