Rayleigh–Schrödinger perturbation expansions for eigenvalues E(λ) of nonlinear Hamiltonians of the form Ĥ(0)+λ〈r p〉rq, p,q≥1 are calculated using hypervirial (HV) and Hellmann–Feynman (HF) theorems. Such Hamiltonians are similar in form to those employed in the study of ‘‘self‐interacting’’ systems, e.g., solute–solvent interactions. The specific cases considered for H(0) are one‐dimensional harmonic oscillators and hydrogen atoms. The eigenvalue expansions for the nonlinear problems are compared with those of the linear problems where p=0, whose large‐order behavior and summability properties are well‐known. Also examined are the perturbation expansions for the expectation values 〈rk〉, which are also products of the HVHF method.
REFERENCES
1.
2.
P. R.
Surjan
and J.
Angyan
, Phys. Rev. A
28
, 45
(1983
), and references therein. 3.
4.
5.
6.
7.
8.
J. J.
Loeffel
, A.
Martin
, B.
Simon
, and A. S.
Wightman
, Phys. Lett. B
30
, 656
(1969
).9.
10.
11.
12.
13.
F. M.
Fernandez
and E. A.
Castro
, Int. J. Quantum Chem.
19
, 521
, 533
(1981
).14.
15.
16.
F. M. Fernandez and E. A. Castro, Hypetvirial Theorems, Lecture Notes in Chemistry, Vol. 43 (Springer, New York, 1987).
17.
18.
19.
J. O.
Hirschfelder
and C. A.
Coulson
, J. Chem. Phys.
36
, 941
(1962
)., J. Chem. Phys.
20.
L. I. Schiff, Quantum Mechanics (McGraw‐Hill, New York, 1968), 3rd ed.
21.
H. Hellmann, Einfuhrung in die Quantenchemie (Franz Deuticke, Leipzig, 1937);
22.
23.
A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956).
24.
E. Borel, Lecons sur les Series Divergentes (Gauthier‐Villars, Paris, 1923).
25.
M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1978), Vol. 4.
26.
27.
W. B. Jones and W. J. Thron, Continued Fractions, Analytic Theory and Applications (Addison‐Wesley, Reading, MA, 1980).
28.
P. Henrici, Applied and Computational Complex Analysis (Wiley, New York, 1977), Vol. 2.
29.
G. Baker, Jr., Essentials of Padé Approximants (Academic, New York, 1975).
30.
B. W. Char, K. O. Geddes, G. H. Gonnet, and S. M. Watt, MAPLE User’s Guide (Watcom, Waterloo, 1985), 4th ed.
31.
32.
C. R. Handy, Phys. Lett. (to appear).
33.
H. S. Wall, Analytic Theory of Continued Fractions (Van Nostrand, Toronto, 1948).
34.
35.
36.
E. R. Vrscay, Theor. Chim. Acta (to appear).
37.
38.
39.
40.
41.
42.
43.
B. G.
Adams
, J.
Cizek
, and J.
Paldus
, Int. J. Quantum Chem.
21
, 153
(1982
).44.
H. J.
Silverstone
, B. G.
Adams
, J.
Cizek
, and P.
Otto
, Phys. Rev. Lett.
43
, 1498
(1979
).
This content is only available via PDF.
© 1988 American Institute of Physics.
1988
American Institute of Physics
You do not currently have access to this content.
