Orbital nodeless

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. 

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... 

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. 

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). 

One not so obvious problem with the shape-consistent REP formalism (or any nodeless pseudoorbital approach) is that some molecular properties are determined primarily by the electron density in the core region (some molecular moments, Breit corrections, etc.) and cannot be computed directly from the valence-only wave function. For Phillips-Kleinman (21) types of wave functions, Daasch et al. (52) have shown that the core electron density can be approximated quite accurately by adding in the atomic core orbitals and then Schmidt orthogonalizing the valence orbitals to the core. This new set of orbitals (core plus orthogonalized valence) is a reasonable approximation to the all-electron set and can be used to compute the desired properties. This will not work for the shape-consistent case because / from Eq. (18) cannot be accurately described in terms of the core orbitals alone. On the other hand, it is clear from that equation that the corelike portion of the valence orbitals could be reintroduced by adding in fy (53),... 

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... 

Uranocene, U( -C8H8) provides an excellent example. The f ionization band, the first of the spectrum, shows two characteristic features, a delayed maximum and a giant resonance. Figure 13(a) shows the cross section of this band. The next two bands are primarily ring C 2p r bands of 62 symmetry. They show the characteristic decay associated with nodeless 2p orbitals (Figure 13(b)). The second band, however, has, superimposed on this decay, a maximum around 40 eV and a double resonance between 95 and 125 eV. This indicates partial 5f character and assigns the orbital to the e2 orbital. The third band is consequently associated with the C2g orbital and its higher IE is indicative of the fact that the U 6d orbitals form more effective bonds that the 5f in this instance. 

Here n denotes "effective quantum number", exponent 5C, is an arbitrary positive number, r, t, y) are polar coordinates for a point with respect to the origin A in which the function (2,3) is centered. Apart from the first two terms that represent a normalizing factor, the function (2,3) is closely related to hydrogen-like orbitals. For the hydrogen Is orbital the function I q q 0 identical with Q q, if we assume Z Z/n, However, it should be recalled that in contrast to hydrogen-like orbitals STO s are not mutually orthogonal. Another essential difference is in the number of nodes. Hydrogen functions have (n -i - 1) nodes, whereas STO s are nodeless in their radial part. Alternatively, the STO may be expressed by means of Cartesian coordinates as follows... 

C-H bond is replaced by calculating the difference in correlation energy between a are then replaced by a set of nodeless pseudo-orbitals. The regular valence orbitals will... 

The n5-orbitals are all spherically symmetrical, being associated with a constant angular factor, the spherical harmonic Too = 1 /V4. They have n — radial nodes—spherical shells on which the wavefunction equals zero. The I5 ground state is nodeless, and the number of nodes increases with energy, in a pattern now familiar from our study of the pai1icle-in-a-box and harmonic oscillator. The 2s orbital, willi its radial node at r = 2 bohr. is alsit shown in Fig. 7.3. 

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