The interaction of rotation and local mode

The interaction of symmetrical

of rotation and local mode tunneling in the overtone spectra hydrides

Kevin K. Lehmann Department

of Chemistry,

Princeton

University,

Princeton,

New Jersey 08544

(Received 1 March 1991; accepted 3 May 1991) In the “local mode limit” where the tunneling time for vibrational energy exchange is long compared to the classical rotational period, one expects that the effective rotational Hamiltonian will reflect the reduced symmetry of the local mode state. Hamiltonians in the local mode basis are given for interaction of rotation and local mode tunneling for molecules of the XH, , XH, , and XH, type. Transformation of these Hamiltonians to a symmetrized basis (which diagonalizes the vibrational problem), produces rotational couplings between the vibrational states. Relations between the spectroscopic constants are derived that are less restrictive than those given earlier by Halonen and Robiette, but reduce to them when the assumptions of their model are met. The present algebraic procedure can be easily extended to include higher order terms. The effect of these couplings is to reduce the size of the pure vibrational splittings. This is due to the fact that in the rovibrational problem, in general, one must reorient the angular momentum vector in the body frame as well as transfer the vibrational action between bonds. This increases the length of the tunneling path and thus decreases the rate of vibrational energy transfer. Model calculations show that a simple semiclassical picture can rationalize the observed trends.

INTRODUCTION The characterization of highly excited vibrational states has become one of the central goals in chemical physics. Such studies tie spectroscopy to intramolecular dynamics (a connection that is always there but is often only implicit) and reaction dynamics. Often, anharmonic interactions, which in traditional spectroscopic theory are “perturbations,” change even the topological character of the motion. Perhaps the best understood of these changes is the transition from normal to local modes of vibration that characterize the stretching overtone bands of almost all symmetric hydrides.’ Despite the extensive literature on local mode vibrations, little is known of the implications of local mode vibrational motion on the rovibrational energy levels, and thus the rotational dynamics. The first systematic study on this topic was the work of Halonen and Robiette on XY,, XY,, and XY, molecules.2 By assuming a vibrational potential equal to a sum of uncoupled Morse oscillators, and then making several very restrictive assumptions, Halonen and Robiette derived a set of relations between the resonant terms of the H,, operator.3 This operator contains terms of the form qiqj J, Jb and dominates the vibrational dependence of the rotational structure. An interesting prediction of this model is that the rotational structure of a XH, molecule should resemble a parallel transition of a symmetric top, including the factor of 2 higher statistical weight for the K = 3n levels. This effect has since been observed in the overtone spectra of GeH, ,4 SiH, ,’ and SnH, .6 An independent derivation of this effect has been given by Michelot er al.’ using an algebraic approach, and this model has been shown to fit the observed first overtone band of SiH, spectrum with an rms 0.003 cm - ’with only two adjustable parameters in the effec-

tive Hamiltonian.

This symmetric top rotational structure can be understood in a simple physical way. In the traditional treatment of the rotational motion, one assumes that the vibrational motion is much faster than the rotation. As a result, the rotational energy level structure is determined by a vibrationally averaged effective moment of inertia (the diagonal terms of H,, ), But in the local mode limit, a molecule has nearly degenerate vibrational energy levels.’ These can be viewed as symmetrized combinations of “local-mode” states wherein each X-H stretch has a constant vibrational action unlike normal mode states which have constant vibrational action in each normal mode. The splitting between the different symmetry representations of the local-mode states is related to the time scale for tunneling to a wave function for which the excitation has swapped bonds. If this tunneling time is much longer than the rotational period, one would expect that the rotational motion and thus energy level structure will reflect the reduced symmetry of a single localmode state. The adiabatic separation works in reverse, since rotational motion is now fast compared with tunneling. The local-mode state of XH, with all the excitation in a single bond will look like a slightly prolate symmetric top since, on average, the excited bond will be longer. It should be remembered that changes in the vibrationally averaged structure are only part of the vibrational dependence of the rotational constants, but such a simplified model correctly predicts the symmetry of the effective rotational Hamiltonian. It is clear that the above argument depends only on a separation of time scales, and thus does not depend upon the restrictive assumptions about mass, structure, and bending force constants that Halonen and Robiette made in their work. Halonen and Robiette had given numerical evidence that in fact the effective constants for overtone levels would obey certain of the constraints of their local-mode limit

0021-9606/91 /I 62361-l 0$03.00 J. Chem. Phys. 95 (4), 15 August 1991 @ 1991 American Institute of Physics 2361 Downloaded 18 Mar 2002 to 128.112.83.42. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

2362

Kevin K. Lehmann: Overtone spectra of hydrides

much more than the fundamentals; in particular, that the effective rotational constants for each of the local-mode states tends to become equal for each of the nearly degenerate local-mode vibrational states and that the effective Coriolis coupling of the states vanish. In this paper, we shall examine the spectroscopic consequences of a model that invokes an effective rotational Hamiltonian for each localmode vibrational state plus vibrational tunneling between local-mode states. The relationship of the present approach to that of Halonen and Robiette is similar to the two independent derivations of the ‘X-K ” relations that allow a local-mode vibrational Hamiltonian to be expressed in a normal mode basis set. Mills and Robiette’ derived the x-K relationships by starting with a simplified model for the potential energy function and then using perturbation theory. My independent derivation” started with the Child and Lawton harmonically coupled, anharmonic oscillator Hamiltonian, ” expressed as a polynomial in the bond mode raising and lowering operators, and by a symmetry transformation of these operators demonstrated the samex-Krelations. Each method has its own advantages. The perturbation method allows one to use a more general force field and estimate the effect of the neglected terms. General perturbation expressions for all possible Darling-Dennison (quartic) coupling terms between nondegenerate modes were recently published. l2 The algebraic method is much easier to implement, however, and proves an exact equivalence for all eigenvalues, which the perturbation method did not. Della Valle has provided general formulas for the x-K relations for any set of equivalent bonds using the algebraic approach. I3 The present study of vibration-rotation interactions will again start in a local-mode basis with a rotation-inversion Hamiltonian operator based upon general physical considerations, much as in Child and Lawton’s” treatment of the vibrational coupling. Transformation of this Hamiltonian to a normal mode basis generates a set of relations between the coefficients of the H22 terms. As with the x-K relationships derived earlier, one does not expect the relationships to hold exactly in real spectra since the starting Hamiltonian is only approximate. But the relationships will be useful in attempts to predict and assign spectra, and as possible constraints to be used in spectral fitting when parameter correlation or limited data sets prevents a free fit of all the possible terms. More importantly, it provides a physical model for interpreting the rotational constants that are derived from a fit of a local-mode spectrum. An interesting result of the present treatment is that rotation of a molecule can greatly reduce the tunneling rate, stabilizing the molecule for a much longer time with vibrational motion localized along a single bond. It will be shown that the qualitative features of this effect can be understood by semiclassical analysis of motion on the rotational energy surface, as used by Harter and Patterson.14 The physical explanation for this effect is that to reach an isoenergetic state, one must not only exchange all of the vibrational action from one bond to another, but in general, one must also reorient the angular momentum in the body fixed frame. Such a rotational suppression of local-mode tunneling appears to have been observed in the first overtone band of

stannane,‘j but it escaped explicit notice though it was correctly predicted by the fitted constants. The present study provides a physical picture for this effect. XH, MOLECULES Consider an XH2 molecule in a local-mode state In, ) which means there are n quanta in bond 1. By symmetry the state In,) will be degenerate with energy Go. These two states will be coupled by tunneling, leading to a term in the Hamiltonian R (n, ) (n, (. Here R is the rate at which the IZ quanta of vibrational excitation tunnels from bond 1 to bond 2. It is known from earlier work on local modes that this tunneling rate will decrease exponentially with increasing nk.’ This tunneling term will lead to the vibrational eigenstates being $,,, = (In,) f In,))/+‘% with Es = Go +A and E, = Go - /2. In each local-mode state, the rotational motion is described by an asymmetric top Hamiltonian. But due to the reduced dynamical symmetry ofthe (n, ) state, the principle axes for this state will be rotated in the molecular plane by an angle 19away from those of the ground state (which are by symmetry the C,, axis and perpendicular to it). We label the axes (X,JJ,Z) as theA,B,Caxes of the equilibrium structure and thus the C,, axis is along y, and the perpendicular to the molecular plane is z. To lowest order, the inertial axis rotation comes about because the H22 operator has a term q,q, (J, Jy + J,, J, ). The operator qsq, will have a nonzero expectation value, despite being antisymmetric, because the local-mode state does not have C,, symmetry. Because of the longer average bond length of the excited bond, one expects that the A axis will rotate towards this excited bond. The state In*) by symmetry should have the same rotational constants, but its principal axes will be rotated by an equal but opposite amount compared to (n, ). Thus we write an effective Hamiltonian to describe the rotationaltunneling dynamics of the In, ) and In, ) states as H

=

[Go +

+AJ:, [Go

+BJ;,

+AJll

+CJ:,]IMn,I

+CJ:>]ln,)(n,I b,>hl]7

+BJ;2

+A [IMn,l+

(1)

where the principal axes are given by Jx, = cos t9J, + sin OJ,,, J,, = - sin OJ, + cos BJ,,, Jx, = cos OJ, - sin OJ,, (2)

(2)

Jy2 = sin OJ, + cos BJ,, Jz, =J+=J*. If we rewrite this Hamiltonian in terms of projections onto the vibrational eigenstates and rotational operators on axes determined by symmetry we find H = [Go+~++A,J~+B,J~+C,Jt]Is>(sl + [Go -A+A,J: ++-L{J,,J,~[ld(~l

+B,J; +

+C,J:]I4(al

Idbll,

where CJx,J,)

= JxJ,

+ J,Jx,

J. Chem. Phys., Vol. 95, No. 4,15 August 1991

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(3)

Kevin K. Lehmann: Overtone spectra of hydrides

~,=A,=A~os~0+Bsin~t4 B, =

B, = A sin’ 0 + B COS’0,

c, = c, = c, d, = (A - B) sin 20.

(4)

This is the same form as the Hamiltonian determined by Halonen and Robiette, but the relationship between the parameters is not as restrictive. In absorption from the ground state, one will have x polarized transition amplitude to the I a) rovibrational basis states, and y polarized transition amplitude to the Is> rovibrational basis states. It is worth pointing out that, while one is tempted to predict the ratio ofx and y matrix elements from the projections of the bonds on the x and y axes (assuming the transition dipole is along the bond), this is generally a poor approximation. Several interactions are not present in this Hamiltonian, and their absence needs to be justified. The first is an absence of Coriolis interaction between states 1n’ ) and 1n, ) . By symmetry, there exists a Caxis Coriolis interaction between the s and a stretching fundamentals. However, the Coriolis coupling constant is small, since without bend-stretch mixing, it arises only from motion of the central atom,” and thus c T3 is on the order of (mn /m, ). The Coriolis interaction matrix between the symmetrized local-mode basis functions will be 2(e-

II (8)

-r;l, J- =J, +I;r,. (9) Once again, the form of the terms is the same as Halonen and Robiette, but the present Hamiltonian is not as restrictive as to the relationship of the parameters. The relations of Halonen and Robiette are recovered by setting B = C and 0 = 125.26” as required by their model. One obvious feature is that the spectrum should, in the limit of small R, be that of an asymmetric top with A,C hybrid band. This was predicted by Ovchinnikova” who derived a model for the high overtone bands of NH, that is similar to the present one. Halonen and Robiette found that in their model, the local-mode rotational structure of XH, should be that of a symmetric top. This is due to Halonen and Robiette’s assumed geometry, which resulted in the molecule being an accidental spherical top in the ground state. As a result, the vibrating bond ensured that one axis would be unique, but the other two retained the same moments of inertia and thus a symmetric top spectrum resulted. Notice, however, even in this limit the quantization axis is one of the bonds, not the symmetry axis and thus the rotational levels will not have the 2: 1 nuclear spin weight alternation of the ground vibrational state. Like before, in the limit of small il, the tunneling between rovibrational levels should be suppressed by a rotational overlap factor. Table II contains the J = 20 rotational-tunneling energy levels of a XH, molecule calculated with A = 3.72, B = 3.61, C= 3.40, il = - 0.01 cm-‘, and 8 = 0 and 12”. These rotational constants, and the latter angle, are approximately those expected for ASH, at n = 3 based upon the analysis of the fundamentals by Olson et aLI9 When 8 = 0, the different local-mode states have their A axes differing by 60” but have the same C axis. As for the XH, case, the states with lowest energy (rotation around the

and the vibrational states are essentially unmixed. The degenerate E vibrational state is, however, split by a small amount. As the energy rises, there is a decreasing tunneling rate which drops to one-third the rotationless value when K, = 13. When the asymmetry splitting grows to be larger than the tunneling rate, the tunneling rate is strongly suppressed further, reaching a rate only 2% as large as the rotationless value for the state (2O,,,, ) localized on the separatix between A and C axis motion. When asymmetry splitting once again becomes negligible, at KP = 16, the tunneling rate is back to 25% of the nonrotating value. With increasing KP, the tunneling splitting again decreases, forming a pattern of A-E-E-A levels, with a splitting 1:2:1. For K, = J = 20, the total splitting of the pattern is only 6% of the bare vibrational value. The rate of decrease of the tunneling rate for K, -J levels will be larger the more asymmetric the top. This is because, while the distance that the A axes must be rotated to overlap is constant at 60”, the more asymmetric the top, the more localized the highest rotational states will be near the A axis. Again, if the tunneling splitting is unresolved, the observed levels will have identical nuclear spin weights unlike the 2-l alternation expected for a C,, symmetric top. This is expected since the local-mode state has no rotational axis of symmetry. When we look at the energy levels for 8 = 12”, the most striking change is that the lowest energy levels (K, w J) now have a suppressed tunneling splitting. This is because the rotation of the Caxis means the rotational wave functions of different local-mode states no longer overlap strongly. By the time K, decreases to a value of 13, the tunneling splitting is close to that of the 8 = 0 case. For such values of K,, the rotational functions are sufficiently spread out on the sphere that a 12” rotation has little effect on the overlap. For the state 201,,, , which is localized on the separatix, the C axis rotation increases the overlap, and the tunneling splitting is a factor of 10 larger than was calculated with no rotation of the Caxis (0 = 0). The state with K,, = J has a further reduced splitting. This can be understood since the A axes of the different local-mode states are now 64”instead of 60”apart. It is the nature of tunneling that such a small change in rotation

C axis) have essentiallyan unperturbedtunneling splitting

angleleadsto a factor of 4 reduction in the tunneling rate.

C, =C,=Asin26+Ccos26, r=(A-C)sin28, q=Ac0s~8+Csin28-B, J,

=J,

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Kevin K. Lehmann: Overtone spectra of hydrides

2366 TABLE II. Rotation-tunneling

energy levels for XH, J = 20.

R = 0.0

Energy

A = - 0.01 (cm-‘) l9= 12’

A = -0.01 (cm-‘) 8=(r rm

Energy 0.161 73 0.191 38 0.191 55

1433.18155

% *iz

Energy

l-N

% A

E +A, E

99 0 0

0.173 15 0.181 58 0.189 90

E E

K. = 20

A, +A,

33 61 6

E E

0.275 77 0.276 68 0.283 53

A, +A, E E

43 40 17

K. = 19

& = 18

I-N

4

95

0.259 22 0.288 10 1 0.288 65

A, +A,

98 2 0

0.823 84 0.851 32 1 0.852 27

A, +A, E E

95 4 1

0.837 54 0.838 41 0.851 49

A, +A,

50 47 3

1461.870 04

0.852 67 0.878 03 I 0.879 41

E A, +A, E

91 7 2

0.865 25 0.866 52 0.878 34

E A, +A, E

49 45 6

K,, = 17

1470.357 34

0.341 75 0.364 24 i 0.366 02

0.354 30 0.354 59 0.363 11

E -4 i-4 E

43 43 14

K. = 16

A, +A,

85 10 4

0.285 65 0.30444 I 0.306 56

A, +A, E E

77 15 8

0.295 31 0.299 13 0.302 21

K, = 15

A, -i-A,

45 33 22

0.676 59 0.690 89 I 0.693 21

A, +A, E

67 20 12

0.681 80 0.688 77 0.690 15

A, +A, E E

50 27 23

K. = 14

0.502 0.511 0.515 0.515

90 63 47 50

E E 4 A,

57 28 15 15

E

1443.278 66

1452.842 48

1478.298 88

1485.686 90

1492.509 99

E E

E

0.503 0.512 0.514 i 0.514

17 29 53 56

4 A,

56 25 18 18

E E

E E

E E

K. = 13

1498.750 26 0.750 92

A, +E A, +E

0.747 0.747 0.751 t 0.753

32 98 12 00

4 A, E E

43 43 31 26

0.741 0.752 0.753 0.757

73 61 27 11

A2 A, E

63 25 25 12

1504.374 62 0.383 99

A, +E A, +E

0.373 0.374 0.383 1 0.384

80 99 17 45

4 E A, E

36 32 36 32

0.365 0.374 0.379 0.389

CO 34 24 01

4 A, E E

65 65 19 16

A, +E A, -I-E

0.290 0.294 0.384 1 0.388

12 41 24 69

A* E A, E

43 29 43 10

0.290 0.297 0.385 0.391

36 50 23 86

E

1509.293 04 0.387 14

4 E A,

44 18 38 18

1513.227 16 0.821 06

A, i-E A, +E

0.224 0.232 0.817 i 0.827

54 37 83 53

E A2 E

0.225 0.230 0.819 0.825

60 21 01 05

E 4 E

A,

42 16 44 12

A,

39 23 40 20

0.026 07 1 0.027 34

E A2

35 30

0.025 94 0.026 78

4 E

35 33

(123)

0.969 01 1 0.969 73

A, E

35 33

0.965 25 0.971 60

A, E

47 26

(13,a)

0.703 16 0.704 57

‘6 E

36 32

0.703 58 0.705 14

E 4

35 30

(13,7)

1516.026 50

1517.969 48

1518.704 10

A, i-E

A, -I-E

A, A-E

K, = 12

K. = 11

K. = 10

K. =9

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Kevin K. Lehmann: Overtone spectra of hydrides TABLE

II. (Continued.) A = 0.0

A = - 0.01 (cm-‘) O=o”

Energy

rN

Energy

A,

57 21 43 28

0.278 0.280 0.396 0.406

52 56 48 39

48 4 34 33

0.062 0.063 0.074 0.080

44 88 a4 67

52 37 24 26

0.393 60 0.396 47 0.400 41 0.40103

A,

37 36 27 32

0.248 0.251 0.256 0.259

64 13 61 40

4

57 45 21 10

0.592 0.595 0.601 0.604

42 36 24 17

4

49 41 25 17

0.404 0.405 0.407 0.408

69 65 56 52

‘42

36 35 32 30

0.661 0.662 0.662 0.662

94 04 25 36

4

0.272 0.282 0.399 0.404

05 a7 94 57

1527.063 68 0.076 02

A, +E A, +E

0.059 0.072 0.075 0.076

35 35 a7 33

E Al E

0.392 0.397 0.400 0.400

03 36 46 68

A,

0.252 0.253 0.254 0.255

87 06 42 70

Al

1538.253 90 0.253 94

A, +E A, +E

A, -I-E A, +E

f 0.594 0.605 0.601 77 0.591 32 a5 17

E 4 E

A2

E E 4

E E 4

4 E E A,

1544.598 29

1551.406 60

1558.662 14

0.401 0.404 0.409 0.411

79 20 01 42

4

0.661 0.661 0.662 0.663

29 72 58 01

4

E E A,

E E A,

XH, MOLECULES We now consider the general XH, molecule with tetrahedral symmetry. We now have four degenerate local-mode states In,) with k = 1,4. Without vibrational excitation, the molecule is a spherical top. In each local-mode state, the molecule will have its symmetry dynamically lowered to C,,. It will thus have a prolate symmetric top rotational Hamiltonian with the A aligned along the excited bond, We set up the tetrahedral molecule such that the four X-H bonds l-4 point in directions (l,l,l), (1, - 1, - l), ( - l,l, - l), and ( - 1, - 1,l 1, respectively. Thus we write an effective tunneling-rotation Hamiltonian as =

i

k=l

[Go

+BJ*+

Energy

% *o

A, +E A, i-E

1532.397 65 0.398 47

A = - 0.01 (cm-‘) e= 120

r,

1522.279 21 0.403 02

H

2367

r,

% *a

E

36 29 55 22

A, A2 E E A, E

*,

Kp = 14

41 33 33 la

A2

K, = 15

47 39 25 25

E E A*

K, = 16

51 43 24 15

E E A,

K, = 17

53 43 23 14

E E A,

K,=la

40 36 30 27

E E A,

K, = 19

34 34 33 33

E E A,

K, = 20

ek are unit vectors pointing along the four bonds. We introduce symmetrized vibrational states by Ia) =i(ln,>

+ In*) + In,> + In,)),

I&) =:(b,)

+ In*) - 14) - ln4)),

IQ =I(br> - 1%) + I%> - In‘s)), 14) =:(I%) - In*>- I%> + In,)).

(12)

Substituting in for Ink) (nk I we find f~ = [Go + 3A + B,J’]

la>(al

(A-B)(Bk.J)*]Ink)(nkI +idcZt

(10)

B, =B,

(&

{Ji9Ji)[la>(rkl

=f(2B+A)

+

Itk)(al

+

Iti>(tjl],

da, ={(A--), (13)

where

U,j,k)

is a cyclic

permutation

of (x,y,z). Introducing

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Kevin K. Lehmann: Overtone spectra of hydrides

2366 TABLE III. Rotation-tunelliug 4 K=O

K=l

cl

E( rot)

% *”

AEX 10’

K,

A, / F, 4 4 1E

839.995 74 40.00123

36 22

549

K=

40.095 72 0.101 29 0.104 38

36 22 14

557 309

40.394 99 0.399 18 0.404 15

37 27 15

419 496

E F2 14

K=2

energy levels in cm - ’ XH, J = 20. r”r

E(rot)

% 4”

AE x 10’

E 6 F,

52.092 02 0.100 42 0.104 93

45 24 13

841 451

54.388 0.399 0.401 0.403

80 43 al 75

53 26 20 16

63 238 194

4

56.898 38 0.898 68 0.902 43

28 28 19

30 375

14 E 4

59.596 30 0.599 66 0.603 97

34 26 15

336 431

4 A2

62.494 0.498 0.501 0.505

89 33 71 02

38 29 21 12

344 340 331

E Fl

65.594 10 0.599 99 0.605 94

40 25 10

589 595

68.896 86 0.900 03 0.903 la

33 25 17

317 316

72.398 76 0.899 60 0.400 43 0.40127

28 26 24 22

a4 a3 a4

11

K=12

1 4E

K=

13

4 Fz AZ t A,

40.897 0.899 0.902 0.908

20 11 51 52

32 27 18 4

191 339 601

E 4 I4

41.594 86 0.600 50 0.602 91

38 24 17

564 241

4 F2 1 E

42.499 03 0.499 16 0.592 71

28 27 la

13 355

F2 A2 8 f A,

43.598 66 0.599 72 OACO 12 0.603 98

28 26 25 16

lo6 40 386

K=7

E F2 I 4

44.894 15 0.899 04 0.904 a7

40 27 13

489 583

K=

la

A, 4 4 i 4

K=8

4 F2 1 E

46.394 85 0.402 54 0.493 a9

38 19 15

769 135

K=

19

F2 E I F*

76.099 90 0.10001 0.100 12

25 25 25

11 11

K=9

4 A, 4 i A2

48.096 0.099 0.100 0.109

78 31 24 67

33 27 24 1

253 93 843

F2 E 1 F1

ao.ooo oo ao.ooo 01 ao.oco 01

25 25 25

1 0

E 4 i F2

49.994 93 50.000 46 0.002 a9

38 24 18

553 243

K=3

K=4

K=5

K=6

K=

10

the tensor operators of Robiette, Gray, and Birss,” one finds the same relationship for their coefficients as found by Halonen and Robiette. This relationship has also been found to be approximately correct for the XH, overtone spectra that show symmetric top rotational structure.4*6*8 When A ( 2K(A - B) , the energy levels will cluster into a groups of eight quasidegenerate states made up of basis functions Ink,& f K >. These basis functions will be connected by tunneling integrals of two different magnitudes A, =Ad;,,(lW) and A, = Ad ;, _ K ( 1090) = Ad & (71”). For K = 3N, the phase factors of the inte-

K=

14

K=15

I

K=16

K=

4

17

i

K=20

4

E 4

grals will be equal, and these eight functions will produce a cluster of four energy levels with shifts 3 (A, f A2 > (A, and A,)and - (A, f/2,) (Fr andF,).ForK=3Nf 1,weget three states F, , F2, and E. The order and splitting of these levels depends upon the size of A, and A,. If we look at the J= K levels, then the tunneling rate will fall off rapidly, since d&(109”) = (1/3jJ and d&(71”) = (2/3)J. This will produce the largest rotational quenching of the localmode tunneling, since the angular momentum vector needs to move the furthest to reach a configuration of equal rotational energy.

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Kevin K. Lehmann: Overtone spectra of hydrides

The rotation-inversion level structure will depend (except for J dependent scaling and shift) only on the ratio of (A - B) and R. In Table III is given the rotation-inversion energy levels calculated for a XH, top with A = 2.1, B = 2.0, andil = - 0.01 cm - ‘. These constants where chosen to emphasize the local-mode limit. For these parameters, the tunneling splitting of the nonrotating molecule is 0.04 cm-’ . Unlike the previous examples, all of the J = 20 levels have a substantial reduction in the tunneling rate, the largest splitting being only about 20% of the rotationless value. It is also shown that the K = J levels have the smallest splitting, predicted above. But when ?i(J + l/2) = cos( 35”) = 0.82 (which implies K = 16.7 for J = 20), the classical trajectories for K with one bond excited and - K with another bond excited overlap at a tangent. The calculations reveal that the K = 16 is a local maximum in the tunneling splitting. If one looks at K/(J + l/2) = cos(54”) = 0.58(K= 11.8forJ= 20),the classical trajectories for K with one bond excited and K with another bond excited now overlap at a tangent. Again, the splitting of the level K = 11 is a local maximum in the calculated splitting. The slow decrease in the splitting as one goes to K = 0 can be understood by the fact that while the classical trajectories continue to cross, they cross at a steeper angle, and the width of the quantum states gets narrower, thus leading to reduced overlap. It is interesting that in the published figure of the stannane overtone,6 the R (6) transition clearly shows a reduction of the tunneling splitting as K increases. This spectrum is only partially in the local-mode limit in that R and A - B are comparable. Since there will be Stark transitions across the tunneling levels (with matrix elements that scale with the size of the vibrationally induced dipole of the local-mode

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state), some type of double resonance experiment would be ideal for determining the splitting pattern. EXTENSIONS This initial work has presented a general procedure for producing rotational-tunneling Hamiltonians for a set of nearly degenerate local mode states. One can clearly extend the work to include quartic and sextic centrifugal distortions terms in the local-mode Hamiltonians. For a quantitative treatment of hydrides, both terms are usually required. The most straightforward way to do a calculation is to use a basis set of unsymmetrized local mode states and use rotational wave functions for each vibrational state aligned with it’s own inertial ellipsoid. Then vibrationally diagonal couplings can be treated by a standard program. The tunneling matrix elements will be izD JK,,K,(#,19,x), where #,19,x are the Euler angles to convert one local-mode principle axis system into another Alternatively, one can use a common set of rotational functions, aligned with the symmetry axes, for all the local-mode vibrational functions, and use the rotation matrices to transform each of the rotational basis function. This is likely the preferred method, since it allows a convenient calculation of transition intensities from the ground state, though the basis functions are not of definite symmetry, and symmetry assignments need to be made by examination of the eigenvectors coefficients of symmetry related basis functions. In order to work in a symmetrized local-mode basis, one needs to convert higher powers of the rotational operators, as give above for the quadratic terms. For the tetrahedral molecule, the terms produced by including DJJ, DJK, and DKK in the local mode, symmetric top, rotational Hamiltonian are given below:

I HD = HD = -

D,,J4+DJKJ2(~k.J)2+D~~(~k.J)4](nk)(nk(,

D,J” -+dJ:

+ J; + J:)][ lh.4

+ 7 \td(b(,

-3

D.xJ* 2 CJi,JiI[ la)(f/cl +

Ifk)(al

+

Iti)(rjl]

(iik)

-- 1 DKK DLl = D.,.,

9

(ijv )

-I- :D.,K

-I- Pm.

SCJf,CJi,JkII

+ 2CJj,JkI +2CJj,J:I

-7CJi,JkI] [ IQ>(fiI + Itt)(al A- Ifj)(fkl]9 (14) I

Only some of these terms appear in the Hamiltonian used by Halonen et al. to fit the spectrum of the first overtone of stannane.6 Such selective inclusion of the terms will destroy the high J,K local-mode energy patterns of the spherical top. In fact, the rovibrational levels predicted by the final fitted constants of Halonen etal. deviate from the expected pattern of falling inversion splitting for the high J = K lines when J become larger than those used in their fit.*l I believe this is due to selective inclusion of terms. Including all of the terms, but imposing the above constraints on the spectroscopic constants, has a firmer theoretical basis and may well prove a

nian is much easier to write (andprogram) in the unsymmetrized basis. As one goes to the sextic distortion terms and beyond, which are often needed for a quantitative fit to the rotational structure of a hyride, the advantage of the unsymmetrized basis only increases. In fitting the stannane overtone spectrum, Halonen et al. determined a diagonal (in the symmetrized basis) Coriolis term, but 2Bc3 was only - 0.0042 cm - ‘. Such terms are easily included in the symmetrized local-mode Hamiltonian since they take the same form as the interaction between the fundamentals. In the unsymmetrized basis, one will have a

practical advantageas well. Clearly, the distortion Hamilto-

vibrational angular momentumbetweeneachpair on local-

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Kevin K. Lehmann: Overtone spectra of hydrides

mode vibrations, pointing in the direction of the cross product of the two bonds. One can use Morse oscillator matrix elements to estimate the size of the expected Coriolis term. Halonen ef aL6 also found that a Hz3 higher order Coriolistype interaction between the A and F, symmetrized localmode states could also be determined from the spectrum. This term does not show up in the present treatment. It could be generated by adding rotational corrections to the Coriolis coupling between the bond modes, but it is desirable to have a physical model for this interaction so that its magnitude can be estimated from a model force field. CONCLUSIONS The present paper extends the earlier work of Halonen and Robiette to provide a systematic treatment of the interaction of rotation and local-mode tunneling. Effective Hamiltonians are developed, both in an unsymmetrized and a symmetrized basis. The unsymmetrized basis provides a natural framework for elucidating how rotational motion can reduce the local-mode tunneling rate. At present, it is only applicable when the interaction of the local-mode polyad with other vibrational states can be treated perturbatively, i.e., by an effective rotational Hamiltonian. Cases for Fermi resonance, for example, will require extension but can likely be handled in the same framework. Also, the treatment can be generalized to cases of the local-mode combination bands, with excitation in more than one bond mode, or with a bending mode excited. Note added inproo$ Independently, M. S. Child and Q. Zhu have shown that for an XH, molecule, the normal mode Hamiltonian [ Eq. ( 131 and the Halonen and Robietie constraints can be transformed to the symmetric top, local mode Hamiltonian [Eq. (lo] thus confirming one aspect of the present work. [ Chem. Phys. Lett. (to be published) I.

ACKNOWLEDGMENTS I want to thank the Department of Physical Chemistry, University of Helsinki, for their hospitality during the initial stage of this project; Lauri and Marjo Halonen and Quingshi Zhu for helpful discussions on rotational structure in the local-mode limit. This work was supported by the National Science Foundation and the Donors of the Petroleum Research Fund, administered by the American Chemical Society.

’M. S. Child and L. Halonen, Adv. Chem. Phys. 57, 1 ( 1984). 2 L. Halonen and A. G. Robiette, J. Chem. Phys. 84,866l ( 1984). ’ F. W. Birss, Mol. Phys. 31,49 1 ( 1976). 4Q. Zhu, B. A. Thrush, and A. G. Robiette, Chem. Phys. Lett. 150, 181 (1988). “Q. Zhu, B. Zhang, Y. Ma, and H. Qian, Chem. Phys. Lett. 564, 596 (1989). bM. Halonen, L. Halonen, H. Burger, and S. Sommer, J. Chem. Phys. 93, 1607 (1990). ‘F. Michelot, J. Moret-Bailly, and A. DeMarino, Chem. Phys. Lett. 148, 52 (1988). *M. Chevalier, A. DeMartino, and F. Michelot, J. Mol. Spectrosc. 131,382 (1988). 9 I. M. Mills and A. G. Robiette, Mol. Phys. 56, 743 ( 1985). ‘OK. K. Lehman, J. Chem. Phys. 79, 1098 (1983). “M S. Child and R. T. Lawton, Faraday Discuss. Chem. Sot. 71, 273 (lb81). “K. K. Lehmann, Mol. Phys. 66,1129 (1989). “R. G. D. Della Valle, Mol. Phys. 63, 611 ( 1988). I4 W. G. Hatter and C. W. Patterson, J. Chem. Phys. 80,424l (1984). I5 See Eq. ( IO) of Ref. 2. 16L. Halonen, J. Chem. Phys. 86, 588 (1987). “J. R. Gillis and T. H. Edwards, J. Mol. Spectrosc. 85, 74 ( 1981). ‘*M. Ya, Ovchinnikova, Chem. Phys. 120,249 ( 1988). 19W. B. Olson, A. G. Maki, and R. L. Sams, J. Mol. Spectrosc. 55, 252 (1975). “A. G. Robiette, D. L. Gray, andF. W. Birss, Mol. Phys, 32,1591, ( 1976). ” M. Halonen and L. Halonen (private communication).

J. Chem. Phys., Vol. 95, No. 4,15 August 1991

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