Potential Energy Matrix Research Articles

Ab initio potential energy surfaces and nonadiabatic collision dynamics in H++O2 system

The adiabatic potential energy surfaces for the lowest five electronic states of (3)A" symmetry for the H(+)+O(2) collision system have been obtained at the multireference configuration interaction level of accuracy using Dunning's correlation consistent polarized valence triple zeta basis set. The radial nonadiabatic coupling terms and the mixing angle between the lowest two electronic states (1 (3)A" and 2 (3)A"), which adiabatically correlate in the asymptotic limit to H((2)S)+O(2) (+)(X (2)Pi(g)) and H(+)+O(2)(X (3)Sigma(g)(-)), respectively, have been computed using ab initio procedures at the same level of accuracy to yield the corresponding quasidiabatic potential energy matrix. The computed strengths of the vibrational coupling matrix elements reflect the trend observed for inelastic vibrational excitations of O(2) in the experiments at collision energy of 9.5 eV. The quantum dynamics has been preformed on the newly obtained coupled quasidiabatic potential energy surfaces under the vibrational close-coupling rotational infinite-order sudden framework at the experimental collision energy of 9.5 eV. The present theoretical results for vibrational elastic/inelastic excitations of O(2) are in overall good agreement with the available experimental data obtained from the proton energy-loss spectra in molecular beam experiments [F. A. Gianturco et al., J. Phys. B 14, 667 (1981)]. The results for the complementary charge transfer processes are also presented at this collision energy.

Ab initio potential energy surfaces and nonadiabatic interactions in the H++NO collision system

Ab initio calculations on the H(+)+NO system have been carried out in Jacobi coordinates at the multireference configuration interaction level employing Dunning's correlation-consistent polarized valence triple zeta basis set to analyze the role of low-lying electronic excited states in influencing the collision dynamics relevant to the experimental collision energy range of 9.5-30 eV. The lowest two adiabatic potential energy surfaces, asymptotically correlating to H(+)+NO(X (2)Pi) and H((2)S)+NO(+)(X (1)Sigma(+)), have been obtained. Using ab initio procedures, the (radial) nonadiabatic couplings and the mixing angle between the lowest two electronic states (1 (2)A' and 2 (2)A') have been obtained to yield the corresponding quasidiabatic potential energy matrix. The strengths of the computed vibrational coupling matrix elements reflect a similar trend, as has been observed experimentally in the magnitudes of the state-to-state transition probability for the inelastic vibrational excitations [J. Krutein and F. Linder, J. Chem. Phys. 71, 559 (1979); F. A. Gianturco et al., J. Phys. B 14, 667 (1981)].

Energy and energy gradient matrix elements with N-particle explicitly correlated complex Gaussian basis functions with L=1

In this work we consider explicitly correlated complex Gaussian basis functions for expanding the wave function of an N-particle system with the L=1 total orbital angular momentum. We derive analytical expressions for various matrix elements with these basis functions including the overlap, kinetic energy, and potential energy (Coulomb interaction) matrix elements, as well as matrix elements of other quantities. The derivatives of the overlap, kinetic, and potential energy integrals with respect to the Gaussian exponential parameters are also derived and used to calculate the energy gradient. All the derivations are performed using the formalism of the matrix differential calculus that facilitates a way of expressing the integrals in an elegant matrix form, which is convenient for the theoretical analysis and the computer implementation. The new method is tested in calculations of two systems: the lowest P state of the beryllium atom and the bound P state of the positronium molecule (with the negative parity). Both calculations yielded new, lowest-to-date, variational upper bounds, while the number of basis functions used was significantly smaller than in previous studies. It was possible to accomplish this due to the use of the analytic energy gradient in the minimization of the variational energy.

A Diabatic Representation Including Both Valence Nonadiabatic Interactions and Spin−Orbit Effects for Reaction Dynamics

A diabatic representation is convenient in the study of electronically nonadiabatic chemical reactions because the diabatic energies and couplings are smooth functions of the nuclear coordinates and the couplings are scalar quantities. A method called the fourfold way was devised in our group to generate diabatic representations for spin-free electronic states. One drawback of diabatic states computed from the spin-free Hamiltonian, called a valence diabatic representation, for systems in which spin-orbit coupling cannot be ignored is that the couplings between the states are not zero in asymptotic regions, leading to difficulties in the calculation of reaction probabilities and other properties by semiclassical dynamics methods. Here we report an extension of the fourfold way to construct diabatic representations suitable for spin-coupled systems. In this article we formulate the method for the case of even-electron systems that yield pairs of fragments with doublet spin multiplicity. For this type of system, we introduce the further simplification of calculating the triplet diabatic energies in terms of the singlet diabatic energies via Slater's rules and assuming constant ratios of Coulomb to exchange integrals. Furthermore, the valence diabatic couplings in the triplet manifold are taken equal to the singlet ones. An important feature of the method is the introduction of scaling functions, as they allow one to deal with multibond reactions without having to include high-energy diabatic states. The global transformation matrix to the new diabatic representation, called the spin-valence diabatic representation, is constructed as the product of channel-specific transformation matrices, each one taken as the product of an asymptotic transformation matrix and a scaling function that depends on ratios of the spin-orbit splitting and the valence splittings. Thus the underlying basis functions are recoupled into suitable diabatic basis functions in a manner that provides a multibond generalization of the switch between Hund's cases in diatomic spectroscopy. The spin-orbit matrix elements in this representation are taken equal to their atomic values times a scaling function that depends on the internuclear distances. The spin-valence diabatic potential energy matrix is suitable for semiclassical dynamics simulations. Diagonalization of this matrix produces the spin-coupled adiabatic energies. For the sake of illustration, diabatic potential energy matrices are constructed along bond-fission coordinates for the HBr and the BrCH(2)Cl molecules. Comparison of the spin-coupled adiabatic energies obtained from the spin-valence diabatics with those obtained by ab initio calculations with geometry-dependent spin-orbit matrix elements shows that the new method is sufficiently accurate for practical purposes. The method formulated here should be most useful for systems with a large number of atoms, especially heavy atoms, and/or a large number of spin-coupled electronic states.

A new approach and some criteria to deal with the theory of the normal modes of vibrations in the elpasolite stoichiometric type systems short range intramolecular interactions

There are a number of stoichiometric elpasolite type systems, for which a considerable amount of experimental data has being accumulated over the last three decades. For these purposes, data from linear and non linear optics is available, though we must emphasize, that the experimental evidence is rather scarce or incomplete, and as a consequence, it is not obvious to claim that many relevant problems in the solid state spectroscopy for these systems have been adequately solved. We have been working on systems such as Cs 2NaLnCl 6 for Ln = Pr, Eu, Tb, Dy, Ho, Er, and Tm, aiming to advance our understanding in those electronic and vibrational factors, upon which the one photon electronic transitions depend upon. We have chosen in this article, to report a novel strategy to rationalize the normal coordinate analysis so as to get a closer and more realistic approach to deal with those contributions which determine, what we regard as a natural potential energy distribution (NPED). It is essential to agree on the conditions upon which, we develop our convergence tests based on a physical view, so as to discern between selections of experimental assignments from a more comprehensive viewpoint. As a starting point, we review the magnificent contribution put forward by Lentz, who may claim priority in this area of the vibrational spectroscopy for the cryolite and elpasolite compounds and subsequently by other authors, whom have introduced some correction terms in the description of the vibrational force field. It is indeed a major task to produce a better description for this interacting potential, based on a maximum of 10 vibrational frequencies. In this work, our description is based upon a total of 72 internal coordinates and an initial number of 98 internal force constants. We show that the potential energy matrix F may be symmetrized by reducing the number of these latter to 81 internal force constants. For each of the elpasolite type systems, we have considered a total of three more representatives F-matrices, and calculated both the diagonal and non diagonal contributions to the observed vibrational wave numbers, λ i = ( ν i / 1303.16 ) for each of the experimental data. Following this procedure, we have been able to understand the sensitivity of the mixing between the symmetry coordinates for the same symmetry species with reference to a given vibrational frequency. It is shown that our current approach is both flexible and general so as to work out the most important contributions to the vibrational factor. The current formalism may be employed for other more complicated systems.

Nonadiabatic effects in C–Br bond scission in the photodissociation of bromoacetyl chloride

Bromoacetyl chloride photodissociation has been interpreted as a paradigmatic example of a process in which nonadiabatic effects play a major role. In molecular beam experiments by Butler and co-workers [J. Chem. Phys. 95, 3848 (1991); J. Chem. Phys. 97, 355 (1992)], BrCH2C(O)Cl was prepared in its ground electronic state (S0) and excited with a laser at 248 nm to its first excited singlet state (S1). The two main ensuing photoreactions are the ruptures of the C-Cl bond and of the C-Br bond. A nonadiabatic model was proposed in which the C-Br scission is strongly suppressed due to nonadiabatic recrossing at the barrier formed by the avoided crossing between the S1 and S2 states. Recent reduced-dimensional dynamical studies lend support to this model. However, another interpretation that has been given for the experimental results is that the reduced probability of C-Br scission is a consequence of incomplete intramolecular energy redistribution. To provide further insight into this problem, we have studied the energetically lowest six singlet electronic states of bromoacetyl chloride by using an ab initio multiconfigurational perturbative electronic structure method. Stationary points (minima and saddle points) and minimum energy paths have been characterized on the S0 and S1 potential energy surfaces. The fourfold way diabatization method has been applied to transform five adiabatic excited electronic states to a diabatic representation. The diabatic potential energy matrix of the first five excited singlet states has been constructed along several cuts of the potential energy hypersurfaces. The thermochemistry of the photodissociation reactions and a comparison with experimental translational energy distributions strongly suggest that nonadiabatic effects dominate the C-Br scission, but that the reaction proceeds along the energetically allowed diabatic pathway to excited-state products instead of being nonadiabatically suppressed. This conclusion is also supported by the low values of the diabatic couplings on the C-Br scission reaction path. The methodology established in the present study will be used for the construction of global potential energy surfaces suitable for multidimensional dynamics simulations to test these preliminary interpretations.

Interpreting Correlated Motions Using Normal Mode Analysis

With the increased popularity of normal mode analyses in structural biology, it is important to carefully consider how to best utilize the results for gaining biological insights without over interpretation. The discussion in this article argues that for the purpose of identifying correlated motions in biomolecules, a case separate from concomitant conformational changes of structural motifs, it is generally important to use a large number of normal modes. This is illustrated through three increasingly complex examples. The simplest case includes two bilinearly coupled harmonic oscillators and serves as a straightforward problem where the important considerations are explicit and transparent. The argument is then generalized to include a system of N-coupled harmonic oscillators and finally to a realistic biomolecule. Although a small number of normal modes are useful for probing structural flexibility, it is clear that a much larger number of modes are required for properly investigating correlated motions in biomolecules.

Direct calculation of coupled diabatic potential-energy surfaces for ammonia and mapping of a four-dimensional conical intersection seam

We used multiconfiguration quasidegenerate perturbation theory and the fourfold-way direct diabatization scheme to calculate ab initio potential-energy surfaces at 3600 nuclear geometries of NH3. The calculations yield the adiabatic and diabatic potential-energy surfaces for the ground and first electronically excited singlet states and also the diabatic coupling surfaces. The diabatic surfaces and coupling were fitted analytically to functional forms to obtain a permutationally invariant 2 x 2 diabatic potential-energy matrix. An analytic representation of the adiabatic potential-energy surfaces is then obtained by diagonalizing the diabatic potential-energy matrix. The analytic representation of the surfaces gives an analytic representation of the four-dimensional conical intersection seam which is discussed in detail.

Higher order (A+E)⊗e pseudo-Jahn–Teller coupling

The pseudo-Jahn-Teller (PJT) coupling of a nondegenerate state A with a twofold degenerate state E by a degenerate vibrational mode e is studied for a general system with a C(3) main rotational axis. The PJT coupling terms up to sixth order are derived by symmetry considerations for this general (A+E) multiply sign in circle e case. The obtained expression for the 3 x 3 diabatic potential energy matrix is found to be closely related to the expression recently developed for the higher order Jahn-Teller case [A. Viel and W. Eisfeld, J. Chem. Phys. 120, 4603 (2004)]. The dynamical PJT coupling, which can arise for states of appropriate symmetry if one of the vibrational modes induces a change of the nuclear point group between D(3h), C(3v), C(3h), and C(3), is discussed. The effect of the higher order PJT coupling is tested by a two-dimensional model study based on the e bending mode of NH(3)(+). The models are analyzed by fitting the two-dimensional potential energy surfaces. The significance of the higher order terms on the nonadiabatic dynamics is demonstrated by quantum wave packet propagations.

Vibronic Structure of the Permanganate Absorption Spectrum from Time-Dependent Density Functional Calculations

The UV absorption spectrum of the permanganate anion is a prototype transition-metal complex spectrum. Despite this being a simple d0 Td system, for which a beautiful spectrum with detailed vibrational structure has been available since 1967, the assignment of the second and third bands is still very controversial. The issue can be resolved only by an elucidation of the intricate vibronic structure of the spectrum. We investigate the vibronic coupling by means of linear-response time-dependent density functional calculations. By means of a diabatizing scheme that employs the transition densities obtained in the TDDFT calculations in many geometries around Re, we construct a Taylor series expansion in the normal coordinates of a diabatic potential energy matrix, coupling 24 excited states. The simulated vibronic structure is in good agreement with the experimental absorption spectrum after the adjustment of some of the calculated vertical excitation energies. The peculiar blurred vibronic structure of the second band, which is a very distinctive feature of the experimental spectrum, is fully reproduced in the calculations. It is caused by the double-well shape of the adiabatic energy surface along the Jahn-Teller active e mode of the allowed 1E state arising from the second 1T2 state, which exhibits a Jahn-Teller splitting into 1B2 and 1E states. We trace the double-well shape to an avoided crossing between two diabatic states with different orbital-excitation character. The crossing can be explained at the molecular orbital level from the Jahn-Teller splitting of the set of 7t2{3d(xy), 3d(xz), 3d(yz)} orbitals (the LUMO + 1), to which the excitations characterizing the diabatic states take place. In contrast to its character in the two well regions, at Re the 2(1)T2 state is not predominantly an excitation to the LUMO + 1, but has more HOMO - 1 --> LUMO (2e = {3d(x2-y2), 3d(z2)}) character. The changing character of the 2(1)T2 - 1E state along the e mode implies that the assignment of the experimental bands to single orbital transitions is too simplistic intrinsically. This spectrum, and notably the blurring of the vibronic structure in the second band, can be understood only from the extensive configurational mixing and vibronic coupling between the excited states. This solves the long-standing assignment problem of these bands.

Electronic Diabatic Framework: Restrictions Due to Quantization of the Nonadiabatic Coupling Matrix

In this article, two issues related to the size of the electronic diabatic potential energy matrix are treated. (a) We frequently mention the fact that the dimension of a diabatic matrix obtained by a unitary transformation from the adiabatic framework is determined by the way the nonadiabatic coupling matrix τ breaks up into blocks. In this article, we prove for the first time that the size of the diabatic matrix as obtained in a direct way is determined in the same way. In other words, if the dimension of the above-mentioned decoupled block is N, then the dimension of any diabatic potential energy matrix with physical relevance has to be N as well, regardless of how it was derived. This number, N, is also equal to the number of coupled diabatic−Schrödinger equations to be solved. (b) The second issue is, consequently, related to the actual required number of coupled Schrödinger equations to be solved to obtain a well-converged solution. Starting with the earlier introduced number N, we show that this number can be reduced, and in fact, it is most likely equal to the number of energetically open adiabatic states (for a given energy). While doing that, we rigorously derived the relevant diabatic potential matrix for this reduced case. We also worked out in detail an example related to a three-state case and derived the relevant 2 × 2 diabatic potential matrix.

Vibronic coupling and double excitations in linear response time-dependent density functional calculations: dipole-allowed states of N2.

The present study serves two purposes. First, we evaluate the ability of present time-dependent density functional response theory (TDDFRT) methods to deal with avoided crossings, i.e., vibronic coupling effects. In the second place, taking the vibronic coupling effects into account enables us, by comparison to the configuration analysis in a recent ab initio study [J. Chem. Phys. 115, 6438 (2001)], to identify the neglect of double excitations as the prime cause of limited accuracy of these linear response based TDDFRT calculations for specific states. The "statistical averaging of (model) orbital potentials (SAOP)" Kohn-Sham potential is used together with the standard adiabatic local-density approximation (ALDA) for the exchange-correlation kernel. We use the N2 molecule as prototype, since the TDDFRT/SAOP calculations have already been shown to be accurate for the vertical excitations, while this molecule has a well-studied, intricate vibronic structure as well as significant double excitation nature in the lowest 1Pi(u) state at elongated bond lengths. A simple diabatizing scheme is employed to obtain a diabatic potential energy matrix, from which we obtain the absorption spectrum of N2 including vibronic coupling effects. Considering the six lowest dipole allowed transitions of 1Sigma(u)+ and 1Pi(u) symmetry, we observe a good general agreement and conclude that avoided crossings and vibronic coupling can indeed be treated satisfactorily on the basis of TDDFRT excitation energies. However, there is one state for which the accuracy of TDDFRT/ALDA clearly breaks down. This is the state for which the ab initio calculations find significant double excitation character. To deal with double excitation character is an important challenge for time-dependent density functional theory.

An application of the interpolating scaling functions to wave packet propagation

The wave packet propagation in the basis of interpolating scaling functions (ISF) is studied. The ISF are well known in the multiresolution analysis based on spline biorthogonal wavelets. The ISF form a cardinal basis set corresponding to an equidistantly spaced grid. They have compact support of the size determined by the order of the underlying interpolating polynomial. In this basis the potential energy matrix is diagonal. The kinetic energy matrix is sparse, and in the 1D case, has a band-diagonal structure. An important future of the basis is that matrix elements of a Hamiltonian are exactly computed by means of simple algebraic transformations efficiently implemented numerically. Therefore, the number of grid points and the order of the underlying interpolating polynomial can easily be varied allowing one to approach the accuracy of pseudospectral methods in a regular manner, similar to the high order finite difference methods. The results for the calculation of the H+H2 collinear collision shows that the ISF provide one with an accurate and efficient representation for use in wave packet propagation method.

An application of the interpolating scaling functions to wave packet propagation

The wave packet propagation in the basis of interpolating scaling functions (ISF) is studied. The ISF are well known in the multiresolution analysis based on spline biorthogonal wavelets. The ISF form a cardinal basis set corresponding to an equidistantly spaced grid. They have compact support of the size determined by the order of the underlying interpolating polynomial. In this basis the potential energy matrix is diagonal. The kinetic energy matrix is sparse, and in the 1D case, has a band-diagonal structure. An important future of the basis is that matrix elements of a Hamiltonian are exactly computed by means of simple algebraic transformations efficiently implemented numerically. Therefore, the number of grid points and the order of the underlying interpolating polynomial can easily be varied allowing one to approach the accuracy of pseudospectral methods in a regular manner, similar to the high order finite difference methods. The results for the calculation of the H+H 2 collinear collision shows that the ISF provide one with an accurate and efficient representation for use in wave packet propagation method.

Variational DVR Calculations

In the discrete variable representation (DVR) method the potential energy matrix \({\cal V}\) has a particular simple form. It is diagonal with values of the interaction potential at the discrete points. However, this simple form is obtained by making approximations in the calculation of the matrix elements of \({\cal V}\). As a consequence the results cannot be considered as variational estimates. We will show how to recover the variational character of the method using the discrete variable representation eigenvectors as trial functions and performing a variational calculation in a restricted Hilbert space.

Photoinduced dynamics of the valence states of ethene: A six-dimensional potential-energy surface of three electronic states with several conical intersections

A six-dimensional analytic potential-energy surface of the three valence states (N, V, Z) of ethene has been constructed on the basis of complete-active-space ab initio calculations and ab initio calculations with perturbation theory of second order based on a complete active reference space. The nuclear coordinate space is spanned by the torsion, the C–C stretch coordinate, the left and right pyramidalization and the symmetric and antisymmetric scissor coordinates. The C–H stretch coordinates and the CH2 rocking angles are kept frozen at their ground-state equilibrium value. A diabatic representation of the valence states of ethene has been constructed within the framework of a Hückel-type model. The diabatic potential-energy elements are represented as analytic functions of the relevant coordinates. The parameters of the analytic functions have been determined by a least-squares fit of the eigenvalues of the diabatic potential-energy matrix to the ab initio data for one-dimensional and two-dimensional cuts of the six-dimensional surface. As a function of the torsion, the analytic potential-energy surface describes the intersections of the V and Z states for torsional angles near 90°, which are converted into conical intersections by the antisymmetric scissor mode. As a function of pyramidalization of perpendicular ethene, it describes the intersections of the diabatic N and Z states, which are converted into conical intersections by displacements in the torsional mode. The analytic potential-energy surfaces can provide the basis for a quantum wave packet description of the internal conversion of photoexcited ethene to the electronic ground state via conical intersections.

Variational calculations of vibrational energy levels for XY4 molecules 1. Stretching states

A variational method for calculating excited stretching states of symmetric tetrahedral pentaatomic molecules is presented based on the use of Radau coordinates and Morse oscillator-like basis functions. Symmetry is used both to reduce the size of the secular matrix to be diag-onalized and to calculate the potential energy matrix elements over a reduced grid of quadrature points. Test results are presented for methane, silane and germane. For CH4, stretch-bend coupling is found to be significant, whereas it is less important for the more strongly local mode SiH4 and GeH4 molecules. Converged results are obtained for stretching states significantly higher than considered in previous calculations. These states will be used to represent stretching motions in a fully coupled stretch-bend calculation.

Variational calculations of vibrational energy levels for XY4 molecules: 2. Bending states of methane

A variational method for calculating excited bending states of symmetric tetrahedral pentaatomic molecules is presented based on the use of Radau coordinates and Jacobi polynomials as the basis functions. Symmetry is used both to reduce the size of secular matrix to be diagonalized and to calculate potential energy matrix elements over a reduced grid of quadrature points. Methods of treating the redundant coordinate are investigated and fitting is found to be more effective than the use of Taylor expansions. Test results are presented for methane, for which stretch-bend coupling and the contribution due to the redundant coordinate are found to be significant. Converged results are obtained for bending states significantly higher than considered in previous calculations. These states will be used as a basis for bending motions in a fully coupled stretch-bend calculation.

Coupled quasidiabatic potential energy surfaces for LiFH

We present high-level ab initio calculations for the global adiabatic potential energy surfaces of the ground state (X̃ 2A′) and several excited states (Ã 2A′, B̃ 2A″, C̃ 2A′, D̃ 2A′, and Ẽ 2A″) of LiFH, including the valleys leading to Li+HF and LiF+H. The ab initio calculations were carried out using the multireference singles and doubles configuration interaction method with 99 reference configuration state functions (CSFs) for the A′2 states and 39 reference CSFs for the A″2 states. The basis set consisted of 140 contracted Gaussian functions, including specifically optimized diffuse functions, and calculations were performed on a dense grid of ∼3500 nuclear geometries which allowed us to construct an accurate analytic representation of the two lowest-energy LiFH potential energy surfaces. An analytic 2×2 quasidiabatic potential energy matrix was obtained by fitting physically motivated functional forms to the ab initio data for the two lowest-energy adiabatic states and explicitly including long-range interactions. The newly presented LiFH fit is compared to several ground-state LiFH fits and one excited-state LiFH fit that have appeared in the literature.

Construction scheme for regularized diabatic states

A simple construction scheme for quasidiabatic electronic states implemented earlier for a Jahn–Teller situation [J. Chem. Phys. 110, 9371 (1999)] is extended to the case of a seam of symmetry-allowed conical intersections. It is based on the idea of removing only the singular part of the nonadiabatic coupling elements, leading to the notion of “regularized” diabatic states. Explicit working equations are given for the resulting potential energy matrix which require only information from the adiabatic potential energy surfaces alone. The approach is tested for the photodissociation of H2S and O3 and very good agreement with reference data from the literature is found.