Effective Hamiltonian Operator Research Articles

Rotational Analysis of the à 2A1–X̃ 2A1 Band System of Yttrium Dicarbide, YC 2

The 000–000 and 3 1 0 bands of the 775-nm electronic transition of YC 2(à 2 A 1←X̃ 2 A 1) have been studied at high resolution, using the laser-induced fluorescence from a supersonic jet expansion. Three types of experiment have been carried out. First, the complete rotational and hyperfine structures of the two bands were recorded. To measure the small asymmetry splittings in the K=2 levels of the X̃ 2 A 1 state, portions of the b-type 3 1 0 band were then recorded in the presence of a weak static electric field. Finally, a number of pure rotational transitions between the K=0 levels of the ground state were recorded by pump/probe microwave optical double resonance. A few small rotational perturbations occur in the upper electronic state but, omitting the perturbed lines, the combined data sets could be modeled using an effective Hamiltonian operator appropriate for the rotation, electron spin, and hyperfine structure of a rigid asymmetric top molecule. The molecule is confirmed as being “T-shaped,” where the Y atom is bonded to the side of a C 2 group; the rotational constants determined are for the à 2 A 1, 3 1 level, A=1.76128, B=0.189949, C=0.170056 cm −1, and for the X̃ 2 A 1, v=0 level, A=1.742731, B=0.201947, C=0.181285 cm −1. Allowing for electron orbital corrections to the rotational constants, the geometrical structures are found to be à 2 A 1 state, r (Y–C)=2.2795 Å, r (C–C)=1.2630 Å, ∠C–Y–C=32.17°; X̃ 2 A 1 state, r (Y–C)=2.1946 Å, r (C–C)=1.2697 Å, ∠C–Y–C=33.63°. A molecular orbital diagram is given for the states of YC 2 and the interpretation of the electron spin and hyperfine parameters is discussed.

Lie-algebraic approach to vibrational spectra of a linear symmetrical tetratomic molecule:C2H2

Using Lie-algebraic techniques and the simpler expressions of the matrix elements of Majorana operators given by us, we obtain an effective Hamiltonian operator which conveniently describes vibrational spectra of linear tetratomic molecules, including both stretching and bending modes. For a linear symmetrical four-atom molecule ${\mathrm{C}}_{2}{\mathrm{H}}_{2},$ the highly excited vibrational levels are obtained by applying the $\mathrm{u}(4)$ algebraic approach. We have found that the spectra are made up of a clustering structure. The number of levels in one cluster depends on the total quantum number of stretching and bending vibrations. In addition, some other properties, such as the level assignment and the labeling of calculated theoretical results, are also discussed.

Relativistic corrections in magnetic systems

We present a weak-relativistic limit comparison between the Kohn-Sham-Dirac equation and its approximate form containing the exchange coupling, which is used in almost all relativistic codes of density-functional theory. For these two descriptions, an exact expression of the Dirac Green's function in terms of the non-relativistic Green's function is first derived and then used to calculate the effective Hamiltonian, i.e., Pauli Hamiltonian, and effective velocity operator in the weak-relativistic limit. We point out that, besides neglecting orbital magnetism effects, the approximate Kohn-Sham-Dirac equation also gives relativistic corrections which differ from those of the exact Kohn-Sham-Dirac equation. These differences have quite serious consequences: in particular, the magnetocrystalline anisotropy of an uniaxial ferromagnet and the anisotropic magnetoresistance of a cubic ferromagnet are found from the approximate Kohn-Sham-Dirac equation to be of order $1/c^2$, whereas the correct results obtained from the exact Kohn-Sham-Dirac equation are of order $1/c^4$ . We give a qualitative estimate of the order of magnitude of these spurious terms.

Iterative solution of the one-electron Dirac equation based on the Bloch equation of the `direct perturbation theory'

The one-electron Dirac equation is solved in an iterative manner starting with the solution of the Schrödinger equation. The method is based on `direct perturbation theory' for relativistic effects generalized to the case of a set of near-degenerate strongly interacting states. Relativistic energies are obtained by solving a Schrödinger-like equation within a non-relativistic model space. The effective hermitian Hamiltonian and symmetric metric operator are expressed with the help of a Møller waveoperator Ω, which generates the complete four-component Dirac wavefunction from the non-relativistic two-component one. The corresponding Bloch equation is solved to infinite order by iteration in a basis of atom-centered Gaussian-type functions for the ground state of hydrogen-like ion Eka Pt 109+ and for a few states of the heavy quasi-molecule Th 179+ 2.

Suppression of the Rabi oscillations in a cavity partially filled with a dielectric having a time-dependent refractive index

In this paper the dynamics of wave fields in a cavity, part of which is filled with a dielectric, are investigated. The wave equation is solved approximately using the method of multiple time scales. Phase modulations of the cavity modes in an adiabatic case are given up to the first order. A simple effective field Hamiltonian operator is derived and then used to analyze the interaction of radiation with a two-level atom passing through the cavity. Corrections to the Rabi oscillations due to the time-dependent frequency of the cavity photons are investigated. It is shown that adiabatic time dependence of the refractive index of the dielectric can lead to suppression of the Rabi oscillations of atomic inversion.

Correlated wave‐function theory for many‐center many‐electron problems

The correlated electronic wave-function theory developed by S. Obara and K. Hirao [Bull. Chem. Soc. Jpn. 66, 3300 (1993)], as applied to two-electron molecular systems, is generalized to many-center many-electron problems. The exact formulas for effective Hamiltonian operators are given. The rules for the calculation of matrix elements with three-electron operators over Slater determinants are formulated. From the energy-minimum principle, the system of master equations is derived for variational coefficients of a trial wave function for the molecules with closed electronic shells. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 639–648, 1998

Microscopic origins of effective charges in the shell model

We use a large-scale $6\ensuremath{\Elzxh}\ensuremath{\Omega}$ calculation for ${}^{6}$Li with microscopically derived two-body interaction to construct the $0\ensuremath{\Elzxh}\ensuremath{\Omega}$ $0p$ shell effective Hamiltonian, electric quadrupole, and magnetic dipole operators. While the E2 and M1 $6\ensuremath{\Elzxh}\ensuremath{\Omega}$ operators are one-body operators with free nucleon charges, the effective operators are two-body operators with substantially different renormalization for the isoscalar and isovector matrix elements, especially for the E2 operator. We show that these operators can be very well approximated by one-body operators provided that effective proton and neutron charges are used. The obtained effective charges are compatible with those used in phenomenological shell-model studies. The two-body part of the effective operators is estimated.

Effective Hamiltonian for near-degenerate states in direct relativistic perturbation theory. I. Formalism

Direct perturbation theory (DPT) for relativistic effects is generalized to the case of a set of near-degenerate strongly interacting states. This situation, where the standard approach breaks down, is quite common in atoms and especially in molecules. We introduce a new partitioning of the Dirac equation and apply the Mo/ller–Bloch approach. An effective Schrödinger-like equation within a nonrelativistic model space of near-degenerate states is derived. The effective Hamiltonian and metric operators are expressed with the help of a Mo/ller wave operator Ω, which generates the complete four-component Dirac wave function from the nonrelativistic Schrödinger wave function in the finite model space. The corresponding Bloch equation can be solved numerically in a basis set to infinite order by iteration. Also explicit formulas are derived for different orders of Heff and Seff. They can be used to determine the relativistic energies to different orders either directly by diagonalization, or by a perturbation approach.

The role of spin-orbit coupling and symmetry in photochemical rearrangements of α,β-unsaturated cyclic ketones

Abstract The spin-orbit coupling parts in the effective one-electron Hamiltonian operator, with the inclusion of symmetry, have been used to formulate spin-inversion mechanisms based on the Shaik and Epiotis theory in triplet photoreactions for the radiationless decay of triplet complexes to singlet ground-state products. This has been applied to investigate the stereochemical behavior of α,β-unsaturated cyclic ketones. It is shown that triplet photoreaction complexes can be classified according to the number and the direction of the atomic orbital rotations required to maximize intramolecular spin-orbit coupling and simultaneously maintain the intramolecular bonding, from which it may produce different stereoisomers. Six spin-inversion reaction mechanisms are suggested to produce the various photoproducts, including aryl-migrated bicyclohexanone (endo and exo), aryl-migrated cyclohexenone, cyclobutanone, lumiketone, cyclopentenone, and type-A rearrangement. It is found that, because of its larger spin-orbit coupling expressions as well as better initial orbital overlap (owing to the less conformational change), the 1,2-aryl-migrated endo-bicyclic ketone is in general the most favorable product formed by 4,4-diarylcyclohexenones in solution, crystal, and polymer matrix photochemistry. The substituents (including their nature and the sites to be occupied), solvent polarity, heavy atom effects and environments must all play a decisive role in predicting the final photoproducts of α,β-unsaturated cyclic ketones. The results obtained are in good agreement with the available experimental results and permit a number of predictions to be made.

The Role of Spin-Orbit Coupling and Symmetry in Oxadi-pi-methane Rearrangements and Some Related Photochemical Reactions.

The spin-orbit coupling components in the effective one-electron Hamiltonian operator, with the inclusion of symmetry, have been used to formulate mechanisms of spin inversion in triplet reactions for the radiationless decay of triplet complexes to singlet ground state products. This has been applied to investigate the photochemical behavior of oxadi-pi-methane rearrangement in beta,gamma-unsaturated carbonyl cyclic and acyclic systems. It is found that the 1,2-acyl-migrated endo isomer is the most favorable product due to its larger spin-orbit coupling expression as well as better initial overlaps (owing to the less atomic motion). Other related photorearrangements of some molecular systems, such as azadi-pi-methane rearrangement, 4-aryl-substituted cyclopentenones, spirocyclic beta,gamma,delta,epsilon-unsaturated ketones, and heavy atom effect, have also been investigated.

Role of Spin−Orbit Coupling and Symmetry in Triplet Carbenic Addition Chemistry

The spin−orbit coupling parts in the effective one-electron Hamiltonian operator, with the inclusion of symmetry, have been used to formulate mechanisms based on the Shaik and Epiotis theory in triplet reactions for the radiationless decay of triplet complexes to singlet ground state products. This has been applied to investigate the stereochemical behavior of additions of triplet carbenes to olefins. It is shown that triplet carbene−olefin complexes can be classified according to the number and the direction of the atomic orbital rotations required to maximize spin−orbit coupling and simultaneously maintain the intermolecular binding, from which it may produce different stereoisomers. It is suggested that six spin-inversion reaction mechanisms produce geometric isomers. It is found that, as a direct of the different symmetries of the three sublevels of the triplet state (Tx, Ty, Tz) and their comparable spin-orbit coupling expressions, the triplet carbene could add to olefins nonstereospecifically. Never...

Mechanism of photochemical rearrangements of 3-substituted cyclopropenes to cyclopentadienes

The spin-orbit coupling parts in the effective one-electron Hamiltonian operators, with inclusion of symmetry, have been used to formulate mechanism of spin inversion in triplet photoreactions for the radiationless decay of triplet complexes to singlet ground state products. This has been applied to investigate the photochemical behavior of 3-vinyl, acyl, and iminocyclopropenes. Two competing reaction mechanisms are suggested and the results obtained agree with the available experimental results.

Fermionic entropy in 2D black hole

The Dirac equation for a Dirac field in a 2D black hole with a Schwarzschild-like metric is solved. Contrary to the bosonic eigensolution, the fermionic one is found to be divergent on the horizon, but it is not too singular and can be normalized in the inner product. 't Hooft's boundary condition, which is suitable for scalar bosons, is found not to be applicable to the fermionic eigensolutions. In order to discretize the spectrum, a “quasi-periodic” boundary condition is first proposed, and the fermionic effective Hamiltonian operator H F is shown to be self-adjoint with this “quasi-periodic” boundary condition. The result shows that the divergence in the fermionic entropy has the same form as that in the bosonic one, except that the coefficient is different. Thus the original divergence in the bosonic entropy cannot be cancelled by a supersymmetric Bose-Fermi cancellation.

The mechanism of photochemical rearrangement of vinylcyclopropanes to cyclopentenes

The spin-orbit coupling parts of the effective one-electron Hamiltonian operators, with the inclusion of symmetry, have been used to formulate the mechanism of spin inversion in triplet photoreaction for radiationless decay to singlet ground state product. This has been applied to investigate the photochemical behaviour of vinylcyclopropane. A reaction mechanism that satisfies the requirements of both spin-inversion and orbital symmetries is suggested.

Mechanism for di-π-methane rearrangements in nonconjugated systems

The spin-orbit coupling components of the effective one-electron Hamiltonian operator, with the inclusion of symmetry, have been used to formulate mechanisms of spin inversion in triplet photoreactions for the radiationless decay of triplet complexes to singlet ground state products. This has been applied to investigate the photochemical behavior of di-π-methane rearrangement in cyclic and acyclic nonconjugated systems. The results obtained agree with the available experimental results and allow a number of predictions to be made.

A combined use of perturbation theory and diagonalization: Application to bound energy levels and semiclassical rate theory

A new method, mixed diagonalization, is introduced in which an effective Hamiltonian operator acting on a reduced dimensional space is constructed using the similarity transformations of canonical Van Vleck perturbation theory (CVPT). This construction requires the characterization of modes into two categories, global and local, which in the bound vibrational problem are tantamount to the large and small amplitude vibrations, respectively. The local modes in the Hamiltonian are projected out by CVPT, and the resulting Hamiltonian operator acts only on the space of global modes. The method affords the treatment of energy levels of bound systems in which some vibrational assignments are possible. In addition, it systematically provides a reduced dimensional Hamiltonian which is more amenable to exact numerical solution than the original full-dimensional Hamiltonian. In recent work, a semiclassical transition state theory (SCTST) rate expression has been written in terms of a Hamiltonian operator parameterized by the imaginary action along the local reaction path in the transition state region [Chem. Phys. Lett. 214, 129 (1993)]. We show that the Hamiltonian constructed by mixed diagonalization has this form, and can be used to obtain more accurate semiclassical rate expressions.

Dipole moment functions of NH and NH + by ab initio effective valence shell Hamiltonian

The ab initio effective valence shell Hamiltonian ( H v) method has been extended to calculate molecular properties of valence states as well as energies. Both the effective Hamiltonian and effective molecular property operator are perturbatively expanded in powers of correlation, and contain contributions from excitations outside the multireference valence space. To demonstrate the validity of this method, dipole moment functions of several low-lying valence states of NH and NH + have been calculated to the lowest nontrivial order in the correlations. Comparisons of H v results with those of other theoretical calculations show that the H v method provides a useful ab initio formalism for dipole moment functions.

Periodic orbits in quantum standard maps.

The importance of periodic orbits, both elliptic and hyperbolic, in analyzing driven quantum systems is now well established. We present a detailed analysis of the quantum mechanics in the vicinity of these orbits for both a piecewise-linear and the original standard maps. We begin by constructing effective Hamiltonian operators valid locally near nondegenerate periodic orbits. Our general formula allows for the calculation of the quasienergy spectrum near elliptic orbits as well as of the strength of scarring on hyperbolic orbits

Optimum rational versions of effective rotational Hamiltonian operator of the symmetric top-type molecule

The construction of optimum versions of a rational expansion of the effective rotational Hamiltonian operator for symmetric top-type molecules in a non-degenerate vibrational state is considered, using as an example molecules with a point group symmetry C3v. The high efficiency of the optimum versions obtained has been demonstrated when applied to the rotational spectrum of PH3 in its ground state.

Description and interpretation of molecular phenomena in solution, using effective Hamiltonian operators related to continuous solvent distributions

The characteristics and potentialities of effective Hamiltonians using continuous distributions of matter (EHCD) are discussed, taking as the object of analysis the solvation model developed at our laboratory. EHCD methods for liquid systems appear to be quite flexible and of limited cost and acceptable accuracy and with limits of applicability not yet defined. The paper presents several extensions of the primitive elaboration of the model, in particular some addressed to (a) the inclusion of electron correlation in the ab initio treatment; (b) the elaboration of simplified procedures for molecules of large size; (c) the extension to nonisotropic descriptions of the medium; and (d) the consideration of dynamical and time-dependent aspects of solvation effects.