Effective Hamiltonian theories

It is a standard exercise in projection operator techniques to rewrite the full Schrodinger equation 

These equations show that if we ignore AH e) and diagonalize Hin the subspace 5 we are in effect diagonalizing the projected Hamiltonian P HP thus the complementary subspace 5 is treated implicitly through the energy-dependent operator AH s). A useful way of thinking about the relationship between Eqs. (2-2) and (2-3) follows from the observation that ifis completely determined by the requirement that it should reproduce the exact eigenvalues E , 

It is obvious from the denominator in the second term that the matrix will become non-Hermitian when the zeroth-order energies of the determinants in So are not the same. Therefore, this recipe only works for (nearly-)degenerate states. One advantage of this perturb-and-then-diagonalize approach is that the length of the wave function expansion remains of the dimension of the model space, and hence, especially suitable for analysis purposes. Multideterminantal perturbation schemes that follow the diagonalize-and-then-perturb approach are described in Sect.4.3.3. 

Effective Hamiltonian theory establishes a connection between accuracy and interpretation. It is used in many fields of chemistry and physics in different variants and sometimes confused with model Hamiltonians. In the scope of this monograph, the latter term is used for simple Hamiltonians that find their origin in physical/chemical intuition, and hence, are phenomenological in nature. These model Hamiltonians 

Similar to what is done in QDPT, a model space So of dimension N is defined as a subspace of the full Hilbert space S of dimension M. Remember that QDPT is used to determine accurate wave functions and energies starting from a limited description of the system based on the model space. However, in the present case, the accurate energies and wave functions are already known and the action goes in the opposite direction the lengthy wave function of length M is mapped on the smaller subspace So ensuring a minimum loss of the information contained in the full solution. 

In the first place, the eigenfunctions of S () have to be projected onto the model space by applying the projection operator 

However, this definition leads to a non-Hermitian Hamiltonian, which may not be the most optimal representation for interpretation. Therefore, one often adopts the 

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In another promising method, based on the effective Hamiltonian theory used in quantum chemistry [19], the protein is divided into blocks that comprise one or more residues. The Hessian is then projected into the subspace defined by the rigid-body motions of these blocks. The resulting low frequency modes are then perturbed by the higher... 

Average or effective Hamiltonian theory, as introduced to NMR spectroscopy by Waugh and coworkers [55] in the late 1960s, has in all respects been the most important design tool for development of dipolar recoupling experiments (and many other important experiments). In a very simple and transparent manner, this method facilitates delineation of the impact of advanced rf irradiation schemes on the internal nuclear spin Hamiltonians. This impact is evaluated in an ordered fashion, enabling direct focus on the most important terms and, in the refinement process, the less dominant albeit still important terms in a prioritized manner. 

IIOb — 0. if is not self-commuting at all times and effective Hamiltonian theory is therefore applied to gain physical insight. 

The Intermediate Hamiltonian Theory is a generalization of the Effective Hamiltonian Theory. The full Cl space of Slater determinants can be divided into three parts,... 

The problem of estimating crystal field parameters can be solved by considering the CFT/LFT as a special case of the effective Hamiltonian theory for one group of electrons of the whole A -electronic system in the presence of other groups of electrons. The standard CFT ignores all electrons outside the d-shell and takes into account only the symmetry of the external field and the electron-electron interaction inside the d-shell. The sequential deduction of the effective Hamiltonian for the d-shell, carried out in the work [133] is based on representation of the wave function of TMC as an antisymmetrized product of group functions of d-electrons and other (valence) electrons of a complex. This allows to express the CFT s (LFT s or AOM s) parameters through characteristics of electronic structure of the environment of the metal ion. Further we shall characterize the effective Hamiltonian of crystal field (EHCF) method and its numerical results. 

Focussing on the HOMO s and the LUMO e+, DFT yields the KS eigenvectors u, and U- (equation (A.3)). Effective Hamiltonian theory [15] allows to reduce the size of matrix V to 2 X 2. Choosing the latter sub-matrix U, written in the subspace... 

The outline of the review is as follows in the next section (Sect. 2) we introduce the basic ideas of effective Hamiltonian theory based on the use of projection operators. The effective Hamiltonian (1-5) for the ligand field problem is constructed in several steps first by analogy with r-electron theory we use the group product function method of Lykos and Parr to define a set of n-electron wavefimctions which define a subspace of the full -particle Hilbert space in which we can give a detailed analysis of the Schrodinger equation for the full molecular Hamiltonian H (Sect. 3 and 4). This subspace consists of fully antisymmetrized product wavefimctions composed of a fixed ground state wavefunction, for the electrons in the molecule other than the electrons which are placed in states, constructed out of pure d-orbitals on the... 

The renormalized theory of the effective Hamiltonian implied by the restriction to some subspace S of the full Hilbert space also imposes a requirement for renonnalisation of expectation values of other operators (Freed ). Suppose that we have some operator B and we require its expectation value in a state 0 of the full Schrddinger Eq. (2-2) in complete analogy with the effective Hamiltonian theory described above we define an effective operator B by the requirement that its expectation value in a state A ) in the subspace should equal the exact expectation value (c.f. Eq. (2-4)),... 

T. S. Untidt and N. C. Nielsen, Closed solution to the Baker-Campbell-Hausdorff problem exact effective Hamiltonian theory for analysis of nuclear-magnetic-resonance experiments. Phys. Rev. E, 2003, 65, 021108-1-021108-17. 

Mathematica routines for Exact Effective Hamiltonian Theory (EEHT) at University of Aarhus, Denmark http //bionmr.chem.au.dk/bionmr/software/index.php. 

RSPT and BWPT are special cases of effective Hamiltonian theory where the dimension of the model space is equal to unity see [7, 8]. 

Calculation of spectroscopic and magnetic properties of complexes with open d shells from first principles is still a rather rapidly developing field. In this review, we have outlined the basic principles for the calculations of these properties within the framework of the complete active space self-consistent field (CASSCF) and the NEVPT2 serving as a basis for their implementation in ORCA. Furthermore, we provided a link between AI results and LFT using various parameterization schemes. More specifically, we used effective Hamiltonian theory describing a recipe allowing one to relate AI multiplet theory with LFT on a 1 1 matrix elements basis. 

The applicability of LFDFT, like LFT itself, is rooted in an effective hamiltonian theory that states that, in principle, it is possible to define precisely a hamiltonian for a sub-system such as the levels of a transition metal in a transition-metal complex or a solid. This condition is possible, because in Wemer-type complexes the metal-ligand bond is mostly ionic and as such allows one to take a spectroscopically justified preponderant electronic d" or f" configuration as well defined ligand-to-metal and metal-to-ligand charge-transfer states are well separated from excitations within this configuration. 

Effective Hamiltonian theory. Following the formalism developed by Bloch, we will define a new operator called the effective Hamiltonian, Hc(t, given by... 

G. Jolicard, Effective Hamiltonian Theory and Molecular Dynamics, Annual Review of Physical Chemistry, 46 (1995), 83-108. 

The relativistic molecular calculations are very difiicult as long as they keep four-component wavefunctions according to the Dirac theory. The reduction to a two-component Pauli-like formalism within the effective Hamiltonian theory allows one to perform standard relativistic variational calculations. 

Both the methodological part and the review of applications have shown the similarities and differences between effective Hamiltonians and pseudo-Hamiltonians. The similarities sometimes concern the purpose of the modelling (for instance the reduction to a minimal basis set) which may be attained in one way or another. They also concern the use of some reduction of information to a definite part (in general the lowest one) of the spectrum. This reduction is explicit in the effective Hamiltonian theory, through the choice of a model space, while in the pseudo-Hamiltonian approach it goes through the choice of a reduced distance between the exact Hamiltonian and the pseudo-Hamiltonian. 

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