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Notes: A class of reference full density matrices and their reduced density matrices is presented. These density matrices are designed to be of value as references from which to describe and measure the effects of electron correlation in atoms, molecules, and solids. A given reference full density matrix is constructed to contain the least possible information consistent with having the (recognized) symmetry properties of - and reducing to the 1-matrix of - a given “true” full density matrix (which in a typical application is constructed from a correlated variational wave function). Therefore, the reduced density matrices derived from are representable and depend only upon the 1-matrix of and the (recognized) symmetry properties of for their construction. Furthermore, the property of containing the least possible information consistent with the given constraints makes these reference density matrices ideally suitable as references from which to describe the electron correlation contained in the “true” full density matrix .

Notes: Representable reference density matrices are applied to the graphical description of the Coulomb hole in a CI wave function for the first excited 1S state of the four-electron boron ion B+. It is found that a satisfactory picture of the Coulomb hole emerges only when the symmetry of the correlated wave function is recognized in the construction of the reference full-density matrix.

Notes: Toward the goal of efficiently computing field lines for molecular vector fields, where each field-point calculation is computationally intensive, a few appropriate algorithms for calculating vector field lines are presented and compared for various representative applications. Among these algorithms are the first-order tangent-line method (TL), the fourth-order Runge-Kutta method (RK4), the infinite-order Bulirsch-Stoer method (BS), the second-order Taylor's series method (TS2), and the second-order “curvature-following” method (CF). The TL and the RK4 are well known. The TS2 and the CF are new. The RK4, The TS2, and the CF are appropriate for obtaining high accuracy with few field-point calculations. The TL is definitely not appropriate for this purpose, and the BS is so appropriate only at the highest level of required accuracy. The CF uses the value of the vector field and its gradient at the given field point in order to locate the center of curvature of the field line at that point and, thereby, to extrapolate the field line, as an arc of a circle, to the next field point. The TS2 uses the same information, but extrapolates the field line as a segment of a parabola whose vertex is at the field point. The BS is an infinite-order extrapolation on successively finer scale iterations of the lowest-order Runge-Kutta method. All of these methods are compared for the velocity field of a rotating disk, for the vector field of a point dipole, and for electric field of a high-speed orbiting charged particle. For all of these fields, the field lines are exactly expressible in analytic form, so the absolute errors of these different algorithms can be appraised. As several of these methods allow a rather large step size, it was found appropriate to use a continuous-(geometrical)-curvature interpolation scheme to interpolate between the field points on the generated field lines. One prefers this scheme to the use of cubic splines when the physical fields should have their “geometrical pictures” (even under approximation) invariant to an arbitrary change of the coordinate system used in the calculation. These methods have also been used to generate field lines for the approximate and the exact fields of a half-wave antenna. In this case, one sees very large differences in the structure of the field lines of the approximate and the exact fields, in the near-field region, and also sees the manner in which each of these differences diminishes to zero as the field point approaches the far-field region; features that would have been very hard to observe by purely analytical methods. It is hoped that these methods for field line generation might have application for the field lines of the gradient of the molecular density (as in Bader's theory of atoms in molecules) and for the field lines of the electric fields of nucleic acid molecules.

Notes: The RPA, SCRPA, Tamm-Dancoff, and full CI methods are compared by analyzing their transition density matrices, oscillator strengths, and energy moments of oscillator strengths for the 1Sground - 1Podd transitions of the 4-electron B+ ion in the frozen K-shell approximation. It is found that the RPA gives transition density matrices that are aligned nearly as well as possible along those of the full CI, but have vector lengths that are significantly too long. The corresponding transition energies are significantly too small. These errors compensate to give oscillator strengths for the dominant transition that, for all forms of the oscillator strength, are within 1.6% of the corresponding full CI values. The SCRPA gives better transition density matrices than the RPA, but poorer oscillator strengths. The Tamm-Dancoff approximation gives very good values for the mixed length-velocity form of the oscillator strength. The RPA gives a static electric dipole polarizability that is nearly 20% larger than that of the full CI. The SCRPA gives a value 15% smaller than - and the Tamm-Dancoff approximation gives a mixed length-velocity value that is 11% larger than - that of the full CI. Other energy moments of oscillator strengths are also reported. Certain other approximations related to the RPA and the SCRPA are reported as well.

Notes: A density operator analysis is made of the particle-ejection (e.g., photoionization) problem in the “generalized sudden approximation” for the case in which the ejected particle is detected but its dynamical properties are not observed. It is shown that the resultant statistical density operator may be derived from that obtained in a particle-conserving Hamiltonian-evolution linear-response model with a random impulse perturbation, and that the physical applicability of the latter model embraces the range of applicability of the former. The initial value of the statistical density operator for the above particle-ejection problem is shown to be the N - (1)-particle reduced density operator of the initial state; and the initial value of the statistical density operator for the excitations in the random impulse model is shown to be the N-particle Hermitian operator whose matrix representation is the G matrix of Garrod and Percus. The significance of the eigenvalue spectrum of these operators to the excitation properties of the system is discussed, especially for the random impulse model, where a large eigenvalue of the G matrix can signal strong preferential excitation to its corresponding particle-hole collective state, even for a random perturbation. Extensions of these ideas to excitations from states with a large eigenvalue of the two-particle reduced density operator (e.g., superconducting states) are mentioned. The The applicability of these density matrices to the description of the excitation spectrum due to a well-defined perturbation is discussed. The relationship of these time-dependent density operators to the one-particle propagator and the particle-hole propagator (polarization propagator) is established.

Notes: A model many-fermion Hamiltonian is presented for which the ground state is asymptotically an Antisymmetrized Geminal Powers (AGP) wave function with largest possible greatest eigenvalue for its two-particle reduced density matrix. Closed analytical expressions and plane-wave expansions are presented for the generating geminal of the AGP ground state and for its one-particle reduced density matrix. The natural orbitals for this generating geminal are plane waves. The generating geminal shows intensely local character in its intracule and corresponds to the formation of a quasi-boson from two fermions. One may appropriately modify this generating geminal to introduce zero occupation numbers of its one-particle reduced density matrix and to make all the nonzero occupation numbers of its one-particle reduced density matrix equal, thus making this geminal a generator of an extreme AGP wave function, with an extreme large eigenvalue for its two-particle reduced density matrix. Closed analytical expressions are also given for this modified geminal and for its one-particle reduced density matrix. The modified generating geminal develops anomolously long “tails” in the intracule function. Nevertheless, both generating geminals introduced here reduce to a renormalized Dirac δ-function of the intracule coordinate, in the appropriate limit. In this limit, both of these generating geminals produce an absolute extreme AGP wave function with the absolute extreme value, N/2, for the Löwdin-normalized occupation number of the generating geminal, and vanishing occupation numbers for all other natural geminals, in the two-particle reduced density matrix of this wavefunction. A demonstration is given showing the formation of quasi-bosons in this ground state and showing their relationship to the generating geminal and to the other natural geminals of the two-particle reduced density matrix of the extreme AGP wave function. The similarities and differences of the features of this model and the accepted models of the superconducting ground state of electrons in metals, and the superfluid ground state of liquid He4 are mentioned.

Notes: In this article we discuss several principles and tools which should expedite description of the electrostatic potentials and electrostatic interactions of molecules, and show that these also lead to some rather remarkable results in the theory of the irreducible representations of the full rotation group SO(3). First, by representing a molecule's charge-density matrix over a basis of atomic-like orbitals (on the various atoms), we observe that outside its charge distribution the molecule's electrostatic potential is exactly the same as if that charge distribution were merely a sum (and in the case of a finite orbital basis, this is a finite sum) of point multipoles on each of the atomic centers and line multipoles on the line segments joining each of those atomic centers. Possible methods of approximating the field of these line charges and line multipoles, as if they were due to point charges and point multipoles, are discussed. The calculation of the interaction of point multipoles of high order, as is necessary for this procedure to successfully calculate the interaction of arbitrarily oriented molecules, motivates our second topic. Here we present a differential operator which, when acting on the 3-dimensional delta function, produces the source density for a scalar field that is exactly an (l,m) multipole field. Using the Hermitian adjoint of this operator, we express the interaction of this (l,m) multipole with an external scalar field as the result of this differential operator acting on that external field at the location of this multipole source. Irreducible representation matrices of the full rotation group are then used, together with these relations, to simplify the interaction of two arbitrarily oriented multipoles of any orders. Finally, we use the representation of the Condon and Shortley “raising and lowering” relations on eigenstates of the z-component of angular momentum, in an orientation that is not aligned with its fundamental basis states, to generate recursion relations that allow simple calculations of the irreducible representation matrices of the full rotation group, SO(3), and the special unitary group, SU(2). From these recursion relations we display some useful symmetry properties of our parameterization of these matrices, that allow the entire matrix to be very simply generated from an explicit calculation of only about 1/8 of its elements. © 1992 John Wiley & Sons, Inc.

Notes: Several schemes are discussed for partitioning the second-order reduced density matrix Γ into two parts, Γ0 and Γ′.The Γ0s are based on the independent particle model and the Γ′s are corrections due to electron correlation. The difficulties of choosing a Γ0 that will serve as a suitable reference point for studying electron correlation are discussed.In order to compare alternative partitioning schemes, an atomic wave function for the 1S ground state of the Be atom in the configuration-interaction approximation was selected. A fifty-two configuration wave function was computed and contour graphs were made of the total pair density Γ(1 2) and of the “correlation pair density” Γ′(1 2) for several choices of the reference Γ0.