Potential Energy Matrices Research Articles

Interpolation of diabatic potential energy surfaces.

A method is presented for constructing diabatic potential energy matrices from ab initio quantum chemistry data. The method is similar to that reported previously for single adiabatic potential energy surfaces, but correctly accounts for the nuclear permutation symmetry of diabatic potential energy matrices and other complications that arise from the derivative coupling of electronic states. The method is tested by comparison with an analytic model for the two lowest energy states of H(3).

Split diabatic representation

A split diabatic representation is proposed as a technique for generating diabatic potential curves with the maximal physical content and favorable computational characteristics. This method is a mixed adiabatic-diabatic representation, in which smoothly varying couplings appear in both the kinetic- and potential-energy matrices. It requires the solution of the first-order differential equation for the transformation matrix of the standard strict diabatic representation for which an efficient numerical scheme is also presented. A transformation propagator, akin to the Cayley-Hamiltonian time evolution operator, is employed to obtain the diabatic states while preserving unitarity. Several examples illustrate the advantages of the proposed split diabatic representation.

Comparison of full multiple spawning, trajectory surface hopping, and converged quantum mechanics for electronically nonadiabatic dynamics

We present calculations employing the simplest version of the full multiple spawning method, FMS-M or minimal FMS, for electronically nonadiabatic quantum dynamics using three model potential energy matrices with different strengths and ranges for the diabatic coupling. We first demonstrate stability of the branching probabilities and final energy distributions with respect to the parameters in the FMS-M method. We then compare the method to a variety of other semiclassical methods, as well as to accurate quantum mechanical results for three-dimensional atom–diatom reactions and quenching processes; the deviations of the semiclassical results from the accurate quantum mechanical ones are averaged over nine cases. In the adiabatic electronic representation, the FMS-M method provides some improvement over Tully’s fewest switches trajectory surface hopping method. However, both methods, irrespective of electronic representation, systematically overpredict the extent of reaction in comparison to the exact quantum mechanical results. The present work provides a baseline for understanding the simplest member of the hierarchy of FMS methods and its relationship to established surface hopping methods.

Nonadiabatic Trajectories at an Exhibition

The present article reviews two classes of semiclassical (mixed quantum mechanical/classical) methods for investigating multielectronic-state dynamics: the trajectory surface-hopping (TSH) method and the time-dependent self-consistent field (TDSCF) method. The recent availability of accurate quantum mechanical dynamics calculations for a variety of realistic three-body two-state potential energy matrices has allowed an assessment of the validity of semiclassical multisurface dynamics methods that are applicable to larger systems. These studies indicate that Tully's fewest switches algorithm is the best available TSH method and that the Ehrenfest method is the best previously available TDSCF method. The fewest switches surface-hopping method has relatively small errors even when it is not the best method while the Ehrenfest TDSCF method tends to have larger errors when it is not the best. However, the fewest switches algorithm involves unphysical discontinuities in momenta, and the results may depend on the choice of representation. Furthermore, the surface-hopping algorithm is frequently frustrated in its attempt to maintain ensemble-average self-consistency. The Ehrenfest method removes all these troublesome aspects but at the cost of producing unphysical mixed states, which are responsible for its larger errors in observables. A recently introduced TDSCF method, the continuous surface-switching method, removes the unphysical mixed states of the Ehrenfest method, and in initial tests it produces results that are systematically better than those calculated by the Ehrenfest method. The present article illustrates several of these aspects of nonadiabatic trajectory methods pictorially.

Do Semiclassical Trajectory Theories Provide an Accurate Picture of Radiationless Decay for Systems with Accessible Surface Crossings?

We present quantum mechanical and semiclassical calculations of Feshbach funnel resonances that correspond to long-lived exciplexes in the A 2B2 state of NaH2. These exciplexes decay to the ground state, X 2A1, by a surface crossing in C2v geometry. The quantum mechanical lifetimes and the branching probabilities for competing decay mechanisms are computed for two different NaH2 potential energy matrices, and we explain the results in terms of features of the potential energy matrices. We compare the quantum mechanical calculations of the lifetimes and the average vibrational and rotational quantum numbers of the decay product, H2, to two kinds of semiclassical trajectory calculations: the trajectory surface hopping method and the Ehrenfest self-consistent potential method (also called the time-dependent self-consistent field method). The trajectory surface hopping calculations use Tully's fewest switches algorithm and two different prescriptions for adjusting the momentum during a hop. Both the adiabat...

The distribution of density matrices over potential-energy surfaces: Application to the calculation of the far-wing line shapes for CO2

Within the formalism developed previously for the calculation of the far-wing line shape for molecular systems, most of the computer resources were used to diagonalize anisotropic potential-energy matrices whose sizes are determined by the number of states included. As this number is increased, one expects the results to converge. However, for some systems of atmospheric interest, e.g., CO2, the convergence is so slow that one is unable to obtain converged results within reasonable computer limitations. In the present paper, a new formalism is presented in which the eigenfunctions of the orientations of the system, not the states themselves, are chosen as the complete set of basis functions in Hilbert space. In this case, the diagonalization procedure is unnecessary and one can include as many states as desired. The main computational task is transformed from a diagonalization procedure to the carrying out of multidimensional integrations over the continuous orientational variables. In practice, the integrals are approximated by multidimensional summations over discrete values, the number of which is determined by the resolution required so that the approximated integrals are close to their true values. By choosing reasonable resolutions based on the smooth functional behavior of the integrands, one is able to evaluate the required integrations within reasonable computer time. Furthermore, by introducing weighting functions which are the distribution of the density matrices over potential-energy surfaces, one can reduce the multidimensional integrations to two-dimensional ones. The calculation of the weighting functions can also be carried out with reasonable CPU time and furthermore needs only to be done once for a given molecular system at a specified temperature. Using these as input data, the remaining calculations of the line shapes and corresponding absorption for given potential parameters become straightforward. The formalism is applied in the present paper for linear molecular systems and sample calculations for CO2–CO2 and CO2–N2 are presented. To our knowledge, these are the first, first-principle calculations for the far-wing line shape of CO2 except for the much simpler CO2–rare gas systems.

A geometrical approach to maximizing a variance

The matrix form for the variance of n independently chosen real numbers, x 1, x 2,…, x n , is observed to be the same as that for the potential energy matrix for n coupled oscillators symmetrically arranged on a circle. Unitary matrices that have previously been developed from the irreducible representations of the circular symmetries to diagonalize the potential energy matrices are given. These transformations are applied to the variance. A geometric setting is provided for the transformations, and the maximum variance is found for real numbers x i subject to linear constraints on the choices of the numbers.

Computations for a vibrating system diagonalize the variance

The transformations to diagonalize potential energy matrices for coupled harmonic oscillators will also diagonalize the variance when written in matrix form. After a brief review of a geometrical interpretation of the variance, the transformations are described and an example is given.

Summation of the Born series for the normal modes of the Earth

SUMMARY We derive an explicit formula for the response of a laterally heterogeneous, self-gravitating, rotating, dissipative, and physically dispersive earth to an earthquake. The long-period spectrum at a single station is given by u(ω) =RH0 [V(ω) +ωW−ω2T]−1·S0(ω) where R0 and S0(ω) are the unperturbed source and receiver vectors and H denotes the Hermitian transpose. The quantities V(ω), W, and T are the potential energy, Coriolis, and relative kinetic energy matrices, and V(ω) +ωW−ω2T is the Lagrangian matrix whose eigenvalues are the perturbed complex eigenfrequencies. This simple result is obtained by formal summation of the Born series.

Coupled free oscillations of an aspherical, dissipative, rotating Earth: Galerkin theory

Variational theory based on self‐adjoint equations of motion cannot fully represent the interaction of the earth's seismic free oscillations in the presence of lateral structure, attenuation, and rotation. The more general Galerkin procedure can model correctly the frequencies and attenuation rates of hybrid oscillations. Implementation of either algorithm leads to a generalized matrix eigenvalue problem in which the potential and kinetic energy interactions are separated into distinct matrices. The interaction of the earth's seismic free oscillations due to aspherical structure, attenuation, and rotation is best treated as a matrix eigenvalue problem. The presence of attenuation causes the matrices to be non‐Hermitian and requires the use of a general Galerkin procedure. Physical dispersion, represented as a logarithmic function in frequency, must be represented by a truncated Taylor series about a fiducial frequency in order to be incorporated in the Galerkin formalism in a numerically tractable manner. The earth's rotation introduces an interaction matrix distinct from the potential and kinetic energy matrices, leading to a quadratic eigenvalue problem. A simple approximation leads to an eigenvalue problem linear in squared frequency. Tests show that this approximation is accurate for calculations using modes of frequencies f ≳ 1 mHz, unless interaction across a wide frequency band is modeled. Hybrid oscillation particle motions are represented by matrix eigenvectors that can be significantly nonorthogonal. The degrees of freedom in the low‐frequency seismic system remain distinct, since source excitation is calculated by using dual eigenvectors. Synthetic seismograms that are constructed from Galerkin coupling calculations without reference to this eigenvector nonorthogonality can be disastrously noncausal.

Determination of molecular symmetry coordinates using circulant matrices

A matrix theoretical method for obtaining molecular symmetry coordinates is described. The method is based on the diagonalization of the circulant submatrices of G kinetic and F potential energy matrices representing interaction of equivalent internal coordinates. The symmetry coefficients can be obtained by using a special polynomial relationship between the basic circulant and the submatrices and can be generalized to the case when matrices G and F are block circulant with circulant blocks. The application of the method has been demonstrated for the molecule Hg(CH 3) 2 including the determination of the symmetry species.

Generalization of the rayleigh theorem to gyroscopic systems: PMM vol. 40, n≗ 4, 1976, pp. 606–610

The Rayleigh theorem on the properties of the spectrum of a linear conservative mechanical system is generalized to embrace the gyroscopic systems, i.e. to the case in which the equations of motion contain, in addition to the kinetic and potential energy matrices, an arbitrary skew-symmetric matrix of gyroscopic forces.

ChemInform Abstract: MOLECULAR DYNAMICS, ROOT‐MEAN‐SQUARE AMPLITUDES, STATISTICAL THERMODYNAMICS, AND MOLECULAR POLARIZABILITY FOR THE ISOTOPIC SPECIES OF DIOXYGEN MONOFLUORIDE

A brief survey of vibrational spectral studies for the four isotopic species of dioxygen monofluoride has been made. On the basis of group theoretical considerations, symmetry coordinates have been constructed and kinetic energy matrices (orG matrix elements), potential energy matrices, and secular equations have been derived to calculate the valence force constants. The mean-square amplitudes and root-mean-square amplitudes for both the bonded and nonbonded atom pairs have been calculated at the room temperature. On the basis of a rigid rotator and harmonic oscillator model, enthalpy function, free enthalpy function, entropy, and heat capacity have been calculated from 200 to 2000 °K for all the four isotopic species. On the basis of a delta-function potential model based on the variational method and delta-function electronic wave functions, the bond parallel components, the bond perpendicular components, the contribution by the nonbonding electrons, and the average molecular polarizability have been calculated. The results obtained from these studies clearly confirm a double bond character for the oxygen—oxygen distance and a bond order of less than one-half for the oxygen—fluorine distance. The results have been discussed in relation to the nature of the two characteristic bonds involved in this molecular system.

Molecular dynamics, root-mean-square amplitudes, statistical thermodynamics, and molecular polarizability for the isotopic species of dioxygen monofluoride

A brief survey of vibrational spectral studies for the four isotopic species of dioxygen monofluoride has been made. On the basis of group theoretical considerations, symmetry coordinates have been constructed and kinetic energy matrices (orG matrix elements), potential energy matrices, and secular equations have been derived to calculate the valence force constants. The mean-square amplitudes and root-mean-square amplitudes for both the bonded and nonbonded atom pairs have been calculated at the room temperature. On the basis of a rigid rotator and harmonic oscillator model, enthalpy function, free enthalpy function, entropy, and heat capacity have been calculated from 200 to 2000 °K for all the four isotopic species. On the basis of a delta-function potential model based on the variational method and delta-function electronic wave functions, the bond parallel components, the bond perpendicular components, the contribution by the nonbonding electrons, and the average molecular polarizability have been calculated. The results obtained from these studies clearly confirm a double bond character for the oxygen—oxygen distance and a bond order of less than one-half for the oxygen—fluorine distance. The results have been discussed in relation to the nature of the two characteristic bonds involved in this molecular system.

Molecular Potential Functions Expressed in Cartesian Coordinates: Application to the Orbital Valency Force Field

Potential energy matrices are derived for molecules of the type XYn in terms of the Heath and Linnett orbital valency force field. Cartesian coordinates are used, and a simple calculation procedure is outlined that makes use of generally available computer subroutines for matrix diagonalization, etc. The method makes no use of symmetry coordinates, and is therefore particularly convenient for problems involving isotopic substitutions. Extension to larger molecules is suggested but not carried out in detail. The general method may be useful for other force field models, also.

On a new method for the calculation of dipole moment derivatives from infrared intensities

The recent method of Girljavallabhan, Joseph and Venkateswarlu for the calculation of dipole moment derivatives from infrared intensities is unsatisfactory because their matrix of intensity constants is not generally diagonalized by the normal coordinate transformation that diagonalizes the potential and kinetic energy matrices.

The normal coordinates of nitromethane-like molecules

The form of the vibrational kinetic and potential energy matrices for molecules with structures similar to nitromethane is discussed, and the kinetic energy matrix elements tabulated. Symmetry aspects of the vibrational problem are discussed in terms of the group defined by Longuet-Higgins. The possible forms of the normal coordinates are derived, and their relevance to the associated internal rotation problem indicated.

Normal co-ordinate analysis of 1–4 trans polybutadiene

The frequencies of the active fundamental vibrations of 1–4 trans polybutadiene and of its symmetrical deutero-derivatives, viz. (CH 2CDCDCH 2) m , (CD 2CHCHCD 2) n and (CD 2CDCDCD 2) m , have been calculated by means of valence force constants derived from cyclo-olefins. The method for obtaining the kinetic and potential energy matrices for an isolated polymer chain is briefly discussed, and compared with a procedure previously presented in the literature. A detailed discussion of the i.r. spectrum of 1–4 trans polybutadiene, based on normal co-ordinate analysis, is given.

Vibrational Symmetry and Selection Rules for an Ethane-Type Molecule

The vibrational symmetry and selection rules for an ethane-type molecule are re-examined. It is demonstrated that the kinetic and potential energy matrices can be fully reduced under either D3h or D3d. However, the dipole moment and polarizability tensors do not, in general, show quite as much symmetry, so that certain infrared and Raman coincidences, forbidden by D3d selection rules, become allowed.

Potential Constants and Thermodynamic Functions of Iodine Pentafluoride

AbstractAn orthonormalized set of symmetry coordinates satisfying the transformation properties has been constructed for the iodine pentafluoride molecule having the tetragonal pyramidal structure with the symmetry point group C4v. Utilizing these symmetry coordinates and a potential function consisting of the ordinary valance force terms and central force terms, the elements of potential and kinetic energy matrices have been obtained. From the observed Raman and infrared fundamental frequencies the potential constants and thermodynamic functions such as heat content, free energy, entropy and heat capacity have been calculated for a rigid rotator, harmonic oscillator approximation.