Hamiltonian decomposition

The former will be called a Hamiltonian decomposition of boenergy surface, and the latter a topological decomposition. It b clear that the Hamiltonian decomposition b more "detailed. It "remembers the structure of the critical submanifolds of the Bott integral. The topological decomposition b rougher. Its elementary blocks have already partially "forgotten the original Hamiltonian picture. In the sequel, we will use one or the other decompositions subject to the problem of interest. 

The first classical trajectory study of iinimoleciilar decomposition and intramolecular motion for realistic anhannonic molecular Hamiltonians was perfonned by Bunker [12,13], Both intrinsic RRKM and non-RRKM dynamics was observed in these studies. Since this pioneering work, there have been numerous additional studies [9,k7,30,M,M, ai d from which two distinct types of intramolecular motion, chaotic and quasiperiodic [14], have been identified. Both are depicted in figure A3,12,7. Chaotic vibrational motion is not regular as predicted by tire nonnal-mode model and, instead, there is energy transfer between the modes. If all the modes of the molecule participate in the chaotic motion and energy flow is sufficiently rapid, an initial microcanonical ensemble is maintained as the molecule dissociates and RRKM behaviour is observed [9], For non-random excitation initial apparent non-RRKM behaviour is observed, but at longer times a microcanonical ensemble of states is fonned and the probability of decomposition becomes that of RRKM theory. 

The ability to assign a group of resonance states, as required for mode-specific decomposition, implies that the complete Hamiltonian for these states is well approxmiated by a zero-order Hamiltonian with eigenfunctions [M]. The ( ). are product fiinctions of a zero-order orthogonal basis for the reactant molecule and the quantity m. represents the quantum numbers defining ( ).. The wavefimctions / for the compound state resonances are given by... 

Fig. 2. The BO model is the adiabatic limit of full QD if energy level crossings do not appear. QCMD is connected to QD by the semiclassical approach if no caustics are present. Its adiabatic limit is again the BO solution, this time if the Hamiltonian H is smoothly diagonalizable. Thus, QCMD may be justified indirectly by the adiabatic limit excluding energy level crossings and other discontinuities of the spectral decomposition.

Note that, in contrast to other forms of intermolecular perturbation theory to be considered below, the NBO-based decomposition (5.8) is based on a full matrix representation of the supermolecule Hamiltonian H. All terms in (5.8) are therefore fully consistent with the Pauli principle, and both /7units(0, and Vunits(mt) are properly Hermitian (and thus, physically interpretable) at all separations. 

A. J. Coleman and 1. Absar, Reduced Hamiltonian orbitals. 111. Unitarily invariant decomposition of Hermitian Operators. Int. J. Quantum Chem. 18, 1279 (1980). 

The unitary decomposition may be applied to any Hermitian, antisymmetric two-particle matrix including the 2-RDM, the two-hole RDM, and the two-particle reduced Hamiltonian. The decomposition is also readily generalized to treat p-particle matrices [80-82]. The trial 2-RDM to be purified may be written... 

In order to get significant results, the initial data must be formed by a set of clearly non-A -representable second-order matrices, which would generate upon contraction a closely ensemble A -representable 1-RDM. It therefore seemed reasonable to choose as initial data the approximate 2-RDMs built by application of the independent pair model within the framework of the spin-adapted reduced Hamiltonian (SRH) theory [37 5]. This choice is adequate because these matrices, which are positive semidefinite, Hermitian, and antisymmetric with respect to the permutation of two row/column indices, are not A -representable, since the 2-HRDMs derived from them are not positive semidefinite. Moreover, the 1-RDMs derived from these 2-RDMs, although positive semidefinite, are neither ensemble A -representable nor 5-representable. That is, the correction of the N- and 5-representability defects of these sets of matrices (approximated 2-RDM, 2-HRDM, and 1-RDM) is a suitable test for the two purification procedures. Attention has been focused only on correcting the N- and 5-representability of the a S-block of these matrices, since the I-MZ purification procedure deals with a different decomposition of this block. 

Consequently, we can carry out the BCH expansion to arbitrarily high order without any increase in the complexity of the terms in the effective Hamiltonian. In practice, the expansion is carried out until convergence in a suitable norm of the operator coefficients is achieved, as illustrated in Table I. Rapid convergence is usually observed. Note that through the decomposition (23), the effective Hamiltonian depends on the one- and two-particle density matrices and therefore becomes state specific, much like the Fock operator in Hartree-Fock theory. 

We now discuss (ii), the evaluation of operator expectation values with the reference ho- We are interested in multireference problems, where may be extremely complicated (i.e., a very long Slater determinant expansion) or a compact but complex wavefunction, such as the DMRG wavefunction. By using the cumulant decomposition, we limit the terms that appear in the effective Hamiltonian to only low-order (e.g., one- and two-particle operators), and thus we only need the one- and two-particle density matrices of the reference wavefunction to evaluate the expectation value of the energy in the energy expression (7). To solve the amplitude equations, we further require the commutator of which, for a two-particle effective Hamiltonian and two-particle operator y, again involves the expectation value of three-particle operators. We therefore invoke the cumulant decomposition once more, and solve instead the modihed amplitude equation... 

To understand this more clearly, consider a simpler model where A consists of single excitations, only single-particle operators are retained in the effective Hamiltonian, and we choose the reference function iho to be a single determinant. Then, from a cumulant decomposition of the two-particle terms, the effective Hamiltonian becomes... 

This decomposition of the 2-RDM is well known from the cumulant theory (see earlier chapters). F is the cumulant matrix of the 2-RDM. Since it arises from interactions in the Hamiltonian, it might also be called the pair correlation matrix. This definition of correlation differs from the traditional one since is the one-matrix of the correlated system and not that corresponding to independent particles. 

The local expressions obtained in the decomposition procedure can then conveniently be employed in a Heisenberg Spin Hamiltonian, which reads for a pair interaction... 

In Peralta et a/. s18,82 method the NJC decomposition into localized molecular orbitals procedure is as follows. Given a perturbative Hamiltonian A, A being either the FC or PSO operators, the corresponding J coupling term can be obtained as... 

The decomposition eq. (2-6) of the spin-free space FSP induces a decomposition on the Pauli-allowed portion of the Hilbert space of the Hamiltonian H of eq. (2-1). The Hamiltonian H which includes spin interactions may operate on any ket of the space Fsp V", where V is the electronic spin space. Here the symbol indicates a tensor product, so that Fsp Va consists of all spatial-spin kets which are composed of linear combinations of a simple product of a spatial ket of FSP and a spin ket of Va. The Pauli-allowed portion of the total A-electron Hilbert space of is... 

The concept of direct photodissociation corresponds to absorption of a photon leading to excitation to a. repulsive excited state followed by decomposition to photofragments. The Hamiltonian of the molecule can be written in the form... 

The formal theory of resonances due to Feshbach begins with the decomposition of the Hamiltonian in terms of a projection operator Q [8]. He defines Q as the projection onto the closed-channel space, just like the example of H discussed around Eqs. (4) and (5). Then, QBSs described well by the eigenfunctions Q4> of Eq. (5) with his Q may be called Feshbach resonances." A simplified picture would be that eigenstates Q are supported by some attractive effective potential approaching asymptotically the threshold energy of a closed channel. If this is the case, then the energies EQ of... 

In actual resonance calculations, explicit decomposition of the Hamiltonian is not always needed. The whole Schrodinger equation (1) is often solved somehow without explicit use of HQ or H. General methods for analyzing results from such a treatment are the main subject of this article. A comprehensive review of computational methods or computational results is outside of its scope. 

In going from static to dynamic descriptions we have to introduce an explicit dependence on time in the Hamiltonian. Both terms of the Hamiltonian (1.2) may exhibit time dependence. We limit our attention here to the interaction term. Formally, time dependence may be introduced by replacing the set of response operators collected into Q(r, r ) with Q(r, r, t) and maintaining the decomposition of this operator we presented in Section 1.1.2. For simplicity we reduce Q(r, r, t) to the dielectric component under the form P(r, t). With this simplification we discard both dielectric nonlocality and nonelec-trostatic terms, which actually play a role in dynamical processes, especially dispersion and nonlocality. 

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