Natural orbital function practical

When natural orbitals are determined from a wave function which only includes a limited amount of electron correlation (i.e. not full Cl), the convergence property is not rigorously guaranteed, but since most practical methods recover 80-90% of the total electron correlation, the occupation numbers provide a good guideline for how important a given orbital is. This is the reason why natural orbitals are often used for evaluating which orbitals should be included in an MCSCF wave function (Section 4.6). 

Rather than find the most perfect MOs which satisfy eqn (10-2.4) (or eqns (10-2.5)), it is common practice to replace them by particular mathematical functions of a restricted nature. These functions will generally contain certain parameters which can then be optimized in accordance with eqn (10-2.4). Since these MOs are not completely flexible, we will have introduced a further approximation, the severity of which is determined by the degree of inflexibility in the form of our chosen functions. Typical of this kind of approximation is the one which expresses the space part of the MOs as various linear combinations of atomic orbitals centred on the same or different nuclei in the molecule. We write the space part of each of the approximate MOs as... 

In Section 8.3.2, we consider the use of atomic natural orbitals as basis functions for correlated wave functions. The atomic natural orbitals constitute a conceptually important class of atomic basis functions, useful for systematic investigations of molecular electronic structure at the correlated level. An alternative class of basis functions, perhaps somewhat more useful in practice, is provided by the correlation-consistem basis sets discussed in Section 8.3.3 (for valence correlation) and in Section 8.3.4 (for core and valence correlation). The correlation-consistent basis sets are used extensively in the remainder of this book. 

It is a truism that in the past decade density functional theory has made its way from a peripheral position in quantum chemistry to center stage. Of course the often excellent accuracy of the DFT based methods has provided the primary driving force of this development. When one adds to this the computational economy of the calculations, the choice for DFT appears natural and practical. So DFT has conquered the rational minds of the quantum chemists and computational chemists, but has it also won their hearts To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations. There has been misunderstanding concerning the status of the one-determinantal approach of Kohn and Sham, which superficially appeared to preclude the incorporation of correlation effects. There has been uneasiness about the molecular orbitals of the Kohn-Sham model, which chemists used qualitatively as they always have used orbitals but which in the physics literature were sometimes denoted as mathematical constructs devoid of physical (let alone chemical) meaning. 

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... 

At this stage, when the nature of the basis is known, we return to the question, how the MME needed to evaluate (i > I FPerm. cj) and (i > I Fp0i. cj) is done in practice. We use a method that takes advantage of basic properties of Gaussian functions, which is also very similar to the distributed multipole analysis of Stone [123,124,125], For an arbitrary pair of basis functions we use the orbital expansion... 

The introduction of the concept of one-electron crystal orbitals (CO s) considerably reduces difficulties associated with the many-electron nature of the crystal electronic structure problem. The Hartree-Fock (HF) solution represents the best possible description of a many-electron system with a one-determinantal wavefunction built from symmetry-adapted one-electron CO s (Bloch functions). The HF approach is, of course, only a first approximation to the many-particle problem, but it has many advantages both from practical and theoretical points of view ... 

They are equivalent to the Wannier functions used by Anderson for periodic lattices (see below) [54], Under these conditions, OMO s derived by Eqs. 42 and 43 are not strictly equivalent, simply due to the fact that the one-electron Hamiltonian of A-B is not the sum of the local Hamiltonians for A and B, considered separately. However, both types of OMO s show the same defect of locaUzation. hi addition, from a practical point of view, the OMO approach leads to much simpler calculations, as shown by Anderson [54], whereas the NMO approach is closer to the real mechanism involved in the nature of interaction and will favour the use of more reahstic molecular integrals. From now and for clarity, magnetic orbitals will be written without the prime (0 notation. 

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