Ensemble operator

A mixture of isomers may be represented by an ensemble operator , which is defined simply as a member a of the group algebra U of . 

Complex a(s) may be interpreted as representing two mixtures simultaneously, one corresponding to the real, one to the imaginary part. This will not cause any inconsistency with the operations we shall be performing with the ensemble operators. Alternatively, we could simply restrict ourselves to real a(s), as the real numbers suffice for the full reduction of representations of [Pg.48]

That a is nonracemic means that it does not identically annihilate all chirality functions, i.e., that at least one of the coefficients , with j different from zero. Qualitative completeness for % means, therefore, that it is not annihilated by any of the ety with j nor by any linear combination of them. This means that % must possess zr components belonging to each 71linearly independent, but also all functions ey must be linearly independent. For example, if zr= 2, it will not do to have xV —etx since in this case the chirality function % = x l + y(2 would be annihilated by the chiral ensemble operator a — e( l + —... 

One can also speak of a partition p being active or inactive with respect to an ensemble operator a. p is said to be a-active if there is some molecule M belonging to p such that the mixture aM is chiral. Expressing a as before in terms of the e-operators. 

Such higher order prerequisites could be fulfilled by ensemble operation of several sites. For example, a dimeric cluster of cuprous ions on silica gel is very active for the oxidation of CO with NzO at room temperature, but isolated cuprous ions are entirely inactive for this reaction 60). More interesting selectivity may be found in the reaction of olefins with methylene complexes the reaction of olefins with mononuclear methylene undergoes an olefin metathesis reaction, but the reaction of ethylene with bridging methylene in /i-CH2Co2(CO)2(Cp)2 61), /<-CH2Fe2(CO)8 (62), and /<-CH2-/i-ClTi(Cp)2Al(Me)2 (65) (Cp = cyclopentadiene) leads to propene formation (homologation reaction). 

Central to the development is an ensemble operator T akin to the one considered in the previous chapter and constructed from the appropriate system hamiltonian H of the reacting system and a bias operator B. A self-adjoint orthogonal operator B is constructed as the difference between a projector onto the product states and a projector Qr onto the reactant states, such that... 

Obviously it holds that B — 1. A suitable ensemble operator is... 

A stationary ensemble density distribution is constrained to be a functional of the constants of motion (globally conserved quantities). In particular, a simple choice is pip, q) = p (W (p, q)), where p (W) is some fiinctional (fiinction of a fiinction) of W. Any such fiinctional has a vanishing Poisson bracket (or a connnutator) with Wand is thus a stationary distribution. Its dependence on (p, q) through Hip, q) = E is expected to be reasonably smooth. Quanttun mechanically, p (W) is die density operator which has some fiinctional dependence on the Hamiltonian Wdepending on the ensemble. It is also nonnalized Trp = 1. The density matrix is the matrix representation of the density operator in some chosen representation of a complete orthononnal set of states. If the complete orthononnal set of eigenstates of die Hamiltonian is known ... 

A diagrannnatic approach that can unify the theory underlymg these many spectroscopies is presented. The most complete theoretical treatment is achieved by applying statistical quantum mechanics in the fonn of the time evolution of the light/matter density operator. (It is recoimnended that anyone interested in advanced study of this topic should familiarize themselves with density operator fonnalism [8, 9, 10, H and f2]. Most books on nonlinear optics [13,14, f5,16 and 17] and nonlinear optical spectroscopy [18,19] treat this in much detail.) Once the density operator is known at any time and position within a material, its matrix in the eigenstate basis set of the constituents (usually molecules) can be detennined. The ensemble averaged electrical polarization, P, is then obtained—tlie centrepiece of all spectroscopies based on the electric component of the EM field. 

Consider an ensemble composed of constituents (such as molecules) per unit volume. The (complex) density operator for this system is developed perturbatively in orders of the applied field, and at. sth order is given by The (complex). sth order contribution to the ensemble averaged polarization is given by the trace over the eigenstate basis of the constituents of the product of the dipole operator, N and = Tr A pp... 

NVT, and in die course of the simulation the volume V of the simulation box is allowed to vary, according to the new equations of motion. A usefid variant allows the simulation box to change shape as well as size [89, 90], It is also possible to extend the Liouville operator-splitting approach to generate algoritlnns for MD in these ensembles examples of explicit, reversible, integrators are given by Martyna et al [91],... 

Since the averaging operator is not normalized and in general (1), 1 for g 7 1, it is necessary to compute Zq to determine the average. To avoid this difficulty, we employ a different generalization of the canonical ensemble average... 

Here is the position operator of atom j, or, if the correlation function is calculated classically as in an MD simulation, is a position vector N is the number of scatterers (i.e., H atoms) and the angular brackets denote an ensemble average. Note that in Eq. (3) we left out a factor equal to the square of the scattering length. This is convenient in the case of a single dominant scatterer because it gives 7(Q, 0) = 1 and 6 u,c(Q, CO) normalized to unity. 

Consider the following operator, defined in any ensemble, and called the density operator ... 

It is worthwhile to consider the same theorem in terms of coordinate space instead of occupation number space. Thus, we may envision an ensemble of systems whose states are X>, and whose distribution probabilities among these states are w(X). We define the density operator... 

This is an example of Eq. (8-197). The ensemble average of positive spin represented by the operator az, or the Pauli spin matrix Q is quite trivially... 

Entropy and Equilibrium Ensembles.—If one can form an algebraic function of a linear operator L by means of a series of powers of L, then the eigenvalues of the operator so formed are the same algebraic function of the eigenvalues of L. Thus let us consider the operator IP, i.e., the statistical matrix, whose eigenvalues axe w ... 

For any linear operator 22 defined in Fock space, we can similarly prove, by following an argument like that leading to Eq. (8-189), that the trace in Fock space of WB is the grand-ensemble-average of 22 ... 

This is Bloch s equation. We note that this equation is essentially of the same form as Schrbdinger s equation, with playing the role of wave function, and ]8 playing the role of time, j8 (i]E)t, and the operator H — E representing deviation of the hamiltonian from the ensemble average. 

As a somewhat exceptional example of this we shall find the rate of change of the ensemble density operator, Eq. (8-186), which we now rewrite more explicitly to bring out its time-dependence ... 

The expectation value of the density operator, and, indeed, all the components of the density matrix, are stationary in time for an ensemble set up in terms of energy eigenstates. IT we use occupation number representation to set up the density matrix, it is at once seen from Eq. (8-187) that it also is independent of time ... 

Now we shall consider the time dependence of the ensemble average of any operator B not explicitly a function of time, Tr FVR. Because the trace is independent of the representation, we choose the one most convenient, which turns out to be the occupation number representation whose eigenvectors are eigenvectors of H. Thus we write... 

In other words, the rate of change of the ensemble average of any observable is zero, even when the operator does not commute with H, provided it is not explicitly a function of time, and provided we have set up the ensemble in terms of the energy eigenstates. 

Now let us use the set, <0> to form a matrix representation of some operator Q at time hi assuming that Q is not explicitly a function of time. The expectation value of Q in the various states, changes in time only by virtue of the time-dependence of the state vectors used in the representation. However, because this dependence is equivalent to a unitary transformation, the matrix at time t is derived from the matrix at time t0 by such a unitary transformation, and we know that this cannot change the trace of the matrix. Thus if Q — WXR our result entails that it is not possible to change the ensemble average of R, which is just the trace of Q. 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

All discussions of transport processes currently available in the literature are based on perturbation theory methods applied to kinetic pictures of micro-scattering processes within the macrosystem of interest. These methods do involve time-dependent hamiltonians in the sense that the interaction operates only during collisions, while the wave functions are known only before and after the collision. However these interactions are purely internal, and their time-dependence is essentially implicit the over-all hamiltonian of the entire system, such as the interaction term in Eq. (8-159) is not time-dependent, and such micro-scattering processes cannot lead to irreversible changes of thermodynamic (ensemble average) properties. 

Electron polarization operators, 539 Electron tube circuit, 373 Elias, P., 220 Elimination, Gaussian, 62 Elliot, N., 757 EUiot, R. J., 745 Energy operator, total, 541 Ensemble average, 2... 

Along the trajectory within the ensemble of A(t) realizations, any molecular operator X of a target molecule satisfies a SLE... 

The expectation values on the right hand side of this equation depend only on the ensemble averages of position and momentum operators, which can be evaluated using the VQRS Monte-Carlo sampling scheme outlined above. 

We finish this chapter by repeating some of the most important results. If we have detailed knowledge of the energy levels for an ensemble of particles (remember that statistical mechanics always operates on the basis of large numbers) it is possible to... 

A different mechanism seems to operate in the case of poison formation from methanol [Herrero et al., 1993]. In this case, modification of the Pt(lll) surface by Bi deposition only causes a linear decrease in the amount of poison formed, indicating the existence of a mere third-body effect. Complete inhibition of the poisoning reaction is achieved for > 0.23, i.e., before the surface is completely covered. This suggests the existence of ensemble requirements for this reaction, which need enough free contiguous Pt sites to take place. 

An exhaustive statistical description of living copolymers is provided in the literature [25]. There, proceeding from kinetic equations of the ideal model, the type of stochastic process which describes the probability measure on the set of macromolecules has been rigorously established. To the state Sa(x) of this process monomeric unit Ma corresponds formed at the instant r by addition of monomer Ma to the macroradical. To the statistical ensemble of macromolecules marked by the label x there corresponds a Markovian stochastic process with discrete time but with the set of transient states Sa(x) constituting continuum. Here the fundamental distinction from the Markov chain (where the number of states is discrete) is quite evident. The role of the probability transition matrix in characterizing this chain is now played by the integral operator kernel ... 

---