Intermediate Hamiltonians

Let us consider a model case for a single bond dissociation process where there is no need to call for an intermediate Hamiltonian in the sense discussed above. The discussion presented in 2.8 applies. Here, some formal aspects are discussed. 

It is worthwhile to emphasize that the intermediate Hamiltonian Hc(ij) defines a geometry that can be used to construct a model of an activated complex. A portrait of it can be obtained at the BO level of theory. For thermally activativated processes, the transition state is the analogous of the intermediate Hamiltonian, while for processes without thermal activation (a number of reactions taking place in gas phase, such as for example, the SN2 reaction between methyl halides and halides ions [168-171] ) the quantum states of this Hamiltonian mediate the chemical interconversion. For particular... 

This is a (minimal) model including the formation of the complex R1-R2, the active precursor complex APC that interconverts to those states belonging to the active successor complex ASC, as discussed in the previous section. The chemical reaction, in this model, ends up with the formation of the products PI and P2. The kinetic parameters k+ and k- hide the effects of quantum interconversions via the intermediate Hamiltonian Hc(ij). Let us introduce this feature in the kinetic model, so that... 

Let us consider now processes where intermediate stationary Hamiltonians are mediating the interconversion. In these processes, there is implicit the assumption that direct couplings between the quantum states of the precursor and successor species are forbidden. All the information required to accomplish the reaction is embodied in the quantum states of the corresponding intermediate Hamiltonian. It is in this sense that the transient geometric fluctuation around the saddle point define an invariant property. 

The reaction channel is open as soon as quantum states of the intermediate Hamiltonian become populated. By hypothesis, such states have two possible different relaxation channels One back to the reactants, the other forward to product via the quantum states of the successor complex. 

Hamiltonian quantum states. Two situations can be envisaged. In the first one, the system moves stepwise from the APC states to the interconversion complex. We assume that the i-th state of the active precursor is populated i f > at a given time to, and if f> is intermediate Hamiltonian state that are coupled by the electro-magnetic field. 

In Eqs. (25) and (26), the summations are over the incremental steps in going from X to Y in the gas phase or in solution. The Hj are the intermediate Hamiltonians (or force fields in a classical treatment). Thus, Hi=0,gas = Hx,gas, Hi=Njgas = HY,gas, etc. It is of course desirable that the molecules X and Y be structurally similar, so that the perturbation of X that produces Y be small. Another option is to let Y be composed of noninteracting dummy) atoms,75 so that its free energy of solvation is zero. Then Eq. (24) gives the absolute free energy of solvation of X ... 

The Intermediate Hamiltonian Theory is a generalization of the Effective Hamiltonian Theory. The full Cl space of Slater determinants can be divided into three parts,... 

A. Landan, E. Ehav, and U. Kaldor, Intermediate Hamiltonian Fock-Space Coupled-Cluster Method and Applications. In R. F. Bishop, T. Brandes, K. A. Gernoth, N. R. Walet, and Y. Xian (Eds.) Recent Progress in Many-Body Theories, Advances in Quantum Many-Body Theories, Vol. 6. (World Scientific, Singapore, 2002), pp. 355-364 and references therein. 

Let us consider the 5s, 5p, 5d orbitals of lead and Is orbital of oxygen as the outercore and the ai, a2, os, tti, tt2 orbitals of PbO (consisting mainly of 6s, 6p orbitals of Pb and 2s, 2p orbitals of O) as valence. Although in the Cl calculations we take into account only the correlation between valence electrons, the accuracy attained in the Cl calculation of Ay is much better than in the RCC-SD calculation. The main problem with the RCC calculation was that the Fock-space RCC-SD version used there was not optimal in accounting for nondynamic correlations (see [136] for details of RCC-SD and Cl calculations of the Pb atom). Nevertheless, the potential of the RCC approach for electronic structure calculations is very high, especially in the framework of the intermediate Hamiltonian formulation [102, 131]. 

D. Nikolic, E. Lindroth, Intermediate Hamiltonian to avoid intruder state problems for doubly excited states, J. Phys. B 37 (2004) L285. 

A. Landau, E. Eliav, U. Kaldor, Intermediate Hamiltonian Fock-space coupled-cluster method, Chem. Phys. Lett. 313 (1999) 399. 

E. Eliav, M.J. Vilkas, Y. Ishikawa, U. Kaldor, Extrapolated intermediate Hamiltonian coupled-cluster approach Theory and pilot application to electron affinities of alkali atoms, J. Chem. Phys. 122 (22) (2005) 224113. 

It is well known from the Bom-Oppenheimer separation [1] that the pattern of energy levels for a typical diatomic molecule consists first of widely separated electronic states (A eiec 20000 cm-1). Each of these states then supports a set of more closely spaced vibrational levels (AEvib 1000 cm-1). Each of these vibrational levels in turn is spanned by closely spaced rotational levels ( A Emt 1 cm-1) and, in the case of open shell molecules, by fine and hyperfine states (A Efs 100 cm-1 and AEhts 0.01 cm-1). The objective is to construct an effective Hamiltonian which is capable of describing the detailed energy levels of the molecule in a single vibrational level of a particular electronic state. It is usual to derive this Hamiltonian in two stages because of the different nature of the electronic and nuclear coordinates. In the first step, which we describe in the present section, we derive a Hamiltonian which acts on all the vibrational states of a single electronic state. The operators thus remain explicitly dependent on the vibrational coordinate R (the intemuclear separation). In the second step, described in section 7.55, we remove the effects of terms in this intermediate Hamiltonian which couple different vibrational levels. The result is an effective Hamiltonian for each vibronic state. 

Recent developments include exact [12-14, 44, 90, 91] and approximate [14, 90, 92-94] iterative schemes to determine Hg, the intermediate Hamiltonian method [21, 24, 95], the use of incomplete model spaces [43, 44] and some multireference open-shell coupled-cluster (CC) formalisms [16-20, 96, 97]. Only some eigenvalues of the intermediate Hamiltonian H, are also eigenvalues of H. The corresponding model eigenvectors of H, are related to their true counterparts as in Bloch s theory. Provided effective operators a are restricted to act solely between these model eigenvectors, the possible a definitions from Bloch s formalism (see Section VI.A) can be used. 

We now perform the transformation, thus constructing an intermediate Hamiltonian H = exp(L (i))if(°). This means that we should... 

Coupled Cluster based size-extensive intermediate hamiltonian formalisms were developed by our group [33-35] by way of transcribing a size-extensive CC formulation in an incomplete model space in the framework of intermediate hamiltonians. In this method, there are cluster operators correlating the main model space. There are no cluster operators for the intermediate space. This formulation thus is conceptually closer to the perturbative version of Kirtman... 

The Fock space multireference CC methods and the intermediate Hamiltonian techniques (see e.g. Refs. [24-29] and references therein), as well as closely related similarity transformed EOMCC [30-33] are methods particularly suited for calculation of excited/ionized states with a multireference character. Recently, a Brillouin-Wigner formulation of Fock space CC has also been derived [34]. 

Debashis Mukherjee is a Professor of Physical Chemistry and the Director of the Indian Association for the Cultivation of Science, Calcutta, India. He has been one of the earliest developers of a class of multi-reference coupled cluster theories and also of the coupled cluster based linear response theory. Other contributions by him are in the resolution of the size-extensivity problem for multi-reference theories using an incomplete model space and in the size-extensive intermediate Hamiltonian formalism. His research interests focus on the development and applications of non-relativistic and relativistic theories of many-body molecular electronic structure and theoretical spectroscopy, quantum many-body dynamics and statistical held theory of many-body systems. He is a member of the International Academy of the Quantum Molecular Science, a Fellow of the Third World Academy of Science, the Indian National Science Academy and the Indian Academy of Sciences. He is the recipient of the Shantiswarup Bhatnagar Prize of the Council of Scientihc and Industrial Research of the Government of India. 

The basic relativistic equations are described in Sec. 2, and the Fock-space coupled cluster method is discussed in Sec. 3. The recently developed intermediate Hamiltonian approach is described and illustrated by several... 

A particular variant of the coupled cluster method, called Fock-space or valence-universal [49,50], gave remarkable agreement with experiment for many transition energies of heavy atoms [51]. This success makes the scheme a useful tool for reliable prediction of the structure and spectrum of superheavy elements, which are difficult to access experimentally. A brief description of the method is given below. A more flexible scheme with higher accuracy and extended applicability, the intermediate Hamiltonian Fock-space coupled cluster approach, is shown in the next section. 

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