Defining an effective Hamiltonian

A different approach to mention here because it has some similarity to QM/MM is called RISM-SCF [5], It is based on a QM description of the solute, and makes use of some expressions of the integral equation of liquids (a physical approach that for reasons of space we cannot present here) to obtain in a simpler way the information encoded in the solvent distribution function used by MM and QM/MM methods. Both RISM-SCF and QM/MM use this information to define an effective Hamiltonian for the solute and both proceed step by step in improving the description of the solute electronic distribution and solvent distribution function, which in both methods are two coupled quantities. There is in this book a contribution by Sato dedicated to RISM-SCF to which the reader is referred. Sato also includes a mention of the 3D-RISM approach [6] which introduces important features in the physics of the model. In fact the simulation-based methods we have thus far mentioned use a spherically averaged radial distribution function, p(r) instead of a full position dependent function p(r) expression. For molecules of irregular shape and with groups of different polarity on the molecular periphery the examination of the averaged p(r) may lead to erroneous conclusions which have to be corrected in some way [7], The 3D version we have mentioned partly eliminates these artifacts. 

The orbital functional derivative here defines an effective Hamiltonian... 

We now define an effective Hamiltonian matrix in the subspace of the barrier states (a TV x TV matrix)... 

We are now prepared to define an effective Hamiltonian in which the excitations have been integrated out through the relation... 

Thus, this solution defines the exact AI parameterization of the ligand field Hamiltonian. To the best of our knowledge, this is a new result. In practice, of course, the FCI equations cannot be solved for even the smallest transition metal complex. Hence one has to resort to approximation that still define an effective Hamiltonian or at least its energies. We next turn our attention to such approximations. 

A straightforward averaging of Z = f Dri Dr exp(—H) using the probability distribution of Eq. (68) defines an effective Hamiltonian Heft such that... 

We can now proceed to define an effective Hamiltonian, which operates only within the model space, but which has the exact energy as its eigenvalue. 

Equation (2.5.2) is exactly equivalent to (2.5.1), but we have reduced the secular problem from one of n dimensions to one of dimensions by defining an effective Hamiltonian to allow for the other functions. 

In the presence of a phase factor, the momentum operator (P), which is expressed in hyperspherical coordinates, should be replaced [53,54] by (P — h. /r ) where VB creates the vector potential in order to define the effective Hamiltonian (see Appendix C). It is important to note that the angle entering the vector potential is shictly only identical to the hyperangle <]> for an A3 system. 

To compensate for the drastic assumptions an effective Hamiltonian for the system is defined in such a way that it takes into account some of the factors ignored by the model and also factors only known experimentally. The HMO method is therefore referred to as semi-empirical. As an example, the Pauli principle is recognized by assigning no more than two electrons to a molecular orbital. 

Effective Hamiltonian. In this framework it is possible to define an effective... 

By adopting the no-pair approximation, a natural and straightforward extension of the nonrelativistic open-shell CC theory emerges. The multireference valence-universal Fock-space coupled-cluster approach is employed [25], which defines and calculates an effective Hamiltonian in a low-dimensional model (or P) space, with eigenvalues approximating some desirable eigenvalues of the physical Hamiltonian. The effective Hamiltonian has the form [26]... 

To sense the effect one cluster would have on the other let us consider the effective hamiltonian approach. There, one cluster is subjected to the field ofthe other. With obvious notations one can define an effective parametric electronuc-lear problem for each partner via the functional ... 

In this chapter an effective Hamiltonian for a static cooperative Jahn-Teller effect is proposed. This Hamiltonian acts in the space of local active distortions only and possesses extrema points of the potential energy equivalent to those of the full microscopic Hamiltonian, defined in the space of all phonon and uniform strain coordinates. First we present the derivation of this effective Hamiltonian for a general case and then apply the theory to the investigation of the structure of Jahn-Teller hexagonal perovskites. 

It is important to note that the Hamiltonian (2.120) contains the terms which produce both the adiabatic and non-adiabatic effects. In chapter 7 we shall show how the total Hamiltonian can be reduced to an effective Hamiltonian which operates only in the rotational subspace of a single vibronic state, the non-adiabatic effects being treated by perturbation theory and incorporated into the molecular parameters which define the effective Hamiltonian. Almost for the first time in this book, this introduces an extremely important concept and tool, outlined in chapter 1, the effective Hamiltonian. Observed spectra are analysed in terms of an appropriate effective Hamiltonian, and this process leads to the determination of the values of what are best called molecular parameters . An alternative terminology of molecular constants , often used, seems less appropriate. The quantitative interpretation of the molecular parameters is the link between experiment and electronic structure. 

Another method, devised by Cohen et al. to determine oxygen-rate gas collision parameters is to define an effective spin-orbit operator that includes r dependence, Zeff/r3, where the value of Zeff is adjusted to match experimental data (76). Langhoff has compared this technique with all-electron calculations using the full microscopic spin-orbit Hamiltonian for the rare-gas-oxide potential curves and found very good agreement (77). This operator has also been employed in REP calculations on Si (73), UF6 (78), U02+ and Th02 (79), and UF5 (80). The REPs employed in these calculations are based on Cowen-Griffin atomic orbitals, which include the relativistic mass-velocity and Darwin effects but do not include spin-orbit effects. Wadt (73), has made comparisons with calculations on Si by Stevens and Krauss (81), who employed the ab initio REP-based spin-orbit operator of Ermler et al. (35). 

If we now consider the generalized Silverstone-Sinanoglu strategy discussed above, then the most natural way to proceed is by way of an effective hamiltonian formalism. We introduce a single (state-universal) wave-operator O, whose action on 4 s produce the functions 4>k, defined by eq. (6.1.1) or (6.1.7), and write Schrodinger equations for 4[Pg.327]

The renormalized theory of the effective Hamiltonian implied by the restriction to some subspace S of the full Hilbert space also imposes a requirement for renonnalisation of expectation values of other operators (Freed ). Suppose that we have some operator B and we require its expectation value in a state 0 of the full Schrddinger Eq. (2-2) in complete analogy with the effective Hamiltonian theory described above we define an effective operator B by the requirement that its expectation value in a state A ) in the subspace should equal the exact expectation value (c.f. Eq. (2-4)),... 

We first study the electronic spectrum of a single C o molecule and then use the resulting low lying many-body states as the basis for an effective Hamiltonian that describes the physics at length scales larger than the size of the molecule. We begin with the simplest possible model of the one-electron spectrum and the simplest possible electron-electron interactions, known as the Hubbard model, but defined on the truncated icosahedral C o lattice. The... 

At this point we have expressed the Hamiltonian, the density operator and the evolution operator in Fourier space. We have introduced an effective Hamiltonian, defined in the Hilbert space of the same dimension 2 as the total time-dependent Hamiltonian itself, and we have shown how to transform operators between the two representations. The definition of the effective Hamiltonian enables us to predict the overall evolution of the spin system, despite the fact that we can not find time-points for synchronous detection, f, where Uint f) = exp —iWe//t In actual experiments the time dependent signals are monitored and after Fourier transformation they result in frequency sideband... 

The essential feature of the o—n separation is that an effective Hamiltonian can be defined for the n electrons in the field of the nuclei and the a core. As was pointed out by Sinanoglu 7>, this separation can be derived under conditions more general than the Lykos-Parr assumption. A slightly different formulation of the o—n separation can be obtained by the methods of second quantization 8>. 

In Chapt. 2 we have recalled how a and it orbitals can be defined in terms of a rigorous theory and what the notions of a and it electrons actually mean. In Chapt. 3 we have introduced the a—it separation and discussed its justification and limitations. If the a—it separation is valid, then an effective Hamiltonian for the it electrons can be constructed into which the a electrons enter only via the effective potential created by their charge distribution. 

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