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Some application of scaling transformations
Volume 62A, number 5
PHYSICS LETTERS
5 September 1977
SOME APPLICATIONS OF SCALING TRANSFORMATIONS J. KILLINGBECK Physics Department, University of Hull, Humberside, England Received 2 July 1977 A simple scaling approach for the perturbed anharmonic oscillator is shown to be superior to more complicated approaches suggested by other authors. The value of calculations using implicit wavefunctions is illustrated for several examples. There are several apparently simple perturbation problems for which the Rayleigh—Schrodinger (RS)
energy series turns out to be divergent asymptotic, so that a direct linear summation of the series is useless at large perturbation strengths. The Hamiltonian (l) H = —D2 +x2 + Xx4 =H0 + XX4, is of this type; ma recent article m this journal [1] it was suggested that the unperturbed ground state function Ø~ for (1) should be used together with the function .
—1 ~1i=
~(e
—
4
Table 1 Energy estimates for the anharmonic oscillator. 0 1 5 10 ____________
From ref. Eq. (3) Eq. (4) Exact _______
[11
_____________
1.0000 1.0000 1.0055 1.0000
_____
___________________
1.395 1 1.4033 1.3924 a 1.3924
2.8744 2.4886 2.4516 2.4492
__________________________
a Fitted at A = 1 (see text).
shown. For small A the method of
3
2.1228 2.0470 2.0196 2.0183
[11(which was de-
H
0) (x (2) in what is essentially a 2 X 2 matrix calculation of the perturbed ground state energy. ~i is a generalized version the cusual RS first order perturbed i~with yofand as variable parameters. For A =function 1,5, and~ 10 this approach gives much better results than several other approximations, and in [1] this is described as an impressive example of the use of the method. In fact, we point out here that for large A a very much better result is obtained by simple scaling of the function ~ which involves no knowledge of the unperturbed excited states! The theoretical reason for this is fairly clear. Use of Ø~ and i~Li as basis cannot possibly give a better result than use of the finite basis consisting of the lowest three even parity unperturbed functions, whereas for large A many excited states are mixed into Ø~by the perturbation. This effect can be allowed for by a scaling procedure. The use of scaling factor k gives for the energy expectation value, —
(H) = 4k2 + 4k2
+
* Ak—4,
(3)
scribed as a strong perturbation method) is slightly better than use of (3). An analytic calculation shows that the use (3) leads the energy series 2...,ofwhereas RStotheory gives E = 1 E + =A1 + ~A AFor problems in which the perturbation links ~A2 the unperturbed ground state function Øo to an infinity of excited states (e.g. in perturbed hydrogen atom problems) even the construction of a ,li of type (2) is difficult, and it is much easier to use scaled versions of the RS ~‘ i function and of Ø~in a 2 X 2 matrix calculation. This would necessarily yield even better results than (3) for the perturbed oscillator problem. It is fairly easy to obtain the exact eigenvalues of the Hamiltonian (1) and also the value of (x4) by using the method of [2]. The virial theorem then permits calculation of (—D2) and
—
— .... -~
(H) = 0.8263k2 + 0.3059k2
+
0.2602Ak4,
(4)
and optimization of k gives the values in table 1, where
and this leads to results which are shown in table I
the exact values and the best results of [1] are also
along with the other estimates. It might seem pointless 285
Volume 62A, number 5
PHYSICS LETTERS
Table 2 12 1 +Ar. Hamiltonian —~V —r Ground state energy _______________
Scaling calculations ____________
A
0
1
2
3
A = 0 base A = 1 base exact
—0.5000 —0.4882 —0.5000
0.6174 0.5779 0.5779
1.4827 1.405 1 1.4031
2.2464 2.1355 2.1298
to use this approach if we already have a method of finding the exact energy at arbitrary A. However, flumerical calculations involve successive refinement of some initial estimate of E, and so this use of scaling as an extrapolation device is of considerable practical utility, since the calculation time is much decreased if a good upper bound is already known. We note that starting from A 1 gives better results than starting from A0. Since the explicit perturbed wavefunction is not used, we refer to the process as an implicit wavefunction calculation, and have used a similar method elsewhere [3].
286
5 September 1977
Our interesting use of the implicit wavefunction scaling approach is in showing that the H ion is bound, starting from He atom results. Taking the values E = —2.904 and (rj’ +r~~)= 3.377 for He, we find from the virial theorem the values (rj~)= 0.947 and (T) = 2.904 for He. Scaling of the He wavefunction then gives
the value E = —0.509 for H, predicting binding, whereas scaling of the unperturbed ls2 product function is well known to give the opposite conclusion. For Li~the method gives E = —7.262, necessarily above the correct value —7.280. Alternatively, given only the E values for He and Li+, we can establish that (rj~’)< 0.970 for He; the exact value is 0.947.
References [1] K. Schonhammer and L.S. Cederbaum, Physics Lett. 51A 325. [2] (1975) J. Kfflingbeck, J. Phys. AlO (1977) L99. [3] J. Killingbeck, Chem. Phys. Lett., submitted.
PHYSICS LETTERS
5 September 1977
SOME APPLICATIONS OF SCALING TRANSFORMATIONS J. KILLINGBECK Physics Department, University of Hull, Humberside, England Received 2 July 1977 A simple scaling approach for the perturbed anharmonic oscillator is shown to be superior to more complicated approaches suggested by other authors. The value of calculations using implicit wavefunctions is illustrated for several examples. There are several apparently simple perturbation problems for which the Rayleigh—Schrodinger (RS)
energy series turns out to be divergent asymptotic, so that a direct linear summation of the series is useless at large perturbation strengths. The Hamiltonian (l) H = —D2 +x2 + Xx4 =H0 + XX4, is of this type; ma recent article m this journal [1] it was suggested that the unperturbed ground state function Ø~ for (1) should be used together with the function .
—1 ~1i=
~(e
—
4
Table 1 Energy estimates for the anharmonic oscillator. 0 1 5 10 ____________
From ref. Eq. (3) Eq. (4) Exact _______
[11
_____________
1.0000 1.0000 1.0055 1.0000
_____
___________________
1.395 1 1.4033 1.3924 a 1.3924
2.8744 2.4886 2.4516 2.4492
__________________________
a Fitted at A = 1 (see text).
shown. For small A the method of
3
2.1228 2.0470 2.0196 2.0183
[11(which was de-
H
0) (x (2) in what is essentially a 2 X 2 matrix calculation of the perturbed ground state energy. ~i is a generalized version the cusual RS first order perturbed i~with yofand as variable parameters. For A =function 1,5, and~ 10 this approach gives much better results than several other approximations, and in [1] this is described as an impressive example of the use of the method. In fact, we point out here that for large A a very much better result is obtained by simple scaling of the function ~ which involves no knowledge of the unperturbed excited states! The theoretical reason for this is fairly clear. Use of Ø~ and i~Li as basis cannot possibly give a better result than use of the finite basis consisting of the lowest three even parity unperturbed functions, whereas for large A many excited states are mixed into Ø~by the perturbation. This effect can be allowed for by a scaling procedure. The use of scaling factor k gives for the energy expectation value, —
(H) = 4k2 + 4k2
+
* Ak—4,
(3)
scribed as a strong perturbation method) is slightly better than use of (3). An analytic calculation shows that the use (3) leads the energy series 2...,ofwhereas RStotheory gives E = 1 E + =A1 + ~A AFor problems in which the perturbation links ~A2 the unperturbed ground state function Øo to an infinity of excited states (e.g. in perturbed hydrogen atom problems) even the construction of a ,li of type (2) is difficult, and it is much easier to use scaled versions of the RS ~‘ i function and of Ø~in a 2 X 2 matrix calculation. This would necessarily yield even better results than (3) for the perturbed oscillator problem. It is fairly easy to obtain the exact eigenvalues of the Hamiltonian (1) and also the value of (x4) by using the method of [2]. The virial theorem then permits calculation of (—D2) and
—
— .... -~
(H) = 0.8263k2 + 0.3059k2
+
0.2602Ak4,
(4)
and optimization of k gives the values in table 1, where
and this leads to results which are shown in table I
the exact values and the best results of [1] are also
along with the other estimates. It might seem pointless 285
Volume 62A, number 5
PHYSICS LETTERS
Table 2 12 1 +Ar. Hamiltonian —~V —r Ground state energy _______________
Scaling calculations ____________
A
0
1
2
3
A = 0 base A = 1 base exact
—0.5000 —0.4882 —0.5000
0.6174 0.5779 0.5779
1.4827 1.405 1 1.4031
2.2464 2.1355 2.1298
to use this approach if we already have a method of finding the exact energy at arbitrary A. However, flumerical calculations involve successive refinement of some initial estimate of E, and so this use of scaling as an extrapolation device is of considerable practical utility, since the calculation time is much decreased if a good upper bound is already known. We note that starting from A 1 gives better results than starting from A0. Since the explicit perturbed wavefunction is not used, we refer to the process as an implicit wavefunction calculation, and have used a similar method elsewhere [3].
286
5 September 1977
Our interesting use of the implicit wavefunction scaling approach is in showing that the H ion is bound, starting from He atom results. Taking the values E = —2.904 and (rj’ +r~~)= 3.377 for He, we find from the virial theorem the values (rj~)= 0.947 and (T) = 2.904 for He. Scaling of the He wavefunction then gives
the value E = —0.509 for H, predicting binding, whereas scaling of the unperturbed ls2 product function is well known to give the opposite conclusion. For Li~the method gives E = —7.262, necessarily above the correct value —7.280. Alternatively, given only the E values for He and Li+, we can establish that (rj~’)< 0.970 for He; the exact value is 0.947.
References [1] K. Schonhammer and L.S. Cederbaum, Physics Lett. 51A 325. [2] (1975) J. Kfflingbeck, J. Phys. AlO (1977) L99. [3] J. Killingbeck, Chem. Phys. Lett., submitted.