Chemical Hamiltonian

There have been attempts to develop methods where the BSSE is excluded explicitly in the computational expressions, an example of this is the Chemical Hamiltonian Approach (CHA)," but such methods are not yet commonly used. 

Prior to choosing the wave-function approximation it is, however, necessary to set up the electronic Hamiltonian H that describes all interactions of elementary particles. Therefore, we start with the derivation of the full semi-classical many-electron Hamiltonian describing all interactions relevant for chemical problems and subsequently discuss approximations to this full-fledged quantum chemical Hamiltonian. 

Starting Hamiltonian is written in the general form of nonrelativistic - chemical Hamiltonian of interacting set of atoms, ... 

There are alternatives to the most commonly used Boys-Bemardi counterpoise scheme. One approach that shows promise, for example, is a chemical Hamiltonian approach (CHA), pioneered by Mayer " ", which attempts to isolate the superposition error directly in the Hamiltonian operator. The Schrodinger equation that is solved is hence a modified one, which yields a wave function that is hopefully free of superposition error. In the case of (HF)2, it was found that this approach mimics rather closely the results of the standard counterpoise scheme for a series of small to moderate sized basis sets ". Later calculations " extended these tests to other small H-bonded systems as well, again limiting their testing to basis sets no larger than 6-31G. A recent test " has extended the method s use-... 

A general efficient implementation of the BSSE-free SCF and MP2 methods based on the Chemical Hamiltonian Approach ... 

Another method for BSSE elimination is the Chemical Hamiltonian Approach (CHA) formulated in 1983 by Mayer, whose basis idea is the a priori exclusion of BSSE (Ref 153 and therein cited) and in which every... 

Mayer, I., Towards a chemical Hamiltonian, Intern. J. Quantum Chem., 2i, 341-363 (1983). Halasz, G.J., Vibok, A., Suhai, S., and Mayer, I., Toward a BSSE-free description of strongly interacting systems, Intern. J. Quantum Chem., 89, 190-197 (2002). 

AMI Austin Model 1. A semi-empirical quantum chemical Hamiltonian originating from the M.J.S. Dewar group. The quality of the AMI results in many cases is beyond the simpler ab initio results and is superior to the MNDO... 

There have been a number of means proposed for circumventing superposition error. Mayer et al. advocated what they term a chemical Hamiltonian approach, which separates the physical part of this operator from that responsible for BSSE using a nonorthogonal second quantization formalism. However, the physical Hamiltonian is no longer variational and the wavefunction is constructed from orthonormalized molecular spin orbitals. Surjan et al. " further developed this approach and performed pilot applications on small complexes. 

The constraint matrices A,- in Eq. (19) contain the mappings among D, Q, and G, the contractions to D and g, and the fixed-trace condition. Semidefinite programs for quantum chemical Hamiltonians have been solved with a variety of algorithms [27, 33, 37, 54, 60]. In this work, we utilize the boundary-point method [26, 37], a type of quadratic regularization method. The floating-point... 

Mayer I (1983) Towards a chemical hamiltonian. Int J Quantum Chem 23 341-363. 

Orlando R, Dovesi R, Ugliengo P, (1999) A quantum mechanical periodic ab initio approach to materials science the CRYSTAL program. Int J Inorg Mater 1 147-155 Oum KW, Lakin MJ, DeHaan DO, Brauers T, Finlayson-Pitts BJ (1998) Formation of molecular chlorine from the photolysis of ozone and aqueous sea-salt particles. Science 279 74-77 Paizs B, Suhd S (1997) Extension of SCF and DFT versiorrs of chemical Hamiltonian approach to N interacting subsystems and an algorithm for their eflficierrt implementatioa J Comput Chem 18 694-701... 

Another, different way of utilizing the atomic hypothesis was realized in the Chemical Hamiltonian Approach [46] which exploits the LCAO model of quantum chemistry. It is important to stress that the LCAO (linear combination of atomic orbitals) expansion of the molecular wave functions is itself, in a certain sense, based on the concept of building blocks . In effect, the idea behind the LCAO method is the chemical experience that the basic building blocks of the molecules are the atoms. Thus the atomic orbitals obtained from atomic wave functions could be suitable for the expansion of molecular orbitals. 

The Chemical Hamiltonian Approach uses the atomic orbital basis set for the definition of an atomic partition of the Hamiltonian operator. An atomic subsystem of a molecule consists of a nucleus and the set of basis functions centered on it. Accordingly, the partition of the finite basis set corresponds to the physical partition of a molecule into atomic subsystems. This raises the problem of non-orthogonality of the basis functions belonging to the different subsystems (atoms, in the present case) and also the problem of basis set superposition error (BSSE), which is a consequence of the finiteness of the basis set [46]. 

Several procedures have been proposed to avoid, or at least moderate, this effect (Boys Bernardi 1970, Daudey et al. 1974b, Mayer 1983). The problem stimulated a lot of controversy (see, e.g. Gutowski et al. 1986, Collins Gallup 1986)—we shall not jump into this jungle here, since no unique scheme has yet been accepted. Most schemes imply an a posteriori adjustment of the interaction energy. For the present purpose, that is in order to derive an expression for the interaction operator, we shall take advantage of the a priori analysis of Mayer (1983) followed in his Chemical Hamiltonian Approach (CHA). 

Hermitian chemical Hamiltonian an alternative ab initio method... 

Abstract Some previous results of the present author are combined in order to develop a Hermitian version of the Chemical Hamiltonian Approach. In this framework the second quantized Bom-Oppenheimer Hamiltonian is decomposed into one- and two-center components, if some finite basis corrections are omitted. (No changes are introduced into the one- and two-center integrals, while projective expansions are used for the three- and four-center ones, which become exact only in the limit of complete basis sets.) The total molecular energy calculated with this Hamiltonian can then presented as a sum of the intraatomic and diatomic energy terms which were introduced in our previous chemical energy component analysis scheme. The corresponding modified Hartree-Fock-Roothaan equations are also derived they do not contain any three- and four-center integrals, while the non-empirical character of the theory is conserved. This scheme may be useful also as a layer in approaches like ONIOM. 

Keywords Chemical Hamiltonian Approach Alternative non-empirical SCF formalism Second quantized Hamiltonian Excluding three- and four-center integrals Projective integral approximation... 

These properties motivated us to call this formalism as Chemical Hamiltonian Approach (CHA). The disadvantage of the formalism was the non-Hermiticity of the... 

The different forms of the chemical Hamiltonian are obtained if one introduces into formulae (21), (26) and (27), the integral approximations discussed previously. 

We shall mention that using the mixed second quantized formalism of Ref. [1], already mentioned, it is possible to present the chemical Hamiltonian (32)-(34) in a form in which each term of the Hamiltonian contains only creation and annihilation operators assigned to the corresponding atom or pair of atoms. To save place, we shall illustrate that only by considering the first term of Eq. (33)— all the other terms can be treated analogously. The first term in question is... 

Mayer, in a series of papers, developed a theory called the chemical Hamiltonian approach (CHA), which was based on observables, such as charge densities. Beginning with two 1983 papers," he developed a non-Hermitian perturbation theory that was able to separate out the energy components that comprise the BSSE. He was able to show that there are both over- and undercorrection effects, and later we demonstrate that overcorrection typically is small in most cases. Using a complete analysis of a four-orbital, two-electron model, Mayer and Turi were able to separate out and display all the BSSE terms. Their paper contains the most extensive discussion of the origins of BSSE and the most elaborate presentation of the theory. 

M. Kieninpr, S. Suhai, and I. Mayer, Chem. Phys. Lett., 230, 483 (1994). The Chemical Hamiltonian Approach in Density Functional Theory. 

This equation differs from the calculation of AH° from standard enthalpies of formation (AHj), although it is equivalent to it. We return to the issue of obtaining AHj in theoretical methods later. At first sight, the notion of the enthalpy of a substance (Ha, Hb, He, and Ho) might be confusing there is after all no such thing as an absolute enthalpy. The quantities Ha, Hb, He, and Ho are relative enthalpies, but they are enthalpies of substances relative to the so-called quantum chemical standard state. The quantum chemical standard state consists of infinitely separated electrons and nuclei of a substance. For example, the quantum chemical enthalpy of the water molecule is the enthalpy relative to an O nucleus, two protons, and ten electrons, all infinitely separated. The quantum chemical standard state is the zero of the potential energy in the usual quantum chemical Hamiltonian. To make a parallel with experimental thermochemistry, the AHj of a substance is its molar enthalpy relative to the enthalpy of its elements in their standard states. 

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