Hamiltonian neglected terms

Let us return to the case where reaction takes place via a nonadiabatic transition. This situation typically occurs when the PES is constructed from a Hamiltonian in which one or more terms have been neglected. These terms then couple the initial and final states, thereby providing a mechanism for reaction to take place. The neglected terms may include, for example,... 

For simplicity, consider the so-called ferrodistortive ordering pattern. We assume that all elementary cells are distorted in exactly the same way. Therefore, explicitly, the averages, Q and a, do not depend on the elementary cell number, i. Assuming the flucmations, q i) and s(i), much smaller than the respective averages, Q and CT, we can neglect terms of second-order in q(i) and s(i) and keep just the linear ones. The JT Hamiltonian (7) decouples into the following sum of translationally identical one-cell terms. 

Combining the KAM techniques with the RWT will allow us to construct effective Hamiltonians in a systematic way and to estimate the order of the neglected terms (see Section III.D). We show that the KAM technique allows us to partition at a desired order operators in orthogonal Hilbert subspaces. We adapt this partitioning technique to treat Floquet Hamiltonians. Its connection with the standard adiabatic elimination is shown at a second-order approximation. 

In the literature a different technique has been widely used to construct effective Hamiltonians, based on the partitioning technique combined with an approximation procedure known as adiabatic elimination for the time-dependent Schrodinger equation (see Ref. 39, p. 1165). In this section we show that the effective Hamiltonian constructed by adiabatic elimination can be recovered from the above construction by choosing the reference of the energy appropriately. Moreover, our stationary formulation allows us to estimate the order of the neglected terms and to improve the approximation to higher orders in a systematic way. 

We remark that this effective Hamiltonian (190) constructed by the combination of a partitioning of the Floquet Hamiltonian, a two-photon RWT, and a final 0-averaging can be seen as a two-photon RWA, which extends the usual (one-photon) RWA [39,40], We have thus rederived a well-known result, using stationary techniques that allow us to estimate easily the order of the neglected terms. This method allows us also to calculate higher order corrections. We apply it in the next subsection to calculate effective Hamiltonians for molecules illuminated by strong laser fields. 

Thus, strictly speaking, the operator (6.6) can be employed for investigating the states of a crystal with two vibration quanta only when the intramolecular anharmonicity of the form given in (6.5) dominates, and the part of the anhar-monicity that is associated with the presence of intermolecular interaction can be neglected.47 Since, by assumption, A/Ml -C 1 and A/Ml -C 1, the total Hamiltonian (6.6) also neglects terms that do not conserve the number of quasiparticles (inessential corrections are introduced if they are taken into account). 

In Section 3.3 it will be shown that, to describe perturbations which result from neglected terms in the Hel + TN (R) part of the Hamiltonian, two different types of BO representations are useful. If a crossing (diabatic) potential curve representation is used, off-diagonal matrix elements of Hel appear between the states of this representation. If a noncrossing (adiabatic) potential curve representation is the starting point, the TN operator becomes responsible for perturbations. 

Initially, we neglect terms depending on the electron spin and the nuclear spin fin the molecular Hamiltonian //. In this approximation, we can take the total angular momentum to be JV(see (equation A1.4,1 )I which results from the rotational motion of the nuclei and the orbital motion of the electrons. The components of JVin the (X, Y, Z) axis system are given by ... 

Approximate Hamiltonians are formed by neglecting terms in eqn (20.27). The normal mode Hamiltonian results by including only the first term in eqn (20.27), whose energy levels are given by the sum ... 

The spin-Hamiltonian for the Mn(II) ion (electronic spin S=5/2, nuclear spin/ = 5/2) in a quadratie erystal field for orthorhombic distortion, neglecting terms of fourth order in eleetronie spin operator, is given by... 

The modified two-electron terms contain all the relativistic integrals, which means that the integral work is no different from that in the full solution of the Dirac-Hartree-Fock equations. It would save a lot of work if we could approximate the integrals, in the same way as we did for the Douglas-Kroll-Hess approximation. To do so, we must use the normalized Foldy-Wouthuysen transformation. The DKH approximation neglects the commutator of the transformation with the two-electron Coulomb operator, and in so doing removes all the spin-dependent terms. We must therefore also use a spin-free one-electron Hamiltonian. The approximate Hamiltonian (in terms of operators rather than matrices) is... 

If we restrict the order of perturbation admitted in (4.241) then we realize a finite order multi-reference Brillouin-Wigner perturbation theory. Specifically, if we neglect terms of order A are higher, we are led immediately to the second-order theory for which the matrix elements of the effective Hamiltonian (4.239) take the form ... 

To anticipate some of the problems involved in energy calculations, we use the helium atom functions already derived to obtain the actual energy levels. The 2-electron Hamiltonian (neglecting spin terms) is... 

Let US consider the simplified Hamiltonian in which the nuclear kinetic energy term is neglected. This also implies that the nuclei are fixed at a certain configuration, and the Hamiltonian describes only the electronic degrees of freedom. This electronic Hamiltonian is... 

The appearance of the (normally small) linear term in Vis a consequence of the use of reference, instead of equilibrium configuration]. Because the stretching vibrational displacements are of small amplitude, the series in Eqs. (40) should converge quickly. The zeroth-order Hamiltonian is obtained by neglecting all but the leading terms in these expansions, pjjjf and Vo(p) + 1 /2X) rl2r and has the... 

Unlike semiempirical methods that are formulated to completely neglect the core electrons, ah initio methods must represent all the electrons in some manner. However, for heavy atoms it is desirable to reduce the amount of computation necessary. This is done by replacing the core electrons and their basis functions in the wave function by a potential term in the Hamiltonian. These are called core potentials, elfective core potentials (ECP), or relativistic effective core potentials (RECP). Core potentials must be used along with a valence basis set that was created to accompany them. As well as reducing the computation time, core potentials can include the effects of the relativistic mass defect and spin coupling terms that are significant near the nuclei of heavy atoms. This is often the method of choice for heavy atoms, Rb and up. 

The Born-Oppenheimer approximation allows the two parts of the problem to be solved independently, so we can construct an electronic Hamiltonian which neglects the kinetic energy term for the nuclei ... 

In the derivation above, we have included the kinetic energy of the nuclei in the Hamiltonian and considered a stationary state. In Eq. II.3, this term has been neglected, and we have instead assumed that the nuclei have given fixed positions. It has been pointed out by Slater34 that, if the nuclei are not situated in the proper equilibrium positions, the virial theorem will appear in a slightly different form. (A variational derivation has been given by Hirschfelder and Kincaid.11)... 

Here, n denotes a number operator, a creation operator, c an annihilation operator, and 8 an energy. The first term with the label a describes the reactant, the second term describes the metal electrons, which are labeled by their quasi-momentum k, and the last term accounts for electron exchange between the reactant and the metal Vk is the corresponding matrix element. This part of the Hamiltonian is similar to that of the Anderson-Newns model [Anderson, 1961 Newns, 1969], but without spin. The neglect of spin is common in theories of outer sphere reactions, and is justified by the comparatively weak electronic interaction, which ensures that only one electron is transferred at a time. We shall consider spin when we treat catalytic reactions. 

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