REDUCED POTENTIAL ENERGY FUNCTION FOR DIATOMIC MOLECULES
April 20, 1954
Re, 73.4; F, 7.51. Found: Re, 74.2, 74.0; F, 7.61, 7.60; atomic ratio Re:F, 1.00. Boiling and Melting Points.-ReOaF, sublimed a second time was refluxed in a glass tube, heated in a paraffin bath open to the atmosphere through a Mg(C104)Z drying tube. I t boiled a t 164’ a t 760 mm. Slow decomposition occurred
[CONTRIBUTION FROM THE
2045
during the measurement, as indicated by the formation of a dark residue. The melting point determined in sealed capillary tubes was 147”.
PHILADELPHIA 40, PENNA.
DEPARTMENT OF CHEMISTRY, NORTHWESTERN UNIVERSITY]
The Possible Existence of a Reduced Potential Energy Function for Diatomic Molecules1 BY ARTHURA. FROST AND BORISMUSULIN RECEIVED AUGUST14, 1953 A reduced potential energy is defined as V/De where Vis the potential energy, taking the zero for infinite separation of the nuclei. A reduced internuclear distance is defined as ( R Rij)/(Re Rij) where R is the actual internuclear distance, Re the equilibrium distance, and Rij a constant for a given molecule formed from atoms i and j . It is shown that there exists an approximate universal relation, in the neighborhood of the minimum, between these variables for the ground states 0f.a set of 23 diatomic molecules, chosen only for the accuracy of their experimental data. Rij is interpreted as a measure of inner shell radii and is closely correlated with Badger’s dij.
-
-
I t is of interest to consider the possibility of a the theoretical calculations of Burrau4 and of reduced potential energy function of diatomic Teller.6 molecules, i . e . , a relation between a “reduced” potential energy and a “reduced” internuclear distance, analogous t o a reduced equation of state. Let V be the potential energy of a diatomic R, at. un. molecule in the ground state or in any attractive 0 excited state taking the zero of energy a t infinite separation of the nuclei. Let R be the internuclear 7; distance, Re being the equilibrium value. At the .+ S potential energy minimum V = -De where De I is the dissociation energy (including the half quan- .-u S tum of vibration). Now the simplest kind of reduced potential energy and distance would be -0.1 defined as 5
\
4-
2
R‘ = R / R e
V’ = V/D,
(1)
in terms of which variables the P.E. minimum would be a t V’ = - 1and R’ = 1 for every molecule. -4 reduced P.E. in this form V/De has been used by Puppi2a while DaviesZbhas used the reduced distance R/Re. Puppi actually derived a reduced functional relation-his reduced distance variable being y=-
R R
- Re ~
~
D
This was obtained by assuming a Morse P.E. function and introducing an empirical relation equivalent to kRe2 = a constant, where k is the force constant. Puppi’s reduced equation is not universal as kRe2 remains constant only within small groups of molecules. Furthermore his variable y is complicated in that i t mixes distance and energy. The value of reduced variables as defined in (1) is shown in Figs. 1 and 2 where in Fig. 1, V is plotted vemus R and in Fig. 2 , V’ v e r m s R’ for the ground states of Hz and H2+. Atomic units are used for Fig. 1. The variables of Fig. 2 are by definition dimensionless. The H Zcurves are plotted according to the modified Morse function of Hulburt and Hirschfelder3while the H2+ results are (1) A portion of this paper was presented at the Los Angeles Meeting of the American Chemical Society, March, 1953. (2) (a) G. Puppi, Nuovo Cimcnlo, 3, 338 (1946); (b) M . Davies, J . Chcm. Phys., 11, 374 (1949). (3) H. M . Hulburt and J. 0. Hirschfelder, ibid., 9, 61 (1941).
-0.2 Fig. 1.-Potential energy curves for the ground states of Hz and Hz+; 1 atomic unit of energy = 27.2 e.v., 1 atomic unit of length = 0.529 A.
A More General Reduced Internuclear Distance.-If P.E. functions for other than hydrogen molecules are plotted in Fig. 2 there is no general coincidence of curves. In particular, the curvature a t the minimum, which is related to the force constant, may vary considerably from molecule to molecule. This is not surprising inasmuch as an inner shell repulsion for all atoms other than hydrogen should influence the internuclear distance a t the minimum. To take this into account the detinitions of equation 1 will be replaced by V’ = V/De R‘ = ( R
- Rij)/(Re- Rij)
(2)
where R i j is a constant for a given molecule and is a measure of inner shell radii of atoms i and j. Here, as before, the minimum is given by V’ = - 1 and R‘ = 1. R’ of (2) is identical with R’ of (1) when R i , = 0 as i t presumably is for HO and H2+, there(4) 0. Burrau, K g l . Danskc Vidcnskab. Sclskab., 7 , 1 (1927). (5) E.Teller, 2. Physik, 61, 458 (1930).
ARTHURA. FROST At~~ BORISMUSCLIK
2046
fore, Fig. 2 gives the functional relation for L” and R‘ of ( 2 ) as well as for (1).
Vol. 76
quantities are considered to be accurately known. Inspection of .the table shows that R,, in general increases with periodic series classification as expected. Also included in the table are values of Badger’s d,, for comparison.* hlthough his rule k,(R, - d 1 1 ) 3= constant
is purely empirical the similarity to our equation 3 leads one to expect a similarity in R,,and dlj. This similarity is demonstrated in Fig. 3 where it appears that to at least a crude approximation Rlj equals dl,. TABLE I DIATOMIC L f O L E C U L E L).4TA” Periodic series
0-0 Fig. 2.-Reduced
potential energy plot for ground states of Hr and H2+.
For other molecules than H2and H2+the reduced P.E. curves cannot be plotted without knowing the R i j values. I t is apparent that an R i j value could always be found which would make the curvature of the V‘(R’) curve at the minimum the same as for H2. Instead of plotting such curves we shall assume that a universal function exists in the neighborhood of the minimum and derive from this a relation by which Ri, can be calculated from experimental results. If the so calculated R i j values are in reasonable agreement with the idea of inner shell interaction this will constitute evidence in favor of the existence of a universal reduced P.E. function. Derived Relation Involving the Force Constant.-Assume 1’ is a universal function of R’ (of equation a), independent of which molecule is being considered. At the minimum
(g2)z,-l K, =
a dimensionless constant
Since the force constant for infinitesimal amplitude ke = (d2V/dR2)R = R ~ it, follows6that k e ( R e - Rij)’/De
or that Rlj
=
Re
K
- (KDJk,)’’?
(3)
(4)
-
To evaluate K let us first assume that for H2 and H2+R i j 0. Then K may be calculated as k e R e 2 / De. The result is K = 4.14 for H2and 3.96 for H2+, the similarity in values confirming the choice of R i j . Now taking K = 4.00 as a rounded average value, R i j may be calculated for each molecule in any state. In Table I are given experimental data of h e , D e and Re for a number of molecules in their ground states and calculated values of Ri, from 4. The data are from Herzberg’s tables7 and include all molecules where values of all three experimental ( 6 ) A referee of this paper has kindly called t h e authors’ attention t o the work of G . B. B. M. Sutherland, f. Chcm. Phys., 8 , 161 (19401, who has previously obtained a relation equivalent to (3). Although Sutherland assumed a special form of the potential energy function (double reciprocal type) his result m a y be considered as adding weight t o t h e present more general argument. ( 7 ) G. Herzberg, “Spectra of Diatomic Molecules,” 2nd Edition, D. Van Xostrand Co., Inc.. h-ew York, A-. Y.,1950.
0-1 0-2
k, .\Iolecule
Hz’ H2 CH OH HCl
tm+
(I-N KI-I ZIIH HBr 0-4 CdH HI 0 4 HgH 1-1 Liz 02
O.+
1-2 2-2
ClF x:t2
x
ACD CALCULATED RESULTS
10 -3,
dynes/ cm.
1.575 5.73 4:G 7.78 -1.602 X IO-!’. Pi Clz
X289
~1>
Additional Evidence from Anharm0nicities.The comparison of the previous section involves only the curvature of the minimum of the potential energy function. This is equivalent to a check up through the second degree term in a Taylor’s series expansion of 17 as a function of R about the point R = Re or of T” as a function of R‘ about the point R’ = 1. X more complete test of the idea of a universal potential energy function is to test for constancy of the coefficients of higher powers such as L/6 and W 2 1 of the third and fourth degree terms of the latter expansion. v = -1 + ~,?K(R’ - 1 1 2 + ~ I , L ( R ‘- 113 , , . l/?&V(R’- 1 ) 4 (5) where
+ +
I t is well known that the third derivative of a ( 8 ) 12 11,Badger, J . C h c m P i i y s . , 2 , 1 2 8 ( l ! l : j 4 ) , 3 , 710 (1035)
REDUCED POTENTIAL ENERGY FUNCTION FOR DIATOMIC MOLECULES
April 20, 1954
TABLE I1 DIATOMIC MOLECULE ANHARMOXICITY DATAAND -
(e) dR3 Re
uex*,
,=e>
Molecule
cm.-'
H2 H2 CH OH HCl HCI+ KH ZnH HBr CdH HI HgH Liz
1.4 2.993 0.534 ,714 .3019 ,3183 ,0673 ,2500 ,226 ,218 .183 ,312 ,00704 .0 1579 ,01984 ,00436 ,00079 ,00142 ,0017 ,000219 ,000275 ,000536 .000117
+
0 2
0 2
+
ClF Naz
PP Clr Kz Brz IC1 12
cm
-1
62 I.17.995 64.3 82.81 52.05 53.5 14,65 55.14 45.21 46.3 39.73 83.01 2.592 12.073 16.53 4.0" 0.726 2.804 4.0 0.354 1.07 1.465 0.6127
x lo-'% ergs cm. - 3 0.712 3.70 2.67 5 44 2.85 2 30 0 147 0.713 2.09 0.566 1.558 0.656 0.0548 8.76 14.30 2.66 0.0314 2.652 1.838 0.0156 1.058 1.123 0 775
CALCULATED RESULTSa
(:::)Re
x
Rij
10-33,
ergs cm
-4
0 2933
2.34 1.363 3.36 1.357 1.092 0 0253 ,1648 ,884 ,1778 ,684 ,1790 0.00854 5.41 10.01 1 506 0.00275 ,954 ,607 000887 2533 121
,2616
-L
19.39 18.76 17.18 17.11 16.61 18.92 15.49 12 06 15.71 11.82 16.11 11.63 13.96
15. 70 17.25 14.26 12.13 11.69 15.42 11.61 12.45 14.38 13.70
2047
M
89.06 86.82 63.68 6 d . 34 60.18 76.77 39.74 17.85 51.92 23.68 57.63 16.21 35,46 51.71 61.17 49.6 17.73 10.74 35. 75 12.17 21.68 41.44 35.40
x
108
from L , cm.
-0.005 +O.Ol .3P ,33 .49 .46 .66 .85 .58 1.02 0.75 1.14 0.87 .64 .60 .96 1.15 1.06 1.24 1.76 1.46 1.48 1.82
R,,X lo8 from M , cm.
-0,005 +o 01 .34 .31 . .44 .43 .43 .65 .52 .88
.70 .97 .64
.m .57 .93 .60 .96 1.12 .91 1.26 1.40 1.71
1: K M ) ' / Z / L
0.972 ,992 ,928 .944 ,931 ,926 .X14 .702 .912 .819 ,943 ,689 .853 .912 ,904 .987 .693 .870 .773 .599
L4/3/M
0.585 ,575 ,695 ,675 ,705 ,657 ,972 1.550 0.758
1.140
.750
0.706 1.627 0.948 ,760 .729 ,698 1.572 0.883 1.073 2.165 1.333
.895 .868
0.927
0.845
15.06 43.48 0.855 0.982 Av . 42.0 13.2 10.3 31.8 % dev. Data for ae atid w.xC from Herzbcrg, rcf. 7. we.v, for ClF 1 akeri as 4.0 since vdlue of 9.9 yields negative fourth derivative.
diatomic molecule potential energy function is related to the spectroscopic constant me and the fourth derivative to both me and to wexeas f01lows.~ If Vis expressed as the power series V = -De ~ o [ ~ [ l ai.$ a?.$' , . . . . . . . Ihc (6) with
+
+
+
+
then al = -1
-
a,w,/6Be2
(7)
and where ue and Be are the usual spectroscopic cons t a n k 6 Also by differentiating (6)
and by comparing with ( 5 ) and ( 2 )
Table I1 lists for the same molecules as in Table I experimental values of ae and wexeand calculated results for L and M using the Rl, values of Table I . The mean deviation of L and Jf from their averages, 13.2 and 42.OyG,respectively, agree with the (9) bee ref 3 and also J L D u n h a m , P k y s K e n , 41, 721 (1932)
existence of an approximate universal relation. Although these deviations are too large to warrant quantitative predictions there is significance to this result as shown by the fact that the experimental quantities k , De, Re,me and wexe individually vary through anywhere from a fivefold range to a thousand-fold range as seen in Tables I and 11. Other columns of Table I1 show further tests of universality. R,,'s may be calculated from equations 11 and 12 assuming constant values of L and M . As done in the calculation for Table I it is only natural to choose L and M such that R,j for Hz and H2+ are near zero. L is taken as -19.0 and M as 88.0 (both dimensionless). The resulting R,, again follow more or less the periodic series classification. The R,, from L indeed show a closer fit to the line of Fig. 3 than those plotted there. Additional tests are obtained by calculating various combinations such as K 2 / M , Ka/2/L, L4/3/M,and (KM)'/s/L. I n all of these the Re R,,cancels out so that no assumption has to be made concerning the R,,. Also for the last quantity ( K M ) ' / z / Lthe De is cancelled so that the quantity depends only upon k , a e and uexe. It is seen from the table that ( K M ) ' / z / Lhas an average deviation of 10.3y0and L4la/Mof 31.80j0.
Discussion I t could hardly be expected that a universal reduced potential energy function would exist with any precision for all molecules and all states. There are too many known complexities such as differing natures of electronic spectroscopic states,
2048
ARTHURA. FROST AND BORISMUSULIN
Vol. 76
-0.6
k 2.0 0.8
1.0
1.2
1.4
R'. Fig. 4.--Plot of fourth degree reduced potential etiergy function : pull curve calculated with averaged cotistanti; dashed curves for extreme cases of H2arid Kz. 1.o
certain striking exceptions. For example, calculation of K for the B , C and E states of Hz assuming Fig. 3.-Plot of R,, versus d l j ; distance in atomic units, 1 R i j = 0, yields 1.59, 4.64 and 5.01, respectively. The B state value differs most strikingly from the atomic unit of length = 0.529 previous value of 4.00. This perhaps is due to the or especially the existence of ionic states where the fact that the B state is ionic. It may be noticed that very few multibonded potential energy a t large distances would behave quite differently. Also perturbations between molecules are included in the list owing to unstates may interfere. Nevertheless, it is evident certain De data. It is possible that the relations that for a large group of states there is a t least an described in this paper might aid in choosing the approximate universal relation, a t least in the appropriate value for a given molecule. Consider the cases of NSand NO, For Nzthe two values of neighborhood of the minimum. It is a t first disturbing that M , the fourth deriva- De of 7.519 and 9.902 e.v. yield owwith equation 4 tive in reduced variables, shows such a wide varia- R i j values of 0.636 and 0.568 A., respectively. tion. However, i t should be realized that this For N O the D, values of 5;idll and 6.61 e.v. yield R i j quantity is rather sensitive to the shape of the values of 0.685 and 0.635 A., respectively. In both potential energy function and that the contribution cases the lower De value produces best agreement of the corresponding fourth degree term in the with the Badger dij value of 0.680 A. However, expression for V' is relatively small. To show this cannot be taken as strong evidence for the that M can vary considerably without greatly lower values of dissociation energies of these affecting the total potential energy there is plotted molecules as opposed to the currently favored in Fig. 4 V' versus R' as calculated from the fourth higher values'O inasmuch as other 1-1 series moledegree equation 5 using K = 4.0 and L and 31 cules in Table I show a wide variation of R i j values. as listed in Table 11. The curves include the case Acknowledgments.-Thank5 are due to K. L. using average values shown as a full line together Burwell, Jr., for suggestions and to Donald Cook for with the most extreme cases of Table 11, Hz and K 2 , aid in calculation. This work was supported by B shown as dashed lines. The close coincidence of research grant from the National Science Foundathe curves is a visual indication of just how good tion. is the concept of a universal reduced potential EVANSTON, ILLINOIS energy function for the set of molecules considered (IO) See review b y G. Glnckler i n "Annual Revielr of T'hysical here. Chemistry," Vol. 3, Annual Revien~s.Inc., Stanford. California, 1052. The treatment has been extended to some ex- p. 158. Also M. H. J . Wijnen a n d 11. A . Taylor, J. Chem. P h y s . . 2 1 , cited states with the same general results with 233 (1953). 0.0
2.0
3.0
du.
a.
4.0
