Self-consistent field method limitation

The Hartree-Fock orbitals are expanded in an infinite series of known basis functions. For instance, in diatomic molecules, certain two-center functions of elliptic coordinates are employed. In practice, a limited number of appropriate atomic orbitals (AO) is adopted as the basis. Such an approach has been developed by Roothaan 10>. In this case the Hartree-Fock differential equations are replaced by a set of nonlinear simultaneous equations in which the limited number of AO coefficients in the linear combinations are unknown variables. The orbital energies and the AO coefficients are obtained by solving the Fock-Roothaan secular equations by an iterative method. This is the procedure of the Roothaan LCAO (linear-combination-of-atomic-orbitals) SCF (self-consistent-field) method. 

The Hiickel molecular orbital (HMO) model of pi electrons goes back to the early days of quantum mechanics [7], and is a standard tool of the organic chemist for predicting orbital symmetries and degeneracies, chemical reactivity, and rough energetics. It represents the ultimate uncorrelated picture of electrons in that electron-electron repulsion is not explicitly included at all, not even in an average way as in the Hartree Fock self consistent field method. As a result, each electron moves independently in a fully delocalized molecular orbital, subject only to the Pauli Exclusion Principle limitation to one electron of each spin in each molecular orbital. 

There is, in principle, nothing which limits the self-consistent field method to any particular form of the exchange-correlation potential, and the procedure outlined above has been used in connection with several approximations for exchange and correlation. Most notable in this respect is SLATER S Xa method [1.4] which has been applied to all atoms in the periodic table, to some molecules, and in the majority of the existing electronic-structure calculations for crystalline solids. 

However, although, starting from this point, many sophisticated methods for wave function expansion, for example, the coupled cluster approach, multi-configuration self-consistent-field method or multi-reference Cl methods, have been developed, the correlation problem faced many computational limitation, some of them almost insurmountable, due to the immense number of integrals to be evaluated. 

In Chapter IX we discussed one particular limit of the self-consistent field method, where the ground state was dominant. We then introduced a ground state wavefunction r) with the following properties ... 

Abstract Hybrid methods, combining the accuracy of Quantum Mechanics and the potency of Molecular Mechanics, the so-called QM/MM methods, arise from the desire of theoretician chemists to study electronic phenomena in large molecular systems. In this contribution, a focus, on the Physics and Chemistry on which theses methods are based on, is given. The advantages, flaws, and limitations of each type of methods are exposed. A special emphasis is put on the Local Self-Consistent Field method, developed in our group. The latest developments are detailed and illustrated by chosen examples. 

Instead of optimization of all the orbitals, only the active orbitals will be varied within the Complete Active Space Self-Consistent Field approximation (CASSCF). In the acronym of this method, the number of active orbitals and active electrons are also provided for the given system. For instance, CASSCF(6,4) denotes the calculations with the expansion including all the possible exchanges of the four electrons within the six active orbitals. The CASSCF approach leads to all possible exchanges in the given active space, and for a moderately sized system, the size of the active spaces can quickly exceed the computational resources. In such a case, the solution can be the Restricted Active Space Self-Consistent Field method (RASSCF), which supplies a way of limiting the size of the active space. 

For the case of intramolecular energy transfer from excited vibrational states, a mixed quantum-classical treatment was given by Gerber et al. already in 1982 [101]. These authors used a time-dependent self-consistent field (TDSCF) approximation. In the classical limit of TDSCF averages over wave functions are replaced by averages over bundles of trajectories, each obtained by SCF methods. 

To overcome the limitations of the database search methods, conformational search methods were developed [95,96,109]. There are many such methods, exploiting different protein representations, objective function tenns, and optimization or enumeration algorithms. The search algorithms include the minimum perturbation method [97], molecular dynamics simulations [92,110,111], genetic algorithms [112], Monte Carlo and simulated annealing [113,114], multiple copy simultaneous search [115-117], self-consistent field optimization [118], and an enumeration based on the graph theory [119]. 

The various response tensors are identified as terms in these series and are calculated using numerical derivatives of the energy. This method is easily implemented at any level of theory. Analytic derivative methods have been implemented using self-consistent-field (SCF) methods for a, ft and y, using multiconfiguration SCF (MCSCF) methods for ft and using second-order perturbation theory (MP2) for y". The response properties can also be determined in terms of sum-over-states formulation, which is derived from a perturbation theory treatment of the field operator — [iE, which in the static limit is equivalent to the results obtained by SCF finite field or analytic derivative methods. 

Two main approaches for osmotic pressure of polymeric solutions theoretical description can be distinguished. First is Flory-Huggins method [1, 2], which afterwards has been determined as method of self-consistent field. In the initial variant the main attention has been paid into pair-wise interaction in the system gaped monomeric links - molecules of solvent . Flory-Huggins parameter % was a measure of above-said pair-wise interaction and this limited application of presented method by field of concentrated solutions. In subsequent variants such method was extended on individual macromolecules into diluted solutions with taken into account the tie-up of chain links by Gaussian statistics [1]. 

Semiempirical methods are widely used, based on zero differential overlap (ZDO) approximations which assume that the products of two different basis functions for the same electron, related to different atoms, are equal to zero [21]. The use of semiempirical methods, like MNDO, ZINDO, etc., reduces the calculations to about integrals. This approach, however, causes certain errors that should be compensated by assigning empirical parameters to the integrals. The limited sets of parameters available, in particular for transition metals, make the semiempirical methods of limited use. Moreover, for TM systems the self-consistent field (SCF) procedures are hardly convergent because atoms with partly filled d shells have many... 

Of the more exact methods, the limited configuration interaction (Cl MO LCAO) method and the self-consistent field (SCF MO LCAO) method will be mentioned. In contrast to the HMO method, both of these explicitly take electron repulsion into account. The Cl method is particularly valuable for the calculation of various physical properties, especially electronic spectra. A more detailed description is beyond the scope of the present review the reader is referred to original papers [Cl,17-20 SCF,21-23 and VESCF24 (variable electronegativity)] and to various reviews and monographs.5 25,26... 

Nondynamical electron correlation effects are generally important for reaction path calculations, when chemical bonds are broken and new bonds are formed. The multiconfiguration self-consistent field (MCSCF) method provides the appropriate description of these effects [25], In the last decade, the complete active space self-consistent field (CASSCF) method [26] has become the most widely employed MCSCF method. In the CASSCF method, a full configuration interaction (Cl) calculation is performed within a limited orbital space, the so-called active space. Thus all near degeneracy (nondynamical electron correlation) effects and orbital relaxation effects within the active space are treated at the variational level. A full-valence active space CASSCF calculation is expected to yield a qualitatively reliable description of excited-state PE surfaces. For larger systems, however, a full-valence active space CASSCF calculation quickly becomes intractable. 

In the first place the Hiickel approximation for the tt electrons has been replaced by a self-consistent field (SCF) procedure, generally in a semiempirical approximation of the Pariser-Parr-Pople type completed with some limited configuration interaction (PPP-CI method).59 00 Second, the a skeleton of the molecules has been treated by the Del Re procedure61 for saturated systems (which is the counterpart for the a electrons of the Hiickel method for 77 electrons) as refined for the a skeletons of conjugated heterocycles by... 

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