Adiabatic perturbation theory for the degenerate case. I. Perturbation of isolated degenerate or quasidegenerate levels

Adiabatic formulae for secular operators and contracted Hamiltonians in an arbirary combination of degenerate or quasidegenerate subspaces are derived. A detailed consideration of the adiabatic limit in the power series is given, and “stability” of proper linear combinations with respect to a transformation produced by the S_α-matrix is proved.     

Adiabatic perturbation theory for the degenerate case. III. Adiabatic formalism by Gell-Mann and low with an arbitrary switching function

An adiabatic formalism in the degenerate or quasidegenerate subspaces, which does not depend on the particular form of the switching function g(α, t), is outlined. A general factorization theorem for the dynamic operator S_α(t, t₀ | g) is proved. This theorem enables one to formulate the perturbation expansion for the effective Hamiltonian and the wave operator which is free from the adiabatic divergencies.     

Topology of conical intersections and Jahn-Teller crossing: Application to the standard model for XY4 molecules in T2 ground states

The topological understanding of a potential function requires a knowledge of its critical points and indices. These concepts are mathematically meaningless if the potential function is not twice continuously differentiable. When conical intersections or Jahn–Teller crossings are encountered, the adiabatic potential, defined as the lowest eigenvalue of a Hamiltonian, can be smoothed everywhere. Moreover, the limiting properties of the smoothed critical points for vanishing values of a smoothing parameter lead to a unambiguous definition of a pseudo critical point of the limiting potential. Therefore, Morse theory is readily applicable to the ground state potential function in spite of the first order discontinuities. The regularization procedure is formally and numerically applied to the simplest Hamiltonian matrix of order 3 describing the Jahn–Teller effect of XY₄ molecules in the T₂ ground states. Various kinds of pseudo-critical points are encountered and are shown to satisfy the Morse theory.     

A new way of analyzing vibrational spectra. I. Derivation of adiabatic internal modes

A new way of analyzing measured or calculated vibrational spectra in terms of internal vibrational modes associated with the internal parameters used to describe geometry and conformation of a molecule is described. The internal modes are determined by solving the Euler–Lagrange equations for molecular fragments ϕ_n described by internal parameters ζ_n. An internal mode is localized in a molecular fragment by describing the rest of the molecule as a collection of massless points that just define molecular geometry. Alternatively, one can consider the new fragment motions as motions that are obtained after relaxing all parts of the vibrating molecule but the fragment under consideration. Because of this property, the internal modes are called adiabatic internal modes, and the associated force constants k_a, adiabatic force constants. Minimization of the kinetic energy of the vibrating fragment ϕ_n yields the adiabatic mass m_a (corresponding to 1/G_nn of Wilson's G matrix) and, by this, adiabatic frequencies ω_a. Adiabatic modes are perfectly suited to analyze and understand the vibrational spectra of a molecule in terms of internal parameter modes in the same way as one understands molecular geometry in terms of internal coordinates. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67 : 1–9, 1998     

Some current problems in quantum chemistry as described by their key words and the International Journal of Quantum Chemistry

A counterexample to the adiabatic approximation theorem is given when degeneracies are present. A formulation of an alternative version is proposed. A complete asymptotic decomposition for n dimensional self-adjoint Hamiltonian systems is restated and used.     

Theoretical study of the static Jahn-Teller effect—I. Two-mode vibronic coupling in octahedral halocomplexes with doubly degenerate electronic states

A two-mode E_g-(a_1g+e_g) vibronic coupling is analyzed for octahedral systems. Analytic formula for the adiabatic potential surface (APS) is obtained considering quadratic vibronic terms and anharmonicities of normal vibrations as well. Potential constants, viz. five elastic force constants and three vibronic constants, are evaluated from the numerical map of the APS applying the non-linear regression analysis. Numerical values are obtained for hexahalocomplexes on the CNDO/INDO level of total energy calculations.     

Canonical statistical adiabatic channel model calculations of the H + CH3 → CH4 recombination reaction on an ab initio potential energy surface. The role of the transitional modes

The canonical formalism of the statistical adiabatic channel model is used to calculate limiting high pressure rate constants for the H + CH₃ → CH₄ recombination reaction on a recently reported analytic potential energy surface based on ab initio calculations. An effective adiabatic channel potential which incorporates the G_?? matrix element of the twofold degenerate H₃C? H transitional bending mode, quartic anharmonicity, and state selected mode coupling effects is implemented. The rate constants calculated over the temperature range 200–1000 K are in very good agreement with recent canonical variational transition state theory calculations performed on the same surface. The comparison with experimental results is also discussed.     

MRCI study on spectroscopic parameters and molecular constants of the ground state of AsP(X1Σ+) molecule

The potential energy curve (PEC) for the ground state of AsP(X¹Σ⁺) has been investigated by the highly accurate valence internally contracted multireference configuration interaction method in the Molpro2008 program package with the correlation consistent basis set. The PEC is fitted to the analytic Murrrell–Sorbie function (M–S function) from which the spectroscopic constants are determined. The present D_e, B_e, α_e, ω_eχ_e, R_e, and ω_e values are of 4.2823 eV, 0.188622 cm^?1, 0.000749 cm^?1, 1.984427 cm^?1, 2.0194 Å, and 598.60 cm^?1, respectively. In addition, by numerically solving the radial Schrödinger equation of nuclear motion in the adiabatic approximation, the total of 96 vibration states is predicted when the rotational quantum number J = 0. The complete vibration levels, classical turning points, inertial rotation, and centrifugal distortion constants are reproduced. Comparison has been made with recent theoretical and experimental data. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Unrestricted CNDO-MO calculations. II. MnO 4 2− and CrO 4 3−

The electronic structures of the tetrahedral molecule ions MnO ₄ ^2– and CrO ₄ ^3– have been investigated within an unrestricted CNDO-MO approximation [Theoret. Chim. Acta (Berl.)20, 317 (1971)]. Calculations assuming the unpaired electron occupies the 3a ₁, 2e, and 4t2 molecular orbitals indicate that the 3a ₁ and2e orbitals have similar orbital energies and that the 4t ₂ orbital is at a higher energy. The experimentally indicated2e orbital for the unpaired electron is obtained with expanded O^1– type atomic orbitals for oxygen and valence metal orbitals of the expanded 3d and plus one ion 4p types. The metal 4s orbitals must be held to the neutral atom type. The optimum valence orbitals above with a slightly contracted 4s type metal orbitals yield the minimum total energy and places the unpaired electron in the 3a ₁ orbital. Since the contracted 4s metal orbital produces results that are not in agreement with experimental data, the method used apparently does not adequately take into account the increased electron-electron repulsions that contracted 4s orbitals produce.     

Lie algebraic approach to the scattering system He + H2

In this article, the Hamiltonian for the scattering of the H_e + H₂ system is given by using the interaction potential V(X, Y, Θ) determined by experiments and the semiclassical method. From this Hamiltonian we find a dynamical algebra h₆. The statistical expectation of the energy and the transition probability of H₂, P_n→m, are derived; therefore, selection rules have been found easily.     

Hylleraas method for many‐electron atoms. I. The Hamiltonian

A general expression for the nonrelativistic Hamiltonian for n‐electron atoms with the fixed nucleus approximation is derived in a straightforward manner using the chain rule. The kinetic energy part is transformed into the mutually independent distance coordinates r_i, r_ij, and the polar angles θ_i, and φ_i. This form of the Hamiltonian is very appropriate for calculating integrals using Slater orbitals, not only of states of S symmetry, but also of states with higher angular momentum, as P states. As a first step in a study of the Hylleraas method for five‐electron systems, variational calculations on the ²P ground state of boron atom are performed without any interelectronic distance. The orbital exponents are optimized. The single‐term reference wave function leads to an energy of ?24.498369 atomic units (a.u.) with a virial factor of η = 2.0000000009, which coincides with the Hartree–Fock energy ?24.498369 a.u. A 150‐term wave function expansion leads to an energy of ?24.541246 a.u., with a factor of η = 1.9999999912, which represents 28% of the correlation energy. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Transition Metal Carbides. Dependence of some thermochemical quantities on the chemical nature of reactants and stoichiometry of the product

Linear relationship were found among the plots of ΔH _{T ad} and ΔG ⁰ _{T ad} vs. the adiabatic reaction temperature (T _ad) for transition metals carbides and silicides characterised by the same stoichiometry. The slope of the straight line depends on the product stoichiometry. This revised version was published online in July 2006 with corrections to the Cover Date.     

SCF and CI Studies of Hund's rules for the effects of electronic correlation and delocalization. I

In connection with the reinterpretation of Hund's multiplicity rules for molecules, a detailed study has been made of the energy differences in the total energy and its components for the triplet and singlet Π_u states of the hydrogen molecule and the analogous states of the four- and six-membered hydrogen atom rings. For the hydrogen molecule, both SCF and CI studies indicated that the outer electron is considerably more contracted in the triplet than in the singlet state. In both approximations, the energy difference is dominated for all bond distances of chemical and physical significance by the electron-nuclear attraction component and not by the electron repulsion component as predicted by simple first-order perturbation theory. Although the correlation energy for each of the states is of the same magnitude as the energy differences considered here, the difference of the correlation energies is much smaller. It had little effect on the qualitative differences between these states of the hydrogen molecule. For the four- and six-membered rings, SCF studies were made on the lowest singlet and triplet states where one electron was promoted from the σ_g to a Π_u orbital. Even though the coupled electrons were more delocalized in these cases, the electron repulsion became relatively more important. However in all cases, the lower state had the highest electron repulsion energy and lower electron-nuclear attraction. The triplet state continued to have the more contracted outer open-shell orbital.     

A reinvestigation of the hypersonic properties and the specific heat capacity of PMMA around the quasi-static glass transition II. High performance caloric investigations

To continue dynamicc _p measurements in the range of smallest temperature rates and non-linear thermal relaxation investigations into the linear range, simultaneousc _p and thermal relaxation measurements were carried out in an adiabatic vacuum calorimeter, using the pulse heating method. The rate-dependentc _p behaviour, known from dynamic measurements, does not continue at small temperature rates. This is confirmed by the relaxation process which is observed. The results suggest an extended interpretation of the glass transition in atactic polymethylmethacrylate (PMMA).     

Nuclear motion on the orbitally degenerate electronic ground state of fully deuterated triatomic hydrogen

Nuclear motion in the vicinity of conical intersections of the degenerate electronic ground state of fully deuterated triatomic hydrogen, D(3), is investigated with the aid of a time-dependent wavepacket approach in hyperspherical coordinates. Vibronic energy level spectra and the eigenfunctions are examined by including, for example, (1) geometric phase (GP) correction, (2) diagonal Born-Huang (BH) correction, and (3) both GP and BH corrections to the Born-Oppenheimer adiabatic Hamiltonian and finally by considering the nonadiabatic coupling between the two electronic surfaces explicitly. It emerges from this study that inclusion of both the GP and BH corrections is insufficient to explain the spectral features observed in the experiment. The latter are recovered by considering the complete two-states coupled Hamiltonian only. This study shows that both the GP and BH corrections constitute a minor part of the surface coupling effects, in particular, on the dynamics of the upper adiabatic sheet. Most importantly, we add that the experimental signature of the GP effect appears only in the observed shift of the eigenlevels of the electronic state when compared to those obtained from a completely Born-Oppenheimer Hamiltonian. The detail fine structure of the observed band of the electronic state is shaped by the off-diagonal derivative coupling elements of the nonadiabatic coupling operator.     

Contracted polarization functions for B to Ar

Using optimal exponents for B through Ne given by Dunning and those for Al through Ar by Woon and Dunning, d-type contracted polarization functions (2d/1d), (3d/1d), and (3d/2d) are generated from natural orbitals of atomic single and double excitation configuration interaction (SDCI) calculations, where the numbers before and after the slash are those of the primitive and contracted Gaussian type functions. The resulting contracted functions are tested on N₂ and P₂ molecules by self-consistent field and SDCI calculations, which clarify characteristics of the present polarization functions. Received: 5 June 1997 / Accepted: 20 August 1997     

Harmonic oscillator tensors. I. The nondegenerate case

It is shown that the Heisenberg Lie algebra of the nondegenerate harmonic oscillator leads to a basis {J₊, J₀, J_?} of LASU (2). The Hamiltonian of the system is proportional to J₀, and the basis elements give rise to irreducible tensors in the associative enveloping algebra of the Heisenberg Lie algebra. The construction of these irreducible tensors is studied with special attention being paid to the case in which they act upon a single vector space spanned by the harmonic oscillator basis functions. A tensor coupling rule is developed, and useful application is made of it in the calculation of general expressions for vibrational operators and their matrix elements. Throughout, the value of the additional algebraic quantum numbers (l, m) is emphasized.     

Correlation problems in atomic and molecular systems. VII. Application of the open-shell coupled-cluster approach to simple π-electron model systems

The coupled-cluster variational-like direct approach to the calculation of ionization and electron attachment energies and of excitation energies is applied to several π-electron model systems, using the PPP Hamiltonian with various parametrizations. A simple approximation, which represents the triexcited clusters in terms of disconnected W₁T₂ terms, is employed. All the necessary diagrams for both excitation energy and ionization potential (electron affinity) calculations are given in the compact Hugenholtz nonoriented form. The results of the calculations for benzene, trans-butadiene, all-trans-hexatriene, and fulvene are compared with the corresponding full CI results, and the conclusions about the validity and efficiency of this approach are drawn.     

Molecular geometry and excited electronic states : Part VII. Theoretical study on conformational geometries and the dual fluorescence of styrene

Equilibrium molecular geometries of styrene in the electronic S₀, S₁, and S₂, states have been calculated within the scope of the adiabatic approximation by minimization of the total energy with respect to all molecular coordinates. Besides the most stable planar structures, some further conformers twisted about the single and/or double bonds have been found. An interpretation of the styrene dual fluorescence is given using these results.     

Ground-state energy of an s-state model of the inhomogeneous electron liquid in relation to an exactly solvable model of He with additional radial correlation

Following Temkin’s proposal of the so-called s-wave model, Amovilli, Howard and March (AHM) displayed an exactly solvable model Hamiltonian in which an additional radial correlation is added. The ground-state wave function for this Hamiltonian, for modelling He-like atomic ions with nuclear charge Ze, is used here to compare and contrast with the Temkin Model. The differences only appear near the critical charges Z_ce at which one electron ionises, Z_c being precisely unity for the AHM model and having the value 0.948768 in Serra’s variational study of the s-wave model. These two models are then compared with He-like ions having the correct e²/r₁₂ interaction.