Orbital Express

Apply the projection-operator method to obtain the molecular orbital expression shown in Fig. 9 and verify the energies. 

Note that, for a non-interacting system of electrons, the kinetic energy is just the sum of the individual electronic kinetic energies. Within an orbital expression for the density, Eq. (8.14) may then be rewritten as... 

The. totally symmetric operators M(r) are of the same form as the operators fi(r). The c s are atomic core orbitals expressed in the full all-electron basis set, and Fval are normal Fock operators defined in the valence space only. The valence orbitals v = cpXv are expressed in an appropriate valence basis set, determined through some optimization procedure, which is considerably smaller than the original all electron basis set. 

Let us notice that for n2l2 = nih and the same radial orbitals expression (25.6) equals zero. Therefore the radial orbitals, depending on the term quantum numbers, must be employed when calculating /c-transitions using the second form (4.13). 

Let us also mention that using a number of functional relations between the products of 3n/-coefficients and submatrix elements (/ C(k) / ), the spin-angular parts of matrix elements (26.1) and (26.2) are transformed to a form, whose dependence on orbital quantum numbers (as was also in the case of matrix elements of the energy operator, see Chapters 19 and 20) is contained only in the phase multiplier. In some cases this mathematical procedure is rather complicated. Therefore, the use of the relativistic radial orbitals, expressed in terms of the generalized spherical functions (2.18), is much more efficient. In such a representation this final form of submatrix element of relativistic Ek-radiation operators follows straightforwardly [28]. 

The evaluation of the expressions (1-67), (1-87), and (1-94) can be simplified by using the density fitting technique225,227. In this approach the product is expanded in terms of auxiliary atomic orbitals k. In such a case the orbital expressions corresponding to Eqs. (1-67), (1-87), and (1-94) greatly simplify. For the dispersion energy Eq. (1-94) takes the form ... 

For the sp3 hybridization encountered in the amines, the orbital expressions are somewhat more complicated but for symmetrical compounds the coupling can be still expressed in a simple form... 

A molecular orbital expression for the contact contribution to the spin-spin coupling between directly bonded nuclei is given by Eq. (7)... 

Jacquemin, D., Champagne, B., Andre, J.-M. Molecular orbital expressions for approximate uncoupled Hartree-Fock second hyperpolarizabilities. A Pariser-Parr-Pople assessment for model polyacetylene chains. Chem. Phys. 197, 107—127 (1995)... 

As an example let us consider the system of four equivalent univalent atoms at the corners of a square, discussed in the previous section by the valence-bond method. The secular equation for a one-electron wave function (molecular orbital), expressed as a, linear combination of the four atomic orbitals ua, ub, uc, and ud, is... 

Since hybrid functionals, Meta-GGAs, SIC, the Fock term and all other orbital functionals depend on the density only implicitly, via the orbitals i[n, it is not possible to directly calculate the functional derivative vxc = 5Exc/5n. Instead one must use indirect approaches to minimize E[n and obtain vxc. In the case of the kinetic-energy functional Ts[ 0j[rr] ] this indirect approach is simply the Kohn-Sham scheme, described in Sec. 4. In the case of orbital expressions for Exc the corresponding indirect scheme is known as the optimized effective potential (OEP) [120] or, equivalently, the optimized-potential model (OPM) [121]. The minimization of the orbital functional with respect to the density is achieved by repeated application of the chain rule for functional derivatives,... 

Fig 4 Relativistic corrections to electron binding energies and mean expectation value for the outermost occupied orbitals (expressed as the ratio to the non relativistic value). 

Orbital. Each allowed combination of n, I, and mi values specifies one of the atom s orbitals. Thus, the three quantum numbers that describe an orbital express its size (energy), shape, and spatial orientation. You can easily give the quantum numbers of the orbitals in any sublevel if you know the sublevel letter designation and the quantum number hierarchy. For example, the 2s sublevel has only one orbital, and its quantum numbers are n = 2, I = 0, and mi = 0. The 3p sublevel has three orbitals one with n = 3, / = 1, and nii = -1 another with n = 3, I = 1, and / / = 0 and a third with n = 3, I = 1, and m/ = -t-1. 

The MOs are themselves expressed as linear combinations of atomic orbitals (AOs) the latter are usually a basis set of known functions. With a given basis set, the problem of variationally optimizing the energy transforms into that of finding the coefficients of the orbitals. Expressed in matrix form in an AO basis, and in the independent particle approximation of Hartree-Fock theory, this leads to the well-known self-consistent field (SCF) equations. 

The point is, that for any linearly independent set of Xi whatever, the singledeterminant function is identical to an equivalent set of orthonormal orbitals, related to the xt by a linear transformation. Thus, if we assume that at every point in the variational procedure this has been done we may at once avoid searching for turning points for which the determinants are identical and stay within the orthogonal orbital expression for eqn ( 2.1). 

Everything so far in this chapter has assumed a single-determinant wavefunction. It is, however, obvious that the electron density function p, when computed using a basis-function expansion method, must always be capable of being written as a sum of quadratic and bilinear terms in the basis functions. That is, for any wavefunction which is expanded as a linear combination of determinants of orbitals expressed as linear combinations of basis functions <)>k we must have ... 

Note that in the above treatment it was necessary to evaluate only two integrals of the crystal field potential, one over the d orbital [Eq. (7a)] and the second over the orbital [Eq. (7b)]. This derives from the group theory of high symmetry crystal fields, where for an octahedral or tetrahedral complex each of the d orbital expressions is a good wave function for the molecular Hamiltonian and gives an energy appropriate for all members of the 2s and g sets, respectively. For a lower symmetry complex Eq. (4) must be expanded into a perturbation secular determination [Eq. (9)1 with 25 matrix elements, H , involving all pairwise combina-... 

In general, the molecular orbitals expressed by eqn (11.2) have complex amplitudes. Real amplitude molecular orbitals are trivially obtained by taking linear combinations of the degenerate pairs. In particular, we define the real amplitude molecular orbitals as... 

---