Iteration/iterative self-consistent

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. 

Since we do not know the value of C in advance, the optimal C and thus the free energy difference A A can be solved in practice by iterating self-consistently (6.65) and (6.66) or (6.67). A convenient way to do so is to record all the perturbation data during the simulation, then compute C and AA in a postsimulation analysis. This method is also referred to as Bennett s method or the acceptance ratio method. 

Table 4 Iterative self-consistent charge calculation at the MP2 level starting from the Gupta-Gupta [14] charge set with fixed charges on Ba and Y ions.

Initially, the main features of the iterative self-consistent solution of the 2-CSE were the following [48] ... 

In conventional practice, as is well known, in this case we linearize the resulting simultaneous equations, to produce the iterative self-consistent field (SCF) scheme of calculation, according to the eigenvalue equation,... 

Further progress in understanding these MOs requires values for the coefficients Ci — Cg. We used an iterative, self-consistent computer calculation to identify the best values. The resulting MOs are shown in Figure 6.25. 

D QM/MM methods of type D are the most refined and sophisticated, including the iterative, self-consistent mutual polarization of the MM and QM regions. 

The solution of Eq. (1.1.2) is done in an iterative, self-consistent way using the variational principle whenever the true wavefunction is approximated by some incomplete function that depends on a number of parameters, the expectation value of the energy, Eq. (1.1.4), is higher than the expectation value that competes to the exact wavefunction. The procedure requires a lot of mathematical manipulation, but the problem has been solved once and for all and the task is nowadays performed by very efficient black-box computer packages. 

D11.8 In ab initio methods an attempt is made to evaluate all integrals that appear in the secular determinant. Approximations are still employed, but these are mainly associated with the construction of the wavefunctions involved in the integrals. In semi-empirical methods, many of the integrals are expressed in terms of spectroscopic data or physical properties. Semi-empirical methods exist at several levels. At some levels, in order to simplify the calculations, many of the integrals are set equal to zero. Density functional theory (DFT) is considered an ab initio method, but it is different from the Hartree-Fock (HF) or self-consistent field (SCF) approach in that DFT focuses on the electron density while HF/SCF methods focus on the wavefunction. They are both iterative self consistent methods in that the calculations are repeated until the energy and wavefunctions (HF) or energy and electron density (DFT) are unchanged to within some acceptable tolerance. 

The above conditions lead to a set of equations for the group functions, which are coupled by the intergroup Coulomb and exchange potentials. The group functions, obtained by an iterative self-consistent solution of these equations, yield also the stationary value of the total energy of the system. 

Calculated via the equivalent of eq 16 (section V) and expressed correct to two decimal places. All relative ring currents quoted here are based on a simple, noniterative with the exception of those for pyracylene (17) and biphenylene (22) in which certain rings appear to be paramagnetic. Ring-current calculations for these species are, therefore, based on an iteratively self-consistent A... 

The empirical (semiquantitative) methods are based on a one-electron effective Hamiltonian and may be considered as partly intuitive extended Hiickel theory (EHT) for molecules [204] and its counterpart for periodic systems - the tight-binding (TB) approximation. As, in these methods, the effective Hamiltonian is postulated there is no necessity to make iterative (self-consistent) calculations. Some modifications of the EHT method introduce the self-consistent charge-configuration calculations of the effective Hamiltonian and are known as the method of Mulliken-Rudenberg [209]. 

The semiempirical methods are based on the simplification of the HF LCAO Hamiltonian and require the iterative (self-consistent) density matrix calculations complete and intermediate neglect of differential overlap (CNDO and INDO - approximations), neglect of diatomic differential overlap (NDDO) and others, using the neglect of differential overlap (NDO) approximation. 

As usual, the CAS wavefunction incorporates complete Cl over a chosen number of electrons (N) and orbitals (M) that constitute the active space [CAS(N,M)]> with iterative self-consistent optimization of all occupied and partially occupied orbitals. Compared with conventional CAS/MO, the following advantages of CAS/NBO may be noted ... 

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