Analysis of chaotic eigenfunctions by the natural expansion method

New Double Many-Body Expansion Potential Energy Surface for Ground-State HCN from a Multiproperty Fit to Accurate ab Initio Energies and Rovibrational Calculations

“…Within the DVR approximation, { d q 2 } and { F q } are, respectively, the eigenvalues and eigenvectors of the matrix where are the values of the wave function at the ( r α , R β ,θ γ ) DVR set of points corresponding to the coordinates ( r , R ,θ); see also ref . As it is well established, − the result d 1 > 1 / 2 implies a dominant function in the natural expansion and indicates that the coordinate is essentially uncoupled from the other two. Values of d 1 2 for the θ natural expansion obtained for the calculated J = 0 states are given (for the fitted levels) in the last columns of Tables and and shown graphically as a function of the vibrationally averaged angle (〈θ〉) for all calculated states in Figure .…”

“…An useful method for characterizing vibrational states is the natural expansion analysis. [74][75][76][77] Using this methodology, the wave function for each state n is expanded as a product of functions, namely when the θ coordinate is singled out, with equivalent expansions holding for the r and R coordinates. Within the DVR approximation, {d q 2 } and {F q } are, respectively, the eigenvalues and eigenvectors of the matrix where ψ n R γ are the values of the wave function at the (r R , R ,θ γ ) DVR set of points corresponding to the coordinates (r, R,θ); see also ref 78.…”

“…Note especially the many delocalized states that lie above the barrier (shown by the solid line), which poses an intricate problem to their assignment and hence to their use for obtaining the potential energy surface by inversion of the spectroscopic data. An useful method for characterizing vibrational states is the natural expansion analysis. − Using this methodology, the wave function for each state n is expanded as a product of functions, namely when the θ coordinate is singled out, with equivalent expansions holding for the r and R coordinates. Within the DVR approximation, { d q 2 } and { F q } are, respectively, the eigenvalues and eigenvectors of the matrix where are the values of the wave function at the ( r α , R β ,θ γ ) DVR set of points corresponding to the coordinates ( r , R ,θ); see also ref .…”

Highly excited vibrational eigenstates of nonlinear triatomic molecules. Application to H2O

“…For Y= 3, the (3,0,0) and ( 1,0,2) states are strongly coupled such that it is hard to distinguish between the two states simple by counting the number of nodes in each mode. Furthermore, the order of energy levels for (4,0,0) and (2 Using our wave functions we have also evaluated the expectation values of R2 and 3, from which the relative lengths of the two bonds are estimated. Namely, Although X, and X2 are not the expectation values of the individual bond lengths, the relative magnitude of Xi and X2 still give useful information regarding the assignments of local mode states among all states of the same Y.…”

Variational calculation of vibrational energies of triatomic molecules using SCF optimized modes

“…Also, the optimization of the coordinates can be improved by using more general transformations than the one considered here. In particular, hyperspherical coordinates have been proved to be very useful in this regard for two-dimensional systems [21,22]. Work along these lines is presently in progress.…”