Sinc-function discrete variable representation

The Chebyshev expansion method [430] is applied to the propagation of the nuclear wavepacket subject to the above Hamiltonian for both the two-and three-state models. Nuclear wavefimctions, the potential functions, the matrix to represent the first and second nuclear derivatives arising from T of Eq. (6.70) are all expressed in the sine discrete variable representation (sinc-DVR) [92]. The time length for one-step integration of nuclear wavepacket is set to 0.02 fs. The 1200 DVR grid points are employed within a range from —3 to 14 Bohrs. For a practical reason, the potential function is cut off in the range shorter than 1.2 Bohrs. 

Forming the matrix representation of the Hamiltonian operator and manipulating the Hamiltonian matrix to obtain the observable of interest can be computationally intensive. A discrete variable representation [53-55] (DVR) can ameliorate both of these difficulties. That is, the construction of the Hamiltonian matrix is particularly simple in a DVR because no multidimensional integrals involving the potential function are required. Also, the resulting matrix is sparse because the potential is diagonal, which expedites an iterative solution [37, 38]. In the present research we use a sinc-function based DVR (vide infra) first developed by Colbert and Miller [56] for use in the 5—matrix version of the Kohn variational principle [57, 58], and used subsequently for 5—matrix calculations [37, 38] in addition to N(E) calculations [23]. This is a uniform grid DVR which is constructed from an infinite set of points. It is... 

This equation gives the dynamics of the quantum-classical system in terms of phase space variables (R, P) for the bath and the Wigner transform variables (r,p) for the quantum subsystem. This equation cannot be simulated easily but can be used when a representation in a discrete basis is not appropriate. It is easy to recover a classical description of the entire system by expanding the potential energy terms in a Taylor series to linear order in r. Such classical approximations, in conjunction with quantum equilibrium sampling, are often used to estimate quantum correlation functions and expectation values. Classical evolution in this full Wigner representation is exact for harmonic systems since the Taylor expansion truncates. 

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). 

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