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A variational treatment of the effects of vibrational anharmonicity on gas-phase electron diffraction intensities
Journal of Molecular THEOCHEM
Structure,
85 (1981)
325-335
Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
A VARIATIONAL TREATMENT OF THE EFFECTS OF VIBRATIONAL ANHARMONICITY ON GAS-PHASE ELECTRON DIFFRACTION INTENSITIES Part II. Temperature dependence
D. A. KOHL Department (U.S.A.)
of Chemistry,
The
University
of Texas
at
Austin, Austin,
Texas
78712
R. L. HILDERBRANDT Department (U.S.A.)
of Chemistry,
North
Dakota
State
University,
Fargo,
North
Dakota
58102
(Received 17 November 1980)
ABSTRACT The vibrational average of the electron diffraction intensity has been calculated for sulfur dioxide and carbon dioxide for T = 0, 300, 600, 800 and 1000 K using variational wavefunctions. Effective electron diffraction structural parameters were determined by fitting the numerical values of the vibrationally averaged cross section. Analytical expressions which relate those parameters to the vibrational wavefunction are also developed. INTRODUCTION
In a previous paper [ 11, variational wavefunctions for several triatomic molecules were used to calculate the vibrational average of gas-phase electron diffraction intensities at room temperature. Except for harnionic treatments, the temperature dependence of the cross section for polyatomic molecules has received very little attention [ 2,3]. Since several experimental studies of the temperature dependence of electron diffraction parameters have been reported [4-71, the methods developed in ref. 1 were extended to permit computation of vibrational averages at elevated temperatures. In addition to comparison with experiment, analytical expressions for the electron diffraction parameters will be presented. CALCULATION
OF THE WAVEFUNCTIONS
The Hamiltonian and had the form
was expressed
OlSS-1280/81/0000-0000/$02.75
in terms of the nominal
coordinates,
o 1981 Elsevier Scientific Publishing Company
qr,
326
Contributions to H which depend on rotational motion and/or the moments of inertia were omitted. For carbon dioxide, force fields determined by Suzuki [ 81 and by Cihla and Chedin [ 91 were employed. Both sets of force constants are similar but the latter force field includes quintic and sextic terms. For sulfur dioxide, force constants reported by Kuchitsu and Morino [lo] were adopted but with one modification. Both the cubic, 4 rrlYand the quartic, q,,, force constants for the bending coordinate were negative. This potential has maxima for both positive and negative displacements. Since this led to questionable results for the upper states, the quartic constant for sulfur dioxide was changed from -1.6 to 0.2. For the purposes of this paper, this modified potential should be sufficiently accurate to document the temperature dependence of the structural parameters. The number of populated vibrational states grows rapidly with increasing temperature so that, in principle, one needs to use an enormous number of basis functions in a variational calculation of a thermal average. For a molecule like SO*, there are roughly three hundred vibrational states which make a significant contribution to the electron diffraction cross section at 600 K so that at least one thousand basis functions would have to be used to insure high accuracy in each of the states in the thermal average. Since such a calculation cannot be done in core, the computation time would be prohibitive. Furthermore, available potential functions are of questionable accuracy for high levels of excitation so that one cannot justify such a calculation anyway. Although only three hundred basis functions could be used for an in-core variational calculation, the following argument is the basis for the method used. The dominant contribution to the vibrational average arises from the harmonic portion of the potential. Although accurate anharmonic wavefunctions are needed for the low lying states, a reasonable description of the density of states should be sufficient for the high vibrational states. To this level of approximation a sufficient number of states was obtained by a three-step process. As in part I [l] , an anharmonic basis set, {@j(qi)}, was constructed for each normal coordinate, qi, by separately diagonalizing all terms in the Hamiltonian which depend only on qi. Enough states were calculated in this step so that the highest energy was at least 14 000 cm-’ above the ground state. Next, a set of product states, { *j(qi)@,z( 9,). * - } was constructed from the single manifold basis functions. The complete Hamiltonian was then diagonalized in two 150 X 150 blocks for SO, and four 150 X 150 blocks for CO* using the set of states with the lowest (unperturbed) energies. If the set of states {x,,} and eigenvalues obtained from the diagonalization was not sufficient for UG11,n = l,%nax the thermal average (to be described below), these sets were augmented by the simple product states which were not included in the 150 X 150 blocks. The rationale for this procedure is based on the following simple case. Suppose that one considers two degenerate, orthogonal product states
321
xp’ and xg’ with energy E (O).If these two states interact only with each other, diagonalizing the Hamiltonian will produce two new states x?) = axi(O)+ b$ and xir(l) = a’~jO) + b’xg) with energies Ei’) and E,,11), respectively. The contribution of these states to the vibrational average of the electron diffraction cross section is through moments of the form (X(;)14f4$.
. . Ixf’))exp
This approximately [(a2 + P)
($‘Iqfq&.*
(xco) Iqfq’,. I
. .
I$))]
(-Ey)/RT) reduces
+ (x’I:)Iqlq~...Ix~:))exp
(-E$‘/RT)
to
1x:0’) + (b2 + b’2)(x’0’qfq& **1x2’) f 2(ab’ + a%) II exp (-E(O)/RT)
if Ef’ and E$k’ are large compared with RT and if the splittings, Ei” - E(O) and E(l)- E(O),are small. From the orthonormality conditions on x:” and onxII,l(Ifi ‘t can be shown that u2 + u12 - 1, b2 + b12 = 1 and ub’ + u’b = 0. Therefore, for small splittings at high temperature, the contribution to the moments can be calculated from the simple product states. The argument presented here is a special case of a more general argument based on the invariance of the trace to a linear transformation. Although this procedure does not quantitatively account for the exact position of each state in the total manifold of states, it should provide a good approximation to the thermal bath of states. For SO2 and C02, it was only necessary to augment the set of states for E, -Eo > 9000 cm-‘. With the states defined in this manner, the thermal averaged moments 2 hlqfq$
- * In)exp (-E,/RT)/(:exp
were calculated VIBRATIONAL
for use in the vibrational AVERAGING
+ F+TCij(s)(jo(srij))
average of the cross section.
OF THE CROSS SECTION
For a gas-phase polyatomic electrons is given by I(s) = Fli(s)
(-EJRT)
molecule,
the intensity,
vib
I(s), of scattered
(2)
where s = (4n/x)sin(e/2) is the momentum transfer, ii(s) is the scattered intensity from the ith atom, Cij is a product of atomic scattering amplitudes for a pair of atoms separated by a distance rij, and j,(x) is the zero-order spherical Bessel function. The ()vib denotes a thermal average over the set of vibrational states. It was shown in part I [l] that
(3) where rs is the equilibrium
internuclear
separation
(4)
328
matrix from massand by and ~2, are functions of the transformation weighted Cartesian to normal coordinates. (See eqns. 13 and 14 of part I [l] .) Once the various thermal averaged moments, (sfsi* - -) are calculated, the evaluation of the vibrational average of j,(srij) is straightforward. The radius of convergence of the sum in eqn. (3) depends on the rate at which the value of Fh decreases with increasing h. Since the rate of decrease of the sequence Fi, Fi+i gets slower as temperature increases (the moments increase rapidly as higher vibrational states are included), some method was needed to analytically continue the sequence of F values. The following proved to be a stable approach. Expressions were given in part I [l] which related the moments (A i-n) = ((r -- r,)” ) to a series expansion in the F’s. For high values of II, those expressions are not optimal for computation since the leading terms cancel. A more appropriate set of expressions for computational purposes is (A r)/r, = ,F, diFi
(5)
(Ar2)/rz = -2iz2diFi (Ar”)/rz = 4iz,diFi
+ X2diFi+l
(A r4)/rz = -giz4diFi
-4 i53diFi+ 1
(Ar’)/rZ
+12,F4diFi+,tiCSdiFi+2
= 16i5,diFi
where d, = l/2 di>l=
@a)
(-1)‘+‘(2i
-
3)/(2’i!)
(6b)
Written in this way, the nth moment of Ar depends on Fi > n and round-off error is minimized. These moments can be used to construct a distribution function, P,(r), which was assumed to have the form P,(r)
= [2n(Ar”)]-‘“exp
(-(Ar)2/2(Dr2j)ii0ciA?
(7)
Since (A?‘) = 7 drP,(r)Ar”
(8)
one caniet up a set of simultaneous equations and solve for the Ci’s. The Fi’S can be expressed in terms of the Ci’s and the moments (Ar”) since Fi = ((2r,Ar
+ Ar2)‘),ib/rzi
Assuming that the distribution and that Ci > 7 can be neglected,
(9) function P,(r) is dominated by the Gaussian then it is possible to calculate the Fi’S for
329
as large a value of i as is necessary for convergence of the sum in eqn. (3). By this method the original set {Fi}, i = 0, 11 was expanded to {Fi), i = 0, 29. The vibrational average of j,(sr) was calculated for the bonded and nonbonded distances in SO* and COZ for T = 0,300, 600,800 and 1000 K. Effective electron diffraction parameters, r,, I, and K were obtained by fitting the calculated curves with the conventional expression (jO(Sr)),ib
‘u
exp (-Z$s2/2)sin[s(r,
-KS’)]
/(sr,)
(10)
Values of the parameters as a function of temperature are presented in Tables 1 and 2 and plots of r, vs. 1: are shown in Figs. 1 and 2. As was found in part I [ 11, the agreement between eqns. (3) and (10) was excellent at all temperatures with typical standard deviations of low6 for 0 < s< 35 8-l. The effect of a finite basis was tested on SO2 by diagonalizing two 50 X 50 blocks (instead of two 150 X 150 blocks) and then augmenting this set of states with product states. Not surprisingly, the largest effect on the structural parameters occurred at the highest temperature. Relative to the results reported in Table 1 for T = 1000 K, r,, 1, and K increased by only O.Ol%, 0.274, and lo%, respectively. Before discussing the calculated results, we will develop a quantitative interpretation of the electron diffraction parameters.
TABLE
1
Temperature distribution
dependence of the electron functionb for SO,
diffraction
parametera
and the vibrational
C0 &)
Eqn. (24)
Eqn. (25)
;.k1,
$)
fi-3)
s-o 0 300 600 800 1000
1.4346 1.4347 1.4353 1.4359 1.4367
0.0352, 0.0353, 0.0371 0.0393 0.0418
0.58 0.60 0.97 1.5 2.3
0.59 0.61 1.0 1.5 2.2
0.60 0.62 1.0 1.6 2.2
0.9953 0.9955 0.9966 0.9976 0.9995
2.61 2.64 2.40 2.10 1.88
0.70 0.38 -1.88 -3.91 -7.30
315.0 320.0 396.0 439.0 421.0
2.4761 2.4762 2.4770 2.4780 2.4792
0.0526 0.0555 0.0655 0.0728 0.0799
0.56 0.76 2.0 3.5 5.6
0.56 0.79 2.1 3.6 5.6
0.56 0.77 2.2 3.5 5.4
0.9917 0.9924 0.9935 0.9939 0.9934
2.50 2.26 1.77 1.59 1.48
2.33 1.80 0.80 0.38 0.15
26.5 26.5 25.9 23.8 20.7
o...o 0 300 600 800 1000
aThe effect of centrifugal distortion was not included 2.4697 A were assumed. bSee eqn. (14).
and re values of 1.4308
A and
330 TABLE
2
Temperature distribution
dependence of the electron functionb for CO,’
diffraction
parametera
lo6 K(K3)
and the vibrational
CL?
Yk--I)
;k--2)
fOA)
Fit
Eqn. (24)
Eqn. (25)
1.1639 1.1640 1.1645 1.1651 1.1657
0.0347 0.0347 0.0353 0.0362 0.0375
0.53 0.53 0.63 0.80 1.0
0.54 0.55 0.65 0.82 1.1
0.55 0.56 0.65 0.82 1.1
0.9944 0.9940 0.9937 0.9935 0.9937
3.0 3.05 3.2 3.3 3.3
0.68 0.88 -0.16 -1.8 -4.2
314.0 315.0 336.0 364.0 390.0
2.3243 2.3240 2.3230 2.3223 2.3218
0.0397 0.0398 0.0416 0.0439 0.0466
0.53 0.54 0.80 1.2 1.9
0.52 0.53 0.77 1.2 1.8
0.48 0.49 0.73 1.1 1.6
0.9947 0.9954 0.9983 0.9992 0.9990
2.5 2.3 1.4 0.67 0.12
2.0 1.6 -0.1 -0.3 0.5
123.0 124.0 142.0 155.0 157.0
(TK) c-o 0 300 600 800 1000
o***o 0 300 600 800 1000
aThe effect of centrifugal distortion was not included and re values of 1.1600 a and 2.3200 A were assumed. bSee eqn. (16) for the definition of thedistribution function. the first four coefficients are shown. CAll results reported below were obtained from the Cihla and Chedin [9] potential.
INTERPRETATION
OF ELECTRON
DIFFRACTION
Only
PARAMETERS
Since it has been established that the discrepancies between the 1.h.s. and r.h.s of eqn. (10) are much less than experimental precision in measuring (jo(sr)),ib, we will consider eqn. (10) to be effectively an equality. The next task is to obtain suitable definitions of r=, I, and K so that they can be deduced directly from the vibrational wavefunction. Define a distribution function, P,(r), such that
_?drP,(r)j,(sr)
= exp (--lzs2/2)sin[s(r,
= (jo(sr)),ib
- KS’)] /(sr,)
(11)
0
Taking the sine transform gives P,(r)/r=
2/(7fr,)
7 ds exp (--bs’)
of s times eqn. (11) and expanding
[l + ~(a*/%
ar,)
+ (~*/2) (a”/%“)
sin [s(r,
- KS’)]
+ .**]x
(12)
sin sr sin sr,
0
where b = 1:/2. Two terms, exp (-(r - r,)*/4b) and exp (k(r result from the integral in eqn. (12). If the latter is neglected P,(r)/r
= [h-,6]-’
exp (-(r
+ (45 - K*/21:) (r-r,)’
- r,)‘/21:)
[l -
15~*/21:
+ (IL/~:) (r - r,)3 - (3~‘/2lA’)
-
+ r,)‘/46)
(3~/1:)
(r-r,)
(r - r,)4 + ***I (13)
331
(82)
103L2 o o 25
30
35
40
45
50
55
60
6 5 2 4795
24790
z _ 0 I P
I4370
24765
I4365
24780
I4360
24775
I4355
24770
14350
24765
4 0 ,c
247M)
I4345 120
125
130
135
140
145 lO3?
150
1.55 160
165
170
175
+JA2)
Fig. 1. Temperature dependence of the structural parameters for SO, for T = 0, 300, 600, 800 and 1000 K. The 54 bond parameters are indicated by 0 and correspond to the scales on the left and on the bottom. Two straight lines were drawn through the S-O points to emphasize the change in slope. The O---O parameters are indicated by 0. Centrifugal distortion corrections were not included.
I03g_,, (A21 1.5 r
1.6
I .?
I.6
I9
20
2.1
22 12.3245
. 2.3240 I .I660
.;
I.1655
-
I 1650
.
7 -”
.2.3225 I 1645
I 1640 t
I I20
I I 25
I30
1.35
I 40
c_o(b*) to312 Fig. 2. Temperature dependence of the structural parameters for CO, for 2’ = 0, 300, 600, 800 and 1000 K. The scales on the left and on the bottom correspond to the bond parameters. The points labeled l ((1) were obtained using the potentials reported by Cihla and Chedin [ 9 1. (Suzuki [ 8 ] .) Centrifugal distortion corrections were not included.
332
Expansion P,(r)
of l/r about ra, results in exp (-(r
= [l,$Zi-’
-
r,)‘/2Zi)
Fei (r -
(14)
rJi
where e. =
1 - 15~~121:
e, = l/r,-
(15a)
3~11~ - 15K2/21ir,
(15b)
e, = -3~jr,lz + 45~~121:
(15c)
e3 = K/l; + 45K2/21Zra
(15d)
Note that P, and P,, eqn, (7), have the same form but differ in the reference point. If one determines the distribution function from the wavefunction, the natural coordinates are r, and I, = ((r -- re)2)1’2. The corresponding natural coordinates for interpreting the diffraction experiment are ra and 1, = ((r - ra)2)1’2. The last step is to obtain the relationship between the two distribution functions. The relation between the coefficients of P,(r) and P,(r) is an extremely complex set of infinite series. A considerable simplification can be effected if one compares P,(r) and Pi< r), where Pi(r) = [Z,Jzl;]-’
exp (-(r
- r,)2/21z)
and the coefficients ci are obtained Since P,(r) = P,(r) = PI, exp [--(r-r,
Pa(r) = [E,$Fi]-’ =
exp [-(r
[Z,fi]-’
r,)‘/2li]
An expansion of eqn. (17a), followed eqn. (17b), results in e, = exp (--3c2/2Zi) (CA + c’,x + cix’
-
r,)i
(16)
by solving a set of simultaneous
+ r. -
i+,cl(r
r,)*/2Z~] Fei(r
iiocl(r -
-
ra + ra -
equations
r,)i
(17a)
rJi
(17b)
by a term-by-term
comparison
with
+ 0 (x3))
e, = exp (--~~/21:)
[c; + x(2ca -
e2 = exp (--~~/21~)
[cl + x(3ch - c;/Ez) + x2(6ck -
e3 = exp (--x2/212)
[c; + x(4&
(18)
cb/Zz) + x2(3c5 - c;/lz)
+ 0(X3)]
(19)
2c;lZz + cbl213 + 0(x”)] (20)
-
c;/Zz) + 3c2(10c; -
3c;lZi + c;/21:)
+
0(x3)1 (21)
where x = r, eqns. (15a-d),
r,. Since the ei’s are also related to ra, I, and K through
it is possible to solve for K in terms of ci, r, and I,. Both
333
e, and e, are related to K but neither of these relationships gave satisfactory values for K. However, the values of K obtained from either e, or e3 were in excellent agreement with the values of K obtained from the fitting procedure. Columns 4, 5, and 6 in Tables 1 and 2 demonstrate the stability of the relationships. Examination of the relative magnitudes of the terms in eqns. (19) and (21) showed that certain simple relations exist. To a very good approximation, at least for SO, and CO*, eqns. (19) and (21) can be replaced by e, = c; -xIl~
(22)
e3 = c;
(23)
When combined l/r, -
with the relationships
3~11: 2 c\ -
eqns. (15b)
(r, -- re)/la
ra -
this gives (24)
K = C;l; Equation
and (15d)
(25) (24) can be rearranged
r, = 3~112 --1:/r,
+ c’,lz
to obtain
either (26)
or ra -
r, =
(27)
since (Arj = c’,l% + 3c;14, + - - - . Equation (27) is the conventional expression for the r, to r, conversion in the absence of rotation. However, eqn. (26) affords an interpretation of the temperature dependence of ra displayed in Figs. 1 and 2. If K were proportional to 1: and if c’, were independent of temperature, then a plot of r, vs. 1: would be a straight line. Figures 1 and 2 show that none of the distance parameters are linear functions of 1: although high experimental precision would be required to detect the deviations from linearity. This is unfortunate since a linear relationship would have permitted a determination of r, by extrapolation of ra to the hypothetical value of 1 = 0. Examination of the individual contributions to ra show that c’,lz is the dominant term and that c’, and ~11: are not independent of temperature. Even though K is observable (with sufficient experimental precision) c’, and r, cannot both be determined from the temperature dependence of r, unless a model is introduced. The importance of cl is particularly evident for the 0. * * 0 distance in CO, where r, decreases with increasing temperature even though c;, c; and K are all positive. A detailed discussion of the origin of this shrinkage effect will be presented in a later paper. It should be emphasized that numerical values of rar 1, and K were obtained from two parallel approaches. The tabulated values of ra, 1, and K (fit) were the result of a least-squares fit to a numerical cross section. The use of a distribution function in an intermediate step only served to minimize a convergence problem with eqn. (3). An independent, analytical approach
334
based on P,(r) and PL (r) led to the same values of ra and 1, and to nearly the same values of K. (Compare columns 4, 5, and 6 of Tables 1 and 2.) Since the analytical method is more suitable for routine applications, the following is a step-by-step summary of the procedure for obtaining the diffraction parameters from the wavefunction: (1) Calculate the Fi’s from the vibrationally averaged moments of the normal coordinates. (See eqn. (4).) (2) Calculate ((r - r,)“) = (A?), n = 1, 2;.. using eqn. (5). (3) Determine ra and I, from (Ar) and
The initial impetus for this study and of Part I [l] was to examine the validity of the conventional expression used by electron diffractionists to interpret their data. To this end, all models other than the vibrational potential were strictly avoided. Numerical values of the vibrationally averaged cross section were fitted with an initially undefined set of parameters for a series of molecules at several temperatures. With the possible exception of hydrides, the conventional expression for the vibrational average of electron diffraction intensities is more accurate than we and probably anyone else suspected. Once the analytic expression was established, the relations between the vibrational wavefunctions, the thermally averaged distribution function, and the diffraction parameters were readily deduced. Although a numerical least-squares procedure was employed as an unbiased test, it has been adequately demonstrated that the analytical relationships presented here permit one to go directly from the vibrational wavefunctions to the diffraction parameters. Two very good potential functions for CO* were used which led to only minor differences in the calculated diffraction parameters. For larger molecules, anharmonic vibrational potential functions are not so well determined. It is quite possible that very precise diffraction measurements of the temperature dependence of ra could, in fact, provide valuable input into the determination of potential functions. While there is no substitute for spectroscopic observation of the fundamentals, combination and overtone bands, electron diffraction data may be an adequate source of information about the upper states. With this in mind, the present computer code is being expanded to permit application of these ideas to larger molecules.
335 ACKNOWLEDGEMENTS
D.A.K. acknowledges the Camille and Henry Dreyfus Foundation for their support of this research. R.L.H. acknowledges the support of The National Science Foundation, Grant No. CHE-7908614. Drs. Ed Shipsey and Peter Pulay provided valuable suggestions in the course of the work. REFERENCES 1 2 3 4
R. K. L. K.
L. Hilderbrandt and D. A. Kohl, J. Mol. Struct., Theochem, 85 (1981) 25 (Part I). Kuchitsu, Bull. Chem. Sot. Jpn., 40 (1967) 505. S. Bartell, J. Mol. Struct., 63 (1980) 259. Hedberg and M. Iwasaki, J. Chem. Phys., 36 (1962) 589; S. Samdal, D. M. Barnhart
and K. Hedberg, J. Mol. Struct., 35 (1976) 67. L. S. Bartell, S. K. Doun and S. R. Goates, J. Chem. Phys., 70 M. Kelley and M. Fink, J. Chem. Phys., in press. J. S. Bogard, Ph.D. Thesis, The University of Texas at Austin, I. Suzuki, J. Mol. Spectrosc., 25 (1968) 479; force field IV of the calculations. 9 Z. Cihla and A. Chedin, J. Mol. Spectrosc., 40 (1971) 337. 10 K. Kuchitsu and Y. Morino, Bull. Chem. Sot. Jpn., 38 (1965) 5 6 7 8
(1979)
4585.
Austin, Texas, 1977. Table 2 was used in
814.
Structure,
85 (1981)
325-335
Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
A VARIATIONAL TREATMENT OF THE EFFECTS OF VIBRATIONAL ANHARMONICITY ON GAS-PHASE ELECTRON DIFFRACTION INTENSITIES Part II. Temperature dependence
D. A. KOHL Department (U.S.A.)
of Chemistry,
The
University
of Texas
at
Austin, Austin,
Texas
78712
R. L. HILDERBRANDT Department (U.S.A.)
of Chemistry,
North
Dakota
State
University,
Fargo,
North
Dakota
58102
(Received 17 November 1980)
ABSTRACT The vibrational average of the electron diffraction intensity has been calculated for sulfur dioxide and carbon dioxide for T = 0, 300, 600, 800 and 1000 K using variational wavefunctions. Effective electron diffraction structural parameters were determined by fitting the numerical values of the vibrationally averaged cross section. Analytical expressions which relate those parameters to the vibrational wavefunction are also developed. INTRODUCTION
In a previous paper [ 11, variational wavefunctions for several triatomic molecules were used to calculate the vibrational average of gas-phase electron diffraction intensities at room temperature. Except for harnionic treatments, the temperature dependence of the cross section for polyatomic molecules has received very little attention [ 2,3]. Since several experimental studies of the temperature dependence of electron diffraction parameters have been reported [4-71, the methods developed in ref. 1 were extended to permit computation of vibrational averages at elevated temperatures. In addition to comparison with experiment, analytical expressions for the electron diffraction parameters will be presented. CALCULATION
OF THE WAVEFUNCTIONS
The Hamiltonian and had the form
was expressed
OlSS-1280/81/0000-0000/$02.75
in terms of the nominal
coordinates,
o 1981 Elsevier Scientific Publishing Company
qr,
326
Contributions to H which depend on rotational motion and/or the moments of inertia were omitted. For carbon dioxide, force fields determined by Suzuki [ 81 and by Cihla and Chedin [ 91 were employed. Both sets of force constants are similar but the latter force field includes quintic and sextic terms. For sulfur dioxide, force constants reported by Kuchitsu and Morino [lo] were adopted but with one modification. Both the cubic, 4 rrlYand the quartic, q,,, force constants for the bending coordinate were negative. This potential has maxima for both positive and negative displacements. Since this led to questionable results for the upper states, the quartic constant for sulfur dioxide was changed from -1.6 to 0.2. For the purposes of this paper, this modified potential should be sufficiently accurate to document the temperature dependence of the structural parameters. The number of populated vibrational states grows rapidly with increasing temperature so that, in principle, one needs to use an enormous number of basis functions in a variational calculation of a thermal average. For a molecule like SO*, there are roughly three hundred vibrational states which make a significant contribution to the electron diffraction cross section at 600 K so that at least one thousand basis functions would have to be used to insure high accuracy in each of the states in the thermal average. Since such a calculation cannot be done in core, the computation time would be prohibitive. Furthermore, available potential functions are of questionable accuracy for high levels of excitation so that one cannot justify such a calculation anyway. Although only three hundred basis functions could be used for an in-core variational calculation, the following argument is the basis for the method used. The dominant contribution to the vibrational average arises from the harmonic portion of the potential. Although accurate anharmonic wavefunctions are needed for the low lying states, a reasonable description of the density of states should be sufficient for the high vibrational states. To this level of approximation a sufficient number of states was obtained by a three-step process. As in part I [l] , an anharmonic basis set, {@j(qi)}, was constructed for each normal coordinate, qi, by separately diagonalizing all terms in the Hamiltonian which depend only on qi. Enough states were calculated in this step so that the highest energy was at least 14 000 cm-’ above the ground state. Next, a set of product states, { *j(qi)@,z( 9,). * - } was constructed from the single manifold basis functions. The complete Hamiltonian was then diagonalized in two 150 X 150 blocks for SO, and four 150 X 150 blocks for CO* using the set of states with the lowest (unperturbed) energies. If the set of states {x,,} and eigenvalues obtained from the diagonalization was not sufficient for UG11,n = l,%nax the thermal average (to be described below), these sets were augmented by the simple product states which were not included in the 150 X 150 blocks. The rationale for this procedure is based on the following simple case. Suppose that one considers two degenerate, orthogonal product states
321
xp’ and xg’ with energy E (O).If these two states interact only with each other, diagonalizing the Hamiltonian will produce two new states x?) = axi(O)+ b$ and xir(l) = a’~jO) + b’xg) with energies Ei’) and E,,11), respectively. The contribution of these states to the vibrational average of the electron diffraction cross section is through moments of the form (X(;)14f4$.
. . Ixf’))exp
This approximately [(a2 + P)
($‘Iqfq&.*
(xco) Iqfq’,. I
. .
I$))]
(-Ey)/RT) reduces
+ (x’I:)Iqlq~...Ix~:))exp
(-E$‘/RT)
to
1x:0’) + (b2 + b’2)(x’0’qfq& **1x2’) f 2(ab’ + a%) II exp (-E(O)/RT)
if Ef’ and E$k’ are large compared with RT and if the splittings, Ei” - E(O) and E(l)- E(O),are small. From the orthonormality conditions on x:” and onxII,l(Ifi ‘t can be shown that u2 + u12 - 1, b2 + b12 = 1 and ub’ + u’b = 0. Therefore, for small splittings at high temperature, the contribution to the moments can be calculated from the simple product states. The argument presented here is a special case of a more general argument based on the invariance of the trace to a linear transformation. Although this procedure does not quantitatively account for the exact position of each state in the total manifold of states, it should provide a good approximation to the thermal bath of states. For SO2 and C02, it was only necessary to augment the set of states for E, -Eo > 9000 cm-‘. With the states defined in this manner, the thermal averaged moments 2 hlqfq$
- * In)exp (-E,/RT)/(:exp
were calculated VIBRATIONAL
for use in the vibrational AVERAGING
+ F+TCij(s)(jo(srij))
average of the cross section.
OF THE CROSS SECTION
For a gas-phase polyatomic electrons is given by I(s) = Fli(s)
(-EJRT)
molecule,
the intensity,
vib
I(s), of scattered
(2)
where s = (4n/x)sin(e/2) is the momentum transfer, ii(s) is the scattered intensity from the ith atom, Cij is a product of atomic scattering amplitudes for a pair of atoms separated by a distance rij, and j,(x) is the zero-order spherical Bessel function. The ()vib denotes a thermal average over the set of vibrational states. It was shown in part I [l] that
(3) where rs is the equilibrium
internuclear
separation
(4)
328
matrix from massand by and ~2, are functions of the transformation weighted Cartesian to normal coordinates. (See eqns. 13 and 14 of part I [l] .) Once the various thermal averaged moments, (sfsi* - -) are calculated, the evaluation of the vibrational average of j,(srij) is straightforward. The radius of convergence of the sum in eqn. (3) depends on the rate at which the value of Fh decreases with increasing h. Since the rate of decrease of the sequence Fi, Fi+i gets slower as temperature increases (the moments increase rapidly as higher vibrational states are included), some method was needed to analytically continue the sequence of F values. The following proved to be a stable approach. Expressions were given in part I [l] which related the moments (A i-n) = ((r -- r,)” ) to a series expansion in the F’s. For high values of II, those expressions are not optimal for computation since the leading terms cancel. A more appropriate set of expressions for computational purposes is (A r)/r, = ,F, diFi
(5)
(Ar2)/rz = -2iz2diFi (Ar”)/rz = 4iz,diFi
+ X2diFi+l
(A r4)/rz = -giz4diFi
-4 i53diFi+ 1
(Ar’)/rZ
+12,F4diFi+,tiCSdiFi+2
= 16i5,diFi
where d, = l/2 di>l=
@a)
(-1)‘+‘(2i
-
3)/(2’i!)
(6b)
Written in this way, the nth moment of Ar depends on Fi > n and round-off error is minimized. These moments can be used to construct a distribution function, P,(r), which was assumed to have the form P,(r)
= [2n(Ar”)]-‘“exp
(-(Ar)2/2(Dr2j)ii0ciA?
(7)
Since (A?‘) = 7 drP,(r)Ar”
(8)
one caniet up a set of simultaneous equations and solve for the Ci’s. The Fi’S can be expressed in terms of the Ci’s and the moments (Ar”) since Fi = ((2r,Ar
+ Ar2)‘),ib/rzi
Assuming that the distribution and that Ci > 7 can be neglected,
(9) function P,(r) is dominated by the Gaussian then it is possible to calculate the Fi’S for
329
as large a value of i as is necessary for convergence of the sum in eqn. (3). By this method the original set {Fi}, i = 0, 11 was expanded to {Fi), i = 0, 29. The vibrational average of j,(sr) was calculated for the bonded and nonbonded distances in SO* and COZ for T = 0,300, 600,800 and 1000 K. Effective electron diffraction parameters, r,, I, and K were obtained by fitting the calculated curves with the conventional expression (jO(Sr)),ib
‘u
exp (-Z$s2/2)sin[s(r,
-KS’)]
/(sr,)
(10)
Values of the parameters as a function of temperature are presented in Tables 1 and 2 and plots of r, vs. 1: are shown in Figs. 1 and 2. As was found in part I [ 11, the agreement between eqns. (3) and (10) was excellent at all temperatures with typical standard deviations of low6 for 0 < s< 35 8-l. The effect of a finite basis was tested on SO2 by diagonalizing two 50 X 50 blocks (instead of two 150 X 150 blocks) and then augmenting this set of states with product states. Not surprisingly, the largest effect on the structural parameters occurred at the highest temperature. Relative to the results reported in Table 1 for T = 1000 K, r,, 1, and K increased by only O.Ol%, 0.274, and lo%, respectively. Before discussing the calculated results, we will develop a quantitative interpretation of the electron diffraction parameters.
TABLE
1
Temperature distribution
dependence of the electron functionb for SO,
diffraction
parametera
and the vibrational
C0 &)
Eqn. (24)
Eqn. (25)
;.k1,
$)
fi-3)
s-o 0 300 600 800 1000
1.4346 1.4347 1.4353 1.4359 1.4367
0.0352, 0.0353, 0.0371 0.0393 0.0418
0.58 0.60 0.97 1.5 2.3
0.59 0.61 1.0 1.5 2.2
0.60 0.62 1.0 1.6 2.2
0.9953 0.9955 0.9966 0.9976 0.9995
2.61 2.64 2.40 2.10 1.88
0.70 0.38 -1.88 -3.91 -7.30
315.0 320.0 396.0 439.0 421.0
2.4761 2.4762 2.4770 2.4780 2.4792
0.0526 0.0555 0.0655 0.0728 0.0799
0.56 0.76 2.0 3.5 5.6
0.56 0.79 2.1 3.6 5.6
0.56 0.77 2.2 3.5 5.4
0.9917 0.9924 0.9935 0.9939 0.9934
2.50 2.26 1.77 1.59 1.48
2.33 1.80 0.80 0.38 0.15
26.5 26.5 25.9 23.8 20.7
o...o 0 300 600 800 1000
aThe effect of centrifugal distortion was not included 2.4697 A were assumed. bSee eqn. (14).
and re values of 1.4308
A and
330 TABLE
2
Temperature distribution
dependence of the electron functionb for CO,’
diffraction
parametera
lo6 K(K3)
and the vibrational
CL?
Yk--I)
;k--2)
fOA)
Fit
Eqn. (24)
Eqn. (25)
1.1639 1.1640 1.1645 1.1651 1.1657
0.0347 0.0347 0.0353 0.0362 0.0375
0.53 0.53 0.63 0.80 1.0
0.54 0.55 0.65 0.82 1.1
0.55 0.56 0.65 0.82 1.1
0.9944 0.9940 0.9937 0.9935 0.9937
3.0 3.05 3.2 3.3 3.3
0.68 0.88 -0.16 -1.8 -4.2
314.0 315.0 336.0 364.0 390.0
2.3243 2.3240 2.3230 2.3223 2.3218
0.0397 0.0398 0.0416 0.0439 0.0466
0.53 0.54 0.80 1.2 1.9
0.52 0.53 0.77 1.2 1.8
0.48 0.49 0.73 1.1 1.6
0.9947 0.9954 0.9983 0.9992 0.9990
2.5 2.3 1.4 0.67 0.12
2.0 1.6 -0.1 -0.3 0.5
123.0 124.0 142.0 155.0 157.0
(TK) c-o 0 300 600 800 1000
o***o 0 300 600 800 1000
aThe effect of centrifugal distortion was not included and re values of 1.1600 a and 2.3200 A were assumed. bSee eqn. (16) for the definition of thedistribution function. the first four coefficients are shown. CAll results reported below were obtained from the Cihla and Chedin [9] potential.
INTERPRETATION
OF ELECTRON
DIFFRACTION
Only
PARAMETERS
Since it has been established that the discrepancies between the 1.h.s. and r.h.s of eqn. (10) are much less than experimental precision in measuring (jo(sr)),ib, we will consider eqn. (10) to be effectively an equality. The next task is to obtain suitable definitions of r=, I, and K so that they can be deduced directly from the vibrational wavefunction. Define a distribution function, P,(r), such that
_?drP,(r)j,(sr)
= exp (--lzs2/2)sin[s(r,
= (jo(sr)),ib
- KS’)] /(sr,)
(11)
0
Taking the sine transform gives P,(r)/r=
2/(7fr,)
7 ds exp (--bs’)
of s times eqn. (11) and expanding
[l + ~(a*/%
ar,)
+ (~*/2) (a”/%“)
sin [s(r,
- KS’)]
+ .**]x
(12)
sin sr sin sr,
0
where b = 1:/2. Two terms, exp (-(r - r,)*/4b) and exp (k(r result from the integral in eqn. (12). If the latter is neglected P,(r)/r
= [h-,6]-’
exp (-(r
+ (45 - K*/21:) (r-r,)’
- r,)‘/21:)
[l -
15~*/21:
+ (IL/~:) (r - r,)3 - (3~‘/2lA’)
-
+ r,)‘/46)
(3~/1:)
(r-r,)
(r - r,)4 + ***I (13)
331
(82)
103L2 o o 25
30
35
40
45
50
55
60
6 5 2 4795
24790
z _ 0 I P
I4370
24765
I4365
24780
I4360
24775
I4355
24770
14350
24765
4 0 ,c
247M)
I4345 120
125
130
135
140
145 lO3?
150
1.55 160
165
170
175
+JA2)
Fig. 1. Temperature dependence of the structural parameters for SO, for T = 0, 300, 600, 800 and 1000 K. The 54 bond parameters are indicated by 0 and correspond to the scales on the left and on the bottom. Two straight lines were drawn through the S-O points to emphasize the change in slope. The O---O parameters are indicated by 0. Centrifugal distortion corrections were not included.
I03g_,, (A21 1.5 r
1.6
I .?
I.6
I9
20
2.1
22 12.3245
. 2.3240 I .I660
.;
I.1655
-
I 1650
.
7 -”
.2.3225 I 1645
I 1640 t
I I20
I I 25
I30
1.35
I 40
c_o(b*) to312 Fig. 2. Temperature dependence of the structural parameters for CO, for 2’ = 0, 300, 600, 800 and 1000 K. The scales on the left and on the bottom correspond to the bond parameters. The points labeled l ((1) were obtained using the potentials reported by Cihla and Chedin [ 9 1. (Suzuki [ 8 ] .) Centrifugal distortion corrections were not included.
332
Expansion P,(r)
of l/r about ra, results in exp (-(r
= [l,$Zi-’
-
r,)‘/2Zi)
Fei (r -
(14)
rJi
where e. =
1 - 15~~121:
e, = l/r,-
(15a)
3~11~ - 15K2/21ir,
(15b)
e, = -3~jr,lz + 45~~121:
(15c)
e3 = K/l; + 45K2/21Zra
(15d)
Note that P, and P,, eqn, (7), have the same form but differ in the reference point. If one determines the distribution function from the wavefunction, the natural coordinates are r, and I, = ((r -- re)2)1’2. The corresponding natural coordinates for interpreting the diffraction experiment are ra and 1, = ((r - ra)2)1’2. The last step is to obtain the relationship between the two distribution functions. The relation between the coefficients of P,(r) and P,(r) is an extremely complex set of infinite series. A considerable simplification can be effected if one compares P,(r) and Pi< r), where Pi(r) = [Z,Jzl;]-’
exp (-(r
- r,)2/21z)
and the coefficients ci are obtained Since P,(r) = P,(r) = PI, exp [--(r-r,
Pa(r) = [E,$Fi]-’ =
exp [-(r
[Z,fi]-’
r,)‘/2li]
An expansion of eqn. (17a), followed eqn. (17b), results in e, = exp (--3c2/2Zi) (CA + c’,x + cix’
-
r,)i
(16)
by solving a set of simultaneous
+ r. -
i+,cl(r
r,)*/2Z~] Fei(r
iiocl(r -
-
ra + ra -
equations
r,)i
(17a)
rJi
(17b)
by a term-by-term
comparison
with
+ 0 (x3))
e, = exp (--~~/21:)
[c; + x(2ca -
e2 = exp (--~~/21~)
[cl + x(3ch - c;/Ez) + x2(6ck -
e3 = exp (--x2/212)
[c; + x(4&
(18)
cb/Zz) + x2(3c5 - c;/lz)
+ 0(X3)]
(19)
2c;lZz + cbl213 + 0(x”)] (20)
-
c;/Zz) + 3c2(10c; -
3c;lZi + c;/21:)
+
0(x3)1 (21)
where x = r, eqns. (15a-d),
r,. Since the ei’s are also related to ra, I, and K through
it is possible to solve for K in terms of ci, r, and I,. Both
333
e, and e, are related to K but neither of these relationships gave satisfactory values for K. However, the values of K obtained from either e, or e3 were in excellent agreement with the values of K obtained from the fitting procedure. Columns 4, 5, and 6 in Tables 1 and 2 demonstrate the stability of the relationships. Examination of the relative magnitudes of the terms in eqns. (19) and (21) showed that certain simple relations exist. To a very good approximation, at least for SO, and CO*, eqns. (19) and (21) can be replaced by e, = c; -xIl~
(22)
e3 = c;
(23)
When combined l/r, -
with the relationships
3~11: 2 c\ -
eqns. (15b)
(r, -- re)/la
ra -
this gives (24)
K = C;l; Equation
and (15d)
(25) (24) can be rearranged
r, = 3~112 --1:/r,
+ c’,lz
to obtain
either (26)
or ra -
r, =
(27)
since (Arj = c’,l% + 3c;14, + - - - . Equation (27) is the conventional expression for the r, to r, conversion in the absence of rotation. However, eqn. (26) affords an interpretation of the temperature dependence of ra displayed in Figs. 1 and 2. If K were proportional to 1: and if c’, were independent of temperature, then a plot of r, vs. 1: would be a straight line. Figures 1 and 2 show that none of the distance parameters are linear functions of 1: although high experimental precision would be required to detect the deviations from linearity. This is unfortunate since a linear relationship would have permitted a determination of r, by extrapolation of ra to the hypothetical value of 1 = 0. Examination of the individual contributions to ra show that c’,lz is the dominant term and that c’, and ~11: are not independent of temperature. Even though K is observable (with sufficient experimental precision) c’, and r, cannot both be determined from the temperature dependence of r, unless a model is introduced. The importance of cl is particularly evident for the 0. * * 0 distance in CO, where r, decreases with increasing temperature even though c;, c; and K are all positive. A detailed discussion of the origin of this shrinkage effect will be presented in a later paper. It should be emphasized that numerical values of rar 1, and K were obtained from two parallel approaches. The tabulated values of ra, 1, and K (fit) were the result of a least-squares fit to a numerical cross section. The use of a distribution function in an intermediate step only served to minimize a convergence problem with eqn. (3). An independent, analytical approach
334
based on P,(r) and PL (r) led to the same values of ra and 1, and to nearly the same values of K. (Compare columns 4, 5, and 6 of Tables 1 and 2.) Since the analytical method is more suitable for routine applications, the following is a step-by-step summary of the procedure for obtaining the diffraction parameters from the wavefunction: (1) Calculate the Fi’s from the vibrationally averaged moments of the normal coordinates. (See eqn. (4).) (2) Calculate ((r - r,)“) = (A?), n = 1, 2;.. using eqn. (5). (3) Determine ra and I, from (Ar) and
The initial impetus for this study and of Part I [l] was to examine the validity of the conventional expression used by electron diffractionists to interpret their data. To this end, all models other than the vibrational potential were strictly avoided. Numerical values of the vibrationally averaged cross section were fitted with an initially undefined set of parameters for a series of molecules at several temperatures. With the possible exception of hydrides, the conventional expression for the vibrational average of electron diffraction intensities is more accurate than we and probably anyone else suspected. Once the analytic expression was established, the relations between the vibrational wavefunctions, the thermally averaged distribution function, and the diffraction parameters were readily deduced. Although a numerical least-squares procedure was employed as an unbiased test, it has been adequately demonstrated that the analytical relationships presented here permit one to go directly from the vibrational wavefunctions to the diffraction parameters. Two very good potential functions for CO* were used which led to only minor differences in the calculated diffraction parameters. For larger molecules, anharmonic vibrational potential functions are not so well determined. It is quite possible that very precise diffraction measurements of the temperature dependence of ra could, in fact, provide valuable input into the determination of potential functions. While there is no substitute for spectroscopic observation of the fundamentals, combination and overtone bands, electron diffraction data may be an adequate source of information about the upper states. With this in mind, the present computer code is being expanded to permit application of these ideas to larger molecules.
335 ACKNOWLEDGEMENTS
D.A.K. acknowledges the Camille and Henry Dreyfus Foundation for their support of this research. R.L.H. acknowledges the support of The National Science Foundation, Grant No. CHE-7908614. Drs. Ed Shipsey and Peter Pulay provided valuable suggestions in the course of the work. REFERENCES 1 2 3 4
R. K. L. K.
L. Hilderbrandt and D. A. Kohl, J. Mol. Struct., Theochem, 85 (1981) 25 (Part I). Kuchitsu, Bull. Chem. Sot. Jpn., 40 (1967) 505. S. Bartell, J. Mol. Struct., 63 (1980) 259. Hedberg and M. Iwasaki, J. Chem. Phys., 36 (1962) 589; S. Samdal, D. M. Barnhart
and K. Hedberg, J. Mol. Struct., 35 (1976) 67. L. S. Bartell, S. K. Doun and S. R. Goates, J. Chem. Phys., 70 M. Kelley and M. Fink, J. Chem. Phys., in press. J. S. Bogard, Ph.D. Thesis, The University of Texas at Austin, I. Suzuki, J. Mol. Spectrosc., 25 (1968) 479; force field IV of the calculations. 9 Z. Cihla and A. Chedin, J. Mol. Spectrosc., 40 (1971) 337. 10 K. Kuchitsu and Y. Morino, Bull. Chem. Sot. Jpn., 38 (1965) 5 6 7 8
(1979)
4585.
Austin, Texas, 1977. Table 2 was used in
814.