Nodeless orbitals

The term Q is difficult to evaluate quantitatively. It does not seem proper to use Slater nodeless orbitals since these are generally good only in the overlap region. If we use hydrogen-like orbitals for 2p and 2s, we obtain... 

Once a nodeless orbital has been generated the one-electron atomic Fock equation is easily inverted to produce a (radially) local operator, the EP, which represents the core-valence interactions (22,23). 

The energies of the depend principally upon the value of L, because this defines the number of angular nodes. The nodeless orbital always lies lowest, followed by... 

Here 1/ is the effective potential and a>i i is a nodeless pseudo-orbital that can be derived from Xi, in several different ways. For first-row atoms, Christiansen, Lee and Pitzer (1979) suggest... 

Why is the VB description relatively more successful for H2 than for F2 From the nodeless, directionless character of Is orbitals, one can judge that both lsA and lsB orbitals of H2 are approaching a common united-atom form as their centers approach, so there is not much change if one switches hA with hB for one of the... 

In agreement with theoretical prediction, the experimental analysis shows the more positive atoms to be contracted. This is explained by the decrease in electron-electron repulsions, or, in a somewhat different language, the decreased screening of the nuclear attraction forces by a smaller number of electrons. This contraction is incorporated in Slater s rules for approximate, single exponential (and therefore nodeless), hydrogen-like orbital functions (Slater 1932). For a 2px orbital of a second-row atom, for example, the orbital function is given by... 

The signs have been chosen to make the orbitals antibonding for positive values of the constants. This choice is based on the assumption of nodeless radial functions for all orbitals. If hydrogen-like radial functions are used, not all constants will be positive. 

Nodeless valence orbitals are used with Goddard-Kahn-Melius type ECP s, while the nodal structure in general is kept in conjunction with Huzinaga-type ECP s. In both cases the valence basis set is determined by some fitting procedure. When the nodal structure of the valence orbitals is kept typically one primitive function is used to describe an inner node. 

The action of the projection operator (-e +A) 0 ><0 is to raise the eigenvalue of the core orbital % to the value A. A new lower bound for the eigenvalue for the pseudo-orbital xl can be shown to be the lower of A and ej. In practice the core eigenvalues are usually shifted so as to be degenerate with the lowest valence eigenvalues of the same symmetry. The coefficients a in equation (37) can now assume values which allow the pseudo-orbital Xt to be nodeless and thus capable of representation by a smaller basis set expansion. 

After orbital optimization, the orbitals za and z./, display a node, while the 4a and 4 orbitals are nodeless, like z and z in Equation 10.5. Note that the two sets of spectator orbitals, v / or i /, are common to two ionic structures. The SL-BOVB calculation is described in Input 10.5, which differs from Input 10.4 by having two supplementary ionic structures (see nstruct keyword) and two more orbitals. The active orbitals 23 and 24 correspond to za and z./, and it is noteworthy that the ionic structures 2 and 4 (3 and 5) share the same set of spectator orbitals, respectively, 9—14 (16—21). 

For B (X3P), the fully symmetric configuration was the obvious (si s2 s3 s4 px py), with si and s2 a pair of strongly overlapping inner orbitals, s3 and s4 more difftise, but nodeless, valence s orbitals, and (px, py) a pair of symmetry-equivalent 2p orbitals. The two symmetry-related configurations were (s i S2 s3... 

A complete review of the characteristics of various types of basis sets has been given recently by Schaefer.44 The radial form of STO s is similar to the (nodeless) hydrogenic atomic orbitals, rn -1e - r where n is the principal quantum number and C is a variable exponent.45 Their angular dependence is described by multiplication by a spherical harmonic The use of (jTO s in molecular calculations was... 

Namely, the circular orbit (l = n — 1), which is a rotating state with a nodeless radial wavefunction, corresponds to a vibrational quantum number v = 0, and the next-to-circular single-node state (l = n — 2) corresponds to v = 1,.... This theoretical possibility of large-/ circular orbits behaving like bound states in a Morse potential seems to have no other natural manifestation than in the present case of metastable exotic helium. This situation is presented in Fig. 2, where the potential as well as the wavefunctions are shown. 

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