Orthogonal collective modes

Motivated by the optimized planar dividing surface VTST presented in the previous section (81), it is natural to define the two orthogonal collective modes p, cr as... 

Another form of array is called a microchannel plate detector. A time-of-flight (TOP) mass spectrometer collects ions sequentially in time and can use a point detector, but increasingly, the TOP instrument uses a microchannel plate, most particularly in an orthogonal TOP mode. Because the arrays and microchannel plates are both essentially arrays or assemblies of small electron multipliers, there may be confusion over their roles. This chapter illustrates the differences between the two arrays. 

The parabolic barrier case demonstrates that the effect of the medium is to replace the original reaction coordinate q by a collective mode p along which the dynamics is trivial. It is useful to define a collective bath mode o orthogonal to the unstable mode p as... 

In principle one should also add the constraint that the collective modes are orthogonal (Eq. 113) however, minimizing the negative eigenvalue in an unconstrained form automatically leads to orthogonality. 

This concept assumes that each fraction (peak) collected in the first dimension further separates in the second dimension with regular spacing and that the entire 2D separation space is evenly covered by eluting peaks. More realistically, the peaks would be distributed randomly over the 2D separation space some peaks are likely to coelute, while some area will remain vacant of peaks. Therefore, Equationl2.1 represents an idealized peak capacity estimate although the real number of resolved peaks is lower. Most importantly, the peak capacity proposed by Equation 12.1 is achievable when the chromatographic modes used for separation are completely orthogonal. 

The proposed estimate has several limitations. When taking into account the limited orthogonality of investigated 2DLC modes, the practical peak capacity is reduced approximately to half. It needs to be also emphasized that a full separation power of the first LC dimension is realized only when the number of collected fractions exceeds its peak capacity (Murphy et al., 1998). If the number of fractions analyzed is low, the achievable chromatographic peak capacity suffers. 

A Q-TOF spectrometer is similar to a triple quadrupole but Q3 is replaced by an orthogonal TOF mass spectrometer. Using a Q-TOF instrument only the product ion scan mode can be collected, but because of its high resolving power, accurate masses for both the precursor ion and product ions can be obtained. (See the section below on accurate mass measurements.)... 

A spectrum can only be phased when the data is collected in a phase-sensitive receiver detection mode. For a spectrum to be phase sensitive, we require either two orthogonal receiver channels digitizing... 

Various totally or partially decoupled forms of these equations are also of interest [33-36]. In the next section we consider the normal representation of the independent collective population displacement modes. Each new representation is associated with a particular orthogonal transformation, O, of the population variables, v(0) = NO. The diagonal elements of the associated "charge-and-bond-order matrix, P(0) = O NO, where Njj = NjSij, Prr(O) s p,(0), can be identified as the occupations of the new collective populational modes, which preserve the global number of electrons in the system ... 

Normal modes are a complete and orthogonal set of collective motions of the atoms in a vibrating molecule. 

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