Split-shell wavefunction

Fig. 1. Electron-electron distribution functions for single-configuration He wavefunction (a) radial probability distribution P(ri2) (b) intracule function h(ri2)- In both graphs, the curve with largest maximum is for the closed-shell wavefunction that of intermediate maximum is for the split-shell wavefunction that of smallest maximum is for the wavefunction containing exp( —yri2).

The interpretation of type 5 wavefunction is somewhat difficult. We have said that it can be described as a two-configuration wavefunction. This is sometimes referred to as a split-shell wavefunction others refer to this wave-function as providing for in-out correlation. All agree that wavefunction type 5 takes account of some of the radial correlation, but ten Hoor has cautioned about the interpretation of the results in terms of the in—out terminology. ... 

In this trial function, f has the same value in each atomic orbital. This is not a necessary restriction. There is no physical reason for not choosing the more general trial function where orbitals with different are used. Symmetry requires that such a function be written 2 lulls 2) + ls"(l)ls (2)]2 F2[, (j) (2) - /3(l)cr(2)]. This type of function is called a split shell wavefunction. It gives a lower energy for He than does the function (7-26). However, for most quanrnm-chemical calculations split shells are not used, the gain in accuracy usually not being commensurate with the increased computational effort. 

The differences between the single-configuration wavefunctions are more clearly illustrated by comparing their plots of the intracule function h(ri2), also shown in Fig. 1. This plot reveals the absence of an electron-electron cusp for both the closed and split-shell functions, but shows that the inclusion of exp( —yri2) causes the distribution to have a minimum at ri2=0, forming a cusp (of the correct sign) at that point. This feature will be important for the description of phenomena that depend upon the coincidence probability. 

The wavefunction described with the optimum a,y corresponds well with a somewhat perturbed lslj 2s configuration. Two of the electrons depend on the electron-nuclear distance with a values (screening parameters) that correspond to a partially screened interaction with the - -3-charged Li nucleus in a split-shell electron distribution. The third electron (that with pre-multiplying r) has an a value somewhat larger than for a hydrogenic 2s orbital, indicative of the fact that the inner-shell electrons do not completely shield the Li nucleus. The electron-electron a values all reflect the existence of electron-electron repulsion, with the effect most pronounced for the Is-ls interaction. All these observations are consistent with the notion that the exponentially correlated wavefunction gives an excellent zero-order description of the electronic structure of Li. 

The atomic orbital wavefunctions come in sets that are associated with four different quantum numbers. The first is the principal quantum number, which takes on positive integer values starting with 1 (n = 1,2,3,...). Anatom s highest principal quantum number determines the valence shell of the atom, and it is typically only the electrons and orbitals of the valence shell that are involved in bonding. Each row in the periodic table indicates a different principal quantum number (with the exception of rf and f orbitals, which are displaced down one row from their respective principal shells). In addition, each row is further split into azimuthal quantum numbers (wi = 0,1,2,3,... alternatively described as s, p, d,f...). This number indicates the angular momentum of the orbital, and it defines the spatial distribution of the orbital with respect to the nucleus. These orbitals are shown in Figure 1.1 for = 2 (as with carbon) as a function of one of the three Cartesian coordinates. 

From the standpoint of chemical reactivity, the 5s and 5p shells of lanthanides can be considered to be core electrons. Indeed arguments to this effect could be made for Ln 4f orbitals, given their extremely contracted nature. Shown in Figure 3 is a plot from a DHF calculation of a 4f spinor for Gd(III). Note the maximum in the wavefunction at 0.57 A that is, a value comparable to a hydrogen Is orbital Dolg et al. - examined various lanthanide core sizes and found essentially no difference in state splittings of the Ce atom between all-electron calculations and those in which a 28-electron core ([Ar]3d °) is used, Satisfactory results are also obtained for a 46-electron core ([Kr]4di°). Inclusion of 5s and 5p into the core (i.e, a 54-electron [Xe] core)... 

Fill in the details of the derivation of the spin-spin coupling formula (11.7.13). Obtain the coupling function Qss(dd ri, rj) for the standard state a with M = S, using a 1-determinant wavefunction with the spins of the open-shell electrons all parallel-coupled and show that this function is then expressible in terms of the spin-density matrix Qs da r, r0- What kind of integrals will you have to evaluate to obtain, in an LCAO approximation, the tensor components that determine the zero-field splitting of Zeeman levels [Hint Closed-shell electrons give no contribution to Qs or Qss, while for electrons in orbitals with a common a factor the 2-electron density matrix tr is related to p as in Section 5.3. 

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