Complex coordinate calculations of resonances in N-electron atoms

We present a discussion of the functional dependences of the wave functions for bound, resonant, and scattering states on the radial coordinate ρ and the rotation angle α in the complex coordinate method. We conclude that for bound states and resonances, ρ and α are constrained to appear in the wave functions only in the for ρ exp(iα). On the other hand, this constraint is not obtained for the scattering states since the energy of the scattering states depends on α. In addition we suggest a partitioning of the resonant wave function into two parts—a boundlike or “Q-space” part and a scattering like or “P-space” part. With these concepts one can incorporate physical insight into the choice of configurations as one does in other methods and can apply the complex coordinate method to many electron systems with an expected rate of convergence similar to other techniques. Its advantages are that a single calculation yields the position and width of the resonance, only square integrable functions are used, only a solution of a straightforward eigenvalue problem is required unlike some methods, arbitrarily accurate target states are easily incorporated, and polarization terms can easily be explicitly included. Variational calculations for the position and width of the lowest ²S resonance in the negative helium ion are reported using trial wave functions containing 39, 43, 55, 24, and 32 “P-space” configurations, respectively. Values of 19.387 eV and 12.13 meV are obtained for the position and width, respectively, for the resonance over a range in the rotation angle of almost two orders of magnitude. One also finds that inclusion of free-particle-like basis functions improves the representation of the scattering states.     

A new method for the direct calculation of resonance parameters with application to the quasibound states of the H2X1Σ g + system

A new method for the direct calculation of resonance parameters is presented. It is based upon searching for poles of the scattering matrix at complex energies. This search is expedited by the use of analytic derivatives of the scattering matrix with respect to the total energy. This procedure is applied initially to a single channel problem, but is generalizable to more complicated systems. Using the most accurate available potential energy data, we calculate resonance parameters for all of the physically important quasibound states of the ground electronic state of the hydrogen molecule. Corrections to the Born-Oppenheimer potential are included and assessed. The new method has no difficulty locating resonances with widths greater than about 1×10^–7 cm^–1. It is easier to find narrow resonances by monitoring the dependence of the imaginary part of the reactance matrix on the real part of a complex energy than to monitor the dependence of the eigenphase sum on energy at real energies.     

Role of resonances in building collision cross-sections: Mittag-Leffler-based study

The method of complex dilation is used to define the partial wave S-matrix in the sector of the fourth quadrant of the complex energy plane. Two ways of obtaining the expansion coefficients—the partial wave S-matrix residues—are studied. The Mittag-Leffler decomposition of the partial wave S-matrix as a sum of residue terms and an integral contribution is used to define the contributions of a number of the partial wave S-matrix poles, related to a 1-D potential, to the corresponding S-wave cross-section. The obtained expansion demonstrates a way of describing the contribution from a single pole to the partial wave S-matrix and thereby to various types of cross-sections. Our model study shows how peaks in a cross-section not only can be attributed to so-called isolated resonances but also to a set of overlapping barrier-type resonances. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Coalescence of metastable states in chemical reactions: double poles of the scattering matrix and exceptional points

Accidental degeneracy of two scattering resonances in chemical reactions is discussed. A novel analytical parameterization for the scattering matrix elements valid in the presence of two interacting resonances is derived and applied to investigate the coalescence of metastable states leading to double poles in the partial-wave amplitude. We show how double poles can manifest themselves in reaction cross-sections and discuss their relationship with exceptional points. The presence of two degenerate resonances interacting with each other can create a peculiar double-peak structure in the reaction probability. At the energy where the two decaying states coalesce the scattering matrix has a double pole and correspondingly the reaction probability reaches its minimum value. This novel formula has been tested with the well-established data obtained for the F + H₂ reaction, for which previous studies have revealed the existence of two interfering resonance pathways. The good agreement of model predictions and quantum scattering calculations shows that the present approach is reliable. Hence, we claim that it provides a way for the observation of interacting resonances in molecular collisions. Resonance positions and partial widths can be obtained by fitting the numerically calculated state-to-state reaction probabilities to the square modulus of the novel expression for the scattering matrix elements derived in the paper.     

A simple classical prediction of quantal resonances in collinear reactive scattering

Quantal collinear reactive scattering computations have shown that in the vicinity of thresholds of reactant or product vibrational states, one finds resonances in the state to state reaction probability. We find that these resonances can be explained classically in terms of energy transfer between adiabatic reactant and product channels. This transfer is attributable to resonant periodic orbits, resonating between reactants and products. The classical condition for a quantal resonance is that the action of the orbit over one period be an integer (in units of h) and that the energy at which this occurs be lower than the adiabatic barrier heights of the resonating states. These conditions suffice for a prediction of the location of the quantal resonance within a 1% accuracy!     

Combination of Resonance and Non-Resonance Chiral Raman Scattering in a Cobalt(III) Complex

Resonance Raman optical activity (RROA) spectra with high sensitivity reveal details on molecular structure, chirality, and excited electronic properties. Despite the difficulty of the measurements, the recorded data for the Co(III) complex with S,S-N,N-ethylenediaminedisuccinic acid are of exceptional quality and, coupled with the theory, spectacularly document the molecular behavior in resonance. This includes a huge enhancement of the chiral scattering, contribution of the antisymmetric polarizabilities to the signal, and the Herzberg-Teller effect significantly shaping the spectra. The chiral component is by about one order of magnitude bigger than for an analogous aluminum complex. The band assignment and intensity profile were confirmed by simulations based on density functional and vibronic theories. The resonance was attributed to the S₀→S₃ transition, with the strongest signal enhancement of Raman and ROA spectral bands below about 800 cm⁻¹. For higher wavenumbers, other excited electronic states contribute to the scattering in a less resonant way. RROA spectroscopy thus appears as a unique tool to study the structure and electronic states of absorbing molecules in analytical chemistry, biology, and material science.     

Quantum wave packet study of N(2D) + H2 reactive scattering

N(²D) + H₂ → NH + H reaction at zero total angular momentum is studied by using a time dependent quantum wave packet method. State‐to‐state and state‐to‐all reactive scattering probabilities for a broad range of energy are calculated. The probabilities show many sharp peaks that ascribed to reactive scattering resonances. The probabilities for J > 0 are estimated by using the J‐shifting method. The integral cross sections and thermal rate constants are then calculated. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Exploiting the analyticity of Schrödinger operators: Theory and the computation of partial cross sections

The idea of dilation, or dilatation, analyticity with respect to complex scaling of the interparticle distances for nonrelativistic atomic or molecular electronic Hamiltonians is now over 20 years old and the first major reviews are just over 10. The method continues to be a fruitful source of new theoretical and computational results. Under the scale transformation r → re^iθ, the usual “spectrum” of bound states is exactly preserved, and scattering continua are “rotated” off the real axis by an angle of −2Re(θ) about their respective thresholds. Useful features of this transformation are (1) that resonances are exposed, and, thus, (complex) resonance eigenvalues are easily calculated as the wave functions are L², and standard results of Kato-type perturbation theory can thus be applied to them; (2) that utilization of this technique to study atoms in ac and dc fields was an early, and still evolving extension of the original theory; and (3) the fact that the “continua” are rotated off the real energy axis allows scattering information to be extracted from computations carried out entirely L² in bases as the usual resolvents of scattering theory are no longer singular on the real axis. After a brief survey of the technique and its applications, these ideas are illustrated by discussion of positive energy-bound states and resonances, the extension of the theory to include the dc Stark effect, and a review of the resolution of the initially perplexing problem of atomic and molecular bound states in continua. These theoretical results are followed by a discussion of some very recent computational results, allowing computation of atomic partial photoionization cross sections with no specific coordinate space enforcement of boundary conditions, a highly advantageous situation for calculation of the partial cross section for three-body breakup, as in the process ℏ ω + He → He²⁺ + e⁻ + e⁻. © 1996 John Wiley & Sons, Inc.     

X-ray spectra and electronic structure of several ternary chalcogenides and their solid solutions

The electronic energy structure of substitution solid solutions CuGa(S_xSe_1−x)₂ is studied within wide limits of sulfur concentration x in the onion sublattice. The SK absorption spectrum is calculated for CuGaS₂ in a high-order multiple scattering approximation using the FEFF7 program. For all concentrations x, partial densities of states are calculated in a full multiple scattering approximation by the local coherent potential method. The calculation schemes for the filled and vacant states are employed, which differ in a choice of the crystalline potential. The effect of a vacancy on the SK level on the density of the free Sp states is considered. The theoretical K absorption spectra and densities of states of CuGaS₂ are compared with the experimental X-ray and X-ray photoelectron spectra. The calculated curves are in good agreement with the experiment. It is established that the densities of the S and Se p states change smoothly with varying concentration of anions. Translated fromZhurnal Strukturnoi Khimii, Vol. 39, No. 6, pp. 1076–1082, November–December, 1998.     

Resonances in the elastic electron-SF6 molecule scattering

In agreement with previous calculations by the authors, three definite resonances in the elastic electron-SF₆ molecule scattering in the energy range 5–40 eV have been obtained with a multiple scattering method, elaborated originally for molecular bound states and modified here for scattering states. Results with various more-reliable one-electron potentials have been compared with previous calculations and available experimental data.     

Direct evaluation of the lifetime matrix by the hyperquantization algorithm: narrow resonances in the F + H2 reaction dynamics and their splitting for nonzero angular momentum

We propose a new method for the direct and efficient evaluation of the Felix Smith's lifetime Q matrix for reactive scattering problems. Simultaneous propagation of the solution to a set of close-coupled equations together with its energy derivative allows one to avoid common problems pertinent to the finite-difference approach. The procedure is implemented on a reactive scattering code which employs the hyperquantization algorithm and the Johnson-Manolopoulos [J. Comput. Phys. 13, 455 (1973); J. Chem. Phys 85, 6425 (1986)] propagation to obtain the complete S matrix and scattering observables. As an application of the developed formalism, we focus on the total angular momentum dependence of narrow under-barrier resonances supported by van der Waals wells of the title reaction. Using our method, we fully characterize these metastable states obtaining their positions and lifetimes from Lorentzian fits to the largest eigenvalue of the lifetime matrix. Remarkable splittings of the resonances observed at J>0 are rationalized in terms of a hyperspherical model. In order to provide an insight on the decay mechanism, the Q-matrix eigenvectors are analyzed and the dominant channels populated during the decomposition of metastable states are determined. Possible relevance of the present results to reactive scattering experiments is discussed.     

Molecular Structure, Vibrational Spectrum and Ionization Energy of m-Dimethoxybenzene

The optimized molecular geometries of the three rotamers of m-dimethoxybenzene in the ground So and electronically excited Sl states were predicted by ab initio and density functional theory （DFF） calculations. Their vibrational spectra in the St state were studied by one color resonant two photon ionization （1C-R2PI） method, and their ionization energies were measured by two color resonant two photon ionization （2C-R2PI） experiment. The optimized molecular geometries showed that the total energy of conformer a was the lowest in the So state. Most of the active vibrations assigned from the 1C-R2PI spectrum were found to be of the in-plane ring modes. The ionization energies （IE） of conformers a, b and c were determined to be 63521, 64487 and 63755 cm^-1, respectively.     

Wavepacket quantum dynamics

The article reviews the use of wavepackets in molecular quantum dynamics. The basic theory concerned with their use in both reactive molecular scattering and photodissociation dynamics is outlined. The great advantage of using wavepackets is that the full S matrix for the scattering problem need not be evaluated, and the numerical effort can be concentrated on those initial molecular quantum states which are of interest. Wavepackets may be used within a time-dependent or a time-independent framework, both are discussed and compared. Some examples of calculations from both reactive scattering and photodissociation theory are given.     

Resonances and antibound states in a Morse potential

We calculate the resonant and antibound state energies for a Morse potential with a centrifugal barrier using Siegert boundary conditions. Starting with a complex wave number k (purely imaginary for bound and antibound states), we integrate numerically from the origin up to a matching point using Numerov's method. The inward integration is performed using the corresponding (first-order) Riccati equation. The complex eigenvalues are found by matching the two logarithmic derivatives. We find narrow shape resonances within the well, above the dissociation limit, and broad resonances above the centrifugal barrier. Antibound states are found even with J = 0. © 1997 John Wiley & Sons, Inc.     

Theoretical investigation of charge transfer excitation and charge recombination in acenaphthylene–tetracyanoethylene complex

Ab initio calculations were performed to investigate the charge separation and charge recombination processes in the photoinduced electron transfer reaction between tetracyanoethylene and acenaphthylene. The excited states of the charge‐balanced electron donor–acceptor complex and the singlet state of ion pair complex were studied by employing configuration interaction singles method. The equilibrium geometry of electron donor–acceptor complex was obtained by the second‐order Møller–Plesset method, with the interaction energy corrected by the counterpoise method. The theoretical study of ground state and excited states of electron donor–acceptor complex in this work reveals that the S₁ and S₂ states of the electron donor–acceptor complexes are excited charge transfer states, and charge transfer absorptions that corresponds to the S₀ → S₁ and S₀ → S₂ transitions arise from π–π* excitations. The charge recombination in the ion pair complex will produce the charge‐balanced ground state or excited triplet state. According to the generalized Mulliken–Hush model, the electron coupling matrix elements of the charge separation process and the charge recombination process were obtained. Based on the continuum model, charge transfer absorption and charge transfer emission in the polar solvent of 1,2‐dichloroethane were investigated. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 23–35, 2003     

Obtaining positions and widths of scattering resonances from a complex multiconfigurational self‐consistent field state using the M1 method

We present a complex multiconfigurational self‐consistent field (CMCSCF)‐based approach to investigate electron‐atom scattering resonances. It is made possible by the use of second quantization algebra adapted for biorthogonal spin orbitals, which has been applied to develop a quadratically convergent CMCSCF method. To control the convergence to the correct CMCSCF stationary point, a modified step‐length control algorithm is introduced. Convergence to a tolerance of 1.0 × 10^?10 a.u. for the energy gradient is found to be typically within 10 iterations or less. A method involving the first block of the M matrix defined in the multiconfigurational spin tensor electron propagator method (MCSTEP) based on the CMCSCF reference state has been implemented to investigate ²P Be^? shape resonances. The position and width of these resonances have been calculated for different complete active space choices. The wide distribution of the position and width of the resonance reported in the literature is explained by the existence of two distinct resonances which are close in energy. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

He++N2 Charge Transfer Reaction: An Ab initio Analysis

An ab initio analysis on the involved potential energy surfaces is presented for the investigation of the charge transfer mechanism for the He⁺+N₂ system. At high collision energy, as many as seven low-lying electronic states are observed to be involved in the charge transfer mechanism. Potential energy surfaces for these low-lying electronic states have been computed in the Jacobi scattering coordinates, applying multireference configuration interaction level of theory and aug-cc-pVQZ basis sets. Asymptotes for the ground and various excited states are assigned to mark the entrance (He⁺+N₂) and charge transfer channels (He+N₂⁺). Nonadiabatic coupling matrix elements and quasi-diabatic potential energy surfaces have been computed for all seven states to rationalize the available experimental data on the charge transfer processes and to facilitate dynamics studies.     

X-Ray Photoelectron Spectra and Electronic Structure of Solid Solutions Based on Cubic Boron Nitride

The electronic energy structure of substitution solid solutions based on boron nitride B _1-x NR _x and BN _1-x R_x (R = C, O) (x=0.25) in a diamond-like modification of ZnS type has been investigated by the local coherent potential method in terms of multiple-scattering theory. The total and partial densities of states were calculated for each element in a solid solution. The crystalline potential was calculated using an MT approximation. The lattice parameter was chosen based on X-ray diffraction data for c-BN: 0.3615 nm. The electronic energy structures of the solid solutions and binary c-BN are compared in the framework of a single approximation. The calculated partial densities of states are compared with the experimental X-ray emission and photoelectron spectra of boron, nitrogen, and oxygen in these compounds. The calculated partial charges of electrons at the top of the valence band show that charge transfer from boron to nitrogen takes place in the solid solutions. An analysis of the electronic structures of the solid solutions of boron nitride indicates that the quasicore resonances inherent in binary c-BN are delocalized and that chemical bonding in the solid solutions of boron nitride is weakened.     

Potential energy surfaces of pseudoaromatic molecules: An MMVB and CASSCF study of pentalene

The S₀ and S₁ potential energy surfaces of pentalene were studied using MMVB—a hybrid force-field/parametrized valence bond (VB) method designed to simulate CASSCF calculations for ground and covalent excited states. The results were calibrated against full CASSCF calculations. Four distinct critical points were optimized: on S₀, a C_2h minimum (with alternating single and double bonds) and a D_2h transition structure; and on S₁, a D_2h minimum and an adjacent S₁/S₀ conical intersection. A VB exchange density matrix (which is independent of the choice of the spin-coupled basis) was used to rationalize the S₀ and S₁ surface topologies. Craig defined pseudoaromatic molecules to be those with nontotally symmetric electronic ground states. For pentalene, this is true for both CASSCF and MMVB calculations: the CASSCF S₀ transition structure is an open-shell B_1x singlet, and the VB ground state is dominated by a spin-coupling which transforms as B_1g. A C_2v minimum and a D_2h transition structure were located on the CASSCF S₂ potential energy surface. This state cannot be represented by MMVB because of the importance of ionic configurations. The characters of the S₁ S₂ states of pentalene are shown to be reverse of the S₁ and S₂ states of benzene. © 1996 John Wiley & Sons, Inc.     

硫代樟脑光化学反应的理论研究

运用量子化学方法优化了硫代樟脑的最低5个电子态(S₀, T₁, S₁, T₂和S₂)的结构, 并计算了它们的相对能量. 计算结果表明: S₁, T₁和T₂态的能量非常接近, 而S₂的能量远远高于T₂态, 这与之前对几种小的硫代羰基化合物的研究结论一致. 确定了硫代樟脑分子在T₁态发生β-插入反应和类Norrish II型反应的机理, 计算的势垒相对于S₀的振动零点分别为314.1和332.6 kJ/mol. 在400 nm波长的光的照射下, 分子被激发到S₁态, 此时分子没有足够的能量发生反应, 只能通过内转换回到基态. 当激发光波长在254 nm时, 硫代樟脑分子被激发到S₂态, 这时候体系有了足够的内部能量使反应发生. 实验上已经观察到此激发光波长下, 气态硫代樟脑可以发生β-插入反应和类Norrish II型反应.