Potential energy curves for the X(sup 1)sigma+ and V(sup 1)sigma+ states of HF and DF have been calculated by the Rydberg-Klein-Rees method. The results calculated from the different sets of data for HF and DF are found to be in very good agreement. The theoretical results of Karo are compared to the experimental results obtained here

Fallon, Robert J.

Mason, Edward A.

Vanderslice, Joseph T.

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Reprinted from the JOURNAL OF CHEMICAL PHYSICS, Vol. 32, No.3, 698-700, March, 1960 
Printed in U. S. A. 
Potential Energy Curves of Hydroge·n Fluoride* 
ROBERT J. FALLON, JOSEPH T. VANDERSLICE, AND EDWARD A. MASON 
Institutefor Molecular Physics, University of Maryland, College Park, Maryland 
(Received November 2, 1959) 
Potential energy curves for the X I~+ and V 11;+ .states of !-IF and DF have been calculated by the Ryd-
berg-Klein-Rees method. The results calculated from the different sets of data for HF and DF are found 
to be in very good agreement. The theoretical results of Karo are compared to the experimental results 
obtained here. 
INTRODUCTION 
QUANTUM mechanical calculations on molecular systems are known to give reliable estimates of the 
total energy of such systems; but, in general, the values 
calculated for the binding energies of diatomic mole-
cules are in poor agreement with experiment. The 
reason for this is that the error in the total energy, 
although small on a percentage basis, is often of the 
same order of magnitude as the binding energy. In 
relatively simple cases, such as H 2, the problem can be 
handled by brute force methods. l For more complicated 
systems containing a large number of electrons, such 
techniques are as yet out of the question. It would seem 
that approximation schemes are the most desirable 
at the present time if one wants to be able to calculate 
accurate potential energy curves for diatomic mole-
cules. 
In order to test the reliability of such approximation 
schemes, it is well to have reliable potential energy 
curves for diatomic syst.ems against which the various 
schemes can be tested. Since a relatively large amount 
of effort has been expanded by theoreticians on the 
hydrogen fluoride molecule,2-8 it was felt that it would 
be worthwhile to have an accurate potential curve for 
this system. 
Recently, Johns and Barrow9 have obtained some 
excellent spectroscopic data on HF and DF, and these, 
together with older data,1O-12 have enabled us to obtain 
* This research was supported in part by the National Aero-
nautics and Space Administration. 
1 W. Kolos and C. C. J. Roothaan, paper presented at the Inter-
. national Conference on Molecular Quantum Mechanics, Boulder, 
Colorado, June 7-15, 1959 (to be published in Revs. Modern 
Phys.). . 
2 A. B. F. Duncan, J. Am. Chern. Soc. 77, 2107 (1955). 
3 D. Kastler, J. chim. phys. 50, 556 (1953). 
4 A. C. Hurley, Proc. Roy. Soc. (London) A24S, 119 (1958). 
6 M. Krauss, J. Chern. Phys. 2S, 1021 (1958). 
6 A. M. Karo and L. C. Allen, J. Am. Chern. Soc. SO, 4496 
(1958). 
7 A. M. Karo, Quarterly . Progress Report,. Solid State and 
Molecular Theory Group, M. I. T., October 15,1957, p. 16; A. M. 
Karo and L. C. Allen, J. Chern. Phys. 31, 968 (1959). 
8 R. A. Ballinger, Mol. Phys. 2, 139 (1959). 
9 J. W. C. Johns and R. F. Barrow, Proc. Roy. Soc. (London) 
A251, 504 (1959). 
10 Talley, Kaylor, and Nielsen, Phys. Rev. 77, 529 (1950). 
11 S. M. Naude and H. Verleger, Proc. Phys. Soc. (London) 
experimental potential energy curves for the Xl ~+ 
and VI ~+ states of these molecules over a range of 
internuclear distance. The actual curves were calculated 
by the Rydberg-Klein-Rees (RKR) method.l3- 16 
The experimental curve for the Xl ~+ state was com-
pared with a Morse curve and with one obtained 
theoretically by Karo.7 
RYDBERG-KLEIN-REES METHOD AND 
SPECTROSCOPIC DATA 
The RKR method is a semiclassical procedure for 
obtaining the potential energy curves for diatomic 
molecules. The curves obtained with this method are 
based directly on the measured vibrational and rota-
tional energy levels of the molecule and do not depend 
on the assumption of any erripirical potential form. The 
method has been adequately described in recent 
articles1S- 16 so that we shall omit the details of the 
procedure. It suffices to say that the method is quite 
rapid and appears to give reliable results. 
The spectroscopic data for the Xl ~+ states of HF 
and DF do not include all the vibrational levels from 
the lowest to the highest. Johns and Barrow9 have 
obtained data for the higher levels, whereas Nielsen 
and co-workers/o.12 and Naude and Verlegerll have 
obtained data on the lower. Johns and Barrow9 have 
carried out what appears to be a reliable interpolation 
between the upper and the lower levels so that the inter-
mediate levels could be inferred from the standard 
spectroscopic formulas. We have used the observed 
data plus results obtained by the interpolation method 
to calculate experimental curves for the Xl ~+ state 
of both systems. 
RESULTS AND DISCUSSION 
The results for the Xl ~+ and VI ~+ states of HF and 
DF are listed in Tables I to IV and are shown as solid 
lines in Fig. 1. The corresponding curves for the two 
systems do not differ by more than ±O.OOl A. This 
indicates that the different sets of data and the inter-
13 R. Rydberg, Z. Physik 73, 376 (1931); 80, 514 (1933). 
14 O. Klein, Z. Physik 76, 226 (1932). 
16 A. L. G. Rees, Proc. Phys. Soc. (London) 59, 998 (1947). 
A63, 470 (1950). 
12 Kuipers, Smith, and Nielsen 
16 Vanderslice, Mason, Maisch, and Lippincott, J. Mol. Spec-
troscopy 3, 17 (1959); Vanderslice, Mason, and Lippincott, J. 
Chern. Phys. 30, 129 (1959); Vanderslice, Mason, and Maisch, 
J. Chern. Phys. 25,275 (1956). ibid. 31, 738 (1959). 
698 
699 POT E N T I ALE N ERG Y CUR V E S 0 F H Y D R 0 G E NFL U 0 RID E 
TABLE I. Potential energy of the X I~+ state of HF." 
v 
o 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
2 047 
6 008 
9 798 
13 419 
16 878 
(20 193) 
(23 351) 
(26 350) 
(29 191) 
(31 875) 
34 335 
36 705 
38 921 
40973 
42 850 
44 543 
46 031 
47 292 
48294 
V (ev) 
0.2537 
0.7448 
1. 215 
1.663 
2.092 
(2.503) 
(2.895) 
(3.266) 
(3.619) 
(3.951) 
4.256 
4.550 
4.825 
5.079 
5.312 
5.522 
5.706 
5.862 
5.987 
'm81 (A) 
1.023 
1.115 
1.187 
1.253 
1.317 
1.379 
1.441 
1.503 
1.567 
1.633 
1.701 
1.772 
1.847 
1.928 
2.018 
2.121 
2.244 
2.397 
2.604 
a Nwnbers in parentheses are interpolated values. 
'min (A) 
0.836 
0.786 
0.755 
0.733 
0.716 
0.701 
0.689 
0.679 
0.670 
0.662 
0.656 
0.650 
0.644 
0.639 
0.633 
0.630 
0.626 
0.623 
0.621 
TABLE II. Potential energy of the X I~+ state of DF.· 
v 
o 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
1 488 
4 395 
7211 
(9 947) 
(12 600) 
(15 169) 
(17 655) 
(20 058) 
(22 377) 
(24 613) 
(26 766) 
(28 834) 
(30 820) 
(32 722) 
(34 541) 
(36 276) 
37 815 
39385 
40 866 
42 257 
43 552 
44 746 
45 832 
46 799 
V (ev) 
0.1845 
0.5448 
0.8938 
(1.233) 
(1.562) 
(1.880) 
(2.189) 
(2.486) 
(2.774) 
(3.051 ) 
(3.318) 
(3.574) 
(3.820) 
(4.056) 
(4.282) 
(4.497) 
4.688 
4.882 
5.066 
5.238 
5.399 
5.547 
5.681 
5.801 
r m81 (A) 
1.005 
1.079 
1.137 
1.188 
1.236 
1.283 
1.328 
1.373 
1.418 
1.462 
1.508 
1.554 
1.601 
1.649 
1.699 
1.750 
1.803 
1.858 
1.918 
1.982 
2.051 
2.129 
2.216 
2.315 
• Numbers in parentheses are interpolated values. 
rmin (A) 
0.847 
0.802 
0.774 
0.753 
0.737 
0.723 
0.711 
0.701 
0.692 
0.684 
0.677 
0.670 
0.664 
0.659 
0.654 
0.650 
0.646 
0.642 
0.638 
0.635 
0.632 
0.630 
0.627 
0.624 
TABLE III. Potential energy of the V I~+ state of HF. 
o 
1 
2 
3 
4 
5 
574.8 
1699 
2788 
3844 
4868 
5862 
0.0713 
0.2106 
0.3456 
0.4764 
0.6035 
0.7266 
10.5813 
10.7206 
10.8556 
10.9864 
11.1135 
11. 2366 
2.275 
2.426 
2.540 
2.641 
2.733 
2.822 
1. 924 
1.811 
1.735 
1.675 
1.625 
1.580 
" T, is the energy difference between the bottom of the potential curve in 
question and the bottom of the Curve for the X l~+ state. The value of T, 
used is 10.5100 ev. The value appears to be uncertain to ±0.OO5 ev. 
TABLE IV. Potential energy of the V I~+ state of DF. 
v V (em-I) V (ev) T,+V(ev) Tm .. (A) Tmin(A) 
0 417.4 0.0517 10.5617 2.244 1.946 
1 1239 0.1536 10.6636 2.369 1.848 
2 2042 0.2532 10.7632 2.462 1.782 
3 2828 0.3506 10.8606 2.542 1.730 
polati6ns are consistent and also that the RKR method 
is capable of this precision. 
The heavy dashed line in Fig. 1 represents a Morse 
curve with constants derived from the spectroscopic 
data. The agreement with the experimental curve is 
surprisingly good in view of the fact that the Morse 
curve is often a rather poor representation of the actual 
curves for the cases where it has been compared with 
the experimental curves.16 
The light dashed line in Fig. 1 is the curve obtained by 
Karo for 'the Xl ~+ state using an LCAO-MO method 
which included a large amount of configuration interac-
tion. Although the calcula'lions of Karo are quite 
. extensive, it can not be said that the results are in too 
good agreement with the actual experimental curve. 
Karo, of course, was aware of this in his original work.7 
In view of the difficulties inherent in an ab initio 
calculation on many-electron systems, various approxi-
mation schemes have been suggested in the past 17 
10 
8 
6 
::; . 
OJ ;;: .. / 
• . / 
4 \ XI::;E+ ,,' , , 
, / 
2 
o~~~~~~~--~~~~~~~ 
.6 1.4 1.8 2.2 2.6 
R(A) 
FIG. 1. Potential energy curves for HF and DF. The solid lines 
are curves derived directly from the spectroscopic data by the 
RKR method. The heavy dashed line represents a Morse curve 
and the light dashed line corresponds to the potential curve ob-
tained theoretically by Karo. 
17 Eyring, WaIter, and Kimball, Quantum Chemistry (John 
Wiley & Sons, Inc., New York, 1944). 
FALLON, VANDERSLICE, AND MASON 700 
for the calculation of interaction energies and un-
doubtedly others will be put forward in the future.I8 
With a knowledge of the experimental curves for 
18 H. Preuss, Technical Note No. 24 from the Quantum Chem-
istry Group, UppsaJa University, Uppsala, Sweden, April 20, 
1959. . . 
hydrogep fluoride, as well as for other systems/6 •J6 it 
is now possible to check the adequacy of approximation 
schemes which may be proposed. 
ACKNOWLEDGMENT 
The authors are grateful to Louis T. Ho for checking 
some of the calculations. 


Potential Energy Curves of Hydrogen Fluoride

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