Approximate inclusion of four-mode couplings in vibrational coupled-cluster theory

The vibrational coupled cluster (VCC) equations are analyzed in terms of vibrational Mo?ller-Plesset perturbation theory aiming specifically at the importance of four-mode couplings. Based on this analysis, new VCC methods are derived for the calculation of anharmonic vibrational energies and vibrational spectra using vibrational coupled cluster response theory. It is shown how the effect of four-mode coupling and excitations can be efficiently and accurately described using approximations for their inclusion. Two closely related approaches are suggested. The computational scaling of the so-called VCC[3pt4F] method is not higher than the fifth power in the number of vibrational degrees of freedom when up to four-mode coupling terms are present in the Hamiltonian and only fourth order when only up to three-mode couplings are present. With a further approximation, one obtains the VCC[3pt4] model which is shown to scale with at most the fourth power in the number of vibrational degrees of freedom for Hamiltonians with both three- and four-mode coupling levels, while sharing the most important characteristics with VCC[3pt4F]. Sample calculations reported for selected tetra-atomic molecules as well as the larger dioxirane and ethylene oxide molecules support that the new models are accurate and useful.     

Vibrational coupled cluster response theory: a general implementation

The calculation of vibrational contributions to molecular properties using vibrational coupled cluster (VCC) response theory is discussed. General expressions are given for expectation values, linear response functions, and transition moments. It is shown how these expressions can be evaluated for arbitrary levels of excitation in the wave function parameterization as well as for arbitrary coupling levels in the potential and property surfaces. The convergence of the method is assessed by benchmark calculations on formaldehyde. Furthermore, excitation energies and infrared intensities are calculated for the fundamental vibrations of furan using VCC limited to up to two-mode and up to three-mode excitations, VCC[2] and VCC[3], as well as VCC with full two-mode and approximate three-mode couplings, VCC[2pt3].     

A second quantization formulation of multimode dynamics

A new formalism for calculating and analyzing many-mode quantum dynamics is presented. The formalism is similar in spirit to the second quantization formulation of electronic structure theory. The similarity means that similar techniques can be employed for calculating the many-mode nuclear wave function. As a consequence a new formulation of the vibrational self-consistent-field (VSCF) method can be developed. Another result is that the formalism opens up for the construction of new methods that go beyond the VSCF level. A vibrational coupled cluster (VCC) theory is constructed using the new formalism. The size-extensivity concept is introduced in the context of multimode wave functions and the size extensivity of approximate VCC methods is illustrated in comparison with the non-size-extensive vibrational configuration interaction method.     

Non-normal Lanczos methods for quantum scattering

This article presents a new complex absorbing potential (CAP) block Lanczos method for computing scattering eigenfunctions and reaction probabilities. The method reduces the problem of computing energy eigenfunctions to solving two energy dependent systems of equations. An energy independent block Lanczos factorization casts the system into a block tridiagonal form, which can be solved very efficiently for all energies. We show that CAP-Lanczos methods exhibit instability due to the non-normality of CAP Hamiltonians and may break down for some systems. The instability is not due to loss of orthogonality but to non-normality of the Hamiltonian matrix. While use of a Woods-Saxon exponential CAP-as opposed to a polynomial CAP-reduced non-normality, it did not always ensure convergence. Our results indicate that the Arnoldi algorithm is more robust for non-normal systems and less prone to break down. An Arnoldi version of our method is applied to a nonadiabatic tunneling Hamiltonian with excellent results, while the Lanczos algorithm breaks down for this system.     

Recent Advances in Recursive Solution of Large Molecular Eigenproblems

In studying highly excited ro-vibrational spectra of polyatomic molecules, the traditional direct diagonalization approach becomes impractical because of the formidable computational resources needed I will in this talk discuss some recent advances in solving large dimensional molecular eigenproblems using sparse matrix techniques. In particular, I will talk about the Lanczos algorithm and the Chebyshev-based filter-diagonalization method. These techniques are based on matrix-vector multiplication and thus amenable to larger problems. Applications to highly excited (ro-)vibrational spectra of tri- and tetratomic systems such as SO₂ and HOOH will be presented.     

Full S matrix calculation via a single real-symmetric Lanczos recursion: the Lanczos artificial boundary inhomogeneity method

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.     

Damped response theory description of two-photon absorption

Damped response theory is applied to the calculation of two-photon absorption (TPA) spectra, which are determined directly, at each frequency, from a modified damped cubic response function. The TPA spectrum may therefore be evaluated for selected frequency ranges, making the damped TPA approach attractive for calculations on large molecules with a high density of states, where the calculation of TPA using standard theory is more problematic. Damped response theory can also be applied to the case of intermediate state resonances, where the standard TPA expression is divergent. Both exact damped response theory and its application within density functional theory are discussed. The latter is implemented using an atomic-orbital based density matrix formulation, which makes the approach especially suitable for studies on large systems. A test preliminary study is presented for the TPA spectrum of R-(+)-1,1'-bi(2-naphtol).     

Full-dimensional quantum calculations of vibrational spectra of six-atom molecules. I. Theory and numerical results

Two quantum mechanical Hamiltonians have been derived in orthogonal polyspherical coordinates, which can be formed by Jacobi and/or Radau vectors etc., for the study of the vibrational spectra of six-atom molecules. The Hamiltonians are expressed in an explicit Hermitian form in the spatial representation. Their matrix representations are described in both full discrete variable representation (DVR) and mixed DVR/nondirect product finite basis representation (FBR) bases. The two-layer Lanczos iteration algorithm [H.-G. Yu, J. Chem. Phys. 117, 8190 (2002)] is employed to solve the eigenvalue problem of the system. A strategy regarding how to carry out the Hamiltonian-vector products for a high-dimensional problem is discussed. By exploiting the inversion symmetry of molecules, a unitary sequential 1D matrix-vector multiplication algorithm is proposed to perform the action of the Hamiltonian on the wavefunction in a symmetrically adapted DVR or FBR basis in the azimuthal angular variables. An application to the vibrational energy levels of the molecular hydrogen trimer (H2)3 in full dimension (12D) is presented. Results show that the rigid-H2 approximation can underestimate the binding energy of the trimer by 27%. Finally, it is demonstrated that the two-layer Lanczos algorithm is also capable of computing the eigenvectors of the system with minor effort.     

A new Lanczos-based algorithm for simulating high-frequency two-dimensional electron spin resonance spectra

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.     

Infrared optical constants, molar absorption coefficients, dielectric constants, molar polarisabilities, transition moments and dipole moment derivatives of liquid N,N-dimethylacetamide-carbon tetrachloride mixtures

Mid-infrared spectra of the DMA-carbon tetrachloride system by transmission and single- and multiple-reflection ATR technique in the whole composition range (0相似文献   

Using nonproduct quadrature grids to solve the vibrational Schrödinger equation in 12D

In this paper we propose a new quadrature scheme for computing vibrational spectra and apply it, using a Lanczos algorithm, to CH(3)CN. All 12 coordinates are treated explicitly. We need only 157'419'523 quadrature points. It would not be possible to use a product Gauss grid because 33 853 318 889 472 product Gauss points would be required. The nonproduct quadrature we use is based on ideas of Smolyak, but they are extended so that they can be applied when one retains basis functions θ(n(1))(r(1))···θ(n(D))(r(D)) that satisfy the condition α(1)n(1) + ··· + α(D)n(D) ≤ b, where the α(k) are integers. We demonstrate that it is possible to exploit the structure of the grid to efficiently evaluate the matrix-vector products required to use the Lanczos algorithm.     

Vibrational structure theory: new vibrational wave function methods for calculation of anharmonic vibrational energies and vibrational contributions to molecular properties

A number of recently developed theoretical methods for the calculation of vibrational energies and wave functions are reviewed. Methods for constructing the appropriate quantum mechanical Hamilton operator are briefly described before reviewing a particular branch of theoretical methods for solving the nuclear Schr?dinger equation. The main focus is on wave function methods using the vibrational self-consistent field (VSCF) as starting point, and includes vibrational configuration interaction (VCI), vibrational M?ller-Plesset (VMP) theory, and vibrational coupled cluster (VCC) theory. The convergence of the different methods towards the full vibrational configuration interaction (FVCI) result is discussed. Finally, newly developed vibrational response methods for calculation of vibrational contributions to properties, energies, and transition probabilities are discussed.     

Infrared optical constants, molar absorption coefficients, dielectric constants, molar polarisabilities, transition moments and dipole moment derivatives of liquid N,N-dimethylformamide-carbon tetrachloride mixtures

Mid-infrared spectra of the N,N-dimethylformamide-carbon tetrachloride system by transmission and single- and multiple-reflection ATR technique in the whole composition range (0相似文献   

Discrete variable representation method applied to the determination of rotation-vibration bound states of NO2

The discrete variable representation method is applied to the determination of the rotation-vibration energy levels of the fundamental electronic state of NO₂. The Hamiltonian is expressed in Johnson hyperspherical coordinates and developed on a DVR basis for each internal coordinate, while parity-adapted linear combinations of Wigner functions are used to describe the rotational motion. The diagonalization of the Hamiltonian matrix is performed using the Lanczos algorithm for large symmetric and Hermitian matrices. Results for rovibrational states up to J = 11 for the first five vibrational energy levels are presented. © 1997 John Wiley & Sons, Inc.     

Vibrational coupled cluster theory

The theory and first implementation of a vibrational coupled cluster (VCC) method for calculations of the vibrational structure of molecules is presented. Different methods for introducing approximate VCC methods are discussed including truncation according to a maximum number of simultaneous mode excitations as well as an interaction space order concept is introduced. The theory is tested on calculation of anharmonic frequencies for a three-mode model system and a formaldehyde quartic force field. The VCC method is compared to vibrational self-consistent-field, vibrational M?ller-Plesset perturbation theory, and vibrational configuration interaction (VCI). A VCC calculation typically gives higher accuracy than a corresponding VCI calculation with the same number of parameters and the same formal operation count.     

Three-dimensional ab initio potential-energy surface and rovibrational spectra of the H2-Kr complex

We report a three-dimensional ab initio potential-energy surface for the H2-Kr complex calculated using a supermolecular method. The electronic calculations were performed at the coupled-cluster singles and doubles level with noniterative inclusion of connected triples levels with a large basis set including midbond functions and the full counterpoise correction for the basis-set superposition error. The intermolecular potential energy between the H2 molecule and the Kr atom were evaluated at five potential-optimized discrete variable representation (DVR) grid points generated from the potential-energy curve of H2. The potential for other bond lengths of H2 could be deduced using polynomial interpolations. The complex is found to have a linear preferred structure with a rather flat energy barrier. The three-dimensional DVR method and the Lanczos propagation algorithm were employed to calculate the rovibrational states without separating the inter- and intramolecular nuclear motions. In addition, the rovibrational spectra from the H2 fundamental vibrational band were calculated. The calculated shift for the band origin is -1.50 cm-1, which is in good agreement with the experimental value of -1.706 cm-1, and the calculated transition frequencies in Q1(0) and S1(0) bands are within 3% of the observed values.     

A new potential energy surface and predicted infrared spectra of He-CO2: dependence on the antisymmetric stretch of CO2

A new potential energy surface involving the antisymmetric Q(3) normal mode of CO(2) for the He-CO(2) van der Waals complex is constructed at the coupled-cluster singles and doubles with noniterative inclusion of connected triple [CCSD(T)] level with augmented correlation-consistent quadruple-zeta (aug-cc-pVQZ) basis set plus bond functions. Two vibrationally adiabatic potentials with CO(2) at both the ground and the first excited vibrational states are generated from the integration of the three-dimensional potential over the Q(3) coordinate. The potential has a T-shaped global minimum and two equivalent linear local minima. The bound rovibrational energy levels are obtained using the radial discrete variable representation/angular finite basis representation method and the Lanczos algorithm. The observed band origin shift of the complex (0.0946 cm(-1)) is successfully reproduced by our calculation (0.1034 cm(-1)). The infrared spectra of the complex are also predicted. The fundamental band is in excellent agreement with the experiment. Most of the transitions corresponding to the observed hot band [M. J. Weida et al., J. Chem. Phys. 101, 8351 (1994)] are assigned reasonably.     

Franck-Condon factors within damped displacement harmonic oscillators: Solvent-enhanced absorption and fluorescence spectra

Damped harmonic oscillators that take into account local-mode nuclear vibrations interacting with solvent molecules are developed into Franck-Condon factors within displaced harmonic oscillator approximation. This is practically done by scaling an unperturbed Hessian matrix that represents local modes of force constants for molecules in a gaseous phase, and then by diagonalizing the perturbed Hessian matrix it results in direct modification of Huang–Rhys factors which represent normal modes of solute molecule perturbed by solvent environment. For highly symmetric polycyclic aromatic hydrocarbon molecules in which hydrogen atom vibrations in a solution can be scaled equally, one-set scaling parameters constructed into damped Franck-Condon factors can reproduce solvent-enhanced absorption and fluorescence spectra in solution. However, for low symmetry molecules with atoms other than hydrogen and carbon atoms, multi-set scaling parameters constructed into damped Franck-Condon factors can also reproduce solvent-enhanced absorption and fluorescence spectra in solution. Examples for high symmetry perylene in benzene solution with one-set scaling parameters and for low symmetry carbazole in n-hexane solution with multi-set scaling parameters are given, in both cases, the present damped Franck-Condon simulation can reproduce solvent-enhanced absorption and fluorescence spectra in solution.     

Quantum and electromagnetic propagation with the conjugate symmetric Lanczos method

The conjugate symmetric Lanczos (CSL) method is introduced for the solution of the time-dependent Schrodinger equation. This remarkably simple and efficient time-domain algorithm is a low-order polynomial expansion of the quantum propagator for time-independent Hamiltonians and derives from the time-reversal symmetry of the Schrodinger equation. The CSL algorithm gives forward solutions by simply complex conjugating backward polynomial expansion coefficients. Interestingly, the expansion coefficients are the same for each uniform time step, a fact that is only spoiled by basis incompleteness and finite precision. This is true for the Krylov basis and, with further investigation, is also found to be true for the Lanczos basis, important for efficient orthogonal projection-based algorithms. The CSL method errors roughly track those of the short iterative Lanczos method while requiring fewer matrix-vector products than the Chebyshev method. With the CSL method, only a few vectors need to be stored at a time, there is no need to estimate the Hamiltonian spectral range, and only matrix-vector and vector-vector products are required. Applications using localized wavelet bases are made to harmonic oscillator and anharmonic Morse oscillator systems as well as electrodynamic pulse propagation using the Hamiltonian form of Maxwell's equations. For gold with a Drude dielectric function, the latter is non-Hermitian, requiring consideration of corrections to the CSL algorithm.     

Calculation of intensity in the resonance Raman and two-photon absorption spectra of polyatomic molecules

A direct method for calculating the resonance Raman and two-photon absorption spectra of polyatomic molecules is described in detail The method is based on the adiabatic model and uses Herzberg-Teller’s approximation. Relations ruling out direct summation over vibrational quantum numbers of excited electronic states and representing the matrix elements of the Green function of a multidimensional oscillator as functions of vibration frequencies and Dushinsky transformation parameters are derived. The relations are convenient for constructing algorithms. Translated from Zhumal Struktumoi Khimii, Vol. 38, No. 2, pp. 248–255, March–April, 1997.