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The watson U(ρ) term in the one-dimensional Semirigid Bender Hamiltonian for the HF dimer
JOURNAL
OF MOLECULAR
SPECTROSCOPY
150,5 1I-520 (199 1)
The Watson U(p) Term in the One-Dimensional Semirigid Bender Hamiltonian for the HF Dimer V. C. EPA AND P. R. BUNKER Herzberg Institute of Astrophysics. National Research Council of Canada, Ottawa. Ontario, Canada KIA OR6 We have recalculated the minimum energy path for the trans-bending tunneling motion of the HF dimer using mass-scaled Cartesian coordinates, and have set up the appropriate Semirigid Bender wave equation. Using Hougen-Bunker-Johns axes that keep Z,, = 0, we can calculate the full Watson U(p) term as a function of the tunneling coordinate p. The rapid change in the magnitude of Rob (the distance between the centers of mass of the two HF fragments), along the minimum energy trans-tunneling path, causes zl(p) to become very large in the vicinity of the equilibrium structure. For the calculation of the K-type rotation and trans-tunneling energies it is possible to set up a one-dimensional model Hamiltonian, as we did previously, by neglecting the variation of Rob with p. We do this in an optimization of the minimum energy path to the currently available term values in order to obtain an eficctive minimum energy path and predicted term values. 0 1991 Academic F’ress. Inc.
1. INTRODUCTION
In previous papers of ours (I-3) on the HF dimer an analytical model was developed for the six-dimensional ab initio potential energy surface, a minimum energy path was determined, and a one-dimensional Semirigid Bender calculation of the K-type rotation and truns-bending tunneling energy levels was made. In the present work we consider the effect of the variation of the intermolecular distance Rab, in the dimer, with the trans-bending tunneling coordinate p, and in the process we calculate the full Watson U(p) term (4, 5). As discussed in Ref. (5) this requires that we use a molecule fixed axis system oriented so that I,, = 0, as prescribed by Hougen, Bunker, and Johns (6). We find that U(p) becomes very large in the region of the equilibrium structure because of the rapid variation of Rab with p. The rapid variation of Rab with p means that it is necessary to treat Rab and p simultaneously in order to calculate accurately the energy levels, particularly for end-over-end rotation which involves the I,, and Z,, moments of inertia. However, for modeling the K-type rotation and trunstunneling energies of the o4 = 0 states ( v4 is the Rob stretch) we can use the onedimensional Semirigid Bender Hamiltonian (with R,b held constant), and are able to achieve a good fitting to the observed term values by optimizing the efictive minimum energy path. We hope that our predicted term values are of use in analyzing the spectrum, although they do not allow for any interaction between the truns-bending and &-StretChing states.
511
0022-2852191 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
512
EPA AND BUNKER
II. THE MINIMUM
ENERGY PATH AND ALTERNATIVE
COORDINATE
AXIS SYSTEMS
We have recalculated the minimum energy path for trans-bending tunneling, from the analytical six-dimensional ab initio potential energy surface for the HF dimer described in previous papers (3)) using Cartesian coordinates scaled by the square root of the atomic masses. One-half of this path is given in Table I, and it connects the linear C,, saddle point structure, the C, equilibrium structure, and the symmetric C&hsaddle point structure. We use mass-scaled Cartesian coordinates in the steepest descent method to determine the minimum energy path since this yields a path independent of any internal coordinate choice. The path is almost negligibly different from those obtained previously (2, 3) using internal coordinates. Figure 1 shows the definition of the six internal coordinates (rr , r2, Rob,4, 02, T)for the ( HFh molecule. As before we introduce the angular coordinates p and G as P =
(02 +
Q =
(02 -&I/%
(1)
&J/2
and
(2)
TABLE I The Values of the Coordinate? along the Minimum Energy Pathb in the Analytical ab initio Six-Dimensional Potential
0.9217 0.9216 0.9216 0.9216 0.9216 0.9219 0.9225 0.9234 0.9235 0.9237 0.9237 0.9238 0.9238 0.9239 0.9239 0.9239 0.9238 0.9237 0.9235 0.9233 0.9231 0.9229 0.9226 0.9224
0.9208 0.9211 0.9212 0.9213 0.9212 0.9212 0.9211 0.9214 0.9216 0.9218 0.9219 0.9221 0.9222 0.9223 0.9226 0.9228 0.9229 0.9231 0.9222 0.9221 0.9222 0.9222 0.9223 0.9224
2.8881 2.8877 2.8869 2.8851 2.8829 2.8803 2.8767 2.8650 2.8286 2.7988 2.7844 2.7661 2.7519 2.7557 2.7543 2.7517 2.7488 2.7446 2.7414 2.7303 2.7356 2.7332 2.7320 2.7316
0.0
0.0
0.0
7.7 13.3 21.0 27.5 33.8 41.1 51.8 54.8 57.5 58.8 60.6 62.1 66.9 69.0 73.1 77.9 85.0 91.2 97.3 103.7 111.5 117.5 122.7
0.0 0.0 0.0
0.0 0.0 2.2 3.6 5.0 5.7 6.7 7.5 10.5 11.9 14.6 18.2 23.6 28.4 33.0 38.1 44.7 50.2 55.3
a Bond lengths in A, angles in degrees, and energies in cm-‘. b Obtained using mass-scaled Cartesian coordinates. c p = (8, + !9,)/2.
180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.9 180.0 180.0 180.0 180.0
337.9
325.0 300.0 250.0 200.0 150.0 100.0 50.0 25.0 10.0 5.0 1.0 0.0 5.0 10.0 25.0 50.0 100.0 150.0 200.0 250.0 300.0 325.0 334.5
THE WATSON U-TERM
513
FIG. 1. The definition of the internal coordinates. (I and b are the HF centers-of-mass, 0, = LH,ab, e2 =: 180” - LH*ba, and 7 is the angle between the H,ab and abHz planes measured in a right-handed sense about a + b from HIab. 0 =SfJ,, & G 180”, and 0 G T < 360”.
where 0” G p G 180”, and p describes the truns-tunneling (geared-bend) motion. Although rl and r2 differ slightly from 0.9220 A along the minimum energy path, in subsequent calculations we hold them fixed at this value. It is noteworthy that Rab changes rapidly in the vicinity of the equilibrium geometry (pe = 34.8”) from a limiting value of 2.888 A at the linear saddle point to a limiting value of 2.732 A at the C2h saddle point. A hyperbolic switching function is one way to fit such behavior (2, .3) and we obtain, J&(A)
= (RG)(l
- tanh[cl(p
-
~211)+ (R1/2)(1 - tanh[cl(r - P - cdl)
+ t&/4)(1 - tanh[cl(cz - ~11) (1 - tanh[cl(p ~ r + c2>1), (3) where p is in radians, R, = 2.888 A, and R2 = 2.732 A; cl = 7.3288 (radian)-’ c2 = 0.5296 radian are the least-squares fitted parameters. We also fit I’mi, and 6 along the minimum energy path as functions of p:
V,,,in(cm-l) = 32.9 - 155.7
cos
and
2p + 162.4 c0s22p + 51.6 COS~~/I + 138.7 cos42p + 105.4 cos’2p,
(4)
and a( radian) = 1.1726 sin p - 0.6042 sin 2p.
(5)
Eqs. ( 3)-( 5 ) are not significantly different from the analogous equations in Refs. (2, 3). Following the usual arguments in deriving the one-dimensional Semirigid Bender Hamiltonian (see Refs. (6-8))) we can write the wave equation for the truns-tunneling and K-type rotation as: $
~K,U(P) = {fi(~) + (~zzl~Lpp)K’ +
(6) [2/(h2/+p)1[~min- EK,~~}~JK,~(P),
where pzz and pPPare the zz and pp elements of the P matrix, the inverse of the 4 x 4 extended moment of inertia matrix, Z (see Eqs. (6)-( 14) in Ref. (2) ; note that Eq. ( 10) in Ref. (2) should have a minus sign on the right hand side) _The functionfi ( p) in Eq. (6) is derived from U’(p) (in the notation of Eqs. (22)-( 24) of Ref. (5)) and it is discussed further in the next section.
EPA AND BUNKER
514
The values of the elements of the CLmatrix will depend on the definition adopted for the molecule fixed axes. Previously (2, 3) the right-handed xyz axis system was chosen with the z-axis connecting the monomer centers of mass a + b, the y-axis in the plane of the dimer with Hi having positive y-coordinate, and the x-axis perpendicular to the plane of the dimer; we call this axis system “the standard axis system.” We investigate choosing coordinate axis systems such that either IYz= 0 or IX, = 0, since these choices remove off-diagonal matrix elements occuring in the calculation of the end-over-end rotational energy levels (see Eq. ( 17) of Ref. ( 7)). It is interesting to compare the orientations of these axis systems with that of the standard axis system. (i) Iyz= 0, The Principal Axis System We define the new axes 9 and Z as being rotated clockwise about the x-axis (looking along the x-axis) by an angle of E from the y- and z-axes. The new X-axis is thus unchanged from the x-axis. In order for IYz= 0 we must have -2 C 1 tan-’ 2 C i mi(Zf
i?ZiyiZi
t = -
-VT)
I ’
and hence
where u = C mi yizi , v = are in the old (standard) respect to p.
- J$ ), the coordinates Xi, yi, zi of the four nuclei axes system, and the primes refer to differentiation with
C mi ( Z:
(ii) I,, = 0, The Hougen-Bunker-Johns Axis System (6) To obtain IX, = 0 we must have de -= dp
C mi(yiz: - ziy:) CRZi(yf +Z:)
(9)
’
which leads to c(p = j) =
s ’
dp
C mi(yiZ: - Ziy:) C Wli(yf +
p=o
Zf)
’
(10)
The angle t is computed from the relationships given above for each of the points of integration from p = 0 to p = a. Figure 2 shows the variation oft with p along the minimum energy path for these two alternate axis systems; the standard axis system has c = 0. We see that t remains small for both of these new axis systems at all p values. Once c is known the transformation to the new coordinates can be made using Ji =
YiCOS
Zi =
-y&l
Xi =
Xi,
t + t +
ZiSill
E,
ZiCOS E,
(1 la) (lib) (1lC)
515
THE WATSON U-TERM
-0.7
FIG, 2. The angle e as a function of p for the cases of Iyz = 0 (dotted line) and Ix,, = 0 (full line)
and I and CLcalculated in the new coordinate system; Eq. (6) can then be set up and solved. III. THE FUNCTION J(p)
AND THE MAGNITUDE
OF U(p)
The function J (p) which appears in Eq. (6) above is derived from the U’(p) term (5, 6) by using the expression f,(P)
= &
(12)
WP).
As discussed in Ref. (5) U’(p) is given by U(p) plus extra terms that result from subjecting the Hamiltonian to the transformation (1P,)-“2H( I,,,) ‘I2 in order to remove terms linear in J, . In previous Semirigid Bender calculations on (I-IF), , the asymptotic approximation (valid for p + 0 or p + 7~)of f,(P)
= -[l/p2
+ l/(r
- PJ21/4
(13)
was used, but this approximation is valid only if the variation of Rob with p can be neglected. In Ref. (2) the variation of Rab with p was also neglected in the calculation of Ppp. However, in the minimum energy path there is a rapid change in Rab in the region around the potential minima (see Table I). Using the part of I!?( p ) called Vi ( p ) (5)) whose asymptotic form is given by Eq. ( 13 ) above, (see Eq. (43 ) of Ref. (6)) we have
516
EPA AND BUNKER
where p is the determinant of the inverse of the 4 X 4 matrix I. Figure 3 compares f,(p) calculated from U:(p) (i.e., Eq. ( 14)) with that of the asymptotic approximation of Eq. ( 13);fi (p) has been evaluated numerically at each point of integration of the Semirigid Bender equation (typically about 3000 points) using a five-point recursion formula for the differentiation. Obviously, while the approximation is very good in the asymptotic limits of p + 0 and p --) ?r, in the neighborhood of p - 30” and 150” it is markedly different. The non-negligible magnitude of the derivative of Rab with respect to p in the vicinity of the C, equilibrium structures has a dramatic effect on the behaviour of I,,, pLpp,and therefore on p and the function f,(p) in this region. Figure 4 illustrates ppP as a function of p, using the Ix, = 0 axis system. At this point it should be noted that although the representation offi by Eq. ( 14) is more accurate than using Eq. ( 13), it does not use the full U’(p) term and only uses U:(p). The complexity of the evaluation of the full U’(p) term led in the past to computing only the U, ( p) portion of it. However, Sarka and Bunker (5) have shown that, if a coordinate axis system such that I,, = 0 is used, evaluation of the full U’(p) term becomes relatively easy, and is given by
30’o I 25.0
T
0.0
-
-
5 ‘% 1
e
V
-25.0
-
-50.0
-
-75.0
-
\
P
u-
-100.0
I’ 0.0
I
15.0
1
I
I
I
30.0
45.0
60.0
75.0
I
90.0.
P
1
105.0
I
120.0
I
135.0
I
150.0
I
165.0
180.0
/deg.
FIG. 3. The functionA calculated from U:(p) (Eq. (14)) (dashed line), from U’(p) (Eqs. (12) and ( 15)) (full line), and from the asymptotic approximation ( Eq. ( 13 )) (chain dots).
THE
0.0
1 0.0
WATSON
I
I
I
I
I
15.0
30.0
45.0
60.0
75.0
517
(I-TERM
I
90.0
I
I
I
I
I
105.0
120.0
135.0
150.0
165.0
J
16 SO.0
P /deg. FK; . 4. The variation of ppLpp( p ) with p (dotted at 2.81 .& I,, = 0 axes are used.
U’(P)
line). The full line indicates
the result if Rd is held fixed
=
(15)
where c’(p) is given by Eq. (24) in Ref. (5). (c’(p) involves elements of the 4 X 4 I matrix and derivatives with respect to p). We have, therefore, also calculated fi ( p) using the full U’(p) but as seen in Fig. 3 the use of the full V’(p) term, instead of just Vi ( p), to calculate f; (p) makes little difference. In Fig. 5 we show I’min( Vmin(P) + U’,(p), and V&n(P) + U’(p) SO that the huge effect of U’(p) around the equilibrium nuclear geometries can be appreciated. The large value of V’(p) results from the lack of commutation of J, with pPPand II, which in turn is due to the strong dependence of pPPon p in the region of p = pe. IV. ENERGIES CALCULATED KEEPING Rob CONSTANT AND FITTING TO EXPERIMENTAL DATA
Clearly, if we keep the value of Rob constant at the mean of Rab( C,,) and Ra6( C2,,) (i.e., making Rla,, = 0)) the anomalous behavior of U( p ) disappears and energies and tunneling splittings close to those in Ref. (2) are recovered (see Table II). These term values are close to the experimental values (9-l 1). Using this fixed value of Rob we have adjusted the parameters in I’min and a(p) in a weighted least-squares fitting to the nine experimentally available ( 9-Z I ) K-type rotation and trans-tunneling energies
518
EPA AND BUNKER
i
E < -
100.0
0.0
-100.0
-200.0
0.0
I
15.0
I
30.0
I
45.0
I
60.0
I
75.0
I
90.0
I
105.0
I
I
120.0
135.0
I
I
150.0 165.0
160.0
P /deg.
FIG. 5. The potentials Vmin(p)(chain dots), V,i”(p) + U;(p) (dashes), and vmin(P) + V’(p) (full line) as functions of p.
all involving u4 = 0 (as wd as vI = v2 = v3 = 216 = 0). Our m&iVation for doing this is to improve the predictions and to aid in explaining some of the unassigned structure in the experimental spectrum using a model that is capable of fitting the experimental term values. A weight of 100.0 was given to fitting the (K = 0, v = 1) term value, while the others were given a weight of 1.0. The standard deviation of the weighted fitting was 0.49 cm-‘. The fitted parameters are: V/in(cm-‘)
= 24.7 - 170.4
cos
2p + 256.8
c0s22p
afi’(radian) = 0.6647 sin p.
+
133.1
c0s32p,
(16)
(17)
The resultant energies are given in Table II and compared with experiment. In Fig. 6 we compare I’mi,( p) and I’ $,,( p). We should view this fitted minimum energy path as an effective minimum energy path in which neglected effects caused by using this approximate kinetic energy operator are compensated for within the framework of this one-dimensional model. V. DISCUSSION AND CONCLUSION
A minimum energy path for the truns-bend tunneling motion in the HF dimer has been obtained using a steepest-descent path in mass-scaled Cartesian coordinates on
THE WATSON U-TERM
519
TABLE II The truns-Tunneling and K-Type Rotation Energy Levels of ( HFb
ab ~nr!.to* 4
3
2
rxpt.d
3 2 1
605.52 620.33 518.53 535.29 378.67 386.90
0
369.31 379.46
3 2
446.21 458.08 383.04 393.49
393.550(u)
1 0
228.62 236.32 224.23 232.40
236.516 232.766
3 2 1 0
319.63 278.25 111.44 109.40
328.02 284.18 118.20 116.28
118.137 116.133
2 1 0
227.66 230.24 203.18 205.02 32.32 36.50 31.29 35.56
36.489 35.425
3 2 1 0
176.58 166.69 162.66 155.17 0.70 0.56 0.0 0.0
0.659 0.0
13
0
fittedc
386.727(I0)
a In cm-‘. b Calculated using the ab initio Vminand c (Eqs. (4) and (5)), with Rnb = 2.8 10 A. ’ Calculated using the fitted I’,,,,, and 0 (Eqs. ( 16) and ( 17)), with Rnb = 2.810 A. d From Ref. (9) unless otherwise indicated.
the analytical six-dimensional ab initio potential surface determined in Ref. (3). The path is only very slightly different from that deduced in Ref. (3) using internal coordinates. The position along the minimum energy path is defined by the angle p, and there is a rapid change in the monomer-monomer separation Rab as the molecule passes through the equilibrium structure along the path (see (Table I). As a result of the rapid variation of Rab with p, the reduced mass pPP varies strongly with p, and, therefore, the commutator [J,, CLJ is significant. This causes the Watson term U(p) (4, 5) to be very much larger than is normally the case. However, we can set up a model Hamiltonian for the K-type rotation and trans-bending motion with Rnb held fixed, and use it in a fitting to v4 = 0 data to determine an efictive minimum energy path and to predict term values. It would be better to treat Rab and p (and K-rotation ) together in a two-dimensional Semirigid Bender calculation, but this is best dealt with in a separate paper. ACKNOWLEDGMENT We are very grateful to Dr. Tucker Carrington, Jr. for advice on determining the minimum energy path and for many discussions.
520
EPA
AND
BUNKER
4oo’o 1 350.0
300.0
‘i
E <
250.0
200.0
Q
Y
150.0 100.0
50.0
0.0
P /deg. FIG, 6. The potentials
RECEIVED:
V,,,in( p) (full) and V pi” ( p ) (dashes)
from Eqs. (4) and ( 16 ) , respectively.
July 23, 1991 REFERENCES
1. P. R. BUNKER,M. KOFRANEK,H. LISCHKA,AND A. KARPF-EN,.I. Chem. Phys. 89,3002-3007 ( 1988). 2. P. R. BUNKER, T. CARRINGTONJR., P. C. G~MEZ, M. D. MARSHALL,M. KOFRANEK,H. LISCHKA, AND A. KARPFEN,J. Chem. Phys. 91,5154-5159 (1989). 3. P. JENSEN,P. R. BUNKER,A, KARPFEN,M. KOFRANEK,AND H. LISCHKA,.I. Chem. Phys. 93,, 62666280 (1990). 4. J. K. G. WATSON, Mol. Whys.15,479-490 (1968). 5. K. SARKA AND P. R. BUNKER, J. Mol. Spectrosc. 122,259-268 ( 1987). 6. J. T. HOUGEN, P. R. BUNKER, AND J. W. C. JOHNS,.I. Mol. Spectrosc. 34, 132-172 ( 1970). 7. P. R. BUNKER, AND J. M. R. STONE,J. Mol. Spectrosc. 41, 310-332 (1972). 8. P. R. BUNKER, AND B. M. LANDSBERG,J. Mol. Spectrosc. 67, 374-385 ( 1977). 9. W. J. LAFFERTY,R. D. SUENRAM,AND F. J. L~VAS, J. Mol. Spectrosc. 123,434-452 (1987). 10. K. VON P~TTKAMER,M. QUACK, AND M. A. SUHM, Mol. Phys. 65, 1025-1045 (1988). Il. M. QUACK AND M. A. SUHM, Chem. Phys. Letts. 171,517-524 (1990).
OF MOLECULAR
SPECTROSCOPY
150,5 1I-520 (199 1)
The Watson U(p) Term in the One-Dimensional Semirigid Bender Hamiltonian for the HF Dimer V. C. EPA AND P. R. BUNKER Herzberg Institute of Astrophysics. National Research Council of Canada, Ottawa. Ontario, Canada KIA OR6 We have recalculated the minimum energy path for the trans-bending tunneling motion of the HF dimer using mass-scaled Cartesian coordinates, and have set up the appropriate Semirigid Bender wave equation. Using Hougen-Bunker-Johns axes that keep Z,, = 0, we can calculate the full Watson U(p) term as a function of the tunneling coordinate p. The rapid change in the magnitude of Rob (the distance between the centers of mass of the two HF fragments), along the minimum energy trans-tunneling path, causes zl(p) to become very large in the vicinity of the equilibrium structure. For the calculation of the K-type rotation and trans-tunneling energies it is possible to set up a one-dimensional model Hamiltonian, as we did previously, by neglecting the variation of Rob with p. We do this in an optimization of the minimum energy path to the currently available term values in order to obtain an eficctive minimum energy path and predicted term values. 0 1991 Academic F’ress. Inc.
1. INTRODUCTION
In previous papers of ours (I-3) on the HF dimer an analytical model was developed for the six-dimensional ab initio potential energy surface, a minimum energy path was determined, and a one-dimensional Semirigid Bender calculation of the K-type rotation and truns-bending tunneling energy levels was made. In the present work we consider the effect of the variation of the intermolecular distance Rab, in the dimer, with the trans-bending tunneling coordinate p, and in the process we calculate the full Watson U(p) term (4, 5). As discussed in Ref. (5) this requires that we use a molecule fixed axis system oriented so that I,, = 0, as prescribed by Hougen, Bunker, and Johns (6). We find that U(p) becomes very large in the region of the equilibrium structure because of the rapid variation of Rab with p. The rapid variation of Rab with p means that it is necessary to treat Rab and p simultaneously in order to calculate accurately the energy levels, particularly for end-over-end rotation which involves the I,, and Z,, moments of inertia. However, for modeling the K-type rotation and trunstunneling energies of the o4 = 0 states ( v4 is the Rob stretch) we can use the onedimensional Semirigid Bender Hamiltonian (with R,b held constant), and are able to achieve a good fitting to the observed term values by optimizing the efictive minimum energy path. We hope that our predicted term values are of use in analyzing the spectrum, although they do not allow for any interaction between the truns-bending and &-StretChing states.
511
0022-2852191 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
512
EPA AND BUNKER
II. THE MINIMUM
ENERGY PATH AND ALTERNATIVE
COORDINATE
AXIS SYSTEMS
We have recalculated the minimum energy path for trans-bending tunneling, from the analytical six-dimensional ab initio potential energy surface for the HF dimer described in previous papers (3)) using Cartesian coordinates scaled by the square root of the atomic masses. One-half of this path is given in Table I, and it connects the linear C,, saddle point structure, the C, equilibrium structure, and the symmetric C&hsaddle point structure. We use mass-scaled Cartesian coordinates in the steepest descent method to determine the minimum energy path since this yields a path independent of any internal coordinate choice. The path is almost negligibly different from those obtained previously (2, 3) using internal coordinates. Figure 1 shows the definition of the six internal coordinates (rr , r2, Rob,4, 02, T)for the ( HFh molecule. As before we introduce the angular coordinates p and G as P =
(02 +
Q =
(02 -&I/%
(1)
&J/2
and
(2)
TABLE I The Values of the Coordinate? along the Minimum Energy Pathb in the Analytical ab initio Six-Dimensional Potential
0.9217 0.9216 0.9216 0.9216 0.9216 0.9219 0.9225 0.9234 0.9235 0.9237 0.9237 0.9238 0.9238 0.9239 0.9239 0.9239 0.9238 0.9237 0.9235 0.9233 0.9231 0.9229 0.9226 0.9224
0.9208 0.9211 0.9212 0.9213 0.9212 0.9212 0.9211 0.9214 0.9216 0.9218 0.9219 0.9221 0.9222 0.9223 0.9226 0.9228 0.9229 0.9231 0.9222 0.9221 0.9222 0.9222 0.9223 0.9224
2.8881 2.8877 2.8869 2.8851 2.8829 2.8803 2.8767 2.8650 2.8286 2.7988 2.7844 2.7661 2.7519 2.7557 2.7543 2.7517 2.7488 2.7446 2.7414 2.7303 2.7356 2.7332 2.7320 2.7316
0.0
0.0
0.0
7.7 13.3 21.0 27.5 33.8 41.1 51.8 54.8 57.5 58.8 60.6 62.1 66.9 69.0 73.1 77.9 85.0 91.2 97.3 103.7 111.5 117.5 122.7
0.0 0.0 0.0
0.0 0.0 2.2 3.6 5.0 5.7 6.7 7.5 10.5 11.9 14.6 18.2 23.6 28.4 33.0 38.1 44.7 50.2 55.3
a Bond lengths in A, angles in degrees, and energies in cm-‘. b Obtained using mass-scaled Cartesian coordinates. c p = (8, + !9,)/2.
180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.0 180.9 180.0 180.0 180.0 180.0
337.9
325.0 300.0 250.0 200.0 150.0 100.0 50.0 25.0 10.0 5.0 1.0 0.0 5.0 10.0 25.0 50.0 100.0 150.0 200.0 250.0 300.0 325.0 334.5
THE WATSON U-TERM
513
FIG. 1. The definition of the internal coordinates. (I and b are the HF centers-of-mass, 0, = LH,ab, e2 =: 180” - LH*ba, and 7 is the angle between the H,ab and abHz planes measured in a right-handed sense about a + b from HIab. 0 =SfJ,, & G 180”, and 0 G T < 360”.
where 0” G p G 180”, and p describes the truns-tunneling (geared-bend) motion. Although rl and r2 differ slightly from 0.9220 A along the minimum energy path, in subsequent calculations we hold them fixed at this value. It is noteworthy that Rab changes rapidly in the vicinity of the equilibrium geometry (pe = 34.8”) from a limiting value of 2.888 A at the linear saddle point to a limiting value of 2.732 A at the C2h saddle point. A hyperbolic switching function is one way to fit such behavior (2, .3) and we obtain, J&(A)
= (RG)(l
- tanh[cl(p
-
~211)+ (R1/2)(1 - tanh[cl(r - P - cdl)
+ t&/4)(1 - tanh[cl(cz - ~11) (1 - tanh[cl(p ~ r + c2>1), (3) where p is in radians, R, = 2.888 A, and R2 = 2.732 A; cl = 7.3288 (radian)-’ c2 = 0.5296 radian are the least-squares fitted parameters. We also fit I’mi, and 6 along the minimum energy path as functions of p:
V,,,in(cm-l) = 32.9 - 155.7
cos
and
2p + 162.4 c0s22p + 51.6 COS~~/I + 138.7 cos42p + 105.4 cos’2p,
(4)
and a( radian) = 1.1726 sin p - 0.6042 sin 2p.
(5)
Eqs. ( 3)-( 5 ) are not significantly different from the analogous equations in Refs. (2, 3). Following the usual arguments in deriving the one-dimensional Semirigid Bender Hamiltonian (see Refs. (6-8))) we can write the wave equation for the truns-tunneling and K-type rotation as: $
~K,U(P) = {fi(~) + (~zzl~Lpp)K’ +
(6) [2/(h2/+p)1[~min- EK,~~}~JK,~(P),
where pzz and pPPare the zz and pp elements of the P matrix, the inverse of the 4 x 4 extended moment of inertia matrix, Z (see Eqs. (6)-( 14) in Ref. (2) ; note that Eq. ( 10) in Ref. (2) should have a minus sign on the right hand side) _The functionfi ( p) in Eq. (6) is derived from U’(p) (in the notation of Eqs. (22)-( 24) of Ref. (5)) and it is discussed further in the next section.
EPA AND BUNKER
514
The values of the elements of the CLmatrix will depend on the definition adopted for the molecule fixed axes. Previously (2, 3) the right-handed xyz axis system was chosen with the z-axis connecting the monomer centers of mass a + b, the y-axis in the plane of the dimer with Hi having positive y-coordinate, and the x-axis perpendicular to the plane of the dimer; we call this axis system “the standard axis system.” We investigate choosing coordinate axis systems such that either IYz= 0 or IX, = 0, since these choices remove off-diagonal matrix elements occuring in the calculation of the end-over-end rotational energy levels (see Eq. ( 17) of Ref. ( 7)). It is interesting to compare the orientations of these axis systems with that of the standard axis system. (i) Iyz= 0, The Principal Axis System We define the new axes 9 and Z as being rotated clockwise about the x-axis (looking along the x-axis) by an angle of E from the y- and z-axes. The new X-axis is thus unchanged from the x-axis. In order for IYz= 0 we must have -2 C 1 tan-’ 2 C i mi(Zf
i?ZiyiZi
t = -
-VT)
I ’
and hence
where u = C mi yizi , v = are in the old (standard) respect to p.
- J$ ), the coordinates Xi, yi, zi of the four nuclei axes system, and the primes refer to differentiation with
C mi ( Z:
(ii) I,, = 0, The Hougen-Bunker-Johns Axis System (6) To obtain IX, = 0 we must have de -= dp
C mi(yiz: - ziy:) CRZi(yf +Z:)
(9)
’
which leads to c(p = j) =
s ’
dp
C mi(yiZ: - Ziy:) C Wli(yf +
p=o
Zf)
’
(10)
The angle t is computed from the relationships given above for each of the points of integration from p = 0 to p = a. Figure 2 shows the variation oft with p along the minimum energy path for these two alternate axis systems; the standard axis system has c = 0. We see that t remains small for both of these new axis systems at all p values. Once c is known the transformation to the new coordinates can be made using Ji =
YiCOS
Zi =
-y&l
Xi =
Xi,
t + t +
ZiSill
E,
ZiCOS E,
(1 la) (lib) (1lC)
515
THE WATSON U-TERM
-0.7
FIG, 2. The angle e as a function of p for the cases of Iyz = 0 (dotted line) and Ix,, = 0 (full line)
and I and CLcalculated in the new coordinate system; Eq. (6) can then be set up and solved. III. THE FUNCTION J(p)
AND THE MAGNITUDE
OF U(p)
The function J (p) which appears in Eq. (6) above is derived from the U’(p) term (5, 6) by using the expression f,(P)
= &
(12)
WP).
As discussed in Ref. (5) U’(p) is given by U(p) plus extra terms that result from subjecting the Hamiltonian to the transformation (1P,)-“2H( I,,,) ‘I2 in order to remove terms linear in J, . In previous Semirigid Bender calculations on (I-IF), , the asymptotic approximation (valid for p + 0 or p + 7~)of f,(P)
= -[l/p2
+ l/(r
- PJ21/4
(13)
was used, but this approximation is valid only if the variation of Rob with p can be neglected. In Ref. (2) the variation of Rab with p was also neglected in the calculation of Ppp. However, in the minimum energy path there is a rapid change in Rab in the region around the potential minima (see Table I). Using the part of I!?( p ) called Vi ( p ) (5)) whose asymptotic form is given by Eq. ( 13 ) above, (see Eq. (43 ) of Ref. (6)) we have
516
EPA AND BUNKER
where p is the determinant of the inverse of the 4 X 4 matrix I. Figure 3 compares f,(p) calculated from U:(p) (i.e., Eq. ( 14)) with that of the asymptotic approximation of Eq. ( 13);fi (p) has been evaluated numerically at each point of integration of the Semirigid Bender equation (typically about 3000 points) using a five-point recursion formula for the differentiation. Obviously, while the approximation is very good in the asymptotic limits of p + 0 and p --) ?r, in the neighborhood of p - 30” and 150” it is markedly different. The non-negligible magnitude of the derivative of Rab with respect to p in the vicinity of the C, equilibrium structures has a dramatic effect on the behaviour of I,,, pLpp,and therefore on p and the function f,(p) in this region. Figure 4 illustrates ppP as a function of p, using the Ix, = 0 axis system. At this point it should be noted that although the representation offi by Eq. ( 14) is more accurate than using Eq. ( 13), it does not use the full U’(p) term and only uses U:(p). The complexity of the evaluation of the full U’(p) term led in the past to computing only the U, ( p) portion of it. However, Sarka and Bunker (5) have shown that, if a coordinate axis system such that I,, = 0 is used, evaluation of the full U’(p) term becomes relatively easy, and is given by
30’o I 25.0
T
0.0
-
-
5 ‘% 1
e
V
-25.0
-
-50.0
-
-75.0
-
\
P
u-
-100.0
I’ 0.0
I
15.0
1
I
I
I
30.0
45.0
60.0
75.0
I
90.0.
P
1
105.0
I
120.0
I
135.0
I
150.0
I
165.0
180.0
/deg.
FIG. 3. The functionA calculated from U:(p) (Eq. (14)) (dashed line), from U’(p) (Eqs. (12) and ( 15)) (full line), and from the asymptotic approximation ( Eq. ( 13 )) (chain dots).
THE
0.0
1 0.0
WATSON
I
I
I
I
I
15.0
30.0
45.0
60.0
75.0
517
(I-TERM
I
90.0
I
I
I
I
I
105.0
120.0
135.0
150.0
165.0
J
16 SO.0
P /deg. FK; . 4. The variation of ppLpp( p ) with p (dotted at 2.81 .& I,, = 0 axes are used.
U’(P)
line). The full line indicates
the result if Rd is held fixed
=
(15)
where c’(p) is given by Eq. (24) in Ref. (5). (c’(p) involves elements of the 4 X 4 I matrix and derivatives with respect to p). We have, therefore, also calculated fi ( p) using the full U’(p) but as seen in Fig. 3 the use of the full V’(p) term, instead of just Vi ( p), to calculate f; (p) makes little difference. In Fig. 5 we show I’min( Vmin(P) + U’,(p), and V&n(P) + U’(p) SO that the huge effect of U’(p) around the equilibrium nuclear geometries can be appreciated. The large value of V’(p) results from the lack of commutation of J, with pPPand II, which in turn is due to the strong dependence of pPPon p in the region of p = pe. IV. ENERGIES CALCULATED KEEPING Rob CONSTANT AND FITTING TO EXPERIMENTAL DATA
Clearly, if we keep the value of Rob constant at the mean of Rab( C,,) and Ra6( C2,,) (i.e., making Rla,, = 0)) the anomalous behavior of U( p ) disappears and energies and tunneling splittings close to those in Ref. (2) are recovered (see Table II). These term values are close to the experimental values (9-l 1). Using this fixed value of Rob we have adjusted the parameters in I’min and a(p) in a weighted least-squares fitting to the nine experimentally available ( 9-Z I ) K-type rotation and trans-tunneling energies
518
EPA AND BUNKER
i
E < -
100.0
0.0
-100.0
-200.0
0.0
I
15.0
I
30.0
I
45.0
I
60.0
I
75.0
I
90.0
I
105.0
I
I
120.0
135.0
I
I
150.0 165.0
160.0
P /deg.
FIG. 5. The potentials Vmin(p)(chain dots), V,i”(p) + U;(p) (dashes), and vmin(P) + V’(p) (full line) as functions of p.
all involving u4 = 0 (as wd as vI = v2 = v3 = 216 = 0). Our m&iVation for doing this is to improve the predictions and to aid in explaining some of the unassigned structure in the experimental spectrum using a model that is capable of fitting the experimental term values. A weight of 100.0 was given to fitting the (K = 0, v = 1) term value, while the others were given a weight of 1.0. The standard deviation of the weighted fitting was 0.49 cm-‘. The fitted parameters are: V/in(cm-‘)
= 24.7 - 170.4
cos
2p + 256.8
c0s22p
afi’(radian) = 0.6647 sin p.
+
133.1
c0s32p,
(16)
(17)
The resultant energies are given in Table II and compared with experiment. In Fig. 6 we compare I’mi,( p) and I’ $,,( p). We should view this fitted minimum energy path as an effective minimum energy path in which neglected effects caused by using this approximate kinetic energy operator are compensated for within the framework of this one-dimensional model. V. DISCUSSION AND CONCLUSION
A minimum energy path for the truns-bend tunneling motion in the HF dimer has been obtained using a steepest-descent path in mass-scaled Cartesian coordinates on
THE WATSON U-TERM
519
TABLE II The truns-Tunneling and K-Type Rotation Energy Levels of ( HFb
ab ~nr!.to* 4
3
2
rxpt.d
3 2 1
605.52 620.33 518.53 535.29 378.67 386.90
0
369.31 379.46
3 2
446.21 458.08 383.04 393.49
393.550(u)
1 0
228.62 236.32 224.23 232.40
236.516 232.766
3 2 1 0
319.63 278.25 111.44 109.40
328.02 284.18 118.20 116.28
118.137 116.133
2 1 0
227.66 230.24 203.18 205.02 32.32 36.50 31.29 35.56
36.489 35.425
3 2 1 0
176.58 166.69 162.66 155.17 0.70 0.56 0.0 0.0
0.659 0.0
13
0
fittedc
386.727(I0)
a In cm-‘. b Calculated using the ab initio Vminand c (Eqs. (4) and (5)), with Rnb = 2.8 10 A. ’ Calculated using the fitted I’,,,,, and 0 (Eqs. ( 16) and ( 17)), with Rnb = 2.810 A. d From Ref. (9) unless otherwise indicated.
the analytical six-dimensional ab initio potential surface determined in Ref. (3). The path is only very slightly different from that deduced in Ref. (3) using internal coordinates. The position along the minimum energy path is defined by the angle p, and there is a rapid change in the monomer-monomer separation Rab as the molecule passes through the equilibrium structure along the path (see (Table I). As a result of the rapid variation of Rab with p, the reduced mass pPP varies strongly with p, and, therefore, the commutator [J,, CLJ is significant. This causes the Watson term U(p) (4, 5) to be very much larger than is normally the case. However, we can set up a model Hamiltonian for the K-type rotation and trans-bending motion with Rnb held fixed, and use it in a fitting to v4 = 0 data to determine an efictive minimum energy path and to predict term values. It would be better to treat Rab and p (and K-rotation ) together in a two-dimensional Semirigid Bender calculation, but this is best dealt with in a separate paper. ACKNOWLEDGMENT We are very grateful to Dr. Tucker Carrington, Jr. for advice on determining the minimum energy path and for many discussions.
520
EPA
AND
BUNKER
4oo’o 1 350.0
300.0
‘i
E <
250.0
200.0
Q
Y
150.0 100.0
50.0
0.0
P /deg. FIG, 6. The potentials
RECEIVED:
V,,,in( p) (full) and V pi” ( p ) (dashes)
from Eqs. (4) and ( 16 ) , respectively.
July 23, 1991 REFERENCES
1. P. R. BUNKER,M. KOFRANEK,H. LISCHKA,AND A. KARPF-EN,.I. Chem. Phys. 89,3002-3007 ( 1988). 2. P. R. BUNKER, T. CARRINGTONJR., P. C. G~MEZ, M. D. MARSHALL,M. KOFRANEK,H. LISCHKA, AND A. KARPFEN,J. Chem. Phys. 91,5154-5159 (1989). 3. P. JENSEN,P. R. BUNKER,A, KARPFEN,M. KOFRANEK,AND H. LISCHKA,.I. Chem. Phys. 93,, 62666280 (1990). 4. J. K. G. WATSON, Mol. Whys.15,479-490 (1968). 5. K. SARKA AND P. R. BUNKER, J. Mol. Spectrosc. 122,259-268 ( 1987). 6. J. T. HOUGEN, P. R. BUNKER, AND J. W. C. JOHNS,.I. Mol. Spectrosc. 34, 132-172 ( 1970). 7. P. R. BUNKER, AND J. M. R. STONE,J. Mol. Spectrosc. 41, 310-332 (1972). 8. P. R. BUNKER, AND B. M. LANDSBERG,J. Mol. Spectrosc. 67, 374-385 ( 1977). 9. W. J. LAFFERTY,R. D. SUENRAM,AND F. J. L~VAS, J. Mol. Spectrosc. 123,434-452 (1987). 10. K. VON P~TTKAMER,M. QUACK, AND M. A. SUHM, Mol. Phys. 65, 1025-1045 (1988). Il. M. QUACK AND M. A. SUHM, Chem. Phys. Letts. 171,517-524 (1990).