Iteration-perturbation method

In our simulations of histone modifying enzymes, the computational approaches centered on the pseudobond ab initio quantum mechanical/molecular mechanical (QM/MM) approach. This approach consists of three major components [20,26-29] a pseudobond method for the treatment of the QM/MM boundary across covalent bonds, an efficient iterative optimization procedure which allows for the use of the ab initio QM/MM method to determine the reaction paths with a realistic enzyme environment, and a free energy perturbation method to take account... 

The majority of polarizability calculations use the FFT, perhaps primarily because it is easy to incorporate into standard SCF computer programs in the presence of a perturbation A which is a sum of one-electron operators, the Hartree-Fock SCF hamiltonian hF—h+G(R)z becomes h+A + G R), and the SCF equations are solved by any standard technique. Thus, all that is involved is to add an extra array into the Hartree-Fock hamiltonian matrix hF every iteration. The method can be extended to higher polarizabilities, and a review by Pople et al.73b gives a good introduction to the method, including a discussion of the computational errors likely to be involved. 

MBPT starts with the partition of the Hamiltonian into H = H0 + V. The basic idea is to use the known eigenstates of H0 as the starting point to find the eigenstates of H. The most advanced solutions to this problem, such as the coupled-cluster method, are iterative well-defined classes of contributions are iterated until convergence, meaning that the perturbation is treated to all orders. Iterative MBPT methods have many advantages. First, they are economical and still capable of high accuracy. Only a few selected states are treated and the size of a calculation scales thus modestly with the basis set used to carry out the perturbation expansion. Radial basis sets that are complete in some discretized space can be used [112, 120, 121], and the basis... 

Rates of non-adiabatic intramolecular electron transfer were calculated in Ref. [331] using a self-consistent perturbation method for the calculation of electron-transfer matrix elements based on Lippman-Schwinger equation for the effective scattering matrix. Iteration of this perturbation equation provides the data that show the competition between the through-bond and through-space coupling in bridge structures. 

Various semi-empirical methods have been compared for all properties in an important review by Klopman and O Leary, whilst Adams et < /. have compared the finite field, the variation, and the second-order infinite sum methods for the calculation of a in DNA bases. They find that the variation-perturbation method gives the most reliable results, but as their calculations were at the iterative extended Hiickel level there is no guarantee as to the generality of their conclusions. 

J. Gorecki, W. Byers-Brown, Iterative boundary perturbation method for enclosed onedimensional quantum systems, J. Phys. B At. Mol. Opt. Phys. 20 (22) (1987) 5953-5957. 

MBPT Many-Body Perturbation Theory An iterative perturbation-based method of... 

The form of Eq. (56) is not convenient in most cases to determine C, since the existence of the solution depends critically on the correct normalization of the source term. However, Usachev (76) has given an ingenious method for obtaining by the iterative scheme commonly used to calculate N (and, for that matter, to calculate TV). Since TV is anyway required, we can obtain TV", Ci, and with certainly not more than twice the effort normally expended in a perturbation method in finding N alone. 

Frequency-dependent response functions can only be computed within approximate electronic structure models that allow definition of the time-dependent expectation value. Hence, frequency-dependent response functions are not defined for approximate methods that provide an energy but no wave function. Such methods include MoUer-Plesset (MP) perturbation theory, multiconfigurational second-order perturbation theory (CASPT2), and coupled cluster singles and doubles with non-iterative perturbative triples [CCSD(T)]. As we shall see later, it is possible to derive static response functions for such methods. 

Non-relativistic calculations using the coupled-cluster method CCSD and CCSD(T), CCSD with a non-iterative perturbative triples correction, were carried out for shielding tensors in HaSe and H2Te molecules. The results of DFT calculations with BLYP, B3LYP, PBE and KT2 functionals were compared to these. As had been found in the earlier comprehensive study of Teale et even the best-performing KT2... 

The eigenvectors are normalized such that the first element is identical for all the three computational method. Therefore, the eigenvectors only differ in the second and the third elements. The absolute value of the errors for the first-order perturbation method and the iterative method is given by... 

Structures with Nonviscous Damping, Modeling, and Analysis, Fig. 2 Percentage errors with respect to the exact state-space eigensolutions in a cross-FRF and the driving-point FRF of the system. Results from the first-order perturbation method and the proposed iterative method are shown... 

Clearly the whole approach is strictly analogous to the iterative solution of the Roothaan equations, and the approximation itself will have a similar status. The coefficients E, E ,... provide analytic derivatives of the energy with respect to the applied perturbations and the method of obtaining them is much superior to the use of numerical differentiation based on repeated solution of the Roothaan equations for a series of finite values of the parameters (the so-called finite perturbation method of Pople et al., 1976). It is a straightforward matter to extend the treatment to higher orders though the results (Dodds et al. 1977a) become more cumbersome. 

At the wavefunction level methods based on coupled cluster (CC) theory are among the most reliable ones. For ground-state energetics the CCSD(T) approach is the gold standard of chemistry, whereas for excited states one can use the equation-of-motion (EOM) CC (EOM-CC) method or CC linear response theory (CC-LRT) [4] approaches. Note that the CC-LRT is size-extensive for both energies and properties such as intensities, however for EOM-CC this is true only for energies (unless one uses the closely related similarity transformed (ST)EOM-CC method [18]). As the computational cost of fully iterative (e.g., CCSD, CCSDT, etc.) methods can quickly become prohibitive, perturbative methods [4]... 

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