Scaled Hamiltonians electronic Hamiltonian

It is worth mentioning that although the original problem (8.1.1) depended on the two field parameters P and w, the scaled equations of motion (8.1.8) as well as the scaled Hamiltonian (8.1.10) depend only on the single control parameter This is a major difference compared with the problem of microwave-driven surface state electrons, but is reminiscent of the classical mechanics of the kicked rotor discussed in Chapter... 

Figure 2. Uniform (1) and nonuniform (2-6) scaling paths of electronic charges in the complementary subsystems of M = (i4 B). The subsystem external potentials of the scaled hamiltonians for the starting and end points of each path are also listed, with the upper (lower) entries corresponding to ti (B).

Here, the scaled Hamiltonian H (m matrix form) is obtained by scaling and shifting the real Hamiltonian H (matrix form in electronic diabatic representation) of an A + BC reaction system with the two parameters as and bs ... 

The CCR idea has been around for a long time, as reviewed in Refs. 389 and 391, and many applications to temporary anion resonances have been reported. Nevertheless, this technique has remained somewhat specialized. Within the context of electronic structure theory, what is required for a CCR calculation is to combine the complex-scaled Hamiltonian in Eq. [63] with the usual wave function ansdtze, and this involves extending quantum chemistry codes to handle complex-valued wave functions and energies and non-Hermitian matrices. CCR implementations of the Hartree-Fock, configuration interaction, and multiconfigurational SCF (MCSCF) models have been reported but are not available in standard... 

The classical dynamics of the FPC is governed by the Hamiltonian (1) for F = 0 and is regular as evident from the Poincare surface of section in Fig. 1(a) (D. Wintgen et.al., 1992 P. Schlagheck, 1992), where position and momentum of the outer electron are represented by a point each time when the inner electron collides with the nucleus. Due to the homogeneity of the Hamiltonian (1), the dynamics remain invariant under scaling transformations (P. Schlagheck et.al., 2003 J. Madronero, 2004)... 

The transition from (1) and (2) to (5) is reversible each implies the other if the variations 5l> admitted are completely arbitrary. More important from the point of view of approximation methods, Eq. (1) and (2) remain valid when the variations 6 in a trial function are constrained in some systematic way whereas the solution of (5) subject to model or numerical approximations is technically much more difficult to handle. By model approximation we shall mean an approximation to the form of as opposed to numerical approximations which are made at a lower level once a model approximation has been made. That is, we assume that H, the molecular Hamiltonian is fixed (non-relativistic, Born-Oppenheimer approximation which itself is a model in a wider sense) and we make models of the large scale electronic structure by choice of the form of and then compute the detailed charge distributions, energetics etc. within that model. 

The relativistic adiabatic connection formula is based on a modified Hamiltonian H g) in which not only the electron-photon coupling strength is multiplied by the dimensionless scaling parameter g but also a g-dependent, multiplicative, external potential is introduced. 

As in the previous section, by connected we mean all terms that scale linearly with N. Wedge products of cumulant RDMs can scale linearly if and only if they are connected by the indices of a matrix that scales linearly with N transvec-tion). In the previous section we only considered the indices of the one-particle identity matrix in the contraction (or number) operator. In the CSE we have the two-particle reduced Hamiltonian matrix, which is defined in Eqs. (2) and (3). Even though the one-electron part of scales as N, the division by A — 1 in Eq. (3) causes it to scale linearly with N. Hence, from our definition of connected, which only requires the matrix to scale linearly with N, the transvection... 

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