Self-consistent field method optimization

Zgid, D., Nooijen, M. The density matrix renormalization group self-consistent field method Orbital optimization with the density matrix renormalization group method in the active space. J. Chem. Phys. 2008, 128, 144116. 

PDDO PRDDO RHF SAMO SCF SOGI STO STO-nG UA UHF VB VIP Projectors of Diatomic Differential Overlap Partial Retention of Diatomic Differential Overlap Restricted Hartree-Fock Simulated ab initio Method Self Consistent Field Spin Optimized GVB method Slater Type Orbital Slater Type Orbital expanded in terms of nGTO United Atom Unrestricted Hartree-Fock Valence Bond Vertical Ionization Potential... 

The MC SCF method usually takes into account a minimum number of configurations capable of assuring some fundamental requirements, this step being followed by the optimization of the basis functions using the self-consistent-field method. For example, in order to describe accurately the dissociation of ground-state H2 it is only necessary to consider the two-configuration wave function... 

Hartree -Fock or Self-Consistent Field (SCF) Method Spin Optimized Self-Consistent Field Method Configuration Interaction Iterative Natural Orbital Method Multi-Configuration SCF Many Body Perturbation Theory Valence-Bond Method Pair-Function or Geminal Method... 

C. Orbital Approximations other than Hartree-Fock The Spin Optimized Self-Consistent Field Method... 

The MC SCF (Multiconfiguration Self Consistent Field) method is similar to the Cl scheme, but we vary not only the coefficients in front of the Slater determinants, but also the Slater determinants themselves (changing the analytical form of the orbitals in them). We have learnt about two versions the classic one (we optimize alternatively coefficients of Slater determinants and the orbitals) and a unitary one (we optimize simultaneously the determinantal coefficients and orbitals). 

The Multi-Configuration Self-Consistent Field method combines the ideas of orbital optimization through a SCF technique as in the Hartree-Fock method, and a multiconfiguration expansion of the electronic wavefunction as in the configuration interaction method. In other words, the electronic wavefunction is still expressed as a linear combination of Slater determinants but now both the coeffi-... 

In the multiconfigurational self-consistent field method both the configuration coefficients C o as well as the molecular orbital coefEcients Cfip are varied until the energy becomes minimal. If one keeps the latter fixed and optimizes the energy only with respect to the configuration coefEcients C o, i -... 

Instead of optimization of all the orbitals, only the active orbitals will be varied within the Complete Active Space Self-Consistent Field approximation (CASSCF). In the acronym of this method, the number of active orbitals and active electrons are also provided for the given system. For instance, CASSCF(6,4) denotes the calculations with the expansion including all the possible exchanges of the four electrons within the six active orbitals. The CASSCF approach leads to all possible exchanges in the given active space, and for a moderately sized system, the size of the active spaces can quickly exceed the computational resources. In such a case, the solution can be the Restricted Active Space Self-Consistent Field method (RASSCF), which supplies a way of limiting the size of the active space. 

Coupled-cluster Theory Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field Geometry Optimization 1 Geometry Optimization 2 Mixed Quantum-Classical Methods Mpller-Plesset Perturbation Theory Rates of Chemical Reactions Reaction Path Following Transition State Theory Unimolecular Reaction Dynamics. 

The next five chapters are each devoted to the study of one particular computational model of ab initio electronic-structure theory Chapter 10 is devoted to the Hartree-Fock model. Important topics discussed are the parametrization of the wave function, stationary conditions, the calculation of the electronic gradient, first- and second-order methods of optimization, the self-consistent field method, direct (integral-driven) techniques, canonical orbitals, Koopmans theorem, and size-extensivity. Also discussed is the direct optimization of the one-electron density, in which the construction of molecular orbitals is avoided, as required for calculations whose cost scales linearly with the size of the system. 

To overcome the limitations of the database search methods, conformational search methods were developed [95,96,109]. There are many such methods, exploiting different protein representations, objective function tenns, and optimization or enumeration algorithms. The search algorithms include the minimum perturbation method [97], molecular dynamics simulations [92,110,111], genetic algorithms [112], Monte Carlo and simulated annealing [113,114], multiple copy simultaneous search [115-117], self-consistent field optimization [118], and an enumeration based on the graph theory [119]. 

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. 

To circumvent problems associated with the link atoms different approaches have been developed in which localized orbitals are added to model the bond between the QM and MM regions. Warshel and Levitt [17] were the first to suggest the use of localized orbitals in QM/MM studies. In the local self-consistent field (LSCF) method the QM/MM frontier bond is described with a strictly localized orbital, also called a frozen orbital [43]. These frozen orbitals are parameterized by use of small model molecules and are kept constant in the SCF calculation. The frozen orbitals, and the localized orbital methods in general, must be parameterized for each quantum mechanical model (i.e. energy-calculation method and basis set) to achieve reliable treatment of the boundary [34]. This restriction is partly circumvented in the generalized hybrid orbital (GHO) method [44], In this method, which is an extension of the LSCF method, the boundary MM atom is described by four hybrid orbitals. The three hybrid orbitals that would be attached to other MM atoms are fixed. The remaining hybrid orbital, which represents the bond to a QM atom, participates in the SCF calculation of the QM part. In contrast with LSCF approach the added flexibility of the optimized hybrid orbital means that no specific parameterization of this orbital is needed for each new system. 

Fu et al. [16] analyzed a set of 57 compounds previously used by Lombardo and other workers also. Their molecular geometries were optimized using the semiempirical self-consistent field molecular orbital calculation AMI method. Polar molecular surface areas and molecular volumes were calculated by the Monte Carlo method. The stepwise multiple regression analysis was used to obtain the correlation equations between the log BB values of the training set compounds and their structural parameters. The following model was generated after removing one outlier (Eq. 50) ... 

In the MQC mean-field trajectory scheme introduced above, all nuclear DoF are treated classically while a quantum mechanical description is retained only for the electronic DoF. This separation is used in most implementations of the mean-field trajectory method for electronically nonadiabatic dynamics. Another possibility to separate classical and quantum DoF is to include (in addition to the electronic DoF) some of the nuclear degrees of freedom (e.g., high frequency modes) into the quantum part of the calculation. This way, typically, an improved approximation of the overall dynamics can be obtained—albeit at a higher numerical cost. This idea is the basis of the recently proposed self-consistent hybrid method [201, 202], where the separation between classical and quantum DoF is systematically varied to improve the result for the overall quantum dynamics. For systems in the condensed phase with many nuclear DoF and a relatively smooth distribution of the electronic-vibrational coupling strength (e.g.. Model V), the separation between classical and quanmm can, in fact, be optimized to obtain numerically converged results for the overall quantum dynamics [202, 203]. 

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. 

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