Eigenvalues of the Potential Function V ~ - z 4 ± B z 2

and the Effect of Sixth, Power Terms

Jaan Laane

Department of Chemistry, Texas A&M University, College Station, Texas 77843

(Received 21 May 1969; revision received 22 July 1969)

The one-dimensional Schr6dinger equation in reduced form is solved for the potential functior, V=z4-ffBz 2 where B may be positive or negative. The first 17 eigenvalues are reported for 58 values of B in the range --50KB < 100. The interval of B between the tabulated values is sufficiently small so that the eigenvalues for any B in this range can be found by interpolation. At the limits of the range of B the potential function approaches that of a harmonic oscillator with only small anharmonicity. The effect of a small Cz 6 term on this potential is studied and it is con- cluded that a previously reported approximation formula is quite applicable but only for positive values of B. The success of the quartic-harmonie potential function for the analysis of the ring- puckering vibration is shown; it is also demonstrated that the same potential serves as a useful approximation for many other systems, especially those of the double minimum type. INDEX HEADINGS: Potential functions for ring-puckering; Inversion, internal rotation, and

double minimum type vibrations; Computer applications; Far infrared.

I N T R O D U C T I O N

In the past several years there has been considerable ac t iv i ty in the s tudy of the v ibra t ional potential- function of r ing-puckering in four-and f ive-membered ring molecules. This ac t iv i ty is due to a large pa r t to the improvemen t in far- infrared ins t rumenta t ion ~ which has enabled researchers to observe directly the transit ions between the different energy s tates of this vibration. However, high-speed computers which have made possible the solution of the one-dimen- sional Schr6dinger equat ion for potent ia l functions of the type V = A (z4--t-Bz2), where B m a y be negat ive or posit ive and where z is the r ing-puckering coor- dinate, have also p layed a major role in the interpre- ta t ion of the observed far- infrared spectra.

In te res t ' in solving the wave equat ion for a potent ia l function containing a quart ic t e rm developed af ter i t was predicted by Bell 2 in 1945 tha t the r ing-pucker- ing v ibra t ion should be represented by a "quar t ic oscillator." The eigenvalues for a potent ia l V = A z 4

cannot be solved exactly and an approximat ion method mus t be used. Ear ly calculations of this type included the numerical difference-equation method of Kimbal l and Shortley, 3 the W K B method of approximat ion, 4 and the numerical in tegrat ion method of Milne. 5 I t was not until 1963, however, t ha t Chan and Ste lman 6 reported accurate energy levels for the first twen ty states of the quart ic oscillator by applying a var ia t ional method using harmonic oscillator basis functions.

The mixed quar t ic-harmonic oscillator,

V = A (z4+Bz2) ,

was first invest igated in detail by McWeeny and Coulson 7 who derived approximate formulas for the

energy levels. Chan and co-workers s studied the same sys tem using quart ic oscillator basis functions and repor ted eigenvalues for nine different positive values of B. Other methods t ha t can be applied to solving the Schr6dinger equat ion for this potent ia l include the very accurate power-series computa t ion of Secrest, Cushion, and Hirschfelder, 9 the technique of Milne ~ as applied by Laane and Lord 1°,11, and the method of continued fractions. 12 Other workers have also had an interest in this system. 13,14 ~ Even though m a n y different methods exist for calculating the eigenvalues for the mixed quar t i c - harmonic oscillator, ve ry few actual values have been tabulated, s and these for only posit ive harmonic contributions. The desirabil i ty for more extensive tables became evident f rom the considerable response to the eigenvalue curves published b y Laane and Lord 1° for the potent ia l V = z 4 - B z 2.

This work tabula tes the first 17 eigenvalues for 58 separate values of B, bo th posit ive and negative. Two separa te methods are used to calculate the eigen- values to assure their accuracy. These tables make i t possible to find the eigenvalues for any potent ia l of the form V = A (z4-ffBz 2) by interpolation. As a result a considerable amoun t of compute r t ime can be saved by the use of these tables ; moreover, researchers who do not have access to large computers will find it possible to determine the eigenvalues for any specific potent ial of this type.

The avai labi l i ty of these tables will also make it possible to determine the effect of a hexic t e rm in the potent ia l on the eigenvalues. An approximat ion formula for the es t imat ion of this effect 15 will be tes ted by calculating some actual eigenvalues of this type.

Volume 24, Number 1, 1970 APPLIED SPECTROSCOPY 73

100 , 16/,

90| <B / . " / " ~/" 13.// / . /

I . ,o . -lOll ~ -- . S

0-20 -1'8 -3'6 d4 -12 -10 -i~ -6 -~1 -2 0 i . k 6 8 10 B IN V=E4+BZ =

Fro. 1. Eigenvalues for the potential V=z~TBz ~.

I. E IGENVALUES

We transform the one-dimensional wave equation

-- (h~/2#) ( d ~ / d x ~) + ( V - E)q, = 0 (1)

for V=ax~Wbx ~ by the use of

z= (2#/h~)~/~a~%, B = (2#/h~)~a-~b,

= (2~/h:)~a-{E. (2)

This results in the dimensionless eigenvalue problem

d~v2/dz~+ (X--z~--Bz~)Vz = 0. (3)

The eigcnvalues X i n Eq. (3) can be calculated for different values of B by one of the methods previously mentioned. ~-~ These eigenvalues can be converted to the energy levels E by use of Eq. (2) and the resulting potential in reduced form becomes

V=A(z~WBz~), where A = (M/2#)~a~. (4)

In most calculations A has the units c m -1, B and z are dimensionless, a is cm-~/h ~, b is c m - V £ ~, and x is in A. The value of (2t~/h ~) is 0.059304 ~ . (cm -~ ~ ) -~ where ~ . is in atomic units (usually 100-200 a.u.).

The previous t ransformat ion is the one tha t has been most popular historically :-~,~-~ but several

'=°~ \ / / , / / / / - / 111 ~ / ) / 4 / 2 1 / 3 1 ~ 0 ~ . 12.151013 811 69 47 25 03

3.0

S IN V=Z%BZ ~

Fro. 2. Eigenvalue separations for the potential V=z~+Bz%

other ones have been u s e d also. Several workers at

Berkley T M have used

z,= (8~/h2)ll~a'16x, ~= (8#/h2)ia-ib, and

X,= (8#/h:)ia-iE. (5)

The relations between the two transformations are

~=41B, X,=41X, and A,=4-~A. (6)

For small anharmonicities ye t another t ransformat ion has been used for which the reduced form of the potential becomes

V~ = A ~ (z~ ~-I- }z~). (7) Here

z~ = (8t~b/hM)ix, ~= (h2/8#b3)~a, X~= (8~/bh2)~E, (8)

and

1 __.a 1 ! ~ = ~ B . , X~=2B-~X, and A~=~B,A. (9)

Finally, Chan and co-workers s have used a transfor- mat ion in terms of the parameter a which varies from 0 for the pure harmonic oscillator to 1 for the qdart ic oscillator. The reduced potential

V, = A,Ea~z,4+ (1 -a)z,~2 ] (10)

results from defining

= [1 +8b3v/a2h2) i] -~ and (11)

For these parameters we have the relations

a= ( l+4~B) -~ and ,y,=4~(l+41B)-~X. (12)

Several values of X, for a ranging from 0.0-1.0 have been tabula ted previously, s

In the present work, the values of X are tabula ted for the potential V = z 4 + B z 2 for 58 different values of B. The eigenvalues were calculated by diagonalizing a 70X70 Hamil tonian matr ix in the representat ion of the harmonic oscillator3 ,~ These numerical values were checked using the numerical integrat ion tech- nique developed by Milne 5 and adapted to the com- puter by Laane and Lord. '°,H The la t ter method has the advantage of requiring less computer t ime and less storage space per calculation. The method has a drawback in tha t it is not very good for calculating near-degenerate eigenvalues and is therefore less useful when B is large and negative.

II. RESULTS

Table I lists the eigenvalues calculated for B in the range 4-100 to - 5 0 . The accuracy of these values is for a large par t =t=0.0001 al though it decreases some- what when n is large or when B is large and negative. When the eigenvalue differences are determined from Table I and multiplied by the appropriate factor A in cm -t from Eq. (4), they correspond to the observed frequencies for a given potential function. Figures

74 Volume 24, Number 1, 1970

1 and 2 show graphs of the individual eigenvalues snd eigenvalue differences, respectively, for the range B = - -20 to + 1 0 . Quantum jumps of three ~re shown in Fig. 2 since these ~re s l lowed transit ions for quartic oscillator.

At B = 0 the eigenv~lues are those of ~ pure quartic oscillator. As B increases in the posit ive direction, the potential funct ion takes on more and more char- acter of the hurmonic oscillator. B = + oo corresponds

Table I. Eigenvalues for V ---- z 4 + Bz ~.

to a pure harmonic oscillator. At large values of B ( ~ 1 0 or greater) the eigenvalues are given approxi- mate ly by

1 1 h,~=2B~(n+~), B>>0, (13)

as is expected for the potential V = Bz 2 since B = h2k/4t~ in terms of k, the normal harmonic force constant . The differences between the actual values and those

n B=I00.0 50.0 30.0 20.0 15.0 12.5 i0.0 9.0 8.0 7.0

0 10.008 7.0857 5.5020 4.5089 3.9215 3.5933 3.2335 3.0786 2.9159 2.7444

1 30.038 21.2876 16.5548 13.5986 11.8582 10.8899 9.8341 9.3814 8.9081 8.4114

2 50.099 35.5495 27.7041 22.8286 19.9762 18.3984 16.6884 15.9597 15.2010 14.4091

3 70.190 49.8690 38.9475 32.1934 28.2649 26.1033 23.7730 22.7850 21.7600 20.6948

4 90.311 64.2464 50.2831 41.6877 36.7152 33.9921 31.0669 29.8358 28.5597 27.2375

5 110.460 78.6815 61.7091 51.3071 45.3194 42.0539 38.5637 37.0950 35.5797 34.0141

6 130.639 93.1714 73.2237 61.0476 54.0701 50.2797 46.2418 44,5476 42.8039 41.0055

7 150.848 107.7187 84.8228 70.9042 62.9615 58.6612 54.0943 52.1832 50.2191 48.1980

8 171.087 122.3211 96.5075 80.8759 71.9884 67.1913 62.1113 59.9897 57.8130 55.5768

9 191.354 136.9769 108.2753 90.9561 81.1466 75.8642 70.2848 67.9602 65.5771 63.1330

i0 211.65 151.6875 120.1232 101.1434 90.4285 84.6745 78.6079 76.0847 73.5025 70.8566

ii 231.97 166.4509 132.0516 111.4347 99.8325 93.6162 87.0761 84.3587 81.5819 78.7398

12 252.32 181.2683 144.0583 121.8259 109.3529 102.6840 95..6803 92.7763 89.8083 86.7747

13 272.71 196.1386 156.1434 132.3184 118.9872 111,8756 104.4180 101.3293 98.1754 94.9553

14 293.13 211.0600 168.3042 142.9078 128.7323 121.1837 113.2827 110.0150 106.6801 103.2762

15 313.58 226.0315 180.5377 153.5866 138.5862 130,6076 122.2703 118.8265 115.3151 111.7329

16 334.06 241.0573 192.8459 164.3661 148.5432 140,1466 131.3773 127.7603 124.0734 120.3192

B=6.0 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0

0 2.5625 2.3682 2.2656 2.1588 2.0473 1.9305 1.8075 1.6778 1.5398 1.3923

i 7.8876 7.3321 7.0410 6.7396 .6.4270 6.1022 5.7636 5.4098 5.0389 4.6488

2 13.5796 12.7076 12.2537 11.7870 11.3062 10.8107 10.2990 9.7700 9.2224 8.6551

3 19.5845 18.4243 17.8239 17.2086 16.5777 15.9305 15.2659 14.5825 13.8799 13.1567

4 25.8651 24.4375 23.7012 22.9491 22.1803 21.3944 20.5898 19.7661 18.9222 18.0577

5 32.3939 30.7147 29.8511 28.9712 28.0737 27.1576 26.2231 25.2688 24.2937 23.2975

6 39.1499 37.2317 36.2479 35.2467 34.2269 33.1889 33.1315 31.0538 29.9554 28.8357

7 46.1163 43.9692 42.8707 41.7534 40.6182 39.4637 38.2895 37.0944 35.8786 34.6411

8 53.2781 50.9124 49.7031 48.4749 47.2279 45.9617 44.6758 43.3687 42.0408 40.6906

9 60.6244 58.0463 56.7308 55.3962 54.0422 52,6685 51.2745 49.8598 48.4235 46.9657

i0 68.1441 65.3612 63.9427 62.5045 61.0472 59.5698 58.0720 56.5526 55.0125 53.4502

ii 75.8296 72.8473 71.3283 69.7901 68.2327 66.6546 65.0556 63.4352 61.7937 60.1302

12 83.6717 80.4966 78.8803 77.2438 75.5886 73.9125 72.2166 70.4978 68.7580 66.9962

13 91.6645 88.2997 86.5890 84.8579 83.1072 81.3365 79.5431 77.7303 75.8948 74.0377

14 99.8018 96.2524 94.4484 92.6258 90.7822 88.9176 87.0321 85.1255 83.1968 81.2455

15 108.0784 104.3476 102.4538 100.5391 98.6046 96.6487 94.6728 92.6742 90.6547 88.6126

16 116.4882 112.5807 110.5979 108.5940 106.5710 104.5256 102.4606 100.3735 98.2645 96.1333

APPLIED SPECTROSCOPY 75

/

Table h (Continued)

_n B=0.5 0.0 -0.5

0 1.2334 1.0604 0.9324

1 4.2367 3.7997 3.3962

2 8.0665 7.&557 6.8841

3 12.4121 11.6447 10.9164

4 17,1711 16.2618 15.3919

5 .22,2795 21.2386 20.2368

6 27.6936 26.5288 25.403].

7 33.3815 32.0991 30.8555

8 39.3185 37.9232 36.5675

9 45.4849 43.9817 42.5170

i0 51.8650 50.2569 48.6881

ii 58.4439 56.7356 55.0648

12 65.2113 63.4037 61.6359

13 72.1568 70.2536 68.3893

14 79.2719 77.2751 75.3176

15 86.5475 84.4601 82.4107

16 93.9788 91.8008 89.6618

n B ~4.5 -5.0 -5.5

__ -i. Q__~____L~5__ ___-~g_0 -2.5 -3.0 -,3,5 , _ Lt~,~_

0.9076 0.9793 1.1378 1.3699 1.6565 1.9726 2 2897

3. 0845 2. 8590 2. 7129' 2. 6390 2. 6277 2. 6690 2. 7520

6. 4139 6. 0458 5. 7825 5. 6292 5. 5954 5. 6946 5. 9413

I0. 2887 9. 7611 9. 3328 9. 0042 8. 7750 8. 6451 8. 6129

14.6224 13.9535 13.3848 12.9162 12.5483 12.2826 12.1216

19.3359 18.5356 17.8354 17.2352 16.7351 16.3357 16.0375

24.3784 23.4543 22.6306 21.9064 21.2828 20.7595 20.3365

29.7131 28.6714 27.7303 26.8894 26.1479 25.5068 24.9660

35.3126 34.1586 33.1048 32 . 151-3 31.2982 30.5444 29.8931

41.1539 39.8922 38.7302 37.6690 36.7077 35.8462 35.0846

47.2203 45,8537 44.5880 43.4221 42.3563 41.3910 40.5245

53.4960 52.0282 50.6609 49.3955 48.2275 47.1610 46.1945

59.9682 58.4024 56.9372 55.5613 54.3069 53.1422 52.0776

66.6269 64.9650 63.4037 61.9428 60.5827 59.3221 58'.1614

73.4618 71.7060 70.0504 68.4974 67.0429 65.6901 64.4362

80.4623 78.6167 76.8709 75.2257 73.6803 72.2364 70.8915

87. 6253 85. 6892 83. 8547 82.1206 80. 4857 78. 9521 77. 5180

-6.0 -6.5 -7,0 -7.5 -8.0 -8.5 -9.0

0 2.5830 2.8398 3.0603

i 2.8655 2.9992 3.1441

2 6.3428 6.8889 7.5493

3 8.6766 8.8313 9.0681

4 12.0699 12.1353 12.3319

5 15.8419 15.7502 15.7649

6 20.0157 19.7979 19.6851

7 24.5260 24.1882 23.9534

8 29.3384 28.8868 28.5372

9 34.4235 33.8633 33.4041

i0 39.7597 39.0942 38.5301

ii 45.3277 44.5613 43.8956

12 51.1128 50.2463 49.4835

13 57.1018 56.1411 55.2814

14 63.2832 62.2292 61.2762

15 69.6473 68.5025 67.4574

16 76.1848 74.9508 73.8171

3.2517 3.4229 3.5791 3.7253 3.8639 3.9967 4.1239

3.2930 3.4421 3.5879 3.7292 3.8654 3.9972 4.1243

8.2760 9.0]24 9.7061 10.3268 ].0.8732 11.3620 11.8083

9.3752 9.7375 10.1380 10.5599 10.9890 11.4155 11.8315

12.6795 13.2043 13.9237 14.8282 15.8677 16.9640 18.0300

15.8873 16.1179 16.4550 16.8929 17.4194 18.0188 18.6689

19.6808 19.7904 20.0217 20.3880 20.9115 21.6253 22.5607

23.8229 23.7993 23.8855 24.0845 24.3991 24.8316 25.3819

28.2905 28.1488 28.1144 28.1899 28.3793 28.6901 29.1302

33.0467 32.7933 32.6444 32.6026 32.6700 32.8502 33.146]

38.0675 37.7065 37.4501 37.2982 37.2535 37.3178 37.4948

43.3306 42.8678 42.5070 42.2508 42.0991 42.0548 42.1194

48.8204 48.2582 47.7976 47.4403 47.1869 47.0390 46.9977

54.5218 53.8635 53.3061 52.8519 52.5000 52.2516 52.1099

60.42]] 59.6707 59.0]97 58.4706 58.0235 57.6801 57.4408

66.5134 65.6701 64.9267 64.2852 63.7460 63.3090 62.9753

72.7842 71.8503 7].0]83 70.2866 69.6569 69.1292 68.704]

76 Volume 24, Number 1, 1970

Table I. (Continued)

n B=-9.5 -i0.0 -i i. 0 -] 2.0 -13.0 -14.0 -15.0 -I 6.0 -17.0 -] 8. U

0 4.2472 4.3666 4.5951 4.8120 5.0200 5.2182 5.4094 5,594 5.772 5.949

1 4.2473 4.3666 4.5951 4.8120 5.0200 5.2182 5.4094 5.594 5.772 5.949

2 12.2254 12.6207 13.3637 14.0587 14.7144 15.3381 15.9363 16.509 17.061 17.600

3 12.2351 12.6247 13.3647 14.0587 14.7144 15.3381 15.9363 16.509 17.061 17.600

4 19.0031 19.8671 21.3472 22.6380 23.8217 24.9306 25.9821 26.983 27.942 28.868

5 19.3476 20.0349 21.3793 22.6425 23.8223 24.9306 25.9821 26.983 27.942 28.868

6 23.7240 25.0645 27.8598 30.2344 32.1705 33.8807 35.4625 36.949 38.362 ?~.714

7 26,0444 26.8075 28.5489 30.4006 32.1990 33.8848 35.4629 36.949 38.362 39.714

8 29.7167 30.4760 32.6722 35.8165 39.1529 41.9215 44.2306 46.308 48.244 50.077

9 33.5626 34.1033 35.5669 37.4998 39.7394 42.0435 44.2496 46.310 48.244 50.077

i0 37.7881 38.2035 39.4383 41.3436 44.2395 48.0805 51.8123 54.849 57.466 ~,9.871

ii 42.2960 42.5885 43.5368 44.9999 47.0043 49.4798 52.2079 54.917 57.475 59.872

12 47.0663 47,2463 47.9575 49.1693 50.9625 53.5545 57.3117 61.779 65.700 68.936

13 52.0755 52,1499 52.6404 53.6017 55.0771 57.1106 59.7109 62.740 65.906 68.967

14 57.3069 57,2809 57.5612 58.3000 59.5310 61.3085 63.7605 67.219 71.917 76.735

15 62.7470 62,6240 62.7063 63.2340 64.2350 65.7435 67.8101 70.475 73.698 77.251

16 68.3832 68~1669 68.0573 68.3861 69.1746 70.4503 72.2600 74.682 77.934 82.485

n 8= -19.0 -20.0 -22.5 - 25.0 -30.0 -35.0 -~0.0 -50.0

0 6.113 6.279 6.666 7.039 7.728 8.342 8.923 9.982

1 6.113 6.279 6.666 7.039 7.728 8.342 8.923 9.982

2 18.112 18.616 ].9.807 20.935 23.018 24.902 26.661 29.864

3 18.112 18.6].6 19.807 20.935 23.018 24.902 26.661 29.864

4 29.754 30.6]8 32.656 34.572 38.092 41.283 44.242 49.621

5 29.754 30.618 32.656 34.572 38.092 41.283 44.242 49.621

6 41.004 42.254 45.192 47.934 52.949 57.477 61.665 69.25

7 4i.004 42.254 45.].92 47.934 52.949 57.477 61.665 69.25

8 51.813 83.485 57.393 61.004 67.575 73.484 78.925 88.75

9 51.813 53".485 57.393 61.004 67.575 73.484 78.925 88.75

i0 62.119 64.259 69.216 73.762 81.959 89.289 96.017 108.12

ii 62.119 64.259 69.2]6 73.762 81.959 89.289 96.017 108.12

12 71.817 74.506 80.633 86.178 96.084 104.888 , 112.941 127.36

13 71.820 74.506 30.633 86.178 96.084 104.888 112.941 127.36

14 80.689 84.101 91.582 98.222 109.943 120.274 129.68 146.5

15 80.773 84.109 91.582 98.222 109.943 120.274 129.68 146.5

~6 ~7.673 92.725 101.990 109.859 123.514 135.441 146.24 165.5

calculated from Eq. (13) are due to the anharmonieity from the z 4 term.

At negative values of B the potential becomes one of the double-minimum type in which the molecules are constrained to a value near Z=Zmin= ::t=: (B/2) i and where the barrier height is B2/4. As the barrier height increases the eigenvalues merge in pairs. First the 0 and 1 levels become near degenerate, then the 2 and

3, etc. At the same time the separation between the 1 and 2, 3 and 4, etc., levels increases. For large negative values of B it is useful to expand the potential function in a Taylor series about the minimum of the potential well z,ni,. If the zero of the potential energy is set at the minimum of the well, our original poten- tial is

V = z 4 + B z 2 + Vo, where V0 = B2/4

APPLIED SPECTROSCOPY 77

and (14) BK<O.

If we expand about Zmi,, where Vmi.=O and define

~Z~Z--Zmin, we have

V = V ( z m i n ) + V"(Zmin)aZ+~V"/ ( Z m i n ) / 2 ] (z~Z) 2

+ i V ' " (zm~,)/6] (Az)3 + iV ' ' ' ' (zm~.)/24] (Az) 4 = - 2 B ( a z ) ~ + ( - 8 B ) ~ ( a z ) ~ + (az) 4. (15)

I t is therefore evident that the eigenvalues for the potential ~Eq. (14)] for large negative B values are also those of a harmonic oscillator perturbed slightly by auharmonicity. Since the well is of the double minimum type, the eigenvMues are doubly degenerate. Ignoring the anharmonicity, the eigenvMues are given by

X~= ( - -2B)~(n+l) , n even, B<<O. (16) Xn = (--2B) n n odd,

If B is finite, the levels will be only near degenerate. I t can be shown ls that the lower limit of the separation AE between the 0 and 1 states is given by

AE = 6B2e -2B2/~, Bf<0, (17)

where e is the separation between the 1 and 2 levels. This approximation is many orders of magnitude too small even at B = - 2 0 and is useful only at much more negative values of B.

The previous approximations for the eigenvMues indicate that the values at B = +100 should be the same as those at B = --50, except for the small effect of anharmonicity and the degeneracy of the latter. This is indeed seen to be the case. The harmonic frequency for each case is 20.0 while the anharmonicity constants, as defined by Herzberg, TM are +0.015 and -0.063, respectively, for B = +100 and -50 . The larger negative anharmonicity constant at B = - 5 0 is due to the cubic term in Eq. (15). At smaller ab- solute values of B these anharmonicities are larger in magnitude since they are inversely proportional to B.

Ill. EFFECT OF h Z 6 TERM

Recently Robinson and Kim 15 have derived the first order energy correction to the quartic-harmonic oscillator resulting from the addition of a hexic term. The eigenvMue correction AE, for state n resulting from the addition of a +Cz 6 term to V=z4+Bz 2 is given by

AE. = ( C / 3 0 ) [ 9 - 8 B X . + (16B2+ 18X~) (OX./OB)]. (18)

In Eq. (18) C is in reduced form but can be trans- formed into dimensioned form (usually cm-I /h 6) by

C= (2#/h2)-ia-tc. (19)

This results in the dimensioned potential

V= ax4+bx~ +cx 6. (20)

Table II lists the corrections expected from a Cz ~ term as cMculated from Eq. (18) and compares these

Table II. Eigenvalue corrections to V : z 4 + Bz ~ resulting from ad- dition of a Cz ~ term.

Calculated Error in corrected vMue b correction a

n f o r C = l C = I O -a C = i O -2 C = I O -I C ~ - - i O - * C = - - I O -2

B =0.0

0 0.492 --0.0001 --0.0002 1 2.215 --0.0002 --0.0010 2 5.585 --0.0002 --0.0017 6 36.76 --0.0006 0.0014

10 96.08 --0.0005 0.0002 16 238.08 --0.0003 0.0929

B =5.0

0 0.117 --0.0001 0.0001 1 0.690 0.0000 0.0O01 2 2.133 O.0000 0.0002 6 20.25 0.0OOO 0.0030

10 61.29 0.0000 0.0149 16 169.37 --0.0O01 0.0633

B = --5.0

0 --3.091 --0.0050 --0.0088 1 +0.693 --0.0037 --0.0313 2 -- 1.522 --0.0033 --0.0106 6 36.11 --0.0257 --0.2144

10 97.66 --0.0517 --0.4198 16 238.28 --0.0997 --0.7561

0.0016 0.0O01 0.0005 0.0138 0.0002 0.0018 0.0535 0.0003 0.0040 0.690 0.0014 0.0225 2.326 0.0024 0.0644 7.080 0.0056 0.1887

O.O000 --0.0001 --0.0001 --O.O001 O.OOOl 0.0000 0.0001 +0.0003 0.0003 0.0170 0.0012 0.0174

0.0051 0.0643 0.0039 0.0523 0.0036 0.0476 0.0268 0.3195 0.0538 0.6621 0.1054 1.3844

CMculated from Eq. (18). b Correct vMue cMculated direct ly by computer.

to the actual corrections as calculated directly by the computer. Calculations are shown for B=O.O, +5.0, and -5 .0 . The approximation given in Eq. (18) is seen to be an excellent one for the first two values of B when C is sufficiently small. The approximation for B = - 5 . 0 , is quite poor. Other calculations also show that for all negative values of B Eq. (18) is not~ very useful, but is quite accurate for all positive B values when C is sufficiently small (less than about 0.1). Even at B = - - 0 . 5 the calculated correction is not very accurate; it becomes increasingly worse as B becomes more negative. Apparently for double minimum potentials the effect of a hexic term is not directly proportional to the value of C, especially for levels below the barrier. This is quite evident in Table II.

The application of Eq. (18) where B is positive, however, makes it possible to use Table I to find the eigenvalues for any potential.

V=z4+Bz2+Cz 6, B>O, C~0.1. (21)

IV. APPLICATIONS

The most important application of the mixed quartic-harmonic potential has been to the ring- puckering vibration. Table I I I lists the molecules for which accurate potential functions have been derived from far-infrared spectra. In each case the fit between observed and calculated spectra is remarkable, especially for a two parameter potential. For instance, for silacyclobutane 2° 33 frequencies are calculated, all of which are in excellent agreement with the recorded far-infrared absorption bands. Quite a range of ring- puckering potentials have been found. 2,5-dihydro- thiophene ~1 has considerable harmonic character for this vibration, for example, silacyclopent-3-ene 22 is

78 Volume 24, Number 1, 1970

Table III . Molecules with the ring-puckering potential function V = A(z 4 +Bz 2) cm -~.

Molecule A (cm -1) B Reference

( ~ S 16.86 21 +5.93

/O 24,74 +2.91 17

\ , \ S Sills 31.45 +2.40 a %/

~ ? S i D 2 12.99 +0.352 22

(~ ;S i , , 14. 23 ~0.163 22

~/O--d, 21.83 --1.674 16

~/~ 28.3 -- 3.44 C

• iH2 20.23 -- 9.333 20

~)SiD, 18.64 --9.723 20

,~ ,I. Laane, J. Chem. Phys . (to be published). b T . R. Borgers and H. L. Strauss, J . Chem. Phys . 45, 947 (1966). ¢ W. H. Green, J. Chem. Phys . 50, 1619 (1969).

nearly pure quartic, and si lacyclobutane has a rela- t ively large barr ier to inversion of 440 cm -1 in an otherwise quart ic well. The value of B in all these molecules has been found between + 6 and - 1 0 and all of these potentials fall well within the range of Table I. The value of A has also had a relatively small range between 12 and 32 cm -1. The da ta of Table I I I in combinat ion with the tabula ted values of Table I make it possible to calculate quite accurately the observed far-infrared spect rum arising from the ring- puckering vibrat ion in each of these molecules.

The type of potent ial function under discussion has also been used by microwave spectroscopists to calculate changes in rotat ional constants for t h e

excited s ta tes as well as to calculate v ibra t ional levels. Methylene cyclobutane 23, cyclopentene, 24 t r imethylene oxide, 16 t r imethylene sulfide, 25 and cyclobutanone 26 have been studied in this manner .

When the magni tude of B, ei ther posit ive or nega- t i ve , becomes large, the potent ia l becomes a good representat ion of a harmonic oscillator with small anharmonici ty . Wi th smaller posit ive values of the harmonic coefficient, the potent ia l seems to be char- acteristic of ve ry few systems other t han the ring- puckering vibration. For negat ive values of B, how- ever, where the potent ia l well has a double minimum, the potent ia l funct ion is applicable to several o ther systems. Hydrogen bonding sys tems with symmet r ic potentials can be well represented, for instance. ~a,27 The same form of the potent ia l has also been applied to the inversion of ni trogen in ethylenimine, is

The potent ial function V = A ( z 4 + B z 2) for B < 0 can be used as an approximat ion to pract ical ly any double-minimum problem even though the best form of the potent ia l m a y be of a different type. The inversion in ammonia and ND~, for instance, m a y be represented b y the quar t i c -harmonic oscillator as shown in Table IV. Only slightly be t te r f requency agreement is obta ined using a harmonic potent ia l with a Gaussian barr ier 2s

V = ½XQ2-t--A e x p - a2Q 2 (22)

and the barr ier heights f rom the two calculations differ by less than 3°7o. In general, i t is seen tha t the three pa ramete r potent ial in Eq. (22), for which extensive tables have been repor ted by Coon and co- workers, 2s cart often be represented quite closely by the quart ic potent ia l with the quadrat ic barrier, especially when the barr ier is sufficiently high as in ammonia .

Another example of a s i tuat ion where the tables of this paper can be used to approx imate a potent ial is tha t of hydrogen peroxide and of D202. H u n t and co2workers 29,3° have studied the internal ro ta t ion of these molecules and have used a three pa rame te r potent ia l :

V(z) = V1 coszq- V2 cos2z-t- V3 cos3z. (23)

Such a potent ial has a period of 2rr and V(z) does not up-

Table IV. Calculated inversion levels for ammonia using V ( N H 3 ) = 142.2(z ~ - - 7.45z ~) cm -1 and V(ND~) = 98.3(z 4 - - 8.90z 2) cm -~.

NH3 levels (cm -1) ND3 levels (cm -1) Level Observed ~ Calculated Observed ~ Calculated

0 0.00 0.00 0.00 0.00 1 0.79 0.63 0.053 0.036 2 932.5 932.6 745.7 754.8 3 968.3 968.6 749.4 757.7 4 1597.6 1569.1 1359 1359 5 1882.2 1869.4 1429 1430 6 2383.5 2367.2 1830 1810 7 2895.5 2894.9 2106.6 2095.4 8 3442 3480 2482 2468 9 • • • 4108 2876 2869

Barrier 2031 1973 1982 1951

See Ref. 30.

APPLIED SPECTROSCOPY 7 9

/

Table V. Calculated internal rotation levels for hydrogen peroxide using V(H~O~) -~ 62.0(z 4 - -5 .0z 2) cm -~ and V(D202) ~ 40.5(z 4 - - 6.1z 2) cm -1.

H~02 levels (cm -1) D202 levels (cm -1) Level Observed b Calculated Observed¢ Calculated

0 0.00 0.00 0.00 0.00 1 11.43 9.34 1.88 1.50 2 254.2 250.5 208.6 208.2 3 370.7 366.7 250.9 249.7 4 569.3 568.6 387.7 384.9 5 775.9 788.6 511.2 512.5

Barrier a 386 387 377 377

a Ba r r i e r a t the cis conf igurat ion; the ba r r i e r a t the t rans configurat ion is 2460 cm- t for H20~ and 2470 cm -l for D~O2 bu t is assumed inf ini tely la rge for the ca lcula t ion.

b See Ref. 31. e See Ref. 32.

proaeh infinity at large values of z although the barrier for the trans configuration (z = ~r) is much smaller than that for the cis structure (z=0, 2~-). In order to ap- proximate the potential [Eq. (23)] by one in the form of Eq. (4), it is necessary to assume that the barrier for the cis configuration is infinitely high whereas it is actually 2460 cm -~ in H202. Table V shows that this approximation does not have a serious effect on the calculation of the first five excited energy levels for the two isotopic hydrogen peroxide molecules. Furthermore, the quartic potential with a quadratic barrier estimates a barrier height for the trans configuration that is in excellent agreement with that calculated from Eq. (23).

5~Iany other double-minimum potentials can be approximated ia this manner including those reported for the excited electronic states of H2CO. 2s The advantage of the approximations is that the assign- ment of the transitions and the estimation of the barrier height are very easily made before a more complicated potential such as that in Eq. (2) or Eq. (23) is used.

V. SUMMARY

The tables reported in this work make it possible to calculate very quickly and accurately the far- infrared frequencies of the transitions of the ring- puckering potential. Even when eigenfunctions are required to determine the intensities or other molecular properties, these tables facilitate the calculation by making it necessary to carry out only one computer calculation for the actual potential. The tables can also be used for studying the potential functions,

especially of the double minimum type of other types of molecules. They also make it possible to determine the effect of a hexic term in the potential V=A(z4+Bz2+Cz6) for B>_0.

ACKNOWLEDGMENT

The computations were carried out on an IBM 360/65 computer at the Texas A&M Data Processing Center.

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8 0 Volume 24, Number 1, 1970