SCF equation

To overcome some of the problems inlierent in the UFIF method, it is possible to derive SCF equations based on minimizing the energy of a wavefiinction fomied by spin projecting a single Slater detemiinant starting... 

The olassio papers in whioh the SCF equations for olosed- and open-shell systems are treated are ... 

For sueh a funetion, the CI part of the energy minimization is absent (the elassie papers in whieh the SCF equations for elosed- and open-shell systems are treated are C. C. J. Roothaan, Rev. Mod. Phys. 23, 69 (1951) 32, 179 (I960)) and the density matriees simplify greatly beeause only one spin-orbital oeeupaney is operative. In this ease, the orbital optimization eonditions reduee to ... 

A mathematician would classify the SCF equations as nonlinear equations. The term nonlinear has different meanings in different branches of mathematics. The branch of mathematics called chaos theory is the study of equations and systems of equations of this type. 

We have encountered oscillating and random behavior in the convergence of open-shell transition metal compounds, but have never tried to determine if the random values were bounded. A Lorenz attractor behavior has been observed in a hypervalent system. Which type of nonlinear behavior is observed depends on several factors the SCF equations themselves, the constants in those equations, and the initial guess. 

Changing the constants in the SCF equations can be done by using a dilferent basis set. Since a particular basis set is often chosen for a desired accuracy and speed, this is not generally the most practical solution to a convergence problem. Plots of results vs. constant values are the bifurcation diagrams that are found in many explanations of chaos theory. 

There is more than one solution to the SCF equations for the system, and the calculation procedure converges to a solution which is not the minimum (often a saddle point in wavefunction space). This indicates an RHF-to-RHF or UHF-to-UHF instability, depending on the wavefunction type. 

In the unrestricted treatment, the eigenvalue problem formulated by Pople and Nesbet (25) resembles closely that of closed-shell treatments.-On the other hand, the variation method in restricted open-shell treatments leads to two systems of SCF equations which have to be connected in one eigenvalue problem (26). This task is not a simple one the solution was done in different ways by Longuet-Higgins and Pople (27), Lefebvre (28), Roothaan (29), McWeeny (30), Huzinaga (31,32), Birss and Fraga (33), and Dewar with co-workers (34). 

Famulari, A., Gianinetti, E., Raimondi, M., Sironi, M. and Vandoni, I. (1998) Modification of Guest and Saunders open shell SCF equations to exclude BSSE from molecular interaction calculations, Theor. Chim. Acta, 99,358-365. 

Although continuum solvation models do appear to reproduce the structural and spectroscopic properties of many molecules in solution, parameterization remains an issue in studies involving solvents other than water. In addition, the extension of these approaches to study proteins embedded in anisotropic environments, such as cell membranes, is clearly a difficult undertaking96. As a result, several theoretical studies have been undertaken to develop semi-empirical methods that can calculate the electronic properties of very large systems, such as proteins28,97 98. The principal problem in describing systems comprised of many basis functions is the method for solving the semi-empirical SCF equations ... 

One consequence of this annihilation algorithm is that the number of atoms involved in a specific LMO increases as a result of mixing the original LMOs. The wavefunction describing the new occupied LMOs not only has intensity on atoms originally in the LMO but also on atoms with which the virtual LMO was associated. If no action is taken, then the number of atoms spanned by a given LMO increases until every LMO includes contributions, albeit extremely small ones in most instances, from every atom in the QM system. As a consequence, after each iteration to solve the SCF equations, the contributions to each LMO from individual atoms are examined, so that if those associated with a specific atom, J, are small, then atom J is deleted from the LMO. In practice, the number of atoms that contribute to LMOs appears to reach a limit of 100-130, as the number of atoms in the molecule increases. 

The basis set is the set of madiematical functions from which the wave function is constructed. As detailed in Chapter 4, each MO in HF theory is expressed as a linear combination of basis functions, the coefficients for which are determined from the iterative solution of the HF SCF equations (as flow-charted in Figure 4.3). The full HF wave function is expressed as a Slater determinant formed from the individual occupied MOs. In the abstract, the HF limit is achieved by use of an infinite basis set, which necessarily permits an optimal description of the electron probability density. In practice, however, one cannot make use of an infinite basis set. Thus, much work has gone into identifying mathematical functions that allow wave functions to approach the HF limit arbitrarily closely in as efficient a manner as possible. 

Symmetry is also tremendously useful in several aspects of solving the SCF equations. A key feature is the degree to which it simplifies evaluation of the four-index integrals. 

In certain favorable instances, one can coax the SCF equations to converge to different determinants of the same electronic state symmetry. For instance, phenylnitrenes have two different closed-shell singlet states, as re-illustrated in Figure 14.3 (cf. Section 8.5.3),... 

The standard method for selecting the 4>j is to ask for the <)>i which maximize the importance of one or more terms in the sum. This gives the self-consistent-field (SCF) or multiconfiguration SCF (MC-SCF) equations. If each < >. is expanded as a linear combination of some fixed set of basis functions f - the coefficients can be found by an extension of the Roothaan SCF equations. 

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