Effective Hamiltonian Operator Research Articles

Optimum rational perturbation theory series when treating rotational spectra of nonlinear molecules

The problem of constructing optimum versions of the perturbation theory series with phenom-enological constants in each order of this series is considered. Optimum versions are obtained for rational expansion of the effective rotational Hamiltonian operator of the asymmetric top type molecule in the nondegenerate state. The high efficiency of these versions is shown by the example of treating the rotational spectrum of the H 2 16O molecule in the ground state.

Shell-model calculations for the energy levels of the N=50 isotones with A=80-87.

The detailed features of the calculated energy-level schemes and of the single-particle, orbit-occupancy properties of the low-lying levels of the N=50 isotones 80Zn, 81Ga, 82Ge, 83As, 84Se, 85Br, 86Kr, and 87Rb are presented and discussed. These results are obtained with a new effective Hamiltonian operator obtained empirically from an iterative fit to experimental energies taken from all experimentally studied (A=82–96) N=50 nuclei. The model space for the calculations consists of active 0f5/2, 1p3/2, 1p1/2, and 0g9/2 proton orbits relative to a nominal 78Ni core. This space is truncated internally by restricting the number of particles excited from the negative-parity orbits into the g9/2 orbit to be no greater than four. The typical structures predicted for these lighter N=50 isotones are found to be dominated by well-mixed combinations of fp-orbit configurations, with the g9/2 orbit playing a minor role in all but a few special cases. The model energy-level spectra are compared with existing experimental information, as are calculated spectroscopic factors for single-proton stripping and pickup reactions.

Tunneling splittings in the water dimer: Further development of the theory

Energy level expressions are derived for the hydrogen-bonded water dimer complex and some of its deuterated species, using a previously reported internal-axis-method-like formalism developed for high barrier tunneling problems involving several large-amplitude motions. First, the feasibility of various tunneling motions is considered, using a potential surface given by D. F. Coker and R. O. Watts ( J. Phys. Chem. 91, 2513–2518 (1987)), and three main tunneling paths for (H 2O) 2 are chosen. We assume that the most feasible path corresponds to a 180° rotation of the acceptor monomer; that the next most feasible path corresponds to a geared-type rotation of the two monomers, leading to an exchange of donor and acceptor monomer roles; and that the third most feasible path corresponds to an exchange of donor hydrogen atoms in the hydrogen bond, accompanied by an umbrella motion of the acceptor monomer. As the second step, the J and K dependence of the purely vibrational tunneling splittings, which arises in the IAM-like formalism because angular momentum is generated during the tunneling motions, is taken into account by computing sets of angles χ, θ, φ for Wigner D ( J) ( χ, θ, φ) K, K′ expressions appearing as factors in the vibrational tunneling matrix elements. Symmetry relations determined for the angles χ, θ, φ greatly simplify the calculations. Finally, the Hamiltonian matrix is set up and diagonalized to obtain the desired energy level expressions (used without proof in an earlier paper treating observed microwave transitions of (H 2O) 2). The J and K dependence of the tunneling splittings arising from the presence of rotational terms in the effective Hamiltonian operator is also discussed, but only for the most symmetric species (H 2O) 2 and (D 2O) 2.

The torsional-wagging tunneling problem and the torsional-wagging-rotational problem in methylamine

A theoretical formalism is presented for fitting rotational energy levels in isolated (unperturbed) vibrational states of methylamine. This formalism is obtained by recasting and extending theoretical studies in the earlier literature, which were undertaken to help analyze the methylamine microwave spectrum. The present formalism is applicable when both the NH 2 umbrella (wagging) motion and the CH 3 internal rotation (torsion) motion take place near the high-barrier limit and leads to the usual Fourier sine and cosine series expansions for molecular energy levels. The derivation is separated into two parts, one treating the large-amplitude vibrational problem (the torsional-wagging problem) by itself, the other treating the torsional-wagging-rotational problem. In both treatments, permutation-inversion group and extended group ideas are used to determine the allowed terms in an effective rotational-tunneling Hamiltonian operator and to block diagonalize the matrix representation of this operator for a near-prolate symmetric top. The resulting energy levels and selection rules are discussed, but application of the method in detail to the methylamine spectrum is planned for a later paper.

An effective hamiltonian formalism for the computation of diabatic surfaces

The diabatic-surface model has been presented in terms of an effective Hamiltonian formalism in which diabatic surfaces for reactant-like and product-like bonding situations are obtained as the diagonal elements of the effective Hamiltonian operator. It is argued that the effective Hamiltonian enables a simple chemical interpretation in terms of a non-orthogonal valence-bond computation where the valence-bond structures are built from non-orthogonal distorted fragment orbitals. Numerical results on the transition structure for the H 2 + CO addition reaction are given. The adiabatic energies reproduce the MC-SCF energies and force constants to an acceptable level of accuracy. An examination of the diabatic surface forces and force constants gives some considerable insight into the nature of the transition structure surface topology.

Microwave spectrum of the hydrogen sulfide molecule H 232S in the ground state

Frequencies, line strengths, and intensities of microwave electric dipole rotational transitions of H 2 32S in the ground state have been calculated. The calculation was based on representation of the effective rotational Hamiltonian operator in the form of a Pade operator. As initial information, all known microwave data on frequencies of rotational transitions of H 2 32S in the ground state were used. Some of these data were obtained in the present paper.

The calculation of the energy levels of an asymmetric top free radical in a magnetic field

The form o the effective Hamiltonian operator for an asymmetric top free radical subject to a magnetic field is presented with an emphasis on machine calculation of the eigenvalues of the operator. Problems associated with the choice of a suitable basis. The calculation and the matrix elements of the Hamiltonian are given in a fully coupled parity conserving basis. The calculation of frequencies, relative intensities and Zeeman tuning rates of gas pahase magnetic resonance transitions are illustrated using the available high resolution data for the formyl radical as an exmple.

Effective pade Hamiltonian operator and its application for treatment of H 216O rotational spectrum in the ground state

Both microwave and infrared data on frequencies of rotational transitions in the H 2 16O ground state are fitted for the first time within the limits of experimental errors. This fitting is based on a representation of the effective rotational Hamiltonian operator in the form of a Pade operator. New experimental data on frequencies of microwave transitions in the submillimeter wave region are obtained.

A new approach to the treatment of rotational spectra of molecules with small moments of inertia applied to the PH 3 molecule in the ground state

A new approach to the treatment of rotational spectra of molecules with small moments of inertia is considered based on a representation of the effective rotational Hamiltonian operator in the form of a Pade operator. An extended set of high-precision data on transition frequencies in the rotational spectrum of the PH 3 molecule in the ground vibrational state is analyzed. The fitting of these data within experimental error demonstrates the advantages of the new approach. Frequencies of the Q-branch forbidden transitions | K| = 3 ← 6, first measured in this work, represent an essential part of the data under consideration.

Effective hamiltonian and effective dipole operator in the principal axis method (PAM)

An alternative to ‘denominator corrections’ is developed using a recently published projector formulation of the Van Vleck transformation. The allowedness of torsional transitions in PAM-type molecules is attributed to higher-order terms in the dipole operator originating from the Van Vleck transformation. A formula which predicts a weak dependence of the molecular dipole moment on the torsional state is obtained. Acetaldehyde is considered as an example.

Effective Operator Formalism in Optical Pumping

Real and virtual absorption of weak monochromatic light is analyzed with semiclassical radiation theory. The influence of the light on the atoms is described by an effective ground-state Hamiltonian operator, and the effect of the atoms on the light is described by a dielectric susceptibility operator. These operators are expressed explicitly in terms of familiar ground-state observables, angular factors, oscillator strengths, and plasma dispersion functions. The theory gives a comprehensive description of optical pumping, light shifts, and light modulation due to real and virtual absorption of light, and several new effects are predicted. Repopulation of the atomic ground state from a polarized excited state is not considered in this paper.

Effective Exchange Integral

The magnetic properties of a linear chain of monovalent atoms are investigated from the point of view of perturbation theory. The many-electron wave functions for the system are expanded as linear combinations of determinantal functions which are eigenfunctions of ${S}^{2}$ and ${S}_{z}$. These determinantal functions are constructed from orthonormal one-electron orbitals of the Wannier type so that the nearest neighbor exchange integral is positive definite and approaches zero at large lattice spacings. The secular equation is set up using the method of the Dirac vector model. By means of the Kramers perturbation technique, the interaction of ionic states with those arising from the ground configuration is represented by means of an effective Hamiltonian operator with its associated matrix. The results of this treatment are analogous to those obtained by Paul in that an analytic expression is found for an effective nearest neighbor exchange integral ${J}^{\ensuremath{'}}$. This quantity is represented as the difference between the positive definite exchange integral and additional terms from ionic states. The present treatment defines in a fairly precise manner the type configurations which contribute to this effective exchange integral and the limits for which this parameterization is valid. The results of this analysis are compared with those obtained from recent calculations on a system of six hydrogen atoms.