Discrete variable representation formulation

In the basis set formulation, we need to evaluate matrix elements over the G-H basis functions. We can avoid this by introducing a discrete variable representation method. We can obtain the DVR expressions by expanding the time-dependent amplitudes a (t) in the following manner ... 

Discrete Fourier transform (DFT), non-adiabatic coupling, Longuet-Higgins phase-based treatment, two-dimensional two-surface system, scattering calculation, 153-155 Discrete variable representation (DVR) direct molecular dynamics, nuclear motion Schrodinger equation, 364-373 non-adiabatic coupling, quantum dressed classical mechanics, 177-183 formulation, 181-183... 

Another way to evaluate the expressions appearing in Eqs. (34 and 37) as well as related partial sums is closely related to the discrete variable representation of reaction probabilities as formulated by Seideman and Miller [148-150]. We have already seen that the sum... 

The format that has proved most useful, and the focus of this chapter, is numerical representation—formulation of the problem in terms of numerically valued decision variables. Discrete decision options are encoded as specific numerical variable values. For example, variable yj = may mean plant j is selected for construction and yj = 0 that it is not. 

The main challenge in short-term scheduling emanates from time domain representation, which eventually influences the number of binary variables and accuracy of the model. Contrary to continuous-time formulations, discrete-time formulations tend to be inaccurate and result in an explosive binary dimension. This justifies recent efforts in developing continuous-time models that are amenable to industrial size problems. 

The expressions eqs. (1.197), (1.199), (1.200), (1.201) are completely general. From them it is clear that the reduced density matrices are much more economical tools for representing the electronic structure than the wave functions. The two-electron density (more demanding quantity of the two) depends only on two pairs of electronic variables (either continuous or discrete) instead of N electronic variables required by the wave function representation. The one-electron density is even simpler since it depends only on one pair of such coordinates. That means that in the density matrix representation only about (2M)4 numbers are necessary to describe the system (in fact - less due to antisymmetry), whereas the description in terms of the wave function requires, as we know n 2m-n) numbers (FCI expansion amplitudes). However, the density matrices are rarely used directly in quantum chemistry procedures. The reason is the serious problem which appears when one is trying to construct the adequate representation for the left hand sides of the above definitions without addressing any wave functions in the right hand sides. This is known as the (V-representability problem, unsolved until now [51] for the two-electron density matrices. The second is that the symmetry conditions for the electronic states are much easier formulated and controlled in terms of the wave functions (Density matrices are the entities of the second power with respect to the wave functions so their symmetries are described by the second tensor powers of those of the wave functions). 

A simple estimate of the computational difficulties involved with the customary quantum mechanical approach to the many-electron problem illustrates vividly the point [255]. Consider a real-space representation of ( ii 2, , at) on a mesh in which each coordinate is discretized by using 20 mesh points (which is not very much). For N electrons, becomes a variable of 3N coordinates (ignoring spin), and 20 values are required to describe on the mesh. The density n(r) is a function of three coordinates and requires only 20 values on the same mesh. Cl and the Kohn-Sham formulation of DFT (see below) additionally employ sets of single-particle orbitals. N such orbitals, used to build the density, require 20 values on the same mesh. (A Cl calculation employs in addition unoccupied orbitals and requires more values.) For = 10 electrons, the many-body wave function thus requires 20 °/20 10 times more storage space than the density and sets of single-particle orbitals 20 °/10x 20 10 times more. Clever use of symmetries can reduce these ratios, but the full many-body wave function remains inaccessible for real systems with more than a few electrons. 

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