Hamiltonian, unperturbed

Each electron in the system is assigned to either molecule A or B, and Hamiltonian operators and for each molecule defined in tenns of its assigned electrons. The unperturbed Hamiltonian for the system is then 0 = - A perturbation XH consists of tlie Coulomb interactions between the nuclei and... 

Assuming that aU of the wavefunctions k and energies E O belonging to the unperturbed Hamiltonian H are known... 

The MoIIer-PIesset perturbation method (MPPT) uses the single-eonfiguration SCF proeess (usually the UHF implementation) to first determine a set of LCAO-MO eoeffieients and, henee, a set of orbitals that obey F( )i = 8i (jii. Then, using an unperturbed Hamiltonian equal to the sum of these Foek operators for eaeh of the N eleetrons =... 

Ei=i N F(i), perturbation theory (see Appendix D for an introduetion to time-independent perturbation theory) is used to determine the Ci amplitudes for the CSFs. The MPPT proeedure is also referred to as the many-body perturbation theory (MBPT) method. The two names arose beeause two different sehools of physies and ehemistry developed them for somewhat different applieations. Later, workers realized that they were identieal in their working equations when the UHF H is employed as the unperturbed Hamiltonian. In this text, we will therefore refer to this approaeh as MPPT/MBPT. 

This Foek operator is used to define the unperturbed Hamiltonian of Moller-Plesset perturbation theory (MPPT) ... 

Proceeding now to the instanton treatment of the Hamiltonian (5.24) we observe that the spectrum of quasienergies differs from that of the unperturbed harmonic oscillator, f Q) = 0, only by a shift independent of n [Bas et al. 1971],... 

Let us give a brief summary of the LSGF method. We will consider a system of N atoms somehow distributed on the underlying primitive lattice. We start with the notion that if we choose an unperturbed reference system which has an ideal periodicity by placing eciuivalent effective scatterers on the same underlying lattice, its Hamiltonian may be calculated in the reciprocal space. Corresponding unperturbed Green s... 

Where // is the complete Hamiltonian operator for the unperturbed system and the usual quantum-mechanical integrals over all space are indicated. 

Interaction Representation.—In many physical problems the hamiltonian of a system that is engaged in interaction with another is of the form H + V,H being the stationary normal ( unperturbed ) hamiltonian and V the interaction. Equation (7-51) then reads... 

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. 

We shall take the Heisenberg and Schrodinger pictures to coincide at time 7 = 0. We next correlate with the complete hamiltonian 27(0) (i.e., the total hamiltonian in the Heisenberg picture at time 1 = 0, which by the above convention is also the total hamiltonian in the Schrodinger picture) an unperturbed hamiltonian 27o(0). We shall write... 

Perhaps the most straightforward method of solving the time-dependent Schrodinger equation and of propagating the wave function forward in time is to expand the wave function in the set of eigenfunctions of the unperturbed Hamiltonian [41], Hq, which is the Hamiltonian in the absence of the interaction with the laser field. 

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, () ... 

The eigenvalues and eigenvectors of the unperturbed hamiltonian are assumed to be known ... 

In the present implementation, the unperturbed functions are not subject to any orthogonality constraint nor are required to diagonalize any model hamiltonian. This freedom yields a faster convergence of the variational expansion with the basis size and allows to obtain the phaseshift of the basis states without the analysis of their asymptotic behaviour. 

The quantity k > is the unperturbed Hamiltonian operator whose orthonormal eigenfunctions and eigenvalues are known exactly, so that... 

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... 

The Hamiltonian operator for the unperturbed harmonic oscillator is given by equation (4.12) and its eigenvalues and eigenfunctions are shown in equations (4.30) and (4.41). The perturbation H is... 

In reality, this term is not small in comparison with the other terms so we should not expect the perturbation technique to give accurate results. With this choice for the perturbation, the Schrodinger equation for the unperturbed Hamiltonian operator may be solved exactly. 

The unperturbed Hamiltonian operator is the sum of two hydrogen-like Hamiltonian operators, one for each electron... 

With the experiment described above in mind, represent the Hamiltonian of the unperturbed system by H° and that of the perturbed system by... 

In fin problem of interest here, the Hamiltonian in Bq. (62) can be decomposed into a time-independent, unperturbed part and a much smaller, time-dependent operator H (t). Then, die Hamiltonian becomes to first oiler... 

The statistical perturbation theory arising from the classical work of Zwanzig34 and its detailed implementation in a molecular dynamics program for computation of free energies is described in detail elsewhere.35 36 We give a very brief description of the method for the sake of completeness. The total Hamiltonian of a system may be written as the sum of the Hamiltonian (Ho) of the unperturbed system and the perturbation (Hi) ... 

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