- Email: [email protected]
A tier model calculation of the absorption spectrum of the fourth CH stretch overtone state CH(v=5) in benzene
29 October 1999
Chemical Physics Letters 312 Ž1999. 561–566 www.elsevier.nlrlocatercplett
A tier model calculation of the absorption spectrum of the fourth CH stretch overtone state CH žÕ s 5 / in benzene S. Rashev
)
Institute of Solid State Physics, Bulgarian Academy of Sciences, Tsarigradsko chaussee 72, 1784 Sofia, Bulgaria Received 7 June 1999; in final form 31 August 1999
Abstract The absorption spectrum of the fourth CH stretch overtone state CHŽ Õ s 5. in benzene has been calculated quantum mechanically using a completely symmetrized vibrational basis set and a combined local mode normal mode model, including all 30 molecular vibrations in the calculations. An artificial intelligence search procedure, based on a tier model, with subsequent time and memory saving Lanczos tridiagonalization was employed in the calculation. The dependence of the spectral shape upon the number of tiers included, has been studied. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Understanding the specifics of vibrational motion in benzene has been one of the central problems in molecular spectroscopy and dynamics for many years w1–18x. This is a problem of considerable difficulty, especially at the higher vibrational excitation energies, because of the large number of molecular vibrational degrees of freedom Ž30.; this difficulty is however relieved to a certain extent by the high molecular symmetry Žsymmetry point group D6h . w1x. The intensive exploration of highly excited vibrational states in ground electronic state benzene, began about 20 years ago, by measuring the absorption spectra of the CH stretch fundamental and overtone states CHŽ Õ s 1–9. w3x. A major point addressed in the work of Reddy, Heller and Berry ŽRHB. w3x, has been the rationalization of the tendency in CHŽ Õ . spectral bandwidths and shapes with increasing over)
Fax: q359-2-975-36-32; e-mail: [email protected]
tone number Õ. In a subsequent theoretical treatment of the overtone states CHŽ Õ s 5–9., based on a tier model with cubic kinetic couplings, Sibert, Rinehardt and Hynes ŽSRH. w4x were able to reproduce correctly the most important tendencies in the spectral bandwidths, experimentally observed by RHB w3x. Subsequent studies, both experimental w5,13,14,16x and theoretical w6–12,15,17,18x, concentrated exclusively on the lower overtone states CHŽ Õ s 1–4., for which a considerable amount of data has been accumulated by now. However, for the higher overtone states Õ ) 4, there exists very scant information. For the theoretical investigation of higher CH stretch overtones in benzene, where the vibrational level density is extremely high, it is of crucial importance to properly take symmetry considerations into account. In our previous work w19,20x, we have introduced a symmetrized vibrational basis in product form and a totally symmetrical representation of the molecular vibrational interaction Hamiltonian terms for benzene, based on a set of complex sym-
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 1 0 0 6 - 4
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metrized vibrational coordinates w19x. Using this formalism, the exploration of vibrational level mixing in benzene can be limited only to the basis states of one definite symmetry type – that of the initially excited Ž‘bright’. state < s : Žwhich is generally taken as E 1ua w1,21,22x.. The density r E 1ua of these states is on the average 1r12 from the full density r of the vibrational states of all possible symmetries w23x. In contrast, when using an unsymmetrized vibrational basis set, the wavefunctions can be regarded as linear combinations from symmetrized states of all possible symmetry types. In this case the entire vibrational level manifold Žof density r . has to be explored Žsince each state might contain an E 1ua contribution., whose density r is by more than one order of magnitude greater than r E 1ua. As a result of all this, the dimension of a selected active space, required for the investigation of a vibrational mixing problem in benzene, would be by more than one order of magnitude lower in the symmetrized approach, than for the unsymmetrized basis. This means that the dimension of the obtained Hamiltonian matrices is reduced by about an order of magnitude in the symmetrized basis, which is of crucial importance for their diagonalization. In this way, the use of a symmetrized basis and interaction Hamiltonian, has made possible the extension of theoretical studies to the higher excited vibrational states in benzene. In the present Letter we employ the symmetrized Žcombined local mode q normal mode ŽLM q NM.. approach and a tier structure model w6,7,19,20x, in order to carry out the first quantum mechanical study of the CHŽ Õ s 5. state in benzene, taking into account all 30 molecular vibrational degrees of freedom.
2. Symmetrized LM H NM vibrational basis set for benzene In the calculations to be discussed below, a set of complex symmetrized curvilinear coordinates qk has been used w19,20x Žlinear combinations of Whiffen’s coordinates w2x., allowing for the construction of a symmetry adapted vibrational basis set < k : in combined LM q NM direct product form. The vibrational basis functions < k : are products of a symmetrized LM CH stretch wavefunction CSÕ Žlinear
combination of Morse oscillator eigenfunctions for the 6 equivalent bond oscillators, each of them excited to the overtone level Õ; S is the symmetry type of CSÕ , which is one of the following: A 1g , E 2gaŽb. , B 1u , E 1uaŽb. w1,19–22x. and a NM wavefunction COH , describing the vibrational excitation in the ‘ring’ Žnon-CH stretch. vibrational modes Žfor the ‘ring’ modes curvilinear NM coordinates are used, which are obtained by normalization of the non-CH stretch Hamiltonian.: < k : s CSÕ COH s < ÕS ; n i , n k , . . . :
Ž 1.
Žwhere n i is excitation quantum number in the ith NM.. Each basis function < k : has a well defined symmetry type, which is determined by simple product symmetry rules w19,20x. The molecular vibrational Hamiltonian is obtained Žin explicit totally symmetric form. as a sum of a zeroth-order Hamiltonian H0 Žwhose eigenfunctions are the basis states < k :. and interaction Hamiltonian parts: quadratic, cubic, quartic, etc. In terms of the curvilinear coordinates employed here, all Hamiltonian terms contain a kinetic contribution Žwhich can be obtained from Wilson’s G-matrix elements w1x, expressed as explicit functions of the coordinates qk ., and a potential contribution, kinetic constants largely prevailing over potential constants in absolute magnitude w4x. The quadratic Hamiltonian interaction terms arise as a result of the combined LM q NM treatment wthey couple each of the CH stretches to NM Žring. vibrations of like symmetryx. All cubic interaction Hamiltonian terms were calculated in detail as described previously w19,20x, as sums of kinetic Žfirst derivatives of G-matrix elements w1,6x. and potential Žanharmonic potential field of Maslen et al. w8x. contributions. The analysis of our own calculations on the cubic terms w19,20x has shown, that the potential contributions to the coupling constants Žas taken from Maslen et al. w8x. amounted on the average to about one-fourth of the kinetic contributions. In the present work beside cubic terms, quartic interaction Hamiltonian terms were also included. Since in w8x only a small number of the calculated quartic anharmonic constants are given, and assuming that for the quartic terms Žas for the cubic terms. kinetic contributions should greatly prevail over potential ones, we have decided to take into account only kinetic contributions to the quartic cou-
S. RasheÕr Chemical Physics Letters 312 (1999) 561–566
pling constants. Coupling constants for the quartic kinetic Hamiltonian terms were calculated from analytical second derivatives of Wilson’s G-matrix elements, expressed as functions of the coordinates qk , in a totally symmetric form, in terms of ladder operators Žanalogously to the cubic terms, described previously w1,19,20x.. The quartic coupling constants obtained in this way were quite small, leading to a minor overall effect of the quartic Hamiltonian terms on the spectrum and mixing pattern at CHŽ Õ s 5..
3. AI search algorithm, based on a tier structure An artificial intelligence ŽAI. search procedure was used to select the active space P, consisting of all important basis states < i :, effectively involved in the mixing with the initially excited Ž‘bright’. Žpurely LM. state < s : s c E51ua, at EÕ s 14 072 cmy1 Žall selected states < i :, having the same symmetry type E 1ua .. The AI search algorithm for successive selection of basis states, involved in the mixing Ždescribed previously w19,20x., employed a perturbation type criterion, based on the values of two parameters Ž C and W .. A new state < k : is selected, if: Ž1. the ratio of coupling matrix element vs. energy spacing L i k s <² i < H X < k :rŽ Ei y Ek . of < k : with a previously selected state < i : exceeds a minimum value of 1rC and Ž2. the cumulative Žproduct. ratio Wk s k Ł is1 L iy1, i Žwhere a factor L iy1, i is replaced by 1, if greater than 1. exceeds the chosen value W, for at least one of the paths Žchains. < s : ™ <1: ™ . . . < k : leading from < s : s <0: to < k :. In order to obtain a representative active space P, C is usually chosen to be G 5, while W is chosen to be F 0.01. Such a selection method, based on a perturbation criterion, has serious drawbacks when, as in the present case, there exists a very strong variation in the magnitude of couplings Žby several orders of magnitude.. Indeed, under the action of strong couplings, some basis states are shifted far away from their original positions Žfrom surrounding initially selected weakly coupled states. and placed in the vicinity of other basis states, which may have however not been selected. In this way, weak couplings are practically invalidated to a great extent, and as a result of this, the calculated density of effectively coupled states reff is underestimated. There are ways
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to remedy this situation, but they lead invariably to the selection of extremely large dimensional active spaces, which cannot be manipulated. However it is possible to assess the role of couplings of various magnitudes in the vibrational mixing pattern, and of weak couplings in particular. For this purpose, another pair of parameters Vmax and Vmin can be introduced in the search algorithm, allowing to take into account only such coupling terms V, lying within an energy range: Vmax ) < V < ) Vmin . Such an analysis will be carried out in forthcoming work. In the present work we shall apply the perturbation criterion to the selection of important basis states, as described above, being fully aware that the true vibrational level density, effectively involved in the vibrational mixing will not be correctly reproduced in the calculated absorption spectra. The selection of important basis states is done by successive tiers, which are a generalization of the tiers, introduced by SRH w4x. As in SRH, each tier consists of the states < Õs ; k :, with a constant Õ-value, i.e. pertaining to a definite overtone manifold Õ and arbitrary number of NM quanta k. The first tier consists of only one state: < s : s < Õ E 1ua; 0:. The successive tiers are designated as < Õ y 1 s ; k :4 , Õ y 2 s ; kX :4 , etc. The states of a new tier are selected in the following order from the states of the preceding tier: first, the states directly coupled to the states of the preceding tier by cubic and quartic Hamiltonian interaction terms H Ž3. and H Ž4., are selected. Next, the remaining states from the tier Žovertone manifold., coupled to the states already selected, are searched for.
4. Lanczos tridiagonalization and subsequent diagonalization Performing the search procedure, starting with the ‘bright’ state < s :, an ‘active space’ P is selected, encompassing all basis states, effectively involved in the mixing with < s :. The dimensionality N of P depends upon the values chosen for the parameters of the search – C and W. The obtained symmetrical Hamiltonian matrix H Ž N = N ., consists of the energies of the selected basis states Ei Žas diagonal elements. and the coupling matrix elements Hi j s Hi j s ² i < H X < j : as nondiagonal elements. A Lanczos
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tridiagonalization procedure with M recursion steps Ž M F N . can be applied to this matrix H, in order to obtain an M = M tridiagonal matrix L w24–27x. Since the mixing is entirely realized through chains of 3rd- and 4th-order interactions, the obtained Hamiltonian matrix H is very sparse Žcontaining a large number of zeros.. This sparsity has allowed to design an economic Žtime and memory saving. procedure, for storing the matrix H in computer memory. Only the nonzero matrix elements Hm n Žin conjunction with the relevant values of m and n., were stored in the computer memory, as an array of Z = 3 dimension, where Z is the number of nonzero Hm n , with m G n. This enables the storage and manipulation of extremely large dimensional matrices. Each multiplication of H = vector, which is in principle the most time consuming step in the Lanczos algorithm, is implemented in one run over the Z values of the first index in this array. This storage and matrix= vector multiplication procedure makes possible the manipulation of very large-dimensional Hamiltonian matrices and leads to a great reduction in computing time. The tridiagonal matrix L, obtained upon implementation of the Lanczos method, was subsequently diagonalized, using a standard program tqliŽ . from ‘‘Numerical Recipes in C’’ w28x. This program yields both eigenvalues E i and eigenvectors. It can however readily be modified to yield eigenvalues and only one or a few rows of the eigenvector matrix. The modified program is much faster and requires much less computer memory, but yields enough information for obtaining the absorption spectrum and survival probability. However, if the detailed energy flow among various Žgroups of. vibrational modes during the relaxation process is required, diagonalization with full eigenvector information has to be carried out.
5. Calculation of the CH( z s 5) spectrum The aim of the present work has been to carry out a computation of the absorption spectrum of the fourth overtone state CHŽ Õ s 5.. Computations on this state have been performed only once in the past, by SRH w4x, taking into account only in-plane vibrations, only three tiers and only the most important
Žkinetic. Hamiltonian interaction terms. From the results of their computations SRH concluded, that the spectral width of the investigated overtone state was determined mainly by the small number of first tier states Ž2nd tier in our present notation., which
Fig. 1. Absorption spectrum of the overtone state CHŽ Õ s 5. at EÕ s14 072 cmy1 in ground electronic state benzene. The spectra displayed in the plots Ža. – Že. are obtained with successively increasing number of tiers included in the calculation Žthe full-tier spectrum is displayed in Že.. All calculations were carried out at the following parameter values Žwhose meaning is explained in the text.: C s 5, W s 0.01. The dimensionality N of the selected active space and the Hamiltonian matrix, obtained for each of the five spectra, is as follows: Ža. CHŽ Õ s 5–4., N s 2 454; Žb. CHŽ Õ s 5–3., N s 7 245; Žc. CHŽ Õ s 5–2., N s15 716; Žd. CHŽ Õ s 5–1., N s 42 112; Že. CHŽ Õ s 5., N s107 239.
S. RasheÕr Chemical Physics Letters 312 (1999) 561–566
are the most strongly coupled ones, while ‘‘the second-, third- and higher-order tiers served to fill in the band envelope’’. In order to verify the validity of this suggestion, starting with the state < s : s c E51ua as the first tier in the AI search, we have carried out computations, including successively increasing number of tiers from 2 to 6 Ži.e. all existing tiers., in the following order: < s : s <5 E 1ua; 0: ™ <4s ; k :4 ™ <3 s ; kX :4 ™ <2 s ; kY :4 ™ <1 s ; kZ :4 ™ <0A ; k IV :4 . 1g wNote that the CH stretch ŽLM. part of the wavefunction can have one of the symmetry types S s A 1g , E 2gaŽb. , B1u , E 1uaŽb. , for the intermediate tiers 2–5, but it must be fixed at E 1ua for the 1st and A 1g for the 6th tiers, respectively.x The results from our calculations are displayed in Fig. 1a–e. The values of the AI search parameters, employed in the calculations, as well as the dimensionalities N of the selected active spaces P for each case, are given in the figure captions. From the spectra displayed in Fig. 1a–e it can be seen, that contrary to what SRH have suggested, the two-tier spectrum wCHŽ Õ s 5–4., Fig. 1ax, has very little to do with the real spectrum. The 3-tier spectrum wCHŽ Õ s 5–3., Fig. 1bx, has still very few features in common with the true spectrum. The 4-tier spectrum wCHŽ Õ s 5–2., Fig. 1cx is already rather satisfactory in width and general appearance, while the 5-tier spectrum wCHŽ Õ s 5–1., Fig. 1dx, looks almost the same as the full-tier spectrum wCHŽ Õ s 5–0. s CHŽ Õ s 5.x, which is displayed in Fig. 1e. However, a closer look at these spectra reveals a substantial distinction in the calculated effective vibrational level density, involved in the vibrational mixing with < s :. This distinction can be seen from the insets in Fig. 1c–e, showing very narrow portions of the corresponding spectra in expanded form, in semilog scale. It is seen, that the 5-tier spectrum ŽFig. 1d., although very similar to the full CHŽ Õ s 5. spectrum ŽFig. 1e., is about two times sparser, while the 4-tier spectrum ŽFig. 1c., is incomparably sparser than both. A general conclusion from Fig. 1 is, that the spectrum, obtained with only a small number of tiers included, is not a good characteristic of the real spectrum. This latter spectrum is well characterized only upon inclusion of all tiers in the calculation. The CHŽ Õ s 5. spectrum ŽFig. 1e., calculated using a symmetrized vibrational basis set, with all 30 molecular vibrations included, is the central result of
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this work. As seen from Fig. 1e, the theoretically obtained absorption spectrum is highly structured. According to the experimental measurements of RHB w3x, this overtone spectrum was the most broadened one, with a FWHMf 111 cmy1 . The absorption spectrum of CHŽ Õ s 5. calculated in this work, whose FWHM can hardly be defined unambiguously Žhowever, its envelope has a FWHM similar to that, measured by RHB., is indeed much broader than the lower overtone spectra and in particular the CHŽ Õ s 4. spectrum, calculated recently w29x. The observed spectral density, which is a monitor of the ‘dark’ vibrational level density around CHŽ Õ s 5., effectively involved in the mixing with the ‘bright’ state < s :, is displayed in the inset of Fig. 1e. The calculated ‘effective’ vibrational level density Ž reff f 50 cmy1 ., representative of the extent of vibrational mixing around CHŽ Õ s 5., is seen to be very low as compared to the estimated very large available vibrational density of suitable symmetry E 1ua Ž r E 1ua f 10 6 cmy1 .. From the inset in Fig. 1e, due to the semilog scale employed, it is seen, that the spectral line intensities vary by several orders of magnitude, which is an evidence for highly nonstatistical vibrational mixing and a number of time scales in the relaxation.
References w1x E.B. Wilson, J.C. Decius, P.C. Cross, Molecular Vibrations, McGraw-Hill, New York, 1955. w2x D.H. Whiffen, Philos. Trans. R. Soc. London, Ser. A 248 Ž1955. 131. w3x K.V. Reddy, D.F. Heller, M.J. Berry, J. Chem. Phys. 76 Ž1982. 2814. w4x E.L. Sibert, W.P. Reinhardt, J.T. Hynes, J. Chem. Phys. 81 Ž1984. 1115. w5x P.H. Page, Y.R. Shen, Y.T. Lee, J. Chem. Phys. 88 Ž1988. 4621. w6x Y. Zhang, S.J. Klippenstein, R.A. Marcus, J. Chem. Phys. 94 Ž1991. 7319. w7x Y. Zhang, R.A. Marcus, J. Chem. Phys. 97 Ž1992. 5283. w8x P.E. Maslen, N.C. Handy, R.D. Amos, D. Jayatilaka, J. Chem. Phys. 97 Ž1992. 4233. w9x D.-H. Lu, W.L. Hase, R.J. Wolf, J. Chem. Phys. 85 Ž1986. 4422. w10x D.-H. Lu, W.L. Hase, J. Chem. Phys. 89 Ž1988. 6723. w11x K.L. Bintz, D.L. Thompson, J.W. Brady, J. Chem. Phys. 85 Ž1986. 1848. w12x J.M.G. Llorente, O. Hahn, H.S. Taylor, J. Chem. Phys. 92 Ž1990. 2762.
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w13x M. Scotoni, C. Leonardi, D. Bassi, J. Chem. Phys. 95 Ž1991. 8655. w14x M. Scotoni, A. Boscetti, N. Oberhofer, D. Bassi, J. Chem. Phys. 94 Ž1991. 971. w15x F. Iachello, S. Oss, J. Chem. Phys. 99 Ž1993. 7337. w16x A. Callegari, H.K. Srivastava, U. Merker, K.K. Lehmann, G. Scoles, M.J. Davis, J. Chem. Phys. 106 Ž1997. 432. w17x R.E. Wyatt, C. Iung, C. Leforestier, J. Chem. Phys. 97 Ž1992. 3458. w18x R.E. Wyatt, J. Chem. Phys. 109 Ž1998. 10732. w19x S. Rashev, J. Chem. Phys. 110 Ž1999. 2489. w20x S. Rashev, M. Stamova, S. Djambova, J. Chem. Phys. 108 Ž1998. 4497. w21x L. Halonen, Chem. Phys. Lett. 87 Ž1982. 221. w22x R. Wallace, Chem. Phys. 11 Ž1975. 189.
w23x S.M. Lederman, J.H. Runnels, R.A. Marcus, J. Phys. Chem. 87 Ž1983. 4364. w24x J.K. Cullum, R.A. Willowghby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations, vols. I, II, Birkhauser, Boston, MA, 1985. ¨ w25x R.E. Wyatt, Adv. Chem. Phys. 73 Ž1989. 231. w26x R. Bigwood, M. Gruebele, Chem. Phys. Lett. 235 Ž1995. 604. w27x R. Bigwood, M. Gruebele, Int. Rev. Phys. Chem. 17 Ž1998. 91. w28x W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes in C, Cambridge University Press, Cambridge, 1988. w29x S. Rashev, J. Chem. Phys. 111 Ž1999. 5617.
Chemical Physics Letters 312 Ž1999. 561–566 www.elsevier.nlrlocatercplett
A tier model calculation of the absorption spectrum of the fourth CH stretch overtone state CH žÕ s 5 / in benzene S. Rashev
)
Institute of Solid State Physics, Bulgarian Academy of Sciences, Tsarigradsko chaussee 72, 1784 Sofia, Bulgaria Received 7 June 1999; in final form 31 August 1999
Abstract The absorption spectrum of the fourth CH stretch overtone state CHŽ Õ s 5. in benzene has been calculated quantum mechanically using a completely symmetrized vibrational basis set and a combined local mode normal mode model, including all 30 molecular vibrations in the calculations. An artificial intelligence search procedure, based on a tier model, with subsequent time and memory saving Lanczos tridiagonalization was employed in the calculation. The dependence of the spectral shape upon the number of tiers included, has been studied. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Understanding the specifics of vibrational motion in benzene has been one of the central problems in molecular spectroscopy and dynamics for many years w1–18x. This is a problem of considerable difficulty, especially at the higher vibrational excitation energies, because of the large number of molecular vibrational degrees of freedom Ž30.; this difficulty is however relieved to a certain extent by the high molecular symmetry Žsymmetry point group D6h . w1x. The intensive exploration of highly excited vibrational states in ground electronic state benzene, began about 20 years ago, by measuring the absorption spectra of the CH stretch fundamental and overtone states CHŽ Õ s 1–9. w3x. A major point addressed in the work of Reddy, Heller and Berry ŽRHB. w3x, has been the rationalization of the tendency in CHŽ Õ . spectral bandwidths and shapes with increasing over)
Fax: q359-2-975-36-32; e-mail: [email protected]
tone number Õ. In a subsequent theoretical treatment of the overtone states CHŽ Õ s 5–9., based on a tier model with cubic kinetic couplings, Sibert, Rinehardt and Hynes ŽSRH. w4x were able to reproduce correctly the most important tendencies in the spectral bandwidths, experimentally observed by RHB w3x. Subsequent studies, both experimental w5,13,14,16x and theoretical w6–12,15,17,18x, concentrated exclusively on the lower overtone states CHŽ Õ s 1–4., for which a considerable amount of data has been accumulated by now. However, for the higher overtone states Õ ) 4, there exists very scant information. For the theoretical investigation of higher CH stretch overtones in benzene, where the vibrational level density is extremely high, it is of crucial importance to properly take symmetry considerations into account. In our previous work w19,20x, we have introduced a symmetrized vibrational basis in product form and a totally symmetrical representation of the molecular vibrational interaction Hamiltonian terms for benzene, based on a set of complex sym-
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 1 0 0 6 - 4
562
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metrized vibrational coordinates w19x. Using this formalism, the exploration of vibrational level mixing in benzene can be limited only to the basis states of one definite symmetry type – that of the initially excited Ž‘bright’. state < s : Žwhich is generally taken as E 1ua w1,21,22x.. The density r E 1ua of these states is on the average 1r12 from the full density r of the vibrational states of all possible symmetries w23x. In contrast, when using an unsymmetrized vibrational basis set, the wavefunctions can be regarded as linear combinations from symmetrized states of all possible symmetry types. In this case the entire vibrational level manifold Žof density r . has to be explored Žsince each state might contain an E 1ua contribution., whose density r is by more than one order of magnitude greater than r E 1ua. As a result of all this, the dimension of a selected active space, required for the investigation of a vibrational mixing problem in benzene, would be by more than one order of magnitude lower in the symmetrized approach, than for the unsymmetrized basis. This means that the dimension of the obtained Hamiltonian matrices is reduced by about an order of magnitude in the symmetrized basis, which is of crucial importance for their diagonalization. In this way, the use of a symmetrized basis and interaction Hamiltonian, has made possible the extension of theoretical studies to the higher excited vibrational states in benzene. In the present Letter we employ the symmetrized Žcombined local mode q normal mode ŽLM q NM.. approach and a tier structure model w6,7,19,20x, in order to carry out the first quantum mechanical study of the CHŽ Õ s 5. state in benzene, taking into account all 30 molecular vibrational degrees of freedom.
2. Symmetrized LM H NM vibrational basis set for benzene In the calculations to be discussed below, a set of complex symmetrized curvilinear coordinates qk has been used w19,20x Žlinear combinations of Whiffen’s coordinates w2x., allowing for the construction of a symmetry adapted vibrational basis set < k : in combined LM q NM direct product form. The vibrational basis functions < k : are products of a symmetrized LM CH stretch wavefunction CSÕ Žlinear
combination of Morse oscillator eigenfunctions for the 6 equivalent bond oscillators, each of them excited to the overtone level Õ; S is the symmetry type of CSÕ , which is one of the following: A 1g , E 2gaŽb. , B 1u , E 1uaŽb. w1,19–22x. and a NM wavefunction COH , describing the vibrational excitation in the ‘ring’ Žnon-CH stretch. vibrational modes Žfor the ‘ring’ modes curvilinear NM coordinates are used, which are obtained by normalization of the non-CH stretch Hamiltonian.: < k : s CSÕ COH s < ÕS ; n i , n k , . . . :
Ž 1.
Žwhere n i is excitation quantum number in the ith NM.. Each basis function < k : has a well defined symmetry type, which is determined by simple product symmetry rules w19,20x. The molecular vibrational Hamiltonian is obtained Žin explicit totally symmetric form. as a sum of a zeroth-order Hamiltonian H0 Žwhose eigenfunctions are the basis states < k :. and interaction Hamiltonian parts: quadratic, cubic, quartic, etc. In terms of the curvilinear coordinates employed here, all Hamiltonian terms contain a kinetic contribution Žwhich can be obtained from Wilson’s G-matrix elements w1x, expressed as explicit functions of the coordinates qk ., and a potential contribution, kinetic constants largely prevailing over potential constants in absolute magnitude w4x. The quadratic Hamiltonian interaction terms arise as a result of the combined LM q NM treatment wthey couple each of the CH stretches to NM Žring. vibrations of like symmetryx. All cubic interaction Hamiltonian terms were calculated in detail as described previously w19,20x, as sums of kinetic Žfirst derivatives of G-matrix elements w1,6x. and potential Žanharmonic potential field of Maslen et al. w8x. contributions. The analysis of our own calculations on the cubic terms w19,20x has shown, that the potential contributions to the coupling constants Žas taken from Maslen et al. w8x. amounted on the average to about one-fourth of the kinetic contributions. In the present work beside cubic terms, quartic interaction Hamiltonian terms were also included. Since in w8x only a small number of the calculated quartic anharmonic constants are given, and assuming that for the quartic terms Žas for the cubic terms. kinetic contributions should greatly prevail over potential ones, we have decided to take into account only kinetic contributions to the quartic cou-
S. RasheÕr Chemical Physics Letters 312 (1999) 561–566
pling constants. Coupling constants for the quartic kinetic Hamiltonian terms were calculated from analytical second derivatives of Wilson’s G-matrix elements, expressed as functions of the coordinates qk , in a totally symmetric form, in terms of ladder operators Žanalogously to the cubic terms, described previously w1,19,20x.. The quartic coupling constants obtained in this way were quite small, leading to a minor overall effect of the quartic Hamiltonian terms on the spectrum and mixing pattern at CHŽ Õ s 5..
3. AI search algorithm, based on a tier structure An artificial intelligence ŽAI. search procedure was used to select the active space P, consisting of all important basis states < i :, effectively involved in the mixing with the initially excited Ž‘bright’. Žpurely LM. state < s : s c E51ua, at EÕ s 14 072 cmy1 Žall selected states < i :, having the same symmetry type E 1ua .. The AI search algorithm for successive selection of basis states, involved in the mixing Ždescribed previously w19,20x., employed a perturbation type criterion, based on the values of two parameters Ž C and W .. A new state < k : is selected, if: Ž1. the ratio of coupling matrix element vs. energy spacing L i k s <² i < H X < k :rŽ Ei y Ek . of < k : with a previously selected state < i : exceeds a minimum value of 1rC and Ž2. the cumulative Žproduct. ratio Wk s k Ł is1 L iy1, i Žwhere a factor L iy1, i is replaced by 1, if greater than 1. exceeds the chosen value W, for at least one of the paths Žchains. < s : ™ <1: ™ . . . < k : leading from < s : s <0: to < k :. In order to obtain a representative active space P, C is usually chosen to be G 5, while W is chosen to be F 0.01. Such a selection method, based on a perturbation criterion, has serious drawbacks when, as in the present case, there exists a very strong variation in the magnitude of couplings Žby several orders of magnitude.. Indeed, under the action of strong couplings, some basis states are shifted far away from their original positions Žfrom surrounding initially selected weakly coupled states. and placed in the vicinity of other basis states, which may have however not been selected. In this way, weak couplings are practically invalidated to a great extent, and as a result of this, the calculated density of effectively coupled states reff is underestimated. There are ways
563
to remedy this situation, but they lead invariably to the selection of extremely large dimensional active spaces, which cannot be manipulated. However it is possible to assess the role of couplings of various magnitudes in the vibrational mixing pattern, and of weak couplings in particular. For this purpose, another pair of parameters Vmax and Vmin can be introduced in the search algorithm, allowing to take into account only such coupling terms V, lying within an energy range: Vmax ) < V < ) Vmin . Such an analysis will be carried out in forthcoming work. In the present work we shall apply the perturbation criterion to the selection of important basis states, as described above, being fully aware that the true vibrational level density, effectively involved in the vibrational mixing will not be correctly reproduced in the calculated absorption spectra. The selection of important basis states is done by successive tiers, which are a generalization of the tiers, introduced by SRH w4x. As in SRH, each tier consists of the states < Õs ; k :, with a constant Õ-value, i.e. pertaining to a definite overtone manifold Õ and arbitrary number of NM quanta k. The first tier consists of only one state: < s : s < Õ E 1ua; 0:. The successive tiers are designated as < Õ y 1 s ; k :4 , Õ y 2 s ; kX :4 , etc. The states of a new tier are selected in the following order from the states of the preceding tier: first, the states directly coupled to the states of the preceding tier by cubic and quartic Hamiltonian interaction terms H Ž3. and H Ž4., are selected. Next, the remaining states from the tier Žovertone manifold., coupled to the states already selected, are searched for.
4. Lanczos tridiagonalization and subsequent diagonalization Performing the search procedure, starting with the ‘bright’ state < s :, an ‘active space’ P is selected, encompassing all basis states, effectively involved in the mixing with < s :. The dimensionality N of P depends upon the values chosen for the parameters of the search – C and W. The obtained symmetrical Hamiltonian matrix H Ž N = N ., consists of the energies of the selected basis states Ei Žas diagonal elements. and the coupling matrix elements Hi j s Hi j s ² i < H X < j : as nondiagonal elements. A Lanczos
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tridiagonalization procedure with M recursion steps Ž M F N . can be applied to this matrix H, in order to obtain an M = M tridiagonal matrix L w24–27x. Since the mixing is entirely realized through chains of 3rd- and 4th-order interactions, the obtained Hamiltonian matrix H is very sparse Žcontaining a large number of zeros.. This sparsity has allowed to design an economic Žtime and memory saving. procedure, for storing the matrix H in computer memory. Only the nonzero matrix elements Hm n Žin conjunction with the relevant values of m and n., were stored in the computer memory, as an array of Z = 3 dimension, where Z is the number of nonzero Hm n , with m G n. This enables the storage and manipulation of extremely large dimensional matrices. Each multiplication of H = vector, which is in principle the most time consuming step in the Lanczos algorithm, is implemented in one run over the Z values of the first index in this array. This storage and matrix= vector multiplication procedure makes possible the manipulation of very large-dimensional Hamiltonian matrices and leads to a great reduction in computing time. The tridiagonal matrix L, obtained upon implementation of the Lanczos method, was subsequently diagonalized, using a standard program tqliŽ . from ‘‘Numerical Recipes in C’’ w28x. This program yields both eigenvalues E i and eigenvectors. It can however readily be modified to yield eigenvalues and only one or a few rows of the eigenvector matrix. The modified program is much faster and requires much less computer memory, but yields enough information for obtaining the absorption spectrum and survival probability. However, if the detailed energy flow among various Žgroups of. vibrational modes during the relaxation process is required, diagonalization with full eigenvector information has to be carried out.
5. Calculation of the CH( z s 5) spectrum The aim of the present work has been to carry out a computation of the absorption spectrum of the fourth overtone state CHŽ Õ s 5.. Computations on this state have been performed only once in the past, by SRH w4x, taking into account only in-plane vibrations, only three tiers and only the most important
Žkinetic. Hamiltonian interaction terms. From the results of their computations SRH concluded, that the spectral width of the investigated overtone state was determined mainly by the small number of first tier states Ž2nd tier in our present notation., which
Fig. 1. Absorption spectrum of the overtone state CHŽ Õ s 5. at EÕ s14 072 cmy1 in ground electronic state benzene. The spectra displayed in the plots Ža. – Že. are obtained with successively increasing number of tiers included in the calculation Žthe full-tier spectrum is displayed in Že.. All calculations were carried out at the following parameter values Žwhose meaning is explained in the text.: C s 5, W s 0.01. The dimensionality N of the selected active space and the Hamiltonian matrix, obtained for each of the five spectra, is as follows: Ža. CHŽ Õ s 5–4., N s 2 454; Žb. CHŽ Õ s 5–3., N s 7 245; Žc. CHŽ Õ s 5–2., N s15 716; Žd. CHŽ Õ s 5–1., N s 42 112; Že. CHŽ Õ s 5., N s107 239.
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are the most strongly coupled ones, while ‘‘the second-, third- and higher-order tiers served to fill in the band envelope’’. In order to verify the validity of this suggestion, starting with the state < s : s c E51ua as the first tier in the AI search, we have carried out computations, including successively increasing number of tiers from 2 to 6 Ži.e. all existing tiers., in the following order: < s : s <5 E 1ua; 0: ™ <4s ; k :4 ™ <3 s ; kX :4 ™ <2 s ; kY :4 ™ <1 s ; kZ :4 ™ <0A ; k IV :4 . 1g wNote that the CH stretch ŽLM. part of the wavefunction can have one of the symmetry types S s A 1g , E 2gaŽb. , B1u , E 1uaŽb. , for the intermediate tiers 2–5, but it must be fixed at E 1ua for the 1st and A 1g for the 6th tiers, respectively.x The results from our calculations are displayed in Fig. 1a–e. The values of the AI search parameters, employed in the calculations, as well as the dimensionalities N of the selected active spaces P for each case, are given in the figure captions. From the spectra displayed in Fig. 1a–e it can be seen, that contrary to what SRH have suggested, the two-tier spectrum wCHŽ Õ s 5–4., Fig. 1ax, has very little to do with the real spectrum. The 3-tier spectrum wCHŽ Õ s 5–3., Fig. 1bx, has still very few features in common with the true spectrum. The 4-tier spectrum wCHŽ Õ s 5–2., Fig. 1cx is already rather satisfactory in width and general appearance, while the 5-tier spectrum wCHŽ Õ s 5–1., Fig. 1dx, looks almost the same as the full-tier spectrum wCHŽ Õ s 5–0. s CHŽ Õ s 5.x, which is displayed in Fig. 1e. However, a closer look at these spectra reveals a substantial distinction in the calculated effective vibrational level density, involved in the vibrational mixing with < s :. This distinction can be seen from the insets in Fig. 1c–e, showing very narrow portions of the corresponding spectra in expanded form, in semilog scale. It is seen, that the 5-tier spectrum ŽFig. 1d., although very similar to the full CHŽ Õ s 5. spectrum ŽFig. 1e., is about two times sparser, while the 4-tier spectrum ŽFig. 1c., is incomparably sparser than both. A general conclusion from Fig. 1 is, that the spectrum, obtained with only a small number of tiers included, is not a good characteristic of the real spectrum. This latter spectrum is well characterized only upon inclusion of all tiers in the calculation. The CHŽ Õ s 5. spectrum ŽFig. 1e., calculated using a symmetrized vibrational basis set, with all 30 molecular vibrations included, is the central result of
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this work. As seen from Fig. 1e, the theoretically obtained absorption spectrum is highly structured. According to the experimental measurements of RHB w3x, this overtone spectrum was the most broadened one, with a FWHMf 111 cmy1 . The absorption spectrum of CHŽ Õ s 5. calculated in this work, whose FWHM can hardly be defined unambiguously Žhowever, its envelope has a FWHM similar to that, measured by RHB., is indeed much broader than the lower overtone spectra and in particular the CHŽ Õ s 4. spectrum, calculated recently w29x. The observed spectral density, which is a monitor of the ‘dark’ vibrational level density around CHŽ Õ s 5., effectively involved in the mixing with the ‘bright’ state < s :, is displayed in the inset of Fig. 1e. The calculated ‘effective’ vibrational level density Ž reff f 50 cmy1 ., representative of the extent of vibrational mixing around CHŽ Õ s 5., is seen to be very low as compared to the estimated very large available vibrational density of suitable symmetry E 1ua Ž r E 1ua f 10 6 cmy1 .. From the inset in Fig. 1e, due to the semilog scale employed, it is seen, that the spectral line intensities vary by several orders of magnitude, which is an evidence for highly nonstatistical vibrational mixing and a number of time scales in the relaxation.
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