Demixing and Cleaning of Wave Functions by Projection, Application

May 11, 2010 - The projection cleans the wave function and decomposes mixtures caused by near-accidental degeneracy. The idea is applied to vibrationa...
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J. Phys. Chem. A 2010, 114, 9693–9699

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Demixing and Cleaning of Wave Functions by Projection, Application to the Assignment of Molecular Vibrations† Christof Jung* Instituto de Ciencias Fisicas, UniVersidad Nacional Autonoma de Mexico, AV. UniVersidad s/n 62251 CuernaVaca, Mexico ReceiVed: February 23, 2010; ReVised Manuscript ReceiVed: April 26, 2010

It is shown how the projection of a quantum mechanical state into an appropriate subspace of the Hilbert space can be useful for the classification and the assignment of states. The projection cleans the wave function and decomposes mixtures caused by near-accidental degeneracy. The idea is applied to vibrational states of the molecule CF3CHFI obtained from an algebraic Hamiltonian. 1. Introduction An important problem of molecular spectroscopy is the classification and assignment of highly excited vibrational states for polyatomic molecules of moderate size. As long as the various modes are involved in a single resonant interaction at most, the Hamiltonian stays integrable and semiclassical methods provide the solution. The problem becomes more complicated and also more interesting if modes are involved in several resonant interactions simultaneously. According to ref 1, overlapping resonances can cause chaos on a large scale in the corresponding classical dynamics. However, in such cases there are also important organization structures in the system, and in many cases they support rather regular ladders of states embedded into the total spectrum, which appears irregular at first sight. Then, the problem is to identify the underlying organization structures and their corresponding regular ladders of states and to assign to the rungs of these ladders quantum numbers relative to the organization structure. One rather efficient method to realize this program for realistic algebraic models of molecular vibrations has been proposed in ref 2. It is based on an investigation of the localization properties of each wave function in angle space constructed as Fourier transformation (here it is a finite Fourier sum) of the action space function, which is the originally given eigenvector of the Hamiltonian matrix in number state basis (see also ref 3). The basic idea of the method is: For any particular state or any group of states based on the same dynamics, not all interaction terms in the Hamiltonian are important, and only some subset of them determines the qualitative structure of a state. As a consequence, the densities of the configuration space wave functions have only in some directions (which we called the transverse directions) a structure that is meaningful for the assignment and are rather simple in the remaining directions (which we called the longitudinal directions). The density is concentrated around some lower dimensional subset of the configuration space that we called the organization center of the group of states. The complete complex wave functions have an approximate factorization into a transverse factor (transverse to the organization center), which is similar to an oscillator function, and into a longitudinal factor (parallel to the organization center), which comes close to a plane wave. From the nodal †

Part of the “Reinhard Schinke Festschrift”. * E-mail: [email protected].

structure of the transverse factor we read off transverse quantum numbers, and from the wave vector of the longitudinal factor we read off longitudinal quantum numbers. Thereby, the group of states belonging to one dynamics and one set of active resonant interactions are ordered naturally and thereby assigned on one ladder of states. The transverse quantum numbers can be interpreted as the quantized actions of the fluctuations around the organization center.4 The longitudinal quantum numbers are interpreted as linear combinations of the quantum numbers of the basis states that are approximately conserved for this particular group of states. Of course, for dynamically different groups of states the set of active resonances and accordingly the organization center and the transverse and longitudinal directions are different. All the details of the method are explained in ref 2. It is clear that this method is easy to apply when the actual eigenstates come close to ideal ladder states of some organization center. Unfortunately, in many cases the eigenstates contain dirt that makes them visually appear different from what we expect for ideal ladder states. Sometimes by near-accidental degeneracy, eigenstates come close to sums and differences of ideal ladder states. In such cases, to understand the nature of these states it is necessary to decompose the given eigenstate into its ideal ladder state content. In the past we have found such cases and demixed them by trial and error, or in the simplest cases we have just been satisfied with forming the sum and the difference of the mixed eigenstates. For an example, see Figure 1 in ref 5 or Figure 1 in ref 2; these two figures are essentially the same. The results of this demixing have sometimes been rather clean and satisfactory; in other cases this unsystematic demixing did not lead to any useful result. Therefore, the basic problem treated in the present paper is how to extract the content of any given eigenstate with respect to some known organizing center. The procedure will be a cleaning and decomposition of the wave function by projection. In many systems we have investigated in the past we found that some classes of states possess approximate symmetries that are not exact symmetries of the Hamiltonian. Usually in physics symmetries are important and should be taken advantage of. Therefore, we will include into the method the use of approximate symmetries of the system. We do it by including into the projection onto some organization center mentioned above also a projection onto one of the representations of the approximate symmetry. As an example, we use an algebraic

10.1021/jp101629s  2010 American Chemical Society Published on Web 05/11/2010

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model for the chromophore dynamics of the H atom in the molecule CF3CHFI; the model will be presented in detail in the next section. 2. CF3CHFI, The Sample System 2.1. Hamiltonian. An algebraic fitted Hamiltonian for the motion of the H atom in the molecule CF3CHFI has been published in ref 6. It has four modes labeled by the indices s, a, b, and f, where s is the stretch mode of the H atom, a and b are the two bend modes of the H atom, and f is the only other mode of the whole molecule that comes into low order resonance with the modes of the H atom. The Hamiltonian is given in terms of harmonic creation and destruction operators a†j and aj. It is convenient to split the Hamiltonian as

H ) H0 + W

(1)

H0 )

∑ ωjaj†aj + ∑ xj,maj†ajam† am j

(2)

jem

which is diagonal in number state representation, where the indices j and m run over all four degrees of freedom and into a part

∑k

ψa ) φa - φs /2,

ψb ) φb - φs /2,

† † s,j,m(asaj am

+ as†ajam)/81/2 + γ(aaaaa†ba†b + a†aa†aabab)/2 +

jem

containing all resonant interactions. In the sum in eq 3 the indices j and m run over the degrees of freedom a, b, f. The numerical values of all the coefficients can be found in ref 6. It is checked easily that the following combination P of number operators is a conserved quantity

P ) as†as + (a†aaa + a†bab + af†af)/2

(4)

P is called the polyad number, and it is a measure of the total excitation of the molecule. For a systematic method to find all the conserved combinations of excitation numbers of algebraic Hamiltonians, see refs 7 and 8. The Hamiltonian matrix in number state basis is block diagonal with one block belonging to each eigenvalue of P. Each block represents one polyad of the system and can be analyzed as if it would be an independent system. The polyads are finite dimensional, that is, they are spanned by a finite set of basis functions. Let us call HP the Hilbert space that belongs to the polyad P. Later we will analyze some eigenstates belonging to the eigenvalue 5 of P. This is the same polyad as the one investigated in ref 9, therefore we can compare results. The corresponding classical Hamiltonian is obtained by the application of the usual correspondence relation10

aj† T √Ijexp(-iφj);

aj T √Ijexp(iφj)

(5)

after bringing all operators into symmetric order. Here Ij and φj are action/angle variables in the classical phase space of

∑ cn(k),n ,n ,n |ns, na, nb, nf〉 s

a

b

f

(7)

where the sum runs over all basis states in the polyad, then the semiclassical wave function according to ref 3 is given as

Ψk(φs, φa, φb, φf) )

∑ cn(k),n ,n ,n exp[i(nsφs + naφa + s

a

b

f

nbφb + nfφf)] (8)

On the reduced configuration space the wave function is given up to a global phase factor by

δ(aaaaaf†af† + a†aa†aafaf)/2 + ε(ababaf†af† + a†ba†bafaf)/2

(3)

ψf ) φf - φs /2 (6)

The reduced configuration space is a three-dimensional torus with the coordinates ψj. From the operator Hamiltonian the Hamiltonian matrix is constructed in the number basis, where we note a basis state as |ns, na, nb, nf〉. If an eigenstate of H is given in number basis as

|Ψk〉 )

into a part

W)

dimension 8. However, in the present paper we will not study the classical dynamics. We can use the existence of the conserved quantity P to reduce the system to one with three degrees of freedom and a six-dimensional phase space. In this reduced phase space we use the action variables Ja ) Ia, Jb ) Ib, Jf ) If, and the conjugate angle variables

Ψk(ψa, ψb, ψf) )

∑ cn(k),n ,n ,n exp[i(naψa + nbψb + nfψf)] s

a

b

f

(9) This function is the object that in the method of ref 2 is studied in order to obtain the classification and the assignment of the state. 2.2. Approximate Symmetry. The experience from ref 9 shows that at the lower end of the polyad 5 the eigenstates of the complete Hamiltonian are almost pure eigenstates of ns ) as†as. This indicates that all the various Fermi resonances of the Hamiltonian have little effect; they are the only ones that could cause a mixing of basis states with different quantum numbers ns. Therefore, for these states at the lower end of the polyad, all the Fermi resonances between mode s and the other three modes only act as small perturbations, but they are not essential for the formation of the qualitative properties of the eigenstates. That is, the qualitative structure of these eigenstates is the same as the one we would find for dropping the Fermi resonances in the Hamiltonian. Dropping these parts of H (or even dropping the mixed Fermi resonances only, i.e., the ones involving two different ones of the modes a, b, and f) has an interesting consequence: The remaining Darling Dennison resonances (and also the nonmixed Fermi resonances, i.e., the Fermi resonances involving only a single one of the modes a, b, and f) are only able to shift the occupation numbers of the modes a, b, and f by two units, that is, they can only connect basis states with even values of these quantum numbers with other even values, and odd values only with other odd values. Accordingly, after dropping the mixed Fermi resonances, the eigenstates become either even or odd in the quantum numbers na, nb, and nf. For the following it is sufficient to indicate the parity of na and nb, since the one in nf follows automatically by the knowledge of

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the polyad number. In the following we label states even/odd in na and even/odd in nb as symmetry sectors ++, +-, -+, and -- respectively. The approximate constancy of ns implies that these states have one-dimensional fibers as organization centers and have a two-dimensional transverse space. In the following we use the symbols HP, s,++/+-/-+/-- for the subspaces of HP which belong to the value s of ns and to the symmetry sectors ++/+-/-+/-- respectively. By Πs,++/+-/-+/-- we denote the projection operators onto these subspaces. Functions that belong exactly to one of the symmetry classes are either invariant or change sign under a corresponding translation of the function in ψa or ψb direction by π. The density is invariant in any case. According to what has been mentioned above we expect that at the lower end of the polyad the eigenstates of the full Hamiltonian are approximately constant in ns and as consequence belong to one of the four symmetry classes with good approximation. Then, a classification and assignment of the states should be done along the following line: First, we check on which one of these subspaces any given state has a high weight. This provides the value of ns and the symmetry type of the state. Second, we project the wave function into the subspace of high weight found in the first step and investigate its properties in the two transverse directions to obtain two transverse excitation numbers ta and tb. In total with P, ns, ta, tb, and the symmetry type, we have a complete assignment for a state of the given system with four degrees of freedom. As shown in ref 9, the polyad 5 of CF3CHFI has also another class of states that shows strong coupling of the stretch with the modes a and f. This other class of states does not have the approximate symmetry described in the present subsection. This demonstrates two properties of the system that are worth being emphasized very clearly. First, it shows that the approximate symmetry is a property of one particular class of states and it is not a property of the functional structure of the Hamiltonian. Second, it shows that the appearance of the approximate symmetry is not a question of total energy or of total excitation, that is, of the polyad number. In one and the same polyad we have the coexistence of states having this approximate symmetry with states that do not have it. 3. Wave Functions and Their Projection 3.1. Cleaning of a Wave Function. A typical eigenfunction Ψk at the lower end of the polyad consists of one dominant part Φk, s, ij with high weight that is an eigenstate of ns to eigenvalue s and in addition belongs to symmetry class ij, where i and j can take the values + or -, respectively, and of a rest which we call the dirt. This dirt is a mixture of contributions from various values of ns and from all the symmetry classes. We have

Φk,s,ij ) Πs,ijΨk

(10)

Now our expectation is that a density plot in transverse directions of Φk is a lot cleaner than the corresponding plot of Ψk and therefore is easier to read off the transverse excitation numbers. In many cases only from Φk is it possible to read off any transverse excitation numbers at all, since the plot of Ψk is too dirty to see anything meaningful. As a typical example, let us have a look at state 41 of polyad 5. A look at the column vector of this state in number state basis shows immediately that most big coefficients come from ns ) 0 and from the symmetry class -+. However, it also shows

Figure 1. Density of the reduced angle space wave function of state 41 in polyad 5 plotted in the cut ψf ) 0. Darker gray means higher density. The boundaries of the square should be glued together to form a 2D torus.

Figure 2. Density of the projected wave function Φ41,0,-+ in the cut ψf ) 0. The nodal lines are plotted as broken lines. Otherwise as Figure 1.

that there is considerable dirt, especially from ns ) 1 and symmetry class +-. Figure 1 shows the density of the function Ψ41(ψ) in the plane ψf ) 0. It is extremely difficult, if not impossible, to see any meaningful structure in this plot. For comparison we show in Figure 2 the corresponding plot for the function Φ41,0,-+. Immediately we see that this plot is invariant under a shift by π in either one of the two coordinate directions. This demonstrates that the projected function is of pure symmetry type. Accordingly, it is sufficient to analyze onequarter of the plot only for the transverse quantum numbers. In Figure 2 we see a clean structure of horizontal and a less-clean one of vertical nodal lines. The next question is, what is the correct center point from which to count nodal structures? Here we must refer to the organization point of the ground state functions for this class of states. As we will see in the next subsection, the ground states are organized in the cut ψf ) 0 around the point (ψa, ψb) ) (0, 0), which by symmetry is

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equivalent to the other three points (0, π), (π, 0), and (π, π). In Figure 2 it is easiest to view the point (π, π) as central point and to count one vertical nodal line and two horizontal nodal lines giving the transverse excitation numbers ta ) 1 and tb ) 2. These nodal lines are included as broken lines in the plot. Remembering that the domain of the plot should be glued together to a torus, it is clear that around the other three equivalent central points we count exactly the same transverse excitation numbers. Therefore, the classification and assignment of state 41 of polyad 5 is the following: Since the three reduced angle variables ψj (and therefore by eq 6 also the original angles φa, φb, and φf) run in rigid phase relation, the organization center is the space diagonal of the reduced configuration space (or a parallel fiber through any one of the other three equivalent points mentioned above). It belongs to symmetry class -+, and its quantum numbers are P ) 5, ns ) 0, ta ) 1, and tb ) 2. As mentioned in the Introduction, the idealized wave function has an oscillator function behavior in transverse direction. For an exact harmonic oscillator wave function the Fourier transform shows the same nodal pattern as the configuration space wave function. In our case, the configuration space is a torus and the quantized action space (the one belonging to a single polyad) is a finite lattice. Each lattice point corresponds to a single basis state in number representation, remember the semiclassical relation Ij ) nj + 1/2 for oscillating degrees of freedom. The representation of a state on this finite lattice is just the column vector of the state in number state basis as it comes out of the diagonalization procedure of the Hamiltonian matrix. The projection of a state in the lattice representation consists just in a restriction to the lattice points that belong to the subspace HP,s,ij. To see the nodal pattern in this algebraic column vector, the following considerations work successfully (they are explained in all detail in ref 11): As mentioned above, the transverse ground state is concentrated in angle space in the cut ψf ) 0 around the point (0, 0). This behavior is given, if all coefficients in the Fourier sum of eq 9 have the same sign. Exactly in this case we find maximal constructive interference between the contributing plane waves at angle values zero. In this sense, the set of coefficients is the discrete counterpart of a single humped function in action space, a behavior we expect for a ground state wave function. For transverse excited states we expect a change of sign of the coefficients along some defect lines of the lattice. Figure 3 shows this behavior for state 41. Remember that state 41 belongs to the symmetry class -+, therefore for the projected state the values of na run over odd numbers only and the values of nb over even numbers only. It is not necessary to indicate the value of nf, as it is taken care of automatically by the polyad number and by the constancy of ns for the projected state. In Figure 3 we plot the sign of the coefficients cn41s,na,nb,nf on the lattice of na and nb values, which span the relevant subspace H5,0,-+. This subspace has dimension 15, since it is spanned by 15 basis functions and correspondingly the lattice consists of 15 points. Included in Figure 3 as broken lines are the defect lines where the sign of the coefficients change. Note how the pattern of these defect lines on the number lattice coincides with the pattern of the nodal lines in angle space as shown in Figure 2. Also from Figure 3 we read off the quantum numbers ta ) 1 and tb ) 2. 3.2. Demixing by the Separation of Symmetry Representations. In the previous subsection we have assumed that the state under study comes close to belonging to one symmetry class only and accordingly having big coefficients on one subspace HP,s,++/+-/-+/-- only. However, a part of the eigenstates shows a strong mixture of contributions from two such

Jung

Figure 3. Signs of the coefficients of state Φ41,0,-+ on the lattice of the quantum numbers na and nb of the basis functions. The defect lines, here lines along which the sign changes, are included as broken lines.

subspaces. Two energetically nearby states mix because of nearaccidental degeneracy. Then, the task is to filter out the contributions coming from the two subspaces and to undo the mixing caused by accidental degeneracy. With the projection method this is rather easy to do. As representative example let us look at states 4 and 5 of polyad 5 of CF3CHFI. In Figure 4 we show the two reduced wave functions again in the cut ψf ) 0. The left part belongs to state 4, and the right part shows state 5. Both parts are constructed exactly the same way as was Figure 1. We can not detect any clear nodal pattern, and it is impossible at all to make any sense out of these two plots. Both of these states have strong contributions in the two subspaces H5,0,-+ and H5,0,++. Therefore, we study next the projected wave functions into these two subspaces. Figure 5 shows the two projections into subspace H5,0,-+, the left part shows Φ4,0,-+ and the right part shows Φ5,0,-+. Both projections are almost identical and are easy to interpret; they show the transverse ground state with ta ) 0 and tb ) 0. Figure 6 shows the two projections into the subspace H5,0,++, the left part shows Φ4,0,++ and the right part shows Φ5,0,++. Again, the two projections are almost identical, this time we see an antidiagonal nodal line through the central point. In many systems we have observed that nodal lines have the tendency to turn their direction away from the coordinate directions into diagonal or antidiagonal directions. In Figure 6, the lines of smallest density are rather close to the horizontal line lying at ψb ) π, whereas the two humps of high density are pushed apart along the diagonal direction such that the separating line between these two humps lies more in the antidiagonal direction. This is the dividing line included into the plots as a broken line. We prefer to interpret this antidiagonal line as a true nodal line because of its coincidence with the defect line of Figure 7, see below. By analyzing the whole ladder of states we see that the antidiagonal nodal lines are the logical continuation of horizontal nodal lines. Accordingly, we assign the transverse quantum numbers ta ) 0 and tb ) 1 to the symmetry sector -+ of the states 4 and 5. Also in this case we obtain the same result by a study of the sign pattern of coefficients. Figure 7 shows the sign pattern of all four projected states. The upper left part gives the sign pattern

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Figure 4. Densities of the reduced angle space wave functions of states 4 (left part) and state 5 (right part) in the cut ψf ) 0. To save space, the axis lettering is omitted. For each of the two frames, it is exactly the same as in Figure 1.

Figure 5. Density of the projections of states 4 (left part) and 5 (right part) onto the subspace H5,0,-+ plotted in the cut ψf ) 0. To save space the axis lettering is omitted. For each one of the two frames it is exactly the same as in Figure 1.

for Φ4,0,-+, the upper right for Φ5,0,-+, the lower left for Φ4,0,++, and the lower right part for Φ5,0,++. Again, the defect lines are included into the figures and again we observe the parallelism between nodal lines in the density in angle space and the defect lines on the discrete quantum number lattice. By the comparison of the various parts of Figure 7 we see immediately that the sum of the eigenstates 4 and 5 has its big weight on the symmetry sector ++ and the difference on the symmetry sector -+ . It should already have become clear that the component of states 4 and 5 belonging to the symmetry sector -+ is the transverse ground state of a ladder of states to which also the state 41, analyzed in the previous subsection, belongs. All states at the lower end of the polyad and many in the middle belong to the same class of states (i.e., are organized along fibers parallel to the space diagonal of the reduced angle space) and can be assigned easily by the same procedure, that is, by a study of the density of the projected wave function in a cut transverse to the organizing fibers and/or by a study of the sign pattern of the coefficients on the 2D lattice of na and nb basis state quantum numbers in the appropriate subspace.

4. Conclusions We have seen how the projection into an appropriate subspace of the Hilbert space cleans eigenstates and decomposes them into the components belonging to various regular ladders of states. Thereby the technique of ref 2 is refined and we obtain clean results also in many cases where the application of the method in its original form, that is, without the projection, did not lead to clear results. Of course, the decomposition into contributions from various symmetry sectors and belonging to different values of ns does not help, if two states of the same symmetry sector are mixed and if in addition both have their main contributions from the same value of ns. that is, it does not work if both ladder states belong to the same subspace HP,s,ij. In such cases one can demix the eigenstates either by a trial and error method or by use of the very sophisticated method of ref 12. In this latter method the dynamics of wave packets is analyzed to find out which eigenstates have to be superimposed with which coefficients to produce wave functions of a particular and pure dynamical behavior. Unfortunately, this method requires some numerical

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Figure 6. Density of the projections of states 4 (left part) and 5 (right part) onto the subspace H5,0,++ plotted in the cut ψf ) 0. The nodal lines are included as broken lines. To save space, the axis lettering is omitted. For each one of the two frames it is exactly the same as in Figure 1.

Figure 7. Signs of the coefficients of the projected states Φ4,0,-+ (upper left part), Φ5,0,-+ (upper right part), Φ4,0,++ (lower left part), and Φ5,0,++ (lower right part). Defect lines are included as broken lines. To save space, the axis lettering is omitted. It should be similar to the one in Figure 3 for each of the four frames. In the upper two frames the horizontal axis runs over odd integers from 1 to 9, in the lower two frames it runs over even integers from 0 to 10. The vertical axis runs over even integers from 0 to 8 in the upper frames and from 0 to 10 in the lower frames.

effort, but it is able to decompose even strong mixing of larger groups of eigenstates.

Demixing of complicated looking wave functions is not always dynamically meaningful. To appreciate this point, the

Demixing and Cleaning of Wave Functions following considerations are clarifying: There is a certain energy distance ∆E over which states have the tendency to mix strongly.13 ∆E depends mainly on the strengths of the resonances in the Hamiltonian, see the parameters ks,j,m, γ, δ, and ε in eq 3. If the average spacing of states in the spectrum is large compared to ∆E, then only occasionally do two states come close, and we recognize this near-accidental degeneracy immediately and know exactly which two states we have to demix from each other. If, however, the average spacing comes into the same order as ∆E or if it even becomes considerably smaller, then we have strong mixing in a large part of the states and, even worse, a state mixes with many other states simultaneously. In such a case, an eigenstate is a random interference of many patterns, this is what one expects for quantum states based on classical chaotic motion.14 Accordingly, we interpret such states as the quantum counterpart of chaos; any intent to demix such states and assign them is dynamically meaningless. The experience from previous work is that realistic molecular Hamitonians with few degrees of freedom rarely show this behavior. See ref 15 for interesting arguments why a large fraction of the molecular vibrational states should stay rather regular even close to the dissociation limit. There is an interesting connection of the assignment by projection of the state with the assignment by the diabatic correlation method of ref 16. The basic idea of the diabatic correlation is: The qualitative properties of a state and therefore the properties relevant for the assignment are given by a simplified Hamiltonian, where the unnecessary resonant interactions are dropped. Therefore, we can assign the states of the simplified Hamiltonian and establish a 1:1 correlation between a group of states of the simplified Hamiltonian and the corresponding group of states of the full Hamiltonian. The correlation transfers the assignment from the simplified Hamiltonian to the full Hamiltonian, at least for the relevant group of states. The connection between the two spectra is obtained by switching off the irrelevant resonances continuously and by following the diabatic energy curves. Also, in the projection method presented in this paper the same simplified Hamiltonian plays an essential role. Namely, the target spaces of our projections are the invariant subspaces of exactly the same simplified Hamiltonian, that is, the subspaces invariant under the resonances kept in the simplified Hamiltonian. They are the polyad subspaces for this simplified Hamiltonian. Switching off interactions means switching off some possibility for the mixing of basis states, and this in turn means the restriction of an eigenstates to a smaller set of basis states that can be interpreted as the projection into some subspace. Here we begin to see the natural parallelism between projection and correlation. In this sense, in our projection method the eigenstates (and not the eigenvalues) are followed under the switching off of irrelevant resonances. This analogy could be made even closer by the following modification of the projection procedure. Before we constructed the projection in one sudden step by the application of the operator Πs,ij. One could imagine to let all coefficients of a state outside of the target space HP,s,ij go to zero continuously and proportional to some parameter λ. The final result and the assignment would be the same, but the correlation aspect of the method would become more evident. Similar connections also exist with other versions of the assignment by correlation, for example with the assignment by an analysis of the level velocity under parameter variations,17,18

J. Phys. Chem. A, Vol. 114, No. 36, 2010 9699 or with the dressed basis method presented in ref 19, or with the correlation technique of ref 20 where the variation of strength parameters also of important interactions and of anharmonicity parameters is allowed in addition. At first these connections might seem surprising. However, there is the following explanation. In a semiclassical interpretation, quantum numbers are the quantum mechanically allowed values of conserved classical actions. Therefore, strictly speaking, a complete assignment only exists for integrable systems with a complete set of conserved actions. Therefore, any attempt to construct an assignment for a nonintegrable system must first construct an integrable approximation for the given system (given Hamiltonian) and then must establish some kind of correlation between the states (or at least some group of states) of the given system with states of the integrable reference system. This correlation then transfers the quantum numbers of the integrable reference systems to the corresponding states of the nonintegrable original system. In this sense all methods to assign nonintegrable systems rely at some point on some variant of correlation, and the very idea of a correlation is the unifying concept behind all attempts to assign nonintegrable systems. The ideas presented in this publication can be applied immediately to improve and supplement the results of all the investigations done by the techniques presented in ref 2. It should be emphasized that these methods are by no means restricted to the analysis of molecular vibrations. The method works without any modification for any system given by an algebraic Hamiltonian with a moderate number of degrees of freedom. Rather promising is its applications to condensates in multiwells; for the applications of the technique of ref 2 to condensates, see ref 21. Acknowledgment. This work has been supported by CONACyT under grant No. 57334 and by DGAPA under grant No. IN-107308. The author thanks Professor H. S. Taylor for stimulating discussions and one of the referees for making him aware of interesting related references. References and Notes (1) Chirikov, B. V. Phys. Rep. 1979, 52, 265. (2) Jung, C.; Taylor, H. S. J. Phys. Chem. A 2007, 111, 3047. (3) Sibert, E. L.; McCoy, A. B. J. Chem. Phys. 1996, 105, 469. (4) Diaz, A.; Jung, C. Mol. Phys. 2008, 106, 787. (5) Jung, C.; Taylor, H. S.; Atilgan, E. J. Phys. Chem. A 2002, 106, 3092. (6) Pochert, J.; Quack, M.; Stohner, J.; Willeke, M. J. Chem. Phys. 2000, 113, 2719. (7) Fried, L. E.; Ezra, G. J. J. Chem. Phys. 1987, 86, 6270. (8) Kellman, M. E. J. Chem. Phys. 1990, 93, 6630. (9) Jung, C.; Mejia-Monasterio, C.; Taylor, H. S. Phys. Chem. Chem. Phys. 2004, 6, 3069. (10) Heisenberg, W. Z. Physik 1925, 33, 879. (11) Jung C.; Taylor H. S. J. Chem. Phys., in press. (12) Davis, M. J. J. Chem. Phys. 1997, 107, 4507. (13) Benet, L.; Seligman, T. H.; Weidenmu¨ller, H. A. Phys. ReV. Lett. 1993, 71, 529. (14) Berry, M. V. J. Phys. A 1977, 10, 2083. (15) Chowdary, P. D.; Gruebele, M. Phys. ReV. Lett. 2008, 101, 250603. (16) Rose, J. P.; Kellman, M. E. J. Chem. Phys. 1996, 105, 7348. (17) Keshavamurthy, S. J. Phys. Chem. A 2001, 105, 2668. (18) Semparithi, A.; Charulatha, V.; Keshavamurthy, S. J. Chem. Phys. 2003, 118, 1146. (19) Kellman, M. E.; Dow, M. W.; Tyng, V. J. Chem. Phys. 2003, 118, 9519. (20) Diaz, A.; Jung, C. Mol. Phys. 2010, 108, 43. (21) Mossmann, S.; Jung, C. Phys. ReV. A 2006, 74, 033601.

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