General set of f orbitals

There is another way of looking at this. In symmetry the f orbitals transform as 2u + Tiu + T2 . Reduce the symmetry by a distortion (compression or elongation, for example) along a fourfold axis so that the symmetry is now 1)4. The f orbitals now transform as (Bi ) + ( 2 + m) + B2U + F ), where the brackets correspond, in order, to the symmetries in 0 given above. It can be seen that two different sets of f orbitals have symmetry. Because they have the same symmetry they can mix and it proves convenient to let them do this and to work with combinations of them. As a result, some differently shaped f orbitals arise. The members of this so-called general set of f orbitals are shown in Fig. 11.3, where both their abbreviated and complete labels are given. 

Fig. 11.3 The general set of f orbitals together with their shortened and detailed Cartesian angular forms.

Fig. 11.4 The nodal patterns of the general set of f orbitals, viewed down the z axis (cf. Fig. 10.2(a)).

In any molecule, the potential close to a nucleus approximates that of the free atom (in the correct valence state) consequently, in these regions the molecular orbitals lie near the free atomic orbitals. Accordingly, it is generally assumed that the molecular orbitals may be expressed as linear combinations of the separate atomic orbitals, the LCAO approximation. The total electronic energy of the molecule may then be minimized with respect to the coefficients in this approximation and the form of the molecular orbitals thus determined. If the molecular orbitals 0, are expressed in terms of the set of atomic orbitals < b. .., N as... 

We assume that the wave functions of a set of d orbitals are each of the general form specified by 9.2-1. We shall further assume that the spin function [jj% is entirely independent of the orbital functions and shall pay no further attention to it for the present. Since the radial function R(r) involves no directional variables, it is invariant to all operations in a point group and need concern us no further. The function 0(0) depends only upon the angle 0. Therefore, if all rotations are carried out about the axis from which 0 is measured (the z axis in Fig. 8.1), (0) will also be invariant. Thus, by always choosing the axes of rotation in this way (or, in other words, always quantizing the orbitals about the axis of rotation), only the function (< ) will be altered by rotations. The explicit form of the 4>(0) function, aside from a normalizing constant, is... 

The general setting of the electronic structure description given above refers to a complete (and thus infinite) basis set of one-electron functions (spin-orbitals) (f>nwave functions, an additional assumption is made, which is that the orbitals entering eq. (1.136) are taken from a finite set of functions somehow related to the molecular problem under consideration. The most widespread approximation of that sort is to use the atomic orbitals (AO).17 This approximation states that with every problem of molecular electronic structure one can naturally relate a set of functions y/((r). // = M > N -atomic orbitals (AOs) centered at the nuclei forming the system. The orthogonality in general does not take place for these functions and the set y/ is characterized... 

The reasons for the loss of degeneracy of sets of atomic orbitals with the same value of n are embedded in the radial distribution functions of the orbitals. The general effect in polyelectronic atoms is for the degeneracy of a set of atomic orbitals with a given n value to break up into subsets such that the s orbital is of lower energy than the p orbitals, the p orbitals are lower than the d orbitals, and the d orbitals are lower than the f orbitals, as shown in Figure 3.1. 

The general setting of the problem of global bifurcations on the disappearance of a saddle-node periodic orbit is as follows. Assume that there exists a saddle-node periodic orbit and that all trajectories which tend to this periodic orbit as i — 00 also tend to it as -f-oo along some center manifold. In other words, assume that the unstable manifold of the saddle-node returns to the saddle-node orbit from the side of the node region. In this case, either ... 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

First, it is possible to simplify the secular equation (2) by means of symmetry. It can be shown by group theory (140) that, in general, the integrals Hi and Si are nonzero only if the orbitals < , and j have the same transformation properties under all the symmetry elements of the molecule. As a simple example, the interaction between an s and a pn orbital which have different properties with respect to the nodal plane of the pn orbital is clearly zero. Interaction above the symmetry plane is cancelled exactly by interaction below the plane (Fig. 13). It is thus possible to split the secular determinant into a set of diagonal blocks with all integrals outside these blocks identically zero. Expansion of the determinant is then simply the product of those lower-order determinants, and so the magnitude of the... 

The total wave function for open-shell systems [8] is, in general, a sum of several antisymmetrized products, each of which contains a closed-shell core and a partially occupied open shell. The combinal set of orbitals is defined by... 

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