Two state approximation

The difficulty of evaluating the quantity in Eq. (17) is that it requires the time-dependent solution of the Schrodinger equation in which the Hamiltonian is a function of all the solvent, ion, and metal nuclei, as well as of the metal electrons. In the two-state approximation, the Hamiltonian can be written as... 

Sum Over States [3] (SOS) approaches constitute one of the most commonly used class of methodologies for theoretical estimation of hyperpolarizabilities. The strength of this approach is related to the fact that for many compounds of interest, only a few excited states make a major contribution. The simplest scheme, proposed by Oudar and Chemla [4—5] to analyze variations of p among push-pull conjugated materials, restricts the summation to a unique excited state. In this resulting two-state approximation (TS A), the static longitudinal electronic first hyperpolarizability, Pl, is given by ... 

Several such approaches for calculating electronic coupling matrix elements Vda have been proven to be useful. Most of them employ a two-state approximation [27, 28] where one assumes that donor and acceptor elec-... 

Guha, S. and Ghosh, A.S. (1981). Positron-lithium-atom collisions using the two-state approximation. Phys. Rev. A 23 743-750. 

The calculations have been performed using the Amsterdam Density Functional (ADF) program package [17-21] with the choice of the functionals described in Ref. [10]. We used a triple zeta basis in all geometry optimizations. For the chosen examples the two-state approximation is valid though using an extended basis set the LUMO is well separated from higher excited MOs of the same symmetry. 

Figure 3.22 Generic electron transfer (ET) system composed of local donor (D) and acceptor (A) sites, the intervening bridge (B), and the surrounding medium (or solvent). In the two-state approximation (TSA), the ET kinetics (e.g., for charge separation (CS) DBA-D+BA ) may be modeled in terms of initial (i) and final (f) states, in which the transferring charge is localized primarily on the D and A sites, respectively.

In connection with the two-state approximation we mention paper [107] giving detailed analysis of the Landau-Zener-Stueckelberg model in which transition probabilities have been calculated for a wide set of parameters of the model, and also papers devoted to extensions of the model [108-110]. The exponential model with the Hamiltonian given by equation (26) was discussed in [111] and extended in [112],... 

Equation [139] is exact for a two-state solute, but differs from the traditionally used connection between the transition dipole and the emission intensity by the factor Vo/Vav." The commonly used combination miiVo/Vav appears as a result of neglect of the frequency dependence of the transition dipole mi2(v) entering Eq. [129]. It can be associated with the condensed-phase transition dipole in the two-state approximation." Exact solution for a two-state solute makes the transition dipole between the adiabatic free energy surfaces inversely proportional to the energy gap between them. This dependence, however, is eliminated when the emission intensity is integrated with the factor... 

The transition dipole mn in Eqs. [136] and [139] is the gas-phase adiabatic transition dipole. Therefore, emission intensities measured in different solvents should generate invariant transition dipoles when treated according to Eqs. [136] and [139]. A deviation from invariance can be used as an indication of the breakdown of the two-state approximation and the existence of intensity borrowing from other excited states of the chromophores (the Murrell mechanism T88,94j Figure 16 illustrates the difference between Eq. [139] and... 

Table III. Parameters for the ionic-covalent interaction in the two-state approximation.

In the avoided crossing region the experimental RKR potential of the state and the "essentially experimental" potential of the state are used to determine the crossing distance R(- and the coupling matrix element T -. in the two state approximation. These quantities are relevant to the evaluation of the total charge transfer cross section at high energy (e.g. in the Landau-Zener model). 

In order to eliminate an unimportant global phase in our two-state approximation, it is convenient to represent the quantum evolution of the electronic wave function by Bloch variable. The Bloch variables x, y and z axe defined via the density matrix pmn — A,tiA according to... 

DFT calculations on model compounds 45-48, where hydrogen atoms replace the alkyl groups of 1,4-R2pipdt and 1,3-R2timdt (Figure 10), allowed elucidation of this behavior. Moreover, analysis of the calculated first molecular hyperpolarizabilities and their components with the two-state approximation accounts for the various NLO responses in terms of the important following components the transition dipole moment between the first excited and the ground states the difference between the dipole moment of the... 

An adequate account of many ET processes of interest is provided by the so-called two-state approximation (TSA), according to which the dynamics of ET is assumed to be adequately accounted for by the two-component space (denoted as the D/A space) spanned by states in which the transferring electron is primarily confined to the D and A sites [6, 60]. (When one or both of the two states are degenerate or near-degenerate, a somewhat more elaborate two-level model must be adopted [60].) In general, the two-state approximation is only valid if the gap... 

A convenient point of departure is provided by the golden-rule non-adiabatic rate-constant expression [8. 60] appropriate under the usual two-state approximation (TSA),... 

While the two-state approximation (TSA) introduced in Section 1.3.1 accounts well for many classes of electron-transfer kinetics, there are, of course, situations in which a high density of electronic states in the initial- and final-state manifolds makes it necessary to generalize the TSA expressions given, for example, by Eqs. 31-37. A paramount example is the case of metal or semiconductor electrodes, where one must deal essentially with an electronic continuum [25, 31, 32, 106]. In spite of this complication, one may still obtain expressions with similar form to those shown above when reaction exothermicity is small (i.e., the difference between the electrode Fermi level and the standard potential of the redox species is small compared to /) [25b]. Nevertheless, in the inverted region , at electrodes is generally observed to approach a constant maximum value with increasing driving force (for an exception, see [107]), in contrast to the fall-off predicted in the case of the TSA (see Eq. 27). 

In the case of thermal ET, which applies either to ground state ET or photo-intitiated ET from a vibrationally relaxed excited state (cf, [15d]), we now define the effective coupling T,f by employing the two-state approximation (TSA) introduced in Eqs. 14-15 ... 

For a single particle in a symmetrical double minimum potential the ground state splits into two sublevels, symmetric 0+) and antisymmetric 0—), respectively. The tunnel splitting is /wj — E0+ = hioo . If the potential barrier is sufficiently high, the wave functions can be written within the two-state approximation as symmetrical and antisymmetrical combinations of harmonic wave functions centered at the potential minima op... 

Applied to the diagonal components of the linear polarizability the two-states approximation of Eq. (43) is... 

Figure 5. Evolution of non-linear optical properties (jS, y and <5) within two-state approximation as a function of k parameter...

It has been shown in a few theoretical studies drat the TPA cross section S of the donor-acceptor TT-conjugated molecules can be correctly described, similarly as the molecular (hyper)polarizabilities by a simple two-state model, involving only the ground and the CT excited state [111-114]. Hence, the solvent effect on S can be discussed within this approximation. It leads to the better understanding of the theoretical results cited in the above papers. Including only the ground and the CT excited state in the Eq. (8), one obtains the two-state approximate equation for dominant component along the molecular axis (in this case chosen as x axis) ... 

The understanding and reliable prediction of the influence of the solute-solvent interactions on the nonlinear optical properties of molecular systems is a significant issue for a width range of theoretical and experimental areas of studies. In this review, it was shown that the simple two-state approximations combined with tlie solvatochromic methods are an effective tools in prediction tlie direction of tlie changes of molecular nonlinear responses as a function of solvent polarity. This methodology based on the description of the solvent effects at the molecular level should be treated as a supporting for the most sophisticated quantum chemical approaches. 

Including only the ground and the CT excited state in the sum-over states (SOS) expressions for the static a and y, one obtains the two-state approximate... 

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