Localized orbital transformation

This problem too has an orbital and a correlation part. The orbital part was solved by Lennard-Jones, > who showed that in an H.F. SCF MO determinant, , electrons are already localized with respect to one another (e.g. in CH4, HgO, Ne,.. . ) as mentioned in Section VII. This relative distribution of electrons with respect to one another is better described by a unitary "equivalent or "localized orbital transformation t which leaves (f>o unchanged ... 

The natural localized orbital transformations provide useful tools for analyzing various molecular properties or energy components in localized bonding terms. We briefly summarize the underlying principles of several such analyses that are implemented in the general NBO program, restricting attention primarily to HF-level treatment. 

There is always a transformation between symmetry-adapted and localized orbitals that can be quite complex. A simple example would be for the bonding orbitals of the water molecule. As shown in Figure 14.1, localized orbitals can... 

In standard quantum-mechanical molecular structure calculations, we normally work with a set of nuclear-centred atomic orbitals Xi< Xi CTOs are a good choice for the if only because of the ease of integral evaluation. Procedures such as HF-LCAO then express the molecular electronic wavefunction in terms of these basis functions and at first sight the resulting HF-LCAO orbitals are delocalized over regions of molecules. It is often thought desirable to have a simple ab initio method that can correlate with chemical concepts such as bonds, lone pairs and inner shells. A theorem due to Fock (1930) enables one to transform the HF-LCAOs into localized orbitals that often have the desired spatial properties. 

The MO and VB methods provide altema ive but equivalent descriptions of the bonding in a molecule. A set of molecular orbitals can always be transformed into a corresponding set of more localized orbitals, and vice versa. For example, according to the MO de-... 

This transformation leaves invariant all observable molecular properties of ground-state norbornadiene that can be derived from our SCF model. Note that the two localized orbitals describing a double bond are two banana LMOs Xb,Up and Xb.down, as shown on the left of Figure 17, Their normalized, out-of-phase linear combination... 

The natural localized orbitals introduced in Section 1.5 provide a useful alternative to the canonical delocalized MOs (CMOs) that are usually employed to analyze chemical bonding. The NAO and NBO basis sets may be regarded as intermediates in a succession of basis transformations that lead from starting AOs /, to the final canonical MOs , ... 

In order to construct localized orbitals for molecules, it is necessary to define a measure for the degree of localization of an arbitrary set of molecular orbitals. The localized orbitals are then defined as that set of orthogonal molecular orbitals obtained by a transformation of the type given in Eq. (5), for which the measure of localization has the maximum value. It is clear that the resulting localized orbitals will depend, at least to some degree, upon the choice of the localization measure. In the present work the localized molecular orbitals are defined as those self-consistent-field orbitals which maximize the localization sum 14)... 

The Eqs. (33) and (34) could be used for a practical determination of the localized orbitals. So far, however, a different procedure has been used which is based on the premise that the canonical orbitals are determined first. From these, the localized orbitals are then obtained by a sequence of 2 X 2 orthogonal transformations which iteratively increase the localization sum until it reaches the maximum. 17>... 

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. 

Let us consider now the application of local-scaling transformations to sets of single-particle functions or orbitals. As it was shown in Sect. 2.1, a set of plane waves gives rise to the transformed orbitals described by Eq. (2). In particular, the application of this transformation to one-dimensional plane-waves leads to Harriman s equidensity orbitals [27], which are given by ... 

We illustrate here a specific example of the application of local-scaling transformations to atomic orbitals [111]. Consider the i is(r) and / 2s( ) orbitals of the Raffenetti type for the beryllium atom [71] ... 

The density associated with the Hartree-Fock-Raffenetti wavefunction is denoted by puVif)- We take this to be the initial density in our local-scaling transformation, i.e., pi r) = puVir)- We take as the final density, that associated with the 650-term Cl wavefunction of Esquivel and Bunge [73], which we call P2ir) = pair). These two densities are practically about the same, as can be seen clearly in Fig. 4, where we have also plotted their difference. The transformed radial orbitals are given by ... 

Table I. Selected values of the Raffenetti-Hartree-Fock orbitals Isg and 2shf for Be, of their locally-scaled transformed functions IsJ, and 2sgr and of their differenees di, = Ish, - Isgj,) and = (2siif — 2sgp). [Reproduced with permission from Table I Ludeiia et al. [Ill]]...

Local-Scaling Transformation Version of Density Functional Theory Talbe 3. Orbital parameters for function... 

Thus, for example, from wavefunction we can generate an orbit which contains - among the infinite number of wavefunctions obtained through the application of local-scaling transformation, the particular wavefunctions and The important aspect of orbits is that the uniqueness of the... 

It follows from the above considerations that local-scaling transformations can be advanced in momentum space on an equal footing with those in position space. In particular, wavefunctions in momentum space can be transformed so as to generate new wavefunctions that have the property of belonging to an orbit . 

It is instructive, however, in order to establish the connection between the usual methods in quantum chemistry - based on molecular orbitals - and the local-scaling transformation version of density functional theory, to discuss Cioslowski s work in some detail. 

It is clear, therefore, that Cioslowski s approach based on density-driven orbitals [74, 75, 77], corresponds to a finite orbital representation of the local-scaling transformation version of density functional theory [38]. 

Thus, for any p(r) e there exists a unique wavefunction generated by means of local-scaling transformation from the arbitrary generating wavefunction The set of all the wavefunctions thus generated, yielding densities p(f) in J g, is called an orbit and is denoted by... 

The uniqueness of the local-scaling transformation guarantees that within an orbit there exists a one-to-one correspondence between one particle... 

Hohenberg-Kohn orbit. Clearly, within the application of local-scaling transformations to any initial wavefunction leads to the exact ground-state wavefunction as well as to the exact ground-state density. 

Let us finish this Section by discussing the relationship between the Kohn-Sham-like equations advanced above and the actual Kohn-Sham equations. From the perspective of local-scaling transformations, we can analyze this relationship as follows. First of all, we assume that, for an interacting system, we are able to select an orbit-generating wavefunction belonging to the... 

Let us note, however, that in spite of the fact that the orbit jumping optimization is carried out at fixed density, the resulting wavefunction, epi[p%) = is not necessarily associated with the fixed density p ]],(r). For this reason, it is then possible to apply a local-scaling transformation to it and produce an optimized wavefunction which, at the same time, is associated with the fixed density. We denote this wavefunction by Moreover, we can... 

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. 

Calculation of Kohn-Sham Orbitals and Potentials by Local-Scaling Transformations... 

We have reviewed here the implementation of the inverse method for going from densities to potentials, based on local-scaling transformations. For completeness, let us mention, however, that several other methods have also been advanced to deal with this inverse problem [101-111]. Consider the decomposition of into orbits Such orbits are characterized by the fact that... 

The way in which local-scaling transformations have been used for the minimization of the kinetic energy functional is as follows [108-111], An arbitrary Slater determinant is selected to be the orbit-generating... 

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