wave functions for anharmonic oscillators by perturbation methods
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Wave Functions for Anharmonic Oscillators by Perturbation MethodsA. M. Shorb, R. Schroeder, and E. R. Lippincott Citation: The Journal of Chemical Physics 37, 1043 (1962); doi: 10.1063/1.1733210 View online: http://dx.doi.org/10.1063/1.1733210 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/37/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Perturbation theory for coupled anharmonic oscillators J. Chem. Phys. 106, 2681 (1997); 10.1063/1.473370 Eigenvalues of anharmonic oscillators from a variational functional method J. Math. Phys. 25, 932 (1984); 10.1063/1.526248 Energetics, wave functions, and spectroscopy of coupled anharmonic oscillators J. Chem. Phys. 78, 1348 (1983); 10.1063/1.444874 Wave Functions and Intensity Calculations for a Rotating Anharmonic Oscillator J. Chem. Phys. 35, 767 (1961); 10.1063/1.1701214 On the Use of Harmonic Oscillator Wave Functions in the Treatment of Anharmonic Crystals J. Chem. Phys. 31, 361 (1959); 10.1063/1.1730359
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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 37, NUMBER 5 SEPTEMBER 1, 1962
Wave Functions for Anharmonic Oscillators by Perturbation Methods
A. M. SHORB* AND R. SCHROEDER t National Bureau of Standards, Washington, D. C.
AND
E. R. LIPPINCOTT
Department of Chemistry, University of Maryland, College Park, Maryland
(Received May 7, 1962)
Approximate wave functions and the corresponding energies computed by perturbation methods to the second order are given in tabular form for anharmonic potentials which have been expanded in polynomial series up to the sixth power in the displacement x.
INTRODUCTION Perturbation Treatment for an Anharmonic Oscillator
SEVERAL methods are available for the construction of reliable internuclear potential curves for
diatomic molecules.! These curves are of considerable value for the understanding of spectral phenomena, stellar structure, chemical kinectics, molecular mechanics, and many associated problems. However, except for curves calculated ab initio from the Schrodinger equation for simple systems the corresponding anharmonic oscillator wave functions are not readily available. If an empirical form is assumed for the internuclear potential function the anharmonic oscillator wave functions may be obtained by solution of the Schrodinger equations. However, except for a few relative simple empirical forms, such as the Morse function, exact solutions have not been obtained. The more reliable internuclear potential functions are thus ones for which solutions to the Schrodinger equation cannot be obtained in closed form. However, potential curves for bound states can be expanded in a power series in the displacement from the equilibrium bond length x. Approximate wave functions for such anharmonic potentials are readily obtained by perturbation methods. The purpose of this paper is to report these functions with the corresponding anharmonic energies for any function which has been expanded in a power series in x up to a power of six.2 These wave functions may be useful for calculations of spectral phenomena such as transition probabilities and for quantum calculations in energy transfer problems.
It is appropriate to discuss briefly the application of perturbation methods to the anharmonic oscillator problem. The following assumptions have been made:
* Present address: Department of Mathematics, Cornell University, Ithaca, New York.
t Present address: Department of Chemistry, University of Southwestern Louisiana, Lafayette, Louisiana.
1 A review of the various methods is given by D. Steele, E. R. Lippincott, and J. T. Vanderslice, Revs. Modern Phys. 33, 239 (1962) .
2 For a brief discussion of the treatment up to the fourth power in x see L. D. Landau and E. M. Lifshitz Qttanlum Mechanics (Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1958),1'. 136.
(A) That our problem
(1)
may be written (2)
TABLE I. First-order perturbation energies.
n B. B6
0 3/4 15/8
15/4 105/8
2 39/4 375/8
3 75/4 945/8
4 123/4 1935/8
5 183/4 3465/8
6 255/4 5655/8
7 339/4 8625/8
8 435/4 12 495/8
9 543/4 17 385/8
10 663/4 23 415/8
where
(n=O, 1, ..• ) (3)
is a problem which may be solved rigorously and where }..XI is small compared with Xo, }.. being a small parameter;
(B) that >/111 and En may be expressed as powcr st'rit'~ 1043
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1044 SHORB, SCHROEDER, AND LIPPINCOTT
TABLE II. Second-order perturbation energies.
-1111 En2
n BaB. BaB• BoB. B"B. B.B6 BoBs
0 11/8 65/8 449/32 21/8 180/8 3735/64
71/8 615/8 5769/32 165/8 3780/16 47 145/64
2 191/8 2425/8 31 529/32 615/8 9270/8 295 095/64
3 371/8 6335/8 107 969/32 1575/8 30 240/8 1 197 945/64
4 611/8 13 185/8 280 449/32 3249/8 76 680/8 3 677 895/64
5 911/8 23 815/8 609 449/32 5841/8 164 430/8 9 317 385/64
6 1271/8 39 065/8 1 170 569/32 9555/8 313 290/8 20 547 735/64
7 1691/8 119 550/16 2 054 529/32 14 595/8 547 020/8 40 837 785/64
8 2171/8 86 785/8 3 367 169/32 21 165/8 893 340/8 74 882 535/64
9 2711/8 120 935/8 5 229 449/32 29 469/8 1 383 930/8 128 791 785/64
10 3311/8 137 325/8 5 846 949/32 39 711/8 2 054 430/8 210 278 775/64
in A:
(4)
(5)
with the 1/In i's and Eni'S independent of A, and that the series converge rapidly;
(C) that
and
1/In2= tbmn1/lmO, m~
'JI:}l)1/InO= tx",n(ll1/lmO, m~
(6)
(7)
(8)
where the 1/Imo,s are the solutions of (3) and the amn, b"m, and x",n(l) are expansion coefficients to be determined;
(D) that the 1/Ino,s are orthonormal and nondegenerate.
With these assumptions, one obtains the following results to second-order by standard perturbation methods3 :
1/In=1/Ino+ALamn1/lmO+A2~)mn1/lmo+ .•• , (9) m m
where
(m¥-n; au,,=O), ( 10)
3 As found, for example, in H. Eyring, J. Walter, and G. Kimball, Quantum Chemistry (John Wiley & Sons, Inc., New York, 1944), pp. 93-96.
and JC k(I)JCk (I)
b _"" m n mn- £..J eE o-E' 0) (E O_£i 0)
k¢n It -'k n -<1m
( 11)
where
x",n\ll= !1/ImoJC(1)1/InOdt.
The solutions of Eqs. (3) are
1/InoW =[(,B/1r)i/2nn!Jl·Hn(~) exp( -e/2) , (13)
where ,8= 21r(mK)!/h,
and Hn(~) are the Hermite polynomials. The energy associated with 1/Ino is
En= (h/21r) (K/m)!(n+!) = (n+!)hl'. (14)
The potential of an anharmonic oscillator may be written
v =!KX2+ tKixi• ;"3
(15)
In the subsequent treatment, it will be assumed that the potential of the system can be described as a polynomial of finite degree, i.e.,
6
V=!KxZ+ LKi:l·i. (16 ) i=3
4 The expression given by Eyringa is incorrect.
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WAVE FUNCTIONS FOR .-\NHARMONIC OSCILLATORS 1045
TABLE III. First-order perturbation parameters.
hvaiO
B. B6 e Ba B. e
0 0 0 0 3/4 15/8 -2+
2 3/2 45/8 -2+/2 3 1/2 5/2 -3+/3
4 1/2 15/4 -6+/4 5 0 1/2 -151/5
6 0 3/2 -5+/6 7 0 0 0
8 0 0 0 9 0 0 0
10 0 () 0 11 0 0 0
12 0 0 0 13 0 0 0
hvail
Ba B. e B. B6 e
0 3/4 15/8 21 0 0 0
2 3 45/4 -1 3 5/2 105/8 -61/2
4 15/2 -31/3 5 1/2 21/4 -30+/4
6 0 3/2 -10+/5 7 0 3/2 -35ij6
8 0 0 0 9 0 0 0
10 0 0 0 11 0 0 0
12 0 0 0 13 0 0 0
hvai2
B. B6 e B, B6 e
0 3/2 45/8 2+/2 3 45/5 2 0 0 0 3 9/4 95/8 -61
4 7 195/4 -3+/2 5 1/2 5 -301/3
6 3/2 81/4 -10+/4 7 0 3/2 -351/5
8 0 3 -351/6 9 0 0 0
10 0 0 0 11 0 0 0 12 0 0 0 13 0 0 0
hvai' B, B. e B. B6 e
0 1/2 5/2 31/3 1 5/2 105/8 61/2 2 9/4 95/8 6' 3 0 0 0
4 6 165/4 -21 5 9 315/4 -51/2
6 1 25/2 -151/3 7 1/2 33/4 -2101/4
8 0 -2101/5 9 0 3 -1051/6
10 0 0 0 11 0 0 0
12 0 0 0 13 0 0 0
hvai'
i B. BG e B, B. e
0 1/2 15/4 6+/4 1 15/2 31/3 2 7 195/4 31/2 3 6 165/4 21
4 0 0 0 5 15/4 255/8 -101
6 11/2 465/8 -301/2 7 1/2 15/2 -1051/3
8 39/2 -105!j4 9 0 3/2 -2101/5
10 0 15/2 -42+/6 11 0 0 0
12 0 0 0 13 0 0 0
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1046 SHORB, SCHROEDER, _\N D LIPPINCOTT
Table III (continued)
ltv ai'
B" B, C B, B, C
0 0 1/2 Istj5 1 1/2 21/4 301/4
2 1/2 5 30!/3 3 CJ 315/4 S'/2
4 15/4 255/8 IOl 5 0 () 0
6 9 365/4 -31 7 13/2 645/8 -42!/2 8 35/2 -421/3 CJ 3 135/2 -211/4
10 0 3 -lOWS 11 0 3/2 -23101/0 12 0 0 0 13 0 0 0
-------.--~--~ ------- ~~----------- -" ~ -~- --_._- .-ltv ail;
B, B6 C B" 13, C
0 0 3/2 5!j6 0 3/2 101/5
2 3/2 81/4 10!j4 3 25/2 15!/3
4 11/2 465/8 301/2 5 CJ 365/4 3!
6 0 0 0 7 21/4 495/8 -141
8 15 855/4 -141/2 9 3 60 -71/3
10 3 153/2 -351/4 11 0 3/2 -7701/5
12 0 3 -11551/6 13 0 0 0
ltvai7
B3 B5 e B, B, e
0 0 0 0 0 3/2 351/6
2 0 3/2 351/5 3 1/2 33/4 2101/4
4 1/2 15/2 1051/3 5 13/2 645/8 421/2
6 21/4 495/8 14! 7 0 0 0
8 24 645/2 -1 9 51 3285/4 -21/2
10 3 135/2 -101/3 11 3 171/2 -55'14
12 0 3 -3301/5 13 0 3 -21451/6
Itvais
B, Bs e B3 Bo e
2 0 3 351/6 3 0 1 2101/5
4 1 39/2 1051/4 5 1 35/2 421/3
6 15 855/4 141/2 7 24 645/2
8 0 0 0 9 81/4 489/8 -21
10 57/2 4095/8 -101/2 11 3/2 150/4 -551/3
12 3/2 189/4 -3301/4 13 0 3/2 -21451/5
14 0 3/2 -15 0151/6 15 0 0 0
Itvai9
B3 B5 e B, B, e
2 0 0 0 3 0 3 1051/6
4 0 3/2 21O!/5 5 3 135/2 211/4
6 3 60 71/3 7 51 3285/4 21/2 8 81/4 2445/8 21 9 0 0 0
10 15 1005/4 -51 11 21/2 1665/8 -1101/2
12 1 55/2 -1651/3 13 1/2 69/4 -4290'/4 14 0 1/2 -300301/5 15 0 15/2 -1001'/6
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WAVE FUNCTIO~S FOR ANHARMONIC OSCILLATORS 1047
Table III (continued) .~--.---~~-~---- .. - ..
Izv Oi10
4
6
/;
10
12
14
16
In this case
o 3
S7/2 o 23
1/2
o
15/2
153/2
4095/8
o 1995/4
75/4
3
6
Je(l)= LKiXi .
i=3
c
421/6 351/4 101/2
o ~331/2
~60061/4
-100101/6
(17)
The value of the Jemn(l)'S, and hence of the amn'S and bm,,'s, can readily be found now without integration, by application of the recursion formula for the Hermite polynomials
~H"W =nHn- 1W+iHn+1W. (18)
REMARKS
There are two approximations involved in this treatment: (1) the perturbation operator is terminated at .1'6; (2) the perturbation energy is obtained to second order. Consequently, it must be recognized that this calculation is only an approximation and as such it is subject to errors, particularly for large values of x.
If the problem of a particle with a polynomial potential is solved rigorously, it has a continuous spectrum in most cases; that is when the leading term is an odd power of x, or if a leading even power of x has a negative coefficient.
Even in the case where the highest power of x is even and has a positive coefficient, the expansion may converge only slowly, and the error may not even decrease monotonically.
Brazley and Fox5 have computed the first eigenvalue Ao of
- y" + .1'2y+J.LX4y= AOY
as a function of the parameter J.L within very narrow limits. This is merely a special case of our problem if the proper changes of variable are made. They found that second-order perturbation theory gave good results for small values of J.L; however, when p. reached 0.25 the second-order approximation was no better than the first, and at p.=0.5, the correction was even of the wrong sign.
Nevertheless, experience shows that despite these shortcomings, perturbation theory yields fairly accurate results.
• N. W. Bazley and D. W. Fox, Phys. Rev. 124,483 (1961).
5
7
9
11
13
15
17
Ba c ._--------- .. -----
o 3
15
33/4
1/2
o o
3
135/2
1005/4
1215/8
15
3/2
o
1051/5
101/3
Sl
-221
~8581/3
-50051/5
o
The range of validity of the results given here will depend on how well the anharmonic potential can be fitted by a polynomial series which is terminated at x6•
Such series can be used to fit diatomic internuclear potential curves rather w@ll to displacements corresponding to energies of about i the dissociation energy.
USE OF THE TABLES
Table I contains the En 1'S of (5), and Table II contains -hI! times the En2's of (5). Table III contains hI! times the amn's of (6) and Table IV contains (hl!)2 times the bmn's of (7). These extra factors of hI! arise from the fact that the energy difference terms in the denominators of the coefficients are equal to (hI!) times an integer. It was found to be much easier to calculate and tabulate the coefficients after they were multiplied by the appropriate power of (hl!).
In order to use the tables, given a perturbation
Jell) = Kax3+ K4x4+ K.X5+ K 6x6,
it is first necessary to change the variable to ~ of Eq. (13) .
Je(l) = (K3~a/{31) + (K4~4/{32) + (K6~5/{36/2) + (K6~6/{33) = Ba~3+ B4~4+ B5~5+ B6~6,
where Ba=K3/fj3/2,
B4=K4/{32, Bb=K./fj6/2,
B6=K6/fj3.
These are the constants which appear in the tables. To obtain the desired coefficient, say bm", first find
the table labeled (hI!) 2bin • In row "m" there will be several numbers in columns headed by B's and one headed by "C." Multiply each number in the "B" column times the letters at the head of its column, then multiply the sum of the results by the number in the "c" column. This answer is (hl!)? times the desired coefficient.
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---_._------
o 2
4
6
8
10
12
14
16
18
20
22
o 2
4
6
8
10
12
14
16
18
20
22
B3B,
29/24
27/16
7/16
1/12
o o o o o o o o
B3B,
73/8
83/2
783/16
137/8
7/4
()
()
o ()
()
o ()
B3B,
155/24
211/16
85/16
137/60
1/10
o o o o o o o
BJ3,
1245/32
7995/32
11 175/32
2913/16
339/8
45/4
o o ()
o o o
BsB5
4447/480
1865/64
515/32
257/24
31/32
9/20
o o o o o o
BsB,
1170/32
3627/16
9723/32
5031/32
297/8
81/8
o o o o o o
B.B.
39/32
75/16
9/4
17/16
3/64
o o o o o o o
BoB,
2781/16
93 603/64
147 585/64
101 295/64
9057/16
5031/16
33/4
o o o o o
TABLE IV. Second-order perturbation parameters.
B,B, BoB,
315/32 8815/128
(hV)2 biO
c
I -2
3165/64 18 495/128 21
485/16 27 825/256 61
51
701
71
2311
o
1377/64 13 165/128
129/64 3735/256
15/16 441/32
o 5/16
o () o () o 0
() ()
o
c
-21
31/3
101/5
351/7 141/9
4621/11
o o o o o
o
o o o o
(hV)2 bit
BJ33
1
3
5
7
9
11
13
251/24
67/8
11/4
1/2
o o o
15 0
17 0
19 0
21 0
3
5
7
9
11
13
15
17
19
21
B3B,
625/8
707/8
859/20
189/10
12/5
o o o o o o
B,B,
49/8
31/8
63/80
1/16
o o o o o o o
BoB,
24 519/160
8025/32
1299/8
441/4
117/4
9/2
o o o ()
o
B3B,
105/4
685/32
1059/160
255/224
1/4
()
()
()
()
o o
B,B,
315/32
195/8
33/2
69/8
9/8
o o o o o o
BsB, BoB,
191/8 7449/64
19 7365/64
939/160 16 421/320
571/560 2829/224
9/W 91/16
o 9/44
o () () 0
() ()
() ()
() 0
B,B, BoB,
3465/32
10 955/32
553/2
6831/32
489/8
75/8
19 985/64
81 795/64
77 385/64
78 465/64
16 965/32
2655/16
15/4 o o o o o o o o o
c
21
31
151
701
351
1541
o ()
o o ()
c
-1/2
61/2
301/4 351/6
701/8 771/10
30031/12
()
o o o
......
*
U"1
::t: o ';it! I:d
U"1
n ::t: ';it! o t!j
t;j t!j
';it!
;.. Z t;j
t-' H
'"d '"d ...... Z n o >-l >-l
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i
o 2
4
6
8
10
12
14
16
18
20
22
o 2
4
6
8
10
12
14
16
BaBa
9/8
965/24
121/4
45/4
1
o o o o o o o
BaB•
BaB.
350/16
9715/24
3322/8
8686/40
482/10
12
o o o o o o
BaB•
3/8 1015/32
1152/16 18 060/32
2227/16 48 930/32
1983/16 42 203/32
37/2 2244/8
35/4 2535/8
o 45/2
o 0
o 0
18 0 o o o
20 0
22 0
B;B. B.B.
2575/32 192/16
508 387/480 1257/32
24 085/16
16 234/16
1372/4
705/4
9/2
o o o o o
B;B.
435/16
16980/32
22 346/16
37 209/32
3853/16
2171/8
117/4
o o o o o
747/8
2496/32
87/4
45/8
o o o o o o
BoB6
24 175/64
276 099/64
974 823/64
827 819/64
110 565/32
103 005/16
6903/8
231/4
o o o o
Table IV (continued)
B.B6 BoB.
8820/64 56 160/128
17 865/32 66 025/32
54 395/32 513 015/64
6339/4 533 475/64
10 233/16 139 125/32
5910/16
75/8
o o o o o
c
-31/3
-6' 2!
15'/3
2101/5
211/7
17'/9 286;/11
o
120 465/32
3105/16
105/4
o o o o
1
3
5
7
9
11
13
15
17
(lIP)2 b'2
C
-2i/2
-1/2
3!/2
101/4
351/6 141/8
4621/10
4291/12
o o o o
(hV)2 b;a
BaBa
-7/8
2499/24
189/4
19/4
1
o o o o
o o
19 0
21 0
o
BaB. BaB.
1 509/8 11 955/32
3 684/16 11 415/32
5 330/8 11 721/32
7 359/16 9255/32
9 21/8 1269/16
11 0 45/4
13 0 0
15 0 0
17 0 0
19 0 0
21 0 0
BaB. BoB. BJJ.
30/16 6575/32 480/16
32 475/24 2 169 657/480 3645/32
32 026/40 56 957/16 1215/8
4382/40 9822/16
586/10 1984/4
4 275/4
o 9/2
o 0
o 0
o o
o o
732/32
105/4
15/8
o o o o o
B;B, B.B. c
5635/16 141 243/64 -1
10 398/32 177 514/64 6'
51 249/160 191 226/64 301/3
39 725/160 193 455/64 35;/5
2742/16
81/8
o o o o o
BJJ.
20 112/16 70i /7
5967/16 771/9
33/4 30031/11
o 0
o 0
o 0
o 0
BoB. C
28 980/64 234 360/128 -6'/2
63 315/32 571 515/64 -1/2
110 061/32 1 272 915/64 5;/2
57 504/64 355 545/64 2101/4
14 337/16 226 335/32 1051/6
2310/16 54 315/32 4624/8
75/8 3555/16 20021/10
o 105/4 21454/12
o 0 0
o o
o o
o o
:::J ;.-<: tr:1
'"xj
c:: Z (j ..., .... o Z rJ'l
'"xj
o ?:1
;.Z
= :»-?:1 ~ o z .... (j
o rJ'l (j .... t"' t"' ;...., o ?:1 rJ'l
~ '.0
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BJ3~
o -13/4
2 -37/4
4 5181/24
6 271/8
8 23/2
10 5/2
12 0
14 0
16 0
18 0
20 0
22 0
i BaB,
BaB. BoB.
-230/8 -450/16
130/8 11 475/16
27 785/8 2 281 149/160
54614/80 115 137/32
6178/20
690/4
12
o o o o o
BJ3.
16 182/8
13 590/8
945/4
63/2
o o o o
BoB.
B.B.
-48/32
915/8
8649/32
1788/16
1632/16
615/8
45/8
o o o o o
BJh
B.B.
1440/64
Table IV (continued)
BoB.
(ltv) 2 bi4
C
39 150/128 -6'/4
70 695/32 712 935/64 -3i /2
178 065/32 1 900 005/64 -1/2
97 356/32 671 445/32 30+/2
45 096/16 1 284 210/64 105+/4
95 715/32 1 724 775/64 42'/6
7950/16
525/8
o o o o
c
161 415/16
28 035/16
210/4
o o o
154'/8
143i /1O 2145+/12
o o o
(ltv)' bi;
BaB, B"B.
1
3
5
7
9
11
13
15
17
19
21
BoB.
1 -17/4 -2138/40 -1938/16
3 -SI/4 -4046/40 14 683/16
BaB• BaB.
-207/16 1595/32
4231/16
1580/16
693/8
529/16
35/8
o o o o o
83 850/32
21 954/16
34 375/32
18 453/32
2955/16
45/4
o o o o
B.s.
B.B. BoB. c
705/32 92 925/64 -3+/3
39 330/16 1 593 783/64 -2'
100 422/80 1 083 231/64 10'
76 353/80 789 741/64 105+/3
79 365/160 525 315/64 210+/5
1257/8
81/8
o o o o
B.s.
124 830/32 23lij7
7839/16 1001./9
231/8 4290+/11
o 0
o 0
o 0
BoB. c
-192/32
1503/8
-1008/64 29 295/64 -301/4 o -107/16 -1221/32 -7055/160 26895/64
2 -138/8 695/32 -7539/160 64 034/64
4 3440/16 82 617/32 193 453/80 943 143/32
-15</5
30'/3
-10+ 5 9339/24 179 955/24 11 611 341/480 17 829/32
143 073/32 1 749 195/64 -5+/2
852 390/64 1 314 915/16 -1/2
313 680/64 1 260 855/32 421/2
99 960/8 3 222 675/32 21'/4
6 2043/8
8 3687/16
10 307/4
12 21/4
14 0
16 0
18 0
20 0
22 0
139 785/32 63 993/16
104 375/32 93 351/32
24 222/16
2025/8
63/4
o o o o
20887/16
8571/40
567/40
o o o o
4 159 563/64 3>
2 750 005/64 42'/3
783 615/32 lOStj5
96 519/16
12 285/16
462/4
o o o
385+/7
1430'/9
858'/11
o o o
7 367/8 17 134/16 208 705/32 615/4
9 81/2
11 1/2
13 0
15 0
17 0
19 0
21 0
4970/4
794/20
12/5
o o o o
74 530/8
3574/8
213/4
63/2
o o o
1596/4
141/8
9/S
o o o o
24 651/32 499 845/64 2310+/6
1794/16
525/8
o o o
53 685/32
31 185/16
105/2
o o
10 0101/S
4291/10
(429·17)'/12
o o
~
o '.It o
en lI: o ~ t:d
en ("'J
ll: ~ o t'1 t)
t'1 ~
:> Z t)
r ...... '"tj
'"tj ...... Z ("'J
o ""l ""l
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Table IV (continued)
(l1P)2 biG
BaE. BaB. BsE. B,B. B.B. BsE. c
o -1/2 -454/20 -1354/8 -75/8 -4887/32 34215/64 -5i /6
2 -63/4 -10306/40-13934/16 -150/4 -21408/6485365/64 -1Olj4
-1, -139/8 -14334/80 12023/32 2250/16 127847/32 1 846965/64 -301/2
6 15301/24 344 275/24 39220571/480 33 135/32 904 995/32 12 710 255/64 -1/2
8 477/4 25 298/8 349325/16 3225/8 457 875/32 8681 445/64 14'/2
10 93/2 32 094/20 108474/8 2016/4 564 528/32 10 162 710/64 351/4-
12 1 898/10 4552/4 159/4 30861/16 695 865/32 1155~/6
14 0 56/10 553/4 21/8 4662/16 154665/32 42901/8
16 0 0 63 0 525/4 34335/8 2861/10
18 0 0 0 0 0 315/2 (143-17)1/12
20 0
22 0
o o
o o
o o
o o
o o
o o
(liV)2 bi7
3
5
7
9
11
13
15
17
19
21
BaE. BaB. BsE. BsE. c BaB, BaB. BsE,
o -7/16 -417/32 -801/80 -6369/32 -701/7 -1/2 - 558/20 -1962/8
2 -491/16 -10 065/32 -10305/32 -189045/64 -351/5 3 -25/4 -5042/40 -8922/16
4 -810/16 -10 089/32 -29298/80 -51/64 -1051/35 5 -211/8 -5642/16 -2265/32
BaB. BaB• BsE, BsE. c
-203/8 -3237/16 -6919/32 -93 105/64 -101/5
-1023/16 -8216/32 -10 701/32 115431/64 -15i /3
5166/8
1231/8
3546/4
699/16
7/8
o o o o
B,B,
-93/8
291 585/32 136401/16 7 784723/64 -31
101 955/32 46687/16 1 206 492/32 141
225555/16 202 869/16 6 703 080/32 71/3
30 771/32 26601/32 1 115895/64 7701/5
759/16
315/4
o o o
820/16
567/8
o o o
45462/32
67977 /16
231/2
o o
BJ3s BoB.
300301/7
1431/9
(143·17)1/11
o o
c
-7803/32 74625/64 -351/6
-672/32 -9172/32 4350/128 -2101/4
1581/8 209 305/32 3 498 465/64 -42l/2
6 3667/8
8 5593/8
118860/16 111 114/16
278040/16 254 592/16
567 651/32 2819979/64
7 268829/64 -14'
11 419263/32 1
7 23395/24 200 905/8 26 151 387/160 56907/32 1 752345/32 3 456325/8 -1/2
10 16695/16
12 196/2
14- 7/4
16 0
18 0
20 0
22 0
19 051/8 33 015/16
843/8 3543/40
90/2 162/4
o o o
o o o
20 645 181/64 101j3
1 531 455/32 3301/5
150151/7
10011/9
9
11
13
15
17
1803/4
105/2
o o
49443/16
10647/4-
99/2
o 340341/11 19 0
21 0
o
107 022/8 1 650 735/16
40 266/20 151 434/8
1002/10 5652/4
12 1305/4
o 9
o o
o o
12 231/8 2020605/32 41 921 235/64 2!/2
2484/4 192 216/8 15286950/64 5S!j4
177/4 37 773/16 937875/32 21451/6
45/8
o o o
11 010/16
75/4
o o
401 265/32
6525/4
45/2
o
20021/8
340341/10
(17 ·19·1001)1/12
o
:E ;.. <: M
>-ri c: Z ("'J
~ ...... o Z en
>-ri o :;>:j
;.Z ~ ;.:;>:j
~ o Z H
("'J
o en ("'J ...... r r ;.. ..., o :;>:j en
..... o t.Ft .....
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..... 0 tJ\ N
Table IV (continued)
(hp)' biS
B3B3 B3B• BoB. B.B. B.B. B,J3, C
0 0 -8/10 -41/4 -3/8 -354/16 -6975/32 - 7()!/8
2 -1 -1324/20 -2692/4 -111/4 -11 421/16 -137085/32 -351/6
4 -29/2 -6958/20 -15 122/8 -126/2 -9016/8 -69 585/32 -1051/4 en ~
6 -297/4 -10 054/8 -35 525/32 4245/8 642 345/32 12 212 505/64 -14i/2 0
8 33 949/24 986 075/24 156 876 071/480 91 875/32 3 154 095/32 13 831 055/16 -1/2 ?::1 Cd
10 2217/8 727 818/80 2 478 891/32 7515/8 1 384 089/32 31 990 455/64 101/2 en
12 117/4 49366/40 204 466/16 1500/4 254 268/16 11 065 635/64 3301/4 (")
14 1/2 1106/20 6874/8 195/8 45 387/32 1 230 735/64 15 0151/6 ~ ?::1
16 0 24 1425/2 45/4 12 030/8 477 945/16 10011/8 0 ~
18 0 0 27/2 0 225/8 17 415/16 34034'/10 t;:I
20 0 0 0 0 0 45/4 (10 010·17·19)'/12 ~
?::1 22 0 0 0 0 0 0 0
>-B3B. BsE. BoB. BrJJs C Z
tj
-7/4 -501/8 -389/8 -17907/16 -35'/7 r 3 -24 -2406/8 -1164/4 -125 335/32 -210'/5
H
'" 5 -2343/16 -38565/32 -42 603/32 -290 415/64 -421/3 '" H
7 19 969/8 730 560/16 341 282/8 25 171 743/32 -1 Z (")
9 4509/8 269 100/16 246 258/16 25 546 047/64 2' 0
11 10 782/16 417 957/32 189 630/16 1 522 784/64 551/3 >-l >-l
13 869/16 46 209/32 40 129/32 2 039 595/64 21451/5
15 105/16 13 905/32 5829/16 446 295/32 2002'/7
17 0 45/4 81/8 11 583/16 34034'/9
19 0 0 0 99/4 (34 034·19)'/11
21 0 0 0 0 0
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Table IV (continued)
(hv) bi.
i B.;B. B.;Bs BsB. BsBe C :s 0 0 -9/4 -81/40 -1053/16 -35t!9
;.-<:
2 -21/8 -1755/16 -1716/20 -36042/16 -70+/7 M
4 -661/16 -19 587/32 -19 177/32 -530 985/64 -210~/5 "!j
c::: 6 -2394/4 -97 005/16 -104067/16 -1 266 240/32 -7W Z
8 19 737/8 1 607 895/32 750 771/16 61 622 952/64 -2~ ()
>-l 10 3050/8 441 591/32 1 009 219/SO 24 089 907/64 51 .....
0 12 8607/16 363 495/32 1 655 007/160 14458 857/64 165+/3 Z
en 14 159/8 9183/16 15 981/32 883 205/16 30 030i /5
"!j
16 35/4 5055/8 4229/8 352 605/16 2oo2i /7 0
18 0 45/2 81/4 12 519/8 17 0171/9 ~
20 0 0 0 33/2 (95-17017)·/11 ;.-Z
22 0 0 0 0 0 ~ ;.-
i B.;Ba B.;B. BfiB. BsBs B.B. BeEs C ~
~
0 -24/10 -147/4 -1266/16 -9/8 -28 845/32 -70+/8 0 Z
3 -1 -766/10 -3544/4 -15 741/16 -129/4 -226 355/32 -105ij6 ..... ()
5 -198/4 -5510/4 -70 790/8 -92 520/16 -1056/4 -86625/32 -21~/4 0
7 -1191/4 -48 426/8 -290 055/16 2 813 685/32 16 515/8 60 015 015/64 -21/2 en ()
9 47 291/24 1 529 555/24 249 326 341/480 5 351 445/32 141 159/32 103 409 315/64 -1/2 ..... r
11 891/8 320 494/80 1 193 897/32 611 421/32 3009/8 15 587 505/64 110+/2 r
13 43/4 19 798/40 89 542/16 109 448/16 594/4 5 173 305/64 4290+/4 ;.->-l
15 5/2 1210/4 41 090/8 268 515/32 1065/8 7 896 525/64 1oo1i /6 0 ~
17 0 8 515/2 4350/8 15/4 187 155/16 17 017~/8 en
19 0 0 9/2 75/8 0 6255/16 (34 034-19)~/1O
21 0 0 0 0 0 105/4 (2-3-5-11-13-17-19)1/12
..... 0 <It W
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.... 0 ~
Table IV (continued)
(hl')2 bilO
BaB3 BaB. B"B. B(B4 B4B6 BsE. C
0 0 0 -9/2 0 -75/8 -2745/16 -7+/10
2 0 -12 -855/4 -45/8 -7350/16 -190 665/32 -W/8 U1
4 -5/2 -870/4 -22 590/8 -735/8 -103 815/32 -1 737 975/64 -42i/6 :z: 6 -111/2 -35 154/20 -104094/8 1404/4 -72 456/8 -1 366 515/32 -35;;4 0
::d 8 -1533/8 -362478/80 -588 771/32 5211/4 987 546/16 23 365 755/32 -10i/2 ttl
10 63 749/24 757 585/8 136 013 745/160 208 269/32 8 648325/32 22 836 625/8 -1/2 (Jl
12 1057/4 82 666/8 1 671 045/16 7095/8 1 576 855/32 43 973 475/64 33'/2 Ci :z:
14 47/4 4699/8 114 990/16 696/4 138 500/16 7 064 625/64 6OO6tj4 ::d
16 1 1314/10 9684/4 231/4 62 721/16 1 988 445/32 10 010i/6 0 trJ
18 0 14/5 333/2 9/4 2814/8 130 365/16 8508W8 t::t
20 0 0 9 0 55/4 6705/8 (1001-17 -19);/10 trJ ::d
22 0 0 0 0 0 1155/4 (2-3-13-17 -19);/12 >
i BaB, BaBe BsE. BsE. C Z t::t
1 0 -45/4 -81/8 -6201/16 -14'/9 t"" .... 3 -35/4 -3345/8 -2631/8 -155 505/H; -21+/7 "d
"d 5 -373/4 -12 759/8 -24 663/16 -795 585/32 -105'/5 .... 7 -12 501/16 -299 355/32 -1 574 919/160 -5 300 271/64 -10+/3
Z Ci
9 16 895/8 1 515 759/32 3 537 531/80 63 939 627/64 -5; 0 >-3
11 1444/8 260 295/32 118 791/16 16 291 014/64 22; >-3
13 2538/8 231 945/32 211 857/32 9999 510/64 858'/3
15 1039/16 64 767/32 56 457/32 3 370 815/64 5005;/5
17 7/4 1095/8 2286/20 41 244/8 85 085./7
19 0 9/4 81/40 2691/16 (10010-17-19)!/9
21 0 0 0 231/4 (2-3-11-13-17-19);/11
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WAVE FUNCTIONS FOR ANHARMONIC OSCILLATORS
Example (1) : Find bal. In Table IV under (hv)2b il , row 3, one finds
1 3 5
Therefore
67/8 707/8 8025/32 195/8 10 955/32 81 795/64
1055
C
(l/2) (6)~.
(hv)2bal = [(67 BaBa/8)+ (707 BaBs/8)+ (8025BsBs/32)+ (195B4B4/8)+ (10 955B4Bs/32)+ (81 795B~6/64)J(1/2)(6)!.
Example (2) : Find asa in Table IV under hvaia. In row 5 one finds
Therefore
5 B6
315/4 C
(-1/2)(5)!.
/zvasa= - [9B4+ (315/4) B6J(1/2) (5)!.
The tables for Enl and En2 are similar, except there is no coefficient "C" by which to multiply.
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