Sep 8, 1999 - Recently, Lee, Martin, and Taylor calculated a very accurate ab initio quartic force field for methane referred to as LMT force field hereafter.11 ...
JOURNAL OF CHEMICAL PHYSICS
VOLUME 111, NUMBER 10
8 SEPTEMBER 1999
A nine-dimensional perturbative treatment of the vibrations of methane and its isotopomers Xiao-Gang Wang Open Laboratory of Bond Selective Chemistry, University of Science and Technology of China, Hefei, 230026 China
Edwin L. Sibert III Department of Chemistry and Theoretical Chemistry Institute, University of Wisconsin-Madison, Madison, Wisconsin 53706
共Received 6 May 1999; accepted 15 June 1999兲 The vibrations of methane isotopomers with T d , C 3 v , and C 2 v symmetry are studied by means of high order Van Vleck perturbation theory. The vibrational states up to 9000 cm⫺1 are investigated by combining the ab initio force field of Lee, Martin and Taylor 关J. Chem. Phys. 95, 254 共1995兲兴 with a fourth order perturbative treatment based on curvilinear normal coordinates. Implementation of the perturbation theory using both analytical and numerical expression of the kinetic energy operator is considered. The quadratic and select cubic and quartic force constants are refined via a nonlinear least squares fit to experimental data The fit force constants reproduce 130 experimental band centers with a root mean squares deviation of 0.70 cm⫺1 . The choice of polyad quantum number is discussed with respect to different molecules. The convergence of the energy levels is discussed by carrying out the perturbation calculation up to eighth order. © 1999 American Institute of Physics. 关S0021-9606共99兲00334-7兴
I. INTRODUCTION
in which the vibrational self-consistent field approach was used to calculate the J⫽0 and/or 1 levels for isotopomers with T d , C 3 v , and C 2 v symmetry. There are also some other works employing the algebraic approach.17 In a series of studies, Sibert and co-workers have shown the utility of high order Van Vleck perturbation theory for describing the highly excited vibrational states of polyatomic molecules.18–20 Here we apply fourth order Van Vleck perturbative approach based on curvilinear normal coordinates to investigate the vibrational structure of methane and its isotopomers with T d , C 3 v , and C 2 v symmetry including CH4 , CD4 , 13CH4 , 13CD4 , CH3 D, 13CH3 D, CHD3 , 13 CHD3 , and CH2 D2 . The essence of this approach is that the Hamiltonian is canonically transformed order by order into the effective Hamiltonian which, when expressed as a matrix, is block diagonal in a form chosen to include the most important resonances for a given molecule. Our initial fourth order calculations based on the LMT force field show that it provides a rather good prediction of transition energies 关the root mean squares deviation 共RMSD兲 is 2.89 cm⫺1 兴, so that only a limited number of the force constants need refining and the fast-converging second order nonlinear least squares fitting method can be employed. It is interesting to note that in addition to the Van Vleck perturbative approach, there is a classical perturbation theory, the Birkhoff–Gustavson perturbation theory, which has also been carried out to very high order by canonically transforming the operators to study the vibrational energy and classical dynamics of some triatomic molecules.21–25 The close similarity between these two types of perturbation theory has been discussed by several researchers.26–28 Our work is particularly relevant to the paper by Venuti
Over the past few years there have been surprisingly rapid developments in the ab initio calculation of potential energy surfaces of spectroscopic accuracy for a variety of molecules. These advances, in turn, have motivated the development of theoretical techniques for calculating rotationvibration spectra from these ab initio surfaces.1–8 The goal of the rotation-vibration studies is to bridge the gap between experiments and the solution of the electronic Schro¨dinger equation, in order to test the quality of potential surfaces away from the equilibrium geometry and gain insights as to how rotation-vibration coupling leads to mixing of the normal modes. In this paper we calculate vibrational energy levels of the methane molecule and its isotopomers using perturbation theory. While variational calculations have been performed on pentatomic molecules,9,10 perturbation theory is particularly suitable for low energy studies, since it is accurate and is computationally less demanding. The vibrations of methane and its isotopomers are known to involve complex couplings between the stretch and bend modes. A full nine-dimensional vibrational study is therefore very demanding. Recently, Lee, Martin, and Taylor calculated a very accurate ab initio quartic force field for methane 共referred to as LMT force field hereafter兲.11 This study stimulated a number of works;12–16 among them are the works12,15 by Halonen and co-workers in which Van Vleck perturbation theory was applied to study isotopomers of T d symmetry, the work14 by Mourbat et al. in which the quadratic and cubic force field and dipole moment derivatives were simultaneously refined to the vibration rotational parameters using an effective Hamiltonian based on the second order perturbation theory, and the work16 by Carter et al. 0021-9606/99/111(10)/4510/13/$15.00
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The vibrations of methane
et al.15 who present results for second order Van Vleck perturbation theory based on symmetry curvilinear internal coordinates. Although their approach and ours are both perturbative, we calculate energies to sixth or eighth order and work in a different coordinate system. They also restricted attention to isotopomers with T d symmetry, while we include additional C 3 v and C 2 v symmetry isotopomers. The outline of the rest of the paper is as follows. In Sec. II, we discuss the construction and expansion of the Hamiltonian, describe how the Van Vleck perturbation theory is realized, and discuss the dependence of the results on the choice of the coordinate systems. In Sec. III, the results for different isotopomers are presented and discussed in conjunction with the choice of polyad quantum number. The fitting procedure and the convergence of the perturbation theory are discussed in Sec. IV. II. CALCULATION OF VIBRATIONAL LEVELS
In this section we outline the perturbative treatment of the molecular vibrations. The distinctive feature of our approach, compared with the majority of similar approaches is the use of curvilinear normal coordinates coupled with a higher order perturbation theory. We are motivated to pursue this higher order approach due the large number of possible resonance interactions and the possible lack of convergence of the lower order approach. We begin this section by describing the coordinates and the form of the Hamiltonian. Having constructed the Hamiltonian, we briefly review the essence of the perturbative approach. A. The Hamiltonian
The Van Vleck calculations follow those of Sibert et al.18 with several modifications. The curvilinear internal coordinates are used to construct and expand the Hamiltonian. By convention they are defined as the stretch and bend extension coordinates. A modification is made to replace the stretch extension coordinates r i with the Morse coordinates y i ⫽1⫺exp(⫺ari). Given a quartic potential in coordinate r i , the Morse parameter a is chosen so that the potential for the pure stretch extension in the Morse coordinates has no cubic contribution, i.e., a⫽⫺ f rrr /3f rr . Taking into account the redundancy among the bend coordinates as detailed in the Appendix, the (J⫽0兲 Hamiltonian can be reexpressed in terms of the symmetry coordinates S and their conjugate momenta as H v⫽
1 2
兺i j P s ⌫ i j 共 S兲 P s ⫹V 共 S兲 ⫹V ⬘共 S兲 , i
j
共1兲
where the ⌫ matrix is a modification of Wilson’s G matrix to take account of the Morse coordinates29 and the redundancy, V the Born–Oppenheimer potential and V ⬘ the potential-like kinetic term. All these terms are functions of the curvilinear symmetry coordinates S. A further linear transformation by means of the L matrix gives rise to the curvilinear normal coordinates Q⫽L ⫺1 S.30,31 The kinetic energy contribution to Eq. 共1兲 is expressed in terms of the normal momenta P i and the ⌫ i j matrix elements which themselves are functions of the normal coordinates.
4511
We determine this dependence by two different methods. The first method uses numerical finite differences following the methods of Wilson, Decius, and Cross.32,29 Details of this procedure are discussed in Ref. 29. The second method starts with analytical expressions for the Wilson G-matrix elements expressed as a function of the internal coordinates R. By defining the redundancy coordinate S r in Eq. 共A2兲 of the Appendix and then substituting S for R via the orthogonal transformation defined in Eq. 共A3兲, the G-matrix elements, become a function of S bearing in mind that S r is not an independent variable but rather depends implicitly on the other symmetry coordinates. This dependence is solved in the form of a power-series expansion as discussed in the Appendix. For an nth order expansion of ⌫ in terms of S, the power-series expression of S r is needed up to nth order only. We use the computer symbolic language Mathematica to compute these analytical expansions and output the results in a Fortran format which is readily inserted into the Fortran main program. For the present study, the expansion of the matrix elements of ⌫ is calculated to fourth order, where we find that there are 4424, 7566, 4547 terms for T d , C 3 v , and C 2 v methane isotopmers, respectively. Having expanded ⌫ in terms of S, the L-matrix transformation is applied to obtain the expansion in curvilinear normal coordinates. Generally speaking, the analytical approach is faster and more accurate than the numerical one, and is chosen in this work. However the numerical approach is probably more suitable for larger molecular systems. We have performed the vibrational calculation using both approaches, and the energy level differences are below 0.001 cm⫺1 . The remaining two contributions to Eq. 共1兲 depend solely on the normal coordinates. They are the potential V of Eq. 共1兲 and the mass dependent contribution V ⬘ . The LMT potential, initially expressed as a quartic expansion in internal extension coordinates, is reexpanded in terms of the curvilinear coordinates described above and then truncated at fourth order. With this choice, both the reexpanded potential and the original potential are identical through fourth order when expanded in the original extension coordinates. We have chosen to work with the reexpanded potential, since this potential, written in terms of the Morse coordinates for the stretch, is expected to be a more faithful representation of the true potential. The V ⬘ contribution is readily evaluated numerically, since it is a known function of the G i j matrix elements and the determinant of the moment of inertia tensor.33 The procedure has been discussed in Ref. 29 for water. For the present study, a new problem arises due to the redundancy. As discussed in the Appendix, the numerical approach to evaluating the redundancy coordinates is more accurate and chosen. But in fact both the numerical and analytical approaches lead to V ⬘ which agrees better than 10⫺3 cm⫺1 . The mass-dependent V ⬘ terms, albeit small, are important for light molecules like methane. They first contribute to the perturbative calculation of transition frequencies at third order, since the lowest order, constant contribution is of order of ប 2 . In our fourth order calculation, it is found that its contribution can be as large as 1.5 cm⫺1 . Hence, the V ⬘
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X. G. Wang and E. L. Sibert III
terms are considered throughout the calculations in this work although many previous studies have neglected them. The ⌫ matrix and the V ⬘ terms are functions of molecular mass and geometry, and are dependent only on two force constants f rr and f rrr which define Morse parameter a. To save time in refining the force field, we expand the ⌫ matrix and V ⬘ in symmetry coordinates first and store the results as files for each isotopomer, with the exception that they are calculated again when force constants F 11 and F 33 are varied. Then in the subsequent refinement phase, these kinetic data are read in and transformed to curvilinear normal coordinates using the L matrix in each cycle.
B. Van Vleck perturbation theory
To carry out the perturbation theory, we follow Nielson,34 and separate the Hamiltonian in the form H v ⫽H (0) ⫹H (1) ⫹ 2 H (2) ⫹•••⫹ n H (n) ,
共2兲
where is the perturbation parameter. To identify terms of different order, we expand H v of Eq. 共1兲 by reexpressing the normal coordinate dependencies of V, V ⬘ and the matrix ⌫ as a Taylor series about the equilibrium configuration. The potential V terms of order n are included in H (n⫺2) , while the ⌫ and V ⬘ contributions of order n are included in H (n) and H (n⫹2) respectively. Having expanded H v , we rewrite it as a function of harmonic oscillator raising and lowering operators. These operators are initially written in Cartesian coordinates for the doubly and triply degenerate modes, and then different symmetrization approaches are applied for the two cases. For the C 3 v isotopomers, the right and left circular raising and lowering operators35 are used, and the symmetrization is readily realized in this case. Details have been described in the work of Pak et al.36 For the T d isotopomers, the Cartesian basis is symmetrized using the vector coupling coefficients from Ref. 37 combined with the Gram-Shmidt orthogonalization method described in Ref. 12. The correctness of the above two symmetrization methods are verified by comparing energy levels obtained using symmetrized and unsymmetrized bases. The transformations are accomplished via a succession of canonical transformations, K v ⫽exp兵 i n 关 S (n) , 兴 其 ¯exp兵 i 2 关 S (2) , 兴 其 ⫻exp兵 i 关 S (1) , 兴 其 H v ,
共3兲
where the S (n) are chosen such that K v has the desired form through order n.18,19 There are many different forms the final Hamiltonian can take. In this work the Hamiltonian was transformed so that its matrix representation has a block diagonal form, characterized by one or more polyad quantum numbers N t⫽
兺i m i v i ,
共4兲
where the integers m i define the block. These integers are chosen such that the resonance interactions are not perturbatively decoupled. The eigenvalues of the individual blocks are obtained by matrix diagonalization.
C. Comparison of the approaches using different coordinates
Before turning to the main results obtained by using the curvilinear normal coordinates, we investigate the dependence of results on the choice of coordinates used 共Table I兲. Particularly we compare with the results using two other coordinates: the rectilinear normal coordinates and the curvilinear symmetry coordinates. In Table II, we show a comparison of theoretical and experimental band origins for all the fundamentals of the CH3 D molecule. The theoretical results of column four are those of Lee et al.11 using second order perturbation theory based on rectilinear normal coordinates. This calculation allows for 2:1 stretch–bend Fermi resonance interaction between the 1 共CH3 symmetric stretch兲 and 5 modes 共CH3 antisymmetric bend兲. This calculation can be compared to the equivalent calculation based on curvilinear normal coordinates shown in column five. These results were obtained as described in last section where the transformed Hamiltonian K v of Eq. 共3兲 has the five polyad numbers given in Table II. The results of the fourth and fifth columns are identical except for the 1 1 state involved in the Fermi resonance. This finding is in accord with the previous work31,38 that has shown that in the absence of resonant interactions both curvilinear and rectilinear calculations give identical results. Moreover, the curvilinear results, as is true here, are better converged generally than are the corresponding rectilinear results. As we shall see, the difference of ⫺23.1 cm⫺1 for the 1 1 state is significantly larger than expected. For this reason, the rectilinear results will not be considered further. When curvilinear symmetry coordinates are used18,15 instead of normal coordinates the quadratic coupling term is included with the cubic terms in H (1) , or in other words treated as first order correction. This can cause the energies to converge more slowly than those using curvilinear normal coordinates, as demonstrated by Sibert in a model system of harmonically coupled Morse oscillators18 and by McCoy and Sibert29 in a perturbative study of the vibrations of water and its deuterated analogs. In this respect, it is interesting to compare our results with those of Venuti et al.15 in which curvilinear symmetry coordinates are employed. Another apparent difference compared to our work is that they implemented the Van Vleck perturbation theory in a harmonic oscillator matrix representation while we canonically transform the operators. These two approaches are, in fact, equivalent after correcting a misprint in Ref. 15. 关As confirmed by authors, the E 0b in Eq. 共2.12兲 of Ref. 15 should refer to zero order energy.兴 We provide one example to quantify the differences. Using the force field of Ref. 15 to calculate the four fundamentals of CH4 to second order we obtain 2916.98, 1534.04, 3015.14, and 1308.75 cm⫺1 , respectively. The discrepancies with the corresponding calculations of Ref. 15 are ⫺0.96, 0.04, ⫺4.81, and ⫺1.75 cm⫺1 , respectively. The largest discrepancies appear in the two F 2 modes. Given that our second order results for the fundamentals are in good agreement with higher order perturbative results, the errors are probably due to the fact that the quadratic cross terms coupling these two F 2 modes are treated perturbatively when the symmetry
J. Chem. Phys., Vol. 111, No. 10, 8 September 1999 TABLE I. Symmetry coordinates for CH4 , CHD3 , CH3 D, and CH2 D2 . CH4 : 1,2,3,4 – Ha 1 A 1 :S 1 ⫽ 2 (r 1 ⫹r 2 ⫹r 3 ⫹r 4 ) 1 E a :S 2a ⫽ (2 ⫺ ⫺ ⫺ ⫺ ⫹2 34) 冑6 12 13 14 23 24 1 E b :S 2b ⫽ 2 ( 13⫺ 14⫺ 23⫹ 24) 1 F 2x :S 3x ⫽ 2 (r 1 ⫺r 2 ⫹r 3 ⫺r 4 ) 1 F 2x :S 4x ⫽ ( ⫺ ) 冑2 24 13 1 F 2y :S 3y ⫽ 2 (r 1 ⫺r 2 ⫺r 3 ⫹r 4 ) 1 F 2y :S 4y ⫽ ( ⫺ ) 冑2 23 14 1 F 2z :S 3z ⫽ 2 (r 1 ⫹r 2 ⫺r 3 ⫺r 4 ) 1 F 2z :S 4z ⫽ ( ⫺ ) 冑2 34 12 CHD3 : 1,2,3 – D, 4 – H A 1 :S 1 ⫽r 4 1 A 1 :S 2 ⫽ 3 (r 1 ⫹r 2 ⫹r 3 ) 1 A 1 :S 3 ⫽ ( ⫹ ⫹ ⫺ ⫺ ⫺ ) 冑6 12 13 23 14 24 34 1 E a :S 4a ⫽ 6 (2r 1 ⫺r 2 ⫺r 3 ) 1 E a :S 5a ⫽ (2 ⫺ ⫺ ) 冑6 14 24 34 E a :S 6a ⫽
1
冑6
(2 23⫺ 13⫺ 12)
The vibrations of methane
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TABLE II. Comparison of second order perturbative energies E(2) for the fundamental band origins 共cm⫺1 ) of CH3 D obtained with rectilinear and curvilinear normal coordinates results. The calculations use the LMT force field. Results D(n)⫽E(expt)⫺E(n) are reported as the difference between experimental energy and nth order perturbation theory. Statea 6 11 31 5 11 21 11 4 11
Sym
E(expt) b
D共2兲c
D共2兲d
D共2兲e
E A1 E A1 A1 E
1161.10 1306.85 1472.02 2200.04 2969.52 3016.71
⫺2.2 ⫺3.0 0.0 5.4 ⫺23.1 4.7
⫺2.2 ⫺3.0 0.0 5.4 ⫺3.4 4.7
⫺2.2 ⫺3.0 0.0 2.6 ⫺2.7 3.0
The notation N lv , used throughout this paper, refers to v quanta of vibration and l quanta of vibrational angular momentum 共if not zero兲 in normal mode number N. b Experimental numbers are from Ref. 40. c From Ref. 11. This calculation used rectilinear normal coordinates. Since 1 ⬇2 5 , the transformed Hamiltonian includes a Fermi resonance interaction between 1 and 5 modes. As a result there are 5 constants: N 1 ⫽2 v 1 ⫹ v 5 , N 2 ⫽ v 2 , N 3 ⫽ v 3 , N 4 ⫽ v 4 , and N 5 ⫽ v 6 . d Curvilinear normal coordinates are used. There are 5 constants, N 1 ⫽2 v 1 ⫹ v 5 , N 2 ⫽ v 2 , N 3 ⫽ v 3 , N 4 ⫽ v 4 , and N 5 ⫽ v 6 . e Curvilinear normal coordinates are used. There are two constants: N 1 ⫽2( v 1 ⫹ v 4 )⫹ v 3 ⫹ v 5 , N 2 ⫽2 v 2 ⫹ v 6 . a
curvilinear coordinates are used; hence the symmetry coordinate results have not yet converged.
1
E b :S 4b ⫽ 2 (r 2 ⫺r 3 ) 1 E b :S 5b ⫽ ( ⫺ ) 冑2 24 34 E b :S 6b ⫽
1
冑2
( 13⫺ 12)
CH3 D: 1,2,3 – H, 4 – Db CH2 Dc2 : 1,2 – H, 3,4 – D 1 A 1 :S 1 ⫽ (r ⫹r ) 冑2 1 2 A 1 :S 2 ⫽ A 1 :S 3 ⫽ A 1 :S 4 ⫽
1
冑2
(r 3 ⫹r 4 )
1
冑20 1
冑30
(4 12⫺ 13⫺ 14⫺ 23⫺ 24)
III. RESULTS AND DISCUSSION
In this section we describe the results of the perturbative calculations for the different isotopomers of methane. The choice of the polyad quantum numbers are motivated for each of the species. Results are given for both the LMT force field and a fit force field given in Table III. The details of the fitting procedure are reserved for Sec. IV. A. T d isotopomers
The T d isotopomers have been studied by many researchers. There are only four normal modes. The harmonic TABLE III. Comparison of ab initioa and fit force constants. See text for a discussion of the procedure used for the fit.
(⫺ 12⫺ 13⫺ 14⫺ 23⫺ 24⫹5 34)
Force Constant
1 A 2 :S 5 ⫽ 2 ( 13⫺ 14⫺ 23⫹ 24)
B 1 :S 6 ⫽
1
F 11 F 22 F 33 F 34 F 44 F 12a2a F 14x4x F 2a3z4z F 3x4y4z F 4x4y4z F 4x4x4x4x F 4x4x4y4y
(r 1 ⫺r 2 )
冑2 1 B 1 :S 7 ⫽ 2 ( 13⫹ 14⫺ 23⫺ 24) 1 B 2 :S 8 ⫽ (r ⫺r ) 冑2 3 4 1 B 2 :S 9 ⫽ 2 ( 13⫺ 14⫹ 23⫺ 24) A 1 :S r ⫽ a
1
冑6
( 12⫹ 13⫹ 14⫹ 23⫹ 24⫹ 34) d
The symmetry coordinates of CH4 are those of Ref. 54. The symmetry coordinates of CH3 D are obtained from those of CHD3 by interchanging S 1 with S 2 and S 5 with S 6 . c The symmetry coordinates of CH2 D2 follow that of Ref. 55. In particular, the S 3 and S 4 coordinates are solved from the orthonormal requirements. d Redundancy coordinates S r are the same for all methane isotopomers. b
a
ab initio
Fit
b
5.473 84 0.577 70 5.376 96 ⫺0.210 57 0.532 25 ⫺0.254 38 ⫺0.225 56 0.180 04 ⫺0.096 16 0.343 91 0.498 76 0.709 77
5.483 898 0.578 927 5.385 664 ⫺0.205 194 0.532 065 ⫺0.259 348 ⫺0.265 963 0.177 684 ⫺0.099 929 0.404 224 0.414 949 0.807 394
0.2% 0.2% 0.2% 2.6% 0.0% 2.0% 17.9% 1.3% 3.9% 17.5% 16.8% 13.8%
The bond length is assumed to be the ab initio bond length of 1.0879 Å . Ab initio force constants are from Tables I and II of Ref. 11. The fit force constants are obtained using fourth order perturbation theory. All force constants not listed are constrained to those of the ab initio force field. b The absolute relative deviation of the fit force constants from the ab initio force constants.
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frequencies of the two stretch modes are approximately twice those of the two bending modes, leading to a clearly identifiable polyad structure where N⫽2 共 v 1 ⫹ v 3 兲 ⫹ v 2 ⫹ v 4 .
共5兲 17
The very recent studies of Wiesenfeld et al. and Mourbat et al.14 provide good overviews of the extensive literature on the spectroscopic studies of the polyads and the use of effective Hamiltonians in describing it. Our perturbative results, as listed in Table IV, are derived from a potential energy surface; hence the present study is most similar to the work of Venuti et al.15 Comparing these two studies we find agreement in many aspects, including agreement between most of the eigenfunction components. The three levels of CH4 which we excluded from the fitting are the only three levels with differences between observed and calculated energies larger than 5 cm⫺1 in Ref. 15. We also agree with Ref. 15 in finding larger discrepancies for many states involving the 2 and 4 bend modes. However, the 1 1 states of CD4 and 13CD4 are better predicted in this work than in former. The region 5820–5900 cm⫺1 , where the assignment of ⌬ v ⫽2 CH stretch band 1 1 3 1 is in disagreement,13,15 is examined as follows. There are five F 2 levels in this region. Table V compares the results of Georges et al.,13 Venuti et al.15 and our work. It can be seen that our results generally agree with those of Venuti et al. The interesting band 1 1 3 1 is severely perturbed and although our assignment is different from that of Georges et al., in both cases the largest percentage of wave function component is only about 50%. Finally, it is pleasing to note that several states of CH4 up to 9000 cm ⫺1 with N⫽6 are also in good agreement with the calculations even though these states are excluded in the fitting. B. CH3 D molecule
For the CH3 D molecule, there are more resonances than the 2:1 stretch–bend Fermi resonance interaction between the 1 and 5 modes as discussed previously in Sec. II C. In the last three columns of Table II we compare three different choices 关cf. Table II for details兴 of the polyad quantum numbers at the level of second order perturbation theory. The choice for the final column, N 1 ⫽2 共 v 1 ⫹ v 4 兲 ⫹ v 3 ⫹ v 5 , N 2 ⫽2 v 2 ⫹ v 6
共6兲
allows for a greater range of resonance interactions than the former two. In particular, the N 1 polyad resonance involves the four normal modes related with the CH3 methyl group motion, while the N 2 polyad resonance involves the CD stretch ( 2 ) and bend modes ( 6 ). Compared to column 4 of Table II, the difference between observed and calculated energies of the fundamental 2 1 decreases from 5.43 to 2.55 due to inclusion the N 2 polyad number; likewise the results for the 1 1 and 4 11 fundamentals slightly improve. In Ref. 39, where the 4 02 (A 1 ) and 4 22 (E) bands are rotationally analyzed, we used another polyad number N 1 ⫽2( v 1 ⫹ v 4 )⫹ v 5 where the 3 is removed as a good approximation, since this mode
being lower in frequency than the 5 is relatively decoupled from the other three methyl group normal modes in the low energy region. For those states that were analyzed this proved to be a good choice of polyad numbers. As seen in Table VI, all experimental numbers are well reproduced with fourth order perturbation theory, including the only observed state of 13CH3 D which was not included in the fitting. Except for the most recent work on the 4 02 (A 1 ) and 4 22 (E) bands,39 no bands over 3000 cm⫺1 are rotationally analyzed to our knowledge.40 As demonstrated by our calculation, the two distinctive polyad numbers of Eq. 共6兲 govern the overtone spectra of CH3 D. Our predictions of the other unobserved bands should be helpful for future rotational study.
C. CH2 D2 molecule
In addition to the low order resonant couplings which usually govern the global resonance structure and determine the polyad quantum numbers, sometimes smaller resonances, which we refer here to as accidental degeneracies, must be included in the perturbative treatment. These degeneracies are more likely to occur for the CH2 D2 molecule, since this molecule has nine different normal mode frequencies. Approximate integer relations between the harmonic frequency values suggest that two global polyad quantum numbers may exist, N 1 ⫽2 共 v 1 ⫹ v 6 兲 ⫹ v 3 ⫹ v 5 , N 2 ⫽2 共 v 2 ⫹ v 8 兲 ⫹ v 4 ⫹ v 7 ⫹ v 9 .
共7兲
The energies of the states within these two polyads are slightly different. For example, the states described by polyad numbers 兵 N 1 ,N 2 其 ⫽ 兵 2,0其 fall in the region 2700– 3000 cm⫺1 while states with the polyad numbers 兵 0,2其 fall in the region 2000–2400 cm⫺1 . In the latter region, there are 3 A 1 (4 2 ⬇2 1 ⬇7 2 ) and 2 B 2 (8 1 ⬇4 1 9 1 ) resonance clusters as treated in the rotational analysis in Ref. 41. The assumed existence of two polyad quantum numbers, however, is too restrictive to cover all the features of the vibrational spectrum of this molecule. There is an energy overlap of the high energy states of the 兵 0,2其 polyad and the low energy states of the 兵 1,1其 polyad. Diagonalization of the fourth order K v with the 2 polyad quantum numbers defined by Eq. 共7兲 shows that the following states might be coupled by quartic resonances: A 1 :3 1 4 1 共 2464兲 ⬇9 2 共 2464兲 , B 1 :3 1 7 1 共 2516兲 ⬇5 1 9 1 共 2610兲 , A 2 :4 1 5 1 共 2364兲 ⬇7 1 9 1 共 2329兲 ,
共8兲
B 2 :4 1 9 1 共 2285兲 ⬇5 1 7 1 共 2422兲 , where the frequencies, in units of wave number, are given in parentheses. All these four resonances destroy the goodness of the N 1 and N 2 polyad quantum numbers even though the interblock couplings are limited. Fortunately even with these resonances one good polyad quantum number is left N⫽N 1 ⫹N 2 .
共9兲
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The vibrations of methane
TABLE IV. Comparison of experimental and Van Vleck perturbation theory band origins 共cm⫺1 ) for the T d symmetry methane isotopomers. The theoretical energies D(n)⫽E(expt)⫺E(n) are reported as the difference between experiment and nth order perturbation theory results. For each eigenvalue, the values of the leading mixing coefficients c i (i⫽1 or 2兲 of the molecular eigenfunction expanded with respect to the dressed normal mode basis are provided.
Na CH4 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6
D(4)
D(4)
D(6)
E(expt) b
LMT
Fit
Fit
F2 E A1 F2 E F2 F1 A1 F2 A1 E F2 A1 F1 F2 E F1 A1 F2 E A2 F2 F2 E F1 A1 F2 F1 F2 E F1 F2 E A2 A1 F2 F2 F2 A1 F2 F2 A1 F2 E F2 F2 F2 F1 A1 F2
1310.76 1533.34 2587.04 2614.26 2624.62 2830.32 2846.08 2916.49 3019.49 3063.65 3065.14 3870.49 3909.18 3920.50 3930.92 4105.15 4128.57 4132.99 4142.86 4151.22 4161.87 4223.46 4319.21 4322.15 4322.58 4322.72 4348.77 4363.31 4379.10 4446.41d 4537.57 4543.76 4592.03 4595.32 4595.56 5587.98 5623.00d 5628.40 5790.25 5819.72 5826.65d 5968.09 6004.69 6043.87 6054.64 6065.32 8906.78 8947.95 8975.34 9045.92
⫺2.08 1.08 0.10 ⫺2.27 ⫺7.87 ⫺1.43 0.07 0.72 2.61 2.28 2.24 4.49 0.14 ⫺6.83 ⫺12.66 4.19 ⫺1.08 ⫺7.36 0.56 ⫺4.76 ⫺5.10 ⫺3.15 0.49 ⫺0.56 ⫺0.98 ⫺0.98 ⫺0.88 0.47 2.85 12.65 3.02 2.97 3.55 3.44 3.59 1.55 6.48 ⫺6.72 1.97 ⫺1.76 ⫺16.15 3.92 4.73 5.64 3.62 3.35 4.68 6.81 5.13 8.00
⫺0.19 ⫺0.08 ⫺0.81 ⫺0.35 ⫺0.39 ⫺0.67 0.95 ⫺0.15 0.33 0.13 ⫺0.04 0.92 0.24 ⫺1.00 ⫺1.00 2.96 ⫺0.12 ⫺1.36 1.59 1.06 1.57 ⫺0.82 0.15 ⫺0.20 0.05 ⫺0.79 ⫺1.06 0.25 2.81 10.60 ⫺0.32 ⫺0.30 0.43 0.10 0.25 ⫺0.56 7.25 0.06 0.22 ⫺2.95 ⫺16.92 0.14 0.36 1.25 ⫺0.62 ⫺0.83 ⫺0.75 0.15 ⫺0.42 1.78
0.39 0.00 2.54 2.10 0.46 0.19 1.72 ⫺0.02 0.33 0.31 0.11 9.64 7.14 2.75 1.38 6.84 3.01 ⫺0.06 4.58 2.41 2.65 0.15 0.76 0.50 0.72 ⫺0.16 0.05 1.33 3.81 10.90 ⫺0.20 ⫺0.19 0.73 0.30 0.51 3.11 10.01 1.10 0.71 ⫺1.87 ⫺15.96 0.33 0.35 1.26 ⫺0.37 ⫺0.57 ⫺0.61 0.06 0.13 1.89
F2 E A1 F2 E F2 A1 A1
1302.78 1533.49 2572.10 2598.64 2608.74 2822.45 2915.44 3063.96
⫺2.06 1.07 ⫺0.03 ⫺2.29 ⫺7.81 ⫺1.41 0.84 2.28
⫺0.22 ⫺0.09 ⫺0.86 ⫺0.41 ⫺0.44 ⫺0.69 ⫺0.18 0.12
0.34 ⫺0.02 2.41 1.98 0.38 0.15 ⫺0.07 0.29
Sym
c 21 4 1 共100%兲 2 1 共100%兲 4 2 共98%兲 4 2 共99%兲 4 2 共100%兲 2 1 4 1 共97%兲 2 1 4 1 共100%兲 1 1 共95%兲 3 1 共96%兲 2 2 共97%兲 2 2 共100%兲 4 3 共96%兲 4 3 共96%兲 4 3 共98%兲 4 3 共99%兲 2 1 4 2 共94%兲 2 1 4 2 共95%兲 2 1 4 2 共94%兲 2 1 4 2 共97%兲 2 1 4 2 共98%兲 2 1 4 2 共100%兲 1 1 4 1 共93%兲 3 1 4 1 共96%兲 3 1 4 1 共95%兲 3 1 4 1 共94%兲 3 1 4 1 共93%兲 2 2 4 1 共92%兲 2 2 4 1 共97%兲 2 2 4 1 共99%兲 1 1 2 1 共93%兲 2 1 3 1 共96%兲 2 1 3 1 共94%兲 2 3 共95%兲 2 3 共100%兲 2 3 共100%兲 3 1 4 2 共91%兲 3 1 4 2 共88%兲 3 1 4 2 共90%兲 1 2 共74%兲 2 1 3 1 4 1 共68%兲 2 1 3 1 4 1 共91%兲 3 2 共48%兲 3 2 共73%兲 3 2 共94%兲 2 2 3 1 共90%兲 2 2 3 1 共90%兲 3 3 共42%兲 3 3 共84%兲 1 1 2 4 共35%兲 3 3 共68%兲
c 22
1 1 共2%兲 3 1 共1%兲 3 1 共3%兲 2 2 共2%兲 2 1 4 1 共3%兲 1 1 共3%兲 1 1 4 1 共3%兲 3 1 4 1 共3%兲 3 1 4 1 共2%兲 3 1 4 1 共1%兲 3 1 4 1 共4%兲 3 1 4 1 共3%兲 3 1 4 1 共4%兲 3 1 4 1 共1%兲 3 1 4 1 共1%兲 4 3 共2%兲 4 3 共2%兲 2 1 4 2 共5%兲 2 1 4 2 共3%兲 2 1 4 2 共5%兲 2 1 3 1 共4%兲 2 1 3 1 共3%兲 1 1 4 1 共1%兲 2 3 共4%兲 2 2 4 1 共3%兲 2 2 4 1 共4%兲 1 1 2 1 共5%兲 1 1 3 1 共3%兲 2 1 4 3 共3%兲 2 1 4 3 共5%兲 3 2 共13%兲 3 2 共12%兲 2 2 4 2 共3%兲 1 1 2 2 共29%兲 1 1 3 1 共15%兲 2 1 3 1 4 1 共4%兲 2 3 4 1 共2%兲 2 3 4 1 共4%兲 1 1 3 2 共16%兲 2 1 3 2 4 1 共5%兲 3 3 共29%兲 1 1 3 2 共9%兲
13
CH4 1 1 2 2 2 2 2 2
4 1 共100%兲 2 1 共100%兲 4 2 共98%兲 4 2 共99%兲 4 2 共100%兲 2 1 4 1 共97%兲 1 1 共95%兲 2 2 共97%兲
c
1 1 共2%兲 3 1 共1%兲 3 1 共3%兲 2 2 共2%兲 1 1 共3%兲
4515
4516
J. Chem. Phys., Vol. 111, No. 10, 8 September 1999
X. G. Wang and E. L. Sibert III
TABLE IV. 共Continued.兲
Na
D(4)
D(4)
D(6)
E(exptg) b
LMT
Fit
Fit
c 21
F2 E A1 F2 E F2 F1 A1 A1 E F2
997.87 1091.65 1965.54 1990.47 1996.83 2083.40 2090.88 2101.37 2182.23 2182.59 2260.08
⫺1.11 0.80 0.20 ⫺1.34 ⫺3.97 ⫺0.69 0.14 0.04 1.71 1.64 2.36
0.20 ⫺0.11 ⫺0.94 0.19 0.59 ⫺0.23 0.64 0.72 ⫺0.01 ⫺0.15 0.25
0.39 ⫺0.09 0.11 1.02 0.88 0.07 0.90 0.83 0.05 ⫺0.10 0.26
4 1 共100%兲 2 1 共100%兲 4 2 共88%兲 4 2 共99%兲 4 2 共100%兲 2 1 4 1 共99%兲 2 1 4 1 共100%兲 1 1 共86%兲 2 2 共98%兲 2 2 共100%兲 3 1 共98%兲
F2 E A1 F2 E F2 F1 A1 A1 E F2
989.38 1091.80 1951.38 1973.47 1979.64 2074.78 2082.36 2098.33 2182.50 2182.88 2245.43
⫺0.97 0.80 ⫺0.04 ⫺1.35 ⫺3.93 ⫺0.66 0.15 0.30 1.68 1.64 2.38
0.29 ⫺0.11 ⫺0.93 0.12 0.52 ⫺0.26 0.59 0.62 ⫺0.04 ⫺0.16 0.32
0.48 ⫺0.09 0.11 0.92 0.79 0.03 0.84 0.71 0.02 ⫺0.11 0.33
4 1 共100%兲 2 1 共100% 4 2 共91%兲 4 2 共99%兲 4 2 共100%兲 2 1 4 1 共99%兲 2 1 4 1 共100%兲 1 1 共88%兲 2 2 共98%兲 2 2 共100%兲 3 1 共98%兲
Sym
CD4 1 1 2 2 2 2 2 2 2 2 2
c 2c 2
1 1 共12%兲 3 1 共1%兲 3 1 共1%兲 4 2 共12%兲 1 1 共2%兲 2 1 4 1 共1%兲
13
CD4 1 1 2 2 2 2 2 2 2 2 2
1 1 共9%兲 3 1 共1%兲 3 1 共1%兲 4 2 共9%兲 1 1 共2%兲 2 1 4 1 共1%兲
N⫽2( v 1 ⫹ v 3 )⫹ v 2 ⫹ v 4 . Experimental numbers are from Ref. 15. c Absolute values less than 1% are not given. d Data excluded from the RMSD. a
b
Using this new polyad number, we performed fourth order Van Vleck perturbation theory. The results are given in Table VII. The resonance coefficients as obtained at the second order are as follows: K 34,99⫽1.73 cm⫺1 ;
K 37,59⫽7.46 cm⫺1 ;
K 45,79⫽⫺0.41 cm⫺1 ;
K 49,57⫽3.56 cm⫺1 .
As seen from Table VII, the last two resonances of Eq. 共8兲 are unimportant, since either the resonance coefficients are small or the energy differences are large. The most significant improvement in terms of energy occurs for the bands of 3 1 4 1 (A 1 ) and 9 2 (A 1 ). The energy shift due to the K 34,99 resonance is about 5.5 cm⫺1 . The bands of 3 1 7 1 (B 1 ) and 5 1 9 1 (B 1 ) are also affected to some extent with a limited mixture of wave function.
It is interesting to examine the region 2000–3100 cm⫺1 where there are a total of 19 bands belonging to the N⫽2 polyad. All these bands have been systematically studied by Ulenikov and co-workers.41–44 Among them 15 band origins obtained by rotational analysis based on sufficient number of assigned levels are well reproduced in our calculation. For the weak band 3 1 7 1 (B 1 ) for which only two rotational levels are observed,42 our calculated band origins are different from the rotationally fitted results by 5.75 cm⫺1 . There are also three other unobserved bands 4 1 5 1 (A 2 ), 5 1 7 1 (B 2 ), and 3 1 5 1 (A 2 ) 关the two A 2 bands are infrared inactive and the 5 1 7 1 (B 2 ) is very weak兴, for which our fourth order results are 2364.41, 2422.34, and 2765.98 cm⫺1 , respectively. In addition, no vibrational mixing is observed for these three states. Ulenikov et al.42 has estimated the band centers of
TABLE V. Comparison of F 2 states in the region 5800–6000 cm⫺1 for CH4 共cm⫺1 ) a. Georges et al.b 5819.72c 5826.65c 5849.30 5867.66 5894.12 a
1 1 3 1 共51%兲 2 1 3 1 4 1 共67%兲 2 1 3 1 4 1 共80%兲 2 3 4 1 共74%兲 2 3 4 1 共89%兲
Venuti et al.d 5821.49 5844.06 5862.74 5871.17 5893.16
2 13 14 1 2 13 14 1 2 34 1 1 13 1 2 34 1
This work 5822.67 5843.57 5862.41 5868.72 5892.07
The two subcolumns are energy, largest basis, and its percentage in parentheses if available. Reference 13. c Experimental values determined from the jet-cooled Fourier transform spectra of Ref. 13 d Reference 15. b
2 1 3 1 4 1 共69%兲 2 1 3 1 4 1 共90%兲 1 1 3 1 共48%兲 2 3 4 1 共44%兲 2 3 4 1 共95%兲
J. Chem. Phys., Vol. 111, No. 10, 8 September 1999
The vibrations of methane
TABLE VI. Comparison of experimental and Van Vleck perturbation theory band origins 共cm⫺1 ) for CH3 D and 13CH3 D. See Table IV caption for details. D(4) N 1 ,N 2 a Sym E(expt) b LMT CH3 D 1,0 1,0 2,0 2,0 2,0 2,0 2,0 2,0 4,0 4,0 0,1 0,2 0,2 0,2 1,1 1,1 1,1
A1 E A1 A1 A1 E E E A1 E E A1 A1 E A1 A2 E
1306.85 1472.02 2597.68 2910.12 2969.52 2776.29 2940.10 3016.71 5980.41d 6022.21d 1161.10 2200.04 2316.28 2323.30 2632.49 2633.46 2466.97
CH3 D 0,2 A 1
2190.05e
-2.05 0.73 ⫺0.64 2.14 1.09 ⫺1.07 1.63 2.61 4.28 5.19 ⫺1.39 2.61 ⫺2.04 ⫺2.78 ⫺1.25 ⫺1.05 ⫺3.88
D(4)
D(6)
Fit
Fit
⫺0.02 0.55 ⫺0.15 ⫺0.03 0.46 2.12 ⫺0.07 0.30 ⫺0.02 0.04 0.25 1.13 ⫺0.10 0.13 0.23 0.24 0.07 0.35 0.74 0.88 ⫺0.06 0.29 ⫺0.10 0.06 ⫺0.20 1.47 ⫺0.14 0.95 ⫺1.47 0.36 ⫺0.57 ⫺0.04 1.13 2.27
c 2c
c1 3 1 共1.00兲 5 11 共1.00兲 3 2 共0.99兲 5 02 共0.77兲 1 1 共0.77兲 3 1 5 11 共1.00兲 5 ⫺2 2 共0.98兲 4 11 共0.98兲 4 02 共0.81兲 4 ⫺2 2 共0.87兲 6 11 共1.00兲 2 1 共0.95兲 6 02 共0.95兲 6 22 共1.00兲 5 1 6 1 共0.95兲 5 1 6 1 共0.95兲 3 1 6 11 共1.00兲
1 1 共⫺0.11兲 1 1 共⫺0.63兲 5 02 共0.64兲 4 11 共0.17兲 5 ⫺2 2 共⫺0.17兲 1 2 共⫺0.44兲 1 1 4 11 共⫺0.40兲 6 02 共⫺0.30兲 2 1 共0.30兲
13
2.56 ⫺0.05 ⫺1.41
2 1 共0.95兲
6 02 共⫺0.30兲
N 1 ⫽2( v 1 ⫹ v 4 )⫹ v 3 ⫹ v 5 ,N 2 ⫽2 v 2 ⫹ v 6 . Experimental numbers are from Ref. 40 unless noted otherwise. c Absolute values smaller than 0.1 are not given. d Reference 39. e Reference 56. a
b
these three bands to be 2362, 2419, and 2765 cm⫺1 , which are not far from our results. Our calculated band origins for the four weak bands discussed above provide a good estimation and should be helpful in the rotational analysis involving these four bands. D. CHD3 molecule
For the CHD3 molecule there exists an approximate integer relation between the harmonic frequencies of the six normal modes; it is 1 : 2 : 3 : 4 : 5 : 6 ⫽3:2:1:2:1:1. This suggests that the six normal modes can be coupled to each other with a polyad quantum number N 1 ⫽3 v 1 ⫹2 共 v 2 ⫹ v 4 兲 ⫹ 共 v 3 ⫹ v 5 ⫹ v 6 兲 .
共10兲
Table VII gives fourth order perturbation results for the refined force field and the polyad number of Eq. 共10兲. It can be seen that this polyad dominates in the low energy region. For example, most of the 12 bands of the N 1 ⫽2 polyad can be successfully predicted compared with a recent preliminary rotational analysis.45,46 In particular, the 5 mode, which is relatively higher than the 3 and 6 modes and which might tune out of the N 1 polyad at high energies 共see discussion below兲, takes part in the strong resonance between the 4 1 and 3 1 5 11 states of the N 1 ⫽2 polyad 共cf. Table VIII兲. Examining the higher energy states of the N 1 ⫽ 8 polyad, we find that the polyad prediction is still more or less good, even though the number of states in the polyad increases dramati-
4517
cally, since all six normal modes are involved. For example, the N 1 ⫽9 polyad involving the 1 3 state has, respectively, 1161 (A 1 ) and 1148(E) states. In this high energy region, another well-established resonance between CH stretch and CH bend becomes relevant,47 because of the strong anharmonicity of the CH stretch mode. The polyad number for this resonance is defined as N 2 ⫽2 v 1 ⫹ v 5 .
共11兲
This second polyad number is contradictory to that of Eq. 共10兲. Nonetheless for the higher CH stretching states the 2:1 resonance must be included, because it is so pronounced. To explore the effects of the additional mixing, the perturbative results were rerun, so that the transformed Hamiltonian has the same structure as that with N 1 polyad of Eq. 共10兲, but with the additional CH stretch–bend coupling terms included. This treatment allows us to examine the magnitude of the stretch-bend coupling term and make comparisons with other investigations. Using the above approach we calculate ⬘ 兩 ⫽66.48 cm⫺1 in the notation the resonance coefficient 兩 K sbb of Ref. 47. This value is different from the experimental estimated value 30⫾15 cm⫺1 of Ref. 47, but close to the fitted value 72 cm⫺1 of Ref. 48 where, similar to our method, the curvilinear Morse coordinates were used. There are two points that should be considered in making this comparison. First, the higher order corrections to the ⬘ , are imporstretch–bend coupling term, proportional to K sbb tant. For example, if we consider the coupling between the 1 2 state and the 1 1 5 2 state, the coupling changes from 66.49 to 58.51 cm⫺1 in going from second to fourth order perturbation theory. These differences will increase as one goes to higher overtones. Second, it is of interest to compare the ⬘ for our fit force potential and kinetic contributions to K sbb field. We find that the kinetic and potential contributions are 89.13 and ⫺22.64 cm⫺1 at the second order, respectively. We conclude this section by describing how the inclusion of the 2:1 stretch bending terms leads to changes from that which is obtained when N 1 is considered to be a good polyad quantum number. To make this comparison, we first need to describe how we calculate the energies when the 2:1 resonance is included, since the Hamiltonian is not blockdiagonal. For the case of the 1 3 state, we find that this state is a member of the N 1 ⫽9 polyad which has 1161 states of A 1 symmetry. If we had instead assumed that the good polyad number was N 2 ⫽6, we would find that there are only 5 A 1 states (1 3 ,12 5 02 ,11 5 04 ,506 ,536 ). These additional 4 states belong to the N 1 ⫽8,7,6 polyads, respectively. In principle, we thus need to diagonalize a matrix containing all the states of the N 1 ⫽6⫺9 polyads. Fortunately, with the exception of the 1 3 state, the remaining states involved in the 2:1 resonance 共i.e. the states 1 2 5 02 ,11 5 04 ,506 ,536 ) are relatively unmixed in their respective N 1 polyads. Hence it is justified to adopt a simplified treatment where only these 4 states are added to those of the N 1 ⫽9 polyad, with an aim to improve on the 1 3 state. The above treatment improved the energy of the 1 3 state from 8619.60 to 8621.93 cm⫺1 . In addition the significant mixture of wave function confirms that the stretch–bend resonance plays an important role in the region higher than
4518
J. Chem. Phys., Vol. 111, No. 10, 8 September 1999
X. G. Wang and E. L. Sibert III
TABLE VII. Comparison of experimental and Van Vleck perturbation theory band origins 共cm⫺1 ) for CH2 D2 . See caption of Table IV for further details.
a
N
N 1 ,N 2
1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
0,1 0,1 0,1 1,0 1,0 0,2 0,2 0,2 0,2 0,2 0,2 0,2 0,2 1,1 1,1 1,1 2,0 1,1 2,0 2,0 2,0
a
b
Sym
E(expt)
A1 B1 B2 A2 A1 A1 B1 A1 A1 B2 B2 A2 A1 A1 B1 B1 A1 B2 A1 A1 B1
1033.05c 1091.18c 1236.28c 1331.36c 1435.13c 2054.16d 2124.68d 2145.69d 2203.22d 2234.69d 2285.98d 2330.70d 2458.80e 2469.20e 2510.21g 2560.55f 2658.34f 2671.68f 2855.66e 2975.48e 3012.3h
D(4)
D(4)
D(6)
LMT
Fit
Fit
c1
⫺0.83 ⫺1.40 ⫺1.78 0.90 0.31 ⫺1.61 ⫺2.02 0.90 ⫺1.30 1.40 ⫺0.59 ⫺4.88 ⫺1.03 ⫺2.97 ⫺6.20 ⫺0.35 2.00 0.24 0.63 1.88 2.86
0.12 ⫺0.04 0.07 ⫺0.17 ⫺0.20 ⫺0.35 ⫺0.17 ⫺0.26 0.66 0.13 0.50 1.16 ⫺0.35j 0.02j ⫺5.75k 0.23k ⫺0.09 0.74 ⫺0.49 0.06 0.36
0.35 0.24 0.52 ⫺0.12 ⫺0.02 0.35 0.80 0.20 1.33 0.65 1.28 1.69 1.53 0.70 ⫺4.95 0.80 0.04 1.77 ⫺0.03 0.01 ⫺0.03
4 1 (1.00) 7 1 (1.00) 9 1 (1.00) 5 1 (1.00) 3 1 (1.00) 4 2 (0.97) 4 1 7 1 (1.00) 2 1 (0.69) 7 2 (.72) 8 1 (.77) 4 1 9 1 (0.77) 7 1 9 1 (1.00) 9 2 (.82) 3 1 4 1 (.82) 3 1 7 1 (0.98) 5 1 9 1 (0.98) 5 2 (1.00) 3 1 9 1 (1.00) 3 2 (0.97) 1 1 (.97) 6 1 (1.00)
c 2i
2 1 共⫺0.24兲 7 2 共0.68兲 2 1 共0.69兲 4 1 9 1 共0.64兲 8 1 共⫺0.64兲 3 1 4 1 共⫺0.57兲 9 2 共0.57兲 5 1 9 1 共⫺0.22兲 3 1 7 1 共0.22兲 1 1 共⫺0.23兲 3 2 共0.24兲
N⫽N 1 ⫹N 2 with N 1 ⫽2( v 1 ⫹ v 6 )⫹ v 3 ⫹ v 5 and N 2 ⫽2( v 2 ⫹ v 8 )⫹ v 4 ⫹ v 7 ⫹ v 9 . Experimental numbers. c Reference 44. d Reference 41. e Reference 43. f Reference 42. g Uncertain level determined from only two rotational levels and excluded from the RMSD. h Reference 54. i Absolute values smaller than 0.1 are not given. j These two levels are strongly coupled by the K 34,99 resonance. Without this resonance, these two levels are in close degeneracy as the band centers are 2464.64 cm⫺1 (9 2 ) and 2463.82 cm⫺1 (3 1 4 1 ). k Levels are moderately coupled by the K 37,59 resonance. This resonance slightly changes these two levels as the band centers are 2515.60 cm⫺1 (3 1 7 1 ) and 2560.70 cm⫺1 (5 1 9 1 ) without this resonance. a
b
9000 cm⫺1 . 共The 8621.90 level is a mixture of 0.96兩 1 3 典 and 0.24兩 1 2 5 02 典 states.兲 The modification due to the N 2 polyad resonance is not considered in the results listed in Table VII. IV. FITTING OF THE POTENTIAL
Using the proper choice of polyad quantum numbers described above, a fourth order Van Vleck perturbation calculation was carried out for all the methane isotopomers using the LMT force field. The results are listed in Tables IV–VIII. Their differences with experimental data are small with a 2.89 cm⫺1 root mean squares deviation 共RMSD兲, indicating that the LMT force field is indeed very accurate. In a recent study we showed that minor changes of the quadratic contribution to a force field of this caliber can lead to significantly better agreement with experiment, and that Newton method which is basically a fast convergent second order nonlinear least squares method works exceedingly well in this regard.49 We therefore applied the Gauss Newton method, where the derivative Hessian is approximated as the product of first derivatives, simultaneously to all the isotopomers to improve on the force constants.49 The following considerations determine what force constants are chosen in the fitting. First of all, the 5 quadratic constants are all included.
Second, since the stretch–bend coupling is strong in most of the isotopomers, four such cubic resonance constants ( f 12a2a , f 14x4x , f 2a3z4z , f 3x4y4z ) are included. Finally, as we can see from the results based on LMT force field 共particularly those of CH4 ), large differences mostly come from the states involving the bending modes. Therefore, three more such force constants ( f 4x4y4z , f 4x4x4x4x , f 4x4x4y4y )are included. Among the observed band origins listed in Tables IV–VII, the data of 13CH3 D and 13CHD3 are excluded from the fitting, since the number of observed transitions is too few. Moreover, some highly excited states are excluded from the fitting. They are the N⬎4 states of CH4 and the N⬎6 states of CHD3 corresponding approximately to the states over 6000 cm⫺1 . A few significantly deviated or experimentally uncertain states are also excluded from the fitting and marked by asterisks in Tables IV–VII. The resulting force field, given in Table III, yields the 130 levels given in Tables IV–VII with a 0.70 cm⫺1 RMSD, not including a few levels marked by asterisks and those aforementioned highly excited levels. This force field is obtained after two iterations, since further iterations do not diminish the RMSD. In fact, the first iteration has already decreased the RMSD from 2.89 cm⫺1 to 0.73 cm⫺1 . This
J. Chem. Phys., Vol. 111, No. 10, 8 September 1999
The vibrations of methane
4519
TABLE VIII. Comparison of experimental and Van Vleck perturbation theory band origins 共cm⫺1 ) for CHD3 and 13CHD3 . See Table IV caption for further details.
N
a
CHD3 1 1 1 2 2 2 2 2 2 2 2 3 4 4 5 5 6 7 8 8 9
b
D(4)
D(4)
D(6)
LMT
Fit
Fit
Sym
E(expt)
c1
A1 E E A1 A1 A1 E E E E E A1 A1 E A1 A1 A1 E A1 A1 A1
1004.55c 1035.92c 1292.50c 1991.08e 2142.58e 2564.68d 2041.44e 2067.58f 2250.83e 2301.16e 2586.04d 2992.75g 4697.10g 4262.10g 5134.90g 5515.70g 5864.98g 7115.48g 8005.48f 8347.10f 8623.32f
⫺1.18 ⫺0.68 0.17 ⫺0.53 0.41 0.82 ⫺2.27 ⫺0.40 2.12 ⫺1.04 0.40 2.62 ⫺1.10 1.60 2.72 1.55 5.91 2.54 6.16 ⫺0.49 13.18
0.08 0.14 ⫺0.13 ⫺0.63 0.60 ⫺0.23 0.36 1.33 0.21 0.23 ⫺0.21 0.22 ⫺1.50 ⫺0.32 0.78 ⫺0.97 0.30 ⫺2.23 1.08 ⫺4.54 3.72 i
0.29 0.34 0.02 0.35 0.69 0.27 0.85 1.90 0.28 0.58 0.18 0.34 ⫺0.70 0.13 1.36 0.06 ⫺0.20 ⫺2.87 0.75 ⫺4.26 1.70
3 1 共1.00兲 6 11 共1.00兲 5 11 共1.00兲 3 2 共0.94兲 2 1 共0.93兲 5 02 共1.00兲 3 1 6 11 共1.00兲 6 02 共0.99兲 4 11 共0.91兲 3 1 5 11 共0.93兲 5 22 共1.00兲 1 1 共0.97兲 2 1 5 02 共0.92兲 1 1 5 11 共0.91兲 1 1 2 1 共0.88兲 1 1 5 02 共0.98兲 1 2 共0.99兲 1 2 5 11 共0.99兲 1 2 2 1 共0.92兲 1 2 5 02 共0.96兲 1 3 共0.99兲
A1 A1 A1
994.71h 1030.24h 1290.54h
⫺1.17 ⫺0.71 0.21
0.05 0.12 ⫺0.14
0.05 0.11 ⫺0.15
3 1 共1.00兲 6 11 共1.00兲 5 11 共1.00兲
c 2j
2 1 共⫺0.27兲 6 02 共0.30兲
3 1 5 11 共0.37兲 4 11 共⫺0.36兲 3 3 共⫺0.23兲 5 02 6 02 共0.31兲 3 2 4 11 共⫺0.23兲 1 1 6 02 共⫺0.28兲
1 2 6 02 共0.31兲
13
CHD3 1 1 1
N⫽3 v 1 ⫹2( v 2 ⫹ v 4 )⫹( v 3 ⫹ v 5 ⫹ v 6 ). Experimental numbers. c Reference 57. d Reference 45. e Reference 46. f Uncertain data excluded from the RMSD. g Reference 47. h Reference 58. i D(4) decreases to 1.39 cm⫺1 when a new constant of motion N⫽2 v 1 ⫹ v 5 is considered. See text for details. j Absolute values smaller than 0.1 are not given. a
b
indicates that the Newton method converges fast for an ab initio force field of this caliber. It would be interesting to try to refine all the force constants with predicate least squares method as in Ref. 15, but the quality of the present fit is sufficiently good that it is currently limited, as we will see below, by the errors in the perturbative calculations. Consequently the more sophisticated procedure was not pursued. It is pleasing to note that the twelve refined force constants are not much different from those of the LMT force field, bearing in mind that no constraints are made during the fitting. We have noted that among the four cubic stretchbend resonance constants, the relative deviation of the f 14x4x is conspicuously large 共17.9%兲 compared to those of the other three 共below 4%兲. To conclude the discussion of the fit, it is essential that we describe the accuracy of our perturbative calculations. To do this test for the fit force field we carried out the perturbation theory to eighth order. In doing so, the kinetic terms are still kept at fourth order, in other words, the Hamiltonian is the same in the sixth and eighth order calculation as that in the fourth order calculation.
The sixth order results are listed in Tables IV–VIII together with those of fourth order. It is seen that most of the levels change no more than 1 cm⫺1 except for those involving high bending modes. Table IX shows the second, fourth, sixth, and eighth order perturbation results for all the states up to N⫽3 polyad and seven 4 4 bend states of the N⫽4 polyad of CH4 molecule. From this table one can see that the second order results are not converged well. Looking at the higher order energies one sees that the energy levels with large stretch excitation appear to convergence rapidly, whereas the bend states converge more slowly. The latter observation indicates that the bend extension coordinates are probably not ideal for the perturbation calculation. It will be worthwhile to try other types of coordinates which might produce better convergence for the bend states. It is also interesting to try to refine the force field using sixth order perturbation theory since at the cost of more calculation time, the energies are more converged at sixth than at fourth order. We plan to report these results in a future publication.
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J. Chem. Phys., Vol. 111, No. 10, 8 September 1999
X. G. Wang and E. L. Sibert III
TABLE IX. Comparison of second, fourth, sixth, and eighth order Van Vleck perturbation theory band origins 共cm⫺1 ) for CH4 ; a check on convergence.a Nb
Sym
1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4
F2 E A1 F2 E F2 F1 A1 F2 A1 E F2 A1 F1 F2 E F1 A1 F2 E A2 F2 F2 E F1 A1 F2 F1 F2 E F1 F2 E A2 A1 A1 F2 E F2 E F1 A1
E(8)⫺E(2)
E(8)⫺E(4)
E(8)⫺E(6)
E(8)
State(c 21 ) c
⫺1.28 ⫺0.55 ⫺8.44 ⫺5.16 ⫺0.63 ⫺1.83 ⫺3.02 ⫺0.96 ⫺0.55 ⫺2.21 ⫺1.28 ⫺19.29 ⫺13.41 ⫺5.97 ⫺2.00 ⫺9.39 ⫺6.68 ⫺1.19 ⫺8.29 ⫺3.66 ⫺3.09 ⫺2.94 ⫺2.38 ⫺0.86 ⫺1.31 ⫺4.29 ⫺3.12 ⫺3.64 ⫺5.82 ⫺1.06 ⫺0.73 ⫺1.57 ⫺4.03 ⫺2.22 ⫺2.19 ⫺43.24 ⫺40.19 ⫺19.78 ⫺14.06 ⫺3.83 ⫺5.42 ⫺0.52
⫺0.55 ⫺0.08 ⫺3.02 ⫺2.27 ⫺0.79 ⫺0.81 ⫺0.74 ⫺0.15 ⫺0.02 ⫺0.18 ⫺0.15 ⫺7.59 ⫺6.22 ⫺3.34 ⫺2.20 ⫺3.41 ⫺2.85 ⫺1.23 ⫺2.73 ⫺1.27 ⫺1.00 ⫺1.03 ⫺0.65 ⫺0.70 ⫺0.69 ⫺0.72 ⫺1.05 ⫺1.02 ⫺0.95 ⫺0.34 ⫺0.14 ⫺0.15 ⫺0.31 ⫺0.21 ⫺0.27 ⫺17.71 ⫺16.96 ⫺8.78 ⫺7.75 ⫺4.41 ⫺4.66 ⫺3.42
0.03 0.00 0.33 0.19 0.05 0.05 0.04 ⫺0.03 ⫺0.02 0.00 0.00 1.13 0.68 0.41 0.19 0.46 0.29 0.06 0.26 0.09 0.07 ⫺0.06 ⫺0.04 0.00 ⫺0.01 ⫺0.09 0.06 0.06 0.05 ⫺0.03 ⫺0.02 ⫺0.03 ⫺0.01 0.00 ⫺0.01 3.52 3.23 1.87 1.18 0.70 0.49 0.28
1310.40 1533.34 2584.83 2612.34 2624.21 2830.18 2844.40 2916.49 3019.14 3063.34 3065.03 3861.98 3902.72 3918.16 3929.73 4098.78 4125.85 4133.11 4138.54 4148.89 4159.30 4223.25 4318.40 4321.65 4321.84 4322.79 4348.78 4362.04 4375.34 4435.47 4537.75 4543.91 4591.29 4595.02 4595.04 5097.79 5120.73 5157.75 5203.80 5227.02 5227.60 5239.64
4 1 共100%兲 2 1 共100%兲 4 2 共98%兲 4 2 共99%兲 4 2 共100%兲 2 1 4 1 共97%兲 2 1 4 1 共100%兲 1 1 共95%兲 3 1 共96%兲 2 2 共97%兲 2 2 共100%兲 4 3 共96%兲 4 3 共96%兲 4 3 共98%兲 4 3 共99%兲 2 1 4 2 共94%兲 2 1 4 2 共95%兲 2 1 4 2 共94%兲 2 1 4 2 共97%兲 2 1 4 2 共98%兲 2 1 4 2 共100%兲 1 1 4 1 共93%兲 3 1 4 1 共96%兲 3 1 4 1 共95%兲 3 1 4 1 共94%兲 3 1 4 1 共93%兲 2 2 4 1 共92%兲 2 2 4 1 共97%兲 2 2 4 1 共99%兲 1 1 2 1 共93%兲 2 1 3 1 共96%兲 2 1 3 1 共94%兲 2 3 共95%兲 2 3 共100%兲 2 3 共100%兲 4 4 共93%兲 4 4 共93%兲 4 4 共94%兲 4 4 共94%兲 4 4 共96%兲 4 4 共96%兲 4 4 共98%兲
a
The refined force field as given in Table III is used. N⫽2( v 1 ⫹ v 3 )⫹ v 2 ⫹ v 4 . c The largest component and their percentage (c 21 ) as calculated by fourth order Van Vleck perturbation theory. b
V. CONCLUSION
In this paper fourth order Van Vleck perturbation theory, based on the curvilinear normal coordinates, is applied to investigate the vibrations of nine methane isotopomers. Implementation of the perturbation theory using both analytical and numerical expression of the kinetic energy operator is considered. The pseudopotential terms V ⬘ are included. We find that these terms can change the fundamental frequencies by about one wave number. Twelve force constants of the ab initio force field11 were refined, and the refined force field reproduce 130 levels with a RMSD 0.70 cm⫺1 . The choices of polyad quantum numbers are discussed for different isotopomers, particularly the CH2 D2 and CHD3 molecules for which the global polyad quantum numbers have to be modified in some energy regions. The conver-
gence of the perturbation theory is examined by carrying out the perturbation theory to the eighth order. Good convergence is observed.
ACKNOWLEDGMENTS
Professor M. S. Child is gratefully thanked for stimulating discussions and many helpful suggestions and comments. Professor Ulenikov is acknowledged for communicating unpublished data to us. X.G.W. wishes to thank the Physical and Theoretical Chemistry Laboratory 共Oxford兲 where the final stage of this work was finished, and the Royal Society K.C. Wong Fellowship for support. E.L.S., who was partially supported by National Science Foundation Grant No. CHE9424115, thanks the Departament de Fisica i Enginyeria
J. Chem. Phys., Vol. 111, No. 10, 8 September 1999
The vibrations of methane
Nuclear at the Universitat Politecnica de Catalunya 共Barcelona兲 where the final stage of this work was carried out. APPENDIX: REDUNDANCY COORDINATES
⫺
The branching redundancies occur for methane and its isotopomers as the vectors of the four bonds connected to the central atom are not geometrically independent in the 3D space. As a result, the six bond angles around the central atom satisfy a determinant relation50,51
冏
1
cos⌰ 12
cos⌰ 12
1
cos⌰ 13
cos⌰ 23
cos⌰ 13
冏
cos⌰ 14 cos⌰ 24 cos⌰ 34
cos⌰ 34 1
⫽0,
共A1兲
where ⌰ i j ⫽⌰ 0 ⫹ i j is the instantaneous bond angle between bonds i and j and ⌰ 0 is the equilibrium tetrahedral angle and i j the displacement from equilibrium. Among the six symmetry coordinates defined in terms of the six bond angles, one symmetry coordinate S r is redundant and depends on the other five from the second order. At least two different approaches have been used to treat the redundancy problem. In the study of Halonen,12 the six bond angles are expressed as a power series expansion of the five independent symmetry coordinates, and the expansion coefficients are obtained using the L tensor method detailed in Ref. 52. We choose to follow the approach by Raynes et al.53 and propose an iterative numerical approach for doing numerical derivatives with respect to symmetry coordinates. To start with, the redundant coordinate is strictly defined as S r⫽
1
冑6
共 12⫹ 13⫹ 14⫹ 23⫹ 24⫹ 34兲 ,
共A2兲
which is in fact equal to zero at first order for tetrahedral equilibrium bond angles. The six bond angles R are readily expressed as a function of the six symmetry coordinates S through an orthogonal transformation, SⴝUR ,
共A3兲
where U is a 6⫻6 matrix as given in Table I for the four types of methane isotopomers XH4 , XH3 D, XHD3 , and XH2 D2 . The redundant coordinate, however, is dependent on the other five symmetry coordinates. This dependence can be determined both analytically and numerically. The analytical approach follows that of Raynes et al.,53 where the S r is approximated by a power series expansion in terms of the other five symmetry coordinates. This is achieved by expanding Eq. 共A1兲 in terms of the bond angles about equilibrium. In doing so, one recovers on the first order Eq. 共A2兲 and the higher order terms give the dependence of S r on the i j 关see, e.g., Eq. 共A3兲 of Ref. 53兴. Replacing i j in the S r power-series expansion with the six symmetry coordinates using the inverse of Eq. 共A3兲, R ⫽UTS,
共A4兲
one obtains the expansion of S r in terms of the other five symmetry coordinates and S r itself. For example, for CHD3 molecule one obtains
11 8 冑3
S r2 ⫺
冑3 8
S 23 ⫺
1 8 冑3
共 S 25 ⫹S 26 兲 ⫹
1 2 冑3
S 5S 6
7 3 3 13 1 S r ⫹ S r S 23 ⫹ S r 共 S 25 ⫹S 26 兲 ⫹ S r S 5 S 6 72 8 24 3
1 7 2 2 2 ⫺ S 33 ⫹ ⫺S 6b S 共共 S 2 ⫺3S 5b 兲 ⫹3 共 S 6a 兲兲 6 144 5a 5a ⫺
cos⌰ 14
cos⌰ 23 cos⌰ 24 1
S r ⫽⫺
4521
5 1 2 2 2 2 S 6a 共共 S 6a ⫺3S 6b ⫺S 5b 兲 ⫹3 共 S 5a 兲兲 ⫹ S 3 共 S 25 ⫹S 26 兲 144 8
1 1 ⫺ S 3 S 5 S 6 ⫹ 共 5S 5a ⫺7S 6a 兲 S 5b S 6b ⫹••• . 4 24
共A5兲
To avoid a lengthy expression, Eq. 共A5兲 is given only to the third order. By using Eq. 共A5兲 as a recursive definition of S r and repeatedly substituting for S r on the RHS of this equation, one finally obtains the expression of S r purely in terms of the other symmetry coordinates,
S r ⫽⫺
冑3 8
S 23 ⫺
1 8 冑3
共 S 25 ⫹S 26 兲 ⫹
1 2 冑3
S 5S 6
1 7 2 2 2 S 共共 S 2 ⫺3S 5b ⫺ S 33 ⫹ ⫺S 6b 兲 ⫹3 共 S 6a 兲兲 6 144 5a 5a ⫺
5 1 2 2 2 ⫺S 5b S 共共 S 2 ⫺3S 6b 兲 ⫹3 共 S 5a 兲兲 ⫹ S 3 共 S 25 ⫹S 26 兲 144 6a 6a 8
1 1 ⫺ S 3 S 5 S 6 ⫹ 共 5S 5a ⫺7S 6a 兲 S 5b S 6b ⫹••• . 4 24
共A6兲
It should be noted that terms involving S r on the RHS of Eq. 共A5兲 contribute to Eq. 共A6兲 only from fourth order which is not given here. In the case of the CH3 D molecule, the dependence of S r on the other symmetry coordinates is readily obtained from that of CHD3 by interchanging S 5 with S 6 . In the case of the CH2 D2 molecule, the S r up to the third order is found to be
S r ⫽⫺ ⫹
⫺
⫺
冑3 40
S 23 ⫺
1 20冑30 1 4 冑30 3 4 冑5
7 40冑3
S 33 ⫹
S 24 ⫹ 1
90冑5
S 3 S 25 ⫺
1 8 冑3
S 34 ⫹
1 12冑5
S 25 ⫺ 1
20冑5
S 4 S 25 ⫹
S 4 共 S 27 ⫺S 29 兲 ⫹••• .
冑3 8
共 S 27 ⫹S 29 兲 ⫹
S 23 S 4 ⫹
1 10冑30
冑2 5
S 3S 4
S 3 S 24
冑3 S 3 共 S 27 ⫺S 29 兲 2 冑10 共A7兲
The expression of S r up to the fourth order for CH4 is given in Eq. 共A5兲 of Ref. 53. Here it is rederived and given as
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J. Chem. Phys., Vol. 111, No. 10, 8 September 1999
S r ⫽⫺ ⫺
⫹ ⫺
1 8 冑3
冑3 2
S 22 ⫺
冑3 8
S 24 ⫹
S 4x S 4y S 4z ⫹
11 512冑3
1 36冑2 13
1536冑3
X. G. Wang and E. L. Sibert III 13
2 2 ⫺3S 2b S 2a 共 S 2a 兲
S 42
4 4 ⫹S 44y ⫹S 4z 兲⫺ 共 S 4x
3 冑3 2 2 S S 256 2 4
63冑3 2 2 2 2 2 ⫹S 4z S 4x 兲 ⫹••• . 共 S 4x S 4y ⫹S 24y S 4z 256
共A8兲
Note that the coefficient of the last term is incorrectly printed in Ref. 53. In the numerical approach S r is evaluated iteratively, given the numerical value of the other five symmetry coordinates. With the redundant coordinate assumed to be zero initially, we calculate 5 bend coordinates which can be any 5 of the 6 bend coordinates using Eq. 共A3兲 and evaluate the final bend coordinate geometrically. This geometric evaluation amounts to implementing the determinant relation of Eq. 共A1兲. The coordinate S r is then evaluated from the new set of 6 bend coordinates using Eq. 共A2兲, and Eq. 共A3兲 is again used to calculate the six bend coordinates with the sixth one being evaluated geometrically. This process is repeated until further iterations do not change the value of S r beyond that given by the precision of the computer. Typically ten to eleven iterations suffice to converge S r to 10⫺16 accuracy. In summary, the numerical approach is more accurate than the analytical one, especially when the magnitude of bend coordinate is large, since higher order terms in the power series are needed when the displacement is large. On the other hand, the analytical approach is fast and indispensable when analytical derivatives of S r are needed. For a typical value of 0.1 rad of the bend symmetry coordinate, the S r of the sixth order analytical approach and the numerical approach agrees to five significant digits and the value of the determinant is 10⫺8 and 10⫺16 for the two approaches, respectively. 1
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