Hamiltonian geometry dependence

Fig. 2. (a) Selected states representing the (alg + blg) 0 Blg PJT and the e 0 E effect excited state JT effect in the square C4H4. The marked points show the CASSCF computed values, the continuous and dashed lines corresponding to the fit with Hamiltonian of equation (8) (left side) (b) the corresponding geometry dependence of spin Hamiltonian parameters that fit the energy spectrum (right side). 

This theory also possesses an ab initio based quantitative version (10,19), in which the parameters are geometry dependent and fitted on accurately calculated potential surfaces of ethylene. Despite its simplicity, the spin-Hamiltonian theory has proven itself to be accurate for predicting ground state, as well as excited state, properties and transition energies. 

They do not depend either on the kinds of atoms or on molecular geometry. The unperturbed effective Hamiltonians Hnfn themselves, of course, depend on all these parameters, so that the energies of bonds even in this simple picture are composition-and geometry dependent, due to the corresponding dependence of the matrix elements of the Hamiltonian, but not the ESVs under consideration. The structure of the problem squeezes the whole multidimensional manifold of matrix elements (and even more dimensional manifold of the parameters defining the matrix elements) into two independent quantities A/TO and Apm. One can see that the invariant values of the ESVs eq. (3.7) are rather close to the exact SLG values which appear from numerical experiments. These are almost independent of the particular parameterization used. 

The construction of the LD theory of the ligand influence evolves in terms of two key objects the electron-vibration (vibronic) interaction operator and the substitution operator. The vibronic interaction in the present context is the formal expression for the effect of the system Hamiltonian (Fockian) dependence on the molecular geometry taken in the lower - linear approximation with respect to geometry variations. It describes coupling between the electronic wave function (or electron density) and molecular geometry. 

The total energy has an explicit geometry dependence in the nuclear-electron and nuclear-nuclear interaction terms, and an implicit geometry dependence in the wave function. In approximate calculations where finite nuclear-fixed basis sets are used, the total energy has an explicit dependence also in the basis set. Using the technique of second quantization, the geometry dependence of the basis set may be transferred to the Hamiltonian. In Section II we describe how the Hamiltonian at X0 + p may be expanded around X0... 

So far we have considered the Hamiltonian at one geometry, as appropriate for single-point calculations. However, if we wish to calculate the derivatives of the energy with respect to variations in the geometry, we must also consider the geometry dependence of the Hamiltonian. This introduces certain complications, which are treated in the remainder of this section. 

The Hamiltonian integrals depend on the molecular geometry in two ways. The first is trivial and arises because the Coulomb interactions between the electrons and the nuclei depend on the geometry. The second is more complicated and arises because the orbitals are themselves functions of the geometry. The reason for this is that the MOs are expanded in a finite set of AOs fixed on the nuclear centers. A consequence of using a finite set of AOs is that we are presented with a different basis set at each geometry. 

The geometry dependence of the Hamiltonian js isolated in the integrals. It only remains to determine the explicit form of h( and g(pqrs. 

In the next step it will be shown how the information on the hindering potential may be inferred from the fine structure of rotational spectra. We start with the pure internal-rotation Hamiltonian Hj [Eq. (5)], which contains one mass-geometry-dependent constant, F, and the potential parameters. First, assume V3 alone is important. Both parameters may be incorporated in a reduced potential barrier s, defined as... 

If the nuclear charges are regarded as classical particles (i.e., the QC approach), the geometry-dependent coefficients By( ) result from the coupling between electronic states in the electromagnetic field, yet contain no information on nuclear masses. Note that equation (21) is not intended to represent a BO wave function, the fact that P](q ) may not converge rapidly to a BO function is irrelevant to the present approach. The GED ansatz is conceived, by construction, as a scheme that diagonalizes the electronic Hamiltonian the standard BO approach does not satisfy this property. 

In equation (1) the Born-Oppenheimer (BO) approximation is employed. This means a standard partition of the Hamiltonian into an electronic and a nuclear part, as well as the factorization of the wave function into an electronic and a nuclear component. In this approximation, equation (I) refers to the electronic wave Junction with the electronic Hamiltonian parametrically dependent on the nuclear geometry, which we shall denote by the set of coordinates R. 

Since the form of the electronic wave functions depends also on the coordinate p (in the usual, parametric way), the matrix elements (21) are functions of it too. Thus it looks at first sight as if a lot of cumbersome computations of derivatives of the electronic wave functions have to be carried out. In this case, however, nature was merciful the matrix elements in (21) enter the Hamiltonian matrix weighted with the rotational constant A, which tends to infinity when the molecule reaches linear geometry. This means that only the form of the wave functions, that is, of the matrix elements in (21), in the p 0 limit are really needed. In the above mentioned one-elecbon approximation... 

Moreover, for the observables depending on external electric field, its specific effect has to be investigated the electric field induces new terms in the nuclear Hamiltonian, due to the change of equilibrium geometry and the nuclear motion perturbation. Pandey and Santry (14) has brought to the fore this effect and calculated the correction which only concerns the parallel component. It is represented by the following expression ... 

There is a general statement [17] that spin-orbit interaction in ID systems with Aharonov-Bohm geometry produces additional reduction factors in the Fourier expansion of thermodynamic or transport quantities. This statement holds for spin-orbit Hamiltonians for which the transfer matrix is factorized into spin-orbit and spatial parts. In a pure ID case the spin-orbit interaction is represented by the Hamiltonian //= a so)pxaz, which is the product of spin-dependent and spatial operators, and thus it satisfies the above described requirements. However, as was shown by direct calculation in Ref. [4], spin-orbit interaction of electrons in ID quantum wires formed in 2DEG by an in-plane confinement potential can not be reduced to the Hamiltonian H s. Instead, a violation of left-right symmetry of ID electron transport, characterized by a dispersion asymmetry parameter Aa, appears. We show now that in quantum wires with broken chiral symmetry the spin-orbit interaction enhances persistent current. 

The present work is firstly focused on the Dih Rh2+ centre in NaCl where active electrons are lying just in the RhClg- complex formed with six closest anions [10]. By means of adiabatic DFT calculations performed for the perfect Oh geometry and also for different values of the Qe coordinate the meaning and weight of parameters involved in the model-Hamiltonian are analysed. To this aim particular attention is paid to the Qe dependence of one electron energies, sa and sb, associated with the antibonding alg ( 3z2 — r2) and blg ( x2 — y2) orbitals. 

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