J . Phys. Chem. 1985, 89, 2540-2543
2540
Because of the small energy gap for the C O F 2 / N 0 transfer, the present data do not unambiguously determine whether attractive forces play a dominant role in- the energy transfer. Acknowledgment. This work was supported by the National Science Foundation under Grant CHE80-23747 and the Joint
Services Electronics Program (US.Army, U S . Navy, and U S . Air Force) under Contract DAAG29-82-K-0080. Equipment support was provided by the Department of Energy undercontract DE-AC02-78ER04940. Registry No. COF2, 353-50-4; NO, 10102-43-9.
Characterization of the Skewed Probability Densities of Anharmonic Oscillators Lawrence S. Bartell Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109 (Received: January 24, 1985)
The applicability of Miller's classical path approximation for calculating the probability density of a thermal ensemble of anharmonic oscillators was investigated at several levels of approximation. What was sought was a tractable method for computing the first three moments of the density to characterize its form with a view of extending the treatment to polyatomic molecules. For a Morse-like anharmonic oscillator it was found that Miller's density, N(x) .exp[-Pff(x)/kT], which allows for some degree of tunneling, begins to yield a satisfactory coefficient of skewness when kT reaches h u / 2 . In comparison, a classical Boltzmann density No exp[-V(x)/kr] requires a temperature about fourfold higher. For a Morse-like oscillator with effective potential energy kT(a2x2- a3x3+ a4x4+ ...) it was also found that approximation of Miller's density by No[exp(-a2x2)](1 + a3x3)gave the three moments virtually as satisfactorily as did the complete expression, over the whole temperature range of chemical interest to which the Miller density applies. Several recursion relations between the moments are given as well as a means of scaling moments for oscillators in corresponding states. Applications to gas-phase electron diffraction are outlined briefly.
Introduction In the past few years electron diffraction has been found to provide a useful source of information about vibrational anharmonicity that has only begun to be exploited.1-8 For example, studies of hot polyatomic uncovered a hitherto unrecognized interrelationship between the various force constants for bending, anharmonic and A problem encountered in such investigations is that, while spectroscopy in favorable cases can yield potential constants fairly directly, diffraction charcterizes probability densities, instead. Although the relation between potential energy surfaces, probability densities, and diffraction patterns was worked out, in principle, decades ago,l2-l9 the resultant expressions are too ponderous for practical
(1) Goates, S . R.; Bartell, L. S. J . d e m . Phys. 1982, 77, 1866. (2) Goates, S. R.; Bartell, L. S . J . Chem. Phys. 1982, 77, 1874. (3) Bartell, L. S.; Vance, W. N.; Goates, S. R. J . Chem. Phys. 1984, 80, 3923. (4) Bartell, L. S.; Stanton, J. F. J . Chem. Phys. 1984, 81, 3792. (5) Gershikov. A. G.;Spiridonov, V. P. J. Mol. Struct. 1982.96. 141; 1983, 101, 315. (6) Gershikov, A. G.; Nasarenko, A. Ya.; Spiridonov, V. P. J. Mol. Strucr.: THEOCHEM 1984, 106, 225. (7) Nasarenko, A. Ya.; Spiridonov, V. P.; Butayev, B. S.; Gershikov, A. G. J. Mol. Struct.: THEOCHEM 1984, 106, 233. (8) Spiridonov, V. P.; Gershikov, A. G.; Butayev, B. S. J . Mol. Struct. 1979, 51, 137; 1979, 52, 53. (9) Bartell, L. S. J . Mol. Struct. 1982, 84, 117. (IO) Bartell, L. S . Croat. Chem. Acta 1984, 57, 927. (11) Bartell, L. S.; Barshad, Y. Z . J. Am. Chem. SOC.1984, 106, 7700. (12) Bartell, L.S. J. Chem. Phys. 1955, 23, 1219. (13) Reitan, A. ET. Nor. Vidensk. Selsk. Skr. 1958, 2. (14) Reitan, A. Acta Chem. Scnnd. 1958, 12, 785. (15) Kuchitsu, K.; Bartell, L. S . J . Chem. Phys. 1961, 35, 1945. (16) Kuchitsu, K.; Bartell, L. S . J . Chem. Phys. 1962, 36 2460, 2470. (17) Kuchitsu, K. Bull. Chem. SOC.Jpn. 1967, 40, 498, 505. (18) Bartell, L. S . J. Mol. Struct. 1981, 63, 259. (19) Hilderbrandt, R. L.;Kohl, D. A. J . Mol. Struct.: THEOCHEM 1981, 85, 25. Kohl, D. A,; Hilderbrandt, R.L. Ibid. 1981, 85, 325.
0022-3654/85/2089-2540$01.50/0
application to a thermal ensemble of polyatomic molecules at elevated temperatures. A very simple approximation, based on Miller's of Feynman's path integral technique,21was recently proposed to handle the problem.8~22It yielded the first three moments of the internuclear radial distribution functions that are needed to complete the treatment. This was accomplished22by taking the cubic term, say k;x3/6kT, of the effective potential energy ratio Pff(x)/kT, as a perturbation and disregarding higher order terms in a key expansion. While this is plausible and led to seemingly reasonable results, it is open to criticism on several counts. At low temperatures the Miller effective potential cannot be relied upon to generate a satisfactory probability density. At higher temperatures the characteristic values of k3ex3/6kT increase because T increases only as (approximately) the mean square of x. Furthermore, when higher order terms were retained in the principal expansion to investigate whether the simple approximation worked satisfactorily over any temperature range, contributions from successive expansion terms oscillated wildly. It became necessary, then, to solve a prototype problem rigorously, over physically realistic ranges of parameters, to evaluate the Miller approach and to investigate whether the simple approximation proposed has any validity. It turns out that the first-order approximation is very good even if the higher orders are not, and that some useful relations connecting the moments were found. Results of the investigation are sketched in the following and compared, where possible, with results of alternative treatments. Treatment Statement of the Problem. The key element in the practical problem of deriving information about potential surfaces is to relate the anharmonicity of the force field to the skew of the internuclear (20) Miller, W. H. J . Chem. Phys. 1971, 55, 3146. (21) Feynman, R. P.; Hibbs, A. R. "Quantum Mechanics and Path Integrals"; McGraw-Hill: New York, 1965; pp 273-279. (22) Bartell, L. S . J. Mol. Strucf. 1984, 116, 279.
@ 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 12, 1985 2541
Anharmonic Oscillators radial distribution peaks. While the greatest recent interest and novelty has been in nonbonded internuclear distances of multidimensional oscillator^,^^^^^" the ill-conditioned mathematical behavior mentioned in the introduction is characteristic of any markedly asymmetric quantum oscillation and can be explored adequately by studying a one-dimensional oscillator. The prototype anharmonic oscillator, for which much of the treatment required has already been worked out, is the Morse oscillator23
-
(la)
+ (7a2/12)x4 - (a3/4)x5 + ...I
(lb)
V(x) = (f/2a2)(e” = (f/2)[xz - ax3
We shall adopt this as a model potential energy either in the “full-Morse” representation ( l a ) or in the series representation “Morse-cubic” or “Morse-quartic” truncated at the cubic or quartic term. I n all cases the essential asymmetry is expressed by the Morse constant a. What is sought is the reliability of the moments (x), (x2), and (x3) calculated at various levels of approximation from the probability density p ( x ) derived via the path integral approach. These moments respectively characterize the anharmonic shift in center of gravity from the equilibrium position, the variance or mean-square amplitude of vibration, and the skewness. Miller’s path integral expression for the probability density of a one-dimensional oscillator iszo p(x) = N ( x ) exp(-IFff/kT)
(2)
N(x) = (2mkT/4~h2)’/2(u/sinh( u ) ) I / ~
(3)
where
and
with u(x) = ( h / k T ) [ V ” ( ~ ) / m ] l / ~
(5)
Equation 2 yields the exact quantum result for a harmonic oscillator and more quickly approaches the quantum density of an anharmonic oscillator, as T increases, than does the classical Boltzmann density P,l(X)
NCI
exp(-V(x)/kT)
(6)
(xn)=
+ ...)
pI(x) = Noe-a2x2(1+ a3x3)
=fwl/2kT a3 = f a ( ]
(8)
+ 3W2/2)/2kT
a4 = (7fa2/24kT)[1 - 13(1 - W1)/7
(9)
+ 27W3/7]
(10)
with WI = [tanh (uo/2)l(uo/2)
(11)
- W,
(12)
+ W?)(u0/2)~
(13)
W2 = sech2 (u0/2)
W3 = -13W2/18 - Wl(Wl
pII(x) = No exp(-a2x2 + a3x3- a4x4)
No(l
(17)
PI&)
= N(x) exp(-a2x2
+ a3x3- a4x4)
(18)
where, in eq 16 and 17, No is a constant. In addition, we include some results based on eq 2 for a full-Morse oscillator, some classical results invoking eq 6, and some previously derived results for a Morse-quartic o ~ c i l l a t o r . ’ ~The ~ ~great ~ simplicity of approximation I, especially when applied to multidimensional oscillators, makes it desirable to test its acceptibility. An exact treatment for a cubic oscillator, of course, is of no utility because the potential energy is pathological at large x. Approximation I1 examines the effect of the quartic term, circumventing the problems associated with the finite expansion in eq 14. Approximation I11 examines whether the x dependence of Miller’s normalization function N(x) plays a significant role. Note that we shall treat only the one-dimensional problem and disregard rotational effects. For treatments including rotation, see ref 17 and 24. Skew Parameter and Relation to Diffracted Intensity. Of greatest importance in relating the shape of the nearly Gaussian probability density to the anharmonicity is the skewness. This index of form can be expressed quantitatively in terms of the dimensionless standard coefficient of skewnessz5 A3
= ( ( x - (X)l3)/((X - (x))2)3/2
(19)
For historical reasons we shall adopt the related parameter
a = A,/((x - (x))2)1/2 - (x3)
- 3(x2)(x)
+ 2(x)3
((x2) - (x)2)2
(20)
Parameter 6, which we shall express in A-1, reduces for a Morse oscillator to the Morse parameter a at 0 K and, at high temperature, varies only slowly with T. I n a subsequent section we shall calculate in several ways the moments needed in eq 20. I n an earlier treatment of a Morse-like oscillator Kuchitsu,17 extending treatments by ReitanI3*l4and other^,^^,^^ used secondorder perturbation theory and summed over a Boltzmann population of vibrational states to arrive at coefficients C1and C3 in the density p(x)
i=
NI(l
+ Clx + C3x3) exp(-x2/2(x2))
(21)
Kuchitsu’s expression corresponds to a value for 6 of ciK = (6C3(x2)3+ ~ ( X ) ~ ) / ( (-X(~x) ) ~ ) ~
+ R’x + ‘/,R”.$+ ...)e-“’”’[l + (a$ ...)
- a4x4+
+ ...I
that yields simple analytical expressions for the moments (23) Morse, P. M. Phys. Rev. 1929, 34, 57.
(16)
approximation 111, with
where uo is u ( x ) evaluated at x = 0. Equation 2 can be expanded into a form p(x) =
(15)
approximation 11, with
(7)
where, from eq 2, 4, and 5
x”p(x) dx
where R’ = aN(x)ax, etc. The moments so calculated, however, produce a skew parameter that gets worse when the quartic constant a4is retained and when higher order terms in the square brackets of eq 14 are included, unless an impractical number of higher order terms are kept. Therefore, alternatives were investigated. The levels of approximation that are considered in this paper are approximation I, taking p as
The effective potential energy can also, of course, be represented as a power series
IFff = kT(a2x2 - a3x3 + a4x4
1,
i=
a[l
+ 8 x / ( 1 + x ) ~ ] ( x ~ ) / I +, , ~~ ( x ) ~ / ( ( x- ~( x) ) ~ )(22) ~
(14) (24) Bonham, R. A.; Su, L.S . J . Chem. Phys. 1966, 45, 2827. (25) Burington, R. S.; May, D. C. “Handbook of Probability and Statistics”; Handbook Publishers: Sandusky, 1953. (26) Morino, Y . ;Iijima, T. Bull. Chem. SOC.Jpn. 1963, 36, 412. (27) Bonham, R. A.; Peacher, J. L. J . Chem. Phys. 1963, 38, 2319.
2542
The Journal of Physical Chemistry, Vol. 89, No. 12, 1985
where x is exp(-hu/kT), and explicit formulas for ( x ) , (x2) I:, and lh2= (x2) - ( x ) are ~ given in ref 17. Note that terms C,kn that are of higher degree than cubic enter a more exact expressim than eq 21 and can contribute significantly t a 2. Nevertheless, Kuchitsu's results for the truncated expreskion based on second-order perturbation theory ace quite good for normal molecules over the entire range of T likely to be encountered. It is ~ e l l - k n o w n that ' ~ the reduced molecular intensity M ( s ) for gas-phase electron diffraction can be expressed as a sum of terms, one for each internuclear distance r,, in the form
= C, exp(-lm2s2/2)[sin s(r, - K S -~ q s 4 - ...)I,/ ra (23) where G,(s) involves electron scattering factors and ra and I , are defined average distances and vibrational amplitudes addrehed in ref 17. Anharmonicity manifests itself in a modest frequepcy modulation of the sinusoidal interference pattern. This modulqtion, induced by the terms K S ~ K ~ S --, ~ has never been observed to require more than the first term, K S ~ , The modulation parameter K is related to the foregoing skew parameter 2 by9 K
+
= ci((x2) - ( ~ ) ' ) ~ / 6
(24)
whatever the order of the terms C,#'in an expression of the form of eq 21 that contribute to 2; prior perturbation treatments have stopped at cubic. Eualuation of the Moments. Computation of the moments is easily carried out analytically for approximation I, yielding the results (X)I = 3CY3/4a22
(25)
($)I = 1/2a,
(26)
(x3)I = 1 5 ~ ~ 3 / 8 ~ ~ 2 ~
(27)
and
+
21 = ( 1 9 2 ~ ~ 2 ~2~1~63~ ~ 2 ~ ~ ~ 3-~9) ~ /(~ 8 ~3~(28) ~2 ~) ~
In the case of approximations I1 and 111, moments were evaluated by numerical integration, as were full-Morse r'esults according to eq 3 and 2. In the latter case (eq 2), correction of Morse force constants to effective force constants corresponding to eq 4 was only carried out through quartic terms (cf. eq 8-10). At low temperatures the higher force constants scarcely matter, while at higher temperatures they approach the actual force constants. Therefore, the error (not checked) may be small. Moments through (2) are available for the Morse-quartic oscillator through the work of Bonham and Su2, who applied the Hellmann-Feynman and hypervirial theorems together with Dunham's WKB energy eigenvalues to generate explicit analytical results. Certain recursion relations among the moments for thermal ensembles can be derived that may be of utility in some applications. For any oscillator, Ehrenfest's theorem2*to the effect that ( d V / a x ) vanishes can be applied. For a Morse oscillator, at any temperature, it yields (x)
- 3a(x2)/2 + 7 a 2 ( x 3 ) / 6- 5a3(x4)/8 + ... = 0
(29)
If the potential energy is truncated, the expression for the moments is correspondingly truncated. Therefore, for example, for a Morse-quartic oscillator ( x ) = 3a(x2)/2
- 7a2(x3)/6
(30)
Any oscillator to which the Miller approach pertains has a probability density, according to eq 2, of form L
p(x) = N(x)
II exp[-(-~)'a/x/J
1=2
~~
(28) See Bartell, L. S. J . Chem. Phys. 1963, 38, 1827.
whence a(x")/aCY, = -(-l)"(x"+")
(32)
If, further, we neglect the modest x dependence of N(x) (an approximation that does little harm, as we shall see), additional relations can be derived by integrating eq 15 parts. This procedure yields (33)
which, for a Morse-quartic oscillator, gives
s M , ( s ) / G , ( s )= x m [ P , ( r ) / r ] sin sr d r
+
Bartell
(31)
( n + l)(x") = ~ C Y ~ ( X- "~+c~Y) ~ ( x+" ~ + c~Y) , ( x " +(34) ~) including, for n = -1, 2a2(x) - 3a3(x2)
+ 4cu4(x3) = 0
(35)
to which expressions eq 32 can be applied, if desired. Reduced Variables. Although the actual moments and skew parameter for a Morse oscillator depend upon the force constant, mass, asymmetry constant a, and temperature, certain correspondences can be made. Let the Miller density of eq 31 for such an oscillator be written as p(z)
= ~ ( z exp[-{(z2 ) - v3z3+ v4z4 +
...)I
(36)
where z = ax and
r' = 2a2Ia2coth ( u 0 / 2 )
(37)
with 1: and uo representing the zero-point mean-square amplitude and ratio hu/kT for a harmonic oscillator of force constantfand mass m (cf. eq 5-13). For this reduced Morse density it can be seen that the coefficients vs, v4, etc. depend only upon the reduced temperature kT/hu. Therefore, corresponding oscillators A and B with aA
(la)A
=
aB
(la)B
(38)
will have equal reduced moments ( z n )= ((ax)") at equal reduced temperatures, whatever the individual force constants, masses, or asymmetry constants.
Illustrative Calculations For the purpose df evaluating the various levels of approximation, several reasonably realistic models were tested that covered the range of parameters encountered in nonbonded distances of molecules recently studied The first was a comparatively high-frequency case roughly simulating a representative nofibonded atom pair in CF4 The others were an intermediateand a low-frequency case approximately simulating SiF, and SnCl,. Force constants, masses, and Morse a values were chosen to reproduce the general magnitudes of the observed mean bending frequencies, zero-point nonbonded mean-square amplitudes, and * * ~ of u (in asymmetry constants ri estimated e l ~ e w h e r e . ~Values cm-I), I, (in A), and a (in A-') adopted were (530, 0.052, 0.7), (310, 0.061, 0.36), and (100, 0.070, 0.2) for the three cases, respectively. Mean displacements (x) and asymmetry constants ci calculated for the CF4-like case over a range of temperatures according to various approximations are listed in Table I. Tables I1 and 111 show calculated ri values for the other two cases. Of less interest are the mean-square amplitudes calculated by the different options inasmuch as they are not strikingly different from those of a harmonic oscillator, and the SHO results are reproduced exactly by Miller's approach.20 For a harmonic oscillator the values of ( x ) and ci are zero, of course. As temperature increases, the effective potential energy of Miller's classical path method approaches the actual potential energy. Plotted in Figure 1 is the ratio of the "effective" to actual force constants, quadratic, cubic, and quartic, for a Morse oscillator, as a function of reduced temperature u0-' = kT/hu. (29) Stanton, J. F.; Bartell, L. S. J . Phys. Chem., following article in this issue.
The Journal of Physical Chemistry, Vol. 89, N o . 12, 1985 2543
Anharmonic Oscillators
TABLE I: Skew Parameter ri and Mean Displacement from Equilibrium of a Morse-Like Oscillator“ Roughly Simulating an F.-F Pair in CFI ri l o w
T,K 254.3 381.5 763.0 1526 3052
ut
B-SC
Id)e
IIdf
3.0 2.0 1.0 0.5 0.25
0.954 1.291 1.806 2.023 2.089
1.278 1.351 1.828 2.085 2.231
1.259 1.347 1.818 2.024 2.049
*
IIIdsg 1.245 1.342 1.816 2.022 2.048
IVd**
A?
1.258 1.354 1.835 2.065 2.167
0.969 1.312 1.850 2.119 2.294
class’ 2.11 2.11 2.12 2.14 2.18
B-Sc 3.14 3.73 6.17 11.69 23.15
Id,‘
IIdC
1.90 2.39 5.27 11.13 22.59
1.89 2.39 5.32 11.25 22.94
IIIdsg 5.03 4.39 6.27 11.72 23.17
class’ 1.90 2.85 5.73 11.55 23.52
“ a in ,&-I, ( x ) in A. h v / k T . ‘Bonham-Su approximation, Morse-quartic, ref 25. dMiller classical path approximation, ref 20. ‘Morse-cubic, approximation I in text. /Morse-quartic, N(x) taken as constant. Morse-quartic. Ir Full-Morse. Kuchitsu approximation, ref 17. jClassical Boltzmann distribution for which ri 3a as T 0.
-
-
TABLE 11: Skew Parameter ri for Morse-Like Oscillator Roughly Simulating an Fa-F Pair in SiF/ ~~
T 223 446 892 1784
UO
B-S
I
ri I1
111
K
2.0 1.0 0.5 0.25
0.663 0.927 1.038 1.071
0.693 0.932 1.050 1.097
0.692 0.930 1.039 1.066
0.691 0.930 1.039 1.065
0.667 0.935 1.055 1.107
“See Table I for units and specifications of the approximations. TABLE 111: Skew Parameter ri for Morse-Like Oscillator Roughly Simulating a CI-CI Pair in SnCL”
a T
~n
B-S
288 900 2000
0.500 0.160 0.072
0.576 0.598 0.601
I 0.576 0.607 0.622
I1 0.577 0.598 0.598
111
0.577 0.598 0.598
K 0.580 0.611 0.630
“See Table I for units and specifications of the approximations.
at small T . This is due, of course, to the undercalculated mean-square amplitude at low T (where the classical oscillator is not restricted to an irreducible zero-point amplitude) and not to an excessive third moment ( x 3 ) . Moment ( x 3 ) is actually more undercalculated than ( x 2 ) . The reason the coefficient of skewness is too large, then, is that the product ( x ) ( x z )is relatively even more undervalued than is ( x 3 ) . Comparisons of results for approximations I1 and I11 show that the dependence of Miller’s normalization factor N ( x ) of eq 15 upon the coordinate has only a minor influence on the value of ii computed. It has a substantial effect on ( x ) only in the lowtemperature range where the path integral method fails badly, anyway, and in this range inclusion of N ( x ) instead of No helps very little. Carrying the expansion of V ( x ) beyond quartic terms begins to alter ii by more than 3% (increasing the skew) only at comparatively high temperatures (in excess of 2000 K for the CF4-like case). At temperatures above 2hv/k, where the Miller classical path values should be accurate, the Bonham-Su Morse-quartic moments through ( x 3 )checked our numerical integrations to within a few parts per thousand until kT exceeded about 4hv for the CF4-like case. At higher thermal excitations the Bonham-Su moments began to worsen. They became ill-behaved at extremely high temperatures (greater, say, than lo4 K, a temperature at which a Morse-quartic model is of little relevance). The most important conclusion is that the simplest approximation of all, approximation I, is not greatly inferior to the best classical path approximation. It is considerably better than the approximation of the same form (based on eq 14) retaining a number of higher order terms, a circumstance that led to the present investigation when magnitudes of truncation errors were (fruitlessly) sought by studying effects of such terms. That the most severe truncation leads to good results is fortunate because it suggests that analogous truncation should be satisfactory in treatments of polyatomic molecules where higher order terms about which little is known proliferate rapidly. this conclusion is also borne out empirically by studies of hot molecules of CF4, SiF4, and SF6 where estimates of ( x ) and ii based on an approximation of type I were able to account for the observed results to within estimated ~ n c e r t a i n t i e s . The ~ . ~ ~degree of simplification of polyatomic molecules made possible by a treatment of the form of approximation I is so enormous that further attention is warranted.
, -
OO
10
20
30
kT/hv
Figure 1. Ratio of force constants of Miller’s effective potential energy to actual force constants, quadratic, cubic, and quartic, for Morse oscillator, as a function of temperature.
Discussion Over the range of thermal excitation covered in Tables 1-111 we can regard the Bonham-Su results as virtually exact for a “Morse-quartic” oscillator. At temperatures above hv/2k, all of the tabulated approximations for ( x ) and ii except those for the classical Boltzmann distribution are in fairly good agreement with the Bonham-Su values.24 At lower temperatures, however, the skew parameter ii corresponding to the various classical path approximations tends to be too large and ( x ) too small according to approximations I and 11. Parameter ii for the classical distribution is nearly independent of T, and, hence, is much too large
Acknowledgment. This research was supported by the National Science Foundation under Grant No. CHE-7926480.
