Numerical Convergence of the Sinc Discrete Variable Representation for Solving Molecular Vibrational States with a Conical Intersection in Adiabatic Representation

Within the Born-Oppenheimer (BO) approximation, nuclear motions of a molecule are often envisioned to occur on an adiabatic potential energy surface (PES). However, this single PES picture should be reconsidered if a conical intersection (CI) is present, although the energy is well below the CI. The presence of the CI results in two additional terms in the nuclear Hamiltonian in the adiabatic presentation, i.e., the diagonal BO correction (DBOC) and the geometric phase (GP), which are divergent at the CI. At the same time, there are cusps in the adiabatic PESs. Thus usually it is regarded that there is numerical difficulty in a quantum dynamics calculation for treating CI in the adiabatic representation. A popular numerical method in nuclear quantum dynamics calculations is the Sinc discrete variable representation (DVR) method. We examine the numerical accuracy of the Sinc DVR method for solving the Schr?dinger equation of a two dimensional model of two electronic states with a CI in both the adiabatic and diabatic representation. The results suggest that the Sinc DVR method is capable of giving reliable results in the adiabatic representation with usual density of the grid points, without special treatment of the divergence of the DBOC and the GP. The numerical uncertainty is not worse than that after the introduction of an arbitrary vector potential for accounting the GP, whose accurate form usually is not easy to obtain.     

Ab initio vibration and ro-vibrational spectrum of the 1A1 state of He2Si2+

The low-lying ro-vibrational states for the ground electronic state (1A1) of HeSi2+ have been calculated using an ab initio variational solution of the nuclear Schr?dinger equation. A 96 point CCSD(T)/cc-pCVQZ potential energy surface (PES) has been calculated and a Ogilvie-Padé (3,6) potential energy function has been generated. This force field was embedded in the Eckart-Watson Hamiltonian from which the vibrational and ro-vibrational eigenfunctions and eigenenergies have been variationally calculated. A 70 point QCISD/aug-cc-pCVTZ discrete dipole moment surface (DMS) was calculated and a 5th order power series expansion (in terms of the two bond lengths and the included bond angle) has been generated. Absolute line intensities have been calculated and are presented for some of the most intense transitions between the vibrational ground state and the low-lying ro-vibrational states of this ion.     

A paradox of grid-based representation techniques: accurate eigenvalues from inaccurate matrix elements

Several approximately variational grid-based representation techniques devised to solve the time-independent nuclear-motion Schrödinger equation share a similar behavior: while the computed eigenpairs, the only results which are of genuine interest, are accurate, many of the underlying Hamiltonian matrix elements are inaccurate, deviating substantially from their values in a variational basis representation. Examples are presented for the discrete variable representation and the Lagrange-mesh approaches, demonstrating that highly accurate eigenvalues and eigenfunctions can be obtained even if some or even all of the Hamiltonian matrix elements in these grid-based representations are inaccurate. It is shown how the apparent contradiction of obtaining accurate eigenpairs with far less accurate individual matrix elements can be resolved by considering the unitary transformation between the representations. Furthermore, the relations connecting orthonormal bases and the corresponding Lagrange bases are generalized to relations connecting nonorthogonal, regularized bases and the corresponding nonorthogonal, regularized Lagrange bases.     

Approximate Eigenvalue and Eigenfunction Solutions for the Generalized Hulthén Potential with any Angular Momentum

An approximate solution of the Schr?dinger equation for the generalized Hulthén potential with non-zero angular quantum number is solved. The bound state energy eigenvalues and eigenfunctions are obtained in terms of Jacobi polynomials. The Nikiforov–Uvarov method is used in the computations. We have considered the time-independent Schr?dinger equation with the associated form of Hulthén potential which simulate the effect of the centrifugal barrier for any l-state. The energy levels of the used Hulthén potential gives satisfactory values for the non-zero angular momentum as the generalized Hulthén effective potential.      

Supersymmetric solutions of <Emphasis Type="Italic">PT</Emphasis>-/non-<Emphasis Type="Italic">PT</Emphasis>-symmetric and non-Hermitian screened Coulomb potential via Hamiltonian hierarchy inspired variational method

The supersymmetric solutions of PT -symmetric and Hermitian/non-Hermitian forms of quantum systems are obtained by solving the Schr?dinger equation for the Exponential-Cosine Screened Coulomb potential. The Hamiltonian hierarchy inspired variational method is used to obtain the approximate energy eigenvalues and corresponding wave functions.      

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 Schro?dinger 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.     

Trigonometrically fitted and exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation

The solution of the one-dimensional time-independent Schr?dinger equation is considered by exponentially fitted symplectic integrators. The Schr?dinger equation is first transformed into a Hamiltonian canonical equation. Numerical results are obtained for the one-dimensional harmonic oscillator and the doubly anharmonic oscillator.     

CVRQD ab initio ground-state adiabatic potential energy surfaces for the water molecule

The high accuracy ab initio adiabatic potential energy surfaces (PESs) of the ground electronic state of the water molecule, determined originally by Polyansky et al. [Science 299, 539 (2003)] and called CVRQD, are extended and carefully characterized and analyzed. The CVRQD potential energy surfaces are obtained from extrapolation to the complete basis set of nearly full configuration interaction valence-only electronic structure computations, augmented by core, relativistic, quantum electrodynamics, and diagonal Born-Oppenheimer corrections. We also report ab initio calculations of several quantities characterizing the CVRQD PESs, including equilibrium and vibrationally averaged (0 K) structures, harmonic and anharmonic force fields, harmonic vibrational frequencies, vibrational fundamentals, and zero-point energies. They can be considered as the best ab initio estimates of these quantities available today. Results of first-principles computations on the rovibrational energy levels of several isotopologues of the water molecule are also presented, based on the CVRQD PESs and the use of variational nuclear motion calculations employing an exact kinetic energy operator given in orthogonal internal coordinates. The variational nuclear motion calculations also include a simplified treatment of nonadiabatic effects. This sophisticated procedure to compute rovibrational energy levels reproduces all the known rovibrational levels of the water isotopologues considered, H(2) (16)O, H(2) (17)O, H(2) (18)O, and D(2) (16)O, to better than 1 cm(-1) on average. Finally, prospects for further improvement of the ground-state adiabatic ab initio PESs of water are discussed.     

New Second-order Exponentially and Trigonometrically Fitted Symplectic Integrators for the Numerical Solution of the Time-independent Schrödinger Equation

The solution of the one-dimensional time-independent Schr?dinger equation is considered by trigonometrically and exponentially fitted symplectic integrators. The Schr?dinger equation is first transformed into a Hamiltonian canonical equation. Numerical results are obtained for the one-dimensional harmonic oscillator, doubly anharmonic oscillator and the exponential potential.     

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.     

Vibrational motion in isotopomers of the HeH+ molecular ion: An application of the electron nuclear dynamics method

The importance of isotopic substitution as a tool for elucidation of chemical reaction events originates in the fact that the Coulombic Hamiltonian is isotopically invariant except for the nuclear kinetic energy term. Thus, in theories of isotope effects based on the Born-Oppenheimer scheme, the basic presumption is the invariance of the potential energy surface (PES). We use, however, a fully dynamic approach, called Electron Nuclear Dynamics (END), which does not require a preconstructed PES. Since the END formalism is rather different from commonly used procedures, we study the anharmonic nuclear vibration in isotopic species of the HeH⁺ molecular ion as a model problem. A single time-dependent complex parametrized determinantal wave function is used for the electrons and the nuclei are treated classically. The time evolution of the nuclear and electronic dynamical variables obtained by integration of equations of motion are reported as bond length, nuclear kinetic energy, and Mulliken populations. The molecule vibrates as a classical object. The product of the reduced mass and the square of the vibrational frequency is isotopomer invariant for any common total energy. The difference between the total energy and the nuclear kinetic energy as a function of the internuclear distance is interpreted as the average dynamic potential. © 1997 John Wiley & Sons, Inc.     

Particular and homogeneous solutions of time-independent wavepacket Schrödinger equations: calculations using a subset of eigenstates of undamped or damped Hamiltonians

A variety of causal, particular and homogeneous solutions to the time-independent wavepacket Schr?dinger equation have been considered as the basis for calculations using Chebychev expansions, finite-τ expansions obtained from a partial Fourier transform of the time-dependent Schr?dinger equation, and the distributed approximating functional (DAF) representation for the spectral density operator (SDO). All the approximations are made computationally robust and reliable by damping the discrete Hamiltonian matrix along the edges of the finite grid to facilitate the use of compact grids. The approximations are found to be completely well behaved at all values of the (continuous) scattering energy. It is found that the DAF–SDO provides a suitable alternative to Chebychev propagation. Received: 29 February 2000 / Accepted: 5 April 2000 / Published online: 18 August 2000     

Calibration-quality adiabatic potential energy surfaces for H3(+) and its isotopologues

Calibration-quality ab initio adiabatic potential energy surfaces (PES) have been determined for all isotopologues of the molecular ion H(3)(+). The underlying Born-Oppenheimer electronic structure computations used optimized explicitly correlated shifted Gaussian functions. The surfaces include diagonal Born-Oppenheimer corrections computed from the accurate electronic wave functions. A fit to the 41,655 ab initio points is presented which gives a standard deviation better than 0.1 cm(-1) when restricted to the points up to 6000 cm(-1) above the first dissociation asymptote. Nuclear motion calculations utilizing this PES, called GLH3P, and an exact kinetic energy operator given in orthogonal internal coordinates are presented. The ro-vibrational transition frequencies for H(3)(+), H(2)D(+), and HD(2)(+) are compared with high resolution measurements. The most sophisticated and complete procedure employed to compute ro-vibrational energy levels, which makes explicit allowance for the inclusion of non-adiabatic effects, reproduces all the known ro-vibrational levels of the H(3)(+) isotopologues considered to better than 0.2 cm(-1). This represents a significant (order-of-magnitude) improvement compared to previous studies of transitions in the visible. Careful treatment of linear geometries is important for high frequency transitions and leads to new assignments for some of the previously observed lines. Prospects for further investigations of non-adiabatic effects in the H(3)(+) isotopologues are discussed. In short, the paper presents (a) an extremely accurate global potential energy surface of H(3)(+) resulting from high accuracy ab initio computations and global fit, (b) very accurate nuclear motion calculations of all available experimental line data up to 16,000 cm(-1), and (c) results suggest that we can predict accurately the lines of H(3)(+) towards dissociation and thus facilitate their experimental observation.     

Vibrations of H+(D+) in stoichiometric LiNbO3 single crystal

A first principles quantum mechanical calculation of the vibrational energy levels and transition frequencies associated with protons in stoichiometric LiNbO(3) single crystal has been carried out. The hydrogen contaminated crystal has been approximated by a model one obtains by translating a supercell, i.e., a cluster of LiNbO(3) unit cells containing a single H(+) and a Li(+) vacancy. Based on the supercell model an approximate Hamiltonian operator describing vibrations of the proton sublattice embedded in the host crystal has been derived. It is further simplified to a sum of uncoupled Hamiltonian operators corresponding to different wave vectors (ks) and each describing vibrations of a quasi-particle (quasi-proton). The three dimensional (3D) Hamiltonian operator of k=0 has been employed to calculate vibrational levels and transition frequencies. The potential energy surface (PES) entering this Hamiltonian operator has been calculated point wise on a large set of grid points by using density functional theory, and an analytical approximation to the PES has been constructed by non-parametric approximation. Then, the nuclear motion Schro?dinger equation has been solved by employing the method of discrete variable representation. It has been found that the (quasi-)H(+) vibrates in a strongly anharmonic PES. Its vibrations can be described approximately as a stretching, and two orthogonal bending vibrations. The theoretically calculated transition frequencies agree within 1% with those experimentally determined, and they have allowed the assignment of one of the hitherto unassigned bands as a combination of the stretching and the bending of lower fundamental frequency.     

Trajectory-guided configuration interaction simulations of multidimensional quantum dynamics

We propose an approach to modelling multidimensional quantum systems which uses direct-dynamics trajectories to guide wavefunction propagation. First, trajectory simulations are used to generate a sample of dynamically relevant configurations on the potential energy surface (PES). Second, the sampled configurations are used to construct an n-mode representation of the PES using a greedy algorithm. Finally, the time-dependent Schr?dinger equation is solved using a configuration interaction expansion of the wavefunction, with individual basis functions derived directly from the 1-mode contributions to the n-mode PES. This approach is successfully demonstrated by application to a 20-dimensional benchmark problem describing tunnelling in the presence of coupled degrees of freedom.     

On the single-electron local kinetic energy

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Solving the Schrödinger and Dirac equations of hydrogen molecular ion accurately by the free iterative complement interaction method

The nonrelativistic Schr?dinger equation and the relativistic four-component Dirac equation of H(2) (+) were solved accurately in an analytical expansion form by the free iterative complement interaction (ICI) method combined with the variational principle. In the nonrelativistic case, we compared the free ICI wave function with the so-called "exact" wave function as two different expansions converging to the unique exact wave function and found that the free ICI method is much more efficient than the exact method. In the relativistic case, we first used the inverse Hamiltonian to guarantee Ritz-type variational principle and obtained accurate result. We also showed that the ordinary variational calculation also gives a nice convergence when the g function is appropriately chosen, since then the free ICI calculation guarantees a correct relationship between the large and small components of each adjacent order, which we call ICI balance. This is the first application of the relativistic free ICI method to molecule. We calculated both ground and excited states in good convergence, and not only the upper bound but also the lower bound of the ground-state energy. The error bound analysis has assured that the present result is highly accurate.     

Approximate analytical solutions for arbitrary l-state of the Hulthén potential with an improved approximation of the centrifugal term

An approximate analytical solution of the radial Schr?dinger equation for the generalized Hulthén potential is obtained by applying an improved approximation of the centrifugal term. The bound state energy eigenvalues and the normalized eigenfunctions are given in terms of hypergeometric polynomials. The results for arbitrary quantum numbers n _r and l with different values of the screening parameter δ are compared with those obtained by the numerical method, asymptotic iteration, the Nikiforov-Uvarov method, the exact quantization rule, and variational methods. The results obtained by the method proposed in this work are in a good agreement with those obtained by other approximate methods.