Unperturbed

As a general comment, it is fortunate for the study of monolayers that dissolving processes are generally slow enough to permit the relatively unperturbed study of equilib-... 

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... 

In addition to [A ] being qiiasi-stationary the quasi-equilibrium, approximation assumes a virtually unperturbed equilibrium between activation and deactivation (equation (A3.4.125)) ... 

When relaxation of the internal motion is slow compared with the fast relative speed v., then T is expanded in temrs of the known unperturbed (diabatic) ortironomral eigenstates j(V> ... 

This method [ ] uses the single-configuration SCF process to detennine a set of orbitals ( ).]. Then, using an unperturbed Flamiltonian equal to the sum of the electrons Fock operators // = 2 perturbation... 

Optical detectors can routinely measure only intensities (proportional to the square of the electric field), whether of optical pulses, CW beams or quasi-CW beams the latter signifying conditions where the pulse train has an interval between pulses which is much shorter than the response time of the detector. It is clear that experiments must be designed in such a way that pump-induced changes in the sample cause changes in the intensify of the probe pulse or beam. It may happen, for example, that the absorjDtion coefficient of the sample is affected by the pump pulse. In other words, due to the pump pulse the transparency of the sample becomes larger or smaller compared with the unperturbed sample. Let us stress that even when the optical density (OD) of the sample is large, let us say OD 1, and the pump-induced change is relatively weak, say 10 , it is the latter that carries positive infonnation. 

The gates referred to above can be created in various ways. For example, suppose that the probe beam goes tlirough the sample, but only half of its physical width (in the sample) is crossed with the pump beam. Now, if we have two photodiodes, one can measure the intensify of the perturbed part of the probe beam, whilst the second measures the unperturbed part as a result of creating spatial gates, the two recorded output signals can be used to measure the... 

Many experimental techniques now provide details of dynamical events on short timescales. Time-dependent theory, such as END, offer the capabilities to obtain information about the details of the transition from initial-to-final states in reactive processes. The assumptions of time-dependent perturbation theory coupled with Fermi s Golden Rule, namely, that there are well-defined (unperturbed) initial and final states and that these are occupied for times, which are long compared to the transition time, no longer necessarily apply. Therefore, truly dynamical methods become very appealing and the results from such theoretical methods can be shown as movies or time lapse photography. 

Vo + V2 and = Vo — 2 (actually, effective operators acting onto functions of p and < )), conesponding to the zeroth-order vibronic functions of the form cos(0 —4>) and sin(0 —(()), respectively. PL-H computed the vibronic spectrum of NH2 by carrying out some additional transformations (they found it to be convenient to take the unperturbed situation to be one in which the bending potential coincided with that of the upper electi onic state, which was supposed to be linear) and simplifications (the potential curve for the lower adiabatic electi onic state was assumed to be of quartic order in p, the vibronic wave functions for the upper electronic state were assumed to be represented by sums and differences of pairs of the basis functions with the same quantum number u and / = A) to keep the problem tiactable by means of simple perturbation... 

The first-order energy correction with respect to the unperturbed problem is then... 

Next, we analyze the P-curl condition with the aim of examining to what extent it is affected when the weak coupling is ignored as described in Section IV.B.l [81]. For this purpose, we consider two components of the (unperturbed) X matrix, namely, the mafiices Xq and Xp, which are written in the following form [see Eq. (43)] ... 

Thus if Tjj+i—the full non-adiabatic coupling term—and the unperturbed non-adiabatic coupling term, xoj,+i, are assumed to be related to each other as... 

In the unperturbed case, say 7 = 70, let the system have two disjoint invariant sets Bi and B2 associated with two eigenmodes and invariant inea-... 

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

The zeroth-order wavefunetions and energies are given in terms of the solutions of the unperturbed problem as follows ... 

This simply means that one must be willing to identify one of the unperturbed states as the best approximation to the state being sought. This, of eourse, implies that one must therefore strive to find an unperturbed model problem, eharaeterized by H that represents the true system as aeeurately as possible, so that one of the Ok will be as elose as possible to /k. 

The first-order energy eorreetion is given in terms of the zeroth-order (i.e., unperturbed) wavefunetion as ... 

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. 

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