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.     

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.     

Diagrammatic analysis of adiabatic and time-independent perturbation theory for degenerate energy levels

Time-ordered folded diagrams are used to represent the effective hamiltonian in the adiabatic formalism. Resummation of the diagrams is shown to give a term-by-term correspondence with time-independent perturbation theory.     

Double perturbation theory and symmetry

A formulation of double perturbation theory is described which allows symmetry to be included in those cases where the zero-order hamiltonian does not contain the full symmetry of the problem. As an example we apply the theory to the Epstein-Johnson spin model.     

Level set topologies and convexity relations for hamiltonians with linear parameters

Level sets of the expectation value functional of hamiltonians depending on linear parameters are convex over a subset of the space spanned by these parameters. When the parameters enter the hamiltonian through a positive definite perturbation term, then properties of convex polyhedra in the parameter space lead to energy inequalities which may be utilized in perturbation theory.     

Resonance by the complex coordinate method with hermitean hamiltonian

Using the complex coordinate method, a new hermitean hamiltonian is formulated, in which the resonance position and width are the “natural” perturbation strength parameters. If the resonance position is known the resonance width obtained by the presented theory is a lower bound to the exact value.     

Ab initio calculations of the pi electron hamiltonian: singlet-triplet splittings

Some results of approximate ab initio calculations of the “correlation” contribution to the true parameters of the pi electron hamiltonian are presented for the ethylene molecule. In particular, by using sum-of-the-pairs type generalized perturbation theory, it is shown that there is a large core “correlation” contribution to singlet-triplet splittings within pi electron theories that results from the difference in the degree of ionicity of the isoconfigurational states.     

The direct reaction field hamiltonian: Analysis of the dispersion term and application to the water dimer

The induction and dispersion terms obtained from quantum-mechanical calculations with a direct reaction field hamiltonian are compared to second order perturbation theory expressions. The dispersion term is shown to give an upper bound which is a generalization of Alexander's upper bound. The model is illustrated by a calculation on the interactions in the water dimer. The long range Coulomb, induction and dispersion interactions are reasonably reproduced.     

A derivation of the exact pi-electron hamiltonian

The exact pi-electron hamiltonian is derived from the full many-electron Schrödinger equation. The derivation employs techniques which also enable the development of open shell generalized perturbation theory. The parameters occurring in the pi-hamiltonian are therefore amenable to calculation using the approximate theories and methods of Sinano?lu, Nesbet, Kelly, etc. To a good approximation, the pi-hamiltonian is found to be generally consistent with its customarily assumed form.     

Approximate and exact quantum mechanical energies and eigenfunctions for a system of coupled oscillators

A non-separable model hamiltonian, which describes the normal vibrations of a pair of coupled oscillators, has been solved by a number of approximation procedures, and approximate solutions compared with the “exact” solutions. The results of a simple perturbation theory calculation are seen to be rather more reliable than those of self-consistent-field model.     

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.     

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.     

Effect of error in the unperturbed wavefunction on electric polarizabilities and shielding factors of atoms. Application of double perturbation theory

The polarizability and the shielding factor of the hydrogen atom are calculated by using an approximate hamiltonian and an approximate unperturbed wavefunction Φ^(0,0) with one parameter. The double perturbation theory is applied for the calculation. The accuracy of Φ^(0,0) is changed by varying the parameter. When the error in Φ^(0,0) is small, the geometric approximation is best among various approximations investigated.     

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.     

Extended average energy perturbation calculation of the 2s2 and 2p21S resonance state energies of two-electron atoms

An extended average energy (EAE) calculation of the 2s² and 2p²¹S resonance state energies of two-electron atoms is carried out. We take the bare-nucleus hamiltonian as the initial approximation and treat the zeroth-order degeneracy by van Vleck perturbation theory, For both H^? and He the lower state energies agree quite well with experiment while the upper state is about 1–2 eV too high. Our remits for He are comparable in accuracy to those obtained by the 1/Z Hartree—Fock perturbation method. A brief concluding discussion of ways to improve the simple EAE technique is presented.     

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.     

Construction of an effective hamiltonian for open-shell molecular systems

An effective Hamiltonian for open-shell molecular systems is constructed. The unrestricted Hartree–Fock orbitals are applied as a basis set of one-particle functions. This effective Hamiltonian is determined as a simple product of the original total Hamiltonian and the spin annihilator. The second-quantization formalism and the Feynman–Goldstone diagrammatic technique are used. The resulting effective hamiltonian is composed of zero- to four-particle terms. A possibility of applying the nondegenerate diagrammatic perturbation theory constructed over this effective Hamiltonian is discussed.     

The fast CI method for second-order properties

Second-order properties are computed by diagonalizing a perturbed hamiltonian in a linear space L of low dimensionality. The choice of the basis vectors generating L is suggested by perturbation theory and previous work on correlation energy. Diagrammatic techniques can be used to compute the relevant matrix elements. Test numerical computations performed on the hydrogen molecule gave satisfactory results for the dipole polarizability and nuclear spin-spin coupling constant.     

Comparison of high-order many-body perturbation theory and configuration interaction for H2O

Diagrammatic many-body perturbation theory, coupled with a recursive computational procedure, is employed to obtain the correlation energy of H₂O within a 39-STO basis set by evaluating all double-excitation diagrams through twelfth order without any approximations. This provides, for the first time, the complete double-excitation diagrams contributions to the correlation energy, which is ?0.28826 hartree, compared with a correlation energy of ?0.27402 hartree obtained from a configuration interaction calculation which includes all double excitations. The difference of 0.0142 hartree includes the “size consistency” correction to the all-double-excitations CI energy, due to the “pathological” unliked-diagram terms remaining in that result, but also involves certain fourth- and higher-order rearrangement diagrams. Contrary to conventional belief, the unshifted, or Møller-Plesset partitioning of the hamiltonian provides a much more rapid convergence of the perturbation series that does the shifted, or Epstein-Nesbet partitioning. In both cases. Padé approximants enhance the convergence of the series considerably. A simple variation-perturbation scheme based on the first-order MBPT wavefunction is sufficient to provide 97.5% of the all-doubles CI correlation energy.     

Comments on the convergence of the ordinary Rayleigh-Schrödinger perturbation expansion

The convergence properties of the ordinary Rayleigh-Schrödinger perturbation expansion are discussed for the polarization expansion of H⁺₂ like systems. It is shown that this expansion may still be useful if the perturbed hamiltonian has an additional symmetry.