Molecular vibrations: Iterative solution with energy selected bases

JOURNAL OF CHEMICAL PHYSICS VOLUME 118, NUMBER 8 22 FEBRUARY 2003 ARTICLES Molecular vibrations: Iterative solution with energy selected bases Hee-Seung Lee a) and John C. Light Department of Chemistry and James Franck Institute, University of Chicago, Chicago, Illinois 60637 Received 5 November 2002; accepted 27 November 2002 An efficient and accurate quantum method for the calculations of many large amplitude vibrational states of polyatomic molecules is proposed and tested on three triatomic molecules; H 2 O, SO 2, and HCN. In this approach we define zero-order reduced dimensional Hamiltonians ĥ k using minimum energy reduced dimensional potentials. The eigenfunctions and eigenvalues of ĥ k, (k) n, and (k) n, are used to form an energy selected basis ESB for the full system including all the product functions k (k) n for which (k) E cut. We show that ESB can be used efficiently in an iterative solution of the Schrödinger equation by the transformation between the ESB and the direct product quadrature grid. Application of the ESB of one-dimensional basis functions is shown to be very efficient for vibrational states of H 2 O and SO 2 up to 30 000 and 23 000 cm 1, respectively. A combined two-dimensional/one-dimensional basis is used very effectively for HCN above the isomerization energy to HNC. The present approach is shown to be substantially more efficient than either the direct product discrete variable representation DVR bases or compact bases from the DVR with the sequential diagonalization/truncation method. 2003 American Institute of Physics. DOI: 10.1063/1.1539037 I. INTRODUCTION Molecular spectroscopy is of vital importance for identification of molecules, determination of their structures, and understanding the dynamics of molecules. 1,2 Molecular structure is determined by analysis of spectra corresponding to different vibration rotation transitions near the energy minima, usually using normal mode harmonic analysis. 1 At higher energies and for floppy molecules such as van der Waals clusters spectra give dynamical information on tunneling, intramolecular vibrational energy transfer, predissociation, and other reactive events. 3,4 The support of these experimental investigations by theoretical calculations is an important subject of chemical physics. The results of these calculations provide us detailed information on the motions of atoms that constitute the molecule of interest. They also make it possible to test and modify the potential energy surface s which govern the dynamics and spectra of the molecule. During the last two decades, many numerical algorithms for the variational calculations of molecular spectra have been developed and tested for triatomic and tetra-atomic systems. 5 10 We note that alternative approaches have also been developed such as the analysis via polyads, 11,12 spectroscopic Hamiltonians, and perturbation theories. 13,14 In addition, fully group theoretical approaches have been applied to some systems. 15,16 In this paper we focus on improving the exact variational approach for highly excited states of molecules with large amplitude motions assuming the potential energy surfaces PES are known. There are three basic goals of such a Electronic mail: seung1@uchicago.edu methods for the theoretical spectroscopy of molecules with large amplitude motions; accuracy of solutions given the PES ; efficiency in terms of CPU time, memory, and program complexity ; and ease of qualitative interpretation. These goals often conflict. However, we believe the method synthesized in the following fulfills them well. The four major components of variational approaches to theoretical spectroscopy are choices of the coordinate system which may enhance separability of motions, the basis or representation, the method of evaluation of the Hamiltonian matrix or its operation on a vector, and the method of solution of the linear algebraic problem. These choices together define the full numerical algorithm and determine both its efficiency and accuracy. Our primary contribution in this paper is to suggest a general approach to choosing better representations given the coordinate system. Other major contributions are to show that these representations can be utilized with very efficient methods of solution, and that they often lead to reasonable qualitative interpretation. In this section we comment on the common choices for each of the above-mentioned components and indicate our choices. A more complete discussion and the algorithms are given in Sec. II. In this paper we make the common choice of orthogonal curvilinear coordinate systems. Separability of the potential depends on the specific choice of coordinates and the system. We examine cases where the separability is both relatively good and relatively poor. The use of nonorthogonal coordinates may improve separability and has been explored and used quite successfully 8,17 at the cost of more complex kinetic energy operators. Our primary concern in this paper is the choice of basis 0021-9606/2003/118(8)/3458/12/$20.00 3458 2003 American Institute of Physics

J. Chem. Phys., Vol. 118, No. 8, 22 February 2003 Molecular vibrations 3459 representation and we briefly discuss the common choices and problems. In any representation, the desirable characteristics of a basis for accurate and efficient solutions for large amplitude vibrational motions are as follows. 1 The basis must be inclusive of all solutions up to the desired energy E max. Another way of saying this is that the basis for each coordinate or group of coordinates must cover the phase space up to E max, 18,19 i.e., have a sufficient number of functions over a sufficient coordinate range to represent the motions of the system in that degree of freedom up to E max. A basis function for the full system consists of a product of functions for each coordinate. 2 In this basis the Hamiltonian or the operation of the Hamiltonian on a vector, including both kinetic and potential energy terms, must be evaluated simply, efficiently, and accurately. 3 The basis must be as small as possible while satisfying 1 and 2. For simplicity we use an orthogonal set of basis functions although alternatives such as distributed Gaussian bases have been used 20 22. For large amplitude motions normal coordinates and the associated harmonic bases are not usually appropriate. The most direct approach is to choose a set of functions often based on orthogonal polynomials for each internal distance coordinate and a coupled angular basis for the angles. The kinetic energy matrix elements are usually given analytically and the potential matrix elements are evaluated by a quadrature approximation. This approach is often called the variational basis representation VBR. 5 8 The resulting dense Hamiltonian matrix, if small enough, is diagonalized by a direct Householder s method. On the other hand, one can use the discrete variable representation DVR. 7,23 25 The major advantage of the DVR is in its simplicity. No explicit integration is necessary to compute the matrix elements. In addition, the DVR for multidimensional systems leads to very sparse and structured Hamiltonian matrices, which facilitates the use of iterative eigensolvers, such as Lanczos. 26 DVRs in distances may be combined with coupled angular bases. 27,28 For both VBR and DVR, a direct product basis is normally employed. However, there is one critical hurdle associated with the direct diagonalization of any direct product basis approach and that is the unfavorable scaling (N 2 and N 3 for memory and CPU time, respectively with respect to the number of basis functions, N n d, where n is some average number of basis function per coordinate and d is the dimensionality of the system. This limits the feasibility of calculations for either the high internal energy regime increasing n or larger molecules increasing d. Thus researchers have made substantial efforts to find ways to overcome these difficulties. These include 1 constructing highly correlated multidimensional basis to reduce the required number of basis functions 8,22,29 33 and 2 developing efficient algorithms to calculate the eigenpairs of large Hamiltonian matrices. 9,34 40 Developing a correlated basis inarguably requires the knowledge of the underlying PES of the system at hand. Well-known procedures associated with DVR include the sequential diagonalization and truncation SDT method 7,17,29,41,42 and the potential optimized DVR PODVR. 9,30,31,43 For the SDT method, however, the size of the matrix eventually precludes the solution by the direct diagonalization method for highly excited states and the compact eigenfunctions are difficult to use in iterative methods. The basic idea behind the PODVR is to reduce the number of DVR points by determining them from eigenfunctions of the model one-dimensional 1D potential for each coordinate. However, the PODVR method is not generally efficient when the potential is poorly separable. Both DVR and PODVR are not strictly variational as the quality of the quadrature depends on the size of the basis. 44 Contracted bases can also be used for a fully variational calculation VBR. One first defines zero-order Hamiltonians for each coordinate or group of coordinates and uses the direct product of truncated sets of eigenfunctions of the zeroorder Hamiltonians as the basis. Since the basis functions are tailored for the system potential, fewer basis functions may be required to achieve the desired level of accuracy. This type of contracted basis scheme has been extensively used by Handy and co-workers 6,8 and more recently with iterative methods by Wang and Carrington. 40 Zou and Bowman very recently 45 determined the vinylidene/acetylene isomerization states by using a direct product of basis functions of two stretch coordinates and a subset of high energy fourdimensional functions one stretch and three angles for the vinylidene/acetylene isomerization energy range. While the states were not converged to spectroscopic accuracy some nevertheless showed clearly the characteristics of vinylidene states. These contracted basis approaches, however, face similar kinds of problems as the SDT method if one is interested in accurate calculations for highly excited vibrational states. The purpose of this paper is to test simple and efficient numerical approaches for the computations of vibrational energies and wave functions of triatomic molecules up to very high internal energies. The method is based on correlated multidimensional basis functions and an iterative eigensolver. The procedures presented here are not limited to the triatomic systems and can be extended to larger systems. In our method we divide the coordinates into subgroups q k which contain one or more degrees of freedom each. We then define a zero-order or model Hamiltonian for each subgroup which contains model or exact kinetic energy operators and a model potential, Vˆ 0 (q k ). One key to this work is that each model potential is defined as the minimum potential for given values of the subgroup coordinates q k when varied over all other coordinates. By this definition the true potential is always greater than or equal to the model potential for a given value of q k. The basis functions for the full system are obtained as products of eigenfunctions of the zero-order Hamiltonians. However we use a selected set of product functions, not the direct product. We use an energy cut-off criterion to select the basis functions. The resulting Hamiltonian matrix is not sparse, but full, mainly due to the residual potential coupling. Diagonalization of the Hamiltonian matrix or extraction of eigenpairs is carried out by the Lanczos method. The Lanczos method has recently been widely used for calcula-

3460 J. Chem. Phys., Vol. 118, No. 8, 22 February 2003 H.-S. Lee and J. C. Light tions of molecular vibrational energy levels, but has been applied relatively rarely to problems where contracted or correlated basis functions are used 32,33,38,40 because of the poor scaling for the matrix-vector product. In this work, the action of the Hamiltonian on a vector is carried out by the transformation to a direct product grid representation. We show in the following that the contracted bases used in the present work, which are neither direct products nor the DVR functions, are much better suited for an efficient matrix vector product than those obtained from the SDT procedure. 32,40 This has, in fact, been used before, mostly for angles with spherical harmonic bases. 46 49 Among the variants of the Lanczos algorithm, we used the implicitly restarted Lanczos method IRLM of Sorensen, 50 which is implemented in ARPACK. 51 In addition to the accelerated convergence, ARPACK requires little extra work to obtain eigenvectors as well as eigenvalues at the cost of more memory. 52 It is very important to have the ability to obtain information on wave functions easily because they are essential for the assignment of spectra. We will show that the present method leads to a fairly compact representation of the Hamiltonian, requires a relatively small amount of core memory and CPU time and leads to high accuracy. We also note that alternative iterative methods to look at specific energy ranges 34 36 or to accelerate convergence by spectral transforms 38,39,53 may also be used if necessary. The main focus of this paper is to provide a better representation both in size and spectral range for iterative solution, not to investigate the various iterative solution methods. The remainder of this paper is organized as follow. In Sec. II, we present the algorithm and relevant theories. We first describe the idea for the general N-atom systems and then explain the details of the algorithms for the triatomic systems. In Sec. III, the applications of the present algorithm to H 2 O, SO 2, and HCN molecules are discussed and the comparisons with other correlated basis set approaches are made. We conclude in Sec. IV. II. THEORY The basic idea we want to pursue is the fact that for a polyatomic system we can divide the vibrational modes into several groups, where the couplings between vibrational modes in different groups are weak. One popular way is to divide the vibrational modes into two groups, stretches and bends, and construct the contracted basis as the direct product of the eigenfunctions of the stretch and bend Hamiltonians. 8,40,54 However, dividing the system into bends and stretches is not necessarily the best way to solve the problem for two reasons. First, the coupling may be strong between the bend and the stretch coordinates, requiring a large direct product basis. Second, if the coupling is weak, a direct product basis is not required. Rather an energy selected basis including only product basis functions, 0 0 m,stretch n,bend m,stretch, for which n,bend E cut should be very accurate. In this section, we present an energy selected basis set contraction scheme and show how it may be used with the Lanczos diagonalization. Some of the ideas have been suggested earlier, but have not been used extensively. Detailed numerical algorithms are given for the applications to triatomic molecules. A. General In the present work, we consider the vibrational Hamiltonian (J 0) for an N-atom system in an orthogonal coordinate system 55,56 of dimension d. We first divide the coordinates into n d groups (n d d) and define a zero-order Hamiltonian for each group. The kth group may be a single dimension or include several coordinates q k that may have both stretch and bend coordinates. The Hamiltonian can be written in terms of the zero-order Hamiltonians as Ĥ k 1 n d ĥ k Tˆ Vˆ, 1 n d Vˆ Vˆ Vˆ 0 q k, 2 k 1 where ĥ k is the zero-order Hamiltonian for the kth group and Vˆ 0 (q k ) is the reference potential corresponding to ĥ k. The Tˆ refers to the remaining terms. Note that for an orthogonal coordinate system the vibrational Hamiltonian does not have any mixed derivative with respect to the radial and angular coordinates. 55,56 Therefore, all the derivatives with respect to the radial coordinates are included in ĥ k and the Tˆ term may include derivatives with respect to the angles only. 40 In order to define basis functions, we first solve the Schrödinger equations for the n d zero-order Hamiltonians. The n th k eigenfunction and eigenvalue for ĥ k are denoted by (k) nk and (k) nk, respectively, i.e., ĥ k k nk q k k nk k nk q k. 3 The eigenfunctions, (k) nk (q k ), can be represented in primitive direct product DVR. The multidimensional correlated basis functions are then defined as energy selected products of eigenfunctions of zero-order Hamiltonians and the ith wave function of the system is expanded as i i C n1,n n 1,n 2,...,n 2,...,n nd nd n d k 1 k nk q k, (i) where C n1,n 2,...,n are expansion coefficients. nd In the above-mentioned expansion, we include only those product basis functions that satisfy the following energy cutoff criterion; n d k 1 k nk E cut. Therefore, the contracted basis functions used in the present work are product bases functions, but only a subset of the direct products of eigenfunctions of the zero-order Hamiltonians. We will refer to this set of basis functions as an energy selected basis ESB. The value of E cut depends on the energy of the highest vibrational state that we want to compute as well as the desired accuracy of the calculation. The energy cut-off criterion, Eq. 5, used for determining the 4 5

J. Chem. Phys., Vol. 118, No. 8, 22 February 2003 Molecular vibrations 3461 ESB excludes less important functions systematically in the expansion of wave function, Eq. 4, which, in turn, reduces the size of the Hamiltonian matrix. Since the excluded functions have unnecessarily high vibrational excitation in some of the coordinates for our energy range of interest, the ESB reduces the spectral range of the representation, which is desirable as far as the convergence of the Lanczos recursion is concerned. In fact, if the couplings among the vibrational modes are weak, the largest eigenvalue of the Hamiltonian matrix given by the ESB is not much different from E cut see Sec. III A. Similar approaches that limit the size of the Hamiltonian matrix have been suggested, 57 59 but they have not been used extensively, especially with iterative diagonalization. One very important factor that affects the efficiency of the basis is the choice of the reference potentials, Vˆ 0 (q k ), in Eq. 2. Ideally, one should choose reference potentials that generate the most efficient multidimensional basis in the sense that it requires the least number of basis functions for a given accuracy. Often, for a PODVR basis or for generating a contracted basis, the reference potential for a given coordinate is obtained from Vˆ by freezing all other coordinates at the equilibrium geometry 8,40 denoted as Vˆ e hereafter. However, we found that a better choice is the minimum value of the potential with respect to all other coordinates, i.e., Vˆ 0 min q k Vˆ q k,q k, 6 where q min k are the values of other coordinates at the minimum of the potential for given values of q k see Sec. III B. By defining the reference potentials in this way, the basis functions are not excluded from any region of the configuration space that might be sampled in the energy range of interest. We should note that we only need to evaluate the values of the potential once at the grid points to obtain the the values of all reference potentials at the primitive DVR points. Using Vˆ 0 instead of Vˆ e for the reference potentials becomes more important as the energy range of interest increases. In fact, the choice of reference potentials as defined in Eq. 6 serves as a zero-order approximation for the derivation of the optimized one-dimensional 1D reference potentials for the direct product PODVR basis in the context of quasiclassical phase space optimization method. 18 The importance of using Vˆ 0 in the definition of a contracted basis was recognized by Carter and Meyer, 58 but has not been used often. Calculations using contracted basis have usually focused on the lower end of the spectrum, where Vˆ e is expected to work reasonably well. The matrix elements for Tˆ and Vˆ can be evaluated in the primitive DVR basis, which is equivalent to utilizing the quadrature approximation. As noted earlier, we used a Lanczos method to diagonalize the Hamiltonian matrix, namely the IRLM for ARPACK. 51 The Lanczos method requires the action of Hamiltonian matrix onto a vector. The matrix vector product involving the Vˆ term can be performed without explicit calculation of the matrix elements. This is done by transforming the Lanczos vector in the ESB representation to a direct product DVR, where the potential matrix elements are diagonal. For the Tˆ term in Eq. 1, we take advantage of the factorizability of the Hamiltonian. 9 In other words, the matrix elements for Tˆ can be expressed as products of terms that depend only on one of the (k) nk (q k ) s. Therefore, we can evaluate the matrix elements for Tˆ from a series of smaller matrices that are calculated before the matrix vector routine is called. For a larger system four and five atoms, the number of primitive DVR points can be very large. This will increase the time required for a matrix vector product dramatically. In those cases, it would be better to use PODVR as the primitive basis. 40 In the following, we consider the applications of the present method to the triatomic systems and present a more detailed discussion on the matrix vector product. B. Triatomics 1. Products of one-dimensional functions The quantum mechanical vibrational Hamiltonian for a triatomic molecule 1 in orthogonal coordinates, such as Jacobi or Radau, is given by Ĥ 1 2 1 2 r 1 2 1 2 2 2 r 2 2 f r 1,r 2 1 sin sin Vˆ r 1,r 2,, 7 where the definitions of reduced masses, ( 1, 2 ), coordinates, (r 1,r 2, ), and f (r 1,r 2 ) depend on the particular choice of coordinate system. 60,61 As the simplest application of the present method, we use 1D zero-order Hamiltonians for all three coordinates for H 2 O and SO 2 molecules see Secs. III A and III B. In this case the zero-order Hamiltonians become ĥ j 1 2 2 Vˆ 0 r 2 j r j j 1,2, j 8 ĥ 3 f r e 1,r e 1 2 sin sin Vˆ 0, where (r e 1,r e 2 ) are the fixed values of (r 1,r 2 ), usually at the equilibrium geometry. For this particular partitioning of the Hamiltonian, we can remove the Tˆ term in Eq. 1 by rearranging the Hamiltonian as Ĥ ĥ 1 ĥ 2 f r 1,r 2 f r e 1,r e 2 ĥ 3 Vˆ, 9 Vˆ Vˆ r 1,r 2, Vˆ 0 r 1 Vˆ 0 r 2 f r 1,r 2 Vˆ f r e 1,r e 0. 2 10 The three-dimensional basis functions are then defined as products of eigenfunctions of zero-order Hamiltonians with the energy cutoff criterion. The eigenfunctions for each coordinate, (k) n (q k ), are represented in the primitive DVR functions and they are denoted as T (k) n, where is the index for the primitive DVR functions. The Hamiltonian matrix has a simple structure and its elements are given by

3462 J. Chem. Phys., Vol. 118, No. 8, 22 February 2003 H.-S. Lee and J. C. Light n,m,l Ĥ n,m,l n n m m l l n 1 m 2 l l l 3 n,m f r 1,r 2 n,m f r 1 e,r 2 e n,m,l Vˆ n,m,l. 11 From Eq. 11, we can divide the matrix vector product into three parts. Since the first term in Eq. 11 is diagonal and is obtained from Eq. 3, its contribution to the cost of matrix vector product is trivial. For the third term, we evaluate the matrix vector product by transforming the vector in the ESB representation to the primitive DVR. For the second term, we may explicitly calculate n,m f (r 1,r 2 ) n,m and save it before the matrix vector product routine is called. Alternatively, we may combine the second and third terms in the matrix vector product routine without explicit calculation of matrix elements. We followed the latter approach as explained in the following. The matrix vector products for the Vˆ and f (r 1,r 2 ) terms are evaluated through the pseudospectral type transformations of the Lanczos vector to the primitive DVR and back. 46,47,62,63 Explicitly, the (n,m,l ) element of the matrix vector product is given by n u n m l 1 n 2 m v n m l 3 f r n,r l f r e 1,r e 2 l max M l 2 T l l 3 T m m N m,l n n 1 2 T n T m T 3 V r l,r, T 1 n v nml, 12 where (n,n,n ) and (r,r, ) are the number and the location of primitive DVR points for (r 1,r 2, ), respectively. Note that the upper limits for the summations over n and m depend on (m,l) and l, respectively, due to the nondirect product nature of the basis. The limit of the summations, N(m,l) and M(l), may be stored in small arrays before the matrix vector routine is called. In practice, the summation is carried out sequentially from the right to the left of Eq. 12. In other words, the summation over n is carried out first for all values of (,m,l), which transform the vector v into the DVR for r 1 and the (2) m (r 2 ) (3) l ( ) representation for (r 2, ). The subsequent summation over m for all values of (,,l) transforms the vector in the (,m,l) representation to that in (,,l) representation. Once the vector is transformed into the primitive DVR for all three coordinates after summing over (n,m,l), the multiplication with the diagonal potential matrix, V(r,r, ), is performed. The results are transformed back to the ESB representation by summing over the primitive DVR points for each (n,m,l ) element of the u vector. Since the ESB is a nondirect product basis, it is not trivial to calculate the exact cost of a matrix vector product. Since the cost is dominated by the Vˆ term, it scales roughly as n n n l max if we preserve the order of transformation as written in Eq. 12. For a more detailed discussion on the possible improvements in the matrix vector product of this kind readers may refer to the work of Carrington and co-workers. 40 2. Products of two-dimensional and one-dimensional functions For a system that has strong bend/stretch couplings, the product basis of 1D eigenfunctions of bend and stretch Hamiltonians becomes inefficient and the spectral range of the Hamiltonian matrix may become very large. In one limiting case, the molecule may isomerize to another geometry within the energy range of interest. One such case is the HCN HNC isomerization See. Sec. III C. For the HCN molecule, we divide the system into two parts, H CN stretch/bend 2D and the C N stretch. The two-dimensional 2D zero-order Hamiltonian for the H CN stretch/bend and the 1D zero-order Hamiltonian for the C N stretch are defined as ĥ 1 1 2 Vˆ 2 r r 2 0 r, ĥ 2 1 2 1 2 R R 2 f r,r e sin sin 13 Vˆ 0 R,, where (R,r, ) are the usual Jacobi coordinates defined as the distance between H and the center of mass of CN R, the CN bond distance distance r and the angle between the two radial vectors where 0 refers to the linear HCN geometry. A similar partitioning of the Hamiltonian was used by Bowman et al. for the HOCl molecule, but they used Vˆ e and the direct diagonalization method. 64 The 2D reference potential, Vˆ 2 0 (R, ), is obtained at the DVR points in (R, ) by minimizing the full potential with respect to r. The ESB functions are then defined using the corresponding energy cut-off criterion; n 1 l 2 E cut, 14 where n (1) and l (2) are eigenvalues of ĥ 1 (r) and ĥ 2 (R, ), respectively. The matrix elements in the ESB become slightly more complicated than Eq. 11 and they are given by n,l Ĥ n,l n n l l n 1 l 2 l 1 sin sin n,l V n,l. n 1 2 r r n 2 l 15 In Eq. 15, the Tˆ term leads to a full matrix as well as the Vˆ term. However, as noted in the previous section, it is a product of two parts and each part depends on either n (1) or l (2). Therefore, they can be evaluated and saved separately before the matrix-vector product routine is called. The matrix-vector product can be performed in a similar fashion as describe in Eq. 12.

J. Chem. Phys., Vol. 118, No. 8, 22 February 2003 Molecular vibrations 3463 TABLE I. Convergence of vibrational energy levels of H 2 O below 30 000 cm 1 261 states. The results for basis set I IV are obtained from the present algorithm, whereas the sequential diagonalization and truncation SDT is used for the calculations with basis set V VII. max is the maximum deviation from the basis set converged results. N basis in SDT calculations is the size of final 3D Hamiltonian matrix in truncated basis. Basis N basis a E cut /cm 1 ) N mv b EV max (/cm 1 ) c max (/cm 1 ) I 1362 50 000 919 52 813.1 0.34 II 1902 55 000 1064 58 932.7 0.11 III 2583 60 000 1074 64 657.3 0.048 IV 3436 65 000 1219 71 102.1 0.024 E1D cut /cm 1 ) d E2D cut /cm 1 ) e V 1789 90 000 40 800 0.36 VI 2103 90 000 43 000 0.10 VII 2604 93 000 46 000 0.046 a The number of basis functions. b The number of matrix vector product performed to obtain the lowest 261 vibrational states of H 2 O. c The largest eigenvalue of the Hamiltonian matrix. d The energy cut-off used to retain eigenstates after 1D diagonalization in SDT. e The energy cut-off used to retain eigenstates after 2D diagonalization in SDT. III. RESULTS AND DISCUSSION A. H 2 O As a first application to demonstrate the efficiency of the present algorithm, we calculated the 261 vibrational energies and wave functions of H 2 Oupto30000cm 1. 65 We used Radau coordinates for (r 1,r 2, ) in Eq. 7 and the Polyansky, Jensen, and Tennyson PJT potential 66 for Vˆ (r 1,r 2, ). We do not take advantage of the symmetry of the molecule See Sec. III B for the symmetry adaptation. Due to the large mass difference between O and H, Radau coordinates closely resemble bond coordinates. In addition, H 2 O is a classic example of a local mode molecule and there is little coupling between the two OH stretches. 67,68 For H 2 O, our 1D reference potentials in Radau coordinates, Vˆ 0, are similar to Vˆ e, which is also an indication of little intermode coupling. This local mode character makes the eigenstates of 1D reference potentials in Radau coordinates excellent bases to represent the vibrational motions of H 2 O. The most important parameter that affects the convergence is E cut. Of course, the larger E cut, the larger the product basis. The other parameters are the numbers of primitive DVR functions for each coordinate. They control the accuracy of the eigenpairs of 1D problems, Eq. 3, and the quadrature error associated with the evaluation of Vˆ matrix. We report the convergence behavior of the calculations with respect to the parameter E cut in Table I. For the 1D eigenvalue problems, Eq. 3, we used 40 Chebyshev DVR bases for the stretches, distributed in 0.55,2.25 Å, and 39 Gauss Legendre DVR basis for the angle. Four different E cut values bases I IV are tested and the results are compared with accurate basis set converged values that are obtained from three-dimensional direct product DVR calculations. The above-mentioned DVR parameters lead to a Hamiltonian matrix of size 62 400 62 400 in the direct product DVR calculation. The diagonalization of this matrix is done by ARPACK without basis set truncation. The max values in Table I are the maximum deviation of the energy levels obtained by the present algorithm from the converged results. Table I also reports the number of ESB functions (N basis ) included in the expansion of wave functions, total number of matrix-vector product performed (N mv ), and the largest eigenvalue of the ESB Hamiltonian matrix (EV max ). We first notice that the spectral range of the Hamiltonian matrix is very small. 69 The largest eigenvalue of the Hamiltonian matrix is close to the E cut value. This reflects the fact that the intermode couplings are weak in H 2 O and, as a result, the perturbation induced by Vˆ term in Eq. 11 is small. In general, a larger spectral range requires more matrix vector products in the Lanczos method. For the largest calculation in Table I set IV, ARPACK requires less than 5 matrix vector product per converged state and the CPU time for the diagonalization of the Hamiltonian matrix is less than 8 min on a Pentium III 833 MHz machine. This indicates an excellent performance of ARPACK, given the fact that there are many states separated by less than 2 cm 1 at energies below 30 000 cm 1. For the direct product DVR calculation, more than three times as many matrix vector products are needed 70 and the spectral range is in the order of 5 10 5 cm 1. The efficiency of the ESB used in the present algorithms was compared to other correlated basis set approaches, the SDT and distributed Gaussian approaches. For triatomic systems, it has been shown that the SDT is a very powerful technique to calculate rovibrational energy levels. The details of algorithms regarding the SDT procedure are described elsewhere. 7,29 To make a direct comparison with the results for basis sets I III of Table I, we performed the SDT calculations using the same DVR parameters as earlier, 40,40,39 DVR points in (r 1,r 2, ), and compare the energies with the basis set converged values. We adjusted the energy cut-off parameters for the 1D and 2D diagonalizations (E1D cut and E2D cut ) in the SDT procedure so that the max values of the SDT calculations are similar to those obtained from the calculations with basis sets I III. The results are shown in the bottom half of Table I. Comparing the SDT calculations bases V VII with those of the present algorithm basis sets I III, we see that the SDT calculations require larger threedimensional 3D Hamiltonian matrices for a given accuracy,

3464 J. Chem. Phys., Vol. 118, No. 8, 22 February 2003 H.-S. Lee and J. C. Light although the difference becomes smaller as the accuracy increases. Thus the 3D ESB used here has comparable efficiency to the basis from the SDT procedure. However, there is a large difference in the CPU cost, with the SDT calculation as much as 10 times slower. The bottleneck in smaller 3D SDT calculations is the repetitive 2D diagonalizations and transformations. For larger SDT calculations, the final diagonalization dominates. Overall, the SDT calculations 3 scale as N DVR with a small prefactor whereas the ESB with 2 Lanczos method scales better than N DVR. Note that if a more efficient grid is used, such as PODVR or the DVR based on Morse eigenfunctions, the SDT benefits relatively more than the ESB with Lanczos. We also compared with a recent quasirandom distributed Gaussian basis QDGB calculation of water, 22 using a Gauss Legendre DVR for the angle and two-dimensional distributed Gaussians for (r 1,r 2 ). Below 30 000 cm 1,we found that the calculations with bases III and I in Table I have similar accuracies to those reported earlier with bases IV and IX in Table III of Ref. 22, respectively. The total numbers of basis functions in the QDGB were 10% and 50% larger than the ESB for basis III and I, respectively. Finally, we comment on convergence with respect to the primitive DVR basis. The numbers of primitive DVR functions used in the present work are large enough to obtain the energy levels of water up to 30 000 cm 1 with accuracies specified in Table I. However, this does not necessarily means that the eigenvalues of the 1D problem, Eq. 3, which are subsequently used for constructing the 3D Hamiltonian matrix, are converged within the comparable accuracies. The (k) values used in Eq. 11 are in general accurate up to the energy range we are interested in i.e., up to E max 30 000 cm 1 for H 2 O), but not up to E cut. In fact, the (k) values between E max and E cut differ from the accurate eigenvalues by as much as several hundreds cm 1. Therefore, the number of primitive DVR functions should be determined by monitoring the convergence of the energy levels from the diagonalization of the 3D Hamiltonian matrix, not by the convergence of (k) in the 1D problem. B. SO 2 Sulfur dioxide (SO 2 ) is a prototypical heavy triatomic molecule. Due to its importance in atmospheric chemistry, extensive experimental 71 74 and theoretical 16,31,75 79 studies have been performed. In contrast to H 2 O, a normal mode description of vibrational motions works pretty well for vibrational states up to the moderate energy, which is an indication of strong coupling between the two local SO stretches. However, above 18 000 cm 1, an indication of irregularity in the spectral pattern has been found, 72,73 which implies that a normal-to-local mode transition occurs in this energy region. 78,79 These observations distinguish SO 2 from H 2 O, and therefore make SO 2 a good candidate to test the present method. In addition, due to the large atomic masses of S and O, the density of vibrational state is very large approximately 4000 states below 25 000 cm 1 and SO 2 is a challenging system for any variational method. The potential energy surface of SO 2 used in this work is FIG. 1. One-dimensional reference potentials of SO 2 for R, r, and coordinates obtained from the Kauppi and Halonen surface Ref. 77. R and r are the symmetrized Radau coordinates Eq. 16 and is the angle in the Radau coordinate system. The solid lines are Vˆ 0 as defined in Eq. 6 and the dashed lines are Vˆ e, which are obtained by freezing other coordinates at their equilibrium positions. the empirical surface of Kauppi and Halonen. 77 For SO 2,we take advantage of the permutation symmetry of O atoms and use symmetrized Radau coordinates, (R,r), which are defined in terms of Radau coordinates, (r 1,r 2 ), as R 1 2 r 1 r 2, r 1 2 r 1 r 2. 16 Note that r should be smaller than R in Eq. 16. The Hamiltonian in symmetrized Radau coordinates preserves the form of Eq. 7. In symmetrized Radau coordinates, r is the symmetry carrying coordinate and the eigenfunctions of 1D Schrödinger equation for r are either even or odd with respect to the exchange of oxygen atoms. Only product basis functions with even odd symmetry are included in the expansion of wave functions, Eq. 4, for the calculations of even odd vibrational states of SO 2. Since the symmetry adaptation only affects the selection of basis functions, the structure of the matrix that is diagonalized is intact and virtually no extra work is needed to implement the symmetry; the matrix vector product routine stays the same whether we use the symmetry or not. Figure 1 depicts the 1D reference potentials (Vˆ 0 )ofso 2 in the symmetrized Radau coordinates. For comparison, we also plot the Vˆ e obtained by freezing other coordinates at the

J. Chem. Phys., Vol. 118, No. 8, 22 February 2003 Molecular vibrations 3465 TABLE II. Convergence of vibrational energy levels of SO 2 below 12 000 cm 1 219 even and 158 odd states. Basis N basis E cut /cm 1 ) N mv EV max (/cm 1 ) max (/cm 1 ) Even states I 1444 22 000 1268 46 903.2 0.245 II 1544 22 500 1275 48 963.5 0.149 III 1644 23 000 1388 50 695.2 0.076 IV 1995 24 500 1411 57 201.5 0.011 V a 1496 25 000 1143 38 395.0 11.1 Odd states VI 1216 22 000 931 46 934.8 0.293 VII 1299 22 500 934 48 623.2 0.146 VIII 1396 23 000 1019 50 808.7 0.083 IX 1710 24 500 1116 57 361.8 0.012 a Calculation is performed by using Vˆ e. TABLE III. Convergence of the lowest 3000 vibrational energy levels of SO 2 1620 even and 1380 odd states. max is the maximum deviation in energy levels from the results of the largest calculation basis IV for the even states and basis IX for the odd states. Basis N basis E cut /cm 1 ) N mv max (/cm 1 ) Even states I 10 845 42 000 14 522 1.00 II 12 581 44 000 17 096 0.13 III 13 501 45 000 18 415 0.03 IV 14 474 46 000 19 296 Odd states V 9 927 42 000 14 080 1.36 VI 11 561 44 000 16 482 0.18 VII 12 429 45 000 17 033 0.05 IX 13 336 46 000 18 676 equilibrium geometry. The 1D potentials obtained from two methods are substantially different, especially for the r coordinate. Note that the 1D potentials for the angle are plotted between 80 and 180. The potential increases sharply as approaches to 0. We show the convergence of vibrational energies of SO 2 up to 12 000 cm 1 219 even and 158 odd states in Table II. For the primitive DVR basis, we used 50 Gauss Chebyshev DVR points for R in 1.5,2.5 Å and 40 Gauss Hermit DVR points for r in 0.47, 0.47 Å. For, we used Jacobi polynomials, P (, ) ( ), with, 199,0 to limit the 62 DVR points to the range 80 180. Four values of E cut, between 22 000 and 24 500 cm 1, are shown and compared with the basis set converged energies, obtained by diagonalizing the Hamiltonian matrix based on the direct product DVR basis with the same DVR parameters (N DVR 62 000). The max value is again the maximum deviation in the vibrational energies from the converged values. Table II shows that the convergence with respect to the number of ESB function is excellent. In order to converge the 219 even states within 0.01 cm 1, we only need 1995 basis functions. This is even better than what we saw in the H 2 O case. For H 2 O, 3411 basis functions were needed to converge 261 states within 0.02 cm 1. Furthermore, the energy levels converge very quickly as the number of basis functions increases. Increasing the number of basis functions by 40% basis IV vs basis I leads to the decrease of max by a factor of 25. In the case of water, we have to increase the size of basis by almost a factor of 3 to reduce max from 0.34 to 0.02 cm 1. Odd states show similar behavior as even states. Even with the symmetry reduction, SO 2 requires more matrix vector product per converged state than H 2 O. This simply reflects the fact that the energy level spacing of SO 2 is much smaller than that of H 2 O even after symmetrization. In addition, the largest eigenvalue obtained from a given ESB increases quite rapidly with E cut and the difference between E cut and EV max is noticeable. This implies that the potential energy surface of SO 2 is not as separable in the symmetrized Radau coordinates as that of H 2 O in Radau coordinates. The larger spectral range found in the SO 2 system is also responsible for the larger number of matrix vector products required per state compared to the H 2 O system. In Table II, we also report the result of a representative calculation with Vˆ e basis V rather than Vˆ 0. The size of basis functions in this calculation is comparable to those of bases I and II, but the E cut value is much larger and the accuracy of the calculation is significantly worse. This clearly shows that the choice of Vˆ 0 as the 1D reference potentials can have a large impact on the efficiency of calculations, especially for higher excited states. For the SO 2 molecule, we extended the calculations up to very high internal energy regime, about 23 000 cm 1. Most of the quantum calculations on SO 2 system 75 77 are limited for vibrational states below 12 000 cm 1 due to the extremely large density of states. One notable exception is the work of Guo and co-workers. 78,79 They calculated vibrational energies of SO 2 below 26 500 cm 1 4690 vibrational states using a direct product DVR basis for three symmetrized Radau coordinates and the filter diagonalization method. 34,37 The convergence of their calculation is on the order of 0.1 cm 1. In the present work, we calculated the energies and wave functions of the lowest 3000 vibrational states 1620 even states and 1380 odd states of SO 2. The energy of 3000th vibrational state is 23 104.4 cm 1. In Table III, we report the convergence of 3000 vibrational states with respect to the E cut values. For the calculations reported in Table III, we used 60,45,82 primitive DVR functions (N DVR 221 400) in (R,r, ). The convergence within 0.05 cm 1 with respect to the number of DVR functions was checked by performing a larger calculation with 64,49,85 DVR functions (N DVR 266 560) and E cut 45 000 cm 1. In Fig. 2, we plot the difference in energies from the results of the largest calculation as a function of E cut value for even vibrational states. The lines are drawn to guide eyes and have no physical meaning. Note that the plots include states between 20 000 and 23 105 cm 1 588 states and each panel has a different scale. There are 1032 even vibrational states below 20 000 cm 1 and the values of these states are smaller than 0.001 cm 1. Although the convergence seems to deteriorate quickly above 22 000 cm 1, most of the states obtained from the calculation with E cut 45 000 cm 1 Fig. 2 c are converged within 0.03 cm 1

3466 J. Chem. Phys., Vol. 118, No. 8, 22 February 2003 H.-S. Lee and J. C. Light FIG. 2. Convergence of the even vibrational energy levels of SO 2 between 20 000 and 23 105 cm 1 588 states with respect to E cut value; a E cut 42 000 cm 1, b E cut 44 000 cm 1, and c E cut 45 000 cm 1.The is the deviation of each state from the results of the largest calculation with E cut 46 000 cm 1. below 23 000 cm 1 and within 0.1 cm 1 below 24 000 cm 1. Only a small portion of the vibrational states shows slow convergence with respect to the E cut value. From Fig. 2, we also notice that the convergence is from above with respect to the number of ESB function or E cut, which implies that the present algorithm is variational. Comparing Tables III and II, we find that the efficiency of the basis functions remains excellent at this high internal energy regime. In order to converge 1620 even vibrational states within 0.1 cm 1 we need 13 501 basis functions basis III, which is comparable to the value in the lower energy regime, below 12 000 cm 1. However, the number of matrix vector products per converged state increases from 6 below 12 000 cm 1 to 10 below 23 105 cm 1 due to the decrease in energy level spacing. The spectral range increase from about 50 000 cm 1 to 3 10 5 cm 1 should also affect the convergence of Lanczos recursion. Although we calculated a very large number of vibrational states, the present algorithm requires relatively small amount of memory. The largest calculations, basis IV in Table III requires only about 500 Mbyte of core memory, where most of the core memory 80% is used for storing Lanczos vectors in ARPACK. C. HCN: A 2DÕ1D example For any system that has significant correlations between coordinates, i.e., the potential is poorly separable, no product of 1D basis functions can represent the system Hamiltonian very efficiently. Isomerizing molecules are usually examples of this. In this section, we investigated the efficiency of the present method for the HCN/HNC system which has strong correlation between the H CN stretch and bend. HCN has been the subject of numerous studies testing new computational procedures. 14,54,80 Although several newer potentials are available, 81,82 we used the Murrell, Carter, and Halonen surface. 83 As expected, the efficiency of the ESB using the three 1D zero-order Hamiltonians is not as good as for the H 2 O and SO 2 cases. In fact, more than 8000 ESB functions were required to obtain all the (J 0) vibrational energies of HCN below 15 000 cm 1 103 states within 0.1 cm 1 accuracy. In addition, the spectral range of the Hamiltonian matrix was increased dramatically, which therefore required more iterations to converge and led to an increase of CPU time for the diagonalization. The inadequacy of using 1D zero-order Hamiltonians for HCN molecule can be understood. First, the characteristics of the H CN stretch are very different in the HCN and HNC potential wells. In particular, the minima of the wells occur at substantially different values of the H CN distance. Therefore it is not possible to describe the HCN potential wells by a sum of 1D reference potentials, which is good at both geometries. This is clearly shown in Fig. 3, which plots the 1D reference potentials for HCN in Jacobi coordinates. As shown in Fig. 3 a, the bottom of the 1D potential for H CN stretch resembles the characteristics of HCN well, but the flat region around 2.0 bohr comes from the T-shape geometry. The location of the potential minimum for the H CN stretch in the HNC well is around R 2.9 a.u. and its energy is more than 3000 cm 1 higher than that of HCN geometry. Therefore, the Vˆ 0 (R), plotted in Fig. 3 a, does not properly represent any of the potential wells. Even though HCN is a floppy molecule, the majority of the vibrational states below 15 000 cm 1 are localized in one of the wells. They will therefore require large numbers of eigenfunctions of the zero-order Hamiltonian which are delocalized in R to represent them. Since the product basis of energy selected 1D functions also contain amplitudes at high energy configurations, the spectral range also increases due to the large V values. For example, the linear HCN configurations with R 2.0 a.u. are energetically inaccessible in our range of interest and Vˆ values are very large for these configurations. However, n (R) can have large probabilities around R 2.0 a.u. due to the large contributions to Vˆ 0 (R) and therefore the contribution to Vˆ from these configurations is quite large. This is much less true for H 2 O and SO 2 for which the potentials are much more separable. These problems can be removed by using a 2D zeroorder Hamiltonian for the H CN bend and stretch and 1D zero-order Hamiltonian for the CN stretch 2D/1D partition. In Table IV, we show the convergence of the vibrational energies of HCN below 20 000 cm 1 298 states obtained from calculations with the 2D/1D partition. For these calculations, we used Gauss Chebyshev DVR quadratures on 1.6,3.6 a.u. for r and 1.0,5.5 a.u. for R. A Gauss

J. Chem. Phys., Vol. 118, No. 8, 22 February 2003 Molecular vibrations 3467 FIG. 3. One-dimensional reference potentials, Vˆ 0, for HCN in Jacobi coordinates obtained from the Murrell, Carter, and Halonen surface Ref. 83. R is the distance between H and the center of mass of CN, r is the C N distance, and is the angle between two radial vectors with 0 referring to the linear HCN geometry. FIG. 4. Convergence of the vibrational energy levels of HCN below 20 000 cm 1 298 states with respect to E cut value; a E cut 40 000 cm 1, b E cut 42 000 cm 1, and c E cut 45 000 cm 1.The is the deviation of each state from the results of the largest calculation with E cut 48 000 cm 1. Legendre DVR was used for quadrature over the angle. For all calculations reported in Table IV, we use 42,40,49 DVR functions in (r,r, ). The maximum error with respect to the results of the largest calculation basis V, max, is monitored. The energy levels obtained from the basis V are converged better than 0.01 cm 1 compared to the direct product DVR calculation. Table IV shows that the efficiency of the 2D/1D ESB functions improves dramatically compared to the ESB based on 1D reference potentials for all three coordinates. To achieve a convergence within 0.1 cm 1 for 298 states, we needed fewer than 3400 basis functions, which is comparable to the H 2 O and SO 2 cases. In fact, the ESB with the 2D/1D partition is more efficient than the basis derived from the DVR/SDT procedure. From a series of SDT calculations with the same DVR, we TABLE IV. Convergence of vibrational energy levels of HCN below 20 000 cm 1 298 states. max is the maximum deviation in energy levels from the results of the largest calculation with basis V. Basis N basis E cut /cm 1 ) N mv EV max (/cm 1 ) max (/cm 1 ) I 2517 38 000 1389 60 117.2 0.398 II 2971 40 000 1558 64 990.0 0.170 III 3469 42 000 1569 70 111.6 0.074 IV 4325 45 000 1742 78 375.9 0.016 V 5348 48 000 1904 90 109.3 find that the size of the final 3D Hamiltonian matrix for the SDT calculation required to achieve the accuracy comparable to basis III in Table IV is about 4300 vs 3500 for the ESB. The convergence behavior of the states below 20 000 cm 1 is also plotted in Fig. 4. Each panel shows the difference,, in energy levels from the results of the largest calculation basis V. Most of the energy levels below 15 000 cm 1 103 states are converged to 0.01 cm 1 with E cut less than 40 000 cm 1 Fig. 3 a. Note that the scale changes by a factor of 10 between Figs. 3 a and 3 c. The use of the 2D zero-order Hamiltonian for the strongly coupled coordinates also removes the spectral range problem. As shown in Table IV, the maximum eigenvalue of the Hamiltonian matrix for a given basis calculation remains relatively small with respect to the E cut values, although it increases sharply as E cut and the basis size increases. The number of matrix vector products required to obtain the 298 converged states is also very small, roughly six matrix vector products per state. The procedure described here for HCN should also be useful for larger molecules. For many four- and five-atom systems, it is probably more efficient to divide the vibrational modes into several groups and define multidimensional zero-order Hamiltonians rather than to use 1D model Hamiltonians for all coordinates. As the dimensionality of

3468 J. Chem. Phys., Vol. 118, No. 8, 22 February 2003 H.-S. Lee and J. C. Light the zero-order Hamiltonians increases, the basis functions become more strongly correlated and lead to a smaller full dimensional Hamiltonian matrix. Recently, Wang, and Carrington 40 presented a bend/stretch contraction scheme with Lanczos diagonalization and calculated the lower end of the vibrational spectrum of the H 2 O 2 molecule. In their approach, 3D bend and 3D stretch zero-order Hamiltonians with Vˆ e not Vˆ 0 ) are first diagonalized and the direct product of bend and stretch eigenfunctions are used to solve the sixdimensional problem. For H 2 O 2, the coupling between the bend and the stretch is relatively weak and therefore the bend/stretch contraction scheme is a reasonable choice. However, as shown in the present work, more general choices of zero-order Hamiltonians, reference potentials, and ESB are advantageous for systems and energy ranges where stronger intermode couplings exist. IV. CONCLUSION In this paper we have developed from known elements a greatly improved numerical algorithm for determining accurate vibrational eigenvalues and eigenfunctions for large numbers of states of polyatomic molecules. The key components of the algorithm are as follows: 1 Determination of reduced dimensionality minimum zero order potentials, Vˆ 0 (q k ), for appropriate groups of coordinates these may be 1D, 2D or higher groupings. 2 Determination of eigenpairs for the zero-order Hamiltonians with each of these potentials. 3 Choice of an energy selected basis ESB from products of the eigenfunctions of the reduced dimensional zeroorder Hamiltonians. 4 Efficient solution of the full dimensional problem via Lanczos iteration using a direct product grid for the evaluation of the residual potential and kinetic energy terms. The advantages of 1 are that the set of eigenfunctions of the reduced dimensionality Hamiltonians below E cut span all configuration space accessible up to E cut. The basis determined in 2 and 3 is therefore inclusive, with all eigenfunctions with energies below E max well represented by the ESB. The ESB of 3 minimizes the spectral range of the Hamiltonian matrix while including all states of interest up to E cut ). This greatly improves the convergence of the Lanczos iteration. Iterative methods such as Lanczos are essential for large systems and thus the ESB with Vˆ 0 (q k ) is more amenable to iterative solution. The ESB is also very efficient with respect to the size of basis for a given accuracy. The combination of 1 3 makes the ESB tailored both to the potential and to the energy range of interest. For the three cases investigated in the present work, the ESB outperforms other correlated basis set methods, such as SDT. The fact that a smaller basis is required for the ESB than the SDT for a given accuracy is somewhat surprising since the SDT is a more aggressive contraction scheme than a direct product of simply contracted basis functions based on Vˆ e however, the SDT always leaves the last coordinate uncontracted. 6,8,40 This implies that the improvement of ESB over the simple contraction scheme is substantial: the ESB with Vˆ 0 (q k ) requires a smaller basis size and is more amenable to iterative solution than other contracted basis methods. Given the fact that a compact representation of the Hamiltonian matrix is essential for the multidimensional problem, where the feasibility depends on the size of basis, the present method should be very useful for larger systems. ACKNOWLEDGMENT We are grateful for the support of this research by a grant from the National Science Foundation, Grant No. NSF CHE- 0203101. 1 E. B. Wilson, J. C. Decius, and P. C. 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