Connection formulae

Renumber formulas 71 and 72 to 71a and 72a. (An arrow should connect formulas 73 and 74. Formula... 

The first compound containing a well-characterized cluster in a metal oxide system was Zn2Mo308, reported in 1957 by McCarroll, Katz and Ward (11,12). In this compound the M03OQ cluster units are constructed such that three Mo08 octahedra share common edges, and share 0 atoms with adjacent cluster units as reflected in the connectivity formula / 2 O3/ 3 ... 

The relativistic adiabatic connection formula is based on a modified Hamiltonian H g) in which not only the electron-photon coupling strength is multiplied by the dimensionless scaling parameter g but also a g-dependent, multiplicative, external potential is introduced. 

Although the adiabatic connection formula of Eq. (8) justifies a certain amount of Hartree-Fock mixing, there are situations in which a should vanish. In a spin-restricted description of the molecule Hj at infinite bond length (Sect. 4) the Hartree-Fock or A = 0 hole is equally distributed over both atoms, and is independent of the electron s position. But the hole for any finite A, however small, is entirely localized on the electron s atom, so no amount of Hartree-Fock mixing is acceptable in this case. 

Nagy, A. (1995). Coordinate scaling and adiabatic connection formula for ensembles of fractionally occupied excited states, Ini. J. Quantum Chem. 56, 225-228. 

A single particle in three dimensions may be described in terms of cartesian coordinates (x, y, z) or polar coordinates (r, 9, (p) with the connection formulas... 

Let us, for a moment, consider a single particle in one dimension with a Hamiltonian of the type// = p2/2m + V(x). This is a second-order differential operator, and this means that the general solution to the inhomogeneous Eq. (3.51)—considered as a second-order differential equation—will consist of a linear superposition of two special solutions, where the coefficients will depend on the boundary conditions introduced. As a specific example, one could think of the two solutions to the JWKB problem, their connection formulas, and the Stoke s phenomenon for the coefficients. 

We shall first briefly describe the phase-integral approximation referred to in item (i). Then we collect connection formulas pertaining to a single transition point [first-order zero or first-order pole of Q2(z) and to a real potential barrier, which can be derived by... 

Connection formulas associated with a single transition point... 

Connection formulas pertaining to a first-order transition zero on the real axis... 

Q2(z) R(z). Below we shall present these general connection formulas. 

The connection formula for tracing a phase-integral solution of the differential equation (4.1) from the classically allowed to the classically forbidden region is... 

The arbitrary-order connection formulas (4.19), (4.21) and (4.22) can in many cases be used for obtaining very accurate solutions of physical problems, when the turning points are well separated, and there are no other transition points near the real axis in the region of the complex 2-plane of interest. Within their range of applicability, these connection formulas are very useful because of their simplicity and the great ease with which they can be used. They have been discussed by Froman and Froman (2002) see Sections 3.10-3.13 and 3.20 there. 

Connection formula for a real, smooth, single-hump potential barrier... 

The barrier connection formula presented in this section is valid uniformly for all energies, below and somewhat above the top of the barrier. We would also like to emphasize that while the connection formulas pertaining to a turning point are one-directional (N. Froman 1966a, Froman and Froman 2002), the barrier connection formula (4.30a,b) along with (4.31)-(4.35) is bi-directional. However, when the energy is close to a resonance energy, a careful discussion is required. 

By means of (4.57) and the connection formula (4.19) one finds that the physically acceptable wave function is given by... 

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