Hamiltonian method

Balint-Kurti G G, Dixon R N and Marston C C 1990 The Fourier grid Hamiltonian method for bound state eigenvalues and eigenfunctions J. Chem. See. Faraday Trans. 86 1741... 

Chu S I 1991 Complex quasivibrational energy formalism for intense-field multiphoton and above-threshold dissociation—complex-scaling Fourier-grid Hamiltonian method J. Chem. Phys. 94 7901... 

According to the effective Hamiltonian method, the energy Ea is complex, that is,... 

In the previous section we discussed the effective Hamiltonian method a main feature of this method is that it results in the appearance of damping operator T in the Liouville equation. However, the damping operator is introduced in an ad hoc manner. In this section we shall show that the damping operator results from the interaction between the system and heat bath. 

The similarity-transformed Hamiltonian method has so far been applied only to two-electron systems. Using closure (i.e., RI) approximations, this technique will be generalized to many-electron systems (IS). 

The ACSE has important connections to other approaches to electronic structure including (i) variational methods that calculate the 2-RDM directly [36-39] and (ii) wavefunction methods that employ a two-body unitary transformation including canonical diagonalization [22, 29, 30], the effective valence Hamiltonian method [31, 32], and unitary coupled cluster [33-35]. A 2-RDM that is representable by an ensemble of V-particle states is said to be ensemble V-representable, while a 2-RDM that is representable by a single V-particle state is said to be pure V-representable. The variational method, within the accuracy of the V-representabihty conditions, constrains the 2-RDM to be ensemble N-representable while the ACSE, within the accuracy of 3-RDM reconstruction, constrains the 2-RDM to be pure V-representable. The ACSE and variational methods, therefore, may be viewed as complementary methods that provide approximate solutions to, respectively, the pure and ensemble V-representabihty problems. 

Both the effective valence Hamiltonian method [31, 32] and unitary coupled cluster [33-35] employ a single two-body unitary transformation. In the effective valence Hamiltonian method [31, 32], the unitary transformation, selected by perturbation theory, is applied to the Hamiltonian to produce an effective... 

J. E. Stevens, R. K. Chaudhuri, and K. F. Freed, Global three-dimensional potential energy surfaces of H2S from the ab initio effective valence shell Hamiltonian method. J. Chem. Phys. 105, 8754 (1996). 

One additional advantage of this connection is that it permits the use of Hamiltonian methods to calculate various dynamical quantities. See the chapter by Poliak in this book for further details. However, it is not generally possible to provide... 

Prior to considering semiempirical methods designed on the basis of HF theory, it is instructive to revisit one-electron effective Hamiltonian methods like the Huckel model described in Section 4.4. Such models tend to involve the most drastic approximations, but as a result their rationale is tied closely to experimental concepts and they tend to be inmitive. One such model that continues to see extensive use today is the so-called extended Huckel theory (EHT). Recall that the key step in finding the MOs for an effective Hamiltonian is the formation of the secular determinant for the secular equation... 

Returning to the SCE formalism of HE theory, one can proceed in the spirit of an effective Hamiltonian method by developing a recipe for the replacement of matrix elements in the HE secular equation, Eq. (4.53). One of die first efforts along these lines was described by Pople and co-workers in 1965 (Pople, Santry, and Segal 1965 Pople and Segal 1965). The complete neglect of differential overlap (CNDO) mediod adopted the following conventions ... 

D A Micha. Effective hamiltonian methods for molecular collisions. Adv. Quantum Chem., 8 231, 1974. 

To analyze the reactivity of organics in systems using zero-valent iron, AH, (standard heat of formation) and ELUMO (energy of the lowest unoccupied molecular orbital) were computed using the PM3 Hamiltonian method. Reductive dechlorination rate constants for five chlorobenzenes in the pres-... 

In parallel, the related activity was in the field of single-electron shuttles and quantum shuttles [143-153]. Finally, based on the Bardeen s tunneling Hamiltonian method [154-158] and Tersoff-Hamann approach [159,160], the theory of inelastic electron tunneling spectroscopy (IETS) was developed [113-116,161-163],... 

Before presenting a rigorous method to treat this problem, a very useful and simple method, called the effective Hamiltonian method [17], shall be discussed as follows. 

The two-state system shall be considered as an example for illustration (see Eqs. (2.1)-(2.7)). According to the effective Hamiltonian method, the energies are complex, i.e.,... 

Using the effective Hamiltonian method, the EOM for the reduced density matrix a and the structure of 1 have been derived. However, how to evaluate F is not provided. 

From the above one can see that a main difference between the effective Hamiltonian method and the above results is T (d). It should be noted that the results given by Eqs. (3.45)—(3.53) can be obtained by using the singular perturbation method [18,20] and other methods [19,21]. 

An Effective Hamiltonian Method from Simulations ASEP/MD... 

Following the rules of the Hamiltonian method, the equations of motion can be written as follows ... 

In this way, a splitting of the A-doublets arises, A-doubling can be described very succinctly by the effective Hamiltonian method. 

Carrington and Miller (235) developed a method called the reaction-surface Hamiltonian for reactions with large amplitudes perpendicular to the reaction path and for some types of reactions with bifurcation of the reaction path. In contrast to the reaction-path Hamiltonian method, in the reaction-surface Hamiltonian method two coordinates are extracted from the complete coordinate set. One coordinate describes motion along the reaction path and the second one describes the large-amplitude motion. Potential energy in space of the remaining 3JV — 8 coordinates perpendicular to the two-dimensional reaction surface is approximated by quadratic functions. It... 

The separation of the PES into a part determined by the reaction coordinate and a part described by a quadratic approximation in a subspace of the remaining coordinates has recently often been used, typically with the WKB approximation (236,237) Yamashita and Miller (238) utilized the reaction-path Hamiltonian method combined with the path-integral method to calculate the rate constant of the reaction of H + H2. 

Finally, the advantages of respectively, the Lagrangian and Hamiltonian methods, are compared below. Two points should be emphasized ... 

A. P. J. Jansen, Compensating Hamiltonian Method for Chemical Reaction Dynamics Xe Desorption from Pd(lOO), J. Chem. Phys., 94 (1991) 8444. 

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