SCF theory

Linear Combination of Atomic Orbitals—Molecular Orbitals (LCAO—MO) 

We must represent the molecular orbitals in our calculations in a manner such that they can be incrementally changed in the SCF process. A common approach is to represent a molecular orbital as a linear combination of atomic orbitals (Eq. 14.33). Here, the subscript / is the same as in the HE equation, where i is a, b, c, etc. The subscript k stands for all different atomic orbitals that are included on the atoms in the molecule. Hence, every MO can potentially have a contribution from every AO in the molecule. 

This is called the linear combination of atomic orbitals method for creating molecular orbitals (the LC AO-MO method). As long as each AO is represented by a single mathematical function (corresponding to a minimal basis set see below), the calculations produce the same number of t/s as 0 s included. This corresponds to the statement that one gets as many molecular orbitals for a molecule as there are atomic orbitals on the individual atoms. Note that LCAO-MO is just one of many possible ways to computationally develop MOs. It is computationally expedient, and it is consistent with our notion that molecules are built up from combinations of atomic orbitals, a conceptual advantage over other possible ways of building up MOs. 

Every molecular orbital y/j starts by including every atomic orbital 0 on every atom in the molecule. The SCF iterative process changes the coefficients (c, ) for each y/y creating new y/i s until a minimum energy for the molecule is achieved. 

The atomic orbitals used in the LCAO-MO procedure are represented by what is known as the basis set. Typically, you must distinguish three types of basis sets  

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Lengsfield B H III 1980 General second-order MC-SCF theory a density matrix directed algorithm J. Chem. Phys. 73 382... 

In the Huckel theory of simple hydrocarbons, one assumes that the election density on a carbon atom and the order of bonds connected to it (which is an election density between atoms) are uninfluenced by election densities and bond orders elsewhere in the molecule. In PPP-SCF theory, exchange and electrostatic repulsion among electrons are specifically built into the method by including exchange and electrostatic terms in the elements of the F matrix. A simple example is the 1,3 element of the matrix for the allyl anion, which is zero in the Huckel method but is 1.44 eV due to election repulsion between the 1 and 3 carbon atoms in one implementation of the PPP-SCF method. 

In the RISM-SCF theory, the statistical solvent distribution around the solute is determined by the electronic structure of the solute, whereas the electronic strucmre of the solute is influenced by the surrounding solvent distribution. Therefore, the ab initio MO calculation and the RISM equation must be solved in a self-consistent manner. It is noted that SCF (self-consistent field) applies not only to the electronic structure calculation but to the whole system, e.g., a self-consistent treatment of electronic structure and solvent distribution. The MO part of the method can be readily extended to the more sophisticated levels beyond Hartree-Fock (HF), such as configuration interaction (Cl) and coupled cluster (CC). 

Inherent in the analytical SCF theory are (1) that the free ends of the chains can sample any position within the brush rather than be constrained to reside at the peripheral surface of the layer and, (2) that there is strong stretching, that is... 

As described in Sect. 2.4, SCF calculations are useful in determining local details of density profiles. A more local examination of profiles is indeed necessary to study the question of interpenetration in more detail. The analytical SCF theory [56, 57] shares with the adapted Alexander model embodied in Eq. 35 the characteristic of impenetrability. The full numerical SCF theory is necessary to... 

No informative experimental data have been obtained on the precise shape of segment profiles of tethered chains. The only independent tests have come from computer simulations [26], which agree very well with the predictions of SCF theory. Analytical SCF theory has proven difficult to apply to non-flat geometries [141], and full SCF theory in non-Cartesian geometry has been applied only to relatively short chains [142], so that more detailed profile information on these important, nonplanar situations awaits further developments. 

On the basis of Roothaan SCF theory, the following expressions hold (58, 62) for matrix elements between doublets ... 

The Fourier transformation method enables us to immediately write the momentum space equations as soon as the SCF theory used to describe the system under consideration allows us to build one or several effective Fock Hamiltonians for the orbitals to be determined. This includes a rather large variety of situations ... 

MC-SCF treatments written in terms of coupled Fock equations [44], The simplest examples are the two-configuration SCF theory [45] used in and pi+2 atomic mixing [46], or bonding-antibonding molecular problems [47], and more generally the Clementi-Veillard electron-pair MC-SCF theory [48],... 

Instead of using repeated solution of a suitable eigenvalue equation to optimize the orbitals, as in conventional forms of SCF theory, we have found it more convenient to optimize by a gradient method based on direct evaluation of the ener functional (4), ortho normalization being restored after every parameter variation. Although many iterations are required, the energy evaluation is extremely rapid, the process is very stable, and any constraints on the parameters (e.g. due to spatial symmetry or choice of some type of localization) are very easily imposed. It is also a simple matter to optimize with respect to non-linear parameters such as orbital exponents. 

Roos, B. J. (1992) The multiconfigurational (MC) self-consistent field (SCF) theory,in Roos, B. J.(eds.), Lecture notes in quantum chemistry, Springer-Verlag, Berlin,pp. 179-254. 

As a consequence, the presentation of the results will also differ from that in a MD or MC box, where a full set of molecules can be depicted (as snapshots). In an SCF model, all properties will be presented in, for example, (average) numbers of molecules per unit area of the membrane, or equivalent, i.e. the (average) densities of molecules as a function of the z-coordinate. The box thus consists, if one insists, only of one coordinate. For this reason, we can refer to this method as a one-gradient SCF theory or simply 1D-SCF theory. Extensions towards 2D-SCF are available, where lateral inhomogeneities in the bilayer can also be examined [80], There are even implementations of 3D SCF-like models, but here the interpretation is somewhat more delicate [78],... 

Equilibrium properties are surprisingly accurately predicted by molecular-level SCF calculations. MC simulations help us to understand why the SCF theory works so well for these densely packed layers. In effect, the high density screens the correlations for chain packing and chain conformation effects to such a large extent that the properties of a single chain in an external field are rather accurate. Cooperative fluctuations, such as undulations, are not included in the SCF approach. Also, undulations cannot easily develop in an MD box. To see undulations, one needs to perform molecularly realistic simulations on very large membrane systems, which are extremely expensive in terms of computation time. 

It would be out of place to give a detailed review of the approximate SCF theory, as developed for 7r-electron systems, at this point. It is, however, necessary to explain the basic equations, and convenient to use a form in which only the charges and bond orders appear (McWeeny, 1956, 1964). The total 7r-electron energy E may then be written... 

Theoretical calculations of minimum energy structures and thermodynamic terms using SCF theory with thermodynamic and solvation corrections have been made of the cyclization of l-amino-8-(acetylamino)naphthalene (136) to give 2-methylperimidine... 

While the (one-particle) Brillouin condition BCi has been known for a long time, and has played a central role in Hartree-Fock theory and in MC-SCF theory, the generalizations for higher particle rank were only proposed in 1979 [38], although a time-dependent formulation by Thouless [39] from 1961 can be regarded as a precursor. 

This is nothing but the Brillouin condition of MC-SCF theory. Explicitly, in an... 

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