Model calculation of weak and medium strong linear hydrogen bonded systems

The model hamiltonian with two coupling constants between the low and high frequency vibrations of a linear AH?B system is studied. The results show that the wave functions and energy levels of this hamiltonian obtained by exact diagonalisation can be adequately approximated by the wave functions and energy levels of a harmonic “effective hamiltonian”. The experimental spectrum of (CH₃)O?HCl in the stretching region was reconstituted by means of this model.     

An ab initio effective potential for two particles in the field of the core: a reduction of (N + 2)-electron problem to two-electron problem

We have used many-body Green function theory and the two-electron Bethe—Salpeter equation to derive an approximate two-electron position space hamiltonian eigenvalue equation for two electrons in the presence of a closed shell core. The resulting effective hamiltonian is nonlocal, energy independent, hermitian and nonadiabatic. It includes all the core—valence, valence—valence exchange effects, core screening effects and electron—electron correlation effects. If a closed form solution of the equation is difficult because of the need to construct the hamiltonian, a semi-empirical approach can be taken which expresses much of the hamiltonian in terms of known properties of the core. A semi-empirical analysis of this effective hamiltonian is shown to give well-known phenomenological effective hamiltonians and the connections to them. Thus this work can also be viewed as a theoretical justification and extension of the two-electron model potential or pseudopotential theories.     

A variational method to calculate static electronic properties

Using a coupled cluster form of the wave function, a variational method is formulated for calculation of static properties of any order. Corresponding to an appropriate perturbed hamiltonian H() including the relevant static property, a size consistent functional is set up. In a hierarchical fashion, properties of different orders may be found out using a variational method.     

Quasi-degenerate rayleigh-schrödinger perturbation theory with hermitean model hamiltonian

A method for the construction of the hermitean model hamiltonian in the framework of the quasi-degenerate Rayleigh-Schrödinger perturbation theory is suggested. The approach of a model hamiltonian is based on the assumption that if it is diagonalized in a chosen finite-dimensional model space it will yield eigenvalues of the original hamiltonian in the entire Hilbert space. For a unitary operator transforming the unperturbed state vectors onto the perturbed state vectors, and a model-interaction operator, the perturbation-expansion formulae are developed.     

A second-quantization representation of the Epstein-Nesbet partitioning of the full hamiltonian

A second-quantization representation of the Epstein-Nesbet partitioning of the total electronic hamiltonian is suggested. In this approach, the unperturbed hamiltonian contains not only the one-particle orbital energies but also the Coulomb and corresponding exchange two-particle terms. Such a hamiltonian can advantageously be used in all branches of the many-body diagrammatic perturbation theory for simple and correct inclusion of the diagonal ladder and ring diagrams in all orders of perturbation theory.     

Quantum investigation of linear triatomic exchange reactions. A computational method

The model hamiltonian for a linear triatomic exchange reaction is derived in natural reaction coordinates, and a method is developed to solved the system of wave equations in close coupling approximation. The interaction zone is divided into subintervals, in each of which the coefficients of the system are assumed to be constant; this assumption provides an exact analystical solution over the interval. Special boundary conditions which account for the derivatives of the coefficients are used to match the solutions for adjacent intervals. The method converges to the exact solution with an increasing number of intervals of decreasing length. Results obtained for certain model system illustrate the merits of the method and its convergence.     

Solution of a molecular statistical model for chiral nematic liquid crystals

A molecular statistical model for the chiral nematic phase of liquid crystals is investigated. The model is treated in the molecular field approximation. The resulting set of coupled integral equations for the order parameters and the pitch determining equation are solved numerically. The model hamiltonian consists of a nematic and a twist producing term. If only the nematic term is present, the model is known to have a first order phase transition. The model containing only the twist producing term is shown to exhibit a second order phase transition. The order parameters and wave length are presented for three ratios of the coupling constants in the case that both interactions are present.     

Perturbational and variational treatments of the Morse oscillator

The Morse oscillator hamiltonian is expressed as an infinite expansion in powers of a natural perturbation parameter, the square root of the anharmonicity constant, relative to the simple harmonic oscillator as zeroth-order hamiltonian. A transformation of variables leads to a hamiltonian which involves terms no higher than second order in this natural perturbation parameter. In both cases, the exact bound state eigenvalues of the Morse oscillator are given by second-order perturbation theory. The Schrödinger equation corresponding to the transformed Morse hamiltonian is solved variationally, via a complete set expansion in simple harmonic oscillator eigenstates. Accurate approximations to the exact eigenvalues and eigenfunctions of bound states of the Morse oscillator can be obtained for all but the very highest levels.     

Fock space perturbation theory

Diagonal and non-diagonal operators in Fock space are defined. With a universal Fock space wave operator W the Fock space hamiltonian H can be transformed to a diagonal operator L containing all relevant information about eigenvalues of H for arbitrary particle number in a simply coded form. W and L are constructed by perturbation theory, even in a spinfree form, and illustrated diagrammatically.     

Relativistic corrections in the multiple-scattering method

We show how to introduce the Foldy-Wouthuysen relativistic corrections in the multiple-scattering method, for the determination of the electronic energy levels of molecules. The present derivation begins from a variational expression where the Foldy-Wouthuysen hamiltonian is inserted, instead of the Schrödinger hamiltonian. The resulting secular equation becomes identical with the standard multiple-scattering secular equation, if one neglects the relativistic correction terms.     

Electronic effect on the rotational energy of a diatomic molecule

The rotational hamiltonian for a diatomic molecule has been rederived from the total classical hamiltonian. This procedure directly introduces the effect of electronic motion which is ordinarily neglected in zero-order approximation. Kronig's rotational hamiltonian is discussed and shown to be an approximation of our findings. Our general result is then specialized to ¹Σ states, and the theory tested by calculating the observed fractional discrepancy between the experimentally determined H³⁵Cl energy level constant Y₀₂ and its predicted value from Dunham's theory. When all corrections are summed, the results are in good agreement with experiment.     

The generalized transition state method

Current theories of unimolecular reaction rates are based on the transition state method which replaces internal reactant dynamics by an assumption of internal equilibrium. The present work is devoted to the development of generalized transition state method which allows effects such as nonergodicity and non-exponential decay to be accounted for within a simple theoretical framework. The derivation is quantum mechanical and not limited by any weak perturbation assumption. An effective hamiltonian is constructed for the reactant dynamics. The loss of amplitude due to reaction is accounted for by a dissipative term in the hamiltonian which is obtained on a phenomenological basis. The diagonalization of the hamiltonian allows the decay of reactant state to be predicted. The decay information is then used to set up a non-markovian master equation which in turn yields the rate coefficient for the reaction. The accuracy of the method is tested in one-dimensional model calculations in which particular attention is paid to decay by quantum mechanical tunneling through a potential barrier.     

Negativity of ground-state interaction energies of atom-excess-electron systems

Any atom—excess-electron system confined to an arbitrary finite region of space that is free of external fields and which can be characterized by a non-relativistic spin-free hamiltonian has, in the limit of infinite nuclear mass of the atom, an exact ground-state expectation value of the atom—electron interaction hamiltonian that is negative-definite.     

The Effective Hamiltonian and Electron Correlation Energy

An effective,hermitian hamiltonian is derived in amodel space. Its perturbation expressions to third order approximation are given.The correlation energy is also given to the third order approximation.The effective hamiltonian deviates form the actual one by the presence of acorrelation operator.The cprrelation operatop is given in an explicit form.     

Quasi-homopolar levels of hydrocarbons with conjugate bonds: Series for projected π-electron operators

These are states that go over to the 2p_z states of the neutral atoms as the latter recede to infinity; they include the ground state and most of the lower excited states. Then Schrödinger's equation and the operators for the physical quantities may be projected on the space of spin functions. A method is given for calculating the projected hamiltonian and operators as a rapidly convergent series in the number of interacting centers. Pair interactions are shown to play the main part in the spin hamiltonian. The convergence is examined for the series for the momentum and spin-density operators. Schrödinger's equation with the spin hamiltonian then gives a complete solution of the problem; as in the valence-bond method, the task is facilitated by the fact that the subspaces of defined system spin may be distinguished in spin space. A method is given for selecting the states from the measured terms for the molecule. It is shown that all absorption lines corresponding to excitation of such states should be weak for alternant hydrocarbons.     

Analysis of exact valence shell hamiltonian: Nonclassical terms and molecular based parameters

The exact valence shell effective hamiltonian is analyzed for one- and two-valence orbital systems using a second quantized formulation. The exact solutions of the exact effective hamiltonian are used, to display the meaning of each of its terms. The well-known α_υ = ?I_p and γ_υυ = I_p ? A_f relations are provided a molecular basis for certain special cases, thereby enabling molecular definitions for molecular “integrals” and allowing the determination of the molecule and bond length dependence of traditional semi-empirical “integrals”. It is shown how the presence of nonclassical terms in the effective hamiltonian destroys the pairing symmetry of alternate hydrocarbons.     

Generalized perturbation theory of effective valence shell hamiltonians

The second quantized effective valence shell hamiltonian of Iwata and Freed is generalized to incorporate valence orbital energies into the perturbative (matrix) energy denominators, eliminating convergence difficulties in calculations of atomic valence shell hamiltonians. When the matrix energy denominators are taken to be the simplest form our generalized effective hamiltonian reduces to Brandow quasi-degenerate theory.     

Theory of NMR shifts and endor spectroscopy in paramagnetic complexes: (H. Eicher,Chem. Phys. 86 (1984) 331)

The nuclear-spin-electron-orbit interaction term in the general hamiltonian for a paramagnetic complex is correctly derived.     

Mössbauer spectra of the tetrakis-(1,8-naphthyridine) iron(II)perchlorate in external magnetic fields. Evidence of slow relaxation ln paramagnetic iron(II)

⁵⁷Fe Mössbauer spectra of [FeL₄] (ClO₄)₂ where L = 1,8-naphthyridine have been measured at 4.2°K in external magnetic fields up to 55 kG parallel to the direction of the γ-rays. The spectra have been fitted in the spin hamiltonian approximation assuming an orbital singlet ground state of the ⁵D multiplet of Fe²⁺. The fit of the spectra is not unique, yet the possible spin hamiltonian parameter sets found lead to a spin doublet ground state split by less than 1 cm^?1. The transition probabilities for spin-lattice relaxation have been calculated for those ground states. Orbach processes via excited spin hamiltonian states cannot be neglected. The results explain the fluctuations observed in the spectra in low external magnetic fields (10 kG).The spin hamiltonian parameters provide information on the orbital energy levels. Therefrom the reduction of the quadrupole splitting by spin—orbit coupling results to be small thus explaining the extremely large quadrupole splitting of 4.54 mm/sec.