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Ground state of hydrogen by the Rydberg-Kelein-Rees method
JOURNAL
OF
Ground
MOLECULAR
State
SPECTROSCOPY
of
3, 17-29 (1253)
Hydrogen by the Method *
Rydberg-Klein-Rees
J. T. VANDERSLICE,E. A. MASON, AND W. G. MAISCH Institute for Molecular
Physics,
University
of Maryland,
College Park,
Maryland
AND E. R. LIPPINCOW Department
of Chemistry,
University
of Maryland,
College Park,
Maryland
A detailed investigation is made of the Rees modification of the RydbergKlein graphical method of calculating potential energy curves without the use of empirical functions (such as Morse functions). The XZ, ground state of HP is chosen for investigation because of the large amount of both experimental and theoretical information available. It is found that the Rees procedure yields an accurate potential curve very rapidly, and hence the use of empirical functions is questionable when data are available and accurate curves are desired. Our calculations for HZ join smoothly with the accurate quantum mechanical calculations of Dalgarno and Lynn, valid at large internuclear separations. The resulting “most likely” potential energy curve agrees between 0.5A and 2.OA with a similar curve given previously by Hirschfelder and Linnett, but at smaller and larger distances differs appreciably from the Hirschfelder-Linnett curve. INTRODUCTION
Interatomic potential energy curves that are accurate over a large range of distance are desirable in a variety of problems involving atomic and ionic collision phenomena. One method of obtaining these potentials is to assume the validity of some empirical curve such as given by the Morse (I), Hulburt-Hirschfelder (2), or Lippincott (3) functions, and obtain the parameters for the curve from spectroscopic data. Although it is well known that substantial errors may be introduced by this procedure (4), it has been the one usually employed to obtain such potential curves (5). * This research tration.
was supported
in part by the National 17
Aeronautics
and Space Adminis-
18
VANDERSLICE,
MASON,
LIPPINCOTT,
AND
MAISCH
It is one of the purposes of this paper to emphasize t,hat spectroscopic data can yield accurate potential curves for bound states and that methods have been outlined which make the procedure quite rapid. In particular, Klein (6) and Rydberg (7) have developed a method for calculating potential curves which makes use of the experimental energy levels themselves and does not depend on some derived formula for these levels. This method has been used with success in a number of cases (8), but great care must be taken with the graphical integrations involved if reliable results are to be obtained (4). This difficulty has been eliminated by Rees (9), who has given an analytical modification of the method and applied it to the &I~+ state of Brz . As modified by Rees, the method is capable of giving interatomic potential curves rapidly once t’he spectroscopic data are available. In view of the potential importance of the Rees method, it seemed desirable to test its accuracy. A comparison has been made with the original graphical procedure of Klein and Rydberg, using the same data as used by Rydberg for the ground state of HZ . The curve has also been recalculated, using the more recent spectroscopic data (10-13), and combined with the quantum mechanical calculations of Dalgarno and Lynn (14). The result is a “most likely” potential energy curve for the ground state of Hz which differs in some respects from that proposed by Hirschfelder and Linnett (15). EVALUATION
OF METHODS
Rydberg (7) used a graphical method involving the action integral, together with another integral related to the spectroscopic rotational constant B, , to calculate the potential energy curves for Hz , CdH, and 02 . The action integral I for a rotating vibrator can be written I=
f
p, ar = (2#
[U -
dr,
V&)1”’
f
where p, is the radial momentum, mass, U the constant total energy,
r the internuclear
U = P,2/(2P)
+
separation,
=
V(r)
p the reduced
Veffk),
@I
and V,,,(r) is the effective potential energy. This effective the actual potential V(r) and the centrifugal potential, 2 V,rr(r)
(1)
+
K/T
potential
is the sum of
)
(3
where (4)
K = Pe2/(2P), and pe is the angular
momentum,
a constant
of the motion.
Quantization
of the
RYDBERG-KLEIN-REES
vibrational tion,
motion
METHOD
then gives the first relation
FOR H2
19
to be used in an actual
calcula-
I = h(u + $5) = (2p)l’2 4 w - Verr(r)J”2dr,
(5)
where h is Planck’s constant and v is the vibrational quantum number. The fact that Eq. (5) arises from a quantization condition on a phase integral shows that this is a WKB approximation (16, 17’). The second relation used in the calculat’ion is obtained from the expression for the rotational energy, E,,, , of a vibrating rotator, E,,t = K(1/T2)v = (K/G) # where r, is the period of vibration. integral leads to
(I/r’)dt
= &/TV)
Quantization
K = (87r*&)-’
(r2p,)-‘dr,
of the angular
J(J
where J is the rotational quantum number. (2) and K from (7) leads to the final relation scopic rotational constant:
i
momentum
(6) phase
+ l),
(7)
Substitution in Eq. (6) for p, from involving the experimental spectro-
h2(2,4-1’2 (S~T’T~)-~ [U - V,rr(r)]-“2 dr/r2 = & . f
In Eq. (S), B, is expressed in energy units rather than wave numbers. The condition (7) is again a WKB approximation. The quantity rV can be obtained from the experimental vibrational frequencies. The experimental values of U, 7, , and B, are inserted into Eqs. (5) and (8) and V(r) is adjusted to give the proper fit, the integrals being evaluated graphically. Since the integrand of Eq. (8) becomes infinite at the classical turning points, graphical integration is not very accurate. Klein (6) modified Rydberg’s procedure so that the classical turning points, rmin and rmax , could be calculated directly in terms of the functions f and g, where lYd(r,,,X 14[(l’/rmin)
and where as
U and
K
-
rdn)
=
(l/rmax)]
are as previously
f
(as/au),
= =
defined.
g
=
(9)
-(as/aK),
The auxiliary
(10)
function
S is defined
S( U,K) = (2?r2p)-“*/I’ [U - E(I,K)]“~ d1,
(11)
0
where E(I,K)
is the vibrational
and rotaCorral
energy
expressed
as a function
of
20
VANDERSLICE,
MASON,
LIPPINCOTT,
AND
MAISCH
the action and the square of the angular momentum, and the upper limit I’ of the integral is the value of I for which the integrand vanishes. As in the Rydberg procedure, the WEB approximation enters through the replacement of I and K by their quantum-mechanical equivalents in Eqs. (5) and (7). Klein de2 veloped the expressions for f and g from the expressions for 7 and (l/r ), and introduced S merely as a mathematically convenient function. It can be shown, however, that 5’ can be given a simple graphical interpretation as one-half the area between the constant total energy U and the potential curve V&r). To show this, we note that according to classical mechanics.
dI = rdE,
(12)
and 7 = (p/2)“2
(E -
(13)
Ve&“*dr.
The expression (11) for S then becomes
S = (2?r)-’ I’ (U - E)“* [f
(E - VJ1’*
dr] dE,
(14)
or, on changing the order of integration, S = (274-l
f
dr
’ (U - E)“’ (E - V,fJ1’* dE, s Veff
(15)
where the lower limit of the integral over dE is V eff since this integral is now evaluated at a constant value of r. This integral is a standard form and is equal to (r/2) ( U- Vefr), so that the expression for S can be written in the following forms:
(16s) (16b) s=x
s0
u [~rmx(E) - rmin(E)l dE,
(16~)
where in (16a) the effective potential is written out explicitly according to Eq. (3). From these expressions it is obvious that S is equal to one-half the area between U and Vefr . Furthermore, Eq. (9) for f follows immediately from (16~) by differentiation, and Eq. (10) for g follows similarly from (Isa). Thus 2f can now be interpreted as the change in the area between U and Verf with respect to a change in U at constant rotational energy (constant K), and 2g can be interpreted
RYDBERG-KLEIN-REES
METHOD
I
r
rmin
21
FOR H2
%a8
.
B
r
rmin
FIG. 1. Schematic
diagram showing in the Rydberg-Klein-Rees method.
rmaX
the graphical
interpretation
of the quantities
used
as the change in this area with respect to a change in K at constant total energy (constant U). These various relations are shown schematically in Fig. 1. In Klein’s procedure, the integration in (11) and the differentiations in (9) and (10) are carried out numerically. This process is fairly laborious and considerable care must be taken to obtain accurate results (4, 8). Rees (9) pointed out that Eqs. (9)-(11) can readily be solved for f and g if E(I,K) can be expressed as a quadratic in I = h(v + 35): E(I,K)
=
W(V
-
a
+
xi>
(v
+
-
36)
CO+
J(J
+
+
34)” 1) + BJ(J
+ 1) + DJ’(J
where w, ox, OJ, B, D, etc. are constants. Usually E(~,K) over the whole experimental range by such an expresssion,’
+ 1)’ + . . . ,
(17)
cannot be represented but it can be expressed
i It should be pointed out that the expression for g given by Rees for a quadratic ,!?(I+) does not behave properly in the limit f + 00, corresponding to dissociation of the molecule. As f ---t m a value of rmin = 0 is obtained rather than a finite value. It is therefore unsafe to use Rees’ expressions forf and g outside the region in which the quadratic expression for E(Z,K) is valid.
22
VANDERSLICE,
MASON,
LIPPIXCOTT,
.4ND
MAISCH
as a quadratic over different regions so that the entire range COILbe covered by a series of quadratics.’ For such a case we have extended Rees’ Eqs. (15) and (Ifi) slightly. In place of Eq. (11) we have
where Ai = h [U zi = w; mi
=
B,J(J
(Yi J(J
+ 1) -
DiJ"(J+ 1)’ + . ..I.
+ I),
(wz)i/h,
In = 0
and
I, = I’.
The summation extends over the vibrational energy levels. The following expressions are then obtained from (18) for f and g for the rotationless (J = 0) state: f = (%r’~~/h)-~” C& g = (27r2/.&C/h)1’2 c:X”=l [f&Q(wz) $-I (Uil’? -
(wr)[l”
In Wi ,
LTi_-llD)
+ (wx)_~” (2B; -
Wi
=
[
w;Wi2
-
ai(wz)i’wi)
w; -wi I[
where
(19)
4(ti.r)JTi 1’L 4(UX);Ua_1
In WJ,
1
2(WZ)i”ZUi_1”2 2(WZJL’W;‘2 *
(20)
(21)
The values of rminand rmnxare then easily calculated from Tmin
=
[(f/g)
+
fzll’z
-
f7
(22)
rmnx
=
[(f/g)
+
f211’*
+
f.
(23)
In Eqs. (19) and (20), c is the velocity of light, so that g, U, W, WX,(Y,and B are in cm -I, and f, r,in , and rmnl are in cm. The energy zero is chosen as the minimum of t.he V(r) curve (i.e., U0 = 0). It should be noted that not only the vibrational constants w and wz are allowed to vary with the energy level, but also the rotational constant B and the coupling constant (Y. The inaccuracy introduced by keeping the same values of B and a for all vibrational levels is tested later. 2 Rees has considered the case where E(I,K) can be expressed as a cubic in I, but the calculations are much more difficult and laborious. Also, the success of the method appears t,o depend on the accuracy with which the second anharmonicity constant wy can be obtained. Values of wy are among the least reliable of the spectroscopic constants, so we have not considered this method further.
RYDBERG-KLEIN-REES
METHOD
FOR
H2
23
TABLE I COMPARISON OF RYDBERG GRAPHICAL AND REES ANALYTICAL METHODS FOR THE GROUND STATE OF HP Graphicalb
Analytical
-VB cm-’
5065 7080 9370 11900 14675 17690 20925 24380 28075 31990 36150
rmin, A
rnlan, A
rmim A
~max, A
0.410 0.415 0.430 0.440 0.450 0.465 0.485 0.505 0.535 0.575 0.650
1.970 1.850 1.730 1.610 1.500 1.450 1.315 1.225 1.135 1.025 0.885
0.437 0.441 0.448 0.457 0.467 0.480 0.496 0.516 0.543 0.580 0.644
1.990 1.854 1.735 1.624 1.518 1.418 1.321 1.225 1.126 1.021 0.892
a References 18 and 19. b Reference 7. To compare the Rydberg-Klein graphical procedure with the Rees analytical modification, we have calculated the potential curve for the ground state of H2 by the analytical procedure using the same data as Rydberg used in his graphical calculation. These data consist of the first eleven vibrational levels (through 2, = 10) and the rotational constants B, = B - (Y(U+ x), as obtained from the experimental work of Hori (18) and of Dieke and Hopfield (19). We used four vibrational levels at a time and evaluated the constants w, or, (Y,and B of Eq. (17) by least squares. The integration of Eq. (18) was carried out only over the middle two levels, however, and then another overlapping group of four levels was taken. This procedure is quite rapid and accurate. The results of the graphical and analytical procedures are compared in Table I. The two methods agree to about 5 percent at the higher vibrational levels and to about 1 percent at the lower levels. This agreement is reasonable, since great care is necessary in the graphical procedure to obtain results reliable to 5 percent (4, 8). To test variations of the Rees analytical method and to obtain the best potential energy curve, we have used the more recent spectroscopic data (l&13), especially the data of Jeppesen (10) and Herzberg (13). These data do not yield any new vibrational energy levels, but do yield more reliable values for the energies of the levels than the older data used by Rydberg. Our results are shown in Table II. The best values are given first, and were calculated by the least-squares procedure described in the preceding paragraph. For many molecules of interest, values of B, = B - (Y(V + 34) may not be available for the higher vibrational
24
VANDERSLICE,
MASON,
LIPPISCOTT,
TABLE RESULTS
OBTAINED
Best values
4972 6999 9292 11837 14619 17634 20874 24344 28037 31962 36122 a References
0.426 0.432 0.439 0.449 0.460 0.473 0.489 0.509 0.534 0.571 0.634
2.001 1.860 1.733 1.618 1.513 1.413 1.315 1.218 1.120 1.013 0.883
AND
MAISCH
II
BY THE REES
~L~ETHOD
Constanta and B rmin, A
~max,A
0.427 0.435 0.443 0.452 0.463 0.475 0.491 0.510 0.535 0.571 0.634
2.003 1.862 1.737 1.622 1.516 1.415 1.317 1.219 1.120 1.014 0.883
Morse curve sections
0.386 0.383 0.410 0.425 0.442 0.457 0.478 0.499 0.526 0.565 0.631
1.962 1.823 1.701 1.587 1.485 1.388 1.297 1.205 1.111 1.008 0.881
10-15.
levels, so that a full least-squares treatment involving B and a is not possible. In such cases it is necessary to assume constant values of B and (Ywhich are obtained from the B, values for the lowest levels. The accuracy of this approximation has been tested for Hz, with the results shown in the next columns of Table II. These results agree within 1 percent with the best calculations, showing that the assumption of constant B and (Yis justified in this case. In the last columns of Table II are given the results obtained by means of a rapid Morse curve method proposed by Rees (9). In this method a Morse curve is evaluated for each pair of vibrational levels and the complete curve built up out of segments of Morse curves. These calculations are very rapid and check the best calculations to about 1 percent for the lower vibrational levels, as Table II shows. The agreement rapidly becomes worse at the higher levels, however, and so is not practical over the whole range. This is also shown in Fig. 2. Rees found very good agreement for the Bbou+ state of Brz with the Morse curve method, but he was not working near the dissociation limit. The experimental data used in constructing Tables I and II of course appear directly as the vibrational energy levels given in the first two columns of the tables. It should be remembered that at r,in and ~~~~, all the energies are equal, V&r) = E(I+) = U. The rotational levels are not listed separately since they are readily obtained from the original references as B, = B - CY(U + 36). Since the Rydberg-Klein-Rees method is a semi-classical one, the possibility exists t,hat the results for the lower vibrational levels could be unreliable (8, 9). To check this point we have calculated the potential curve in the vicinity of the minimum by the Dunham method (2~3, which is known to be accurate in this
RYDBERG-KLEIN-REES
METHOD
25
FOR Hz
0
I t
0 X
TE 2 ?3
4
,
I
0.5
1.5
1.0
2.0
r, A FIG. 2. Potential energy curve for the ground state of Hz at small internuclear separations. The solid curve is our “most likely” curve. The circles represent the results of an approximation involving sections of Morse curves. The heavy dashed line is the Lippincott function, and the light dashed line is the Morse and Hulburt-Hirschfelder functions.
region. The results are in agreement with the best values of Table II to one part in the last figure. The Rees method is therefore reliable even for the lower levels. It is perhaps worth pointing out the differences between the present method and the Dunham method, which is also based on a WKB approximation. The Dunham method assumes a polynomial representation of the potential energy curve around the minimum. Since such an expansion cannot converge at very large r, the Dunham method must fail for the higher vibrational levels even though it is very accurate for the lowest levels. Thus the Dunham method has basically the same disadvantage as most methods, in that it assumes an algebraic form for the potential energy curve. ‘MOST
LIKELY”
CURVE
FOR
HYDROGEN
Hirschfelder and Linnett (15) have proposed a “most likely” potential curve for the ground state of Hz , based on Rydberg’s results for r < 2.1A, on their own quantum-mechanical calculations for 2.4A 5 r 5 4.5A, and on Pauling and
26
VANDEHSLICE,
MASON,
LIPPINCOTT,
TABLE C~~~~~ARISON
I
0F
C.~LCUI,.~TED
ENERGIES
MAISCH
III IN
c3fe1
Hirschfelder
Dalgarno and Lynn*
AND
AT
IARCX
and Linnetth
SEr.4~i~TInxs
Pauling and BeachC
A --__
2.12 2.38 2.65 3.18 3.70 4.23 5.29 6.35 a Reference b Reference c Reference
- 3627 -867.3 -198.5 -47.9 -13.5 -2.03 -0.579
-2921 - 1394 -650.2 -139.8 -32.2 -8.78 -1.54 -0.444
- 1002 -407.3 -186.7 -51.0 -17.7 -7.31 -1.72 -0.545
14. 15. 21.
r,A FIG. 3. Potential energy curve for the ground state of HP at intermediate separations. The curves are labelled as follows: a, present “most likely” potential; b, Morse potential; c, Hulburt-Hirschfelder potential; d, Pauling-Beach calculations; e, Hirschfelder-Linnett calculations; f, Lippincott potential.
RYDBERG-KLEIN-REES
METHOD
27
FOR Hz
Beach’s calculations (21) for r 2 -1.5A. More recently, Dalgarno and Lynn (14) have made a careful second-order perturbation calculation of the energy of Hz for r > 2.lA. They allowed for the identity of the nuclei by using properly antisymmetrized electronic wave fun&ions, so that their results are probably superior to both bhe Hirschfelder-Linnett and Pauling-Beach calculations. At the largest internuclear separations, where the effect of the identity of the nuclei is negligible, the Dalgarno-Lynn and Pauling-Beach results merge together as they should. ,411 three sets of values are compared in Table III. The Dalgnrno-Lynn calculations, together with the best experimental values of Table II, constitute the “most likely” potential curve for the ground state of Hz . The join between the two sets of results is quite smooth. In Figs. 24 this “most likely” curve is compared with the well-known empirical curves of Morse, Hulburt-Hirschfelder, and Lippincott. In Figs. 3 and 4 the Hirschfelder-Linnett and Pauling-Beach curves are also included. For T > 2.OA, our “most likely” curve differs from the Hirschfelder-Linnett “most likely” curve by about 50 percent, and for r < 0.5A the difference is about 15 percent in the energy. As ex-
4
5
6
7
0 FIG. 4. Potential
curves
are labelled
energy curve as in Fig. 3.
for the ground
state
of H* at large separations.
The
28
VANDERSLICE,
MASON,
pected, none of the empirical potential tion of the curve over the entire range.
LIPPINCOTT,
functions
AND
MAISCH
gives an adequate
representa-
DISCUSSION
It is evident that the Rees modification of the Rydberg-Klein method will generate accurate interatomic potential energy curves quite easily. The results agree with the older graphical procedure within the limits of accuracy of the latter. The ease and rapidity with which potential functions can be generated by this method make the use of empirical functions highly questionable when the dat,a are available and accurate results are desired. Using recent spectroscopic data on Hz , we have calculated a number of points on the ground state curve which, when combined with t,he accurate calculations of Dalgarno and Lynn, yield a “most likely” potential energy curve. This curve differs appreciably from the Hirschfelder-Linnett “most likely” curve for r > 2.OAand r < GA. The absolute accuracy of the potential energy curves obtained by the Rydberg-Klein-Rees method is not certain because of the fact that it is a WKB method. On the other hand, the fact that is it a WKB method does not necessarily mean that the results are inherently inaccurate, since it is well known that the WKB method often gives excellent results for energy levels even t.hough the corresponding approximate wave functions are of lower accuracy (16, 1’7). Indeed, the pattern of energy levels as given by Eqs. (5) and (7) is exactly correct for the harmonic oscillator-rigid rotator model. By analogy with Dunham’s work on higher terms (8,.20), it might reasonably be expected that small cowections to the present method would be necessary for Hz , but would probably be insignificant for other molecules. The fact that the present results agree with t,he Dunham results near the minimum, and with the Dalgarno-Lynn calculations near the dissociation limit, is an indication that the absolute accuracy is indeed good, although a definite proof of this through the development of higher correction terms would be valuable. RECEIVED:
April 15, 1958 REFERENCES
1. P. M. MORSE, Phys. Rev. 34, 57 (1929). 2. H. M. HULBURT AND J. 0. HIRSCHFELDER,J. Chem. Phys. 9, 61 (1941). 3. E. R. LIPPINCOTT,J. Chews. Phys. 21,2070 (1953); E. R. LIPPINCOTTAND R. SCHROEDER,ibid. 23, 1131 (1955). /t. A. S. COOLIDGE,H. M. JAMES, AND E. L. VERNON, Phys. Rev. 64, 726 (1938). 6. G. HERZBERQ,“Spectra of Diatomic Molecules,” pp. 101-103. Van Nostrand, Princeton, 1950. 6. 0. KLEIN, 2. Physik 76, 226 (1932). 7. R. RYDBERG, 2. Physik 73, 376 (1931); 60, 514 (1933).
RYDBERG-KLEIN-REES
METHOD
FOR Hz
29
8. A. G. GAYDON, “Dissociation Energies and Spectra of Diatomic Molecules,” Chap. 2. Dover, New York, 1950. 9. A. L. G. REES, Proc. Phys. Sot. 69, 998 (1947). 10. C. R. JEPPESEN, Phys. Rev. 44, 165 (1933). 11. G. K. TEAL AND G. E. MACWOOD, J. Chem. Phys. 3, 760 (1935). 12. H. C. UREY AND G. K. TEAL, Revs. Modern Phys. 7, 34 (1935). 19. G. HERZBERG, Can. J. Research A%, 144 (1950). f4. A. DALGARNO AND N. LYNN, Proc. Phys. Sot. A69, 821 (1956). 15. J. 0. HIRSCHFELDER AND J. W. LINNETT, J. Chem. Phys. 18, 130 (1950). See also J. 0. Theory of Gases and HIRSCHFELDER, C. F. CURTISS, AND R. B. BIRD, “Molecular 16. 17. 18. 19. 20. 21.
L. J. T. G. J. L.
Liquids,” pp. 1054-1062. Wiley, New York, 1954. pp. 178-187. McGraw-Hill, New York, 1949. I. SCHIFF, “Quantum Mechanics,” C. SLATER, “Quantum Theory of Matter,” pp. 34-39. McGraw-Hill, New York, 1951. HORI, Z. Physik 44, 834 (1927). H. DIEKE AND J. J. HOPFIELD, Z. Physik 40,302 (1927). L. DUNHAM, Phys. Rev. 41, 713, 721 (1932). PAULING AND J. Y. BEACH, Phys. Rev. 47, 686 (1935).
OF
Ground
MOLECULAR
State
SPECTROSCOPY
of
3, 17-29 (1253)
Hydrogen by the Method *
Rydberg-Klein-Rees
J. T. VANDERSLICE,E. A. MASON, AND W. G. MAISCH Institute for Molecular
Physics,
University
of Maryland,
College Park,
Maryland
AND E. R. LIPPINCOW Department
of Chemistry,
University
of Maryland,
College Park,
Maryland
A detailed investigation is made of the Rees modification of the RydbergKlein graphical method of calculating potential energy curves without the use of empirical functions (such as Morse functions). The XZ, ground state of HP is chosen for investigation because of the large amount of both experimental and theoretical information available. It is found that the Rees procedure yields an accurate potential curve very rapidly, and hence the use of empirical functions is questionable when data are available and accurate curves are desired. Our calculations for HZ join smoothly with the accurate quantum mechanical calculations of Dalgarno and Lynn, valid at large internuclear separations. The resulting “most likely” potential energy curve agrees between 0.5A and 2.OA with a similar curve given previously by Hirschfelder and Linnett, but at smaller and larger distances differs appreciably from the Hirschfelder-Linnett curve. INTRODUCTION
Interatomic potential energy curves that are accurate over a large range of distance are desirable in a variety of problems involving atomic and ionic collision phenomena. One method of obtaining these potentials is to assume the validity of some empirical curve such as given by the Morse (I), Hulburt-Hirschfelder (2), or Lippincott (3) functions, and obtain the parameters for the curve from spectroscopic data. Although it is well known that substantial errors may be introduced by this procedure (4), it has been the one usually employed to obtain such potential curves (5). * This research tration.
was supported
in part by the National 17
Aeronautics
and Space Adminis-
18
VANDERSLICE,
MASON,
LIPPINCOTT,
AND
MAISCH
It is one of the purposes of this paper to emphasize t,hat spectroscopic data can yield accurate potential curves for bound states and that methods have been outlined which make the procedure quite rapid. In particular, Klein (6) and Rydberg (7) have developed a method for calculating potential curves which makes use of the experimental energy levels themselves and does not depend on some derived formula for these levels. This method has been used with success in a number of cases (8), but great care must be taken with the graphical integrations involved if reliable results are to be obtained (4). This difficulty has been eliminated by Rees (9), who has given an analytical modification of the method and applied it to the &I~+ state of Brz . As modified by Rees, the method is capable of giving interatomic potential curves rapidly once t’he spectroscopic data are available. In view of the potential importance of the Rees method, it seemed desirable to test its accuracy. A comparison has been made with the original graphical procedure of Klein and Rydberg, using the same data as used by Rydberg for the ground state of HZ . The curve has also been recalculated, using the more recent spectroscopic data (10-13), and combined with the quantum mechanical calculations of Dalgarno and Lynn (14). The result is a “most likely” potential energy curve for the ground state of Hz which differs in some respects from that proposed by Hirschfelder and Linnett (15). EVALUATION
OF METHODS
Rydberg (7) used a graphical method involving the action integral, together with another integral related to the spectroscopic rotational constant B, , to calculate the potential energy curves for Hz , CdH, and 02 . The action integral I for a rotating vibrator can be written I=
f
p, ar = (2#
[U -
dr,
V&)1”’
f
where p, is the radial momentum, mass, U the constant total energy,
r the internuclear
U = P,2/(2P)
+
separation,
=
V(r)
p the reduced
Veffk),
@I
and V,,,(r) is the effective potential energy. This effective the actual potential V(r) and the centrifugal potential, 2 V,rr(r)
(1)
+
K/T
potential
is the sum of
)
(3
where (4)
K = Pe2/(2P), and pe is the angular
momentum,
a constant
of the motion.
Quantization
of the
RYDBERG-KLEIN-REES
vibrational tion,
motion
METHOD
then gives the first relation
FOR H2
19
to be used in an actual
calcula-
I = h(u + $5) = (2p)l’2 4 w - Verr(r)J”2dr,
(5)
where h is Planck’s constant and v is the vibrational quantum number. The fact that Eq. (5) arises from a quantization condition on a phase integral shows that this is a WKB approximation (16, 17’). The second relation used in the calculat’ion is obtained from the expression for the rotational energy, E,,, , of a vibrating rotator, E,,t = K(1/T2)v = (K/G) # where r, is the period of vibration. integral leads to
(I/r’)dt
= &/TV)
Quantization
K = (87r*&)-’
(r2p,)-‘dr,
of the angular
J(J
where J is the rotational quantum number. (2) and K from (7) leads to the final relation scopic rotational constant:
i
momentum
(6) phase
+ l),
(7)
Substitution in Eq. (6) for p, from involving the experimental spectro-
h2(2,4-1’2 (S~T’T~)-~ [U - V,rr(r)]-“2 dr/r2 = & . f
In Eq. (S), B, is expressed in energy units rather than wave numbers. The condition (7) is again a WKB approximation. The quantity rV can be obtained from the experimental vibrational frequencies. The experimental values of U, 7, , and B, are inserted into Eqs. (5) and (8) and V(r) is adjusted to give the proper fit, the integrals being evaluated graphically. Since the integrand of Eq. (8) becomes infinite at the classical turning points, graphical integration is not very accurate. Klein (6) modified Rydberg’s procedure so that the classical turning points, rmin and rmax , could be calculated directly in terms of the functions f and g, where lYd(r,,,X 14[(l’/rmin)
and where as
U and
K
-
rdn)
=
(l/rmax)]
are as previously
f
(as/au),
= =
defined.
g
=
(9)
-(as/aK),
The auxiliary
(10)
function
S is defined
S( U,K) = (2?r2p)-“*/I’ [U - E(I,K)]“~ d1,
(11)
0
where E(I,K)
is the vibrational
and rotaCorral
energy
expressed
as a function
of
20
VANDERSLICE,
MASON,
LIPPINCOTT,
AND
MAISCH
the action and the square of the angular momentum, and the upper limit I’ of the integral is the value of I for which the integrand vanishes. As in the Rydberg procedure, the WEB approximation enters through the replacement of I and K by their quantum-mechanical equivalents in Eqs. (5) and (7). Klein de2 veloped the expressions for f and g from the expressions for 7 and (l/r ), and introduced S merely as a mathematically convenient function. It can be shown, however, that 5’ can be given a simple graphical interpretation as one-half the area between the constant total energy U and the potential curve V&r). To show this, we note that according to classical mechanics.
dI = rdE,
(12)
and 7 = (p/2)“2
(E -
(13)
Ve&“*dr.
The expression (11) for S then becomes
S = (2?r)-’ I’ (U - E)“* [f
(E - VJ1’*
dr] dE,
(14)
or, on changing the order of integration, S = (274-l
f
dr
’ (U - E)“’ (E - V,fJ1’* dE, s Veff
(15)
where the lower limit of the integral over dE is V eff since this integral is now evaluated at a constant value of r. This integral is a standard form and is equal to (r/2) ( U- Vefr), so that the expression for S can be written in the following forms:
(16s) (16b) s=x
s0
u [~rmx(E) - rmin(E)l dE,
(16~)
where in (16a) the effective potential is written out explicitly according to Eq. (3). From these expressions it is obvious that S is equal to one-half the area between U and Vefr . Furthermore, Eq. (9) for f follows immediately from (16~) by differentiation, and Eq. (10) for g follows similarly from (Isa). Thus 2f can now be interpreted as the change in the area between U and Verf with respect to a change in U at constant rotational energy (constant K), and 2g can be interpreted
RYDBERG-KLEIN-REES
METHOD
I
r
rmin
21
FOR H2
%a8
.
B
r
rmin
FIG. 1. Schematic
diagram showing in the Rydberg-Klein-Rees method.
rmaX
the graphical
interpretation
of the quantities
used
as the change in this area with respect to a change in K at constant total energy (constant U). These various relations are shown schematically in Fig. 1. In Klein’s procedure, the integration in (11) and the differentiations in (9) and (10) are carried out numerically. This process is fairly laborious and considerable care must be taken to obtain accurate results (4, 8). Rees (9) pointed out that Eqs. (9)-(11) can readily be solved for f and g if E(I,K) can be expressed as a quadratic in I = h(v + 35): E(I,K)
=
W(V
-
a
+
xi>
(v
+
-
36)
CO+
J(J
+
+
34)” 1) + BJ(J
+ 1) + DJ’(J
where w, ox, OJ, B, D, etc. are constants. Usually E(~,K) over the whole experimental range by such an expresssion,’
+ 1)’ + . . . ,
(17)
cannot be represented but it can be expressed
i It should be pointed out that the expression for g given by Rees for a quadratic ,!?(I+) does not behave properly in the limit f + 00, corresponding to dissociation of the molecule. As f ---t m a value of rmin = 0 is obtained rather than a finite value. It is therefore unsafe to use Rees’ expressions forf and g outside the region in which the quadratic expression for E(Z,K) is valid.
22
VANDERSLICE,
MASON,
LIPPIXCOTT,
.4ND
MAISCH
as a quadratic over different regions so that the entire range COILbe covered by a series of quadratics.’ For such a case we have extended Rees’ Eqs. (15) and (Ifi) slightly. In place of Eq. (11) we have
where Ai = h [U zi = w; mi
=
B,J(J
(Yi J(J
+ 1) -
DiJ"(J+ 1)’ + . ..I.
+ I),
(wz)i/h,
In = 0
and
I, = I’.
The summation extends over the vibrational energy levels. The following expressions are then obtained from (18) for f and g for the rotationless (J = 0) state: f = (%r’~~/h)-~” C& g = (27r2/.&C/h)1’2 c:X”=l [f&Q(wz) $-I (Uil’? -
(wr)[l”
In Wi ,
LTi_-llD)
+ (wx)_~” (2B; -
Wi
=
[
w;Wi2
-
ai(wz)i’wi)
w; -wi I[
where
(19)
4(ti.r)JTi 1’L 4(UX);Ua_1
In WJ,
1
2(WZ)i”ZUi_1”2 2(WZJL’W;‘2 *
(20)
(21)
The values of rminand rmnxare then easily calculated from Tmin
=
[(f/g)
+
fzll’z
-
f7
(22)
rmnx
=
[(f/g)
+
f211’*
+
f.
(23)
In Eqs. (19) and (20), c is the velocity of light, so that g, U, W, WX,(Y,and B are in cm -I, and f, r,in , and rmnl are in cm. The energy zero is chosen as the minimum of t.he V(r) curve (i.e., U0 = 0). It should be noted that not only the vibrational constants w and wz are allowed to vary with the energy level, but also the rotational constant B and the coupling constant (Y. The inaccuracy introduced by keeping the same values of B and a for all vibrational levels is tested later. 2 Rees has considered the case where E(I,K) can be expressed as a cubic in I, but the calculations are much more difficult and laborious. Also, the success of the method appears t,o depend on the accuracy with which the second anharmonicity constant wy can be obtained. Values of wy are among the least reliable of the spectroscopic constants, so we have not considered this method further.
RYDBERG-KLEIN-REES
METHOD
FOR
H2
23
TABLE I COMPARISON OF RYDBERG GRAPHICAL AND REES ANALYTICAL METHODS FOR THE GROUND STATE OF HP Graphicalb
Analytical
-VB cm-’
5065 7080 9370 11900 14675 17690 20925 24380 28075 31990 36150
rmin, A
rnlan, A
rmim A
~max, A
0.410 0.415 0.430 0.440 0.450 0.465 0.485 0.505 0.535 0.575 0.650
1.970 1.850 1.730 1.610 1.500 1.450 1.315 1.225 1.135 1.025 0.885
0.437 0.441 0.448 0.457 0.467 0.480 0.496 0.516 0.543 0.580 0.644
1.990 1.854 1.735 1.624 1.518 1.418 1.321 1.225 1.126 1.021 0.892
a References 18 and 19. b Reference 7. To compare the Rydberg-Klein graphical procedure with the Rees analytical modification, we have calculated the potential curve for the ground state of H2 by the analytical procedure using the same data as Rydberg used in his graphical calculation. These data consist of the first eleven vibrational levels (through 2, = 10) and the rotational constants B, = B - (Y(U+ x), as obtained from the experimental work of Hori (18) and of Dieke and Hopfield (19). We used four vibrational levels at a time and evaluated the constants w, or, (Y,and B of Eq. (17) by least squares. The integration of Eq. (18) was carried out only over the middle two levels, however, and then another overlapping group of four levels was taken. This procedure is quite rapid and accurate. The results of the graphical and analytical procedures are compared in Table I. The two methods agree to about 5 percent at the higher vibrational levels and to about 1 percent at the lower levels. This agreement is reasonable, since great care is necessary in the graphical procedure to obtain results reliable to 5 percent (4, 8). To test variations of the Rees analytical method and to obtain the best potential energy curve, we have used the more recent spectroscopic data (l&13), especially the data of Jeppesen (10) and Herzberg (13). These data do not yield any new vibrational energy levels, but do yield more reliable values for the energies of the levels than the older data used by Rydberg. Our results are shown in Table II. The best values are given first, and were calculated by the least-squares procedure described in the preceding paragraph. For many molecules of interest, values of B, = B - (Y(V + 34) may not be available for the higher vibrational
24
VANDERSLICE,
MASON,
LIPPISCOTT,
TABLE RESULTS
OBTAINED
Best values
4972 6999 9292 11837 14619 17634 20874 24344 28037 31962 36122 a References
0.426 0.432 0.439 0.449 0.460 0.473 0.489 0.509 0.534 0.571 0.634
2.001 1.860 1.733 1.618 1.513 1.413 1.315 1.218 1.120 1.013 0.883
AND
MAISCH
II
BY THE REES
~L~ETHOD
Constanta and B rmin, A
~max,A
0.427 0.435 0.443 0.452 0.463 0.475 0.491 0.510 0.535 0.571 0.634
2.003 1.862 1.737 1.622 1.516 1.415 1.317 1.219 1.120 1.014 0.883
Morse curve sections
0.386 0.383 0.410 0.425 0.442 0.457 0.478 0.499 0.526 0.565 0.631
1.962 1.823 1.701 1.587 1.485 1.388 1.297 1.205 1.111 1.008 0.881
10-15.
levels, so that a full least-squares treatment involving B and a is not possible. In such cases it is necessary to assume constant values of B and (Ywhich are obtained from the B, values for the lowest levels. The accuracy of this approximation has been tested for Hz, with the results shown in the next columns of Table II. These results agree within 1 percent with the best calculations, showing that the assumption of constant B and (Yis justified in this case. In the last columns of Table II are given the results obtained by means of a rapid Morse curve method proposed by Rees (9). In this method a Morse curve is evaluated for each pair of vibrational levels and the complete curve built up out of segments of Morse curves. These calculations are very rapid and check the best calculations to about 1 percent for the lower vibrational levels, as Table II shows. The agreement rapidly becomes worse at the higher levels, however, and so is not practical over the whole range. This is also shown in Fig. 2. Rees found very good agreement for the Bbou+ state of Brz with the Morse curve method, but he was not working near the dissociation limit. The experimental data used in constructing Tables I and II of course appear directly as the vibrational energy levels given in the first two columns of the tables. It should be remembered that at r,in and ~~~~, all the energies are equal, V&r) = E(I+) = U. The rotational levels are not listed separately since they are readily obtained from the original references as B, = B - CY(U + 36). Since the Rydberg-Klein-Rees method is a semi-classical one, the possibility exists t,hat the results for the lower vibrational levels could be unreliable (8, 9). To check this point we have calculated the potential curve in the vicinity of the minimum by the Dunham method (2~3, which is known to be accurate in this
RYDBERG-KLEIN-REES
METHOD
25
FOR Hz
0
I t
0 X
TE 2 ?3
4
,
I
0.5
1.5
1.0
2.0
r, A FIG. 2. Potential energy curve for the ground state of Hz at small internuclear separations. The solid curve is our “most likely” curve. The circles represent the results of an approximation involving sections of Morse curves. The heavy dashed line is the Lippincott function, and the light dashed line is the Morse and Hulburt-Hirschfelder functions.
region. The results are in agreement with the best values of Table II to one part in the last figure. The Rees method is therefore reliable even for the lower levels. It is perhaps worth pointing out the differences between the present method and the Dunham method, which is also based on a WKB approximation. The Dunham method assumes a polynomial representation of the potential energy curve around the minimum. Since such an expansion cannot converge at very large r, the Dunham method must fail for the higher vibrational levels even though it is very accurate for the lowest levels. Thus the Dunham method has basically the same disadvantage as most methods, in that it assumes an algebraic form for the potential energy curve. ‘MOST
LIKELY”
CURVE
FOR
HYDROGEN
Hirschfelder and Linnett (15) have proposed a “most likely” potential curve for the ground state of Hz , based on Rydberg’s results for r < 2.1A, on their own quantum-mechanical calculations for 2.4A 5 r 5 4.5A, and on Pauling and
26
VANDEHSLICE,
MASON,
LIPPINCOTT,
TABLE C~~~~~ARISON
I
0F
C.~LCUI,.~TED
ENERGIES
MAISCH
III IN
c3fe1
Hirschfelder
Dalgarno and Lynn*
AND
AT
IARCX
and Linnetth
SEr.4~i~TInxs
Pauling and BeachC
A --__
2.12 2.38 2.65 3.18 3.70 4.23 5.29 6.35 a Reference b Reference c Reference
- 3627 -867.3 -198.5 -47.9 -13.5 -2.03 -0.579
-2921 - 1394 -650.2 -139.8 -32.2 -8.78 -1.54 -0.444
- 1002 -407.3 -186.7 -51.0 -17.7 -7.31 -1.72 -0.545
14. 15. 21.
r,A FIG. 3. Potential energy curve for the ground state of HP at intermediate separations. The curves are labelled as follows: a, present “most likely” potential; b, Morse potential; c, Hulburt-Hirschfelder potential; d, Pauling-Beach calculations; e, Hirschfelder-Linnett calculations; f, Lippincott potential.
RYDBERG-KLEIN-REES
METHOD
27
FOR Hz
Beach’s calculations (21) for r 2 -1.5A. More recently, Dalgarno and Lynn (14) have made a careful second-order perturbation calculation of the energy of Hz for r > 2.lA. They allowed for the identity of the nuclei by using properly antisymmetrized electronic wave fun&ions, so that their results are probably superior to both bhe Hirschfelder-Linnett and Pauling-Beach calculations. At the largest internuclear separations, where the effect of the identity of the nuclei is negligible, the Dalgarno-Lynn and Pauling-Beach results merge together as they should. ,411 three sets of values are compared in Table III. The Dalgnrno-Lynn calculations, together with the best experimental values of Table II, constitute the “most likely” potential curve for the ground state of Hz . The join between the two sets of results is quite smooth. In Figs. 24 this “most likely” curve is compared with the well-known empirical curves of Morse, Hulburt-Hirschfelder, and Lippincott. In Figs. 3 and 4 the Hirschfelder-Linnett and Pauling-Beach curves are also included. For T > 2.OA, our “most likely” curve differs from the Hirschfelder-Linnett “most likely” curve by about 50 percent, and for r < 0.5A the difference is about 15 percent in the energy. As ex-
4
5
6
7
0 FIG. 4. Potential
curves
are labelled
energy curve as in Fig. 3.
for the ground
state
of H* at large separations.
The
28
VANDERSLICE,
MASON,
pected, none of the empirical potential tion of the curve over the entire range.
LIPPINCOTT,
functions
AND
MAISCH
gives an adequate
representa-
DISCUSSION
It is evident that the Rees modification of the Rydberg-Klein method will generate accurate interatomic potential energy curves quite easily. The results agree with the older graphical procedure within the limits of accuracy of the latter. The ease and rapidity with which potential functions can be generated by this method make the use of empirical functions highly questionable when the dat,a are available and accurate results are desired. Using recent spectroscopic data on Hz , we have calculated a number of points on the ground state curve which, when combined with t,he accurate calculations of Dalgarno and Lynn, yield a “most likely” potential energy curve. This curve differs appreciably from the Hirschfelder-Linnett “most likely” curve for r > 2.OAand r < GA. The absolute accuracy of the potential energy curves obtained by the Rydberg-Klein-Rees method is not certain because of the fact that it is a WKB method. On the other hand, the fact that is it a WKB method does not necessarily mean that the results are inherently inaccurate, since it is well known that the WKB method often gives excellent results for energy levels even t.hough the corresponding approximate wave functions are of lower accuracy (16, 1’7). Indeed, the pattern of energy levels as given by Eqs. (5) and (7) is exactly correct for the harmonic oscillator-rigid rotator model. By analogy with Dunham’s work on higher terms (8,.20), it might reasonably be expected that small cowections to the present method would be necessary for Hz , but would probably be insignificant for other molecules. The fact that the present results agree with t,he Dunham results near the minimum, and with the Dalgarno-Lynn calculations near the dissociation limit, is an indication that the absolute accuracy is indeed good, although a definite proof of this through the development of higher correction terms would be valuable. RECEIVED:
April 15, 1958 REFERENCES
1. P. M. MORSE, Phys. Rev. 34, 57 (1929). 2. H. M. HULBURT AND J. 0. HIRSCHFELDER,J. Chem. Phys. 9, 61 (1941). 3. E. R. LIPPINCOTT,J. Chews. Phys. 21,2070 (1953); E. R. LIPPINCOTTAND R. SCHROEDER,ibid. 23, 1131 (1955). /t. A. S. COOLIDGE,H. M. JAMES, AND E. L. VERNON, Phys. Rev. 64, 726 (1938). 6. G. HERZBERQ,“Spectra of Diatomic Molecules,” pp. 101-103. Van Nostrand, Princeton, 1950. 6. 0. KLEIN, 2. Physik 76, 226 (1932). 7. R. RYDBERG, 2. Physik 73, 376 (1931); 60, 514 (1933).
RYDBERG-KLEIN-REES
METHOD
FOR Hz
29
8. A. G. GAYDON, “Dissociation Energies and Spectra of Diatomic Molecules,” Chap. 2. Dover, New York, 1950. 9. A. L. G. REES, Proc. Phys. Sot. 69, 998 (1947). 10. C. R. JEPPESEN, Phys. Rev. 44, 165 (1933). 11. G. K. TEAL AND G. E. MACWOOD, J. Chem. Phys. 3, 760 (1935). 12. H. C. UREY AND G. K. TEAL, Revs. Modern Phys. 7, 34 (1935). 19. G. HERZBERG, Can. J. Research A%, 144 (1950). f4. A. DALGARNO AND N. LYNN, Proc. Phys. Sot. A69, 821 (1956). 15. J. 0. HIRSCHFELDER AND J. W. LINNETT, J. Chem. Phys. 18, 130 (1950). See also J. 0. Theory of Gases and HIRSCHFELDER, C. F. CURTISS, AND R. B. BIRD, “Molecular 16. 17. 18. 19. 20. 21.
L. J. T. G. J. L.
Liquids,” pp. 1054-1062. Wiley, New York, 1954. pp. 178-187. McGraw-Hill, New York, 1949. I. SCHIFF, “Quantum Mechanics,” C. SLATER, “Quantum Theory of Matter,” pp. 34-39. McGraw-Hill, New York, 1951. HORI, Z. Physik 44, 834 (1927). H. DIEKE AND J. J. HOPFIELD, Z. Physik 40,302 (1927). L. DUNHAM, Phys. Rev. 41, 713, 721 (1932). PAULING AND J. Y. BEACH, Phys. Rev. 47, 686 (1935).