Self-Consistent Field Convergers

Although all this development had been going on, other research groups had been tackling various technical difficulties. A particularly vexing problem was the occasional failure of the notionally self-consistent equations to actually become self-consistent. 

At the present time, four very different methods are in use to solve this problem. The use of one or more of these convergers will usually yield an SCF. Let us look briefly at these four techniques. 

In the simple mixing scheme the density matrix used in construaing the next Fock matrix is composed of a combination of the two previous density matrices i.e., 

The scalar a is chosen to ensure convergence. Because this converger involves a negligible amount of computation, it is normally the converger of choice. In general, this converger results in a considerable increase in the speed at which an SCF is achieved. Many instances exist, however, when this converger fails to produce an SCF. 

The most common cause of a failure of the SCF equations to converge is autoregenerative charge oscillation. The system is so easily polarizable that any asymmetry in the charge distribution is overcompensated on the next iteration. This can be prevented by decreasing the polarizability of the system. As mentioned above, the polarizability is related to the HOMO-LUMO gap. By increasing this gap the polarizability can be reduced. This is most conveniently achieved by adding to the Fock matrix elements a constant times the square of the unoccupied MOs, e.g., let 

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FIXING SELF-CONSISTENT FIELD CONVERGENCE PROBLEMS... 

Donald B. Boyd (Eli Lilly and Company), David W. Smith, James J. P. Stewart (U.S. Air Force Academy), and Erich Wimmer (Cray Research) Importance of Criteria for Self-Consistent Field Convergence and Geometry Optimization in AMI, MNDO, and MINDO/3 Molecular Orbital Calculations. ... 

Method development (basis sets, ° gradient methods, i methods for calculating correlation energy,self-consistent field convergence,direct methods,semiempirical methods,etc.)... 

Srivastava GP (1984) Broyden s method for self-consistent field convergence acceleration. J Phys A 17 L317-L321... 

Rabuck AD, Scuseria GE (1999) Improving self-consistent field convergence by varying occupation numbers. J Chem Phys 110 695-700... 

The basic self-consistent field (SCF) procedure, i.e., repeated diagonalization of the Fock matrix [26], can be viewed, if sufficiently converged, as local optimization with a fixed, approximate Hessian, i.e., as simple relaxation. To show this, let us consider the closed-shell case and restrict ourselves to real orbitals. The SCF orbital coefficients are not the... 

Fixing Self-Consistent Field A A Convergence Problems... 

Ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Normally, calculations are approached by the Hartree-Fock closed-shell approximation, which treats a single electron at a time interacting with an aggregate of all the other electrons. Self-consistency is achieved by a procedure in which a set of orbitals is assumed, and the electron-electron repulsion is calculated this energy is then used to calculate a new set of orbitals, which in turn are used to calculate a new repulsive energy. The process is continued until convergence occurs and self-consistency is achieved." ... 

The Multi-configuration Self-consistent Field (MCSCF) method can be considered as a Cl where not only the coefficients in front of the determinants are optimized by the variational principle, but also the MOs used for constructing the determinants are made optimum. The MCSCF optimization is iterative just like the SCF procedure (if the multi-configuration is only one, it is simply HF). Since the number of MCSCF iterations required for achieving convergence tends to increase with the number of configurations included, the size of MCSCF wave function that can be treated is somewhat smaller than for Cl methods. 

The physical reasoning for why these densities were frequently employed in the earlier days of density functional theory was that in this way the degeneracy of the partially filled d-orbitals could be retained. A technical reason why these densities still have to be employed in some recent investigations is that calculations with integral orbital occupations simply do not converge in the self consistent field procedure (see, e. g., Blanchet, Duarte, and Salahub, 1997). Such densities correspond to a representation of a particular state 2S+1L with Mg = S and a spherical averaging over ML. 

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... 

The method of many-electron Sturmian basis functions is applied to molecnles. The basis potential is chosen to be the attractive Conlomb potential of the nnclei in the molecnle. When such basis functions are used, the kinetic energy term vanishes from the many-electron secular equation, the matrix representation of the nnclear attraction potential is diagonal, the Slater exponents are automatically optimized, convergence is rapid, and a solution to the many-electron Schrodinger eqeuation, including correlation, is obtained directly, without the use ofthe self-consistent field approximation. 

Yanai, T., Kurashige, Y., Ghosh, D., Chan, G.K-L. Accelerating convergence in iterative solutions of large active-space self-consistent field calculations. Int. J. Quantum Chem., 2009, 109, 2178-90. 

Since the electric field, computed from the filled / is used to construct the Coulomb operator in T, the electric field that is used to constmct T from the converged orbitals is the same as the electric field that is computed from the orbitals that solve Eq. 3 for this Fock operator. Therefore, at convergence, both the orbitals and the electric field computed from them are self-consistent. Consequently, HF theory is also known as self-consistent field (SCF) theory. 

MCSCF calculations in practice require much more technical expertise than do singleconfiguration HF analogs. One particularly difficult problem is that spurious minima in coefficient space can often be found, instead of the variational minimum. Thus, convergence criteria are met for die self-consistent field, but the wave function is not really optimized. It usually requires a careful inspection of the orbital shapes and, where available, some data on relative energetics between related species or along a reaction coordinate to ascertain if this has happened. 

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