Zero-order Hamiltonian

The ability to assign a group of resonance states, as required for mode-specific decomposition, implies that the complete Hamiltonian for these states is well approxmiated by a zero-order Hamiltonian with eigenfunctions [M]. The ( ). are product fiinctions of a zero-order orthogonal basis for the reactant molecule and the quantity m. represents the quantum numbers defining ( ).. The wavefimctions / for the compound state resonances are given by... 

As a naive or zero-order approximation, we can simply ignore the V12 term and allow the simplified Hamiltonian to operate on the Is orbital of the H atom. The result is... 

According to the coherent averaging theory,3,4,53 the zero-order average Hamiltonian can be obtained straightforwardly... 

Since the zero-order average Hamiltonian of each RF field j]m(m Q) and the first-order average Hamiltonians between any two RF fields (m, n O)... 

The zero-order average Hamiltonian can be obtained straightforwardly... 

Here 2tt Aft = trm is imposed to eliminate the effect by the zero-order interaction of the RF field, otherwise a term containing Ix will remain and it will distort the longitudinal magnetization, transverse magnetization, as well as the BSPS of the 13C . For the on-resonance condition, 5 = 0, all the higher-order average Hamiltonians vanish since [7f(/ ), = 0 for arbitrary l and... 

The zero-order Hamiltonian is a function of the actions alone. It therefore corresponds to uncoupled modes whose actions are conserved (since dljdt = - <)H/(), ). From Section 7.5 on we will express the classical limit of algebraic Hamiltonians in terms of variables i = 1,..., n. These are related to the action-angle variables by t, = I112 exp(/0), = I112 exp(-i0). Loosely... 

Since electrons are much faster than nuclei, owing to Wg Mj, ions can be considered as fixed and one can thus neglect the //ion-ion contribution (formally Mion-ion Hee, where Vion-ion is a Constant). This hrst approximation, as formulated by N. E. Born and J. R. Oppenheimer, reflects the instantaneous adaptation of electrons to atomic vibrations thus discarding any electron-phonon effects. Electron-phonon interactions can be a-posteriori included as a perturbation of the zero-order Hamiltonian Hq. This is particularly evident in the photoemission spectra of molecules in the gas phase, as already discussed in Section 1.1 for nJ, where the 7T state exhibits several lines separated by a constant quantized energy. 

Let us now study the effect of including a perturbation //per to the zero-order Hamiltonian Hq, so that the new Hamilton operator H will be given by // = Hq + //per. We further proceed with our example of the linear chain of period a with //at = 1, for the sake of simplicity, and continue to extract fundamental information. Figure 1.32 shows the band dispersion of this half-filled system. 

In this case the zero-order electronic wave functions are, in principle, referred to a Hamiltonian that contains the potential from the ions at their actual positions, i.e., the electrons follow the ionic motion adiabatically. Since both these approximations are sometimes referred to as the Born Oppenheimer approximation, this has led to confusion in terminology for example, Mott (1977) refers to the Born-Oppenheimer approximation, but gives wave functions of the adiabatic type, whereas Englman (1972) differentiates between the two forms, but specifically calls the static form the Bom Oppenheimer method. [We note that, historically, the adiabatic form was first suggested by Seitz (1940)—see, for example, Markham (1956) or Haug and Sauermann (1958)]. In this chapter, we shall preferentially use the terminology static and adiabatic. [Note that the term crude adiabatic is also sometimes used for the static approximation, mainly in the chemical literature—see, for example, Englman (1972, 1979).]... 

Moller-Plesset perturbation theory (MPPT) aims to recover the correlation error incurred in Ilartree- Fock theory for the ground state whose zero-order description is ,. The Moller-Plesset zero-order Hamiltonian is the sum of Fock operators, and the zero-order wave functions are determinantal wave functions constructed from HF MOs. Thus the zero-order energies are simply the appropriate sums of MO energies. The perturbation is defined as the difference between the sum of Fock operators and the exact Hamiltonian ... 

What is a polyad A polyad is a subset of the zero-order states within a specifiable region of Evib (typically a few hundred reciprocal centimeters) that are strongly coupled by anharmonic resonances to each other and negligibly coupled to all other nearby zero-order states. If approximate constants of motion of the exact vibration-rotation Hamiltonian exist, then the exact H can be (approximately) block diagonalized. Each subblock of H corresponds to one polyad and is labeled by a set of polyad quantum numbers. For the C2H2S0 state, a procedure proposed by Kellman [9, 10] identifies the three polyad quantum numbers... 

In tetra-atomic molecules that are linear, the number of degrees of freedom is F =7, which again considerably complicates the analysis. Nevertheless, the dynamics of such systems can turn out to be tractable if the anharmonicities may be considered as perturbations with respect to the harmonic zero-order Hamiltonian. In such cases, the regular classical motions remain dominant in phase space as compared with the chaotic zones, and the edge periodic orbits of subsystems again form a skeleton for the bulk periodic orbits. 

It is worthwhile to examine the Hamiltonian in some detail because it enables one to discuss both intramolecular and intermolecular perturbations from the same point of view. To do so, we start from a zero-order Hamiltonian that contains just the spherical part of the field due to the core (which need not be Coulombic as it includes also the quantum defect [42]) and add two perturbations. U due to external effects and V due to the structure of the core. Here, U contains both the effect of external fields (electrical and, if any, magnetic [1]) and the role of other charges that may be nearby [8, 11, 12, 17]. The technical point is that both the effect of other charges and the effect of the core not being a point charge are accounted for by writing the Coulomb interaction between two charges, at points ri and r2, respectively, as... 

Here, Q is the projector on the bound subspace and P projects onto the open, continuum channels. The intramolecular coupling is written as V+ U so that, as before, U is any additional coupling brough about by external perturbations. The equation H = Hq + V+U, where Ho is the zero-order Hamiltonian of the Rydberg electron and so includes only the central part of the potential due to the core plus the motion (vibration, rotation) of the core, uncoupled to the electron. The perturbations V + U can act within the bound subspace, as the operator Q(V+l/)Q is not necessarily diagonal and is the cause of any intramolecular dynamics even in the absence of coupling to the continuum. The intramolecular terms can also couple the bound and dissociative states. 

The zero-order, spin-free Hamiltonian HSF commutes with the symmetric group SnF of permutations on the N different spatial electronic indices,... 

A time-dependent process, such as radiative absorption, internal conversion, intersystem crossing, unimolecular isomerization, or collision, may be treated in terms of a zero-order Hamiltonian H0 and a perturbation T. An unperturbed eigenstate of H0 evolves in time, since it is not an eigenstate to the perturbed Hamiltonian... 

If electrons are localized in different parts of a molecule, exchange terms between these electrons are small. In this localized state several different permutation states may be degenerate or near degenerate in zero order. Consequently, these states are highly mixed under the full Hamiltonian. 

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