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Toward black-box-type full- and reduced-dimensional variational (ro)vibrational computations

A black-box-type algorithm is presented for the variational computation of energy levels and wave functions using a (ro)vibrational Hamiltonian expressed in an arbitrarily chosen body-fixed frame and in any set of internal coordinates of full or reduced vibrational dimensionality. To make the required numerical work feasible, matrix representation of the operators is constructed using a discrete variable representation (DVR). The favorable properties of DVR are exploited in the straightforward and numerically exact inclusion of any representation of the potential and the kinetic energy including the G matrix and the extrapotential term. In this algorithm there is no need for an a priori analytic derivation of the kinetic energy operator, as all of its matrix elements at each grid point are computed numerically either in a full- or a reduced-dimensional model. Due to the simple and straightforward definition of reduced-dimensional models within this approach, a fully anharmonic variational treatment of large, otherwise intractable molecular systems becomes available. In the computer code based on the above algorithm, there is no inherent limitation for the maximally coupled number of vibrational degrees of freedom. However, in practice current personal computers allow the treatment of about nine fully coupled vibrational dimensions. Computation of vibrational band origins of full and reduced dimensions showing the advantages and limitations of the algorithm and the related computer code are presented for the water, ammonia, and methane molecules.

Vibrational energy levels with arbitrary potentials using the Eckart-Watson Hamiltonians and the discrete variable representation

An effective and general algorithm is suggested for variational vibrational calculations of N-atomic molecules using orthogonal, rectilinear internal coordinates. The protocol has three essential parts. First, it advocates the use of the Eckart-Watson Hamiltonians of nonlinear or linear reference configuration. Second, with the help of an exact expression of curvilinear internal coordinates (e.g., valence coordinates) in terms of orthogonal, rectilinear internal coordinates (e.g., normal coordinates), any high-accuracy potential or force field expressed in curvilinear internal coordinates can be used in the calculations. Third, the matrix representation of the appropriate Eckart-Watson Hamiltonian is constructed in a discrete variable representation, in which the matrix of the potential energy operator is always diagonal, whatever complicated form the potential function assumes, and the matrix of the kinetic energy operator is a sparse matrix of special structure. Details of the suggested algorithm as well as results obtained for linear and nonlinear test cases including H(2)O, H(3) (+), CO(2), HCNHNC, and CH(4) are presented.

On the variational computation of a large number of vibrational energy levels and wave functions for medium-sized molecules

In a recent publication [J. Chem. Phys. 127, 084102 (2007)], the nearly variational DEWE approach (DEWE denotes Discrete variable representation of the Watson Hamiltonian using the Eckart frame and an Exact inclusion of a potential energy surface expressed in arbitrarily chosen coordinates) was developed to compute a large number of (ro)vibrational eigenpairs for medium-sized semirigid molecules having a single well-defined minimum. In this publication, memory, CPU, and hard disk usage requirements of DEWE, and thus of any DEWE-type approach, are carefully considered, analyzed, and optimized. Particular attention is paid to the sparse matrix-vector multiplication, the most expensive part of the computation, and to rate-determining steps in the iterative Lanczos eigensolver, including spectral transformation, reorthogonalization, and restart of the iteration. Algorithmic improvements are discussed in considerable detail. Numerical results are presented for the vibrational band origins of the (12)CH(4) and (12)CH(2)D(2) isotopologues of the methane molecule. The largest matrix handled on a personal computer during these computations is of the size of (4x10(8))x(4x10(8)). The best strategy for determining vibrational eigenpairs depends largely on the actual details of the required computation. Nevertheless, for a usual scenario requiring a large number of the lowest eigenpairs of the Hamiltonian matrix the combination of the thick-restart Lanczos method, shift-fold filtering, and periodic reorthogonalization appears to result in the computationally most feasible approach.

Rotating full- and reduced-dimensional quantum chemical models of molecules

A flexible protocol, applicable to semirigid as well as floppy polyatomic systems, is developed for the variational solution of the rotational-vibrational Schrödinger equation. The kinetic energy operator is expressed in terms of curvilinear coordinates, describing the internal motion, and rotational coordinates, characterizing the orientation of the frame fixed to the nonrigid body. Although the analytic form of the kinetic energy operator might be very complex, it does not need to be known a priori within this scheme as it is constructed automatically and numerically whenever needed. The internal coordinates can be chosen to best represent the system of interest and the body-fixed frame is not restricted to an embedding defined with respect to a single reference geometry. The features of the technique mentioned make it especially well suited to treat large-amplitude nuclear motions. Reduced-dimensional rovibrational models can be defined straightforwardly by introducing constraints on the generalized coordinates. In order to demonstrate the flexibility of the protocol and the associated computer code, the inversion-tunneling of the ammonia ((14)NH(3)) molecule is studied using one, two, three, four, and six active vibrational degrees of freedom, within both vibrational and rovibrational variational computations. For example, the one-dimensional inversion-tunneling model of ammonia is considered also for nonzero rotational angular momenta. It turns out to be difficult to significantly improve upon this simple model. Rotational-vibrational energy levels are presented for rotational angular momentum quantum numbers J = 0, 1, 2, 3, and 4.

Assigning quantum labels to variationally computed rotational-vibrational eigenstates of polyatomic molecules

A procedure is investigated for assigning physically transparent, approximate vibrational and rotational quantum labels to variationally computed eigenstates. Pure vibrational wave functions are analyzed by means of normal-mode decomposition (NMD) tables constructed from overlap integrals with respect to separable harmonic oscillator basis functions. Complementary rotational labels J(K(a)K(c)) are determined from rigid-rotor decomposition (RRD) tables formed by projecting rotational-vibrational wave functions (J not equal 0) onto products of symmetrized rigid-rotor basis functions and previously computed (J=0) vibrational eigenstates. Variational results for H(2)O, HNCO, trans-HCOD, NCCO, and H(2)CCO are presented to demonstrate the NMD and RRD schemes. The NMD analysis highlights several resonances at low energies that cause strong mixing and cloud the assignment of fundamental vibrations, even in such simple molecules. As the vibrational energy increases, the NMD scheme documents and quantifies the breakdown of the normal-mode model. The RRD procedure proves effective in providing unambiguous rotational assignments for the chosen test molecules up to moderate J values.

Molecular structure calculations: A unified quantum mechanical description of electrons and nuclei using explicitly correlated Gaussian functions and the global vector representation

We elaborate on the theory for the variational solution of the Schrödinger equation of small atomic and molecular systems without relying on the Born-Oppenheimer paradigm. The all-particle Schrödinger equation is solved in a numerical procedure using the variational principle, Cartesian coordinates, parameterized explicitly correlated Gaussian functions with polynomial prefactors, and the global vector representation. As a result, non-relativistic energy levels and wave functions of few-particle systems can be obtained for various angular momentum, parity, and spin quantum numbers. A stochastic variational optimization of the basis function parameters facilitates the calculation of accurate energies and wave functions for the ground and some excited rotational-(vibrational-)electronic states of H(2) (+) and H(2), three bound states of the positronium molecule, Ps(2), and the ground and two excited states of the (7)Li atom.

Bridging Theory with Experiment: A Benchmark Study of Thermally Averaged Structural and Effective Spectroscopic Parameters of the Water Molecule †

Extending our previous study on the equilibrium structures of the major isotopologues of the water molecule (Csaszar et al. J. Chem. Phys. 2005, 122, 214305), temperature-dependent averaged structural parameters (for example, r(g)- and r(a)-type distances, their related root-mean-square amplitudes, and moments corresponding to the probability distribution functions of interatomic distances), effective rotational constants, and low-order vibration-rotation interaction constants have been determined for two major symmetric isotopologues of water, H(2)(16)O and D(2)(16)O. The nuclear motion treatments employed full quantum mechanical variational procedures which utilized the accurate adiabatic semiglobal PESs of water isotopologues named CVRQD (Barletta et al. J. Chem. Phys. 2006, 125, 204307). The temperature-dependent molecular structural parameters are based on expectation value computations and Boltzmann averaging in the temperature range 0-1500 K. The precise computed average internuclear, inverse internuclear, rms amplitude, and anharmonicity parameters could support a future gas electron diffraction (GED) investigation, though water isotopologues are far from being ideal species for GED analyses. Using a clearly defined and general formalism applicable to molecules of any size, we have evaluated vibrationally averaged effective rotational constants as expectation values using inertia tensor formulas in the Eckart frame for vibrational states of H(2)(16)O and D(2)(16)O. While such variationally determined rotational constants do not correspond strictly to constants resulting from fits performed by spectroscopists, the expected good agreement is found for the A and B rotational constants for both isotopologues. Low-order vibration-rotation interaction constants, the so-called alpha- and gamma-constants, have also been determined from the computed rotational constants; the latter were derived for the first time.

Numerically constructed internal-coordinate Hamiltonian with Eckart embedding and its application for the inversion tunneling of ammonia.

It is shown that the use of an Eckart-frame embedding with a kinetic energy operator expressed in curvilinear internal coordinates becomes feasible and straightforward to implement for arbitrary molecular compositions and internal coordinates if the operator is defined numerically over a (discrete variable representation) grid. The algorithm proposed utilizes the transformation method of Dymarsky and Kudin to maintain the rotational Eckart condition. In order to demonstrate the applicability and flexibility of our approach the non-rigid ammonia molecule is considered and the corresponding rotational-vibrational energy levels and wave functions are computed using kinetic energy operators with three different embeddings. Two of them fulfill the Eckart conditions corresponding to a trigonal pyramidal (C3v) and a trigonal planar (D3h) reference structure and the third one is a non-Eckart frame. The computed energy levels are, of course, identical, and the structure of the three different wave-function representations are analyzed in terms of the rigid rotor functions for a symmetric top. The possible advantages of one frame representation over another are discussed concerning the interpretation of the rovibrational states in terms of the traditional rigid rotor labels.

Chapter 9 An Active Database Approach to Complete Rotational-Vibrational Spectra of Small Molecules

Publisher Summary This chapter discusses the computation of complete rotational–vibrational spectra of small molecules. Neither experiments nor first-principles computations can determine the complete rotational–vibrational spectra of even small molecules with the required accuracy. The most practical approach to overcome most of the difficulties is through an active database approach. This requires building two databases linked together through a unique assignment scheme, one containing energy levels and the other the related transitions. This way the advantage of the strengths of the two main sources of spectroscopic information can be taken. Variational computations can yield all the possible energy levels with various assignment possibilities and with all the possible labeled transitions, though with limited accuracy deteriorating as the level of excitation increases. Experimental transitions and the energy levels obtained through an appropriate inversion procedure have a much higher accuracy but are limited in number even in the spectroscopically most easily accessible regions.

On the calculation of resonances in pre-born-oppenheimer molecular structure theory

The main motivation for this work is the exploration of rotational–vibrational states corresponding to electronic excitations in a pre-Born–Oppenheimer quantum theory of molecules. These states are often embedded in the continuum of the lower-lying dissociation channel of the same symmetry and thus are thought to be resonances. To calculate rovibronic resonances, the pre-Born–Oppenheimer variational approach of [J. Chem. Phys. 2012, 137, 024104], based on the usage of explicitly correlated Gaussian functions and the global vector representation, is extended with the complex coordinate rotation method. The developed computer program is used to calculate resonance energies and widths for the three-particle positronium anion, Ps(–), and the four-particle positronium molecule, Ps(2). Furthermore, the excited bound and resonance rovibronic states of the four-particle H2 molecule are also considered. Resonance energies and widths are estimated for the lowest-energy resonances of H(2) beyond the b (3)∑(u)(+) continuum.