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#### Effective Hamiltonians in Chemistry

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##### Citation

Neese, F., Lang, L., & Chilkuri, V. G. (2020). Effective Hamiltonians in Chemistry.
In E. Pavarini, & E. Koch (*Topology, Entanglement,
and Strong Correlations. Modeling and Simulation, Vol. 10*. Jülich, Germany: Forschungszentrum Jülich.

Cite as: http://hdl.handle.net/21.11116/0000-0007-82B2-C

##### Abstract

Effective Hamiltonians (EHs) occupy an important place in quantum chemistry. EHs serve a
multitude of different purposes. On the one hand, they are vital in the formulation of new
approximate methods that lead to new computationally efficient tools. On the other hand, they
allow one to recast highly complex problems that are difficult to understand into seemingly
simpler problems that are amenable to human analysis. In this latter application, they unfold
their full power by creating models that can be used to highlight the physical essence of the
problem at hand. In many cases, the emerging model Hamiltonians are of low dimension and
can be solved by hand or with very little computational effort. The crucial step is that the matrix
elements of the EH can be recast in terms of effective parameters. The latter are adjustable and
can, for example, be fitted to experiments or higher-level calculations. The benefit of EH theory
is here that: a) the model Hamiltonians are derived from more complete Hamiltonians; hence
the theory provides explicit and concrete expressions that allow the calculation of the model
parameters and b) being derived from first principles, these model Hamiltonians are solidly
grounded in fundamental physics. Thus, the effective Hamiltonians derived in this way do not
just represent a curve fitting exercise of uncertain physical content and interpretation as would
be the case for model Hamiltonians that are only based on physical intuition or conjecture.
The effective Hamiltonian concept can even be taken a step further and effective Hamiltonians
can be derived from more elaborate effective Hamiltonians that themselves are derived from
first-principles physics. The important point is that there is an unbroken chain of logic that
leads by pure deduction from first physical principles to a simple, intuitively appealing, and
physically sound model that can be used to interpret the results of measurements or even the
behavior of entire classes of substances or materials. The benefit of having model Hamiltonians
derived in this way can hardly be overemphasized since the theory does not only provide a concrete and unambiguous way to compute the model parameters, but it also makes it clear under
which conditions the model Hamiltonian is valid and when it is expected to break down. A concrete example for such a situation that will be discussed in more detail in section 3.2 is the spin
Hamiltonian (SH) used to interpret magnetic measurements (electron paramagnetic resonance,
EPR or nuclear magnetic resonance, NMR) on molecules or solids. The SH is derived conveniently from the time-independent relativistic many-particle Schrodinger equation (technically ¨
the Dirac-Coulomb-Breit (DCB) Hamiltonian). It leads to a low-dimensional model Hamiltonian that contains the SH parameters as adjustable parameters. These are the g-matrix, the
hyperfine coupling (HFC) matrix, the zero-field splitting (ZFS, in physics often referred to as
the magnetic anisotropy), the quadrupole splitting, the chemical shift and the nuclear spin-spin
coupling. For interacting magnetic systems, the familiar Heisenberg exchange Hamiltonian is
obtained from the theory.
However, once there are enough spins in a given system (for example in clusters containing a dozen to a few dozen open-shell transition metal ions), even the SH dimensions become unmanageable and can reach dimensions of hundreds of millions. Quite frequently, one is only interested in the lowest few eigenstates of such a system that are thermally accessible over a given temperature range. In this case, one wants to describe these few magnetic sublevels with an effective Hamiltonian that can be derived from the enormous SH of the entire system. The parameters that enter this secondary SH are then functions of all the spins and SH parameters of the full system. A very simple concrete example would be an S = 5/2 system with strong ZFS (relative to external magnetic fields). Such a system contains 2S+1 = 6 magnetic sublevels that, by means of Kramers degeneracy, form three so-called “Kramers doublets”. The latter can each be described by an effective Hamiltonian with spin S = 1/2. Hence, three pseudo S = 1/2 systems substitute for the entire S = 5/2 system. We refer to the specialist literature for further details [1, 2].
In this chapter, we will provide an introduction into the theory of effective Hamiltonians. We will cover formal aspects in section 2 before proceeding to actual chemical applications in
section 3 that will discuss both computational tools for the calculation of static (“strong” in
physics language) and dynamic electron correlation as well as EHs derived to parameterize and understand magnetic properties. An example for the combination of both strategies will conclude our chapter