摘要:

Quantum representations of dynamical systems: new bending modes of acetylene Symmetry species are labelled as Ls, where s = g ± / u± (ref. 5). For clarity, we adopt the notation where Our interest is in the dynamical analysis of the eigenstates of vibrational bends of acetylene, eqn. (2). In particular, we want to establish if, for certain values of the coefficients the normal mode superposition could lead to a new regular state, |P Ls; n1 n2ñ. In such a state, two quantum numbers n1 and n2 could thus be attributed in place of the normal mode quantum numbers u and l . The notion of quantum number implies that n1, n2 should arise as eigenvalues of two commuting operators They should constitute a complete set of commuting operators (CSCO)6 for the four-dimensional system, along with the operators, P�, L� (whose eigenvalues are P, L).However, one does not expect that the fitted Hamiltonian diagonalizes exactly in the new representation. The question is rather to determine which could be the specific regular state that would be better suited to describe a given eigenstate. The general idea of our method is to find unitary transformations u~ , such that the normal mode CSCO , l� } transforms in a new CSCO { P�, L�, N�1, N�2} { P�, L�, u� defined by This problem can be addressed in a group theoretical framework. We express the normal mode CSCO { u�4, l�4, 5, l�5} in terms of the four normal mode creation– u� annihilation boson operators, J.-M. Champion, M. Abbouti Temsamani and S.Oss Department of Physics, University of Trento and 1stituto Nazionale di Fisica della Materia, 38050 Povo, Trento, Italy. E-mail: oss@alpha.science.unitn.it Received 10th March 2000, Accepted 24th March 2000, Published 3rd April 2000 Through well-established methods of group theory, we suggest how to construct new basis sets in the four-dimensional system of bending modes in acetylene. A novel, comprehensive family of vibrational modes is found allowing one to improve the partial analyses available for this molecule in terms of local/normal quantum numbers. We provide the general features of our technique for future extensions to other, more complex systems of current interest. In this article, we address a problem of general interest in molecular physics, i.e., the attribution of quantum numbers to eigenstates, or more generally, the determination of the most suitable zero-order states in a vibrational system.In a highly non-linear dynamical system, where the approximation of small amplitude vibrations (normal modes) is not suited anymore, this is a central problem, which is often treated in reference to the underlying classical dynamics of the Hamiltonian.1–3 We present here a novel, purely quantum approach applied to the analysis of acetylene bending dynamics. We describe 13 new kinds of vibrational modes. The general features of this method are such that it can be applied to other vibrational systems, in principle without limitations due to the number of degrees of freedom. .h point In acetylene, C2H2, pure bending vibrational motions in the electronic ground state are decoupled from stretching vibrations to a good approximation.4 Bending degrees of freedom of this linear molecule are thus treated as a fourdimensional system transforming under the D symmetry.The projection L of the total angular momentum about the molecular axis is a good quantum number. Moreover, it is known that the dynamic behavior of pure bending modes, up to the highest available energies, approximately conserves a polyad number P (ref. 4). This is usually written in terms of the number of quanta in the doubly degenerate bending normal modes, (1) The Hamiltonian operator contains a diagonal part in the normal modes, as well as off-diagonal coupling terms.This operator has been fitted to the available experimental levels, taking also into account the relative fluorescence intensity within each P polyad, which provides extended information about normal mode couplings.4 In the diagonalization process of such an effective Hamiltonian, one obtains an expansion of the eigenstates in the normal mode states, This sum runs over just two quantum numbers, since P and L = l4 + l5 are conserved. Furthermore, only states transforming under D.h are included in the expansion. DOI: 10.1039/b001934h PhysChemComm, 2000, 2 (2) (3) 1, 2. (4) (5) (for index notations 4d, 4g, 5d, 5g see ref. 7). Consider first the operators u�4 and l�4, which can be directly seen as two generators of the U (2) Lie algebra in the Schwinger realization.8 Moreover, �u4 is the invariant operator N�of U (2), i.e., it commutes with all the generators of the algebra (the four generators of the full U (2) algebra requires J�+ = a�†4d a�4g and J�– = a�†4g a�4d operators as well) and l�4 is the invariant operator J�0 of the O (2) subalgebra of U (2).At the same footing, u�5 and l�5 are invariant operators of another (independent) U (2) algebra and of its O (2) subalgebra, respectively. So, the normal mode CSCO is given by the four invariant operators of the algebras belonging to the subalgebra chain (6) This is quite a general fact. Whenever the space of physical states (Hilbert space of the system) can be realized as an irreducible representation (irrep) of a given Lie algebra (the dynamical algebra), one obtains a CSCO for the system in terms of invariant operators of a chain of subalgebras.9 In the present case, the Hilbert space is constituted by all the normal mode states, with u4 and u5 ranging between 0 and ¥ .In other words, it contains all the irreps [ u4] Ä [ u5] of U (2)Ä U (2). Such a space can be thought of as an infinite dimensional irrep of a Sp (8) dynamical algebra.10 So, chain (6) appears as derived from Sp (8) (7) .h The invariant operators associated with each subalgebra are expressed in the notation of eqn. (3). Note that the polyad number operator P �is the invariant operator of the U (4) algebra. The meaning of the last term in the chain is such that the invariant operators must commute with all the D group operations G �i (ref.5). This is necessary in order that .h, as explained before. The CSCO the basic states |P Ls; u u ñ also belong to a specific representation Ls of D must commute also with time reversal T�11 The key point of our approach is that condition (4) can be expressed as a condition on the chain of subalgebras defining the new CSCO (8) All the chains including the algebra U (4) (for the conservation of P) and the group D.h (for the conservation .h. However, the corresponding invariant operators differ of L and molecular symmetry) provide a solution to our problem. Since there exist techniques for finding such chains, the problem is in principle solved.A given chain is often referred to as dynamical symmetry9,12 (not to be confused with dynamical algebra). Following the principles described in ref. 13, different dynamical symmetries (different CSCO of the chains) are related through unitary transformations. Chains arising from the U (4) algebra may be classified in two classes, depending whether they are related to chain (7) through an outer or an inner (unitary) automorphism of U (4). Outer automorphisms are identified by Dynkin’s diagrams.8 Conversely, inner automorphisms preserve the algebraic structure of the normal mode chain (7), Sp (8) ÉU (4) ÉU¢ (2) Ä U¢ (2) É O¢ (2) Ä O¢ (2) É Dfrom normal mode ones. A particular class of inner automorphisms can be found as related to linear transformations of normal mode boson operators (or, equivalently, to linear canonical transformations in normal coordinates).The well-known transformation from normal to local modes7 belongs to this class. These techniques have been applied to find solutions for chain (8). Considering the aforementioned constraints, the linear transformations, which can be applied to normal mode boson operators, constitute a (finite) group. That is, if one applies successively different transformations (or their inverses) one obtarom one kind of boson operator { a�lg, a�2g, a�ld, a�2d} to a new set { a��lg, a��2g, a��ld, a��2d} can be generated by three basic automorphisms.The first one, l, relates normal mode { a�4g, a�5g, a�4d, a�5d} to local mode { a�Ag, a�Bg, a�Ad, a�Bd} boson operators.7 The second basic pæ pö - 2 2 transformation, 2, introduces a phase shift of between the g (d) kind of boson operators. Because of their linearity in the boson operators, they can be given in matrix form: o~ ~ 1 l�2 1 + > 1 ÷ø The application of the local mode transformation l to the shifted normal modes yields new kinds of modes for acetylene bends. We call them orthogonal modes, since both bonds vibrate in orthogonal planes. We introduce the notation for local and orthogonal mode invariant operators: (10) The third inner automorphism simply exchanges the d boson operators, 3( a�1d) = a�2d, 3( a�2d) = – a�1d.This corresponds to a phase–space transformation u « l, originating new modes when combined with the outer automorphism. There is one possible outer automorphism, o~1, related to the new chain Sp (8) ÉU (4) É U (2) Ä U (2) É U (2) É O (2) É D operator, C�1 = o † ¥1. The U (2) subalgebra defines a new invariant 1, whose diagonalization leads to the new quantum number, V1 = P + 1 – < C� P – max (|L|, | u |) allowing one to define the new basis set (i.e. vibrational modes) çè (9) ..., , 2 , 0 = . The analytical expression of o~1 is non-linear in the normal coordinates. It mixes both configuration space and conjugate momenta coordinates. It is better viewed in the abstract phase space of action-angles coordinates, see ref.3. We postpone these expressions to a future publication dealing with classical analysis. Yet, o~1 can be expressed analytically through the coefficients relating the normal basis set to the 1 basis set. The transformation from O (2) Ä O (2) to U (2) É O (2) algebras indeed properly defines the Clebsch–Gordan coefficients Table 1 Characterization of the 15 basis sets, or modes, of acetylene bends. A given basis set is obtained through the transformation acting on normal modes. The symmetrized CSCOs are given (see ref. 11). Their eigenvalues are used to label states of the basis sets Symbol 1234 12341234 By successive application of the transformations 1, 2, 3, o~1 to the normal mode CSCO, one finally obtains 15 different modes (including normal and local ones).They are listed in Table 1, together with the transformation applied to normal modes and with the labelling of the corresponding states. In particular, the application of inner automorphisms to the U (2) É O (2) chain yields four different U (2) invariant operators. In terms of boson operators (see eqn. (3), (5), (10)) they are given by Some dynamical features of acetylene bending modes will be now analyzed in the language of the regular modes introduced in this article. Below 5000 cm–1, and polyad number P £ 6, it is commonly admitted that eigenstates are very close to normal modes. More precisely, all eigenstates can be assigned in the normal mode representation, according to the Hose–Taylor criterion.14 According to eqn.(2), an eigenstate | yE ñ can be assigned whenever there exists a coefficient such as Obviously, such a criterion depends upon the basis set adopted to expand the eigenstate. Generally speaking, a given state can be assigned in several basis sets. What is the physical meaning of such a multiple choice? Which quantum numbers should be assigned to the state? Let us, for example, analyze the quite regular åg+ state of polyad P = 4 with energy 2684 cm–1. The projection of this state in the normal basis leads to the unambiguous assignment |4 0g+; u = 0 l = 0ñ , with an overlap of 62%. If one projects the same state on the 2 basis (see Table 1), one obtains the assignment |4 0g+; u = 0 CSCO Transformation E 2 2 3 o~ 1 o~ 1 o~ 1 3 o~ 1 P�, L�l� P�, L�2 P�, L�2 P�, L�l� P�, L�l� 1o~ 1 1 o~ 1 1 3 2 o~ 1 1 2 o~ 1 1 3 P�, L�n� l� P�, L�2 P�, L�2 P�, L�2 P�, L�2 2 1 2 0 2 0 = o~ 1 0 3 o~ 1 o~ 1 0 o~ 1 0 3 P�, L�n� l� P�, L�2 P�, L�2 P�, L�2 P�, L�2 (11) .Labels n�n� 1 2 3 ñ |PLs; v l |PLs; v1ñ |PLs; v V2ñ |PLs; l V3ñ |PLs; l V4ñ , l� 2 , C�1 , C�2 , C�3 , C�4 4 , l� 2 , C�1 , C�3 ñ |PLs; v l |PLs; v V1ñ 1 |PLs; l V2ñ 2 |PLs; v V3ñ 3 |PLs; l V4ñ 4 , l� 2 1 , C�2 2 , C�3 3 n� l�2 l�2n� 2 n� 2 l� 2 , C�2 n� 2 l�2 , C�4 n� 2 l� 2 , C�1 n� 2 n� 2 l�2 , C�4 4 2 2 states are similar: ñ |PLs; v l |PLs; l V1ñ |PLs; v V2ñ |PLs; v V3ñ |PLs; l V4ñ V2 = 0ñ , with an overlap of 98%.So, the density of probability of such an eigenstate is more properly described by the regular 2 state. The double assignment can be explained as these (particular) and Yet, one can go further in the physical interpretation of the assignment of this eigenstate. The invariant operators of the 1 ( 2) basis set [see Table 1 and eqn. (11)], in terms of normal mode bosons are given by Apart from the diagonal contribution, these operators carry the vibrational l-coupling [Amat–Nielsen (AN)], which connects states through selection rules Dl4 = –Dl5 = ±2, i.e., = ±4 (ref. 4).Since, by definition, C�1 ( C�2) is diagonal in the 1 ( 2) basis set, one could roughly say that the corresponding quantum number V1 (V2) diagonalizes the AN coupling. This is in fact the unique relevant coupling for acetylene bends at low energies. So, one could say that the eigenstates also diagonalize this coupling in such a way that they are naturally assigned to the 2 quantum numbers. Note that C�1 and C�2 differ only in the sign of the off-diagonal contribution. The sign of the AN term in the Hamiltonian does not change with energy. However, the sign of the diagonal part is not predetermined, because of diagonal anharmonic contributions. Consequently, the assignment of a state according to V1 or V2 provides information concerning the relative position of the zeroorder (uncoupled) normal states.Dl At higher energies, diagonal anharmonicities drastically change the pattern of zero-order states within the P polyads and other couplings start to get more important (see ref. 7). The aforementioned analysis can be nonetheless applied to several eigenstates up to roughly 15 000 cm–1. One can, infact, project the eigenstates on the different modes of Table 1. For instance, C�3 and C�4 diagonalize the Darling– Dennison (DD) coupling, D u ± 4, since in terms of normal boson operators one can write Other CSCOs lead to the simultaneous diagonalization of two or more couplings. For example, local modes arise as a particular combination of AN, DD, and DD II (ref.4) couplings [as it can be seen by writing the operators u� 2 and l�2 in terms of normal boson operators, see eqn. (10)]. As a preliminary case study, we consider the å polyad P = 22. This is more regular than lower polyads, dominated by local mode behavior.7 We assigned 89 states (over 144) in terms of the 15 sets of quantum numbers. Moreover, 30 states (34% of the present assignments) cannot be assigned at all in the local modes, . In Fig. 1, the largest value of eigenstate projections on the most adequate regular basis is plotted versus energy. Most of states belong to 2 and 3 modes. A smooth transition between these modes is also observed. From a physical standpoint, 3 states mix local bend states at the bottom of the polyad in the vibrational degree of freedom (D u = ±4).At the top of the polyad, 2 states mix the local counter-rotation states in the rotational degree of freedom (Dl = ±4).7 Some oftion as clearly seen in certain lower polyads. The set of different modes can provide an efficient framework for the understanding of the evolution of dynamics with energy and polyad quantum numbers. Such a detailed analysis will be presented in a forthcoming paper. Fig. 1 Largest projection on a regular mode state versus its energy for each of the 144 states of the P = 22, å-polyad of acetylene. The corresponding basis set is indicated with a symbol, see Table 1.The 50% threshold for assignment is drawn. To summarize. we present a method leading to a systematic book-keeping of regular modes of a dynamical system. The fundamental hypothesis (or approximation) from which we derive the results is the identification of the quantum Hilbert space with a group representation. One can then use the power of group theory to identify different quantum numbers. Despite its abstract background, this method has deep physical implications for the quantum dynamics analysis, as shown in the application to acetylene bending modes. Contrary to semi-classical assignments2,3 the present ones are exact, they do not require visual inspection, and they furnish global quantum numbers (as opposed to those associated with a localized region of phase space).However, the quantum vibrational modes of acetylene are actually related to the relevant periodic orbits of the classical system.3 We defer this important question to a future publication. The method can be generalized to many other vibrational systems. In principle, it is not affected by the number of degrees of freedom of the system. Quite often, the relevant dynamical couplings can be expressed through several n : m resonances. Such cases can be addressed within the framework here discussed.15 These techniques can also be applied without special modifications to algebraic (vibron) models.12,16 Acknowledgement We thank R.W. Field and M. P. Jacobson for valuable suggestions, constructive criticism and continuous support in the development of this work. References 1 J. P. Rose and M. E. Kellman, J. Chem. Phys., 1996, 105, 10743. 2 M. Joyeux, S. Y. Grebenshchikov and R. Schinke, J. Chem. Phys., 1998, 109, 8342. 3 M. P. Jacobson, C. Jung, H. S. Taylor and R. W. Field, J. Chem. Phys., 1999, 111, 600. 4 M. P. Jacobson, J. P. O’Brien, R. J. Silbey and R. W. Field, J. Chem. Phys., 1998, 109, 121. 5 G. Hemberg, Infrared and Raman Spectra, Van Nostrand, New York, 1945. 6 P. A. M. Dirac, The Principles of Quantum Mechanics, Clarendon Press, Oxford, 4th edn., 1958. 7 M. P. Jacobson, R. J. Silbey and R. W. Field, J. Chem. Phys., 1999, 110, 845. 8 B. G. Wybourne, Classical Groups for Physicists, Wiley, New York, 1975. 9 W.-M. Zhang, D. H. Feng, and J.-M. Yhan, Phys. Rev. A, 1990, 42, 7125. 10 The Hilbert space is the direct product of irreps [½] and [–½] of Sp (8), giving all the normal mode representations with P even or odd, respectively. .h symmetry, both However, due to the exact D representations are physically disconnected, so that the discussion is still valid. 11 Operators l� and L�anticommute with T� and with certain operationsG�. However, one can use products of these operators to construct a "symmetrized" CSCO, { P, u , L�l� }. 12 F. Iachello and A. Arima, The Interacting Boson Model, Cambridge University Press, Cambridge, 1987; S. Oss, Adv. Chem. Phys., 1996, 93, 455. 13 D. Kusnezov, Phys. Rev. Lett.,1997, 79, 537. 14 G. Hose and H. Taylor, J. Chem. Phys., 1982, 76, 5356. 15 See also 'the 2D Analysis', C. C. Martens and G. S. Ezra, J. Chem Phys., 1987, 87, 284. 16 M. Abbouti Temsamani, J.-M. Champion and S. Oss, J. Chem. Phys., 1999, 110, 2893. PhysChemComm © The Royal Society of Ch

ISSN:1460-2733    

DOI:10.1039/b001934h

出版商:RSC    

年代:2000

数据来源: RSC