An effective Hamiltonian

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. 

By introducing the resolution of the identity (2.5), the Schrodinger equation (2.1) can be written 

Operating on eq. (2.12) from the left with the projector Q and using the idempotency of Q, i.e. (2.7), and the orthogonality relation (2.8) gives 

Substituting (2.18) into the second term on the left-hand side of (2.14) gives 

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Physically, why does a temi like the Darling-Dennison couplmg arise We have said that the spectroscopic Hamiltonian is an abstract representation of the more concrete, physical Hamiltonian fomied by letting the nuclei in the molecule move with specified initial conditions of displacement and momentum on the PES, with a given total kinetic plus potential energy. This is the sense in which the spectroscopic Hamiltonian is an effective Hamiltonian, in the nomenclature used above. The concrete Hamiltonian that it mimics is expressed in temis of particle momenta and displacements, in the representation given by the nomial coordinates. Then, in general, it may contain temis proportional to all the powers of the products of the... 

There has been a great deal of work [62, 63] investigating how one can use perturbation theory to obtain an effective Hamiltonian like tlie spectroscopic Hamiltonian, starting from a given PES. It is found that one can readily obtain an effective Hamiltonian in temis of nomial mode quantum numbers and coupling. 

Another group of approaches for handling the R-T effect are those that employ various forms of effective Hamiltonians. By applying pertuibation theory, it is possible to absorb all relevant interactions into an effective Hamiltonian, which for a particular (e.g., vibronic) molecular level depends on several parameters whose values are determined by fitting available experimental data. These Hamiltonians are widely used to extract from high-resolution [e.g.. 

The treatment developed here is based on the density matrix of quantum mechanics and extends previous work using wavefunctions.(42 5) The density matrix approach treats all energetically accessible electronic states in the same fashion, and naturally leads to average effective potentials which have been shown to give accurate results for electronically diabatic collisions. 19) The approach is taken here for systems where the dynamics can be described by a Hamiltonian operator, as it is possible for isolated molecules or in models where environmental effects can be represented by terms in an effective Hamiltonian. 

The QM/MM interactions (Eqm/mm) are taken to include bonded and non-bonded interactions. For the non-bonded interactions, the subsystems interact with each other through Lennard-Jones and point charge interaction potentials. When the electronic structure is determined for the QM subsystem, the charges in the MM subsystem are included as a collection of fixed point charges in an effective Hamiltonian, which describes the QM subsystem. That is, in the calculation of the QM subsystem we determine the contributions from the QM subsystem (Eqm) and the electrostatic contributions from the interaction between the QM and MM subsystems as explained by Zhang et al. [13],... 

In strongly coupled systems, it is not possible to eliminate chemical shifts by refocusing nor is it possible to describe the evolution in terms of an effective Hamiltonian.44 A 90° or a 180° pulse leads to coherence transfer between various transitions, and a multitude of new effective precession frequencies may appear in the F1 dimension. A detailed analysis shows artefacts resulting of strong coupling induced by the 180° pulse applied on the H channel can be efficiently removed by applying a LPJF before acquisition.42 Likewise, artefacts present in HMBC with a terminal LPJF are suppressed by an LPJF in the beginning of the sequence as in conventional HMBC. 

Prior to an effective Hamiltonian analysis it is, in order to get this converging to the lowest orders, typical to remove the dominant rf irradiation from the description by transforming the internal Hamiltonian into the interaction frame of the rf irradiation. This procedure is well established and also used in the most simple description of NMR experiments by transforming the Hamiltonian into the rotating frame of the Zeeman interaction (the so-called Zeeman interaction frame). In the Zeeman interaction frame the time-modulations of the rf terms are removed and the internal Hamiltonian is truncated to form the secular high-field approximated Hamiltonian - all facilitating solution of the Liouville-von-Neumann equation in (1) and (2). The transformation into the rf interaction frame is given by... 

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. 

Warshel and Levitfs study of lysozyme initiated the field of QM/ MM methods [29]. Many groups [10, 30-33] have covered the basic theory of QM/MM methods in detail, so we will outline the theoretical basis only briefly. In accordance with Field et al. [30], the energy of the whole system, E, is written in terms of an effective Hamiltonian, Heff, and the electronic wavefunction of the QM atoms, y/ ... 

The inter/intramolecular potentials that have been described may be viewed as classical in nature. An alternative is a hybrid quantum-mechanical/classical approach, in which the solute molecule is treated quantum-mechanically, but interactions involving the solvent are handled classically. Such methods are often labeled QM/MM, the MM reflecting the fact that classical force fields are utilized in molecular mechanics. An effective Hamiltonian Hefl is written for the entire solute/solvent system ... 

Using a valence bond scheme parametrized with an effective Hamiltonian technique, it was shown that the mechanistic preference for a synchronous pathway with an aromatic transition state versus a non-synchronous mechanism via biradicaloid intermediate can be controlled by two factors (1) the stability of the long bond in the Dewar valence bond structure, and (2) the softness of the Coulomb interaction between the end methylene groups in the 1,5-diene chain. This means that the mechanism of rearrangement (equation 153) can strongly depend on substituents218. 

The second kind of semiempirical procedure mentioned here is even cruder. In the extended Hueckel method (EHM 64<65)) the electronic structure of the molecule is simulated by an effective Hamiltonian. The total energy of the molecule is represented by a sum of one electron energies and even the nuclear repulsion terms are not taken into account explicitly. This type of approach can be shown to give an approximate idea of electronic structures and relative energies of unpolar molecules like hydrocarbons, but it fails inevitably when applied to structures with appreciable polarity 66>. Therefore any application of EHM calculations to interactions between polar molecules or ions should be regarded with a good deal of scepticism. 

Dunham achieved inter-relations between term coefficients l i through use of an intermediate radial function V(J ) in an effective hamiltonian for motion of atomic nuclei of this form. 

As a point of departure we assume, within a conventional separation of nuclear and electronic motions, an effective Hamiltonian for the motion of two atomic nuclei and their associated electrons both along and perpendicular to the internuclear vector, directly applicable to a molecule of symmetry class for which magnetic effects are absent or negligible [25] ... 

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]... 

Equivalently, the unitary operator can be viewed as transforming the reference Hamiltonian. This yields an effective Hamiltonian that has as an eigenfunction and the exact eigenenergy E of the target state as its eigenvalue. Thus... 

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