Free-electron Hamiltonian

The spin free electronic Hamiltonian of the stem,, is partitioned according to the usual Moller-Plesset form (129),... 

The eigenvectors of the corresponding free-electron Hamiltonian are the traveling-wave exponential functions exp(+inx / d) and exp(—i nx/d), or the standing-wave sine functions sin(7o /a) and cosine functions cos(nx/a). Since... 

It is to be noted that the dependence of the electronic basis wavefunctions S/( rJ R ) on the nuclear coordinates R is a parametric one. Since the field-free electronic Hamiltonian is time independent, the basis set is independent of time. Yet, the expansion coefficients T /( R, f) in this basis are time-dependent and have to obey the reduced coupled Schrodinger equations ... 

With the free-electron Hamiltonian of Eq. [52] written as a 4 x 4 matrix, ( 41... 

The masses of the nuclei, m, are at least three orders of magnitude larger than the mass of an electron. We can therefore assume that the electrons will instantaneously adjust to a change in the positions of the nuclei and that we can find a wavefunction for the electrons for each arrangement of nuclei. In the Born-Oppenheimer approximation the total molecular Hamilton operator Hnuc,e from Eq. (2.1) is thus partitioned in the kinetic energy operator of the nuclei, field free electronic Hamiltonian defined as... 

All these basis sets are essentially optimized for the calculation of electronic energies and are therefore able to represent the operators included in the field-free electronic Hamiltonian reasonably well. However, in the calculation of molecular electromagnetic properties it is necessary also to represent other operators such as the electric dipole operator, the electronic angular momentum operator, the Fermi-contact operator and more. Most of these basis sets are a priori not optimized for this and have to be extended. 

In standard notation (and in atomic units), the nonrelativistic, spinless, and field-free electronic Hamiltonian operator is given by... 

There are at least two ways forward, and the first was proposed by Schrddinger. Instead of the non-relativistic Hamiltonian for a free electron, he started from the correct relativistic expression... 

In a seminal paper [4], Su, Schrieffer, and Heeger discussed the energetics of solitons using a simple model of free electrons interacting with the lattice described by the Hamiltonian ... 

In the Hamiltonian Eq. (3.39) the first term is the harmonic lattice energy given by Eq. (3.12). It depends only on A iU, i.e., the part of the order parameter that describes the lattice distortions. On the other hand, the electron Hamiltonian Hcl depends on A(.v), which includes the changes of the hopping amplitudes due to both the lattice distortion and the disorder. The free electron part of Hel is given by Eq. (3.10), to which we also add a term Hc 1-1-1 that describes the Coulomb interne-... 

This treatment differs from the usual approach to molecule-radiation interaction through the inclusion of the contribution from the electric field from the beginning and by not treating it as a perturbation to the field free situation. The notation 7/ei(r R, e(f)) makes the parametric dependence of the electronic Hamiltonian on the nuclear coordinates and on the electric field explicit. 

The Mu spin Hamiltonian, with the exception of the nuclear terms, was first determined by Patterson et al. (1978). They found that a small muon hyperfine interaction axially symmetric about a (111) crystalline axis (see Table I for parameters) could explain both the field and orientation dependence of the precessional frequencies. Later /xSR measurements confirmed that the electron g-tensor is almost isotropic and close to that of a free electron (Blazey et al., 1986 Patterson, 1988). One of the difficulties in interpreting the early /xSR spectra on Mu had been that even in high field there can be up to eight frequencies, corresponding to the two possible values of Ms for each of the four inequivalent (111) axes. It is only when the external field is applied along a high symmetry direction that some of the centers are equivalent, thus reducing the number of frequencies. 

In the high-field limit (F > 1 atomic unit meaning that it is greater than the binding potential) the smoothed Coulomb potential in Eq. (2) can be treated as a perturbation on the regular, classical motion of a free electron in an oscillating field. So, let us first consider the Hamiltonian for the one-dimensional motion of a free electron in the... 

Hamiltonian in c-operator form For a free-electron, the Schrodinger equation takes the form ... 

Ri,R2,. ..,Rk denotes the nuclear coordinates. The first two terms in equation (1) describe, respectively, the electronic kinetic energy and electron-nuclear attraction and the third term is a two-electron operator that represents the electron-electron repulsion. These three operators comprise the electronic Hamiltonian in free space. The term V(r) is a generic operator for an external potential. One of the common ways to express V(f), when it is affecting electrons only, is to expand it as a sum of one-electron contributions... 

Table 4. The isotropic indirect spin-spin coupling constant of calculated at various levels of theory. LL refers to the Levy-Leblond Hamiltonian, std refers to a full relativistic calculation using restricted (RKB) or unrestricted (UKB) kinetic balance, spf refers to calculations based on a spin-free relativistic Hamiltonian. Columns F, G and whether quaternion imaginary parts are deleted (0) or not (1) from the regular Fock matrix F prior to one-index transformation, from the two-electron Fock matrix G...

So far we have assumed that the electronic structure of the crystal consists of one band derived, in our approximation, from a single atomic state. In general, this will not be a realistic picture. The metals, for example, have a complicated system of overlapping bands derived, in our approximation, from several atomic states. This means that more than one atomic orbital has to be associated with each crystal atom. When this is done, it turns out that even the equations for the one-dimensional crystal cannot be solved directly. However, the mathematical technique developed by Baldock (2) and Koster and Slater (S) can be applied (8) and a formal solution obtained. Even so, the question of the existence of otherwise of surface states in real crystals is diflBcult to answer from theoretical considerations. For the simplest metals, i.e., the alkali metals, for which a one-band model is a fair approximation, the problem is still difficult. The nature of the difficulty can be seen within the framework of our simple model. In the first place, the effective one-electron Hamiltonian operator is really different for each electron. If we overlook this complication and use some sort of mean value for this operator, the operator still contains terms representing the interaction of the considered electron with all other electrons in the crystal. The Coulomb part of this interaction acts in such a way as to reduce the effect of the perturbation introduced by the existence of a free surface. A self-consistent calculation is therefore essential, and the various parameters in our theory would have to be chosen in conformity with the results of such a calculation. 

The individual components, the electronic Hamiltonian the free photon Hamiltonian Hy and the electron-photon coupling Hamiltonianare given by [37]... 

Because often only the field-free Pauli Hamiltonian is presented in literature, we shall briefly sketch the derivation of the Hamiltonian hPauh(i) within an external field. For this, we start with the elimination of the small component in the one-electron Dirac equation by substitution of the small component of Eq. (15) to obtain an expression of the large component only... 

The spin Hamiltonian is an artificial but useful concept. It is possible that more than one spin Hamiltonian will fit the data. Further, we should note that in solving Eq. (48), we start with pure + and — spin functions and talk about the upper and lower states as being pure spin states. This is not the true case for the ion, as has already been noted in Eq. (38). As regards the Zeeman interaction, however, the final state behaves as a pure spin state, except that we must assign g values different from that of the free electron. 

Hamiltonian matrix for the cubic ligand held, spin-orbit coupling and the electronic Zeeman interaction in the real cubic bases of a 2D term. The g-factor for the free electron has been set to two for clarity (Table A.l). 

We see that the electron-vibron Hamiltonian (145) is equivalent to the free-particle Hamiltonian (154). This equivalence means that any quantum state IV a), obtained as a solution of the Hamiltonian (154) is one-to-one equivalent to the state ) as a solution of the initial Hamiltonian (145), with the same matrix elements for any operator... 

The full Hamiltonian is the sum of the free system Hamiltonian H the intersystem electron-electron interaction Hamiltonian He, the vibron Hamiltonian Hy including the electron-vibron interaction and coupling of vibrations to the environment (dissipation of vibrons), the Hamiltonians of the leads Hr, and the tunneling Hamiltonian Ht describing the system-to-lead coupling... 

In order to obtain the Hamiltonian for the system of an atom and an electromagnetic wave, the classical Hamilton function H for a free electron in an electromagnetic field will be considered first. Here the mechanical momentum p of the electron is replaced by the canonical momentum, which includes the vector potential A of the electromagnetic field, and the scalar potential O of the field is added, giving [Sch55]... 

The result for a free electron in an electromagnetic field can be transferred to the Hamiltonian H of an atom by using the same approach. Because the electromagnetic field depends on time, one starts with the time-dependent Schrodinger equation... 

Traditionally, the available microwave frequency determines the resonance field For g = ge = 2.0023 (the isotropic g factor of a free electron) and a frequency of 9 GHz, the resonance field equals He = 0.3211 T (X-band), while 35 GHz correspond to He = 1.2489 T (Q-band). Note that EPR spectra are usually evaluated using a spin Hamiltonian, and orbital contributions are effectively accommodated in the g tensor. 

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