1466
J. Phys. Chem. 1996, 100, 1466-1474
Vibration-Pseudorotation Coupling in Symmetric Triatomic Molecules Jae Shin Lee† Department of Chemistry, College of Natural Sciences, Ajou UniVersity, Suwon, Korea 441-749 ReceiVed: June 21, 1995; In Final Form: September 22, 1995X
The spectral and dynamical changes induced by vibration-pseudorotation coupling in a symmetric triatomic molecule AB2 with total angular momentum of zero (J ) 0) were investigated in an exact quantal study which employed a full vibration-rotation Hamiltonian with the model potential functions representing the linear, quasilinear, and bent type of molecule, respectively. For each type of molecule, the vibration-rotation coupling energy and vibrational structures of wave functions were examined up to the levels of high energy, about 15 000-20 000 cm-1 above the bottom of the well. The strong mode dependence of coupling energy is found for all types of potential energy surfaces, and its magnitude becomes nonnegligible in some highenergy levels of many excited vibrational quanta. Although the effect of vibration-pseudorotation coupling on the mode structure of vibrational wave function was not significant for most states, it also was found that the vibration-rotation coupling could play a nonnegligible role in destroying nodal structure of the wave functions in certain excited combination levels of high energy even for J ) 0. Preliminary results for quasilinear CH2+ generally agree with the results for model potentials, supporting the validity of the model potential functions employed in this study.
I. Introduction The dynamics of polyatomic vibrations and vibrationrotation interaction in highly excited states have been the subject of many theoretical and experimental studies for the past decade. In the lower energy region near equilibrium, the dynamics of most polyatomic molecules is usually simple and nearly separable. The assignment of the vibrational modes at lower energy can be done effectively, for example, using the normalmode technique based on small-amplitude vibration of nuclei and harmonic approximation of potential energy surface. At high energies, however, anharmonicity of the potential energy surface and intense vibrational and vibration-rotation interaction as well as potential surface crossings, if present, could make the dynamics extremely complicated. As a result, the spectra and dynamical structure in highly excited regime could be often chaotic and difficult to understand. It is by now well-known that vibration-rotation interaction could play an important role for understanding the vibrational spectra and dynamical behavior in polyatomic molecules as the total angular momentum increases.1-4 For the states with little or no angular momentum, it is generally considered that effects of vibration-rotation coupling terms in the kinetic energy operator are negligible,5-7 and many treatments on vibrationrotation spectra and dynamics of polyatomic molecules often do not include these coupling terms in the calculation.8-10 For example, in the study of the relation between bending vibrational spectral pattern and barrier height to linearity in the potential surface of triatomic molecules. Dixon used the Hamiltonian which did not include any vibrational and vibration-rotation coupling terms in the kinetic energy operator.9 It was found that the barrier height deduced experimentally based on Dixon’s analysis was often not in accord with actual barrier height of the molecule.11-13 Since the angular momentum of the system can also be generated by simultaneous vibrations as well as rotation of the molecule, there exists a coupling between vibration and rotation even for the states with zero total angular † Present address: Noyes Laboratory, Department of Chemistry, University of Illinois, Urbana, IL 61801. X Abstract published in AdVance ACS Abstracts, January 1, 1996.
0022-3654/96/20100-1466$12.00/0
momentum (J ) 0) which would be nonnegligible in some highly excited vibrational levels. There has been no systematic investigation on the effects of these kinetic coupling terms on the spectra and dynamics of polyatomic molecules for J ) 0 as the molecular vibrational energy approaches to the highly excited regime. With the rapid experimental progress in the development of laser spectroscopic techniques over the past two decades, there also has been remarkable progress in the development of the effective theoretical formalism and methodology for obtaining accurate energies and wave functions in highly excited vibration-rotation states of polyatomic molecules.14-16 However, regardless of the formalism and methodology used in the calculation of energies and wave functions, it is always necessary to have a genuine global potential energy function which can represent the surface accurately and reliably to understand the vibrational-rotational structures and dynamics of the molecule. Although it may be claimed that the ab initio potential surface could be the most accurate and reliable one for most molecular systems, it still is a very expensive and difficult task to obtain enough ab initio potential points which are needed for global fitting of the surface.17 Thus, it would be a good idea to use model potential functions rather than real ones which can simulate the real molecular potential surface reasonably well in cases when we are only interested in the general features of spectral pattern and dynamics of polyatomic molecules. In this paper, we investigate the effect on the vibrational spectra and structure of vibration-rotation interaction with total angular momentum of zero (vibration-pseudorotation interaction) in symmetric triatomic molecules AB2 by using three different types of model potential energy functions which represent a linear, quasilinear, and bent molecule, respectively. In section II we explain the basic theoretical aspects of the problem, and in section III we present and discuss the model potential energy functions employed in the study. The results of vibrational calculations for model potentials are discussed and compared with the results for CH2+ in section IV. The summary and final conclusion are in section V. © 1996 American Chemical Society
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Vibration-Pseudorotation Coupling in Triatomic Molecules
J. Phys. Chem., Vol. 100, No. 5, 1996 1467
xB
II. Theoretical Method The derivation of the vibration-rotation Hamiltonian and relevant theoretical technique for calculating eigenvalues and eigenvectors of a polyatomic molecule have been explained in detail in previous papers.18-20 Therefore, here, we only present the essential theoretical aspects and methodology employed in this study. The adiabatic vibration-rotation Hamiltonian operator of a symmetric triatomic molecule in an arbitrary bodyfixed internal coordinate system defined from a reference linear configuration has been derived in ref 18 and successfully used for investigating ro-vibrational structures of various triatomic systems.19-21 For a symmetric triatomic molecule A-B-C with C ) A, the Hamiltonian operator can be written as
H ) H0 + TV-R where
H0 ) -
mA ∂2 1 ∂2 mA ∂η 2 mBM ∂η 2
(
s
)
a
mA ∂2 1 ∂ 1 ∂2 + + + V(ηs,ηa,ηb) mBM ∂η 2 ηb ∂ηb η 2 ∂γ2 b b and
TV-R )
[
(
)
2 1 ∂ 1 ∂2 1 2 2 ∂ 2 J + + + J η Z a mA(2r0 + ηs)2 ∂ηb2 ηb ∂ηb ηb2 ∂γ2
ηb
]
∂ ∂ ∂2 ∂2 - ηb2 + 2η + 2η η + a a b ∂ηb ∂ηa ∂ηa ∂ηb ∂η 2
[{( a
)
}
∂ i ∂ 1 ∂ J+ ηa + ηb + ηb ∂γ ∂ηb ∂ηa mA(2r0 + ηs)2
{(
J- ηa -
)
∂ i ∂ ∂ - ηb + ηb ∂γ ∂ηb ∂ηa
[][ yA
zA
mB η 2mA b ) 0 mB 1 -r0 - ηs + η 2 2mA a
]
yB ) 0
[] [
zB
xc
yc
zc
-ηa
mB η 2mA b ) 0 mB 1 r0 + ηs + η 2 2mA a
]
The origin of the body-fixed Cartesian coordinate is at the center of mass of the molecule at its linear reference configuration. This choice of vibrational coordinate system confines the internal molecular motion in the body-fixed xz plane, and the bodyfixed z axis becomes the figure axis of the molecule. The model potential energy function for a symmetric triatomic molecule was developed in the same vibrational coordinate using the minimum number of expansion terms. Rather than using a vibrational coordinate as the expansion variable, we represented the potential energy function in terms of power expansion of appropriate functions of vibrational coordinates, which behave like a coordinate itself for small values of η and become constant at large values. These types of functions have been used for diatomic and triatomic systems before.19,22 The model potential function is expanded as
V(ηs,ηa,ηb) ) csfs2 + cafa2 + x1fb2 + x2fb4 + x3fsfb2 where
fs )
}]
Here, ηs, ηa, and ηb represent the symmetric stretch, asymmetric stretch, and bend vibrational coordinate, respectively. J and Jz are the total angular momentum operator and its body-fixed z component which can be expressed in terms of the Euler angles R, β, γ, and J+ and J- are the raising and lowering operators for the total angular momentum around the body-fixed z axis, respectively. r0 is the distance between atom A and atom B in the linear reference configuration, and M represents the total mass (M ) 2mA + mB). V(ηs,ηa,ηb) is the potential energy function. The Hamiltonian was expressed in two parts, H0 and TV-R. H0 is the unperturbed (pure) vibrational Hamiltonian including the potential energy function, and TV-R accounts for the vibration-rotation interaction in the kinetic energy operator caused by the centrifugal distortion and Coriolis interaction. For J ) 0 calculation, the total angular momentum operator and its body-fixed z component as well as ladder operators (J+ and J-) are set to zero in the expression, thus making the calculation of Hamiltonian matrix elements simple. The relation between vibrational coordinate η and body-fixed Cartesian coordinate x, y, z is given as
xA
-ηb
1 - exp(-k1ηs) tanh(k2ηa) tanh(k3ηb) , fa ) , fb ) k1 k2 k3
Although this is an oversimplified model of a potential surface for triatomic molecules, it has been successfully used to model the bound state region of the potential energy surface for the triatomic molecule before.23 It also has the advantage of enabling us to analyze the dynamical effects of individual potential coupling terms clearly and simplify the variational calculations involved. The coefficients cs and ca, which are related to symmetric and asymmetric harmonic frequencies, and barrier height to linearity (V0) of the molecule are given initially with the geometries of equilibrium and linear reference configuration. Masses of center and end atoms are also given as the vibrational coordinates are expressed in mass-weighted coordinate system. k1, k2, and k3 are the parameters which are optimized for the model potential function to give the reasonable behavior along the vibrational coordinates. For a linear molecule, the potential energy function contained only quadratic terms in the expansion, thus making the potential energy surface harmonic in the physically important region as the expansion variable becomes a coordinate itself. All expansion coefficients including bending were guessed initially in this case. Meanwhile, in the case of quasilinear and bent molecules, the expansion coefficients for the pure bend terms, x1 and x2, and symmetric stretch-bend mode coupling term, x3, are determined from the conditions which the model potential function should satisfy. If the potential barrier height from equilibrium bent to linear reference geometry of the molecule is represented as V0 with ηs0, ηa0, and ηb0 indicating the vibrational coordinates at equilibrium position of the molecule (ηa0 was set to zero for convenience, and ηa ) ηb ) ηs ) 0 corresponds to the linear reference configuration at which the potential is zero), then x1,
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Lee
TABLE 1: Expansion Coefficients for the Model Potential Functions of an ABA Triatomic Molecule Potential Parameter: k1 ) 0.8, k2 ) 1.4, k3 ) 6.0 Linear Reference Configuration:a r0 ) 1.065 Å expansion coeffb cs ca x1 x2 x3 re (Å) θe (deg)
barrier height to linearity (cm-1) MP(I) 0 MP(II) 1000 MP(III) 10000 0.3484798 65.17453 3.608279 0 0
0.3484798 65.17453 -2.965045 1364.669 35.06913
Equilibrium Configurationc 1.065 1.070 180 136
0.3484798 65.17453 -29.65045 5706.064 35.06913 1.070 136
a 0 r is the bond length between atom B and atom A in the linear saddle point geometry. b In units of hartree/(Å)n, where n ) 2 for cs, ca, x1; n ) 4 for x2; and n ) 3 for x3. c re is the equilibrium bond length between atom B and atom A.
x2, and x3 are obtained from the following equations:
( ) ∂V ∂ηs
) 0, ηs0ηa0ηb0
( ) ∂V ∂ηb
) 0, V(ηs0,ηa0,ηb0) ) -V0 ηs0ηa0ηb0
(1)
The vibration-rotation wave function, which is chosen to be an eigenfunction of the total angular momentum operator and its space-fixed z component, was expanded using the following basis functions
ψJM )
csabKDJMK(Rβγ) Φs(ηs)Φa(ηa) ΦbK(ηb) ∑ sabK
Here, DJMK(Rβγ) is the rotation matrix element24 with M and K representing the quantum number of space-fixed and bodyfixed z-axis component of the total angular momentum, respectively. The vibrational basis function for the symmetric and asymmetric stretch mode was chosen to be an eigenfunction of the one-dimensional harmonic oscillator, while the radial part solution of two-dimensional harmonic oscillator eigenfunction was used for the bend mode which is a generalized Laguerre polynomial.25 III. Model Potential Energy Surface for an AB2 Molecule Three model potential energy functions with different barrier heights to linearity were generated to represent the potential energy surfaces for a linear, quasilinear, and bent symmetric triatomic molecule. (Hereafter model potential functions are called MP(I), MP(II), and MP(III).) For the quasilinear and bent molecule, the barrier height to linearity was set to 1000 and 10000 cm-1, respectively, in eq 1. All other parameters except barrier height were the same for all model potentials. The masses of center and end atom were chosen to be the masses of carbon (12C) and hydrogen (1H) atom, respectively, for comparison with the real molecule. In Table 1, we give the data for the three model potential functions. In Figure 1 a onedimensional plot of the model potential surfaces is shown during the bending vibration of the molecule with stretching coordinates fixed at equilibrium. The potential energy surface becomes steeper and more anharmonic as the equilibrium configuration of the molecule changes from linear to bent configuration. In Figures 2 and 3, we plot the contour surfaces of model potential functions in the symmetric stretch-bend and asymmetric stretch-bend coordinates with the remaining coordinate fixed at equilibrium, respectively. For MP(II) and MP(III), double
Figure 1. Bending surfaces of three model potential functions with stretching coordinates fixed at equilibrium. The energy and massweighted vibrational coordinate are in units of cm-1 and Å, respectively, in all figures.
wells develop along the bend coordinate which correspond to two bent equilibrium positions of the molecule. It is also shown that although the bending vibration becomes difficult by the introduction of the barrier to the linear configuration, it has the effect of smoothing the surface from the bottom of the well along the asymmetric stretch coordinate as the energy goes higher. In Figures 4 and 5, we plot the corresponding surfaces of realistic ab initio potential of CH2+,21 which can be considered quasilinear, and the empirical potential of NO2,26 which is bent, with respective barrier height to linearity of about 1200 and 12100 cm-1. The qualitative agreement is well demonstrated between model and real potentials, although the surface along bending vibration behaves rather steep for MP(III). The longrange behavior of model potential functions is quite different from real molecular potential functions, especially in the contour surface of asymmetric stretch-bend coordinate as the model potential function does not show the proper dissociation channel along these coordinates. However, this would not present any serious problem in studying the vibrational states in bound state region. Although the model potential functions do not show the correct dissociation channels, we found that all model potential functions behave reasonably (positive behavior) in the asymptotic region and do not show any serious unphysical behavior (like spurious negative wells) in the entire surface which could affect the results of the variational calculations. IV. Results and Discussion In Table 2 we give the vibrational energies with and without kinetic vibration-rotation coupling terms for three model potential functions. All energies were converged to the point that adding further basis functions changed the energy by less than 0.5 cm-1. As expected from intuition, addition of coupling terms increases the total energy of the system in all cases. However, the magnitude of coupling energy (which is the difference between E and E0) and its change according to vibrational mode excitation show clear mode dependence for all three types of molecules. From the results of the linear molecule with a very harmoniclike potential, it is clear that both the bend and asymmetric stretch mode excitation increase the coupling energy with the effect more pronounced for bend mode excitation. Meanwhile, the excitation of symmetric stretch mode makes little difference in coupling energy from the value in the ground state as expected from the representation of the symmetric stretch mode coupling
Vibration-Pseudorotation Coupling in Triatomic Molecules
Figure 2. Contour plots of model potential functions: (a) MP(I), (b) MP(II), and (c) MP(III) in the symmetric stretch and bend coordinate with asymmetric stretch coordinate fixed at the equilibrium position. The contours are at energies of 100 × 2n cm-1 (n ) 0, 1, 2, 3, ...) in all figures of potentials.
in the Hamiltonian. As a result, the excitation of symmetric stretch mode does not affect the coupling energy, even in highly excited levels. For example, the magnitude of coupling energy for the (310) level, which is at about 11 000 cm-1 above zeropoint energy, is almost same as for the (010) level. Meanwhile, combination levels of asymmetric stretch and bend mode exhibit an additional interaction between vibrational modes which significantly increases the coupling energy from the values in the pure bend or asymmetric stretch levels. Thus, the coupling energy for the (040) and (001) state is only 145 and 32 cm-1, respectively, while for the (041) levels much larger coupling
J. Phys. Chem., Vol. 100, No. 5, 1996 1469
Figure 3. Contour plots of model potential functions: (a) MP(I), (b) MP(II), and (c) MP(III) in the asymmetric stretch and bend coordinate with symmetric stretch coordinate fixed at the equilibrium position.
energy (∼456 cm-1) is obtained than the sum of coupling energies in the (040) and (001) levels. Overall, our results are consistent with the classical picture of the Coriolis coupling effect in a linear symmetric triatomic molecule.27 In the case of simultaneous motion of rotation and vibration, there exists a strong interaction between asymmetric stretching and bending vibration, whereas symmetric stretching vibration does not produce any coupling with other vibrations. It is clear that the effect on total energy of vibration-rotation coupling terms is strongly affected by the change in the potential energy surface of the molecule. For the quasilinear molecule with slight barrier to linearity, the tendency in the change of
1470 J. Phys. Chem., Vol. 100, No. 5, 1996
Figure 4. Contour plots of potential function for CH2+ (a) in the symmetric stretch and bend coordinate with asymmetric stretch coordinate fixed at the equilibrium position and (b) in the asymmetric stretch and bend coordinate with symmetric stretch coordinate fixed at equilibrium position.
coupling energy according to mode excitation shows a similar behavior of mode dependence which was found in the results for the linear molecule. However, the coupling effect is more significant for the levels of the excited asymmetric stretch mode according to mode excitation than the bend mode, and excitation of the symmetric stretch mode decreases the coupling energy from the value in the ground state. One reason for this change would be the reduced bending vibrational amplitude of nuclei as the potential surface along bend coordinate becomes steeper in a quasilinear molecule than in a linear molecule. It is also shown from Figures 2 and 3 that addition of a symmetric stretch-bend coupling term and introduction of a small barrier to linearity in the potential surface for the quasilinear molecule allow nuclei to move along the asymmetric stretch coordinate more easily by providing a low-energy path in the surface. As the molecule is getting excited in the asymmetric stretch mode, the stretching amplitude of nuclei and the coupling energy increase more rapidly for the quasilinear molecule compared to the linear molecule. For the model potential surface of a bent-type molecule with significant barrier height to linearity, it is found that the coupling energy in the excited pure bend states becomes smaller as the levels go higher. It appears that the barrier effect is maximized at the energy level close to the top of the barrier, which corresponds to the (040) level in this case. (The total energy at the (040) level is about 9100 cm-1 above zero-point energy.
Lee
Figure 5. Contour plots of potential function for NO2 (a) in the symmetric stretch and bend coordinate with asymmetric stretch coordinate fixed at the equilibrium position and (b) in the asymmetric stretch and bend coordinate with symmetric stretch coordinate fixed at equilibrium position. The mass-weighted bend and asymmetric stretch coordinates in this case were scaled to correspond to 1H-12C-1H system in other figures. The conversion factor is 4.8375.
Addition of a zero-point bending contribution (0.5νb) to this energy makes it about 10 290 cm-1, which is close to the height of the barrier.) The reason for this reduction of coupling energy for bend mode excitation in MP(III) could be attributed to the existence of a high potential barrier to linearity as well as the steeper outer surface behavior along the bending motion compared to MP(I) or MP(II). It is not clear which factor in potential energy surface (barrier to linearity or steeper outer wings in MP(III)) is more responsible for reduction of coupling energy. To clarify this point, we performed several test calculations employing model potentials with smoother outer surface along the bend mode than MP(III). The results generally showed the same trend of decreased coupling energy according to bend mode excitation. Thus, the reduction of vibrationrotation coupling energy according to bend mode excitation appears to be the general property of bent triatomic molecules with significant barrier height to linearity. Meanwhile, the excitation of the asymmetric stretch mode increases the coupling energy most significantly for the bent molecule. This appears to be related to the feasibility of large-amplitude motion of asymmetric stretching vibration as the molecule is getting excited in the asymmetric stretch mode in a bent molecule compared to a linear or quasilinear molecule. These kind of features are also shown in the potential surfaces of real molecules.
Vibration-Pseudorotation Coupling in Triatomic Molecules
J. Phys. Chem., Vol. 100, No. 5, 1996 1471
TABLE 2: Vibrational Energy Levelsa for the Model Potential Functions of an ABA Triatomic Moleculeb MP(I)
MP(II)
MP(III)
vib statec
E
E0d
E - E0
E
E0
E - E0
E
E0
E - E0
(000) (010) (020) (030) (040) (050) (060) (070) (080) (100) (200) (300) (001) (002) (003) (110) (120) (130) (140) (150) (011) (021) (031) (041) (051) (101) (111) (121) (131)
4094.04 1575.91 3094.94 4557.28 5963.13 7312.68 8606.19 9843.90 11026.33 3157.07 6271.29 9342.83 3431.58 6861.56 10290.05 4732.63 6251.30 7713.26 9118.69 10467.82 5073.64 6661.05 8194.26 9673.75 11100.04 6588.34 8229.43 9815.79 11347.93
4084.78 1541.88 3025.58 4451.09 5818.41 7127.55 8378.50 9571.26 10706.49 3157.16 6271.47 9342.99 3399.21 6797.63 10195.25 4699.04 6182.74 7608.25 8975.57 10284.71 4941.09 6424.79 7850.30 9217.62 10526.76 6556.37 8098.25 9581.95 11007.46
9.26 34.03 69.36 106.19 144.72 185.13 227.69 272.64 319.84 -0.09 -0.18 -0.16 32.37 63.93 94.80 33.59 68.56 105.01 143.12 183.11 132.55 236.26 343.96 456.13 573.28 31.97 131.18 233.84 340.47
3991.38 700.35 1608.74 2708.78 3932.98 5245.87 6626.46 8059.15 9533.81 3501.59 6858.34 10117.63 3537.20 7050.52 10545.71 4283.64 5329.28 6534.43 7846.14 9233.78 4271.55 5257.44 6429.38 7722.76 9104.12 6988.66 7841.29 8960.55 10236.37
3919.36 690.67 1559.76 2622.52 3811.04 5088.67 6429.01 7835.94 9276.24 3531.76 6904.53 10171.76 3399.21 6797.63 10195.25 4281.12 5288.95 6463.67 7731.16 9084.04 4089.87 4958.97 6021.72 7210.24 8487.85 6930.97 7680.33 8688.16 9862.85
72.02 9.68 48.98 86.26 121.94 157.20 197.45 223.21 257.57 -30.17 -46.19 -54.13 137.99 252.89 350.46 2.52 40.33 70.76 114.98 149.74 181.68 298.47 407.66 512.52 616.27 57.69 160.96 272.39 408.30
5041.70 2456.84 4805.78 6992.01 9061.72 11183.84
4938.63 2458.00 4810.70 7005.19 9075.98 11197.42
103.07 -1.16 -4.92 -13.18 -14.26 -13.58
3873.61 7658.95 11344.78 3619.12 7235.34 10848.58 6182.58 8293.63 10397.67 12675.49
3881.28 7674.60 11366.43 3399.21 6797.63 10195.25 6194.04 8310.87 10407.11 12680.27
-7.67 -15.65 -21.65 219.91 437.71 653.33 -11.46 -17.24 -9.44 -4.78
6073.27 8414.02 10583.19 12652.67 14774.78 7477.21 9778.01 11878.43 13998.57
5857.21 8209.91 10404.40 12475.18 14596.63 7280.49 9593.25 11710.08 13806.31
216.06 204.11 178.79 177.49 178.15 196.72 184.76 168.35 192.26
a All excited levels are above the ground state energy in each case of E and E , in units of cm-1 b . Masses of A and B atom are masses of 1H 0 and 12C, respectively. c Vibrational quantum number assignment is (sym, bend, asym). For the linear molecule of MP(I), the bend quantum number should be read as twice the number written here. Therefore, (010) level is actually (0200). d E0 is the energy obtained from unperturbed Hamiltonian (H0).
The effect of vibration-pseudorotation coupling terms on vibrational structure was examined by plotting and comparing the wave functions obtained with and without coupling terms of the kinetic energy operator in the Hamiltonian. Since the effect of coupling terms can be correlated with the potential energy surfaces employed, we devised another potential function which implemented the model potential function MP(II) by adding a symmetric-asymmetric stretch coupling term in the expansion.
MP(IV): V(ηs,ηa,ηb) ) VMP(II) + λfsfa2 The coefficient λ, which was -52.139 62 in this case, was determined to simulate the potential energy surface for a real molecule of CH2+ more accurately with reasonable (positive) asymptotic behavior. In Figure 6 we present the potential energy surface for MP(II), MP(IV), and real CH2+ molecule in the symmetric-asymmetric stretch coordinate with bending coordinate fixed at equilibrium configuration. The MP(IV) surface is getting closer to the real surface of CH2+ than MP(II). The other surfaces for MP(IV) were very similar to the ones for MP(II). This MP(IV), together with other model potentials, was employed for investigating the changes in the vibrational structure induced by vibration-pseudorotation coupling. For most levels with J ) 0 it was found that inclusion or omission of the vibration-rotation coupling terms in the Hamiltonian did not affect the vibrational structure, regardless of the potential surfaces employed. However, we have found that the vibration-rotation coupling terms could noticeably affect the mode structure in some high-energy states for MP(IV) even with J ) 0. For example, the contour plot of the (121) state wave function for MP(IV) without vibration-rotation coupling terms shows a more distinguishable mode structure
in the symmetric-asymmetric stretch coordinate in Figure 7a, while it is shown in Figure 7b that the breakdown of the symmetric stretch mode structure is more obvious as the coupling is switched on to the Hamiltonian. In these figures, the common logarithmic values of the square of the wave function integrated over bend coordinate and rotational angles were plotted. The validity of the model potentials and the results obtained through them were checked by performing the same calculation and analysis for the realistic case of a quasilinear CH2+ molecule for which an accurate SPF (Simons-Parr-Finlan) type potential function for ground electronic state was available for variational calculation.19 In Table 3, we give the results of vibrationpseudorotation interaction energy according to mode excitation for CH2+. The results for CH2+ generally agree with the results for quasilinear MP(II) potential, exhibiting the clear mode dependence of the coupling energy according to mode excitation. The reduced coupling energy observed for (010) and (110) states could possibly be related to slightly higher barrier compared to MP(II) as well as other differences in the potential energy surfaces. The excitation of the asymmetric stretch mode increases the coupling energy most significantly as in the case of model potentials, but it does so much more rapidly in the case of CH2+ according to mode excitation. This could be explained again by noticing the smoother potential surface for asymmetric stretching motion in CH2+ compared to MP(II). The effect of kinetic coupling terms on the dynamics of the CH2+ molecule was also found to be negligible in most states for J ) 0. No significant changes were observed in the nodal structure of the vibrational wave function by inclusion or omission of the vibration-rotation coupling terms in the Hamiltonian. However, as we already have observed in some
1472 J. Phys. Chem., Vol. 100, No. 5, 1996
Lee
Figure 7. Contour plots of the square of the wave functions of the (121) state for model potential MP(IV) in the symmetric and asymmetric stretch coordinate: (a) without and (b) with vibration-rotation coupling terms in the Hamiltonian. Values of the common logarithm are plotted with a contour interval of 0.5 in these figures. Dotted lines represent the negative values.
introduced. It is interesting to note that vibration-pseudorotation coupling not only could mix up vibrational modes (Figure 9a) but also change the mode structure along one vibrational coordinate differently (Figure 9b). For both the model and real potentials, the coupling effect on the structure of wave function appeared to be more pronounced for combination levels rather than levels of single-mode excitation as expected. It has to be noted, however, that the details of vibrational spectra and structure of CH2+ molecule would be strongly affected by the vibronic interaction like Renner-Teller effect.28 Figure 6. Contour plots of potential functions for (a) MP(II), (b) MP(IV), and (c) CH2+ in the symmetric and asymmetric stretch coordinate with bend coordinate fixed at equilibrium position.
combination levels of higher energy for MP(IV) in the model potential, inclusion of the vibration-rotation coupling terms in the Hamiltonian made a clear difference in the nodal structure in some high-energy states for CH2+, even with J ) 0. For example, the contour plot of the (141) level without coupling terms shows a rather clear nodal structure along the symmetric stretch and bend coordinate in Figure 8a. The corresponding plot in the asymmetric stretch and bend coordinate is shown in Figure 8b, which exhibits loosened but distinguishable nodal structure along the bend mode. However, in Figure 9a,b, the breakdown of nodal structure along the bend coordinate becomes more obvious as the vibration-pseudorotation coupling is
V. Summary and Conclusion We have investigated the effect on vibrational spectra and structure of vibration-rotation coupling terms in symmetric triatomic molecules for J ) 0 up to the levels of high vibrational energies with model potential functions which represent the linear, quasilinear, and bent type of molecule, respectively. The mode dependence of coupling energy was clearly demonstrated, which varied according to potential energy surfaces employed. The magnitude of this vibration-pseudorotation coupling was found to be nonnegligible in some highly excited levels (up to about 5% of the total energy), with the effect most pronounced for excitation of asymmetric stretch and/or bend mode. Meanwhile, excitation of the symmetric stretch mode has a generally negligible effect on coupling energy. For the bent molecule with a significant barrier to linearity, the coupling energy
Vibration-Pseudorotation Coupling in Triatomic Molecules
J. Phys. Chem., Vol. 100, No. 5, 1996 1473
TABLE 3: Vibrational Energy Levelsa,b for CH2+ vib state
E
E0
E - E0
(000) (010) (020) (030) (040) (050) (060) (100) (200) (300) (001) (002) (003) (110) (120) (130) (140) (150) (011) (021) (031) (041) (051) (101) (111) (121) (131)
3552.69 838.81 1738.50 2864.65 4150.84 5519.98 6931.94 2937.99 5857.90 8847.79 3226.43 6659.52 10413.60 3756.79 4645.45 5768.48 7073.67 8497.02 4058.90 4928.80 6021.82 7271.24 8608.25 6130.26 6948.58 7808.85 8899.27
3475.92 849.78 1729.55 2831.88 4095.02 5444.05 6844.32 2938.68 5857.37 8842.16 3033.13 6206.40 9611.72 3767.80 4635.97 5734.11 7008.79 8395.41 3904.81 4729.45 5744.18 6925.70 8196.89 5950.09 6799.18 7618.37 8627.26
76.77 -10.97 8.95 32.77 55.82 75.93 87.62 -0.69 0.53 5.63 193.30 453.12 801.88 -11.01 9.48 34.37 64.88 101.61 154.09 199.35 277.64 345.54 411.36 180.17 149.40 190.48 272.01
a
All excited levels are above the ground state energy, in each case of E and E0, in units of cm-1. b Variationally calculated levels with the potential function in ref 21.
Figure 9. Contour plots of the square of the wave functions of the (141) state for CH2+ with vibration-pseudorotation coupling switched on: (a) in the symmetric stretch and bend coordinate; (b) in the asymmetric stretch and bend coordinate.
Figure 8. Contour plots of the square of the wave functions of the (141) state for CH2+ without vibration-pseudorotation coupling: (a) in the symmetric stretch and bend coordinate; (b) in the asymmetric stretch and bend coordinate. Values of the common logarithm are plotted with contour intervals of 1.0 and 0.5 in (a) and (b), respectively.
according to excitation of the bend mode was found to decrease until the bending energy of the state reaches to the top of the barrier. This appears to be related to the existence of the high barrier to linearity in the potential surface of the bent molecule which could make bending and asymmetric stretching motion difficult as the molecular configuration changes from equilibrium bent to linear according to the excitation of the bend mode. Interestingly, these results are in accord with previous observations on vibrational spectra of bent triatomic molecules which found the decrease of level spacing between successive bending levels until the energy reaches to the top of the potential barrier.9,23 The effect of the kinetic vibration-rotation interaction on vibrational structure of the wave function was usually not significant for most levels with J ) 0. However, we have found that the vibration-rotation interaction could play a nonnegligible role to loosen up the regular mode structure of the wave function in some combination states of high energy, even for J ) 0 in a certain type of potential function. This implies that this vibration-pseudorotation coupling effect could be even more noticeable in some states of high energy depending upon molecular systems considered. Of course, at further higher energies, the nodal structure of the wave function would be broken mainly by the potential coupling (anharmonicity). The results for CH2+ with a slight barrier to linearity generally agree with the results for model potentials, exhibiting a clear mode dependence of vibration-rotation coupling energy and nonnegligible nodal structure changes in some highly excited
1474 J. Phys. Chem., Vol. 100, No. 5, 1996 combination levels. This supports the validity of the models employed in this study, despite their simplicity and quantitative differences from real molecular potential functions. Inclusion of additional coupling terms and variation of parameters in the model potential function would make it possible to simulate the real potential surfaces of triatomic molecules more closely. Acknowledgment. The author thanks Professor Don Secrest for several helpful discussions. The author also acknowledges the System Engineering Research Institute in Daeduk, Korea, for an allocation of computer time on a Cray C90, where most of calculations were performed. This work was supported by a grant from the Korea Science and Engineering Foundation. References and Notes (1) Tennyson, J.; Sutcliffe, B. T. J. Mol. Spectrosc. 1983, 101, 71. (2) Dai, H. L.; Korpa, C. L.; Kinsey, J. L.; Field, R. W. J. Chem. Phys. 1985, 82, 1688. (3) Silbert, E. L. J. Chem. Phys. 1989, 90, 2672. (4) Aoyagi, M.; Gray, S. K. J. Chem. Phys. 1991, 94, 195. (5) Darling, B. T.; Dennison, D. M. Phys. ReV. 1940, 57, 128. (6) Carney, G. D.; Sprandel, L. L.; Kern, C. W. AdV. Chem. Phys. 1978, 37, 305. (7) Tennyson, J.; Sutcliffe, B. T. J. Chem. Phys. 1982, 77, 4061. (8) Dunn, K. M.; Boggs, J. E.; Pulay, P. J. Chem. Phys. 1986, 85, 5838.
Lee (9) Dixon, R. N. Trans. Faraday Soc. 1964, 60, 1363. (10) Tennyson, J.; Sutcliffe, B. T. Mol. Phys. 1982, 46, 97. (11) Herzberg, G.; Johns, J. W. C. Proc. R. Soc. London, Ser. A 1966, 295, 107. (12) Bender, C. F.; Schaefer III, H. F.; Franceschetti, D. R.; Allen, L. C. J. Am. Chem. Soc. 1972, 94, 6888. (13) Xie, W.; Harkin, C.; Dai, H. L. J. Chem. Phys. 1990, 93, 4615. (14) Bacic, Z.; Light, J. C. J. Chem. Phys. 1986, 85, 4594. (15) Chang, B. H.; Secrest, D. J. Chem. Phys. 1991, 94, 1196. (16) Bowman, J. M.; Gazdy, B. J. Chem. Phys. 1990, 93, 1774. (17) Bernstein, R. B. Atom-Molecule Collison Theory; Plenum: New York, 1979. (18) Estes, D.; Secrest, D. Mol. Phys. 1986, 59, 569. (19) Lee, J. S.; Secrest, D. J. Chem. Phys. 1986, 85, 6565. (20) Chang, B. H.; Lee, J. S.; Secrest, D. Comput. Phys. Commun. 1988, 51, 195. (21) Lee, J. S.; Secrest, D. J. Phys. Chem. 1988, 92, 1821. (22) Huffaker, J. M. J. Chem. Phys. 1976, 64, 3175. (23) Lee, J. S. J. Chem. Phys. 1992, 97, 7489. (24) Edmonds, A. R. Angular Momentum in Quantum Mechanics; Princeton University: Princeton, 1960. (25) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover: New York, 1972. (26) Tashkun, S. A.; Jensen, P. J. Mol. Spectrosc. 1994, 165, 173. (27) Herzberg, G. Infrared and Raman Spectra; D. Van Nostrand: New York, 1945. (28) Renner, R. Z. Phys. 1934, 92, 172.
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