Gamow orbital

Equation (31) shows a characteristic feature of the resonance wavefunc-tions, namely that the integral over radial space (not over the spherical harmonics), involves the square of the function itself and not of its absolute value. However, due to the asymptotic behavior (25a), the integral (31) (as well as the one of Eq. (30)), is infinite, since the integrant goes like where a is the imaginary part of the complex momentum of the free Gamow orbital. 

In the CESE approach, the coordinates of fhe Hamiltonian are real. Special attention is given to the accurate, state-specific, single- or multi-state calculation of ho. Complex coordinates are used only in certain orbitals of fhe function space, which is chosen appropriately so as to follow the two-part form of fhe resonance eigenfunction discussed above. We have called these orbitals "Gamow orbitals." They may be chosen to have particular forms in harmony wifh fhe corresponding open channels, or fhey can be expanded in standard forms, such as fhe one of Slater orbitals, albeit with a complex coordinate. Specifically, in fhe case of one-electron decay, the simplest version of fhe SSA square-infegrable resonance wavefuncfion for an N-electron atom is (r symbolizes collectively the real coordinates of fhe bound orbitals). 

Instead, Nicolaides and Beck [37b, p. 506] considered the exactly solvable problems of the harmonic oscillator and of hydrogen and found that in fact, a single function can diagonalize H(rd ). This is the rotated function of the unrotated solution, i.e., [H(rd ) — e ] (rd ) = 0, for each state n > and real e . (The proof is straight-forward). In fact, we stated two "theorems" that are basic to the development of the practical implementation of the CESE-SSA [37b, p. 505], since, in practice, they allow the difficult electron correlation calculations to be done only once on the real axis, and then continue the computation in the complex energy plane where the Gamow orbitals are optimized until the complex energy is stabilized. 

It is difficult theoretically to calculate a tunnelling barrier of Gamow type (not the same as activation barrier) from first principles. This paper summarizes methods that employ molecular orbital (MO) approaches to calculate the "electronic factor" which accounts for the transmittance in the forbidden region. The electronic factor will be structure dependent and there is no reason why it should be isotropic in space. 

The final state nucleus (deuterium in its ground state) in the reaction p + p —> d + e+ + z/e, has JJ = 1 +, with a predominant relative orbital angular momentum If = 0 and Sf = 1 (triplet S-state). For a maximally probable super-allowed transition, there is no change in the orbital angular momentum between the initial and final states of the nuclei. Hence for such transitions, the initial state two protons in the p + p reaction must have /j = 0. Since the protons are identical particles, Pauli principle requires S) = 0, so that the total wave-function will be antisymmetric in space and spin coordinates. Thus, we have a process Si = 0, Z = 0 >—>, S y- 1,/y- 0 >. This is a pure Gamow-... 

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