Density matrix formalism

NMR signals are the response of a quantum mechanic system, the spin systems, to a sequence of rf pulses. Since the recorded signal is only the macroscopic expectation value of an observable quantity, knowledge of the quantum mechanical background is necessary for a complete understanding of NMR. To study the overall effect of a pulse sequences it is necessary to understand how the spin systems behave under the influence 

The spin system, the source of the NMR signal, consists of a multitude of spins and in a pure state each spin state can be described as the superposition of n wave functions whose contributions are scaled by  

Thus the expectation value A of a macroscopic observable quantity A, such as the transverse coherence, is given by 1 2-2] 

For a given basis set, i.e. a set of defined wavefunctions, Amn are constants and the product c Cn determines the observable quantity A for a particular state. Both Aj n and CmCn represent a NxN matrix. The product of the coefficients Cj c can be represented 

For a macroscopic sample it is necessary to define a different set of wave functions because the spins are in a mixed state. The mixed state indicates that the wave functions of a particular nuclear spin are subject to additional molecular contributions that might differ over the whole sample. The expectation value of a mixed state now uses the averaged coefficients and is 

---

It is clear that the density matrix formalism renders a considerable simplification of the basis for the quantum theory of many-particle systems. It emphasizes points of essential physical and chemical interests, and it avoids more artificial or conventional ideas, as for instance different types of basic orbitals. The question is, however, whether this formalism can be separated from the wave function idea itself as a fundament. Research on this point is in progress, and one can expect some interesting results within the next few years. 

When other relaxation mechanisms are involved, such as chemical-shift anisotropy or spin-rotation interactions, they cannot be separated by application of the foregoing relaxation theory. Then, the full density-matrix formalism should be employed. 

For in-situ studies of reaction mechanisms using parahydrogen it is desirable to compare experimentally recorded NMR spectra with those expected theoretically. Likewise, it is advantageous to know, how the individual intensities of the intermediates and reaction products depend on time. For this purpose a computer simulation program DYPAS2 [45] has been developed, which is based on the density matrix formalism using superoperators, implemented under the C++ class library GAMMA. 

Nuclear spin relaxation is considered here using a semi-classical approach, i.e., the relaxing spin system is treated quantum mechanically, while the thermal bath or lattice is treated classically. Relaxation is a process by which a spin system is restored to its equilibrium state, and the return to equilibrium can be monitored by its relaxation rates, which determine how the NMR signals detected from the spin system evolve as a function of time. The Redfield relaxation theory36 based on a density matrix formalism can provide... 

Wangsness and Bloch16>17 were the first to give a quantum mechanical treatment of spin relaxation using the density matrix formalism. The system considered is a spin interacting with an external magnetic field (which we suppose here to be constant) and with a heat bath. The corresponding Hamiltonian is... 

Separability can be exploited even with admission of relativistic effects, by using the standard density matrix formalism with a simple extension to admit 4-component Dirac spin-orbitals this opens up the possibility of performing ab initio calculations, with extensive d, on systems containing heavy atoms. 

A formal expression for the resonant nonlinear susceptibility can be obtained by describing the light-matter interactions in a density matrix formalism (Boyd 2003 Mukamel 1995), which is beyond the scope of this chapter. A third-order perturbative expansion of the system s density matrix yields the following form for the nonlinear susceptibility ... 

Now we generalize this density matrix formalism [19,20] for the degenerate Hubbard Hamiltonian which, with usual notations, reads ... 

The theoretical background has been treated by Mukamel [14] in detail. An approach to describe the experiments according to the density matrix formalism has been given by Dantus and Mukamel [15]. 

S. A. Rice I agree with Prof. Kohler that the use of a density matrix formalism by Wilson and co-workers generalizes the optimal control treatment based on wave functions so that it can be applied to, for example, a thermal ensemble of initial states. All of the applications of that formalism I have seen are based on perturbation theory, which is less general than the optimal control scheme that has been developed by Kosloff, Rice, et al. and by Rabitz et al. Incidentally, the use of perturbation theory is not to be despised. Brumer and Shapiro have shown that the perturbation theory results can be used up to 20% product yield. Moreover, from the point of view of generating an optimal control held, the perturbation theory result can be used as a first guess, for which purpose it is very good. 

A two-proton exchange model using a density matrix formalism has been used to analyze the unusual structure of the PMR spectra in the diamagnetic Ni(R2Dtc)2 complexes (R = n-Pr, Et,/-Pr,/-Bu, and Bz) (273). 

The coherence transfer provides cross peaks which are antiphase for the various 7//-split components. The antiphase nature of the cross peaks then leads to partial or total cancellation of the cross peaks themselves, especially if they are phased in the absorption mode. This behavior can be simulated (Fig. 8.15) using appropriate treatments of the time evolution of the spin system, for instance using the density matrix formalism [17,18]. It is quite common that signals in paramagnetic systems... 

A quantum mechanical treatment combined with the density matrix formalism extends the description to include the dynamic spectra of spin-coupled systems. (34-38) Further developments in the theory and presentation thereof, in a form suitable for computer calculations, are due to Binsch et al., and to Kaplan et al. (14, 15, 39) However, even the recent theories are not rigorous in certain aspects and contain some errors. This is particularly true in the case of the intermolecular exchange of spins. 

It can be seen that, in the average density matrix formalism which is based on spin system model, the scalar couplings and the exchange processes are handled simultaneously. Thus they cannot be separated and a larger atomic basis (spin system) is required for their description. Meanwhile, the Monte Carlo method based on spin sets separates the two interactions, and thus spin systems can be reduced to smaller spin sets. 

From the physical point of view, we are representing a many-electron state by an antisymmetrized product of one-electron states. The density matrix formalism [4,11-13] allows one to analyse in the same footing calculations resulting from different levels of approximation. The density matrix is called reduced when is formed from a pure state ... 

Since it is the dynamics of the system that is of interest, it would be convenient to preaverage over the environment variables and obtain an equation of motion for ps(t), the system component of the density matrix. Formal work of this kind [161,. .. 162] yields the so-called generalized master equation. Deriving the generalized f master equation, and extracting the various approximations utilized, goes well "astray of the central focus of this book. For this reason we just sketch the models id direct the reader to suitable review articles [161, 162] that provide an appropriate pview. 

We assume that the molecule is in a stationary state initially, the wave function of which is describable by HF. In the density matrix formalism [9, 10] (which is equivalent to the usual operator form), the Fock F(0) and density matrices D(Cl> satisfy the time-independent equation... 

Particularly lucid explanations of many fundamental aspects of NMR theory are given in the third edition of Principles of Magnetic Resonance by C. P. Slichter.30 This book is directed toward physicists and assumes that the reader has a good mathematical background and a grounding in the density matrix formalism of the sort we provide in Chapter 11. However, it also includes among the equations excellent qualitative discussions of various topics. 

I think there is still a gap between several very good introductory books that take a rather empirical approach to NMR and books written for more sophisticated users. In particular, product operator and density matrix formalisms are virtually ignored in the more elementary books but are treated as almost self-... 

The general quantum chemical description of the nonlinear susceptibilities and hyperpolarizabilities in the density matrix formalism was developed by Bloembergen and Shen [36]. A simplification of this model for dipolar organic molecules, by only considering the transition between the ground state and the first excited state, led to the well-known two-state expression for the first hyperpolarizability [37] (Eq. (23)). 

Recently, it was theoretically shown, on the basis of standard density-matrix formalism of nonlinear optics, that chiral isotropic media can possess an electrooptic response [174]. Such materials would be inherently stable and could therefore be extremely useful for the development of electro-optic devices. Contrary to usual electro-optic materials, index (absorption) modulation in such media is due to the imaginary (real) part of the electro-optic susceptibility. The response relies on the damping of the material response. 

Besides aetive research he very much enjoys teaching. In the Physical-Teehnieal Institute of Moseow he taught (1966-1992) general courses on Molecular Dynamics and Chemical Kinetics. In the Teehnion (since 1992) he has taught and still teaehes graduate eourses on different subjeets Advanced Quantum Chemistry, Theory of Moleeular Collisions, Kinetie Proeesses in Gases and Plasma, Theory of Fluetuations, Density Matrix Formalisms in Chemical Physics etc. 

To calculate the nutation spectrum the density matrix formalism is used, in which the Hamiltonian is diagonalised either numerically (Kentgens et al. 1987) or analytically (Pandey et al. 1986, Janssen and Veeman 1988). 

Specific aspects of the quantum chemical concept of local electron densities and functional groups of chemistry have been discussed, with emphasis on the Additive Fuzzy Density Fragmentation (AFDF) Principle, on the Adjustable Density Matrix Assembler (ADMA) Method of using a local density matrix formalism of fuzzy electron density fragments in macromolecular quantum chemistry, and on the fundamental roles of the holographic electron density theorem, local symmetry, and symmetry deficiency. 

Since we are primarily interested in the conditional quantum state of the atomic ensemble, we can use the density-matrix formalism to calculate the fidelity with which we create the desired atomic state, Ins ) ... 

The non-linear susceptibilities of the expansion (2) can be evaluated by means of the density matrix formalism. Recall that the non-linear polarization of a non-polar medium is equal to the induced dipole moment of a unit volume which, in turn, can be easily represented by standard methodology of quantum mechanics as an expectation value of the dipole moment. After some straightforward derivations we get ... 

Baev et al. review a theoretical framework which can be useful for simulations, design and characterization of multi-photon absorption-based materials which are useful for optical applications. This methodology involves quantum chemistry techniques, for the computation of electronic properties and cross-sections, as well as classical Maxwell s theory in order to study the interaction of electromagnetic fields with matter and the related properties. The authors note that their dynamical method, which is based on the density matrix formalism, can be useful for both fundamental and applied problems of non-linear optics (e.g. self-focusing, white light generation etc). 

An example The time evolution of a two-level system in the density matrix formalism... 

Time evolution of two-level system in density matrix formalism The Hamiltonian is taken in the form... 

---