Effective Hamiltonian for the R-system

As mentioned earlier, it is highly desirable to get rid of the energy dependence of the effective Hamiltonians describing the subsystems. In order to do so we reconsider the general derivation of an effective Hamiltonian and specify it for the R-system. 

To get the effective Hamiltonian for the R-system which is necessary to calculate / and the corresponding ground and excited state energies, we consider contributions to the effective Hamiltonian eq. (1.232). It is important from the point of view of the further separation of the Hamiltonians into unperturbed parts and perturbations. The bare Hamiltonians for the R-system Hr and for the M-system HM defined by eq. (1.224) on the basis of attribution of the fermi-operators to the R- and M-systems turn out to be not a good starting point for developing a perturbational picture as the 

Hamiltonians thus defined contain large one-electron terms describing the attraction of electrons to the unscreened atomic cores in an alien subsystem  

The real physical situation is much better described by the bare operators where expectation values of the Coulomb operator ((WC))R and Wn))M ensure the screening of the alien core charges in the effective Hamiltonians eq. (1.224)  

Using the definitions of the bare Hamiltonians in eq. (1.234) and of the effective Hamiltonians for the subsystems in eq. (1.235), we get an alternative break down of the effective Hamiltonian for the R-system  

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Estimates of the electronic energy of the complex system employed in the expressions eqs. (1.254), (1.256) for its PES can be further improved. For this let us notice that the solutions of the self consistent system eq. (1.246) are used as multipliers in the basis functions eq. (1.216) of the subspace Im/. It turns out that the effective Hamiltonian Heff eq. (1.232) has nonvanishing matrix elements between the ground state of eq. (1.246) and the basis product states of the subspace ImP, differing from it by two multipliers simultaneously by the wave function for the R-system and by that for the M-system ( A p. p f 0). Indeed ... 

Tully has discussed how the classical-path method, used originally for gas-phase collisions, can be applied to the study of atom-surface collisions. It is assumed that the motion of the atomic nucleus is associated with an effective potential energy surface and can be treated classically, thus leading to a classical trajectory R(t). The total Hamiltonian for the system can then be reduced to one for electronic motion only, associated with an electronic Hamiltonian Jf(R) = Jf t) which, as indicated, depends parametrically on the nuclear position and through that on time. Therefore, the problem becomes one of solving a time-dependent Schrodinger equation ... 

Formally it applies to any atom, but it is nontrivial only for the frontier ones. The condition which specifies the distribution of the core charge ZA between the R- and M-systems is that the cores of the R-system must be as much as possible screened by the electrons of the R-system i.e. the effective Hamiltonian must be as close as possible to the Hamiltonian of the free M-system H°M. This reduces to the electron counting rules based on the concept of the formal oxidation state (see [60] for details). With this we arrive at the possibility of distributing not only the electronic density, but also the total effective charges between the R- and M-systems. This is done by the formulae ... 

In the frame of the target hybrid QM/MM procedure, only the electronic structure of the R-system is calculated explicitly. For this reason, we consider its effective Hamiltonian eq. (1.235) in more detail. It contains the operator terms coming from (1) the Coulomb interaction of the effective charges in the M-system with electrons in the R-system 5VM and (2) from the resonance interaction of the R- and M-systems. 

The reformulation of the theory of interaction between the R- and M-systems in terms of observables pertinent to the M-system assumes certain procedure for evaluating either the wave functions 4>,)/ or directly the necessary expectation values taken over it. To do so, we notice that according to eq. (1.235) the effective Hamiltonian H fj for the M-system in the presence of R-system, defining is close to the Hamiltonian Hfor the free M-system. The assumption that the M-system is inert implies that its characteristic excitation energies are large, thus the reduced interac-... 

In Section IV it was noted that for a given spin system with Hamiltonian and a given multiple-pulse sequence with. f(t), it is always possible to calculate the total propagator U r ), a corresponding effective Hamiltonian and the evolution of the density operator ait) (see Fig. 5). However, because the relationship between t) + [ t) and U(t )... 

A rigorous derivation of the form of the hybrid potentials treated in this chapter requires the construction an effective Hamiltonian, Her, for the system. This Hamiltonian can then be used as the Hamiltonian for the solution of the time-independent Schrodinger equation for the wavefunc-tion of the electrons on the QM atoms, P, and for the potential energy of the system, qm/mm- If R are the coordinates of the MM atoms, the Schrodinger equation (equation 1) becomes ... 

In practice, the search for the optimum density Pa, defined by Eq. 13.7, is performed by exploiting the Kohn-Sham formulation [67] of DPT [68] to solve Eq. 13.4, in which is the environment-free Hamiltonian of the isolated system A and = X iVgmfc(r() is the potential energy operator describing the effect of environment B on system A, where has the form of a local, orbital-... 

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. 

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