Optimization natural orbital functionals

Presently, the widely used post-Hartree-Fock approaches to the correlation problem in molecular electronic structure calculations are basically of two kinds, namely, those of variational and those of perturbative nature. The former are typified by various configuration interaction (Cl) or shell-model methods, and employ the linear Ansatz for the wave function in the spirit of Ritz variation principle (c/, e.g. Ref. [21]). However, since the dimension of the Cl problem rapidly increases with increasing size of the system and size of the atomic orbital (AO) basis set employed (see, e.g. the so-called Paldus-Weyl dimension formula [22,23]), one has to rely in actual applications on truncated Cl expansions (referred to as a limited Cl), despite the fact that these expansions are slowly convergent, even when based on the optimal natural orbitals (NOs). Unfortunately, such limited Cl expansions (usually truncated at the doubly excited level relative to the IPM reference, resulting in the CISD method) are unable to properly describe the so-called dynamic correlation, which requires that higher than doubly excited configurations be taken into account. Moreover, the energies obtained with the limited Cl method are not size-extensive. 

There are several types of basis functions listed below. Over the past several decades, most basis sets have been optimized to describe individual atoms at the EIF level of theory. These basis sets work very well, although not optimally, for other types of calculations. The atomic natural orbital, ANO, basis sets use primitive exponents from older EIF basis sets with coefficients obtained from the natural orbitals of correlated atom calculations to give a basis that is a bit better for correlated calculations. The correlation-consistent basis sets have been completely optimized for use with correlated calculations. Compared to ANO basis sets, correlation consistent sets give a comparable accuracy with significantly fewer primitives and thus require less CPU time. 

All calculations were carried out with the software MOLCAS-6.0 [16]. Scalar relativistic effects were included using a DKH Hamiltonian [14,15]. Specially designed basis sets of the atomic natural orbital type were used. These basis sets have been optimized with the scalar DKH Hamiltonian. They were generated using the CASSCF/CASPT2 method. The semi-core electrons (ns, np, n — 3,4, 5) were included in the correlation treatment. More details can be found in Refs. [17-19]. The size of the basis sets is presented in Table 1. All atoms have been computed with basis sets including up to g-type function. For the first row TMs we also studied the effect of adding two h-type functions. 

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... 

Well-known procedures for the calculation of electron correlation energy involve using virtual Hartree-Fock orbitals to construct corresponding wavefunctions, since such methods computationally have a good convergence in many-body perturbation theory (MBPT). Although we know the virtual orbitals are not optimized in the SCF procedure. Alternatively, it is possible to transform the virtual orbitals to a number of functions. There are some techniques to do such transformation to natural orbitals, Brueckner orbitals and also the Davidson method. 

Hartree -Fock or Self-Consistent Field (SCF) Method Spin Optimized Self-Consistent Field Method Configuration Interaction Iterative Natural Orbital Method Multi-Configuration SCF Many Body Perturbation Theory Valence-Bond Method Pair-Function or Geminal Method... 

If it is necessary to drastically reduce the number of terms to be handled in a Cl process, it seems to be profitable to optimize the orbitals expansion produce the maximum efficiency as concerns correlation. This optimization is the aim of the multi-configuration (MC)—SCF (in which a limited set of functions < k is constructed and the expansion coefficients of the determinants and those of orbitals are optimized) and of alternative methods based on the use of natural orbitals. 

CASSCF wave function may be constmcted for virtually any type of electronic stmcture, closed or open shell, ground or excited state, neutral or ionic, etc. The only limitation is the size of the active space. The wave function is invariant to orbital transformations within each subspace, which simplifies optimization and makes it possible to construct the same wave function from the natural orbitals, or to use localized orbitals. 

The choice of = g ni,n = nirij implies the Hartree-Fock model, and optimization of the occupation numbers and natural orbitals with this choice indeed returns the HF wave function, i.e. the HF wave function cannot be improved by allowing fractional occupation numbers. The /(n /iy) function is usually set equal to n,ny, since this is just the Coulomb interaction, and the exchange-correlation part is modelled by g(ni,rij)P Modelling the exchange-correlation energy by a g(tii/ij) function is stiU in its infancy. 

Hence, after a decade of false starts, chemists finally learned that the correct basis set should consist of functions that could represent the atomic Hartree-Fock orbitals plus allow for contraction and polarization corrections in the region where they are largest. Similarly it was realized that the Hartree-Fock virtual molecular orbitals were too diffuse for representing the correction to the SCF wavefunction due to electron correlation. Rather, correlation effects are best represented using excitations to nonphysical molecular orbitals that are of the same size as the occupied MOs. Initially this was learned by transforming existing wavefunctions to natural orbital form. Later, MCSCF orbital optimizations were used to obtain these localized correlating orbitals. 

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