Orbital transformations

To illustrate sueh symmetry adaptation, eonsider symmetry adapting the 2s orbital of N and the three Is orbitals of H. We begin by determining how these orbitals transform under the symmetry operations of the C3V point group. The aet of eaeh of the six symmetry operations on the four atomie orbitals ean be denoted as follows ... 

The first line indicates that the syrrumetry could not be determined for this state (the symmetry itself is given as Sym). We will need to determine it ourselves. Molecular symmetry in excited states is related to how the orbitals transform with respect to the ground state. From group theory, we know that the overall symmetry is a function of symmetry products for the orbitals, and that only singly-occupied orbitals are... 

If we want to determine the specific type of orbital transformation for this transition, we will need to examine the molecular orbitals for the largest components of the transition, indicated by the largest wavefunction coefficients. In this case, this is the relevant entry ... 

C2 and change signs under C2 E, axz, or ayz operations). Although it may not be readily apparent, the py orbital transforms as B2. Using the four symmetry operations for the C2 point group, the valence shell orbitals of oxygen behave as follows ... 

If we now consider a planar molecule like BF3 (D3f, symmetry), the z-axis is defined as the C3 axis. One of the B-F bonds lies along the x-axis as shown in Figure 5.9. The symmetry elements present for this molecule include the C3 axis, three C2 axes (coincident with the B-F bonds and perpendicular to the C3 axis), three mirror planes each containing a C2 axis and the C3 axis, and the identity. Thus, there are 12 symmetry operations that can be performed with this molecule. It can be shown that the px and py orbitals both transform as E and the pz orbital transforms as A, ". The s orbital is A/ (the prime indicating symmetry with respect to ah). Similarly, we could find that the fluorine pz orbitals are Av Ev and E1. The qualitative molecular orbital diagram can then be constructed as shown in Figure 5.10. 

Table 5.5 Central Atom s and p Orbital Transformations under Different Symmetries. ...

Having seen the development of the molecular orbital diagram for AB2 and AB3 molecules, we will now consider tetrahedral molecules such as CH4, SiH4, or SiF4. In this symmetry, the valence shell s orbital on the central atom transforms as A, whereas the px, py, and pz orbitals transform as T2 (see Table 5.5). For methane, the combination of hydrogen orbitals that transforms as A1 is... 

It is thus found that the one-electron energy H, the Coulomb energy C, and the Exchange energy X are separately invariant under the orbital transformation given by Eqs. (5) and (6). 

Thus, the orbitals uk and vk satisfy Hartree-Fock equations which are identical in form and differ only in the numerical values of the constants X/Jt and Ajk respectively. But since the latter are unknowns in the equation, and since 7(p) is itself invariant as shown in Eq. (21), we can say that the Hartree-Fock self-consistent-field equations are invariant under the orbital transformation given by Eqs. (5) and (6). This means in effect, that the energy integral ( H "X11/0 is minimized by the vk s as well as by the uk s — a circumstance which is in agreement with the invariance of and ( 1 under the transformation (5). 

If there is a molecular symmetry group whose elements leave the hamiltonian 36 invariant, then the closed-shell wavefunction belongs to the totally symmetric representation of both the spin and symmetry groups.8 It is further true that under these symmetry operations the molecular orbitals transform among each other by means of an orthogonal transformation, such as mentioned in Eq. (5) 9) and, therefore, span a representation of the molecular symmetry group. In general, this representation is reducible. 

Orbital Transformations (generation of monomer-like orbitals)... 

Many other methods, of course, take advantage of invariances with respect to orbital transformations to obtain alternative representations of wavefunctions, such as for example the commonly employed localization procedures for the doubly-occupied MOs from Hartree-Fock calculations [11-15]. We give here a brief account of procedures that particularly seek a valence bond representation of MO wavefunctions. 

Both in the work of Macke [53] and of March and Young [55], the orbital transformation is restricted to plane waves. A more general formulation, how-... 

The general Jacobian problem associated with the transformation of a density Pi(r) into a density p2(r) (where these densities differ from that of the free-electron gas) was discussed by Moser in 1965 [58]. This work was not performed in the framework of orbital transformations - which might have interested chemists, nor was it done in the context of density functional theory - which might have attracted the attention of physicists. It was a paper written for mathematicians and, as such, it remained unknown to the quantum chemistry community. In the discussion that follows, we use the more accessible reformulation of Bokanowski and Grebert (1995) [65] which relies heavily on the work of Zumbach and Maschke (1983) [61]. Let us define as ifjy = the space of... 

We describe in this Subsection the application of local-scaling transformations to the calculation of the energy for the lithium and beryllium atoms at the Hartree-Fock level [113]. (For other reformulations of the Hartree-Fock problem see [114] and referenres therein.) The procedure described here involves three parts. The first part is orbital transformation already discussed in Sect. 2.5. The second is intra-orbit optimization described in Sect. 4.3 and the third is inter-orbit optimization discussed in Sect. 4.6. 

In the point-group Td, p-orbitals transform in the same way as dxy, dyt, and dtx, owing to the absence of a center of symmetry. This has the important consequence that all six orbitals are mixed together and the electronic transitions have some d-p character they are therefore not forbidden and have higher intensities than the nearly pure d-d transitions of octahedral complexes. 

The electrons which are important for the bonding in the water molecule are those in the valence shell of the oxygen atom 2s22p4. It is essential to explore the character of the 2s and 2p orbitals, and this is done by deciding how each orbital transforms with respect to the operations associated with each of the symmetry elements possessed by the water molecule. 

In the example under discussion the 2s and 2p, orbitals transform as a, the 2px orbital transforms as b, and the 2p, orbital transforms as b2. In this context transform refers to the behaviour of the orbitals with respect to the symmetry operations associated with the symmetry elements of the particular group. 

By referring to the diagrams in Figure 3.1 it may be seen that the orbital <)), transforms as Gg+, and that the orbital < )2 transforms as ou+. That two molecular orbitals are produced from the two atomic orbitals is an important part of molecular orbital theory a law of conservation of orbital numbers- The two molecular orbitals differ in energy, both from each other and from the energy of the atomic level. To understand how this arises it is essential to consider the normalization of the orbitals. 

The classification of the orbitals of the oxygen atom is a matter of looking them up in the C2v character table, a full version of which is included in Appendix 1. The 2s(0) orbital transforms as an a, representation, the 2px(0) orbital transforms as a b, representation, the 2p (0) orbital transforms as a b2 representation and the 2p.(0) orbital transforms as another aj representation. 

The classification of the 2s and 2p atomic orbitals of the central oxygen atom. The 2s(0) orbital transforms as Gg, the 2px and 2p orbitals transform as the doubly degenerate nu representation, and the 2pz orbital transforms as om+. In some texts the + and superscripts are omitted, but throughout this one there is strict adherence to the use of the full symbols for all orbital symmetry representations. Unlike the 90° case, the 2s and 2p7 orbitals have different symmetry properties so there is no question of their mixing. 

Figure 5.21 The 2 -68, and 5ag+-7a1 orbital transformations which occur when N02+ bends...

The MO theory is straightforward. The hydrogen Is orbital transforms as ag+ within the point group. The 2p orbitals of the fluorine atoms (since the ion is set-up with the molecular axis coincident with the Cartesian z axis) may be arranged in the linear combinations ... 

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