Self consistent field iteration

Lindh, R., J. W. Krogh, M. Schutz, and K. Hirao. Semidirect parallel self-consistent field The load balancing problem in the input/output intensive self-consistent field iterations. Theor. Chem. Acc. 110 156-164,2003. 

Referred to as the DIIS, this method has been developed by Pulay in the context of ah initio molecular orbital calculations [173, 174, 175], On the minimization, further improvement is possible only after updating the iterative subspace of (rt) by new vectors introducing new dimensions that cannot be reduced to a linear combination of the previous ones. The basis vectors are then being updated by several conventional self-consistent field iterative loops [173] or through the approximate Fock matrix at the minimum point (4.A.29) [174, 175]. 

The method achieves self-consistency within a similar number of self-consistent field iterations as eigensolver-based approaches. However, the replacement of the standard diagonalization at each self-consistent iteration by a polynomial filtering step results in a significant speedup over methods based on standard diagonalization, often by more than an order of magnitude. Algorithmic details of a parallel... 

One feature of relativistic self-consistent field calculations to which attention should be drawn is the fact that the Breit interaction can be easily included in the self-consistent field iterations once the algebraic approximation has been invoked. This should be contrasted with the situation in atomic calculations using numerical methods in which the Breit interaction is treated by first-order perturbation theory. 

Coulomb and exchange operators depend on the occupied (j) orbitals and therefore self-consistent field iterations are performed until the occupied orbitals are eigenfunctions of the Fock operator, F. 

In the late 1920s and early 1930s, a team led by Hartree [9] formulated a self-consistent-field iterative numerical process to treat atoms. In 1930, Fock [10] noted that the Hartree-SCF method needed a correction due to electron exchange and the combined method was known as the Hartree-Fock SCF method. It was not until 1951 that a molecular form of the LCAO-SCF method was derived by Roothaan [11] as given in Appendix B but we can give a brief outline here. The Roothaan method allows the LCAO to be used for more than one atomic center and so the path was open to treat molecules Now, all we have to do is to carry out the integral for the expectation value of the energy as... 

The second step determines the LCAO coefficients by standard methods for matrix diagonalization. In an Extended Hiickel calculation, this results in molecular orbital coefficients and orbital energies. Ab initio and NDO calculations repeat these two steps iteratively because, in addition to the integrals over atomic orbitals, the elements of the energy matrix depend upon the coefficients of the occupied orbitals. HyperChem ends the iterations when the coefficients or the computed energy no longer change the solution is then self-consistent. The method is known as Self-Consistent Field (SCF) calculation. 

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." ... 

Both the Fock matrix—through the density matrix—and the orbitals depend on the molecular orbital expansion coefficients. Thus, Equation 31 is not linear and must be solved iteratively. The procedure which does so is called the Self-Consistent Field... 

The Hartree-Fock equations form a set of pseudo-eigenvalue equations, as the Fock operator depends on all the occupied MOs (via the Coulomb and Exchange operators, eqs. (3.36) and (3.33)). A specific Fock orbital can only be determined if all the other occupied orbitals are known, and iterative methods must therefore be employed for determining the orbitals. A set of functions which is a solution to eq. (3.41) are called Self-Consistent Field (SCF) orbitals. 

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 last term in Eq. 11.47 gives apparently the "average one-electron potential we were asking for in Eq. 11.40. The Hartree-Fock equations (Eq. 11.46) are mathematically complicated nonlinear integro-differential equations which are solved by Hartree s iterative self-consistent field (SCF) procedure. 

McWeeny, R., Proc. Roy. Soc. [London) A235, 496, (i) The density matrix in self-consistent field theory. I. Iterative construction of the density matrix." Beryllium atom is studied. Steepest descent method is described. 

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. 

A more efficient way of solving the DFT equations is via a Newton-Raphson (NR) procedure as outlined here for a fluid between two surfaces. In this case one starts with an initial guess for the density profile. The self-consistent fields are then calculated and the next guess for density profile is obtained through a single-chain simulation. The difference from the Picard iteration method is that an NR procedure is used to estimate the new guess from the density profile from the old one and the one monitored in the single-chain simulation. This requires the computation of a Jacobian matrix in the course of the simulation, as described below. 

The most simple approach is the Hartree-Fock (HF) self-consistent field (SCF) approximation, in which the electronic wave function is expressed as an antisymmetrized product of one-electron functions. In this way, each electron is assumed to move in the average field of all other electrons. The one-electron functions, or spin orbitals, are taken as a product of a spatial function (molecular orbital) and a spin function. Molecular orbitals are constructed as a linear combination of atomic basis functions. The coefficients of this linear combination are obtained by solving iteratively the Roothaan equations. 

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. 

SCF. Self Consistent Field. An iterative procedure whereby a one-electron orbital is determined under the influence of a potential made up of all the other electrons. Iteration continues until self consistency. Hartree-Fock, Density Functional and MP2 Models all employ SCF procedures. 

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