Secular approximation

In the secular approximation [89], we can eliminate the coherence terms [e.g., pr, (x)(u / u")] in Eq. (III.9) such that the only diagonal terms contribute to the vibrational transitions through which the vibrational populations in various states are coupled. By applying the ladder model [89] to the interaction between the vibrational and heat-bath modes, the vibrational population decay constant is expressed as... 

In the secular approximation and with the ladder model, the time evolution of the vibrational coherence, pw(x), is determined by... 

The symbol (Oaa denotes the energy difference between the two eigenstates, converted into angular frequency. The first term on the right-hand side (rhs) of Eq. (18) vanishes for the populations (oo a = 0) and describes the preces-sional motion for coherences. Rota pp is an element of the relaxation matrix (also called relaxation supermatrix) describing various decay and transfer processes in the spin system. Under certain conditions (secular approximation), one neglects the relaxation matrix elements unless the condition = pp is fulfilled. 

To account for the radiative decay of CC excited states we consider the density operator p, Eq. (35), reduced to the CC solvent states. It is a standard task of dissipative quantum dynamics to derive an equation of motion for p with a second order account for the CC-photon coupling, Eq. (24) (see, for example, [40]). Focusing on the excited CC-state contribution, in the most simple case (Markov and secular approximation) we expect the following equation of motion... 

Fig. 3 Absorption spectra of one sample of B850 calculated with different methods for an Ohmic spectral density with / = 3.37 and usc = 0.027 eV. Left Methods with Markov approximation, i.e. Redfield theory with and without secular approximation and TL method. Right TL with and without Markov approximation, TNL and modified Redfield method. (Reproduced from Ref. [37]. Copyright 2006, American Institute of Physics.)...

It has been found that the short-range interaction model can be applied to study the vibrational relaxation of molecules in condensed phases. This model is applied to treat vibrational relaxation and pure dephasing in condensed phases. For this purpose, the secular approximation is employed to Eq. (129). This assumption allows one to focus on several important system-heat bath induced processes such as the vibrational population transition processes, the vibrational coherence transfer processes, and the vibronic processes. 

Except for some quadrupolar effects, all the interactions mentioned are small compared with the Zeeman interaction between the nuclear spin and the applied magnetic field, which was discussed in detail in Chapter 2. Under these circumstances, the interaction may be treated as a perturbation, and the first-order modifications to energy levels then arise only from terms in the Hamiltonian that commute with the Zeeman Hamiltonian. This portion of the interaction Hamiltonian is often called the secular part of the Hamiltonian, and the Hamiltonian is said to be truncated when nonsecular terms are dropped. This secular approximation often simplifies calculations and is an excellent approximation except for large quadrupolar interactions, where second-order terms become important. 

If the molecule is in a solid, tr can also be described in terms of Cartesian space-fixed (laboratory) coordinates in which B0 is normally taken along the z axis, and it is the shielding along this axis that alters the resonance frequency. Within the secular approximation, it is only cra that contributes, and it may be related to the three principal components via their direction cosines relative to B0 ... 

For an 5 = 1 spin we now consider the presence of a zero-field term along with an axially symmetric g tensor leading in the secular approximation to the following Hamiltonian ... 

The Hamiltonian describing this spin system in the secular approximation is given by Eq. (3.25) of Section III.B. The evolution equation of the operator S+ is given by... 

The expression of the spin Hamiltonian, in the secular approximation, is given by... 

We find the density matrix elements from the master equation of the system. In the frame rotating with the laser frequency a>L and within a secular approximation, in which we ignore all terms oscillating with (colrf - a>L) and ((]>2d — the master equation for the density operator of the system is given by... 

The smallest distance at which the atoms could experience opposite phases corresponds to rj2 = A-o/2. However, at this particular separation the dipole-dipole interaction parameter Hi2 is small (see Fig. 1), and then all of the transitions between the collective states occur at approximately the same frequency. In this case the secular approximation is not valid, and we cannot separate the transitions at AL + H 2 from the transitions at AL — fii2. 

Equations (120) contain time-dependent terms that oscillate at frequencies exp( /Af) and exp[ 2/((05 — ffiojt + <))]. If we tune the squeezed vacuum field to the middle of the frequency difference between the atomic frequencies, namely, v — oo0)t + < )] become stationary in time. None of the other time-dependent components is resonant with the frequency of the squeezed vacuum field. Consequently, for A > f, the time-dependent components oscillate rapidly in time and average to zero over long times. Therefore, we can formulate a secular approximation in which we ignore the rapidly oscillating terms, and find that Eqs. (120) give us the following steady-state solutions [64] ... 

In most applications of the theory to date, the solution of the Redfield equation has required first the explicit calculation of the Redfield tensor elements [Eq. (11)] given these, Eq. (10) could be solved as an ordinary set of linear differential equations with constant coefficients, either by explicit time stepping [41, 42] or by diagonalization of the Redfield tensor [37,38]. Since there are such tensor elements for an A -state subsystem, the number of these quantities can become quite large. Because of this, until recently most applications of Redfield theory have been limited to small systems of two to four states, or else assumptions, such as the secular approximation, have been used to neglect large classes of tensor elements. 

The distortions in the measured cross-correlated relaxation rates due to violations of secular approximation and differential effects of the non-symmetri-cal coherence transfer periods flanking the relaxation measurement delay can be minimised with the symmetrical reconversion approach introduced in the previous review period. In this approach four experiments are recorded with all combinations of the coherence transfer periods, producing automatic correction of the measured relaxation rate. The method was applied to the measurement of cross-correlated relaxation between CO CSA and long-range CO-HA DD interactions that depends on the backbone angle ip. The cross-correlated rate is evaluated from the relaxation of 2C yNz and 4H zC yNz coherences, recorded separately. The sequence is based on HNCO and HN(CO)CA experiments. The rates measured for ubiquitin show good correlation with the theoretical values. 

The last step in the equation above is known as secular approximation and it is a generally appropriate simplification valid as consequence of the much larger magnitude of the Zeeman interaction with the external magnetic field as compared to the chemical shift one [5]. 

This expression can be rearranged by grouping the terms involving each combination of the Cartesian components of the operators Ii and I2, associated with the two spins. When this is done, only the dominant terms are retained in the secular approximation. For the heteromclear case (i.e., the interaction between two unlike nuclei with ym Yn2) this leads to the simple expression ... 

For application of the perturbation theory, it is necessary to transform this expression to the laboratory frame, where the dominant Hamiltonian Hz is proportional do the spin operator 7. When such transformation is conducted, using Wigner matrices and Euler angles [10], the resulting form for Hq in the laboratory frame, assuming axial symmetry for the EFG tensor (q = 0) and using the secular approximation to keep only terms that commute with Hz, is ... 

Because of the large heat capacity of the lattice relative to that of the nuclear spins, the lattice may be considered at all times to be in thermal equilibrium, while the time-varying spin states, in the absence of a r.f. field, evolve to thermal equilibrium because of the spin-lattice interactions. When the exponential argument (/ —/ —a +a) in Eq. (5.21) is significantly larger than the spin relaxation rates, the exponential term oscillates rapidly in comparison with the slow variation in the density matrix due to relaxation. As a consequence, the impact of these terms becomes zero. The so-called secular approximation ua a = /3 /3) effectively simplifies the equation of motion to... 

In a high magnetic field, the hyperfine Hamiltonian can be truncated after the dominant Zeeman interaction with the magnetic field (secular approximation). Only the terms in 7 are thus retained, and the spin-operators reduce to... 

---