Hamiltonian perturbation

The perturbations in this case are between a singlet and a triplet state. The perturbation Hamiltonian, H, of the second-order perturbation theory is spin-orbital coupling, which has the effect of mixing singlet and triplet states. 

Backward Analysis In this type of analysis, the discrete solution is regarded as an exact solution of a perturbed problem. In particular, backward analysis of symplectic discretizations of Hamiltonian systems (such as the popular Verlet scheme) has recently achieved a considerable amount of attention (see [17, 8, 3]). Such discretizations give rise to the following feature the discrete solution of a Hamiltonian system is exponentially close to the exact solution of a perturbed Hamiltonian system, in which, for consistency order p and stepsize r, the perturbed Hamiltonian has the form [11, 3]... 

In general the transitions appearing between the unperturbed states in such perturbation theories are of no physical significance they are simply a result of our attempt to express the true eigenstates of the true perturbed hamiltonian in terms of convenient but erroneous eigenstates of the unperturbed erroneous hamiltonian. If we were able to find the true eigenstates—mid this is, of course, possible in principle— no such transitions would be discovered and the apparent time-dependence would disappear. 

In order to define the notation which we will use from now on, let us consider the application of the perturbation theory to a system which has a perturbed hamiltonian H composed by an unperturbed one, H", plus a perturbation operator A.V, where A, () ... 

Let us write a perturbed hamiltonian by a set of k independent perturbation operators using the following expression involving a NSS ... 

The operator k is called the perturbation and is small. Thus, the operator k differs only slightly from and the eigenfunctions and eigenvalues of k do not differ greatly from those of the unperturbed Hamiltonian operator k The parameter X is introduced to facilitate the comparison of the orders of magnitude of various terms. In the limit A 0, the perturbed system reduces to the unperturbed system. For many systems there are no terms in the perturbed Hamiltonian operator higher than k and for convenience the parameter A in equations (9.16) and (9.17) may then be set equal to unity. 

In many applications there is no second-order term in the perturbed Hamiltonian operator so that zero. In such cases each unperturbed... 

Higher-order terms. .. in the perturbed Hamiltonian operator do... 

An alternative approach to calculating the free energy of solvation is to carry out simulations corresponding to the two vertical arrows in the thermodynamic cycle in Fig. 2.6. The transformation to nothing should not be taken literally -this means that the perturbed Hamiltonian contains not only terms responsible for solute-solvent interactions - viz. for the right vertical arrow - but also all the terms that involve intramolecular interactions in the solute. If they vanish, the solvent is reduced to a collection of noninteracting atoms. In this sense, it disappears or is annihilated from both the solution and the gas phase. For this reason, the corresponding computational scheme is called double annihilation. Calculations of... 

For a pseudo-axial ligand field the appropriate perturbation Hamiltonian, H, is, in the absence of a magnetic field, (c.f. Section 2)... 

In other words, the diagonal elements of the perturbing Hamiltonian provide the first-order correction to the energies of the spin manifold, and the nondiagonal elements give the second-order corrections. Perturbation theory also provides expressions for the calculation of the coefficients of the second-order corrected wavefunctions l / in terms of the original wavefunctions (p)... 

In second order perturbat i+PJf h e a r y with the perturbing Hamiltonian H = e E r cos u>t, and both the fundamental and created combined frequencies below electronic resonances but well above vibrational and rotational modes, can be expressed as... 

The actual form of the Hamiltonian operator hp does not have to be defined at this moment. As in standard perturbation theory, it is assumed that the solution of the electronic structure problem of the combined Hamiltonian HKS +HP can be described as the solution y/(0) of HKS, corrected by a small additional linear-response wavefunction /b//(,). Only these response orbitals will explicitly depend on time - they will follow the oscillations of the external perturbation and adopt its time dependency. Thus, the following Ansatz is made for the solution of the perturbed Hamiltonian HKS +HP ... 

Thus, the goal is to find frequencies co and corresponding orbitals y/(m) for which the DFPT equation without the external perturbation Hamiltonian ... 

Turning to the question of an interacting system, we define the perturbed Hamiltonian as... 

Consider a spin system whose spin Hamiltonian consists of a time-independent Hamiltonian H0 and a stochastic perturbation Hamiltonian H,(t) due to a small spin-lattice coupling,... 

Weak crystalline field //cf //so, Hq. In this case, the energy levels of the free ion A are only slightly perturbed (shifted and split) by the crystalline field. The free ion wavefunctions are then used as basis functions to apply perturbation theory, //cf being the perturbation Hamiltonian over the / states (where S and L are the spin and orbital angular momenta and. 1 = L + S). This approach is generally applied to describe the energy levels of trivalent rare earth ions, since for these ions the 4f valence electrons are screened by the outer 5s 5p electrons. These electrons partially shield the crystalline field created by the B ions (see Section 6.2). 

The Hamiltonian H of a system in a radiation field, in the absence of interaction with the field and a perturbation Hamiltonian H describing the interaction with the field, is given by... 

In Peralta et a/. s18,82 method the NJC decomposition into localized molecular orbitals procedure is as follows. Given a perturbative Hamiltonian A, A being either the FC or PSO operators, the corresponding J coupling term can be obtained as... 

In principle, then, to calculate A, one simply must know the wave functions describing the states, the density of final states, and the nature of the perturbative Hamiltonian. In practice, however, this is an exceedingly difficult problem. 

Of course, this is an approximation, and (6.105) is not an exact eigenfunction of the molecular Hamiltonian //, but is an eigenfunction of some operator which we shall call H°. (We shall not specify H° explicitly.) A better approximation to the true molecular wave function is obtained by applying perturbation theory, taking the perturbation Hamiltonian H as... 

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