Additional Terms in the Hamiltonian

Given an approximation to the ground state energy of the BO-Hamiltonian by some method, one needs to introduce the smaller field- and spin-dependent terms in the Hamiltonian fliat give rise to the interactions one actually probes by EPR spectroscopy. These terms can be derived through relativistic quantum chemistry, which is outside the scope of this chapter. Among the many terms that arise, we will mainly need the following interactions  

a = c in atomic units is the fine structure constant ( 1/137), f ,P ,S are the position, momentum, and spin operators of the zth electron, and 1 =(f -R )xp is the angular momentum of the zth electron relative to nucleus A. The vector = f,. -R of magnitude is the position of the zth electron relative to atom A. Likewise, the vector of magnitude is the position of 

P is the total eharge density matrix ealeulated by some flieoretieal method. Essentially like the HF approximation, whieh gives 99% of the total moleeular energy, the SOMF operator eovers around 99% of die two-eleetron SOC operator. It will be exclusively used below in order to approximate the SOC terms fliat will arise in the equations for die SH parameters. 

is the nuclear magneton, is the g-value of the th nucleus, and is the spin-operator for the nuclear spin of the th nucleus. While the isotropic Fermi contact term is frequently introduced as a separate operator, it arises naturally as a boundary term in the partial integration of the singular operator in Eq. (30). 

From the fully relativistic treatment there arises a kinetic energy correction (relativistic mass correction) to the spin-Zeeman energy that is given by [45] 

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The most common way of including relativistic effects in a calculation is by using relativisticly parameterized effective core potentials (RECP). These core potentials are included in the calculation as an additional term in the Hamiltonian. Core potentials must be used with the valence basis set that was created for use with that particular core potential. Core potentials are created by htting a potential function to the electron density distribution from an accurate relativistic calculation for the atom. A calculation using core potentials does not have any relativistic terms, but the effect of relativity on the core electrons is included. 

The spin magnetic moment Ms of an electron interacts with its orbital magnetic moment to produce an additional term in the Hamiltonian operator and, therefore, in the energy. In this section, we derive the mathematical expression for this spin-orbit interaction and apply it to the hydrogen atom. 

If the spin-quantum number, Ik, of a spin k is larger than 1/2, we have an additional term in the Hamiltonian, the quadmpolar coupling, hPk. The quadmpolar Hamiltonian arises from the interaction between the electric-field gradient and the nuclear spin. The first-order quadrupolar Hamiltonian is given by ... 

As a consequence of the transformation, the equation of motion depends on three extra coordinates which describe the orientation in space of the rotating local system. Furthermore, there are additional terms in the Hamiltonian which represent uncoupled momenta of the nuclear and electronic motion and moment of inertia of the molecule. In general, the Hamiltonian has a structure which allows for separation of electronic and vibrational motions. The separation of rotations however is not obvious. Following the standard scheme of the various contributions to the energy, one may assume that the momentum and angular momentum of internal motions vanish. Thus, the Hamiltonian is simplified to the following form. 

A more accurate description is obtained by including other additional terms in the Hamiltonian. The first group of these additional terms represents the mutual magnetic interactions which are provided by the Breit equation. The second group of additional terms are known as effective interactions and represent, to second order perturbation treatment, interaction with distant configurations . These weak interactions will not be considered here. 

So far in this book we have only discussed non-relativistic Hamiltonian operators but when atomic or moleoular spectra are considered it is necessary to account for relativistic effects. These lead to additional terms in the Hamiltonian operator which can be related to the following phenomena ... 

We now derive the additional terms in the Hamiltonian which arise from the application of an external magnetic field. The magnetic vector potential will be a sum of the contributions from the electrons (Af) and from the additional term... 

The degeneracy of the three potential minima is lifted by including additional terms in the Hamiltonian ... 

The beauty of Ham s theory is its simplicity. The Ham factors for any particular problem can be classified by symmetry. This means that any other operator which is a function of the Ua2 will be reduced by the same factor. Note that Ham s treatment implies that that the linear coupling is large compared to the additional terms in the Hamiltonian that are subsequently evaluated as perturbations of the ground vibronic energy levels. 

Let us assume that we have computed a single-determinant wavefunction for a molecule and we wish to investigate its properties in the presence of some small additional term in the Hamiltonian. We have used the LCAO method that is we have divided the finite-dimensional space into an occupied space and a virtual space each spanned by (say) the canonical MOs which diagonalise the HF matrix. 

We can remove the degeneracy of each level by applying an external magnetic field (the Zeeman effect). If B is the applied field, we have the additional term in the Hamiltonian [Eq. (6.131)]... 

Additional terms in the Hamiltonian to describe the free radiation as well as interaction with the transverse photons (giving rise to the Breit interaction). 

The electron and the hole are not necessarily localized on the same atom and the electron hole-pair can be localized on a series of equivalent sites of the system. Both possibilities give rise to additional terms in the Hamiltonian that do not exist in the atomic case. 

Before we can derive the additional terms in the Hamiltonian operator due to the interaction with external fields we should recall how in general one constructs the Schrodinger equation for a given system. The standard approach starts from the classical Hamiltonian H for the system that is a function of the position coordinates... 

Letting the orientahon of the field define the z-axis in space means that the field components in the x- and y-directions are zero. Thus, the additional term in the Hamiltonian needed to account for an exfemal field is... 

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