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By doing calculations on the methane–water van der Waals complex, we demonstrate that highly converged energy levels and wavefunctions can be obtained using Wigner D basis functions and the Symmetry-Adapted Lanczos (SAL) method. The Wigner D basis is a nondirect product basis and, therefore, efficient when the kinetic energy operator has accessible singularities. The SAL method makes it possible to exploit symmetry to label energy levels and reduce the cost of the calculation, without explicitly using symmetry-adapted basis functions. Line strengths are computed, and new bands are identified. In particular, we find unusually strong transitions between states associated with the isomers of the global minimum and the secondary minimum.

Two-dimensional electron-electron double resonance (2D-ELDOR) provides extensive insight into molecular motions. Recent developments permitting experiments at higher frequencies (95 GHz) provide molecular orientational resolution, enabling a clearer description of the nature of the motions. In this work, simulations are provided for the example of domain motions within proteins that are themselves slowly tumbling in solution. These show the nature of the exchange cross-peaks that are predicted to develop in real time from such domain motions. However, we find that the existing theoretical methods for computing 2D-ELDOR experiments over a wide motional range begin to fail seriously when applied to very slow motions characteristic of proteins in solution. One reason is the failure to obtain accurate eigenvectors and eigenvalues of the complex symmetric stochastic Liouville matrices describing the experiment when computed by the efficient Lanczos algorithm in the range of very slow motion. Another, perhaps more serious, issue is that these matrices are "non-normal," such that for the very slow motional range even rigorous diagonalization algorithms do not yield the correct eigenvalues and eigenvectors. We have employed algorithms that overcome both these issues and lead to valid 2D-ELDOR predictions even for motions approaching the rigid limit. They are utilized to describe the development of cross-peaks in 2D-ELDOR at 95 GHz for a particular case of domain motion.

We presented the first three-dimensional (3D) ab initio intermolecular potential energy surface (PES) for the H3O+-Ar complex. The electronic structure computations were carried out at the explicitly correlated coupled cluster theory-F12 with an augmented correlation-consistent triple zeta basis set. Analytic 3D PES was obtained by least-squares fitting the multi-dimensional Morse/Long-Range (mdMLR) potential model to interaction energies, where the mdMLR function form was applied to the nonlinear ion-atom case for the first time. The 3D PES fitting to 1708 points has root-mean-square deviations of 0.19 cm-1 with only 108 parameters for interaction energies less than 500 cm-1. With the 3D PES of the H3O+-Ar complex, we employed the combined radial discrete variable representation/angular finite basis representation method and Lanczos algorithm to calculate rovibrational energy levels. The rotational profiles of the O-H anti-stretching vibrational bands of v3 +(S)←0+ and v3 -(A)←0- for the H3O+-Ar complex were predicted and were in good agreement with the experimental results.

An accurate ab initio ground-state intermolecular potential energy surface (PES) was determined for the CO–CO2 van der Waals dimer. The Lanczos algorithm was used to compute rovibrational energies on this PES. For both the C-in and O-in T-shaped isomers, the fundamental transition frequencies agree well with previous experimental results. We confirm that the in-plane states previously observed are geared states. In addition, we have computed and assigned many other vibrational states. The rotational constants we determine from J = 1 energy levels agree well with their experimental counterparts. Planar and out-of-plane cuts of some of the wavefunctions we compute are quite different, indicating strong coupling between the bend and torsional modes. Because the stable isomers are T-shaped, vibration along the out-of-plane coordinates is very floppy. In CO–CO2, when the molecule is out-of-plane, interconversion of the isomers is possible, but the barrier height is higher than the in-plane geared barrier height.

The rovibrational energy levels and intensities of the CH3F-H2 dimer have been obtained using our recent global intermolecular potential energy surface [X.-L. Zhang et al., J. Chem. Phys. 148, 124302 (2018)]. The Hamiltonian, basis set, and matrix elements are derived and given for a symmetric top-linear molecule complex. This approach to the generation of energy levels and wavefunctions can readily be utilized for studying the rovibrational spectra of other van der Waals complexes composed of a symmetric top molecule and a linear molecule, and may readily be extended to other complexes of nonlinear molecules and linear molecules. To confirm our method, the rovibrational levels of the H2O-H2 dimer have been computed and shown to be in good agreement with experiment and with previous theoretical results. The rovibrational Schrodinger equation has been solved using a Lanczos algorithm together with an uncoupled product basis set. As expected, dimers containing ortho-H2 are more strongly bound than dimers containing para-H2. Energies and wavefunctions of the discrete rovibrational levels of CH3F-paraH2 complexes obtained from the direct vibrationally averaged 5-dimensional potentials are in good agreement with the results of the reduced 3-dimensional adiabatic-hindered-rotor (AHR) approximation. Accurate calculations of the transition line strengths for the orthoCH3F-paraH2 complex are also carried out, and are consistent with results obtained using the AHR approximation. The microwave spectrum associated with the orthoCH3F-orthoH2 dimer has been predicted for the first time.

We report a theoretical study on the rotational spectra of Ar‐D232S. The intermolecular potential energy surface was transformed from our latest ab initio three‐dimensional potential of Ar‐H232S. The rotational energy levels and wavefunctions of the complex were calculated by using the radial discrete variable representation/angular finite basis representation method and the Lanczos algorithm. The calculated rotational transition frequencies and structural parameters were found to be in good agreement with the available experimental values.

Full-dimensional vibrational spectra are calculated for both X (H2O) and X (D2O) dimers (X = F, Cl, Br, I) at the quantum-mechanical level. The calculations are carried out on two sets of recently developed potential energy functions (PEFs), namely, Thole-type model energy (TTM-nrg) and many-body energy (MB-nrg), using the symmetry-adapted Lanczos algorithm with a product basis set including all six vibrational coordinates. Although both TTM-nrg and MB-nrg PEFs are derived from coupled-cluster single double triple-F12 data obtained in the complete basis set limit, they differ in how many-body effects are represented at short range. Specifically, while both models describe long-range interactions through the combination of two-body dispersion and many-body classical electrostatics, the relatively simple Born-Mayer functions employed in the TTM-nrg PEFs to represent short-range interactions are replaced in the MB-nrg PEFs by permutationally invariant polynomials to achieve chemical accuracy. For all dimers, the MB-nrg vibrational spectra are in close agreement with the available experimental data, correctly reproducing anharmonic and nuclear quantum effects. In contrast, the vibrational frequencies calculated with the TTM-nrg PEFs exhibit significant deviations from the experimental values. The comparison between the TTM-nrg and MB-nrg results thus reinforces the notion that an accurate representation of both short-range interactions associated with electron density overlap and long-range many-body electrostatic interactions is necessary for a correct description of hydration phenomena at the molecular level.

Exact variational calculations of vibrational energies of CH4 and CH3D are carried out using a two-layer Lanczos algorithm based on the ab initio potential energy surface of D. W. Schwenke and H. Partridge, Spectrochim. Acta, Part A 57, 887 (2001). The convergence of the calculated vibrational energies is discussed in detail. In addition, we report all well converged vibrational energy levels up to 6600 cm(-1) for CH4, and those up to 5000 cm(-1) for CH3D, respectively. These results clearly outperform previous theoretical calculations. And a comparison with experimental results available is also made.

Spectral difference methods represent the real-space Hamiltonian of a quantum system as a banded matrix which possesses the accuracy of the discrete variable representation (DVR) and the efficiency of finite differences. When applied to time-dependent quantum mechanics, spectral differences enhance the efficiency of propagation methods for evolving the Schrodinger equation. We develop a spectral difference Lanczos method which is computationally more economical than the sinc-DVR Lanczos method, the split-operator technique, and even the fast-Fourier-Transform Lanczos method. Application of fast propagation is made to quantum control theory where chirped laser pulses are designed to dissociate both diatomic and polyatomic molecules. The specificity of the chirped laser fields is also tested as a possible method for molecular identification and discrimination.

We present a method for computing bend energy levels of a five-atom molecule. We use polyspherical coordinates in terms of which both the kinetic energy operator and its associated volume element are simple. The basis functions we use are new. They are parity-adapted combinations of products of spherical harmonics and an associated Legendre function. We show that in the parity-adpated basis it is possible to evaluate matrix-vector products efficiently, despite the fact that the parity-adapted functions are not products of functions of a single variable. The method is applied to compute bend levels of methane. Within each parity block we use the symmetry-adapted Lanczos algorithm to compute levels labeled by irreps of a four-member group that contains the parity operator and the permutation of two of the four hydrogen atoms. In a future publication we shall use the bend wave functions as contracted basis functions to compute numerically exact vibrational levels of methane.

An approach for carrying out vibrationally adiabatic mixed quantum-classical molecular dynamics simulations is presented. An appropriate integration scheme is described for the vibrationally adiabatic equations of motion of a diatomic solute in a monatomic solvent and an approach for calculating the adiabatic energy levels is presented. Specifically, an iterative Lanczos algorithm with full reorthogonalization is used to solve for the lowest few vibrational eigenvalues and eigenfunctions. The eigenfunctions at one time step in a mixed quantum-classical trajectory are used to initiate the Lanczos calculation at the next time step. The basis set size is reduced by using a potential-optimized discrete variable representation. As a demonstration the problem of a homonuclear diatomic molecule in a rare gas fluid (N2 in Ar) has been treated. The approach is shown to be efficient and accurate. An important advantage of this approach is that it can be straightforwardly applied to polyatomic solutes that have mult...

It is shown how the quasiminimal residual algorithm (QMR), based on the Lanczos algorithm, can be modified to compute cross-correlation functions Cnm(E)=〈Ψn|(E−H)−1|Ψm〉 without any diagonalization by recursively updating a small number of scalars. Only three Lanczos vectors need to be stored. Several left-hand side vectors 〈Ψn| and multiple shifts E can be considered simultaneously. The new method is termed the quasiminimal recursive residue generation method (QM-RRGM) and is applied to the collinear H+H2 problem to illustrate its convergence behavior. The properties of two different formulations of the Lanczos algorithm, the usual three-term and a coupled two-term recursion, are also discussed. The QM-RRGM exhibits smooth convergence behavior, and it is shown that the stopping criteria used in the QMR algorithm can also be used for computing correlation functions.

A two-layer Lanczos algorithm is suggested to calculate the rovibrational energy levels of polyatomic molecules in terms of a partitioned Hamiltonian. Such a Hamiltonian is formed in a set of orthogonal polyspherical coordinates. This algorithm solves the full dimensional eigenvalue problem in a reduced-dimensional (RD) way. By splitting the coordinates into radial and angular groups, one obtains two small RD Hamiltonians in each coordinate group. The eigenstates of each RD system are computed using either a standard or a guided spectral transform Lanczos method. These two subsystems are exactly coupled via a set of diabatic basis functions in the angular degrees of freedom without any dynamical approximation. The two-layer Lanczos algorithm is illustrated in detail using an example of the variational calculation of the vibrational energies of pentatomic molecules. An application to methane is given. Numerical results show that the two-layer Lanczos method is substantially more efficient, compared to the conventional Lanczos algorithm.

We propose new methods for using contracted basis functions in conjunction with the Lanczos algorithm to calculate vibrational (or rovibrational) spectra. As basis functions we use products of eigenfunctions of reduced-dimension Hamiltonians obtained by freezing coordinates at equilibrium. The basis functions represent the desired wave functions well, yet are simple enough that matrix-vector products may be evaluated efficiently. The methods we suggest obviate the need to transform from the contracted to an original product basis each time a matrix-vector product is evaluated. For HOOH the most efficient of the methods we present is about an order of magnitude faster than a product basis Lanczos calculation.

Two simple algorithms for the diagonalization of a set of sparse symmetric matrices of the form A+{Δk}k=1L for large values of L are proposed and investigated. The numerical strategies economize computer resources by requiring the reconstruction of the Lanczos basis for a small number of times compared to L. Each member of the set {Δk}k=1L is assumed to have a smaller number of nonzero elements compared to A. Both numerical procedures are derived from the Lanczos algorithm and use periodically a recursion to obtain the Lanczos vectors. Tests are conducted with both random symmetric matrices and with DVR Hamiltonians containing parametric potentials. The performance of the algorithms in terms of numerical accuracy, stability, and CPU time is studied as a function of two properties of the matrix set {Δk}k=1L.

An efficient Lanczos subspace method has been devised for calculating state-to-state reaction probabilities. The method recasts the time-independent wave packet Lippmann–Schwinger equation [Kouri et al., Chem. Phys. Lett. 203, 166 (1993)] inside a tridiagonal (Lanczos) representation in which action of the causal Green’s operator is affected easily with a QR algorithm. The method is designed to yield all state-to-state reaction probabilities from a given reactant-channel wave packet using a single Lanczos subspace; the spectral properties of the tridiagonal Hamiltonian allow calculations to be undertaken at arbitrary energies within the spectral range of the initial wave packet. The method is applied to a H+O2 system (J=0), and the results indicate the approach is accurate and stable.

We present a complex-symmetric version of the preconditioned inexact spectral transform (PIST) method, for calculating resonance energies and widths. The PIST method uses an iterative linear solver to compute inexact Lanczos vectors for (EI−H)−1, and then diagonalizes the Hamiltonian in the inexact Lanczos representation. Our new version requires complex-symmetric variants of: (1) the Lanczos algorithm, (2) the linear solver, (3) the preconditioner we introduced in a previous paper [J. Chem. Phys. 114, 9254 (2001)]. The new method works extremely well for HCO, enabling us to calculate the first 17 dissociative resonances in less then 90 second of CPU time.

We present a four-dimensional ab initio potential energy surface of the H2—HCCCN complex at the coupled-cluster singles and doubles with noniterative inclusion of connected triples [CCSD(T)]-F12 level with a large basis set including an additional set of bond functions. The artificial neural networks method was extended to fit the intermolecular potential energy surface. The complex has a planar linear global minimum with the well depth of 199.366 cm−1 located at R = 5.09 A, φ = 0°, θ1 = 0°, and θ2 = 180°. An additional planar local minimum is also found with a depth of 175.579 cm−1 that is located at R = 3.37 A, φ = 0°, θ1 = 110°, and θ2 = 104°. The radial discrete variable representation/angular finite basis representation and the Lanczos algorithm were employed to calculate the rovibrational energy levels for four species of H2—HCCCN (pH2—HCCCN, oH2—HCCCN, pD2—HCCCN, and oD2—HCCCN). The rotational frequencies and spectroscopic parameters were also determined for four complexes, which agree well with th...

We report a rigorous quantum mechanical study of the rovibrational energy levels of vinyl radical C2H3. The calculations are carried out using a real two-component multi-layer Lanczos algorithm in a set of orthogonal polyspherical coordinates based on a recently developed accurate ab initio potential energy surface of C2H3. All well converged 158 vibrational bands up to 3200 cm−1 are determined, together with a comparison to previous calculations and experimental results. Results show a remarkable multi-dimensional tunneling effect on the vibrational spectra of the radical. The vibrational tunneling splitting is substantially different from that of previous reduced dimensional calculations. The rotational constants of the fundamental vibrational bands of C2H3 are also given. It was found that the rovibrational states are strongly coupled, especially among those bending vibrational modes. In addition, the perturbative iteration approach of Gruebele has been extended to assign the rovibrational energy level...

The recently proposed single Lanczos propagation method [J. Chem. Phys. 111, 9944 (1999); ibid. 114, 1467 (2001)] is extended to complex–symmetric Hamiltonians. It is shown that the complex–symmetric Lanczos algorithm possesses several useful numerical properties similar to those observed in real-symmetric cases, which enable one to compute multiple transition amplitudes with a single Lanczos propagation. The usefulness of the method is illustrated in calculating the S-matrix elements for the collinear H+H2 reaction.

We report accurate calculations of vibrational energy levels of HOOH, DOOD, and HOOD up to 10 000 cm−1 above the zero-point energy levels on a high-quality ab initio potential energy surface. These energies were determined by the Lanczos algorithm based on repetitive matrix-vector multiplication. The six-dimensional vibrational Hamiltonian in the diatom–diatom Jacobi coordinate system was discretized in a mixed basis/grid representation. A direct product potential optimized discrete variable representation was used for the radial coordinates, while nondirect product spherical harmonics were employed for the angular degrees of freedom. The calculation and storage of the potential matrix in the angular finite basis representation were avoided by using a series of one-dimensional pseudo-spectral transformations to a direct product angular coordinate grid. The diatom–diatom exchange symmetry, when applicable, was incorporated into the basis, which significantly enhanced the efficiency for symmetric isotopomer...

A reduced dimensionality (RD) reactive quantum scattering model has been presented for the general type AB+XCD3⇌ABX+CD3 reaction, where the CD3 fragment has C3V symmetry. The model Hamiltonian and quantum scattering formulas are derived in hyperspherical coordinates, together with a single hyperradius hyperspherical projection method for boundary conditions. The time-independent coupled channel equations are propagated using a log-derivative method. The surface functions are produced by a guided spectral transform (GST) Lanczos algorithm. We also describe a GST Lanczos method to solve the generalized eigenvalue problem in a nonorthogonal basis. The RD model has been applied to the OH+CH4⇌H2O+CH3 reaction. A comparison of calculated results with previous theoretical and experimental ones is made.

The recently proposed single Lanczos propagation method [J. Chem. Phys. 111, 9944 (1999)] for calculating multiple transition amplitudes is made more efficient in several aspects. It is shown that the amplitudes can be calculated without the explicit calculation and storage of the Lanczos eigenvectors, thus significantly reducing the computational costs, particularly for long propagation. It is also shown that symmetry adaptation can be implemented in the Lanczos propagation in a straightforward manner. In particular, eigenspectra in multiple irreducible representations can be determined from a single recursion by projecting the symmetrized Lanczos states at each step. The accuracy and efficiency of the improved methods are ascertained by numerical tests of realistic triatomic and tetratomic systems.

We present a symmetry-adapted Lanczos method that uses projection operators to calculate energy levels with different symmetries from a single sequence of matrix–vector products. Compared with the conventional Lanczos method, this method has the advantage that energy levels are computed more efficiently and with symmetry labels. High-lying stretching energy levels of a coupled Morse oscillator Hamiltonian describing 28SiH4 are calculated to demonstrate the advantage of the symmetry-adapted method.

We compare the numerical performance of three recursive methods for calculating collisional resonances, which are characterized by complex eigenenergies of an optical potential augmented Hamiltonian. The first approach involves a modified Chebyshev propagation of a real wave packet, followed by low-storage filter-diagonalization. A similar filter-diagonalization scheme replaces the Chebyshev propagation with a more general Faber recursion associated with a specific conformal mapping in the complex plane. The complex resonance eigenenergies are also obtained using a complex-symmetric version of the Lanczos algorithm. Numerical tests for a realistic triatomic system (HCO) indicate that the Lanczos method and the low-storage filter-diagonalization based on the Chebyshev propagation are much more efficient than the Faber approach.

We establish rigourously the scaling properties of the Lanczos process applied to an arbitrary extensive Many-Body System which is carried to convergence n to infinity and the thermodynamic limit N to infinity taken. In this limit the solution for the limiting Lanczos coefficients are found exactly and generally through two equivalent sets of equations, given initial knowledge of the exact cumulant generating function. The measure and the Orthogonal Polynomial System associated with the Lanczos process in this regime are also given explicitly. Some important representations of these Lanczos functions are given, including Taylor series expansions, and theorems controlling their general properties are proven., Comment: 25 pages Latex2e with amsmath

An iterative algorithm is developed for calculating nonlinear optical polarizabilities using a series of generalized sum rules that resemble the Lanczos algorithm and connect spectral moments of the driven single‐electron density matrix to ground state charge distributions and bonding network. The size scaling and saturation of off‐resonant polarizabilities (up to seventh order) of polyacetylene oligomers with up to 300 carbon atoms is analyzed in terms of collective electronic oscillators. Simple analytical expressions for size and bond‐length alternation dependence of off‐resonant polarizabilities are derived using a single‐oscillator approximation.

We present an efficient and robust method for the calculation of all S matrix elements (elastic, inelastic, and reactive) over an arbitrary energy range from a single real-symmetric Lanczos recursion. Our new method transforms the fundamental equations associated with Light's artificial boundary inhomogeneity approach from the primary representation (original grid or basis representation of the Hamiltonian or its function) into a single tridiagonal Lanczos representation, thereby affording an iterative version of the original algorithm with greatly superior scaling properties. The method has important advantages over existing iterative quantum dynamical scattering methods: (a) the numerically intensive matrix propagation proceeds with real symmetric algebra, which is inherently more stable than its complex symmetric counterpart; (b) no complex absorbing potential or real damping operator is required, saving much of the exterior grid space which is commonly needed to support these operators and also removing the associated parameter dependence. Test calculations are presented for the collinear H+H(2) reaction, revealing excellent performance characteristics.

In this study, a two loop iteration scheme, similar to one developed recently [Phys. Rev. E 51, 3643 (1995)], is applied to the computation of high energy vibrational eigenstates in 21‐mode planar C6H5D. The computational method is based upon the use of a spectral filter to extract a small number of eigenpairs (near the test input energy E) from the interior of the dense energy spectrum. In the outer iteration loop, a very effective filter, the Green function G(E)=(E1−H)−1, is used to drive the Lanczos recursion algorithm through a small number of steps (frequently

Four lowest‐lying transition state resonance energies and lifetimes in the three‐dimensional ClH+Cl←ClHCl→Cl+HCl reaction are reported in this paper. This is the first application of the complex coordinate method to a three‐dimensional, triatomic molecule with a double dissociation continuum, which has been handled by means of the hyperspherical coordinates. Two numerical strategies have been considered in order to make the calculations feasible. The first one consists in minimizing the dimension of the Hamiltonian matrix by prediagonalization of the basis set. This has been achieved in terms of the successive adiabatic reduction method of Bacic and Light [J. Chem. Phys. 85, 4594 (1986)], holding the hyperradius fixed. The second strategy is to compute a reduced set of eigenvalues corresponding to the resonances by using the complex Lanczos algorithm. The number of Lanczos recursions required to achieve convergence is dramatically reduced by applying the iterative scheme to the complex scaled resolvant o...

A new formalism was recently proposed to improve random phase approximation (RPA) correlation energies by including approximate exchange effects [B. Mussard et al., J. Chem. Theory Comput. 12, 2191 (2016)]. Within this framework, by keeping only the electron-hole contributions to the exchange kernel, two approximations can be obtained: An adiabatic connection analog of the second order screened exchange (AC-SOSEX) and an approximate electron-hole time-dependent Hartree-Fock (eh-TDHF). Here we show how this formalism is suitable for an efficient implementation within the plane-wave basis set. The response functions involved in the AC-SOSEX and eh-TDHF equations can indeed be compactly represented by an auxiliary basis set obtained from the diagonalization of an approximate dielectric matrix. Additionally, the explicit calculation of unoccupied states can be avoided by using density functional perturbation theory techniques and the matrix elements of dynamical response functions can be efficiently computed by applying the Lanczos algorithm. As shown by several applications to reaction energies and weakly bound dimers, the inclusion of the electron-hole kernel significantly improves the accuracy of ground-state correlation energies with respect to RPA and semi-local functionals.

In this Note, we discuss the convergence scaling law of the filter-diagonalization method based on the Chebyshev recursion, as applied to molecular spectroscopic problems. Comparison with the scaling law observed earlier for the Lanczos algorithm indicates that the two recursive methods have very similar convergence behaviors.

A nonperturbative but absolutely convergent algorithm is applied to the determination of the eigenfunctions of Hamiltonian with a singular potential of the form H=−d2/dx2+x2+λ/xα in the domain [0, ∞] which obey Dirichlet boundary conditons. The formal structure of the algorithm is identical to that of the Lanczos algorithm when it is formally extended to self‐adjoint operators.

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 ...

In a preceding paper [J. Chem. Phys. 92, 247 (1990)], we reported a preliminary three‐dimensional quantum calculation of the dissociation of the ozone molecule in the Hartley band. We present here a fully converged calculation of the autocorrelation function on the excited 1B2 potential energy surface of Sheppard and Walker. The study has been reformulated in terms of hyperspherical coordinates, in order to make use of the permutation‐inversion symmetry of the ozone molecule. The wave‐function has been discretized on a three‐dimensional hyperspherical grid. The autocorrelation function 〈φ0‖φt〉 has been computed within the time formulation, by means of the Lanczos algorithm. A calculation performed for the total angular momentum value J=0 shows a good agreement with the experimental results of Johnson and Kinsey, except for the overall intensity of the recurrence peaks. The calculated photodissociation spectrum reproduces the same oscillation pattern as observed experimentally.

We present an exact three‐dimensional quantum study of the dissociation of the ozone molecule on a single potential‐energy surface (1B2). The wave function has been discretized on a three‐dimensional grid, in conjunction with a mixed pseudospectral scheme in order to represent the action of the Hamiltonian operator on it. The correlation function (Fourier transform of the total cross section) has been computed within the time formulation. This function was obtained by means of the Lanczos algorithm, without any actual propagation of the initial wave packet on the upper surface. A calculation performed for the total angular momentum value J=0, reproduces the characteristic features of the experimental correlation function (recurrence times).

In this paper, we report vibration-rotation-tunneling levels of the van der Waals clusters N2–H2O and N2–D2O computed from an ab initio potential energy surface. The only dynamical approximation is that the monomers are rigid. We use a symmetry adapted Lanczos algorithm and an uncoupled product basis set. The pattern of the cluster’s levels is complicated by splittings caused by H–H exchange tunneling (larger splitting) and N–N exchange tunneling (smaller splitting). An interesting result that emerges from our calculation is that whereas in N2–H2O, the symmetric H–H tunnelling state is below the anti-symmetric H–H tunnelling state for both K = 0 and K = 1, the order is reversed in N2–D2O for K = 1. The only experimental splitting measurements are the D–D exchange tunneling splittings reported by Zhu et al. [J. Chem. Phys. 139, 214309 (2013)] for N2–D2O in the v2 = 1 region of D2O. Due to the inverted order of the split levels, they measure the sum of the K = 0 and K = 1 tunneling splittings, which is in excellent agreement with our calculated result. Other splittings we predict, in particular those of N2–H2O, may guide future experiments.

We investigate the Jahn–Teller (JT) effects in the doubly degenerate Hubbard model by using the exact diagonalization (Lanczos algorithm) technique. This is in connection with the newly discovered “colossal magnetoresistance” perovskite-based La1−xAxMnO3 compounds. We first obtain the parameter region where the ground state of the doubly degenerate Hubbard model is ferromagnetically ordered analytically for a dimer, and numerically for larger clusters. Then the effects of static JT distortion in the magnetic ordering of the model are studied numerically.

We present an extension of a previously reported implementation of a Lanczos-driven coupled-cluster (CC) damped linear response approach to molecules in condensed phases, where the effects of a surrounding environment are incorporated by means of the polarizable embedding formalism. We are specifically motivated by a twofold aim: (i) computation of core excitations in realistic surroundings and (ii) examination of the effect of the differential response of the environment upon excitation solely related to the CC multipliers (herein denoted the J matrix) in computations of excitation energies and transition moments of polarizable-embedded molecules. Numerical calculations demonstrate that the differential polarization of the environment due to the first-order CC multipliers provides only minor contributions to the solvatochromic shift for all transitions considered. We thus complement previous works by confirming numerically the validity of the routinely invoked neglect of the J matrix contribution as well as motivating future use of the approximation that offers a reduction of the dimensionality of the eigenvalue problem. Preliminary applications to K-edge absorption of liquid water and aqueous acrolein are presented and highlight the importance of the environment that gives rise to transition-specific shifts.

A new intermolecular potential energy surface, rovibrational transition frequencies, and line strengths are computed for CO2–CS2. The potential is made by fitting energies obtained from explicitly correlated coupled-cluster calculations using an interpolating moving least squares method. The rovibrational Schrodinger equation is solved with a symmetry-adapted Lanczos algorithm and an uncoupled product basis set. All four intermolecular coordinates are included in the calculation. In agreement with previous experiments, the global minimum of the potential energy surface (PES) is cross shaped. The PES also has slipped-parallel minima. Rovibrational wavefunctions are localized in the cross minima and the slipped-parallel minima. Vibrational parent analysis was used to assign vibrational labels to rovibrational states. Tunneling occurs between the two cross minima. Because more than one symmetry operation interconverts the two wells, the symmetry (−oo) of the upper component of the tunneling doublet is different from the symmetry (−ee) of the tunneling coordinate. This unusual situation is due to the multidimensional nature of the double well tunneling. For the cross ground vibrational state, calculated rotational constants differ from their experimental counterparts by less than 0.0001 cm−1. Most rovibrational states were found to be incompatible with the standard effective rotational Hamiltonian often used to fit spectra. This appears to be due to coupling between internal and overall rotation of the dimer. A simple 2D model accounting for internal rotation was used for two cross-shaped fundamentals to obtain good fits.

A recently implemented asymmetric Lanczos algorithm for computing (complex) linear response functions within the coupled cluster singles (CCS), coupled cluster singles and iterative approximate doubles (CC2), and coupled cluster singles and doubles (CCSD) is coupled to a Stieltjes imaging technique in order to describe the photoionization cross section of atoms and molecules, in the spirit of a similar procedure recently proposed by Averbukh and co-workers within the Algebraic Diagrammatic Construction approach. Pilot results are reported for the atoms He, Ne, and Ar and for the molecules H2, H2O, NH3, HF, CO, and CO2. Funding Agencies|Italian PRIN2009 funding scheme||FP7-PEOPLE-2009-IEF funding scheme|254326

In this paper, we report a new intermolecular potential energy surface and rovibrational transition frequencies and line strengths computed for the OCS dimer. The potential is made by fitting energies obtained from explicitly correlated coupled-cluster calculations and fit using an interpolating moving least squares method. The rovibrational Schroedinger equation is solved with a symmetry-adapted Lanczos algorithm and an uncoupled product basis set. All four intermolecular coordinates are included in the calculation. On the potential energy surface we find, previously unknown, cross-shaped isomers and also polar and non-polar isomers. The associated wavefunctions and energy levels are presented. To identify polar and cross states we use both calculations of line strengths and vibrational parent analysis. Calculated rotational constants differ from their experimental counterparts by less than 0.001 cm(-1).

A variational quantum mechanical protocol is presented for the computation of rovibrational energy levels of semirigid molecules using discrete variable representation of the Eckart-Watson Hamiltonian, a complete, "exact" inclusion of the potential energy surface, and selection of a vibrational subspace. Molecular symmetry is exploited via a symmetry-adapted Lanczos algorithm. Besides symmetry labels, zeroth-order rigid-rotor and harmonic-oscillator quantum numbers are employed to characterize the computed rovibrational states. Using the computational molecular spectroscopy algorithm presented, a large number of rovibrational states, up to J = 50, of the ground electronic state of the parent isotopologue of ketene, H(2) (12)C=(12)C=(16)O, were computed and characterized. Based on 12 references, altogether 3982 measured and assigned rovibrational transitions of H(2) (12)C=(12)C=(16)O have been collected, from which 3194 were validated. These transitions form two spectroscopic networks (SN). The ortho and the para SNs contain 2489 and 705 validated transitions and 1251 and 471 validated energy levels, respectively. The computed energy levels are compared with energy levels obtained, up to J = 41, via an inversion protocol based on this collection of validated measured rovibrational transitions. The accurate inverted energy levels allow new assignments to be proposed. Some regularities and irregularities in the rovibrational spectrum of ketene are elucidated.

Unbiased open-ended methods for finding transition states are powerful tools to understand diffusion and relaxation mechanisms associated with defect diffusion, growth processes, and catalysis. They have been little used, however, in conjunction with ab initio packages as these algorithms demanded large computational effort to generate even a single event. Here, we revisit the activation-relaxation technique (ART nouveau) and introduce a two-step convergence to the saddle point, combining the previously used Lanczós algorithm with the direct inversion in interactive subspace scheme. This combination makes it possible to generate events (from an initial minimum through a saddle point up to a final minimum) in a systematic fashion with a net 300-700 force evaluations per successful event. ART nouveau is coupled with BigDFT, a Kohn-Sham density functional theory (DFT) electronic structure code using a wavelet basis set with excellent efficiency on parallel computation, and applied to study the potential energy surface of C(20) clusters, vacancy diffusion in bulk silicon, and reconstruction of the 4H-SiC surface.

The Lanczos algorithm (LA) is a useful iterative method for the reduction of a large matrix to tridiagonal form. It is a storage efficient procedure requiring only the preceding two Lanczos vectors to compute the next. The quasi-minimal residual (QMR) method is a powerful method for the solution of linear equation systems, Ax = b. In this report we provide another application of the QMR method: we incorporate QMR into the LA to monitor the convergence of the Lanczos projections in the reduction of large sparse matrices. We demonstrate that the combined approach of the LA and QMR can be utilized efficiently for the orthogonal transformation of large, but sparse, complex, symmetric matrices, such as are encountered in the simulation of slow-motional 1D- and 2D-electron spin resonance (ESR) spectra. Especially in the 2D-ESR simulations, it is essential that we store all of the Lanczos vectors obtained in the course of the LA recursions and maintain their orthogonality. In the LA-QMR application, the QMR weight matrix mitigates the problem that the Lanczos vectors lose orthogonality after many LA projections. This enables substantially more Lanczos projections, as required to achieve convergence for the more challenging ESR simulations. It, therefore, provides better accuracy for the eigenvectors and the eigenvalues of the large sparse matrices originating in 2D-ESR simulations than does the previously employed method, which is a combined approach of the LA and the conjugate-gradient (CG) methods, as evidenced by the quality and convergence of the 2D-ESR simulations. Our results show that very slow-motional 2D-ESR spectra at W-band (95 GHz) can be reliably simulated using the LA-QMR method, whereas the LA-CG consistently fails. The improvements due to the LA-QMR are of critical importance in enabling the simulation of high-frequency 2D-ESR spectra, which are characterized by their very high resolution to molecular orientation.

We report a new three-dimensional potential energy surface for Ne-CO(2) including the Q(3) normal mode for the υ(3) antisymmetric stretching vibration of the CO(2) molecule. The potential energies were calculated using the supermolecular method at the coupled-cluster singles and doubles level with noniterative inclusion of connected triples [CCSD(T)], using a large basis set supplemented with midpoint bond functions. Two vibrationally averaged potentials with CO(2) at both the ground (υ=0) and the first (υ=1) vibrational υ(3) excited states were generated from the integration of the three-dimensional potential over the Q(3) coordinate. Each potential was found to have a T-shaped global minimum and two equivalent linear local minima. The radial DVR/angular FBR method and the Lanczos algorithm are applied to calculate the rovibrational energy levels. Comparison with the available observed values showed an overall excellent agreement for the microwave and infrared spectra. The calculated band origin shifts were found to be 0.1306 and 0.1419 cm(-1) for Ne-CO(2) and Ne-C(18)O(2), respectively, which are very close to the experimental values of 0.1303 and 0.1432 cm(-1).

Algorithms for parallel unconstrained minimization of molecular systems are examined. The overall framework of minimization is the same except for the choice of directions for updating the quasi-Newton Hessian. Ideally these directions are chosen so the updated Hessian gives steps that are same as using the Newton method. Three approaches to determine the directions for updating are presented: the straightforward approach of simply cycling through the Cartesian unit vectors (finite difference), a concurrent set of minimizations, and the Lanczos method. We show the importance of using preconditioning and a multiple secant update in these approaches. For the Lanczos algorithm, an initial set of directions is required to start the method, and a number of possibilities are explored. To test the methods we used the standard 50-dimensional analytic Rosenbrock function. Results are also reported for the histidine dipeptide, the isoleucine tripeptide, and cyclic adenosine monophosphate. All of these systems show a significant speed-up with the number of processors up to about eight processors.

A means of evaluating the action of Hamiltonian operators on functions expanded in orthogonal compact support wavelet bases is developed, avoiding the direct construction and storage of operator matrices that complicate extension to coupled multidimensional quantum applications. Application of a potential energy operator is accomplished by simple multiplication of the two sets of expansion coefficients without any convolution. The errors of this coefficient product approximation are quantified and lead to use of particular generalized coiflet bases, derived here, that maximize the number of moment conditions satisfied by the scaling function. This is at the expense of the number of vanishing moments of the wavelet function (approximation order), which appears to be a disadvantage but is shown surmountable. In particular, application of the kinetic energy operator, which is accomplished through the use of one-dimensional (1D) [or at most two-dimensional (2D)] differentiation filters, then degrades in accuracy if the standard choice is made. However, it is determined that use of high-order finite-difference filters yields strongly reduced absolute errors. Eigensolvers that ordinarily use only matrix-vector multiplications, such as the Lanczos algorithm, can then be used with this more efficient procedure. Applications are made to anharmonic vibrational problems: a 1D Morse oscillator, a 2D model of proton transfer, and three-dimensional vibrations of nitrosyl chloride on a global potential energy surface.

For the purpose of calculating (ro-)vibrational spectra, rate constants, scattering cross sections, etc. product basis sets are very popular. They, however, have the important disadvantage that they are unusably large for systems with more than four atoms. In this paper we demonstrate that it is possible to efficiently use a basis set obtained by removing, from a product basis set, functions associated with the largest diagonal Hamiltonian matrix elements. This is done by exploiting the fact that for every factor of every term in the Hamiltonian, there is a basis-set order in which the matrix representation of the factor is block diagonal. Due to this block diagonality the Lanczos algorithm can be implemented efficiently. Tests with model Hamiltonians with as many as 32 coordinates illustrate the merit of the ideas.

In a previous paper [Beyvers et al., J. Chem. Phys. 124, 234706 (2006)], the possibility to mode and state selectively excite various vibrational modes of a CO molecule adsorbed on a dissipative Cu(100) surface by shaped IR pulses was examined. Reduced-dimensionality models with stretching-only coordinates were employed to do so. This model is now extended with the goal to include rotational modes. First, we present an analysis of the bound states of the adsorbed CO molecule in full dimension; i.e., six-dimensional eigenstates are obtained by diagonalizing the six-dimensional Hamiltonian containing the semiempirical potential of Tully et al. [J. Vac. Sci. Technol. A 11, 1914 (1993)]. This is achieved by using a contracted iterative eigensolver based on the coupled two-term Lanczos algorithm with full reorthogonalization. Reduced-dimension subsystem eigenvectors are also computed and then used to study the selective excitation of the molecule in the presence of dissipation within the density matrix formalism for open systems. In the density matrix propagations, up to four degrees of freedom were included, namely, r (the C-O distance), Z (the molecule-surface distance), and phi and theta (the azimuthal and polar angles of the molecular axis with respect to the surface). Short, intense laser pulses are rationally engineered and further refined with optimal control theory, again with the goal for mode and state selective excitation. Also, IR-laser induced desorption is studied. For the calculations, the previous two-mode (r,Z) dipole surface is extended to include the angular dependence and the model for the coupling of the molecule to the surface electronic degrees of freedom is refined.