Continuum orbitals

Many kinds of transition probabilities depend on DOs. Photoionization cross sections, are proportional to the absolute squares of matrix elements between DOs and continuum orbitals, or... 

Any molecule has an infinity of excited orbitals in the continuum above the first ionization energy. The electric dipole polarizability is connected partly with a few of these continuum orbitals and partly with the valence orbitals (7). If the simultaneous formation of empty orbitals of X, but with the continuum, it is reasonable to think of M being polarized by X. The population of the continuum orbitals of X is expected to be the more... 

Johnson and Rice used an LCAO continuum orbital constructed of atomic phase-shifted coulomb functions. Such an orbital displays all of the aforementioned properties, and has only one obvious deficiency— because of large interatomic overlap, the wavefunction does not vanish at each of the nuclei of the molecule. Use of the LCAO representation of the wavefunction is equivalent to picturing the molecule as composed of individual atoms which act as independent scattering centers. However, all the overall molecular symmetry properties are accounted for, and interference effects are explicitly treated. Correlation effects appear through an assigned effective nuclear charge and corresponding quantum defects of the atomic functions. 

Fig. 30. The total differential oscillator strength for benzene including the structure factor but neglecting any vibrational effects. The cross-hatched portion of the figure represents the transition to a continuum orbital of e2ll symmetry, while the remainder is for a transition to an elu orbital. The positions of higher ionization thresholds are indicated.

E. Brandas, P. Froelich, Continuum Orbitals, Complex Scaling, and the Extended Virial Theorem, Phys. Rev. A16 (1977) 2207. 

For simplicity, the constant of proportionality which takes care of different normalizations of the radial functions involved has been omitted, the symmetry of the wavefunctions with respect to an interchange of r2 and r2 is not incorporated explicitly, and the presence of continuum orbitals is indicated only through the integral symbol.) The dntrmtr 2 coefficients then describe the quality of this expansion if only certain truncated sets of basis functions are selected, because for a complete basis set... 

The //-electron target wave function is coupled to a continuum orbital continuum electron does not modify the effective Hamiltonian Q that acts on occupied target orbitals (nt = 1). Q also acts on d>K because 0 cancels out of the functional derivatives in -%j. This implies that exchange equations with a nonlocal correlation potential vc. 

From Eq. (5.7), and Janak s theorem [185], the contribution of correlation energy to the mean energy of the continuum orbital within the //-matrix boundary is... 

On the right hand side, the ion state is still antisymmetric in the first N — 1 electrons, but the continuum orbital is occupied exclusively by the iVth. That it is possible to write this may be seen by expanding the left hand side as a sum of N terms, in which the continuum orbital is occupied by a different electron in turn. Each term must have the same value, since all electrons are equivalent in the state t>. Thus we have N equal terms, with a normalization factor of N K Replacing the ground state i in Eq. (10) by its expansion according to Eqs. (8) and (9), and using the fact that all the ionised states are orthogonal, we find ... 

Though many of the features of molecular PE cross sections can be attributed to atomic effects, some features, such as shape resonances, are critically dependent on the form of the molecular potential. This itself is determined in part by the physical shape of the molecule, from which the name of the effect derives. Shape resonances manifest themselves as increases in cross section, which are frequently sharp and generally at relatively low PE kinetic energies. They are generally viewed as attributable to momentary trapping of the electron in a quasi-bound continuum orbital created by an effective potential of the molecular ion, or, excitation into antibonding orbitals located in the continuum. ... 

Note that continuum orbitals are normalised in the sense discussed in section 6.4. If two orbitals < ) are identical the determinant vanishes because it has two equal columns. No two electrons can be in the same orbital. The iV-electron state p) therefore obeys the Pauli exclusion principle. We call it a configuration. 

Note that the commutation rules (3.137—3.139) and the symmetric operators (3.142,3.149) have been derived from properties of determinants. We have not assumed that the orbitals p), v) are orthogonal. In evaluating matrix elements care must be taken to keep track of the scalar products of orbitals that are not orthogonal, such as bound orbitals and plane waves. The iV-electron target configurations are conveniently normalised by (3.123). The normalisation of the continuum orbitals is discussed in chapter 6. 

The direct reduced potential matrix element in j j coupling is calculated by substituting the full potential matrix elements into (7.37) with the continuum orbitals given by (7.46), the bound orbitals by (7.49), the two-electron potential by (7.60) and the reduced density-matrix element by (7.59). 

The N 4- l)-electron collision problem is solved in a sphere of radius a, which is chosen to be larger than the distance beyond which the radial orbitals of a chosen set of bound states become negligibly small. The solution is essentially a configuration-interaction method where the basis configurations consist of determinants of bound orbitals representing the N target electrons and continuum orbitals for the continuum electron. The radial continuum orbitals are solutions of a potential problem... 

The quantities are Lagrange multipliers ensuring that the continuum orbitals are orthogonal to bound orbitals of the same symmetry. The basis is given further flexibility by including short-range (N 4- l)-electron functions to allow for possible resonant states of the (N 4- l)-electron system. 

The wide applicability of (10.30) justifies showing its computational form. Formally (10.30) is a potential matrix element (6.88) in the distorted-wave representation for a three-body collision with the bound orbital i) replaced by the continuum orbital (distorted wave) lx (ks))- The direct matrix element is written in a form analogous to (7.62) using the distorted-wave form (4.58) of (7.45) for the continuum orbitals, (7.49) for the bound orbital a) and (7.60) for the electron—target potential. Note that the term l/ro in (7.60) vanishes if we require >f (ks)) to be orthogonal to a). This requirement is implicit in (10.10,10.11) and is normally imposed in implementing (10.30). 

In the next section we will discuss the approach we have developed for obtaining the molecular Hartree-Fock continuum orbitals. We will discuss how our approach is based on the Schwinger variational method and how in its present form it can be viewed as a hybrid method that uses both the basis-set expansion techniques of quantum chemistry and the numerical single-center expansion techniques of atomic collision physics. We will then discuss the results of applications of this approach to study shape resonances in the photolonlzatlon of several molecules, e.g., N2, CO, CO2, C2H2, and C2N2. These results will also be compared with available experimental data and with the results of studies of these same systems by different methods and models. 

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