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  • 1
    Electronic Resource
    Electronic Resource
    Efficient calculation of highly excited vibrational energy levels of floppy molecules: The band origins of H+3 up to 35 000 cm−1 (1994)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 100 (1994), S. 6175-6194 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Recent testing of a discrete variable representation (DVR) Lanczos product-basis method to calculate polyatomic vibrational energy levels [M. J. Bramley and T. Carrington, J. Chem. Phys. 99, 8519 (1993)] suggested that, for increasingly floppy molecules, its efficiency will be increasingly competitive with that of contracted-basis explicit-diagonalization methods if one can overcome the problem of poor Lanczos convergence caused by kinetic energy singularities. This may be accomplished through the realization that nondirect product finite basis representations (FBRs) (and the related DVRs) can be used efficiently in dynamics calculations for which the rate-determining step is the evaluation of Hamiltonian matrix–vector products, as is the case with Lanczos recursion [J. W. Tromp and G. C. Corey, J. Chem. Phys. (to be submitted); D. Lemoine and G. C. Corey, J. Chem. Phys. (to be published)]. A synthesis of these two procedures provides a near-optimally efficient variational vibrational method for molecules for which good basis contraction schemes cannot be designed, and for which the inevitable coordinate singularities require ideally a nondirect product basis. To substantiate this claim, we have performed hybrid DVR/FBR Lanczos calculations of vibrational energies of the classic floppy triatomic molecule H+3 up to near dissociation with unprecedently good convergence and unprecedently low computational cost.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.467273
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  • 2
    Electronic Resource
    Electronic Resource
    Calculation of triatomic vibrational eigenstates: Product or contracted basis sets, Lanczos or conventional eigensolvers? What is the most efficient combination? (1994)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 101 (1994), S. 8494-8507 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Numerous practical methods have been described for exact quantum calculations of vibrational eigenstates (energy levels and wave functions) for three- and four-atom molecules. Many descriptions are accompanied by bold claims of efficiency. Such claims are, unfortunately, difficult to test in the absence of fair comparisons on a single computer. The efficiency of these calculations depends above all (once the most appropriate coordinate system has been chosen) on clever choices of (i) the multidimensional basis set, and (ii) the Hamiltonian matrix eigensolver. In the first category come techniques such as the discrete variable representation (DVR) and basis contraction (also known as sequential adiabatic reduction or diagonalization truncation). In the second category, the Lanczos recursion is being increasingly applied. In a recent study taking the HCN/HNC molecule as a test case [R. A. Friesner, J. A. Bentley, M. Menou, and C. Leforestier, J. Chem. Phys. 99, 324 (1993)], reductions in computational effort of one to three orders of magnitude were found for a method combining basis contraction and Lanczos recursion, compared to one widely considered to be state of the art in which the Hamiltonian matrix is diagonalized conventionally [Z. Bacic and J. C. Light, J. Chem. Phys. 86, 3065 (1987)]. We have investigated this finding by developing a computer program which permits choosing both between direct product and two kinds of contracted basis (all derived from DVRs), and between Lanczos and conventional eigensolvers. It has been applied to the calculation of vibrational frequencies both of HCN/HCN up to 12 000 cm−1 and of H2O up to 22 000 cm−1, with a strict convergence criterion of 1 cm−1 in each case. We find the conclusions of Friesner et al. to be exaggerated: while a contracted/Lanczos method is consistently most efficient, other combinations, even the rather simple direct-product Lanczos [M. J. Bramley and T. Carrington, J. Chem. Phys. 99, 8519 (1993)], are never as much as a factor of 5 more costly.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.468110
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  • 3
    Electronic Resource
    Electronic Resource
    Calculation of vibrational (J=0) excitation energies and band intensities of formaldehyde using the recursive residue generation method (1996)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 104 (1996), S. 7807-7820 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: We use the recursive residue generation method (RRGM) with an exact kinetic energy operator to calculate vibrational excitation energies and band intensities for formaldehyde. The basis is a product of one-dimensional potential optimized discrete variable representation (PO-DVR) functions for each coordinate. We exploit the symmetry by using symmetry adapted basis functions obtained by taking linear combinations of PO-DVR functions. Our largest basis set consists of 798 600 functions (per symmetry block). The Lanczos tridiagonal representation of the Hamiltonian is generated iteratively (without constructing matrix elements explicitly) by sequential transformations. We determine a six-dimensional dipole moment function from the ab initio dipole moment values computed at the QCISD level with a 6-311++G(d,p) basis set. We converged all A1, B2 and B1 vibrational states up to the combination band with two quanta in the C–O stretch and one quantum in a C–H stretch at about 6 350 cm−1 above zero point energy. We present a simulated (J=0) infrared spectrum of CH2O for transitions from the ground state. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.471529
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  • 4
    Electronic Resource
    Electronic Resource
    Efficient calculation of rovibrational eigenstates of sequentially bonded four-atom molecules (1993)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 98 (1993), S. 1378-1397 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: A new efficient and fully symmetry-adapted finite-basis variational method for calculating the rovibrational eigenstates (J≥0) of any sequentially bonded four-atom molecule is presented. The exact kinetic-energy operator TˆVR in valence coordinates is used in a scheme of successive basis-set contractions. The success of the method is demonstrated with new results for the molecules HCCH, HOOH, and HCNO, respectively linear, nonlinear, and quasilinear. The complexity of TˆVR contributes little to the computational cost, yielding a Hamiltonian matrix whose elements can all be cheaply calculated from products of arbitrarily accurate one-dimensional integrals; the dominant cost is that of matrix diagonalization. Matrices of up to 6000×6000 are used to obtain low-lying levels converged to small fractions of 1 cm−1, even for the difficult case of HOOH. The generalization to J≥0 allows the calculation of Π states for HCCH and HCNO and effective rotational constants for HCCH, all usefully converged. For the first time nine-dimensional rovibrational wave functions for four-atom systems are calculated without dynamical approximation and with basis sets well completed in all degrees of freedom. For HCCH it is straightforward to obtain hundreds of rovibrational (J=0,1,2) levels converged to better than 1.5 cm−1, opening up the possibility of the systematic refinement of the nuclear potential function VˆN.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.464305
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  • 5
    Electronic Resource
    Electronic Resource
    A general discrete variable method to calculate vibrational energy levels of three- and four-atom molecules (1993)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 99 (1993), S. 8519-8541 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: We present a general variational method to calculate vibrational energy levels of polyatomic molecules without dynamical approximation. The method is based on a Lanczos algorithm, which does not require storage of the Hamiltonian matrix. The rate-determining step of each Lanczos iteration is the evaluation of the product of the matrix and a trial vector. We use simple product basis functions and write the Hamiltonian as a sum of factorizable terms. With n one-dimensional functions in each of f dimensions, the matrix-vector product requires no more than cnf+1 multiplications for a single term involving c coordinates. Choosing a (potential optimized) discrete variable representation (DVR) in each dimension, the potential energy matrix is diagonal. The rate-determining step is now the multiplication of a vector by the kinetic energy matrix and c is effectively (with rare exceptions) at most two. The nf+1 scaling holds for both diagonal and mixed second derivative operators. The method is directly applicable to any three-atom and any nonlinear four-atom molecule. We use a variety of coordinate systems (Jacobi, Radau, a hybrid of the two, and bond), for which the total number of factorizable terms in the exact kinetic energy operator is never large, to calculate very well-converged band origins of H2O up to 22 000 cm−1, of H+3 up to 18 000 cm−1, and of CH2O up to 5700 cm−1; and low-lying levels of H2O2. The results for CH2O are new, and those for H+3 clarify the causes of discrepancies in published work. The product basis results in very large matrices (up to 500 000×500 000 for four atoms), but the cost is within an order of magnitude of that of contracted-basis approaches using explicit diagonalization. While contracted basis approaches are molecule and Hamiltonian specific, it was possible to apply the DVR-Lanczos method to all the examples presented here with a single computer program. The principal advantage of our method is thus its generality, and in this context it is efficient, with the cost scaling slowly with basis size. It is also easily parallelized.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.465576
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  • 6
    Electronic Resource
    Electronic Resource
    On the Meyer–Botschwina–Burton potential energy surface for H3+ (1993)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 98 (1993), S. 10104-10105 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: The 87GTO/corrected H3+ potential energy surface of Meyer, Botschwina, and Burton [J. Chem. Phys. 84, 891 (1986)] has been widely used for studies of the H3+ system. It transpires that two slightly different implementations of this surface are in common use. This observation accounts for a number of small discrepancies noted in studies of the higher vibrational states of the ion.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.464402
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