Diagrammatic techniques

Various one-dimensional processes can be expressed by conneeting these diagrams and can be described by combining the appropriate semiclassical matrices. This technique is called diagrammatic technique [48,52]. When we write the semiclassical wave function on the adiabatic potential E x) with a as a reference point as 

V is a column vector with components V and V , and U is the similar vector that represents the wave function with as a reference point. 

FIGURE 2.2 Diagram for wave reflection at the turning point J. 

FIGURE 2.3 Diagram for potential barrier transmission and reflection. 

Here we have used the comparison equation method for the case of quadratic potential barrier. This transfer matrix M is the same as Equation (2.24), and ij) and e are given by Equations (2.25) and (2.69). 

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The connected stmcture of the CSE has also been explored by Yasuda [23] using Grassmann algebra, by Kutzelnigg and Mukheijee [27] using a cumulant version of second-quantized operators, and by Herbert and Harriman [30] using a diagrammatic technique. 

In recent years the diagrammatic technique of the perturbation theory found wide application in solving the stochastic differential equations, e.g., see a review article by Mikhailov and Uporov [68]. 

The presented form of the master equation (2.3.67) permits us to employ the diagrammatic technique of the perturbation theory [44, 108-110, 113]. The free Hamiltonian could be written as... 

A procedure similar to the condensate separation in the imperfect Bose gas was employed by Lifshitz and Pitaevski [78]. The diagrammatic technique allows us to calculate the reaction rate and steady-state joint correlation functions. A separation of a condensate from terms with k = 0 cannot be done without particle production (p = 0), in which case nA, tiq —> 0 as t —> oo. In this respect the formalism presented by Lushnikov [111] for the non-stationary processes is of certain interest. 

More recently, Caves and Karplus71 have used diagrammatic techniques to investigate Hartree-Fock perturbation theory. They developed a double perturbation expansion in the perturbing field and the difference between the true electron repulsion potential and the Hartree-Fock potential, V. This is compared with a solution of the coupled Hartree-Fock equations. In their interesting analysis they show that the CPHF equations include all terms first order in V and some types of terms up to infinite order. They propose an alternative iteration procedure which sums an additional set of diagrams and thus should give results more accurate than the CPHF scheme. Calculations on Ha and Be confirmed these conclusions. 

To describe consistently cotunneling, level broadening and higher-order (in tunneling) processes, more sophisticated methods to calculate the reduced density matrix were developed, based on the Liouville - von Neumann equation [186-193] or real-time diagrammatic technique [194-201]. Different approaches were reviewed recently in Ref. [202]. 

One can ask what is the advantage to use the more complex two-time Green functions instead of density matrices There are several reasons. First of all, NGF give, as we shall see below, a clear description of both density of states and distribution of particles over this states. Then, the equations of motion including interactions and the influence of environment can be obtained with the help of a diagrammatic technique, and (very important) all diagrammatic results of equilibrium theory can be easily incorporated. Retardation effects are conveniently taken into account by two-time Green functions. And,. .. finally, one can always go back to the density matrix when necessary. 

The name lesser originates from the time-ordered Green function, the main function in equilibrium theory, which can be calculated by diagrammatic technique... 

Now we are able to define contour or contour-ordered Green function - the useful tool of Keldysh diagrammatic technique. The definition is similar to the previous one... 

As in the ordinary diagrammatic technique, the important role is played by the integration (summation) over space and contour-time arguments of Green functions, which is denoted as... 

The aim of this section is to familiarize the reader with the second quantization and the many-body diagrammatic techniques which are now widely used in up-to-date quantum chemistry. These techniques are very efficient since they permit the formulation of the problem by means of diagrams from which the explicit formula can be obtained. Another advantage is that the problem of spin can be handled very simply. This approach also permits us to have a microscopic view of the problem (as will be seen in the study of ionization potentials, excitation energies, interaction of two molecular systems etc.). 

The whole problem of calculating fc/ (at least up to the third order) is now reduced to the calculation of individual terms (55)—(58). The second quantization formalism has the advantage that these terms can be calculated easily by making use of the diagrammatic technique which will be demonstrated in Section IV.B. 

Coulson181 thought that it would take 15 years for the impact of diagrammatic techniques to be fully realized in theoretical chemistry. It is therefore not surprising that the first half of this period has been devoted mainly to the development of new methods and algorithms. Although this development will undoubtedly continue, it is clear that, armed with these new techniques, theoretical chemists will be able to attack problems with an accuracy which was not previously attainable. For example, it will be possible to calculate rotation constants for small molecules more precisely. Although the accuracy of calculated rotation... 

In this section we construct working equations for the coupled cluster singles and doubles (CCSD) method. Beginning from the approximation 7 = 7 + T2, we use algebraic and diagrammatic techniques to obtain programmable... 

Recognition of this relationship between coupled cluster theory and MBPT has inspired research efforts to construct perturbation-based corrections to the CCSD energy to account for higher excitation contributions. Undoubtedly, the most successful and popular of these is the (T) correction first described for closed-shell molecular systems by Raghavachari et al. " In the next section, we will describe the structure of this correction using diagrammatic techniques. 

Diagrammatic techniques also provide a route to the construction of efficient coupled cluster intermediates. " " Kucharski and Bartlett, " for example, described a particularly clever approach by which one uses matrix elements of the similarity transformed Hamiltonian as the desired intermediates. Consider the matrix element of H between the reference (on the left) and a singly excited determinant (on the right). Diagrammatically, this matrix element is resolved into two terms as... 

IV. Density Matrix Approach, the Diagrammatic Technique, and the Microscopic Expressions for Wp (r) and... 

The nonlinear interaction of light with matter is useful both as an optical method for generating new radiation fields and as a spectroscopic means for probing the quantum-mechanical structure of molecules [1-5]. Light-matter interactions can be formally classified [5,6] as either active or passive processes and for electric field based interactions with ordinary molecules (electric dipole approximation), both may be described in terms of the familiar nonlinear electrical susceptibilities. The nonlinear electrical susceptibility represents the material response to incident CW radiation and its microscopic quantum-mechanical formalism can be found directly by diagrammatic techniques based on the perturbative density matrix approach including dephasing effects in their fast-modulation limit [7]. Since time-independent (DC) fields can only induce a... 

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