- Email: [email protected]
Calculations of low-lying vibrational states of cis- and trans-HOCO1
Journal of Molecular Structure ŽTheochem. 461᎐462 Ž1999. 71᎐77
Calculations of low-lying vibrational states of cis- and trans-HOCO 夽 Joel M. BowmanU , Kurt Christoffel 1, Gabe Weinberg 2 Department of Chemistry, Cherry L. Emerson Center for Scientific Computation, Emory Uni¨ ersity, Atlanta, GA 30322, USA Received 21 July 1998; received in revised form 4 August 1998; accepted 4 August 1998
Abstract We report variational calculations of low-lying vibrational states of non-rotating cis-HOCO and trans-HOCO, using the code ‘MULTIMODE’ and a realistic six degree-of-freedom potential surface. The results of these variational calculations are compared with previous five degree-of-freedom calculations wD.H. Zhang and J.Z.H. Zhang, J. Chem. Phys. 103, 6512 Ž1995.x in which the full six degree-of freedom potential is averaged over the ‘spectator’ CO stretch. This approximation is found to work better for cis-HOCO than trans-HOCO, and a simple explanation for this is presented. State-dependent rotation constants are calculated from exact rotationrvibration calculations for J s 1 for low-lying states of cis- and trans-HOCO. 䊚 1999 Elsevier Science B.V. All rights reserved. Keywords: Calculations of vibrational states; Non-rotating cis-HOCO; Non-rotating trans-HOCO
1. Introduction The calculation of bound states of polyatomic molecules is severely hampered by the dimensionality of the problem, which scales exponentially with the number of degrees of freedom in both
夽 Dedicated to Professor Keiji Morokuma in celebration of his 65th birthday. U Corresponding author. Fax: q1-404-7276586; e-mail: [email protected] 1 Permanent address: Department of Chemistry, Augustana College, Rock Island, IL 61201, USA. 2 Current address: Massachusetts Institute of Technology, Cambridge, MA, USA.
spectral and grid approaches. In the familiar spectral approach multidimensional integrals over the potential become very time consuming and are barely feasible even for tetraatomics. For grid methods the size of the Hamiltonian can easily be of the order of 10 6 or greater for a tetraatomic. Thus, the possibility of performing reduced dimensionality calculations on polyatomics seems reasonable. Recently, Zhang and Zhang ŽZZ. reported such a calculation for low-lying bound states of cis- and trans-HOCO, for zero total angular momentum, J w1x. They used a Hamiltonian in diatom᎐diatom, i.e. OH᎐CO, Jacobi coordinates. For zero total angular momentum there are six degrees of freedom, and ZZ reduced this
0166-1280r99r$ - see front matter 䊚 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 1 2 8 0 Ž 9 8 . 0 0 4 2 8 - X
72
J.M. Bowman et al. r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 71᎐77
number to five, by treating the ‘spectator’ CO in a decoupled fashion. Specifically, they used a single CO vibrational wavefunction in their calculations. This function was obtained by solving a one-dimensional Schrodinger equation for the CO vi¨ bration, with a potential given by a cut through the trans-HOCO minimum in which only the CO stretch is varied. Clearly, bound states of cis- and trans-HOCO are restricted to the ground vibrational state of the spectator CO stretch in this approach. However, even for these states the accuracy of these reduced dimensionality calculations has not been determined. We do that in this paper by reporting calculations of low-lying vibrational states of cis- and trans-HOCO using the same potential w2,3x as ZZ used but treating all six degrees-of-freedom. We also present calculations for J s 1 and extract state-dependent rotational constants. In the next two sections we describe the methods used and the numerical details of our calculations. Results and discussion are then given followed by a summary and conclusions.
terms, etc. In our applications to date w4᎐6x, the maximum n-mode representation is n equal to 4. wNote that the separable harmonic approximation is represented by the harmonic approximation to Vi Ž1. Ž Q i ..x This representation of the potential can be used in the complete Watson Hamiltonian w7x, where the kinetic energy operator is given by Tˆs
1 2
Ý ž Jˆ␣ y ˆ␣ / ␣  ž Jˆ y ˆ /
y
1 2
␣
⭸2
1
Ý ⭸ Q 2 y 8 Ý ␣␣ , ␣
k
k
where
ˆ␣ s yi Ý k␣,l Qk k ,l
⭸ , ⭸ Ql
is the effective reciprocal inertia tensor, and k␣,l are the Coriolis coupling constants, related to the vectors of the normal coordinates by
2. Methods and calculations In order to make calculations for many-mode molecules feasible we have suggested the following hierarchical representation of the potential in terms of mass-weighted normal modes w4᎐6x: Ž Q i ,Q j . V Ž Q1 ,Q2 , . . . ,Q N . s Ý Vi Ž1. Ž Q i . q Ý Vi Ž2. j i
k␣,l s ⑀␣ ␥ Ý l i ,k l␥ i ,l . i
The mass-scaled normal coordinates Q k are related to the Cartesian coordinates r␣ i , by Ž r␣ i y r␣oi . , Q k s Ý l␣ i ,k m1r2 i i
ij
Ž . q Ý Vi Ž3. jk Q i ,Q j ,Q k ijk
Ž . q Ý Vi Ž4. jk l Q i ,Q j ,Q k ,Q l q . . . ijkl
where Vi Ž1. Ž Q i . is given by a cut through the full potential where only the coordinate Q i is not Ž Q i ,Q j . is similarly defined but where zero, Vi Ž2. j two coordinates are not zero, etc. The one-mode representation of the potential is given by the above expansion but truncated at the sum containing Vi Ž1. Ž Q i . terms, the two-mode representaŽ Q i ,Q j . tion contains those terms plus the Vi Ž2. j
where r␣oi are the equilibrium Cartesian coordinates in the principal axis system. Finally, Jˆ␣ and Jˆ are the rotational operators, which depend on the Euler angles. We have implemented the above approach in a new code, ‘MULTIMODE’. This code can perform vibrational self-consistent field ŽVSCF. calculations plus several types of ‘Cl’ calculations for many-mode systems, using the above hierarchical representation of the potential and for J ) 0 w4᎐6x. An efficient, approximate treatment of rotation, the so-called ‘adiabatic rotation approximation’ w8,9x can also be done with this code. In the VSCF approach w10,11x the many-mode
J.M. Bowman et al. r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 71᎐77
vibrational wavefunction is written as a ‘Hartree product’ N
Ž Q1 ,Q2 , . . . Q N . s Ł nŽ i. Ž Q i . . ⌿nVSCF 1 ,n 2 , . . . ,n N i is1
Applying the variational principle Žfor the J s 0 Watson Hamiltonian. to the modals yields the VSCF equations N
N
i/l
i/l
Tl q - Ł nŽ i.i < V q Tc < Ł nŽ i.i ) y⑀ nŽ ll .
73
potential Ž n s 1,2,3,4., Tc is the Coriolis coupling operator, and Tl is the Cartesian kinetic energy operator in the Watson Hamiltonian. These equations are solved by a standard iteration method beginning with a zero-order, normal-mode, harmonic oscillator approximation. Convergence is based on convergence of the eigenvalues ⑀ nŽ ll .. This method is quite fast and scales linearly with the number of modes, for a given order of mode-representation of the potential. 3. Configuration interaction
nŽ ll . Ž Q l . s 0,l s 1, N, where V is the n-mode representation of the
In order to incorporate mode᎐mode correlation, and to obtain variational results, which con-
Fig. 1. Normal mode eigenvectors of cis-HOCO.
74
J.M. Bowman et al. r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 71᎐77
verge to exact results Žfor a given representation of the potential. several CI procedures can be done in ‘MULTIMODE’. In the one we use in this paper, the many-mode wavefunction is expanded in terms of the eigenstates of a given VSCF Hamiltonian Žalso known as the virtual states .. Usually the ground state VSCF Hamiltonian is used. This basis is orthonormal and results in a standard eigenvalue problem. The resulting eigenvalues and eigenfunctions are true variational upper bound approximations to the exact eigenvalues and eigenfunctions. We refer to this type of CI as ‘VCI’. In the present calculations the CI matrix was generated by fixing the total quanta of mode excitations to be less than or equal to a maximum value, denoted n max .
In this paper we consider three and four-mode representations ŽMR. of the potential, and the basis sets for the bound states of cis- and transHOCO consist of 9᎐12 primitive harmonic functions for the normal modes. A sufficient number of Gauss᎐Hermite quadrature points were used for these bases. The largest 4-MR calculations were done for trans-HOCO with n max equal to 8. This resulted in a CI matrix of order 3003. The largest 3-MR calculations for trans-HOCO were done with n max equal to 9; this resulted in a CI matrix of order 5005. For cis-HOCO the largest CI matrices were of order of 4501. The calculations were done for J s 0 and 1, and we determined state-dependent rotation constants by fitting the difference in rovibrational
Fig. 2. Normal mode eigenvectors of trans-HOCO.
J.M. Bowman et al. r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 71᎐77
75
energies to a symmetric top expression Žwhich is quite accurate since HOCO is a near prolate symmetric top..
Table 2 Zero-point and fundamental energies Žcmy1 . of trans-HOCO
4. Results and discussion The normal-mode eigenvectors. in mass-scaled coordinates, of cis- and trans-HOCO are shown in Figs. 1 and 2, respectively. As seen, mode 5 in cis-HOCO resembles a CO-stretch very closely; however, the analogous mode in trans-HOCO looks less like a simple CO-stretch. The results of the VCI calculations for J s 0 are shown in Tables 1 and 2 for the zero-point state and the six fundamentals of cis- and transHOCO, respectively, along with previous five degree-of-freedom calculations of Zhang and Zhang w1x. As seen, there is very good agreement between the 3 and 4-MR calculations for the zero-point energy, and good agreement generally. The agreement with the 5-d.o.f. calculations is much better for cis-HOCO than for trans-HOCO. This seems reasonable based on the above discussion about the normal modes of cis- and trans-HOCO. Calculations were also done for J s 1 for transand cis-HOCO using an exact treatment, again within the approximation of a given mode representation of the potential w4᎐6x. These calculations yield rovibrational energies which can be labeled by the usual spectroscopic quantum numbers K a and K c . Both cis- and trans-HOCO are Table 1 Zero-point and fundamental energies Žcmy1 . of cis-HOCO ¨ 1¨ 2¨ 3¨ 4¨ 5¨ 6
3-MRa
4-MRa
4-MRb
4-MRc
5-d.o.f.d
ZPE 100000 010000 001000 000100 000010 000001
4375 423 542 1042 1155 1834 3591
4380 434 549 1013 1164 1837 3591
4379 427 548 1011 1162 1836 3587
4379 422 547 1010 1161 1836 3587
4383 413 556 1026 1081 ᎐ ᎐
Harmonic ZPE 4455
4455
4455
4455
a
Cl matrix of order 1674. Cl matrix of order 2385. c Cl matrix of order 4501. d See Zhang and Zhang w1x. b
3-MRa
3-MRb
4-MRb
5-d.o.f.c
ZPE 100000 010000 000100 001000 000010 000001
4589 547 670 1084 1159 1835.1 3638.8
4589 548 673 1085 1162 1844.7 3638.6
4589 551 671 1093 1183 1847.6 3636.2
4637 570 667 1122 1216
Harmonic ZPE
4680
4680
4680
¨ 1¨ 2¨ 3¨ 4¨ 5¨ 6
a
Cl matrix of order 5005. Cl matrix of order 3003. c See Zhang and Zhang w1x. b
near prolate symmetric tops, and as a result the asymmetry splittings were found to be quite small, e.g. of the order of a few hundredths of a wavenumber, for the low-lying states we considered. ŽThe rigid rotor A e , Be and C e constants are 4.063, 0.355, and 0.327 cmy1 , respectively, for cis-HOCO, and 5.467, 0.339, and 0.319 cmy1 , respectively for trans-HOCO. The average values of Be and C e are 0.341 cmy1 and 0.329 cmy1 for cis- and trans-HOCO, respectively.. Thus, we have chosen to fit the differences in the J s 1 and J s 0 energies by the standard expression Ž i. ⌬ E JK s Bi J Ž J q 1 . q Ž A i y Bi . K a2 a
and determine the rotation constants for each vibrational state i. Note in this expression Bi corresponds to the average of the B and C constants. The results of these fits are shown in Figs. 3 and 4 for the first 30 vibrational states for J s 0. As seen the state-dependent constants fluctuate considerably and differ from the rigid rotor results by 5᎐10%. 5. Summary and conclusions We have applied the code ‘MULTIMODE’ to calculate the energies of the fundamental vibrational states of non-rotating cis- and trans-HOCO for J s 0 and 1. These calculations were done using a variational configuration interaction
76
J.M. Bowman et al. r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 71᎐77
Fig. 3. Rotation constants for cis-HOCO.
method together with three and four-mode representations of the full six degree of freedom potential. Good agreement was found for the energies of the three and four-mode representations. Comparisons were also made with previous five degree-of-freedom calculations, for J s 0, of Zhang and Zhang, in which the CO ‘spectator’ mode is uncoupled. Agreement with those calculations is significantly better for cis-HOCO than for trans-HOCO. This was rationalized based on an examination of the normal mode eigenvectors of cis- and trans-HOCO, where the CO spectator mode is much closer to a normal mode in cisHOCO than in trans-HOCO. Finally, rovibrational energies were calculated for J s 1 and based on a simple symmetric top expression state-dependent rotation constants were presented. These
Fig. 4. Rotation constants for trans-HOCO.
showed significant fluctuations with vibrational state relative to the rigid rotor rotation constants. Noted added in proof We just became aware of exact six degree-offreedom calculations of energies of cis and transHOCO for J s 0 even parity states wR.B. Lehoucq, S.K. Gray, D.-H. Zhang, J.C. Light, Comp. Phys. Comm. 109 Ž1998. 15x. The energies reported here are within 1᎐2 cmy1 of these exact ones for trans-HOCO and within 3᎐8 cmy1 for cis-HOCO. Acknowledgements JMB and GW thanks the Department of Energy ŽDE-FG02-97ER14782. for financial support.
J.M. Bowman et al. r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 71᎐77
KC thanks the Cherry L. Emerson Center for Scientific Computation for a visiting fellowship. References w1x D.H. Zhang, J.Z.H. Zhang, J. Chem. Phys. 103 Ž1995. 6512. w2x G.C. Schatz, M.S. Fitzcharles, L.B. Harding, Faraday Disscus. Chem. Soc. 84 Ž1987. 359. w3x K. Kudla, G.C. Schatz, A.F. Wagner, J. Chem. Phys. 95 Ž1991. 1635. w4x S. Carter, S.J. Culik, J.M. Bowman, J. Chem. Phys. 107 Ž1997. 10458.
77
w5x S. Carter, J.M. Bowman, J. Chem. Phys. 108 Ž1998. 4397. w6x S. Carter, J.M. Bowman, N. Handy, Extensions and Tests of ‘MULTIMODE’: A Code to Obtain Accurate VibrationrRotation Energies of Many-Mode Molecules, Theoretical Chem. Accnts., accepted. w7x J.K.G. Watson, Mol. Phys. 15 Ž1968. 479. w8x J.M. Bowman, Chem. Phys. Lett. 217 Ž1994. 36. w9x For the semi-classical version, see C.W. McCurdy, W.H. Miller, ACS Symp. Ser. No. 56, in: P.R. Brooks, E.F. Hayes ŽEds.., American Chemical Society, Washington, DC, 1977, pp. 239᎐242. w10x For reviews, see J.M. Bowman, Acc. Chem. Res., 19 Ž1986. 202. w11x M.A. Ratner, R.B. Gerber, J. Phys. Chem. 90 Ž1986. 20.
Calculations of low-lying vibrational states of cis- and trans-HOCO 夽 Joel M. BowmanU , Kurt Christoffel 1, Gabe Weinberg 2 Department of Chemistry, Cherry L. Emerson Center for Scientific Computation, Emory Uni¨ ersity, Atlanta, GA 30322, USA Received 21 July 1998; received in revised form 4 August 1998; accepted 4 August 1998
Abstract We report variational calculations of low-lying vibrational states of non-rotating cis-HOCO and trans-HOCO, using the code ‘MULTIMODE’ and a realistic six degree-of-freedom potential surface. The results of these variational calculations are compared with previous five degree-of-freedom calculations wD.H. Zhang and J.Z.H. Zhang, J. Chem. Phys. 103, 6512 Ž1995.x in which the full six degree-of freedom potential is averaged over the ‘spectator’ CO stretch. This approximation is found to work better for cis-HOCO than trans-HOCO, and a simple explanation for this is presented. State-dependent rotation constants are calculated from exact rotationrvibration calculations for J s 1 for low-lying states of cis- and trans-HOCO. 䊚 1999 Elsevier Science B.V. All rights reserved. Keywords: Calculations of vibrational states; Non-rotating cis-HOCO; Non-rotating trans-HOCO
1. Introduction The calculation of bound states of polyatomic molecules is severely hampered by the dimensionality of the problem, which scales exponentially with the number of degrees of freedom in both
夽 Dedicated to Professor Keiji Morokuma in celebration of his 65th birthday. U Corresponding author. Fax: q1-404-7276586; e-mail: [email protected] 1 Permanent address: Department of Chemistry, Augustana College, Rock Island, IL 61201, USA. 2 Current address: Massachusetts Institute of Technology, Cambridge, MA, USA.
spectral and grid approaches. In the familiar spectral approach multidimensional integrals over the potential become very time consuming and are barely feasible even for tetraatomics. For grid methods the size of the Hamiltonian can easily be of the order of 10 6 or greater for a tetraatomic. Thus, the possibility of performing reduced dimensionality calculations on polyatomics seems reasonable. Recently, Zhang and Zhang ŽZZ. reported such a calculation for low-lying bound states of cis- and trans-HOCO, for zero total angular momentum, J w1x. They used a Hamiltonian in diatom᎐diatom, i.e. OH᎐CO, Jacobi coordinates. For zero total angular momentum there are six degrees of freedom, and ZZ reduced this
0166-1280r99r$ - see front matter 䊚 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 1 2 8 0 Ž 9 8 . 0 0 4 2 8 - X
72
J.M. Bowman et al. r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 71᎐77
number to five, by treating the ‘spectator’ CO in a decoupled fashion. Specifically, they used a single CO vibrational wavefunction in their calculations. This function was obtained by solving a one-dimensional Schrodinger equation for the CO vi¨ bration, with a potential given by a cut through the trans-HOCO minimum in which only the CO stretch is varied. Clearly, bound states of cis- and trans-HOCO are restricted to the ground vibrational state of the spectator CO stretch in this approach. However, even for these states the accuracy of these reduced dimensionality calculations has not been determined. We do that in this paper by reporting calculations of low-lying vibrational states of cis- and trans-HOCO using the same potential w2,3x as ZZ used but treating all six degrees-of-freedom. We also present calculations for J s 1 and extract state-dependent rotational constants. In the next two sections we describe the methods used and the numerical details of our calculations. Results and discussion are then given followed by a summary and conclusions.
terms, etc. In our applications to date w4᎐6x, the maximum n-mode representation is n equal to 4. wNote that the separable harmonic approximation is represented by the harmonic approximation to Vi Ž1. Ž Q i ..x This representation of the potential can be used in the complete Watson Hamiltonian w7x, where the kinetic energy operator is given by Tˆs
1 2
Ý ž Jˆ␣ y ˆ␣ / ␣  ž Jˆ y ˆ /
y
1 2
␣
⭸2
1
Ý ⭸ Q 2 y 8 Ý ␣␣ , ␣
k
k
where
ˆ␣ s yi Ý k␣,l Qk k ,l
⭸ , ⭸ Ql
is the effective reciprocal inertia tensor, and k␣,l are the Coriolis coupling constants, related to the vectors of the normal coordinates by
2. Methods and calculations In order to make calculations for many-mode molecules feasible we have suggested the following hierarchical representation of the potential in terms of mass-weighted normal modes w4᎐6x: Ž Q i ,Q j . V Ž Q1 ,Q2 , . . . ,Q N . s Ý Vi Ž1. Ž Q i . q Ý Vi Ž2. j i
k␣,l s ⑀␣ ␥ Ý l i ,k l␥ i ,l . i
The mass-scaled normal coordinates Q k are related to the Cartesian coordinates r␣ i , by Ž r␣ i y r␣oi . , Q k s Ý l␣ i ,k m1r2 i i
ij
Ž . q Ý Vi Ž3. jk Q i ,Q j ,Q k ijk
Ž . q Ý Vi Ž4. jk l Q i ,Q j ,Q k ,Q l q . . . ijkl
where Vi Ž1. Ž Q i . is given by a cut through the full potential where only the coordinate Q i is not Ž Q i ,Q j . is similarly defined but where zero, Vi Ž2. j two coordinates are not zero, etc. The one-mode representation of the potential is given by the above expansion but truncated at the sum containing Vi Ž1. Ž Q i . terms, the two-mode representaŽ Q i ,Q j . tion contains those terms plus the Vi Ž2. j
where r␣oi are the equilibrium Cartesian coordinates in the principal axis system. Finally, Jˆ␣ and Jˆ are the rotational operators, which depend on the Euler angles. We have implemented the above approach in a new code, ‘MULTIMODE’. This code can perform vibrational self-consistent field ŽVSCF. calculations plus several types of ‘Cl’ calculations for many-mode systems, using the above hierarchical representation of the potential and for J ) 0 w4᎐6x. An efficient, approximate treatment of rotation, the so-called ‘adiabatic rotation approximation’ w8,9x can also be done with this code. In the VSCF approach w10,11x the many-mode
J.M. Bowman et al. r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 71᎐77
vibrational wavefunction is written as a ‘Hartree product’ N
Ž Q1 ,Q2 , . . . Q N . s Ł nŽ i. Ž Q i . . ⌿nVSCF 1 ,n 2 , . . . ,n N i is1
Applying the variational principle Žfor the J s 0 Watson Hamiltonian. to the modals yields the VSCF equations N
N
i/l
i/l
Tl q - Ł nŽ i.i < V q Tc < Ł nŽ i.i ) y⑀ nŽ ll .
73
potential Ž n s 1,2,3,4., Tc is the Coriolis coupling operator, and Tl is the Cartesian kinetic energy operator in the Watson Hamiltonian. These equations are solved by a standard iteration method beginning with a zero-order, normal-mode, harmonic oscillator approximation. Convergence is based on convergence of the eigenvalues ⑀ nŽ ll .. This method is quite fast and scales linearly with the number of modes, for a given order of mode-representation of the potential. 3. Configuration interaction
nŽ ll . Ž Q l . s 0,l s 1, N, where V is the n-mode representation of the
In order to incorporate mode᎐mode correlation, and to obtain variational results, which con-
Fig. 1. Normal mode eigenvectors of cis-HOCO.
74
J.M. Bowman et al. r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 71᎐77
verge to exact results Žfor a given representation of the potential. several CI procedures can be done in ‘MULTIMODE’. In the one we use in this paper, the many-mode wavefunction is expanded in terms of the eigenstates of a given VSCF Hamiltonian Žalso known as the virtual states .. Usually the ground state VSCF Hamiltonian is used. This basis is orthonormal and results in a standard eigenvalue problem. The resulting eigenvalues and eigenfunctions are true variational upper bound approximations to the exact eigenvalues and eigenfunctions. We refer to this type of CI as ‘VCI’. In the present calculations the CI matrix was generated by fixing the total quanta of mode excitations to be less than or equal to a maximum value, denoted n max .
In this paper we consider three and four-mode representations ŽMR. of the potential, and the basis sets for the bound states of cis- and transHOCO consist of 9᎐12 primitive harmonic functions for the normal modes. A sufficient number of Gauss᎐Hermite quadrature points were used for these bases. The largest 4-MR calculations were done for trans-HOCO with n max equal to 8. This resulted in a CI matrix of order 3003. The largest 3-MR calculations for trans-HOCO were done with n max equal to 9; this resulted in a CI matrix of order 5005. For cis-HOCO the largest CI matrices were of order of 4501. The calculations were done for J s 0 and 1, and we determined state-dependent rotation constants by fitting the difference in rovibrational
Fig. 2. Normal mode eigenvectors of trans-HOCO.
J.M. Bowman et al. r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 71᎐77
75
energies to a symmetric top expression Žwhich is quite accurate since HOCO is a near prolate symmetric top..
Table 2 Zero-point and fundamental energies Žcmy1 . of trans-HOCO
4. Results and discussion The normal-mode eigenvectors. in mass-scaled coordinates, of cis- and trans-HOCO are shown in Figs. 1 and 2, respectively. As seen, mode 5 in cis-HOCO resembles a CO-stretch very closely; however, the analogous mode in trans-HOCO looks less like a simple CO-stretch. The results of the VCI calculations for J s 0 are shown in Tables 1 and 2 for the zero-point state and the six fundamentals of cis- and transHOCO, respectively, along with previous five degree-of-freedom calculations of Zhang and Zhang w1x. As seen, there is very good agreement between the 3 and 4-MR calculations for the zero-point energy, and good agreement generally. The agreement with the 5-d.o.f. calculations is much better for cis-HOCO than for trans-HOCO. This seems reasonable based on the above discussion about the normal modes of cis- and trans-HOCO. Calculations were also done for J s 1 for transand cis-HOCO using an exact treatment, again within the approximation of a given mode representation of the potential w4᎐6x. These calculations yield rovibrational energies which can be labeled by the usual spectroscopic quantum numbers K a and K c . Both cis- and trans-HOCO are Table 1 Zero-point and fundamental energies Žcmy1 . of cis-HOCO ¨ 1¨ 2¨ 3¨ 4¨ 5¨ 6
3-MRa
4-MRa
4-MRb
4-MRc
5-d.o.f.d
ZPE 100000 010000 001000 000100 000010 000001
4375 423 542 1042 1155 1834 3591
4380 434 549 1013 1164 1837 3591
4379 427 548 1011 1162 1836 3587
4379 422 547 1010 1161 1836 3587
4383 413 556 1026 1081 ᎐ ᎐
Harmonic ZPE 4455
4455
4455
4455
a
Cl matrix of order 1674. Cl matrix of order 2385. c Cl matrix of order 4501. d See Zhang and Zhang w1x. b
3-MRa
3-MRb
4-MRb
5-d.o.f.c
ZPE 100000 010000 000100 001000 000010 000001
4589 547 670 1084 1159 1835.1 3638.8
4589 548 673 1085 1162 1844.7 3638.6
4589 551 671 1093 1183 1847.6 3636.2
4637 570 667 1122 1216
Harmonic ZPE
4680
4680
4680
¨ 1¨ 2¨ 3¨ 4¨ 5¨ 6
a
Cl matrix of order 5005. Cl matrix of order 3003. c See Zhang and Zhang w1x. b
near prolate symmetric tops, and as a result the asymmetry splittings were found to be quite small, e.g. of the order of a few hundredths of a wavenumber, for the low-lying states we considered. ŽThe rigid rotor A e , Be and C e constants are 4.063, 0.355, and 0.327 cmy1 , respectively, for cis-HOCO, and 5.467, 0.339, and 0.319 cmy1 , respectively for trans-HOCO. The average values of Be and C e are 0.341 cmy1 and 0.329 cmy1 for cis- and trans-HOCO, respectively.. Thus, we have chosen to fit the differences in the J s 1 and J s 0 energies by the standard expression Ž i. ⌬ E JK s Bi J Ž J q 1 . q Ž A i y Bi . K a2 a
and determine the rotation constants for each vibrational state i. Note in this expression Bi corresponds to the average of the B and C constants. The results of these fits are shown in Figs. 3 and 4 for the first 30 vibrational states for J s 0. As seen the state-dependent constants fluctuate considerably and differ from the rigid rotor results by 5᎐10%. 5. Summary and conclusions We have applied the code ‘MULTIMODE’ to calculate the energies of the fundamental vibrational states of non-rotating cis- and trans-HOCO for J s 0 and 1. These calculations were done using a variational configuration interaction
76
J.M. Bowman et al. r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 71᎐77
Fig. 3. Rotation constants for cis-HOCO.
method together with three and four-mode representations of the full six degree of freedom potential. Good agreement was found for the energies of the three and four-mode representations. Comparisons were also made with previous five degree-of-freedom calculations, for J s 0, of Zhang and Zhang, in which the CO ‘spectator’ mode is uncoupled. Agreement with those calculations is significantly better for cis-HOCO than for trans-HOCO. This was rationalized based on an examination of the normal mode eigenvectors of cis- and trans-HOCO, where the CO spectator mode is much closer to a normal mode in cisHOCO than in trans-HOCO. Finally, rovibrational energies were calculated for J s 1 and based on a simple symmetric top expression state-dependent rotation constants were presented. These
Fig. 4. Rotation constants for trans-HOCO.
showed significant fluctuations with vibrational state relative to the rigid rotor rotation constants. Noted added in proof We just became aware of exact six degree-offreedom calculations of energies of cis and transHOCO for J s 0 even parity states wR.B. Lehoucq, S.K. Gray, D.-H. Zhang, J.C. Light, Comp. Phys. Comm. 109 Ž1998. 15x. The energies reported here are within 1᎐2 cmy1 of these exact ones for trans-HOCO and within 3᎐8 cmy1 for cis-HOCO. Acknowledgements JMB and GW thanks the Department of Energy ŽDE-FG02-97ER14782. for financial support.
J.M. Bowman et al. r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 71᎐77
KC thanks the Cherry L. Emerson Center for Scientific Computation for a visiting fellowship. References w1x D.H. Zhang, J.Z.H. Zhang, J. Chem. Phys. 103 Ž1995. 6512. w2x G.C. Schatz, M.S. Fitzcharles, L.B. Harding, Faraday Disscus. Chem. Soc. 84 Ž1987. 359. w3x K. Kudla, G.C. Schatz, A.F. Wagner, J. Chem. Phys. 95 Ž1991. 1635. w4x S. Carter, S.J. Culik, J.M. Bowman, J. Chem. Phys. 107 Ž1997. 10458.
77
w5x S. Carter, J.M. Bowman, J. Chem. Phys. 108 Ž1998. 4397. w6x S. Carter, J.M. Bowman, N. Handy, Extensions and Tests of ‘MULTIMODE’: A Code to Obtain Accurate VibrationrRotation Energies of Many-Mode Molecules, Theoretical Chem. Accnts., accepted. w7x J.K.G. Watson, Mol. Phys. 15 Ž1968. 479. w8x J.M. Bowman, Chem. Phys. Lett. 217 Ž1994. 36. w9x For the semi-classical version, see C.W. McCurdy, W.H. Miller, ACS Symp. Ser. No. 56, in: P.R. Brooks, E.F. Hayes ŽEds.., American Chemical Society, Washington, DC, 1977, pp. 239᎐242. w10x For reviews, see J.M. Bowman, Acc. Chem. Res., 19 Ž1986. 202. w11x M.A. Ratner, R.B. Gerber, J. Phys. Chem. 90 Ž1986. 20.