Projection method, cranking condition and collective energy

A treatment of molecular excited states in the second-quantization formalism, using the dual-space operator technique, is given. The open-shell configurations give degenerate multiplets which are easily described in what we call the occupation-branching-number representation. Equivalent monoelectronic and bielectronic operators are defined and a diagrammatic technique is developed. The method is at present restricted to molecules with Abelian symmetry groups. A linked-cluster theorem is finally proved.

Structure of the O16 nucleus

We extend to k \ensuremath{\ne} 0 the work in a previous paper on the solution of the Schr\"odinger equation in a periodic potential. We begin with a perturbation method, and expand the wave function and energy in powers of $k$. The equations determining the wave function to any given order in $k$ are then just inhomogeneous linear equations, which are practical to solve. We give formulas for evaluating the energy from these wave functions: For bands which are not degenerate at k = 0 these involve straightforward summations over the reciprocal lattice; for bands which are degenerate at k = 0, we show how to derive the secular determinant which gives the splitting of these bands. The elements of this determinant can also be calculated as sums, which are practical to evaluate, over the reciprocal lattice. We present a simple numerical example to show how some of the above results work out in practice. Next, we discuss tight-binding formulas in the present formalism, and the practical problem of evaluating these formulas. We give simple expressions, involving the radial derivatives of the atomic momentum space wave functions, for evaluating the energy in powers of $k$; these formulas do not make near-neighbor approximations. Then, for those cases for which neither the tight-binding approximation nor an expansion in powers of $k$ is appropriate, we discuss the direct numerical solution of the equations and give a convenient partial wave expansion for this purpose. Finally, we briefly discuss the relation to each other of the various approximations presented in this paper.

In Z-dependent perturbation theory, the lowest-order wave functions for a polyatomic molecule are not only independent of the nuclear charges, but also of the total number of nuclear centers and electrons in the molecule. The complexity of the problem is then determined by the highest order retained in the calculation. Choosing the simplest possible unperturbed Hamiltonian, we describe an n-electron, m-center polyatomic molecule as n ‘‘hydrogenic’’ electrons on a single center perturbed by electron–electron and electron–nucleus Coulomb interactions. With this H0, the first-order wave function for any polyatomic molecule will be a sum of products of hydrogenic orbitals with either two-electron, one-center or one-electron, two-center first-order wave functions. These first-order wave functions are obtained from calculations on He-like and H+2-like systems. Similarly, the nth-order wave function decouples so that the most complex terms are just the nth-order wave functions of all the p-electron, q-center subsystems (p+q=n+2) contained in the molecule. We illustrate applications of this method with some results, complete through third order in the energy, for H+3-like molecules. These are compared with accurate variational results available in the literature. We conclude that, through this order, this perturbation approach is capable of yielding results comparable in accuracy to variational calculations of moderate complexity. The ease and efficiency with which such results can be obtained suggests that this method would be useful for generating detailed potential energy surfaces for polyatomic molecules.

The nuclear potential, the transfer-induced dissipation, and the mass diffusion coefficient in heavy-ion collisions are investigated in a proximity formulation. An energy-dependent nuclear potential is calculated in the frozen wave function approximation using two slabs of symmetric nuclear matter, each described by Hartree-Fock single-particle wave functions. Corrections to the inertia parameter are also evaluated from this potential. The flux entering the window formula for the friction between two heavy ions is calculated in a simple barrier penetration model. The classically forbidden flux is found to make a significant contribution. The transfer flux arising from both the relative motion and finite temperature of the nuclei is calculated and the latter is used to estimate the mass diffusion coefficient. Using the mean trajectories from time-dependent Hartree-Fock calculations the charge variance is calculated for the reaction 84 Kr(712 MeV) + 209 Bi and is found to be in agreement with experiment.

As the flow becomes rarefied it has been seen that predictions of continuum formulations, such as the Navier–Stokes equations, become inaccurate. These inaccuracies stem from the linear approximations to the stress and heat flux in the viscous flux terms in the Navier–Stokes equations. Hence, it has long been conjectured that the inclusion of higher-order terms in the constitutive relations for the stress and heat flux may improve the predictive capabilities of such continuum formulations. Following this approach, second-order systems of hydrodynamic equations, such as the Burnett and Woods equations, were applied to the shock structure problem. While it was observed that these equations afforded a better description of the shock structure on coarse grids, they were prone to small wavelength instabilities when the grids were refined. The cause of this instability was subsequently traced to the fact that these equations can potentially violate the second law of thermodynamics when the local Knudsen number exceeds a critical limit. This leads to the fundamental question: is entropy consistency achievable in a system of second-order hydrodynamic equations? To answer this question, a novel set of equations, known as the BGK-Burnett equations, is constructed by taking moments of the Boltzmann equation for the second-order distribution function. The formulation of second-order hydrodynamic equations by moment methods is beset by three hurdles: (i) the highly nonlinear collision integral in the Boltzmann equation needs to be evaluated, (ii) the second-order distribution function does not satisfy the moment closure criterion and (iii) identification of the approximations to the material derivatives in the second-order distribution function that will correctly account for the difference in time scales between the first- and second-order fluxes. The first three terms of the Chapman–Enskog expansion, that defines the second-order distribution function, and the Bhatnagar–Gross–Krook model of the collision integral, form the basis of the BGK-Burnett equations. The entropy-consistent behaviour of the equations depend on the moment closure coefficients and the approximations to the material derivatives in the second-order fluxes. The requirement of moment closure alone, however, results in non-unique closure coefficients and a family of BGK-Burnett equations, from which an entropy-consistent set must be identified. From this family, two sets of BGK-Burnett equations have been considered, and this paper presents the details of the formulation of these two sets of equations, the identification of entropy-consistent approximations to the material derivatives by a novel entropy consistent relaxation technique, and shock structure computations in a monatomic hard sphere gas for a range of Mach numbers.

We achieve an explicit construction of the lowest Landau level (LLL) projected wave functions for composite fermions in the periodic (torus) geometry. To this end, we first demonstrate how the vortex attachment of the composite fermion (CF) theory can be accomplished in the torus geometry to produce the "unprojected" wave functions satisfying the correct (quasi-)periodic boundary conditions. We then consider two methods for projecting these wave functions into the LLL. The direct projection produces valid wave functions but can be implemented only for very small systems. The more powerful and more useful projection method of Jain and Kamilla fails in the torus geometry because it does not preserve the periodic boundary conditions and thus takes us out of the original Hilbert space. We have succeeded in constructing a modified projection method that is consistent with both the periodic boundary conditions and the general structure of the CF theory. This method is valid for a large class of states of composite fermions, called "proper states," which includes the incompressible ground states at electron filling factors $\nu=\frac{n}{2pn+ 1}$, their charged and neutral excitations, and also the quasidegenerate ground states at arbitrary filling factors of the form $\nu=\frac{\nu^*}{2p\nu^*+ 1}$, where $n$ and $p$ are integers and $\nu^*$ is the CF filling factor. Comparison with exact results known for small systems for the ground and excited states at filling factors $\nu=1/3$, 2/5 and 3/7 demonstrates our LLL-projected wave functions to be extremely accurate representations of the actual Coulomb eigenstates. Our construction enables the study of large systems of composite fermions on the torus, thereby opening the possibility of investigating numerous interesting questions and phenomena.

The quantum theory of nucleon correlation in finite nuclei. I. Hartree-Fock approximation and independent-pair model

By using the classically solvable minimal massless scalar minisuperspace model we study two examples of wave functions of the universe in the semiclassical limit. The first one consists of a Lorentzian component and a Euclidean component and admits a clear semiclassical interpretation as a superposition of universes. The second one consists of a Lorentzian (or Euclidean) component and another oscillatory component which corresponds to (neither Lorentzian nor Euclidean) complex classical solutions. This example has some resemblance to Hawking's minimal massive scalar minisuperspace model. We suggest a possible way of recovering the classical interpretation in such a case.

A projection formalism is proposed for describing scattering in a system of several particles on the basis of separation of a finite-dimensional internal subspace. For a sufficiently large internal subspace the kernels of the modified integral equations of Faddeev-Yakubovskii type become small, and this justifies the use of perturbation theory. Expressions are obtained for the amplitudes in the zeroth order, which are completely determined by the inhomogeneous terms of the equations and tend to the exact amplitudes when the internal subspace is enlarged.

A highly optimized code for calculating atomic data at neutron star magnetic field strengths using a doubly self-consistent Hartree–Fock–Roothaan method

A theory of a Fermi superfluid is formulated based on the condensate wave function of the configuration space. First, two basic ingredients are extracted from the wave function, i.e. the effective condensate wave function v ( x 1 , x 2 ) and the quasiparticle field γ( x ) with x denoting the space and spin coordinates r α. After transforming a Hamiltonian into a normal-ordered product of γ, a perturbation expansion with respect to the quasiparticle interaction is formulated for both the thermodynamic potential Ω and the one-particle density matrix ρ. The thermodynamic potential thus obtained includes the normal-state expression of Luttinger and Ward as the limit of v → 0. The density matrix is written as a sum of the coherent part ρ ( c) and the quasiparticle part ρ ( q) , where ρ ( c) is expressed only with respect to v ( x 1 , x 2 ). It is shown that v and γ are responsible for the coherent and thermodynamic properties respectively. Thus the present formalism gives a microscopic foundation for the two-fluid model. Connections with the Gor'kov equations and the Bogoliubov-de Gennes equations are also clarified. It is found that the one-particle density matrices of those formalisms are not equivalent to ρ of the present formalism. The origin of the angular momentum paradox of 3 He - A is traced to this difference.

A new trial wave function is proposed for nuclear physics, in which an exact solution to the long-standing center-of-mass problem is given. In the new approach, the widths of the single-nucleon Gaussian wave packets and the widths of the relative Gaussian wave functions describing correlations of nucleons or clusters are treated as variables in the explicit intrinsic wave function of the nuclear system. As an example, this new wave function was applied to study the typical ${^{20}{\rm Ne}}$ ($\alpha$+${{^{16}}{\rm O}}$) cluster system. By removing exactly the spurious center-of-mass effect in a very simple way, the energy curve of ${^{20}{\rm Ne}}$ was obtained by the variational calculations with the width of $\alpha$ cluster, the width of ${{^{16}}{\rm O}}$ cluster, and the size parameter of the nucleus. They are considered as the three crucial variational variables in describing the ${^{20}{\rm Ne}}$ ($\alpha$+${{^{16}}{\rm O}}$) cluster system. This shows that the new wave function can be a very interesting new tool for studying many-body and cluster effects in nuclear physics.

We study the wide latitude available in choosing the reference space and the zeroth order Hamiltonian H0 for complete reference space multireference perturbation theory. This effective Hamiltonian Heff method employs a general one-body form of H0 which is varied by using different molecular orbitals and orbital energies. An energy gap is imposed between the zeroth order reference and secondary space states by forcing the valence orbitals to be degenerate. The forced valence orbital degeneracy removes the occurrence of detrimentally small perturbation energy denominators. Extensive computations are provided for the nitrogen molecule, where calculated ground state spectroscopic constants are compared with full configuration interaction computations and calculated vertical excitation energies are compared with multireference coupled cluster computations. It is demonstrated that the forced reference space degeneracy can lead to certain perturbation denominators becoming too small for practical convergence. This characteristic is illustrated by a simple two-orbital model which stresses the need for important zeroth order excitation energies (equivalent to the factors appearing in the perturbation energy denominators) to exceed those in an Epstein–Nesbet perturbation partitioning. This simple model illustrates the general behavior found in all the more extensive Heff computations. In many cases where zeroth order excitation energies are too small for satisfactory third order results, improvements are obtained by using an H0 which redefines the orbital energies in order to increase problematic zeroth order excitation energies. The necessary orbital energy shifts are identified by examining the first order wave functions from larger reference spaces and the zeroth order energies. Frequently, fractional occupancy Fock-type operators are employed to provide the requisite orbital energy shifts. Some of the reference spaces investigated deviate extremely from quasidegeneracy and, thus, appear to be beyond the range of applicability of the forced degeneracy Heff method. Novel techniques are employed for properly treating some of these cases, including the use of orbitals which optimize the quasidegeneracy of the reference space and minimize energy denominator problems. By considering reference spaces of varying sizes, we describe the tradeoff between employing large reference spaces, which provide excellent first order descriptions, and the difficulties imposed by the fact that larger reference spaces severely violate the quasidegeneracy constraints of the Heff method. The same tradeoff exists when the optimal first order CASSCF orbitals are compared with orbitals generated by a VN−1 potential. The VN−1 potential orbitals, which produce relatively quasidegenerate reference spaces, are equivalent to the sequential SCF orbitals used in previous Heff computations, but are more simply obtained by a unitary transformation. The forced degenerate valence orbital energy εv̄ is computed from an averaging scheme for the valence orbital energies. The ground state N2 computations contrast two averaging schemes—populational and democratic. Democratic averaging weighs all valence orbitals equally, while populational averaging weighs valence orbitals in proportion to their ground state populations. Populational averaging is determined to be useful only in situations where core–core and core–valence correlation are unimportant. A Fock-type operator used by Roos and co-workers is employed to uniquely define CASSCF orbitals within their invariant subspaces. This operator is found to be more compatible with populational than democratic averaging, especially when the reference space contains high lying orbitals.

We examine various perturbation–variational approximations for molecular electronic energy when the fifth‐order perturbational energies are available. Such theories require very little additional computation once the sequence of perturbation energies are known yet supply a bound even when the peturbation sequence is poorly convergent. We choose for computational examples results obtained very rapidly from a zeroth order wave function consisting of doubly occupied localized bonds and examine polarization within these bonds, delocalization, and bond breaking. In general, we find that the fifth‐order energy itself, and especially the [2, 1] Padé approximant on this sequence, are especially accurate in estimating the total energy and more accurate than any variational scheme when the zeroth order localized wave function is a good description of the electronic structure. The variational results, however, are nearly as accurate, and a [1, 0] Padé on the sequence of variational results is remarkably robust, even in those cases where the perturbation sequence is poorly defined.We also examine several scaling techniques, or partitionings of the Hamiltonian. Although these scaling techniques do accelerate convergence of the perturbation sequence, none that we examine give better results, than the [2, 1] padé, which is independent of any scaling.